Atomtronics deals with matter-wave circuits of ultracold atoms manipulated through magnetic or laser-generated guides with different shapes and intensities. In this way, new types of quantum networks can be constructed in which coherent fluids are controlled with the know-how developed in the atomic and molecular physics community. In particular, quantum devices with enhanced precision, control, and flexibility of their operating conditions can be accessed. Concomitantly, new quantum simulators and emulators harnessing on the coherent current flows can also be developed. Here, the authors survey the landscape of atomtronics-enabled quantum technology and draw a roadmap for the field in the near future. The authors review some of the latest progress achieved in matter-wave circuits' design and atom-chips. Atomtronic networks are deployed as promising platforms for probing many-body physics with a new angle and a new twist. The latter can be done at the level of both equilibrium and nonequilibrium situations. Numerous relevant problems in mesoscopic physics, such as persistent currents and quantum transport in circuits of fermionic or bosonic atoms, are studied through a new lens. The authors summarize some of the atomtronics quantum devices and sensors. Finally, the authors discuss alkali-earth and Rydberg atoms as potential platforms for the realization of atomtronic circuits with special features.
Quantum technologies are enabling important innovations in the 21st century with applications in areas as diverse as computation, simulation, sensing, and communication. The core of these new technological developments is the ability to control quantum systems all the way from the macroscopic scale down to the single quantum level. The latter has been achieved in physical systems ranging from atomic and spin systems to artificial atoms in the form of superconducting circuits.1,2
In this article, we mostly focus on cold atom systems, where recent technological developments have delivered a collection of magnetic or laser-generated networks and guides in which atomic matter-waves can be controlled and manipulated coherently.3,4 Atomtronics exploits the state of the art in this field to realize matter-wave circuits of ultracold atoms.4,5 Some key aspects of this emerging field give atomtronic circuits great promise as a quantum technology. First, since atomtronic circuits employ matter-waves of neutral atoms, spurious circuit-environment interactions, which might, e.g., lead to decoherence, are expected to be less serious than in networks employing electrically charged fluids sensitive to Coulomb forces. Second, atomtronic networks can realize new types of circuits with current carriers having bosonic and/or fermionic quantum statistics along with tunable particle–particle interactions ranging from short-range to long-distance and from attractive to repulsive. Third, recent progress in the manipulation of optical guiding potentials enables engineering of time-dependent circuits whose topology can be reconfigured while they operate.3,6–9
The name Atomtronics is inspired by the analogy between circuits with ultracold atomic currents and those formed by electron-based networks of conductors, semiconductors, or superconductors. For example, a Bose–Einstein condensate (BEC) confined in a linear optical lattice with a suitable abrupt variation of the particle density can exhibit behavior very similar to that of an electronic diode.10,11 As another example, a BEC in suitable optical ring trap is the atomic counterpart of the superconducting SQUID of quantum electronics,11–13 displaying the SQUID's defining characteristics of quantum interference15 and hysteresis.16 It is important to note that because atomtronics is entirely based on flexible potential landscapes and not limited to material properties, it is expected to be possible to create quantum devices and simulators with new architectures and functionalities that have no analog in conventional electronics.
The quantum nature of ultracold atoms as coherent matter waves enables interferometric precision measurements and new platforms for quantum information processing with applications in fundamental science and technology.17,18 At the same time, atomtronic circuits can serve as powerful probes of many-body quantum regimes: analogous to solid state I-V characteristics, and many-body cold atom systems can be probed by monitoring the current flowing in them while changes are made to external parameters and applied (effective) fields. In this way, atomtronic platforms can be thought of as extensions to the scope of conventional quantum simulators, revisiting textbook scenarios in many-body physics, such as frustration effects, topological constraints, and edge state formation, with the advantages of tunable boundary conditions and minimal finite size effects. Another interesting domain in which atomtronics can play an important role is mesoscopic physics.19–21 Important themes in the field of mesoscopic physics, such as persistent currents in ring-shaped structures and problems of quantum coherent transport, can be explored with a new twist.
For the implementation of the program sketched above, an important challenge to face in the years to come is to optimize the control of the matter-wave currents in complex networks as, for example, optical lattices, guiding circuits for matter waves based on optical or magnetic fields, or cold atoms-solid state hybrid circuits. On one hand, such a step would be instrumental to harness current and transport for investigations on quantum many-body physics and artificial matter in both the static and dynamic conditions. In particular, Rydberg atoms and ultracold fermionic systems with SU(N) symmetry provide novel interesting directions to go to. Experimental challenges for this goal are to design improved schemes for controlling the resulting matter-wave interactions and for including advanced schemes for their detection. On the other hand, the control of complex quantum networks would be opening the way to work out new types of devices based on integrated atomtronic circuits. In particular, new chips integrating different technologies, for example, silicon-based electronics and the various atomtronics approaches, would provide a milestone in quantum technology. Concerning potential applications, a certainly important direction pursued in the current research in Atomtronics is devoted to interferometry and inertial sensing with enhanced performance, but quantum simulation and computation as well as all other aspects of quantum technology are accessible. In this context, stabilizing the atomic coherence on small-to-intermediate spatial scales, for example, by smoothing the wave guides, are important challenges to be solved in order to harness the full power of cold-matter-wave quantum technology.
In this review, we summarize recent activities in Atomtronics and discuss the future of the field. In Secs. II–IV, we review fabrication principles for atomtronic platforms, ranging from reconfigurable optical potentials employing acousto-optic deflectors, digital micromirror devices, and liquid-crystal spatial light modulators to micro-optical systems and hybrid solid state–cold atom systems circuits where a scanning focused laser beam modifies the current density of a superconducting chip to create the desired trapping potential. These new capabilities open the way to addressing the dynamics of many-body systems, as described in Secs. V and VI. Sections VII and VIII deal with persistent currents in toroidal and ring-shaped condensates. These systems, the simplest atomtronic circuits with a closed architecture, enable the study of basic questions in many-body physics in a variety of new and different conditions. Atomtronic quantum sensors and devices are discussed in Sec. IX. Ring-shaped bosonic circuits are investigated as ideal platforms for matter-wave SQUIDs [the Atomtronic QUantum Interference Device (AQUID)] and flux qubits in Sec. X. These studies have also touched upon a number of fundamental questions, such as macroscopic quantum coherence, the nature of superfluidity in restricted geometries, and vortex dynamics. Transport in fermionic and bosonic circuits are discussed in Sec. XI and XII, respectively. Section XIII deals with bosonic ladders. In addition to their potential relevance to basic research in many-body physics, we envisage that they will be instrumental to the fabrication of coupled atomtronic circuits. In Sec. XIV, we discuss atomtronic circuits that exploit bright solitons both for studying fundamental questions in many-body quantum dynamics and for realizing quantum devices with enhanced performances. Sections XV and XVI deal with alkali-earth atoms with SU(N) symmetry and Rydberg atoms. To date, the latter have received little attention, but we believe that they offer great promise as an atomtronic quantum technology.
The present article was inspired by the Atomtronics@Benasque conference series. The Benasque staff is warmly acknowledged for their invaluable help in the organization of these workshops, and we thank the Benasque director Jose-Ignacio Latorre for his constant support of this line of research.
II. DYNAMICALLY SCULPTED LIGHT
M. Baker, G. Gauthier, T. W. Neely, H. Rubinsztein-Dunlop, F. Tosto, R. Dumke, P. Ireland, D. Cassettari
In recent years, many experiments have been carried out with cold neutral atoms in arbitrary, reconfigurable optical potentials. Single atoms have been trapped in arbitrary-shaped arrays,22–26 which have subsequently led to the demonstration of topological phases of interacting bosons in one-dimensional lattices.27 Various configurations of atomtronic circuits have been demonstrated, namely, closed waveguides and Y-junctions,28 oscillator circuits,29 atomtronic transistors,30 rings, and atomtronic SQUIDs (AQUIDs).12,31,32 Reconfigurable optical potentials have also been used to realize Josephson junctions in rubidium condensates12 and in fermionic lithium superfluids in the BCS-BEC crossover.33 They have even been used for the optimization of rapid cooling to quantum degeneracy.34 Finally, another area of interest is the realization and study of quantum gases in uniform potentials.35,36 Some of these experiments are described in detail in Secs. II A–II E.
Static holographic potentials, as opposed to reconfigurable, also play an important role in atomtronics and have been implemented with great success.37–42 In particular, static holograms can provide substantial advantages for the generation of Laguerre–Gaussian and higher order Hermite–Gaussian modes.40,43,44 Static hologram techniques, such as optical nanofiber evanescent wave trapping,45 structured nanosurfaces to create trapping potentials,46,47 and the use of engineered quantum forces48 (also know as London–van der Waals or Casimir forces), are promisingly emerging technologies that will benefit the field of atomtronics. However, this section focuses on recently adopted dynamic technologies that have opened new avenues of research.
More generally, we note that sculpted light has many more applications beyond cold atom physics, e.g., to microscopy, optical tweezers, and quantum information processing with photonic systems.3 In this section, we review the tools and techniques that underpin all these experiments: scanning acousto-optic deflectors (AODs), digital micromirror devices (DMDs), and liquid-crystal spatial light modulators (SLMs).
A. Fast-scanning AODs
By rapidly scanning a trapping laser beam much faster than the trapping frequencies for the atoms, the atoms experience the time-average of the optical potential. Under these conditions, despite the modulated scanning action of the beam, the density of the atom cloud remains constant in time. The spatial location of the beam can be scanned in arbitrary 2D patterns, “painting” the potential landscape, simply by modulating the RF frequencies driving the crystal.7 The control over the RF power at each scan location allows local control over the potential depth. This feature can be used to error-correct, ensuring smooth homogeneous potentials, or can be deliberately engineered to implement barriers, wells, or gradients in the trap. The trapping geometry can be dynamically changed with the use of deep-memory arbitrary waveform generators or field programmable gate array (FPGA) technology, which combined with nondestructive measurement allows for the real-time correction of the potential. Given the weak axial confinement provided by the scanned beam, this is best used in conjunction with an orthogonal light sheet, which provides tight confinement along the axis of the scanned beam, and ensuring excitation and phase fluctuations in the axial dimension is minimized.7
1. Feed-forward control
The diffraction efficiency of AODs can change with the drive frequency. In order to correct this, it is generally necessary to use feed-forward to compensate by adjusting the RF power of the AOD crystal and, hence, the beam intensity at each (x,y) location. To correct imperfections in other elements of the trapping potential, one can measure the atomic density distribution in the trap using absorption imaging and apply iterative correction to the RF power at each (x,y) location.32
2. Phase evolution in time-averaged potentials
A full treatment time-averaged potentials need to include the phase evolution of the condensate under the effect of the scanning beam. The time-varying potential acts to imprint a phase with the evolution . For sufficiently fast scan rates, the imprinted phase effect is negligible, but at slower scan rates, this phase imprinting action can accumulate local phase, leading to residual micromotion in the condensate, and the signatures of which have been observed.49 This is an important consideration for atomtronic applications where the phase is an observable of interest, such as for guided Sagnac interferometry.50
3. Atomtronics with time-averaged optical traps
The time-averaged optical dipole traps are extremely versatile, allowing a variety of geometries to be generated, and dynamically changed in structure by real-time adjustment of the scanning pattern. In the context of atomtronic geometries, BECs have been trapped into flat bottom line-traps, rings,32,51 lattices,52 and dumbbell reservoirs (Fig. 1). Additionally, single mode matter-wave propagation and coherent phase splitting have been demonstrated in circuit elements such as waveguides and beamsplitters.28 The time-averaged optical beams can be used to introduce multiple repulsive barriers and stirring elements to study persistent currents and superfluid transport in atomtronic circuits.12
B. Optical potentials with liquid-crystal SLMs
A liquid-crystal SLM spatially modulates the phase of the light. The phase pattern on the SLM acts as a generalized diffraction grating so that in the far field, an intensity pattern is formed, which is used to trap atoms. In practice, the far field is obtained by focusing the light with a lens so that the intensity pattern that traps the atoms is created in a well-defined “output plane” coinciding with the lens focal plane. The SLM acts effectively as a computer-generated hologram, and the light field in the output plane is the Fourier transform of the light field in the SLM plane.
The first experiments with these holographic traps go back over ten years ago.53,54 A reason for the use of phase-only SLMs, rather than amplitude modulators, is that the former does not remove light from the incident beam. This is advantageous from the point of view of light-utilization efficiency. Moreover, as is shown below, a phase-only SLM allows the control of both the amplitude and phase on the output plane.
The calculation of the appropriate phase modulation to give the required output field is a well-known inverse problem, which, in general, requires numerical solution. Iterative Fourier Transform Algorithms (IFTAs) are commonly used, and variants that control both phase and amplitude have been recently demonstrated.55,56 The removal of the singularities (e.g., vortices), which particular pattern optimization techniques can introduce, is widely researched due to their importance for controlled beam shaping57–60 and, in particular, to confine BECs in uniform potentials.35,61 One such example is a conjugate gradient minimization technique that efficiently minimizes a specified cost function.60,62 The cost function can be defined to reflect the requirements of the chosen light pattern, such as removing optical vortices from the region of interest.
The intensity patterns obtained with this method are shown in the first row of Fig. 2. They are taken at a wavelength of 1064 nm, i.e., red-detuned relative to the rubidium transition, causing rubidium atoms to be trapped in the regions of high intensity. The SLM light is focused on the atoms by f = 40 mm lens, giving a diffraction limit of the optical system of 6 μm at 1064 nm.
Going from left to right in Fig. 2, shown are a simple waveguide, a waveguide with a potential barrier halfway across, a ring trap, and a crosslike pattern. The latter has been proposed for the study of the topological Kondo effect.63 In all these light patterns, the phase is constrained by the algorithm. For the simple waveguide, the ring and the cross, a flat phase is programmed across the whole pattern. Controlling the phase this way leads to a well maintained intensity profile shape as it propagates out of the focal plane for up to times the Rayleigh range. By comparison, a pattern with random phase loses its shape much sooner.
Differently from the other three patterns, for the waveguide with the barrier, a sharp π phase change halfway across the line was programmed. In the resulting intensity profile, this phase discontinuity causes the intensity to vanish, hence creating the potential barrier whose width is close to the diffraction limit.
The second row of Fig. 2 shows Rb BECs trapped in the potential created by the SLM light patterns combined with an orthogonal light sheet that provides tight confinement along the axis of propagation of the SLM light.64,65 The clouds are imaged after a 2 ms time of flight and undergo mean-field expansion during this time, leading to a final density distribution that is more spread out compared to the transverse size of the SLM traps.
Controlling the phase of the light pattern opens new possibilities for the trapping and manipulation of ultracold atoms. Here, we have shown that the phase control gives an alternative way to create barriers close to the diffraction limit by using discrete phase jumps. Liquid-crystal SLMs were also used to transfer phase structure in a four-wave-mixing process in rubidium vapor, in particular trans-spectral orbital angular momentum transfer from near-infrared pump light to blue light.66 Additionally, they have enabled research into uniform 3D condensates. More recently, they were used in the realization of bottle beams that have been used to create 3D optical trapping potentials for confining Rydberg atoms.26 In addition to this, the phase control can also be useful for many atomtronics applications, for instance, phase imprint via a Raman transition,67 and the realization of artificial gauge fields.68,69
C. Direct imaged DMD optical potentials
A recent addition to the spatial light modulator family is the digital micromirror device (DMD). Developed for digital light processing (DLP) applications, DMDs consist of millions of individually addressable, highly reflective mirrors. Each hinged mirror, of typical size 7.56–10.8 μm, is mounted on a silicon substrate on top of control electrodes. The application of a control voltage tilts the mirrors between two “on” or “off” angles, typically . The mirror array acts as a dynamical configurable amplitude mask for light reflected from their surface. The DMDs can be placed in the Fourier plane of the imaging/project system, similar to typical phase-based SLMs, where it can modulate both the phase and the amplitude of the light.72 If phase modulation is not required, the DMD can be used as a binary amplitude mask in the object plane, similar to its DLP applications.9,73 In the “DC” mode, the mirrors are fixed to the on angle and a static pattern can be projected. The true versatility of the device, however, lies in its dynamical (“AC”) capability with full frame refresh rates exceeding 20 kHz.
1. Half-toning and time-averaging
The projected image from the DMD is binary in nature. Although this would appear as a significant limitation in producing arbitrary optical potentials, a number of techniques exist to overcome this issue. The first of these is half-toning, or error-diffusion, which takes advantage of the finite optical resolution of the projection optical system to increase the amplitude control. With suitable high magnification, such that the projected mirror size is smaller than the resolution, multiple mirrors contribute to each resolution spot in the projected plane.74 In this way, half-toning can be used to create intensity gradients in the light field, as shown in Fig. 3(a). Same as in the case of time-averaged AOD traps, feed-forward using the atomic density32,70,71 can be performed to correct for imperfections in the projection potential, as shown in Fig. 3(b).71
One can also make use of the high-speed modulation of the mirrors to further improve the intensity control. The mirror array of the DMD is capable of switching speeds from DC to 20 kHz. By varying the on/off time of individual mirrors (pulse-width modulation), the time-average of the resulting light field can be utilized to improve the smoothness of the projected potentials.71
2. Atomtronics with DMDs
Atomtronics studies how to use neutral atom currents to create circuits that have properties similar to existing electrical devices. The advances in control and increased resolution of trapping potentials have been instrumental in the development of this field. The dynamic control over the potential given by DMDs have allowed time dependent implementations. Combined with other techniques such as the optical accordion lattice,73 which allows smooth transitions between quasi-2D and 3D systems, they open up further avenues of control for future studies. The high resolution projection of DMD optical potentials enables the creation of complex masks. These have facilitated the study of superfluid transport in a variety of traps. Figure 4 shows three relevant geometries for superfluid transport studies.
3. Turbulence with DMDs
The dynamic properties of DMDs can be used for the creation of turbulence. As shown in Fig. 5, these techniques have been used to study Onsager vortices and their emergence in superfluids,75,76 the creation of tunable velocity solitons,77 and equilibration of chiral vortex clusters.78
D. Hybrid atomic-superconducting quantum systems
Superconducting (SC) atom chips have significant advantages in realizing trapping structures for ultracold atoms compared to conventional atom chips.4,80–85 These advantages have been extended further by the development of the ability to dynamically tailor the superconducting trap architecture. This is done by modifying the current density distribution in the SC film through local heating of the film using dynamically shaped optical fields. This allows for the creation of desired magnetic trapping potentials without having to change the chip or the applied electrical field.
Typically, a high-power laser and a DMD are used to create and shape the light field used to destroy the superconductivity and influence the shape and structure of a trap. Various trapping potentials have been realized using this technique, in particular, to split a single trap (see Fig. 6) or to transform it into a crescent or a ringlike trap (see Fig. 7). Since the atomic cloud evolves with the trapping potential, cold atoms can be used as a sensitive probe to examine the real-time magnetic field and vortex distribution. Simulations of the film heating, the corresponding redistribution of sheet current density, and the induced trapping potentials have been found to agree closely with experiments. Such simulations help us to better understand the process and can be used to design traps with the needed properties.
More complex structures can be achieved by increasing the heating pattern resolution. This method can be used to create magnetic trap lattices for ultracold atoms in quantum computing applications and, in particular, optically manipulated SC chips open new possibilities for ultracold atoms trapping and design of compact on-chip devices for investigation of quantum processes and applications in atomtronics.86–92
E. Concluding remarks and outlook
In this section, we have described the suite of technologies available to the experimenter for creating configurable optical potentials for ultracold atoms, primarily discussing AOMs, SLMs, and DMDs. A hybrid technique utilizing optical configurable potentials to shape magnetic potentials through superconducting quantum chips intermediary were also discussed. These technologies have drastically improved the control and manipulation of ultracold neutral atoms.
Although previously available static holograms technologies provided great control for the creation of optical potentials and are still usually better for 3D trapping potentials, the dynamic manipulation capabilities presented here have enabled new classes of experiments with ultracold atoms. For example, dynamically modulated DMDs have facilitated new studies of two-dimensional-quantum turbulence75,76,78 and condensate evolution in response to rapidly quenched trapping potentials.31,93 AOMs have enabled steerable arrays of single atoms,23,52,94 facilitating quantum simulation experiments. Furthermore, the rapid reconfigurability of DMD traps has enabled groundbreaking studies in the emerging field of atomtronics, where the system parameters can be easily tuned.29,33,77,95,99
As the technology behind optical manipulation continues to mature and evolve through the increase in SLMs pixel array sizes and switching frequencies, these sculpted light and hybrid techniques are sure to have an even bigger impact on the development of atomtronics.
The UQ group has been funded by the ARC Centre of Excellence for Engineered Quantum Systems (Project No. CE1101013) and ARC Discovery Projects under Grant No. DP160102085. G.G. acknowledges support of ARC Discovery Project under No. DP200102239, and T.W.N. acknowledges the support of ARC Future Fellowship under No. FT190100306. The St Andrews group acknowledges funding from the Leverhulme Trust (No. RPG-2013-074) and from EPSRC (Nos. EP/G03673X/1 and EP/L015110/1).
III. IMPLEMENTING RING CONDENSATES
M. Baker, T. A. Bell, T. W. Neely, A. L. Pritchard, G. Birkl, H. Perrin, L. Longchambon, M. G. Boshier, B. M. Garraway, S. Pandey, W. von Klitzing
The many interesting properties of degenerate quantum gases, such as phase coherence, superfluidity, and vortices, naturally make the geometry of these systems of great interest. Ring systems are of particular interest, as the simplest multiply connected geometry for coherent matter-wave guiding and as a potential building block for circuital atomtronic devices. In addition, ring systems have interesting properties such as persistent flow, quantum hall states, and the potential for Sagnac interferometry.
Advances in the control of quantum gases have seen the development of atom waveguides formed from both magnetic trapping and magnetic resonance, and optical dipole trapping, and more recent implementations using hybrids of both. These approaches satisfy the criteria needed for coherent quantum matter-wave flow: i.e., the waveguides are smooth and can form loops that are dynamically controllable.
A. General features of ring traps
Irrespective of the mechanism of trapping, magnetic or optical, some common parameters for ring traps can be described. We will restrict our discussion to ring traps that can be considered approximately harmonic; in cylindrical co-ordinates, the ring potential with radius R is expressed in terms of radial and vertical trapping frequencies and ωz, respectively,
Considering now a trapped gas within this potential, the connected geometry of the ring trap results in modifications to the usual derivation for the condensate critical temperature Tc for a 3D harmonically trapped gas, yielding97
where N0 is the atom number. For sufficiently elongated geometries, such as cigar traps, or ring traps with long azimuthal length, a regime of thermally driven phase-fluctuations in the condensate can exist98,101 even at temperatures below Tc. These phase-fluctuations are suppressed when the correlation length is larger than the system size, which for a ring geometry is half the azimuthal circumference, or πR. As we are typically interested in fully phase coherent ring traps, we can define this transition temperature ,97
Finally, in the Thomas–Fermi approximation, where the interaction energy dominates, the chemical potential in the ring trap can be expressed in terms of the trapping parameters,100
and as is the s-wave scattering length.
In this section, we will discuss the experimental and theoretical developments in all three types of waveguide approach. In what follows, in Sec. III B, we discuss approaches primarily involving magnetic and radio-frequency fields; and in Sec. III C, we we will discuss optical and hybrid approaches to implementing ultracold atoms and condensates in rings before concluding in Sec. III E.
B. Techniques based on magnetic traps
Experimental techniques for trapping atoms in magnetic fields are well developed since the first BECs, and it is natural to consider such an approach, and build on that approach, to make ring waveguides. Nevertheless, this brings particular challenges because of the need to satisfy Maxwell's equations for fields trapping in a ring geometry, the need to avoid the loss of atoms from Majorana spin flips, occurring in the vicinity of field zeros, and the desire, for some experiments, to have trapping systems with high symmetry.
The earliest examples of waveguides for ultracold atoms were produced using static magnetic fields, where DC current carrying wires were used to create large area ring101 and stadium102 geometries which initially trapped thermal atoms. With Ref. 111, we had the first demonstrations of a ring waveguide with a Bose-condensed gas. Subsequent experimental developments can be divided into systems which principally use macroscopic coils for generating the magnetic trap, and those systems which employ microfabricated structures in an atom chip to generate the spatially varying potentials. We will briefly discuss the latter next and the former in Secs. III B 1–III B 3.
The appeal of atom-chip traps is their compact footprint, potential portability, and the ability to fabricate quite complex geometries, switches, and antenna components into a compact package.104,105 Additionally, the close proximity of the wires allows high trapping frequencies to be achieved, even for modest currents. However, trapping in close proximity to a surface brings with it its own challenges. Foremost of these are the corrugations in the magnetic guiding potential, which arise from imperfectly directed currents in the conducting material. An additional challenge is the perturbing effect of the end connections to supply current in and out of the conducting ring. Although these problems can be alleviated to some degree by the use of AC fields,106 which provides a smooth time-averaged current in the wire, as well as switching elements at the end connections to minimize the perturbative phase effects on the ring condensate,107 they cannot be removed completely. A comprehensive survey on the implementation of ring traps based on atom-chips, and their applications, is covered in detail in Ref. 108.
Here, we will focus our attention on ring traps derived from a combination of static magnetic traps with RF and modulated fields. Using macroscopically large conducting elements requires the use of high currents and occupies a greater size, but there are significant gains in the resulting trap smoothness, as the conducting elements are far from the trapping region. This makes such magnetic traps ideal for producing corrugation free toroidal waveguides for coherent matter, which is detailed in this section.
However, the complexity of the fields requires an atom-chip approach to a pure magnetic waveguide system104,105 and this brings a difficult problem for the perfect ring waveguide because of the need to get the currents into, and out of, the wires that define the waveguide. We can try to live with this,108 but asymmetry seems inevitable. We can think of tricks, for example, as the atoms go around the ring, and we can switch the current between different sets of 130 wires as in Ref. 107. This would avoid the bumps and humps in the waveguide, which occur in the places where current enters and leaves the defining structures at the expense of potential losses and heating as the guides are switched over.
1. RF dressing and bubbles
It is not obvious that micrometer-scale trapping structures for ultracold atoms can be created using macroscopic scale magnetic coils. However, by means of the addition of radio-frequency coils, magnetic traps with a simple trapping geometry can be transformed into ring traps and other topologies. The theoretical basis is to treat the atom and radio-frequency field with adiabatic following and the dressed-atom theory.109 Originally introduced in the optical domain by Cohen-Tannoudji and Reynaud,109 we adapt it here in the radio-frequency domain where it has found several applications (see also Secs. III B 2, III B 3, and III D). The approach is suitable for ultracold atoms in magnetic traps where the trap potential is governed by the spatially varying Zeeman energy and the spatially varying energy difference between Zeeman levels can be in the radio-frequency range.110,111 The method relies on the adiabatic following of local eigenstates, and it is notable that the superpositions of Zeeman states can provide some resilience to temporal noise and surface roughness.106 The combination of static magnetic fields and radio-frequency fields with their different spatial and vector variation allows flexibility in the resulting potentials for the creation of shell potentials, rings, tubes, and toroidal surfaces among others.110,111
As a simple example, we can consider a simple spatially varying static field and a uniform radio-frequency field. A simple spatially varying magnetic field (obeying Maxwell's equations) is the quadrupole field,
where represents the gradient of the field in the x-y plane. This field is often generated by a pair of coils with current circulating in opposite directions. When an atom interacts with this static field via its magnetic dipole moment , we obtain the ubiquitous interaction energy,
responsible for magnetic potentials and the Zeeman energy splitting. The second form for has the integer or half integer , which arise from the quantization of the energy along with the Landé g-factor gF and Bohr magneton μB. For our example static field [Eq. (5)], the resulting potential is , where .
In the next step toward radio-frequency dressed potentials, we add the RF field. The interaction is still given by Eq. (6) but with the replacement . The oscillating radio-frequency field is, in general, off-resonant to the local Larmor frequency or local Zeeman energy spacing , and we define a spatially varying detuning of the RF field as
Those locations defined by typically define a surface in space where RF resonance is found, and correspondingly, there is a minimum in the interaction energy overall.110,111 In the linear Zeeman regime, the local Larmor frequency is given by
which is derived from the static potential . The oscillating field yields an interaction energy110,111 in terms of a Rabi frequency ,
where the factor of two arises from the rotating wave approximation in the case of linear polarization (more general polarizations are discussed in Ref. 111), and is the component of perpendicular to the local static field . Finally, by combining the energies (6) and (9) through diagonalization of the Hamiltonian in a full treatment,110,111 we obtain the local eigenenergies, or dressed potentials,
where are a set of integers, or half-integers, similar to the mF described above.
The result of this is that slow atoms are confined by the potential (10), which in a typical configuration, and to a first approximation, confines atoms to an iso-B surface defined by , which approximately reduces the value of in Eq. (10). The term also plays a role, and in particular, it can be zero at certain locations on the trapping surface allowing the escape of atoms. This latter effect prevents the trapping of atoms in a shell potential by using the static quadrupole field (5). However, shell potentials are possible with different field arrangements such as those arising from the Ioffe–Pritchard trap and variations,110–115 which have become candidates for experiments on the International Space Station.116 The requirement is simpe for a local extremum in the magnitude of the field together with a nonzero . The reason for the interest in shell potentials in the earth orbit is that on the earth's surface a gravitational term mgz should be added to Eq. (10), which plays an important role for larger and interesting shells (e.g., see Sec. III B 3).
Although the matter-wave bubbles produced by shell potentials have become an object of great interest, the shell potentials themselves are the building blocks for other potentials of interest such as ring traps: we will see an example in Sec. III D. Another example is in Sec. III B 2, where a modulated bias field is used to make a ring trap: then and is a field varying in space, and time, but typically at a frequency rather lower than the radio-frequency case.
2. Waveguides formed from time-averaged adiabatic potential (TAAP)
Time averaged adiabatic potentials (TAAPs) allow the generation of extremely smooth matterwave guides,117 which can be shaped into a half-moon or ring (see Fig. 8). They are an excellent candidate for matterwave optics, long-distance transport experiments, and interferometry in an atomtronic circuit.117–119 TAAPs are formed by applying an oscillating homogeneous potential to the adiabatic bubble traps described in Sec. III B 1. If the modulation frequency is small compared to the Larmor frequency, but fast compared to the trapping frequency of the bubble trap, then the effective potential for the atoms is the bubble potential time-averaged over one oscillation period.120 Let us consider TAAP potentials formed from a quadrupole bubble trap and an oscillating homogeneous field of the form . The modulation field simply displaces the quadrupole (and thus the bubble trap) by at an instant in time. In order to find the effective potential that the atoms are subjected to by this method, one calculates the time-average. Time-averaging of a concave potential increases the energy of the bottom of the trap, as is readily illustrated by taking the time average of a harmonic potential jumping between two positions: the curvature does not change since it is everywhere the same; however, the energy of the trap bottom increases since it is at exactly the crossing point between the two harmonic potentials. Returning to the modulated bubble trap, one notices that the modulation is orthogonal to the shell at the poles of the shell , but tangential to the shell on the equator (z = 0). Therefore, the time averaging causes a larger increase in the trapping potential at the poles rather than the equator and, therefore, creates a ringlike structure.
Assuming that is modulated such as to stay resonant on the ring and to keep constant, the vertical and radial trapping frequencies can be controlled via the relative amplitude of the modulation as and , where the radial trapping frequency of the bare bubble trap is with the mass of the atom m, the is the Landé g-factor of the considered hyperfine manifold, is the Bohr magneton, and the Rabi frequency of the dressing RF. In order to achieve large RF field strengths (–1 G) and Rabi frequencies, , one usually has to use RF-resonators, which make it very difficult to tune the RF frequency, and which results in a somewhat weaker confinement in the axial (i.e., vertical) direction. Trapping frequencies of the order of a hundred hertz are readily achieved.
In many cases, it is also desirable to confine the atoms azimuthally. This is readily achieved either by tilting the ring away from being perfectly horizontal or by modifying the polarization of the rf-field. The half-moon shaped BEC in Fig. 8(b) was formed this way. A gravito-magnetic trap results from tilting the direction of the and thus tilting the ring against gravity.118 The gravito-magnetic potential forms a single minimum much like a tilted rigid pendulum. One can also create a trap by changing the polarization of the dressing RF: tilting a linear polarization from the z-axes will cause due to its projection on the local B-field, and a sinusoidal modulation of the Rabi frequency along the ring resulting in a two minima on opposite sides of the ring. Alternatively an elliptical RF polarization creates a single minimum. Combining these modulation techniques permits the creation of two arbitrary placed traps along the ring or, more generally, any longitudinal confinement of the form , where is the azimuthal angle and and are phase offsets. Note that there are no angular spatial Fourier components higher than present in the system.
Thermal atoms and BECs are readily loaded into the gravito-magnetic TAAPs from a trapping-frequency-matched dipole trap. This can be done fully adiabatically by ramping down the dipole confinement and at the same time ramping up the TAAP trap. With a sufficiently high level of control on the rf-fields, one can also load them from a TOP trap via a tilted dumbbell-shaped trap.118 Once in the ring, one can then manipulate the atoms with a simple manipulation of the time-averaging fields: The depth of the azimuthal trap can be changed by modifying the degree of tilt applied to the modulation field . By changing the direction of the tilt (i.e., the phase between the modulation fields in the x and y directions), one can move the trap along the ring. This can be used, e.g., to accelerate the atoms along the ring with angular momenta of per atom being readily achieved.117 They can then travel in the waveguides over distances of tens of centimeters without any additional heating associated with the propagation. One can also remove the azimuthal confinement and allow the condensates to expand around the ring. Viewed in the corotating frame at high angular momenta, the atoms see an exceptionally flat potential with the largest resulting density fluctuations corresponding to an energy difference of a few hundred picokelvin: this is equivalent to a few nanometers in height.117 Current experiments have been performed with BECs in the Thomas–Fermi regime with about 20 transverse vibration modes occupied. The 1D regime is readily accessible simply by reducing the atom number and increasing the radius of the ring.
The complete lack of any roughness combined with a picokelvin level control of the trapping parameters make the TAAP waveguides a very good candidate for guided matterwave interferometry and the study of ultralow energy phenomena such as long-distance quantum tunneling. A remaining challenge is to completely fill the ring with a phase coherent condensate. Current experiments allowed a condensate to expand along the ring, which converts the chemical potential of the BEC into kinetic energy. When the condensate touches itself at the opposite side of the ring, the two ends have a finite velocity in opposite directions, resulting in a spiral BEC, i.e., a BEC wrapped around itself. Using atom-optical manipulation of the expansion process, kinetic energies in the pico-kelvin range (a few hundred micrometers per second) can readily be achieved. It will be interesting to study the very low energy collisions that will lead to a thermalization of this system. A promising approach for a fully phase-coherent ring-shaped condensate is to first fill a small ring and then increase its radius. This should not induce any additional phase fluctuations, despite the fact that the lowest excitation has an energy of , which for a ring of 1 mm radius is 3 fK.
3. Dynamical ring in an rf-dressed adiabatic bubble potential
There is a formal analogy between the Hamiltonian of a neutral gas in rotation and the one of a quantum system of charged particles in a magnetic field. This makes rotating superfluids natural candidates to simulate condensed matter problems such as type II superconductors or the quantum Hall effect.121,122 For a quantum gas confined in a harmonic trap of radial frequency ωr and rotating at angular frequency Ω approaching ωr, the ground state of the system reaches the atomic analog of the lowest Landau level (LLL) relevant in the quantum Hall regime.123–125 Reaching these fast rotation rates is experimentally challenging in a harmonic trap because the radial effective trapping potential in the rotating frame vanishes due to the centrifugal force. To circumvent this limit, higher-order confining potentials have been developed,126 which allow to access the regime where Ω even exceeds ωr.
The adiabatic bubble trap has many features that make it a very good candidate to explore this regime. Indeed, it is very smooth, and easy control of its anisotropy is possible through the dressing field polarization.127 This allows us to deform the bubble and rotate the deformation around the vertical axis in a very controlled way, allowing us to inject angular momentum into the cloud. The curved geometry of the bubble provides naturally the anharmonicity required to rotate the atoms faster than the trapping frequency ωr at the bottom.
In the experiment at LPL,128 the atoms are placed in a quadrupole magnetic field of symmetry axis z dressed by a radio-frequency (rf) field of maximum coupling at the bottom of the shell. Here, the equilibrium properties in the absence of rotation (Ω = 0) are well known:127 the minimum of the trapping potential is located at r = 0 and z = z0, and around this equilibrium position, the potential is locally harmonic with vertical and radial frequencies and without measurable in-plane anisotropy. This trap is loaded with a pure BEC of 87Rb atoms with no discernible thermal fraction. This atomic cloud has a chemical potential of , which is much greater than ωr and ωz and well in the three-dimensional Thomas-Fermi (TF) regime. In addition to the dressing field, a radio-frequency knife with frequency is used to set the trap depth to approximately by outcoupling the most energetic atoms in the direction transverse to the ellipsoid.129,130
In a frame rotating at frequency Ω, the effective dressed trap potential is the usual trap described above with the addition of a term taking into account the centrifugal potential. In this frame, the atomic ground state consists of an array of vortices of quantized circulation, each vortex accounting for of angular momentum per atom. When only a few vortices are present, the velocity field differs strongly from the one of a classical fluid, but for a sufficiently large number of vortices, the superfluid rotates as a solid body with a rotation rate Ω. When , the equilibrium position remains on the axis r = 0 at z = z0, and the only difference is a renormalization of the radial trapping frequency: . Of course, as this frequency decreases, the trap anharmonicity becomes more important in the determination of the cloud shape.
For , the trap minimum is located at a nonzero radius. In this situation, a hole grows at the trap center above a critical rotation frequency Ωh,131 leading to an annular two-dimensional density profile [Fig. 9(a)], which we will refer to as a “dynamical ring.”128 Moreover, the velocity of the atomic flow is expected to be supersonic,132 i.e., exceeding by far the speed of sound. For increasing Ω, one expects the annular gas to sustain vortices in its bulk up to a point where the annulus width is too small to host them. The gas should then enter the so-called “giant vortex” regime132,133 where all the vorticity gathers close to the center of the annulus.
The experimental sequence is the following: angular momentum is injected into the cloud by rotating the trap with an ellipsoidal anisotropy at a frequency . The trap rotation is then stopped and isotropy is restored. At this moment, which we take as t = 0, the cloud shape goes back to circular with an increased radius due to its higher angular momentum. An additional evaporation process, selective in angular momentum, continuously accelerates the superfluid and increases its radius.128 Due to this size increase, the chemical potential is reduced and the gas enters the quasi-2D regime . After a few seconds, a density depletion is established at the center of the cloud which is a signature of Ω now exceeding ωr. After a boost in selective evaporation due to a lowering of the frequency of the rf knife, a macroscopic hole appears in the profile, indicating that Ω is now above Ωh and that a fast rotating dynamical ring with a typical radius of has formed as can be seen in Fig. 9(b). The rotation keeps increasing, and a ring is still observable after . Rotational invariance is critical in that regard and is ensured at the level by a fine tuning of the dressing field polarization and of the static magnetic field gradients.111
A Thomas–Fermi profile convoluted with the imaging resolution is much better at reproducing the experimental density profile than a semiclassical Hartree–Fock profile, demonstrating that the samples are well below the degeneracy temperature. Using the Thomas–Fermi model, we can estimate the properties of the cloud. For example, the ring obtained at has a chemical potential of and an averaged angular momentum per particle . Interestingly, the estimated peak speed of sound at the peak radius is much smaller than the local fluid velocity : the superfluid is, therefore, rotating at a supersonic velocity corresponding to a Mach number of 11. Moreover, due to the continuous acceleration of the rotation, the dynamical ring radius grows gradually with time which results in a decrease in the chemical potential and an increase in the Mach number. For , the chemical potential is below and the highest measured Mach number is above 18.
Superfluidity in the dynamical ring has also been evidenced by the observation of quadrupolelike collective modes. After the ring formation, the rotation rate, while accelerating, crosses a value where the quadrupole collective mode is at zero frequency, such that any elliptical static anisotropy can excite it resonantly. A very small bubble anisotropy is enough to excite this mode characterized by an elliptic ring shape rotating with a period of approximately in the direction opposite to the superfluid flow (Fig. 10). This counterpropagating effect is not predicted by a mean-field theory and has been confirmed by resonant spectroscopy of the quadrupole mode during the ring acceleration.128
The persistence of superfluidity at such hypersonic velocity raises fundamental questions about the decay of superfluidity in the presence of obstacles, and how superfluidity can be preserved at such speeds: nonlinear effects, the presence of vortices and the dependence on temperature would be particularly interesting to study experimentally and compare with theoretical predictions.134–138 This hypersonic superflow is not yet a giant vortex, but it is an important step toward this long-sought regime whose transition rotation frequency is not theoretically clearly identified. Moreover, the well-known elementary excitation spectrum for a connected rotating superfluid is strongly modified when the ring appears, and the important discrepancies observed between the experimental results and a mean-field theoretical approach for a quadrupolelike collective mode highlight the need to refine the description of fast rotating superfluids in anharmonic traps.
An alternative way of generating large angular momentum states in rf-dressed adiabatic bubble potentials is to first generate them in a TAAP ring and then reduce the vertical modulation, thus adiabatically transferring the atoms into the bubble.
C. Trapping in rings with optical potentials
Potentials for ultracold atoms can be formed through the use of focused far-detuned optical beams.139 Since the potential is directly proportional to the intensity of the optical field, ring-shaped condensates may be created through the implementation of ring-shaped optical patterns. The most significant advantage in optical dipole ring traps is the insensitivity to the hyperfine state, allowing multicomponent and spinor BECs to be trapped. Additional advantages include the imprinting of superfluid flow, either through phase imprinting or through Raman transitions that can directly transfer angular momentum to the cloud. The advent of spatial light modulator technologies means the optical ring trap has become highly configurable, allowing more complex geometries to be generated.
1. Optical trapping
The light–matter interaction can be parameterized through the complex polarizability, where the real part is associated with the dipole trapping potential and the imaginary component results in the absorptive scattering of photons. Trapping cold atoms requires that absorption is minimized to avoid scattering loss of atoms from the trap. Defining , the detuning of the trapping laser from resonance, the scattering loss rate reduces as while the trapping potential reduces as . Thus, sufficient detuning of the optical field will result in an optical potential that is approximately conservative. The potential arising for far-detuned dipole trapping light is given by
where I(R) is the intensity profile of the light and Γ is the transition linewidth. Since the trapping force is determined through the gradient of Eq. (11), a trapping potential requires a nonuniform optical intensity, obtained by shaping and focusing the intensity profile . Ring traps, can either be created from attractive (red-detuned) or repulsive (blue-detuned) light, usually by combining the ring shaped intensity profile with a perpendicular light sheet that provides confinement along the propagation direction of the projected ring pattern.
2. Optical ring traps
We begin by looking at some of the optical beam techniques for ring traps that are in use and outline their potential for atomtronic applications.
a. Laguerre–Gauss beams
One of the first proposed methods for a ring optical dipole trap was the use of Laguerre–Gaussian (LG) modes having circular symmetry.140 For far-off-resonance light, these provide the spatial structure for a toroidal trap. An additional advantage of such LG modes is that they also carry orbital angular momentum. With pulses of near-resonant light, the LG modes can be tailored to provide two-photon Raman transitions that transfer exact quanta of circulation to the condensate.
The LG0N modes are typically generated by phase transformation of a Gaussian TEM00 mode, which transforms the spatial profile of the beam into a donut mode carrying units of orbital angular momentum. A number of methods exist, including spiral phase plates, computer generated holograms, or through the use of phase based spatial light modulators. The toroidal intensity profile of the LG01 mode is given by141
where PLG is the total laser power in the LG beam and r0 is the radius at the peak intensity of the LG mode. Correction for imperfections in the spatial structure, and obtaining sufficient power in higher order modes, is typically a challenge. Ring traps and circulating currents using LG modes have been demonstrated in both single state and multicomponent spinor gases and were early demonstrations of all-optical trapping of BEC in a ring geometry.67,142,143 To date, they have been used to realize small optical rings for the study of quantized superfluid flows.
b. Painted optical traps
An alternative to projecting a ring shaped beam is to build a time-averaged potential with a moving, red-detuned, focused laser beam. By rapidly steering a Gaussian beam in a circular orbit, a ring trap can be generated. This is achieved through the use of two acousto-optical deflectors (AOD) controlling the two axes of the painting beam by driving the deflectors with lists of frequency points that are repeatably iterated at high speed.7,32 This approach was used to create the first ring BEC,7 as shown in Fig. 11.
The advantages of this technique is that it allows adapting the intensity locally to create desired features in the potential landscape and to flatten imperfections due to possible laser inhomogeneity;28 the available laser power is used in an efficient way as only the relevant trapping locations are illuminated; the painting laser itself can be used as a stirrer to set the quantum fluid into rotation and demonstrate quantized superfluid flows;144 the technique also enables more complex geometries. As an example, the atomtronic analog of a Josephson junction has been demonstrated and used to realize a DC atomtronic SQUID.12 More recently, the dynamic potentials possible with painting were used to show that the atomtronic SQUID exhibits quantum interference.15
The painting approach also comes with specific technical constraints that may need to be addressed. The phase of the time-averaged beam loop plays a role on the fine details of the potential, which results in imprinting of the condensate phase, and has to be compensated for.49 This is particularly relevant for the application of such traps in atom interferometry schemes.
c. Conical refraction
A novel approach to generating ring traps has been demonstrated with the use of conical refraction occurring in biaxial crystals. A focused Gaussian beam passing along the optical axis of the crystal transforms, at the focal plane, into one or more concentric rings of light. In the case of a double-ring, the light field encloses a ring of null intensity, called the Poggendorff dark ring.145 For a blue-detuned laser field, the atoms are trapped between the bright rings. The advantage of this configuration is that it minimizes spontaneous scattering of photons responsible for heating when the laser beam is not very far detuned from resonance. Further advantages include the high conversion efficiency of the incoming Gaussian beam to the ring-trap light field and the access to different ring configurations. The ring diameter is defined by the refractive indices of the biaxial crystal and its length. The width of each ring is given by the focal waist of the focused Gaussian beam. A variation of the ratio of these numbers (e.g., by changing the focal waist) allows for a variation of the resulting light field topology from a single bright ring to a bright ring with a central bright spot and further to bright double rings of increasing diameter. As with LG modes, there are challenges in alignment of the optical beams through the biaxial crystal. On the other hand, the conversion efficiency from a Gaussian TEM00 mode to the ring pattern can be close to unity. The first results on BECs transferred into a Poggendorff ring have been reported.145 Ongoing work is directed toward implementing quantum sensors (e.g., Sagnac interferometers) for rings with large diameter and atomtronic SQUIDs for small rings.
d. Digital micromirror direct projection
Direct imaging of digital micromirror devices (DMDs) has recently emerged as a powerful tool for the all-optical configuration of BECs.9,73,146,147 Ring traps can be created by directly projecting the DMD-patterned light onto a vertically confining attractive light-sheet potential,9,146 similarly to Fig. 12, or onto a vertically oriented accordion lattice.73 This can be accomplished using a relatively simple optical system, usually consisting of an infinite conjugate pair. Due to the large magnification factors required to reduce the DMD image to the typical m scale of the BEC, the final element in the imaging system is typically an infinity corrected microscope objective.9,146 DMDs may also be used in the Fourier plane of the imaging system,72 where the DMD implements an amplitude-only hologram. A detailed discussion of holographic techniques is beyond the scope of this section, and the reader is referred to more complete reviews of the subject.148
In Fig. 13, direct imaging of a DMD is used to create a ring trap, along with a central phase-uncorrelated reference BEC. By introducing a stirring barrier with the DMD, and circulating the barrier around the ring, a 21-quanta persistent current results, corresponding to an angular momentum of per atom. The winding number of the current is visualized through interference with the reference central BEC after a short 5 ms time of flight.149 The DMD technology can also be used to phase imprint an azimuthal light gradient such that angular momentum can be imparted to the atoms150 and a circulating current created.151
e. Microfabricated optical elements
An approach combining flexibility, integrability, and scalability can be based on the application of microfabricated optical elements for the generation of complex architectures of dipole traps and guides.152 It draws its potential from the significant advancement in producing diffraction-limited optical elements with high quality on the micro- and nanometer scale. Lithographic manufacturing techniques can be used to produce many identical systems on one subtrate for a scalable configuration.153 On the other hand, state-of-the art direct laser writing gives high flexibility in producing unique integrated systems and allows for fast prototyping.154 Applications range from integrated waveguides and interferometer-type structures155 to arrays of dipole-traps for quantum information processing156 and single-atom atomtronics implementations.157 In combination with DMD-based control of the light field (see Sec. III C 2 d), access to dynamic reconfiguration becomes possbile. Integrability is not limited to the generation of light fields for dipole potentials but can be extended to the integration of light sources and detectors or even complex quantum-optical systems such as an entire magneto-optical trap.152
3. Imperfections in optical traps
Defects in the optical potential will influence the ability to sustain superfluid flow without dissipation, or may introduce unwanted phase perturbations on the condensate if the optical potential is time-varying. We can gain some measure of the significance, and the level of control required for optical traps useful in atomtronics, by considering the density of the BEC in a ring potential. In the Thomas–Fermi limit, with sufficient atom number in the trap, the interaction energies dominate over kinetic energy terms, leading to a simplified GPE equation , giving the density , where μ is defined by Eq. (4). The density occupies the spatial profile of the ring trap. In the context of the intensity of the optical potential, assuming a fixed light sheet, the trap depth scales directly scales with the ring optical intensity I0, while the chemical potential of the BEC more weakly follows as . This means that for a typical condensate, the chemical potential is on the order of tens of nanokelvin, and is only weakly effected by the trapping intensity, while large optical trap depths on the order of K or larger may be easily achieved and utilized. Since the density of the condensate closely follows the optical potential, small perturbations in the optical field can result in significant fluctuations on at the energy scale of the condensate, and thus significant density fluctuations; variations in the optical intensity must typically be limited to less than 1% in order to avoid unwanted perturbations. The precision of the optical projection is thus a key consideration when implementing configurable optical potentials. These aspects, however, also mean that the condensate density provides a very sensitive probe of the optical potential, and the atom density can be used to feedforward corrections to the optical potential.32
D. Hybrid traps: RF bubble plus light sheet(s)
One can also combine optical potentials and magnetic trapping to produce a hybrid trap and exploit the advantages of each technique for ring trap generation. As mentioned above, optical potentials can achieve large trapping frequencies, while magnetic traps are very smooth due to the macroscopic size of the coils generating them. The bubble geometry described in Secs. III B 1 and III B 3 is particularly suited to create a ring trap: by combining the rf-dressed bubble trap and an optical light sheet as in Sec. III C 2, one can create a toroidal trap. The principle is depicted in Fig. 14: a horizontal light sheet is superimposed with a bubble trap which is rotationally invariant around the vertical direction.100,158 The light sheet is designed to achieve a strong optical confinement in the vertical direction, and the radial confinement is ensured by the bubble trap itself, made with the same rf-dressed quadrupole trap as in Sec. III B 3. Maximum radial trapping and maximum radius will be attained if the light sheet is located at the equator of the bubble, a situation which also ensures maximum decoupling between the vertical trapping frequency ωz and the radial trapping frequency ωr.
Experimentally, the optical trap is formed between two horizontal light sheets, which are made repulsive by their large blue detuning from the atomic transition. The bubble radius is significantly smaller than the light sheets width and also the vertical Rayleigh length to minimize the azimuthal potential variations. The choice of a small radius also comes with a higher critical temperature and a larger chemical potential, which reduces the relative density fluctuations around the ring due to optical imperfections from residual light scattering of the vacuum glass cell (see Sec. III C 2). One then creates a trapped toroidal degenerate gas of approximately atoms [Fig. 14(b)]. With further reduction of optical imperfections in the light sheets, one could enter with 104 atoms the quasi-1D condensate regime,159 where large-scale correlations and solitons play an essential role in the dynamics.
The gas can be set into rotation by different procedures, using either magnetic or optical means. The first method, used in our experiment in Ref. 160, consists in slightly deforming the bubble trap with an ellipsoidal anisotropy, rotate this magnetic deformation at a given fixed frequency and finally restore the circular symmetry. In a second method (Fig. 15), the rotation is induced by a rotating optical defect160,161 driven by a dual-axis acousto-optic modulator system as described in Sec. III B 1. Well-controlled circulation could also be imparted by direct optical phase imprinting onto the ring trap.150
Above some critical rotation frequency depending on the excitation strength, one observes, after a time-of-flight imaging procedure, a hole in the atomic distribution. The hole is absent when the ring is nonrotating, and is, thus, evidence for a nonzero circulation of the superfluid in the ring trap (Fig. 15). The hole area grows for increasing rotation rates and shrinks with time when one lets the cloud rotate freely in the trap. In future experiments, optical barriers created by spatial light modulators could be imposed onto the ring and dynamically modulated in height and position. This would create the equivalent of Josephson junctions in superconductors and allow us to simulate models of nonequilibrium quantum systems and emulate new setups in mesoscopic superconductivity.16,28 This hybrid ring is very promising for the study of 1D superfluid dynamics, for example, shock waves induced by rotation in the presence of a static barrier.162,163 Increasing again the ring confinement toward the one-dimensional regime with fermionization of the atoms164 could lead to NOON states more robust against decoherence.165
E. Concluding remarks and outlook
The development of technology for controlling electronic systems and generating complex optical fields is giving ever greater control of ultracold atoms and condensates of atoms. The ring trap remains of particular interest because of the topology, the possibility for self-interference, circuital currents, Sagnac interferometry and so on. In a way, it is its own primitive atomtronic circuit. For optical ring traps, painted optical potentials and digital micromirror devices have demonstrated high level of configurability and dynamic control over the condensate, allowing state-independent trapping, and the ability to introduce junctions, moveable barriers into the atomtronic ring. The ring systems based on RF dressed magnetic traps are also extremely flexible because of the level of electronic control. Atoms can be accelerated and rotated around ultrasmooth waveguides, simply by varying or introducing additional control frequencies with time. The future challenges for the technology, after this development, will be to create particular atomtronic applications and test the limits of technology for creating large scale structures and structures, which possibly have some 3D features. In the future, we will undoubtedly see more control complexity and more hybrid approaches. Where surface interactions are less of a problem, we can also envisage atomtronic circuits based on atom-chip technology, where rings, and complex guided circuits, may be enabled by the design of wire structures and the fields they produce from static and AC currents.
The UQ group acknowledges funding by the ARC Centre of Excellence for Engineered Quantum Systems (Project No. CE1101013) and ARC Discovery Projects under Grant No. DP160102085. W.K. would like to acknowledge the contribution of the AtomQT COST Action CA16221 and of HELLAS-CH (No. MIS 5002735) implemented under “Action for Strengthening Research and Innovation Infrastructures,” funded by the Operational Programme “Competitiveness, Entrepreneurship and Innovation” (No. NSRF 2014-2020) and cofinanced by Greece and the European Union (European Regional Development Fund). H.P. and L.L. acknowledge financial support from the ANR Project SuperRing (Grant No. ANR-15-CE30-0012) and from the Région Île-de-France in the framework of DIM SIRTEQ (Science et Ingénierie en Région Île-de-France pour les Technologies Quantiques), project DyABoG. B.M.G. would like to acknowledge support from the UK EPSRC Grant No. EP/M013294/1. M.G.B. acknowledges support from the U.S. DOE through the LANL LDRD Program.
IV. ATOMTRONIC CHIPS AND HYBRID SYSTEMS
C. Hufnagel, M. Keil, A. Günther, R. Folman, J. Fortagh, R. Dumke
During the last decade atom chip approaches to quantum technology have become a powerful platform for scalable atomic quantum-optical systems,104,105,166 with applications ranging from sensor and imaging technologies to quantum processing and memory. Atom chips coupled to solid state-based quantum devices, e.g., superconducting qubits or nitrogen vacancy centers, are thereby paving the way for promising quantum simulation and computation schemes.167–169 Along this research line, several groups around the world have developed versatile atom chip configurations, which allow trapping of ultracold atomic clouds and degenerate Bose-Einstein condensates (BECs) close to chip surfaces and well-defined manipulation of their internal and external degrees of freedom. Atom chips provide a very relevant technology for the emerging field of atomtronics,5,30,105,170–173 for which dynamic tunneling barriers are required.174,175 Such barriers may be formed on atom chips with μm-scale widths, matching the length scale dictated by the atomic deBroglie wavelength. The atom chip offers the ability to realize guides and traps with virtually arbitrary architecture and a multitude of novel architectures,176 with a high degree of control over atomic properties, like interactions and spin, enabling new quantum devices.5,105
Here we review progress in our groups in Beer Sheva, Tübingen and Singapore on recent developments in atom chip technologies.
A. Progress toward on-chip interferometry
The Ben-Gurion University of the Negev (BGU) Atom Chip Group (http://www.bgu.ac.il/atomchip) is promoting the idea of atomtronics without light. This entails circuits for atoms based on electric and magnetic traps, guides and tunneling barriers. The vision is for a complete circuit, including particle sources and detection, that makes no use of gravity, e.g., no time of flight for the development of interference fringes. This requirement means that a future technological device could work at any angle relative to gravity.
As a basis for this effort we use the Atom Chip technology developed over the past 20 years.105,166 An example of a circuit design we plan to implement is a continuous-wave, high-finesse Sagnac interferometer, where the multiple turns enabled by the guiding potential allow miniaturization of the loop while maintaining sensitivity to rotation.174 In the following we briefly present some of the work that has been done to advance the atomtronics technology.
To begin with, a stable tunneling barrier (in terms of instabilities of tunneling rates) should be no wider than the de Broglie wavelength, which is on the order of 1 μm. Since the resolution with which we can tailor fields is on the order of the distance from the field source, one must construct the atomic circuit at a distance of no more than a few micrometers from the surface of the chip.175 At these very small atom-surface distances, several problems must be avoided:
Johnson noise. This is a hindering process as it may cause spin-flips (reducing the trap/guide lifetime), as well as decoherence. In several papers we have shown ways to combat both effects either by the geometry or by the choice of material.84,175,177–179 We have also measured Johnson noise and calculated its interplay with phase diffusion caused by atom-atom interactions.170
Finite size effects. As the atom-surface distance becomes smaller, so should the current-carrying wire width, or else the magnetic gradients will be severely undermined. Narrow wires require high-resolution fabrication175 or thin self-assembled conductors such as carbon nanotubes.179
Casimir-Polder and van der Waals forces. As the atom-surface distance becomes smaller, the magnetic barrier between the atoms and the surface should be strong enough to avoid tunneling of the atoms to the surface. This has been calculated for submicrometer distances.175,179
Fragmentation. Due to electron scattering in the current-carrying wires (e.g., due to rough wire edges), the minimum of the trap, or guide, is not smooth and the atomic ensemble may split and exhibit a nonuniform density along the wire axis. This was studied by us both experimentally and theoretically.180–182
High aspect ratios. As the atom-surface distance becomes smaller, the trap or guide exhibits much higher transverse frequencies compared to the longitudinal frequency. This brings about low dimensionality and can cause different problems such as phase fluctuations in a 1D BEC. Alternative wire configurations allow more flexibility for adjusting the trap aspect ratio.
In a proof-of-principle experiment183 we were able to avoid all the above hindering effects, and showed that spatial coherence could be maintained for at least half a second at an atom-surface distance of just 5 μm.
Another important problem that needs to be overcome is that of atom detection at very small atom-surface distances. At these distances of a few micrometers, the stray light from the nearby surface makes it very hard to achieve a reasonable signal-to-noise ratio for in situ detection with typical optical elements. As a solution, and also to avoid on-resonance spin-flips and decoherence, we studied the possibility of off-resonant atom detection with high-Q microdiscs.184–186
With the above tools we are now preparing to go forward with our vision for a Sagnac circuit,174 where as a first stage we have the goal of observing spatial coherence of atoms after one, and then several, turns in a guiding loop. The guiding potential is made in two alternative ways. The first method, using RF potentials, is being led by Thomas Fernholz of the University of Nottingham. It requires multilayer chips (4 layers of currents), which are fabricated at BGU. Two such layers are shown in Fig. 16. The second effort also requires a unique chip. The guiding potential will be based on a repulsive permanent magnet potential in combination with an attractive electric field produced by a charged wire. The first experiments will be pulsed, whereby a BEC will be loaded onto the loop at the beginning of every cycle. Later on we will move toward realizing a continuous-wave version. We will first conduct the experiment in the pulsed mode by loading a thermal cloud, and later on use a 2D MOT as a continuous source.
Finally, let us note that quite a few groups around the world have realized free-space matter wave interferometry. It is now an important challenge to adapt these interferometers to the framework of atomtronics. Specifically, the BGU Atom Chip group has made significant steps in this direction by realizing, in the last 5 years, several types of novel interferometers which are not based on light. These interferometers are based on the magnetic splitting force (Stern-Gerlach) and they have already enabled the observation of spatial fringes,187,188 spin population fringes,189 unique T3 phase accumulation,190 clock interferometry,191,192 and the observation of geometric phase.193
B. Precision sensing
Precise sensors are one of the most important elements in applied and fundamental science. The use of quantum properties in sensing applications promises a new level of sensitivity and accuracy.194 Using cold atoms on atom chips as probes will enable many interesting applications.
In the laboratories at the University of Tübingen we are working with atom chips that host one or two layers of lithographically implemented wire patterns. They allow the creation of spatially and temporally varying magnetic fields, as used for three-dimensional positioning and manipulation of cold atomic quantum matter.195 We typically use wire patterns made of gold in room temperature environments196 and superconducting patterns of Niobium in 4 K and mK surroundings.197
With such a “carrier chip” for cold atoms on hand, we established a dual-chip process, where a second chip hosting nanostructured solid state systems is attached on top of the carrier chip.195 In this way, cold atoms can be efficiently coupled to other quantum systems and hybrid systems can be realized.
We have used this scheme to develop a novel cold-atom scanning probe microscope (CASPM), which uses ultracold atoms and BECs as sensitive probe tips for investigating and imaging nanoscale systems.198 Similar to an atomic force microscope (AFM), the probe tip is scanned across the surface of interest, while static and dynamical properties of the probe tip are monitored. Evaluating changes within the cold-atom tip density and motion then gives access to basic interactions and serves as a novel imaging and sensor technique. In contrast to conventional AFMs with their “heavy and rigid” solid state tips, our CASPM uses a dilute gas of atoms, which not only allows for nondestructive measurements, but also for much higher sensitivity to external forces and fields. Inspired by conventional AFMs, we have been able to demonstrate several modes of operation.199 These include not only a contact mode, where we measure position-dependent losses of the probe tip, but also a dynamic mode, where we initiate a center-of-mass oscillation of the cold-atom tip and monitor the position-dependent changes of the probe tip oscillation frequency.198 Based on the latter, we have used cold-atom force spectroscopy to unveil anharmonic contributions in near-surface potentials. As in atomic force microscopy, this may be used to reconstruct the surface potentials. Moreover, we have developed a novel operation mode, not accessible to conventional AFMs, where we bring the dilute probe tip into direct overlap with the nano-object of interest. By measuring time-dependent probe tip losses, we have then been able to deduce the underlying van der Waals (Casimir-Polder) interactions.200,201 We have demonstrated and characterized all different operating modes of CASPM by measuring individual free-standing carbon nanotubes grown on a silicon chip surface. Here we have shown that CASPM extends the force sensitivity of conventional AFMs by several orders of magnitude down to the yN regime, and the working distance up to several micrometers.199 This makes CASPM a powerful tool for investigating fragile nano-objects with ultrahigh force sensitivity.
While first measurements with CASPM suffered from long measurement times, we have just lately extended the microscope by a powerful single atom detection scheme.202,203 It is based on continuous subsampling of the probe tip via a multiphoton ionization process in conjunction with temporally resolved ion detection and high quantum efficiency. This allows real-time monitoring of the probe tip dynamics and density while losing only few atoms from the probe tip.203,204 This not only speeds up probe tip oscillation frequency measurements by at least three orders of magnitude,203 but also enables new applications for CASPM.
In one of these applications we proposed a quantum galvanometer to detect local currents and current noise in nanoscale mechanical quantum devices.205,206 Measuring the current noise would then give access to the quantum properties of the device. We successfully demonstrated the principal operating scheme of this galvanometer by coherently transferring artificially generated magnetic field fluctuations via a Bose-Einstein condensate onto an atom laser and investigating its single-atom statistics.207,208 Employing second-order correlation analysis, we could not only extract the microwave power spectral density (current noise spectrum) but also the noise correlations within the bandwidth of the BEC, which will give access to the quantum noise properties of the current source. This will extend CASPM to a promising quantum sensor, not only for detecting local forces and force gradients, but also for currents as well as electric and magnetic fields (AC and DC), including their specific noise spectra.
C. Cryogenic atom chips and hybrid quantum systems
Atom chips made from superconducting circuits offer certain advantages over normal metal devices. The coherence properties of trapped atoms are improved by orders of magnitude due to reduction of magnetic noise coming from the surface of the chip. Moreover, superconductors can be operated in the mixed state, where vortices can be used to generate self-sufficient atom traps. In addition to that, working in cryogenic environments offers the possibility to interface atoms with solid state devices to form hybrid quantum systems.84
Besides atom chip experiments in room-temperature environments, the group in Tübingen also operates superconducting atom chips with trapped BECs of rubidium atoms.197,209,210,214 As shown in Fig. 17, condensates are routinely transferred into coplanar cavity structures211 and the measured coherence time between hyperfine ground state superpositions reaches several seconds. Microwave dressing is used to suppress the differential shift of state pairs with the “double-magic point” being the optimum working point for quantum memories.212 We have successfully demonstrated coherent coupling of a hyperfine state pair through a driven superconducting coplanar microwave cavity,213 which paves the way for future cavity-based quantum gate operations.
In addition to manipulating ground-state atoms we have successfully implemented two-photon Rydberg excitation in a cryogenic environment near the superconducting chip.215 We have developed techniques for optical detection of Rydberg populations and coherences216 and measured the increased lifetime of Rydberg states in cryogenic environments.215 In preparation for interfacing Rydberg atoms with superconducting circuits, we have obtained high-resolution spectra of rubidium Rydberg states in a field-free vapor cell as reference,217 and in precisely controlled electrostatic fields218 near surfaces at room and cryogenic temperatures. These studies add to our understanding of electrostatic fields of surface adsorbates that build up during experiments with cold atoms at chip surfaces.88,219,220 Based on the measured data, quasi-classical quantum defect theory,221 Stark-map calculations,218 suitable dressing techniques,212,222 and numerical methods developed for simulating quantum operations in the presence of thermal cavity photons,89 we are currently focusing on the coherent manipulation of Rydberg atoms and quantum computation schemes in the presence of inhomogeneous fields at the surface of superconducting coplanar cavities.
The realization of hybrid quantum systems based on atoms and superconducting qubits requires truly cold temperatures in the 10 mK range, as dictated by the otherwise fast decoherence of the superconducting qubit. The great advantage is that at this temperature the number of microwave photons in the cavities that mediate the interaction between the solid state and the atomic system is near zero. The price to pay is a highly complex experimental system combining cold-atom technologies with a 3He/4He dilution refrigerator.223,224 Our dilution refrigerator consists of several temperature-shielded volumes (stages), of which we use the 6 K-stage and the 1 K-stage for cold atom experiments. The 1 K-stage includes a cold plate with a nominal base temperature of 25 mK.209 We routinely operate a magneto-optical trap at the 6 K-stage, from which we transport magnetically trapped, ultracold rubidium clouds at 100 μK to the 1 K-stage. The 1 K-stage has a sufficiently large volume (several liters) to accommodate microwave cavities, such as coplanar waveguide cavities, and has convenient optical access for optical traps and laser beams for spectroscopic measurements. This experimental setup is currently being extended for studying the fully quantum regime of cold-atom superconductor hybrid systems.
In the Singapore group we are working in two directions. One is the exploration of superconducting atom chips using high-temperature superconductors and another is the development of coherent interfaces between superconducting circuits and ultracold atoms.
High temperature superconductors have various distinct properties when implemented as atom chips. First of all, the technical demands are lower due to the higher working temperatures, which can be reached with liquid nitrogen instead of liquid helium. Moreover, high temperature superconductors are type-II superconductors and allow the storage of magnetic fields in the remanent state. We have shown experimentally and in simulations, that these trapped fields can be used to generate novel traps for ultracold atoms.225,226 Ramping a magnetic field perpendicular to a planar structure of YBCO we were able to generate various magnetic traps for cold atoms [see Fig. 18(a)].227 These traps can be generated either by using external magnetic fields together with vortices or in a completely self-sufficient way, where the trap is solely created by vortices. In the latter case, low noise potentials can be generated, as there is no technical noise coming from external power sources and the noise coming from the movement of vortices is expected to be an order of magnitude less than Johnson noise in normal conductors.84
Another property of vortices in superconducting thin films is that their distribution can be manipulated with light. Heating parts of the superconductor will result in a force on the vortices, which shifts the position of the vortices and consequently changes the vortex distribution. We have used this effect to generate various trap patterns with a thin square of superconducting YBCO, using light patterns generated by a spacial light modulator.228 The advantage of this technique is that multiple trap geometries can be generated with the same chip architecture in-situ, without the need of changing the chip and breaking the vacuum of the ultrahigh vacuum chamber.
Aside from using superconducting chips exclusively to manipulate cold atoms, we are also working on interfaces between cold atoms and superconducting qubits fabricated on the superconducting chip. What we envision here is the coherent transfer of quantum states between cold atoms and qubits made of superconducting integrated circuits. These hybrid systems will have many application, like the transduction of quantum states between the microwave and optical regime or the creation of universal quantum computing devices.
As mentioned before in this article, the practical implementation of a hybrid atom-superconducting qubit system is technically challenging. In Singapore we decided to bring cold atoms inside the dilution refrigerator by magnetically transporting them from a room temperature vacuum chamber directly to the mK stage of the refrigerator. With this technique we are able to bring clouds of 5 × 108 87Rb atoms close to the mK stage, at a base temperature of 70 mK.224 Trapped inside the mK stage, the atomic cloud exhibits an exceptional lifetime of 13 minutes, which is a promising starting point for future experiments.
In order to couple atoms and superconducting circuits a few scenarios are possible, which can be categorized in indirect and direct coupling. Also, the state of the atoms, i.e., ground state or highly excited (Rydberg), will have a significant influence on the experimental parameters. When indirectly coupled, the qubit and atom are individually coupled to a resonator, which mediates the interaction. In this case the coupling of the resonator to the qubit is easily implemented and can reach the strong coupling regime. Coupling ground state atoms to a planar resonator is an ambitious task. It was shown that the coupling strength of a single atom is only 40 Hz at a resonator-atom distance of 1 μm.229 In order to reach strong coupling one consequently needs to collectively couple an ensemble of 106 atoms to the waveguide, which is experimentally challenging. Using Rydberg states can considerably relax these requirements. We have shown that for Rydberg states strong coupling can be achieved with even a single atom.90,230 The strong coupling can even be reached with atom-resonator distances of tens of micrometers, when using the fringe field of the capacitive part of the resonator to couple the atom.
When using Rydberg atoms, even directly coupling of atoms to charge qubits can be realized. A neutral atom placed inside the gate capacitor of a charge qubit acts as a dielectric medium and affects the gate capacitance, resulting in a modulation of the charge-qubit energy bands. Moreover, the local quasi-static electric field strongly depends on the charge-qubit state, leading to different DC Stark shifts of atomic-qubit states. We have shown that in such a setup quantum states can be transferred between the two qubits and CNOT and Hadamard gates can be realized.231 Schemes for Rydberg atoms interacting with flux qubits have been theoretically proposed to realize quantum memories.232
We think that we now have the tools at hand to interface cold atoms with superconducting circuits. In the near future we would like to first couple atoms to 3D transmons, see Fig. 18(b). For this we designed and tested superconducting 3D cavities that have free space access for the transport and optical manipulation of cold atoms. First experiments to transport atoms inside the cavity are under way. At the same time we are developing our own fabrication for superconducting qubits. First chips have already been fabricated and tested. With both systems at hand we can then go forward to build hybrid systems of cold atoms and superconducting circuits.
D. Concluding remarks and outlook
In this review we have described applications of atom chips in atomtronics, precision sensing and quantum information. We illustrated the state of the art in these topics and touched upon future prospects and utilizations. In this zoomed-in view, we omitted many other excellent activities in the field, due only to unavoidable space limitations. Here, we would like to bring up other achievements that will shape the future of the atom chip platform.
Most of the experimental studies described above used bosonic rubidium atoms. In fact, many other species, like fermions, molecules and ions are used in atom chips.105 Fermions are another one of the fundamental building blocks of matter and therefore highly interesting objects to study, including low dimensional physics, the interaction of fermions with different species, or spin physics.233
Molecules, as the bridge between physics and chemistry, are an additional compelling candidate for many studies. Implementations range from fundamental science, like the measurement of the electric dipole moment and parity violation, to applied science in quantum processing. A “Lab on a Chip” for molecules is thus a sought-after goal. Recently, the trapping of simple molecules on microchips was realized,234,235 opening the way for many interesting investigations.
Trapped ions are one of the most promising candidates for practical quantum computing. In order to control and measure a large number of ions it will be necessary to fabricate surface-electrode traps on miniaturized microchips. The development and integration of these chips is currently ongoing and will be a major part in the future development of scalable quantum computer architectures with ions.236
Using the wave nature of atoms, atom chips will in future be used as precise sensors for material research and fundamental science. So-called “quantum gas microscopes” have been developed for room-181 and cryogenic237-temperature environments and are ready to be used in the nontrivial studies of unique materials. At the same time, matter waves are being employed for precision measurements in atomic interferometers. By analogy to the optical interferometer, the splitting and recombination of matter waves on atom chips are, for instance, being used to test theories in quantum thermodynamics,238 quantum many-body physics,239 and find applications in gravitational sensing.240
Intimately connected with precision sensing is the field of fundamental science. Many studies will be possible with atom chips, including tests of the Weak Equivalence Principle,241 interactions of antihydrogen with matter and gravity,242 non-Newtonian gravity, and the search for a fifth fundamental force.
All these examples show that atom chip technology has a bright future ahead. Combined with further integration and miniaturization, atom chips will play a role in many areas, both in fundamental research, as well as practical measurements.
All groups are very thankful to their colleagues who have been working with them on the experiments and their interpretation.
The BGU work has been supported by the Israel Science Foundation, the Deutsche Forschungsgemeinschaft German-Israeli DIP program, the FP7 program of the European Commission, the Israeli Council for Higher Education, and the Ministry of Immigrant Absorption (Israel).
The Tübingen research team gratefully acknowledges financial support from the Deutsche Forschungsgemeinschaft through SPP 1929 (GiRyd) and through DFG Project No. 394243350 and 421077991.
The Singapore team is grateful for administrative assistance and financial support from the Centre for Quantum Technologies (CQT) in Singapore.
V. QUENCH DYNAMICS OF INTEGRABLE MANY-BODY SYSTEMS
N. Andrei, C. Rylands
The study of nonequilibrium quantum physics is currently at the intellectual forefront of con- densed matter physics. One-dimensional systems in particular provide an exciting arena where over the last decade significant advances in experimental techniques have allowed very precise study of an array of nonequilibrium phenomena and where a number of powerful theoretical tools were developed to describe these phenomena. Here we give a brief account of a few systems that are described by one dimensional integrable Hamiltonians, the Lieb-Liniger model and the Heisenberg chain and how integrability gives access to the study of some of their local and global nonequilibrium properties.
While the principles of equilibrium statistical mechanics are well understood and form the basis to describe a variety of phenomena, there is no corresponding framework for the nonequilibrium dynamics, although efforts to fully understand the underlying principles extend back to Boltzmann and beyond. Solving particular models numerically or analytically and comparing to experiments may illuminate bits of the puzzle.
Here, is an extended version of talks given by the first author at Atomtronics 2019 at Benasque where some aspects of the questions were discussed. It is based on a review article243 written with Colin Rylands and builds on work carried out with several collaborators: Deepak Iyer, Garry Goldstein, Wenshuo Liu, Adrian Culver, Huijie Guan and Roshan Tourani to whom we are very grateful for many enlightening and useful discussions.
A. Quench dynamics
A convenient protocol to observe a system out of equilibrium is to prepare it in some initial state , typically an eigenstate of an initial Hamiltonian Hi, and then allow it to evolve in time using another Hamiltonian, H for which is not an eigenstate.244–246 One then follows the correlations of local observables,
as they evolve. One may be interested to know what new properties characterize the system, whether a dynamical phase transition occurs at some point in time247 or how its entanglements evolve. A particularly important question that arises in this context is whether the system thermalizes. Namely, can the system act as a bath to a small subsystem, here the small subsystem is the segment that contains the local operators . In the long time limit (to thermalize it is necessary that where L is the size of the system, and v a typical velocity) one needs to show
with the final inverse temperature β determined by the initial energy, .
Also global properties are of interest. These are commonly studied via the Loschmidt amplitude (LA), the overlap between the initial state with its time evolved self, conveniently expressed using a complete set of energy eigenstates, :
and its Fourier transform,
which measures the work distribution done during the quench.248
B. Evolution under integrable Hamiltonians
We shall consider evolutions effectuated by post-quench Hamiltonians that are integrable, namely Hamiltonians admitting a complete set of eigenstates and eigen-energies En given by the Bethe ansatz. The ability to obtain these follows from the existence of an infinite set of local charges, , that commute with the Hamiltonian and constrain the time evolution leading to a generalized Gibbs ensemble with the final inverse temperatures βn determined by the initial values .249
Thus some features of integrable time evolution are nongeneric. It turns out however that many features observed in integrable models can also be observed when integrability is broken. An example is the “dynamical fermionization” of repulsively interacting bosons in the integrable Lieb-Liniger Hamiltonian, discussed below. We showed this feature can be also observed in the bose-Hubbard model, the lattice version of the Lieb-Liniger model, which is not integrable.250 Further, many systems, in particular ultracold atom systems, are actually described by integrable Hamiltonians and can therefore be studied as such. Here we discuss two of them.
1. The Lieb-Liniger model
The model describes systems of ultracold gases of neutral bosonic atoms moving in one dimensional traps and interacting with each other via a local density interaction of strength c which can be repulsive c > 0 or attractive c < 0. Aside from being an excellent description of the experimental system, it is one of the simpler Hamiltonians for which there exists an exact solution via Bethe ansatz. The Lieb-Liniger Hamiltonian reads
(setting ). Here create and annihilate bosons of mass m. The exact N-particle eigenstate is given by251,252
Here is the two particle scattering matrix, is the phase shift. The single particle momenta kj are unrestricted in the infinite volume limit while with periodic boundary condition on a line segment L they must satisfy the Bethe ansatz equations: , with the integers ni being the quantum numbers of the state. The single particle momenta are related to the conserved charges by , in particular the energy is given by, .
This set of eigenstates allows the study of time evolution through the partition of the unity, . Here is a normalization factor.
In terms of the partition identity, the time evolved wavefunction is given by,
with the initial state encoded in the overlaps . These overlaps have been studied by many groups, e.g.253 and are typically very difficult to calculate. Once these overlaps are known they can be put in exponential form and combined with matrix elements of a given operator to yield a quench action which is typically evaluated in the saddle point approximation.254
Beyond overlaps: One may get around the difficulty of computing overlaps by choosing an alternate form of the partition identity obtained by exchanging the ordering in momentum space for ordering in coordinate space leading to the Yudson representation of the partition of the unity,255,256 or equivalently choosing appropriate trajectories for integrating over the momenta, see250
Here we have introduced the notation (notice the parenthesis replacing the ket) to denote the Yudson state,
with denoting a Heaviside function which is nonzero only for the ordering . The Yudson state is simpler to work with than the full eigenstates of the model and its overlaps with the initial state can be readily calculated, particularly if the initial state is ordered in coordinate space.
The domain wall initial state: As an example we consider an initial state in the form of a domain wall and quench it with c > 0 Lieb-Liniger Hamiltonian. Its time evolution can be studied analytically and several interesting phenomena will be shown to emerge:
Nonequilibrium Steady state (NESS)
RG flow in time
Evolution along space-time rays
Hanbury Brown-Twiss effect
The initial state, as depicted in Fig. 19, consists N cold atom bosons held in a very deep optical lattice of length L with and held fixed. The lattice site , are filled with one boson per site in the left half of the lattice: j = − ∞ to j = 0, and none in the half to the right,
The quench consists of suddenly releasing the trap and allowing the bosons to interact and evolve under the Lieb-Liniger Hamiltonian. Time evolving the system and using the Yudson representation we find,
When the lattice is removed the gas expands and the particle density will become nonzero between the lattice sites and also to the right of the domain wall. In the vicinity of the domain wall particles will begin to vacate the left hand side of the system and populate the right hand side, see Fig. 20. The effects of this quench can only be felt within a “light-cone” centered at the edge and determined by a finite effective velocity, which depends upon ω. On the right, the density will remain zero while to the left, , the average density will remain - the effects of the quench are still felt as the initially confined bosons will expand and begin to interact with each other.
We first examine the local portion of the quench around the domain wall. Since to the left there is an infinite particle reservoir and to the right an infinite particle drain the system will never equilibrate, however at long times a nonequilibrium steady state (NESS) consisting of a left to right particle current is established. This can be investigated by computing the expectation value of the density . Utilizing the known formulae for the matrix elements of the density operator with Bethe eigenstates257 this can be calculated exactly. To the right of the domain wall, at long times and to leading order in three regions emerge:256
where and . Far to the right we see that the density vanishes while closer to the light-cone some complicated crossover behavior occurs. Since the model is Galilean rather than Lorentz invariant the light-cone is not sharp giving instead this crossover regime. Most interesting is the region deep inside the light-cone in which the density becomes independent of x, t, signifying the appearance of the NESS, Nonequilibrium Steady State. We note that the particle density in this regime, , is reduced as compared to the equilibrium value, , the value a closed system would have reached after a quench from a domain wall state. This is a nonequilibrium effect of an open system which allows the bosons to expand further to the right in response to the repulsive interactions among the bosons. It follows from the order of limits with the size of the system L satisfying , to be contrasted with the behavior in a closed system, with the opposite order of limits. Within this region all local properties of the system can be calculated by taking the expectation value with respect to this NESS, where can be determined by taking the appropriate limit of (23).
On the left portion of the lattice we are outside the light-cone, the system is unaffected by the domain wall portion of the quench and the lattice translational invariance is restored. At long times the density within this region is
which describes small oscillation about a uniform density of .
This result coincides with what one would expect for a quench from a lattice initial state of the Tonks-Girardeau (TG) gas, the limit of the LL model. To understand this one should go beyond the density and compute the normalized noise correlation function where
This correlation function is related to the Hanbury Brown-Twiss effect and will detect the nature of the interactions between particles, a peak indicating bosons while a dip indicates fermions.258,259 Computing the noise correlation function by inserting two copies of the identity and evaluating the integrals at long time by saddle point method260 one finds it becomes a function only of the ray variables (measured with respect to see Fig. 20). For sufficiently long times a distinct fermionic dip is seen for arbitrary c > 0 while c = 0 shows a bosonic peak, the turn over to the dip occurring on the time scale, , see Fig. 21. This turn over results from an increase in time of the effective coupling constant c - starting from any initial repulsive value it will flow to strong coupling in the long time limit.261,262 This follows elegantly from the Yudson representation of the time evolving wave function:260 rewriting the dynamic factor in Eq. (18), , as , we note it tends to upon rescaling . Therefore, the product of bosonic fields with the dynamic factors, , behaves fermionically. The physical argument underlying the mathematical manipulations is also simple. In the long time limit only the slow bosons remain around in the noise correlation function and they interact via the effective S-matrix . The system in the long time limit will then behave as if it consisted of noninteracting fermions. This dynamical fermionization, the development of fermioniclike correlations, was subsequently observed in experiment both in the integrable Lieb-Liniger system (the Weiss group 2020) and previously in the corresponding lattice version, the Bose-Hubbbard model (the Greiner group 2015).263
The flow of the coupling naturally leads to the concept of renormalization group (RG) flow in time t. By analogy with conventional RG ideas, increasing time plays the role of reducing the cut off with being a strong coupling fixed point. For comparison we recall that in the usual RG picture c has scaling dimension 1 and so also flows to strong coupling. Subsequently, similar behavior was also seen in strongly coupled impurity models.264,265 Extending the dynamical RG analogy one can envisage that other Hamiltonians close to the Lieb-Linger will flow close the neighborhood of the same strong coupling fixed point, prethermalize in other words, only to end up thermalized on longer time scales if the model is not integrable, see Fig. 22. An example is provided by the lattice version of the Lieb-Liniger model, the nonintegrable Bose-Hubbard model which also exhibits dynamical fermionization.250
We turn now to study the global properties of the post quench system through the Loschmidt amplitude (15) and the work distribution function (16) focusing on the experimentally relevant case of a cold atom gas initially held in a deep optical lattice which is then removed entirely in the quench, see Fig. 23. The system is translationally invariant and described by the Lieb-Liniger model.
We consider N bosons on a circle of length L initially described by the state (22) with N consecutive sites filled, with so that the unfilled part of the lattice is taken to be much larger than the filled portion to avoid complications arising from the boundary conditions. Employing the Yudson resolution of the identity, the Loschmidt amplitude can be determined to be,266
where and is a Heaviside function. Using the same expansion as before the Fourier transform of this can be explicitly found and analytic expressions for the work distribution, obtained. We plot this for both noninteracting and strongly interacting bosons in Fig. 24 for different particle number and see some commonalities as well as striking differences. Notice that the average work in both cases is the same, as is the large behavior. The former statement can be understood from the fact that bosons are initially in nonoverlapping wavefunctions and . In comparison, the small behavior is strongly affected by the presence of interactions. Large resonant peaks are present in the interacting work distribution and can be attributed to the scattering of strongly repulsive excitations in the post quench system. Those peaks which are closest to involve fewer scattering events while those W = 0 involve more. As the particle number is increased these fluctuations are suppressed like .267,268 For large systems of bosons the most interesting behavior therefore occurs in the region of where the effects of the interaction are most keenly felt. In this region it can be shown that the distribution decays as a power law with the exponent drastically differing between the free and interacting cases. For the former we have whereas in the latter it is , the presence of interactions in the system causing a dramatically faster decay of the work distribution. Behavior such as this will be seen in Sec. V B 2 also when the excitations are gapped as well as interacting.
We can use our knowledge of to investigate the global behavior of the post quench system. As a consequence of the large W agreement between the distributions for the interacting and noninteracting systems we can determine that at short times is independent of the interactions. This corresponds to the initial period of expansion from the lattice in which the particles do not encounter one another. On the other hand, small W behavior provides insight to the long time dynamics, the power law decay of near the origin translating to the long time power law decay of the LE. Fourier transforming the distribution for free bosons we find that as while in the interacting case we have instead , a much faster decay. We attribute this dramatic difference in the decay away from the initial state to the fact that the large repulsive interactions acting on each other forcing them to spread out into the one dimensional trap, thereby decreasing their overlap with . We should note that this is true regardless of the strength of the interactions and highlights the strongly coupled nature of even weakly interacting systems in low dimensions. As we saw earlier, in the long time limit any repulsive coupling flows in time strong coupling, therefore the exponent is independent of the initial strength of c, in the TG limit () one finds the same power law behavior at long times as for the finite c case. This is the dynamical fermionization discussed in Sec. V A.
The attractive regime is of significant interest. The properties of the attractive model both in and out of equilibrium are much less studied than its repulsive counterpart. This dearth of theoretical results stems from the increased complexity of the Bethe ansatz solution in the attractive model. When c < 0 the model supports bound states and the ground state consists of a single bound state of all N particles.270 While the eigenstates given by (18) remain valid, complex values of k which correspond to bound states are allowed. The resolutions of the identity (20) also remains formally valid provided these complex valued solutions are accounted for. A stumbling block however remains as the normalization of the Bethe states in the attractive regime is not known in closed form.
The contours of integration, Γ, lie on the real line for repulsive interactions and are spread out in the imaginary direction for the attractive case with Im .
Making use of this here in conjunction with the same expansion we find that the work done in the attractive regime separates into two contributions,
The first term is the contribution from particles which do not form bound states, it is identical to the expression in repulsive case only now c < 0. The major difference imposed by this is that the effective distance between the particles is smaller , the attractive interactions promoting the clustering of particles.
The simple analytic continuation to negative coupling of the first term is reminiscent of the super Tonks-Girardeau gas. This highly correlated state of the LL (Lieb-Liniger) model is created by preparing a repulsive LL gas in the Tonks-Girardeau limit, and then abruptly changing the interaction strength from the being large and positive to large and negative. The result is a metastable nonequilibrium state which exhibits enhanced correlations. Many of the properties of this state can be shown to emerge from a simple analytic continuation of the coupling to large negative values. In effect the negligible overlap of each particle of our initial state mimics the density profile of the TG gas and so super-TG like behavior is not unexpected. We should stress that the expression (28) is valid at arbitrary negative values c and so not limited to super-TG regime.
The second term is entirely different. It is due to the bound states and is calculated by deforming the contours in (28) to the real line and picking up contributions due to the poles at present in (18). An n-particle bound state can be shown to contribute with factors from multiple bound states being multiplicative.
This exponential factor means that the probability that the initial state transitions to one containing bound states is highly suppressed and in the true super-TG limit vanish entirely. Despite this, for finite the bound states have a strong signature in work distribution function. Since forming a bound state will lower the energy of the system270 the work distribution becomes nonvanishing at negative values of W. There is a nonzero probability that work can be extracted from the system. Importantly this does not violate the 2nd law of thermodynamics as the average work remains positive .271,272 In fact, it has been observed recently that the probability of extracting work from a single electron transistor can be as high as 65% whilst still satisfying the 2nd law.273
To see this we examine the leading term of which arises due to the formation of a single two particle bound state
which is nonvanishing for . Determining the full bound state contribution is a straightforward yet involved calculation that we we will not deal with here.
2. The XXZ Heisenberg spin chain
The XXZ Heisenberg chain provides another example of an experimentally relevant integrable model. The Hamiltonian
models a linear array of spin interacting via anisotropic spin exchange. The isotropic case Δ = 1 is SU(2) invariant and enjoys the distinction of being the first model solved by Bethe by means of the approach that bears his name.274 The generalization to the anisotropic case was given by Orbach.275 The eigenstates are again characterized by a set of Bethe momenta describing the motion of M down-spins in a background of N – M up-spins, and are given by
where mj the position of the jth down-spin is summed from 1 to N (the length of the chain), and the S-matrix is given by
The Heisenberg chain exhibits a complex spectrum which includes bound states in all parameter regimes. To carry out the quench dynamics for the model one needs to construct the appropriate Yudson representation and use it to time evolve any initial state.276 Here we display in Fig. 25 the time evolving wavefunction of two adjacent flipped spins in the background of an infinite number of unflipped spins and compare it to the experimental results (no adjustable parameters are involved.) The time evolution of the magnetization from an initial state of three flipped spins for different values of the anisotropy Δ is given in Fig. 26. We see that excitations propagate outward after the quench forming a sharp light-cone in contrast to the Lieb-Liniger model. The boundary of the light-cone arises from the propagation of free magnons which travel with the maximum velocity allowed by the lattice. Rays within the light-cone are the propagation of spinon bound states. As the anisotropy Δ is increased the bound states slow down and more spectral weight is shifted to them. Due to the integrability of (31) these excitations have infinite lifetime which prevents any dispersion of these features. The introduction of integrability breaking terms can therefore be expected to alter this picture, for example through spinon decay.277
C. Concluding remarks and outlook
In this chapter we have explored some aspects of the far from equilibrium behavior of integrable models. After a broad overview of the current status of the field we investigated some particular phenomena through a number of illustrative examples. We saw that the Bethe ansatz solution of the Lieb-Liniger and Heisenberg models provided us with a powerful tool with which to study both the local and global, nonequilibrium behavior of these strongly coupled systems. The quench dynamics of more complex models such as the Gaudin-Yang model279,280 describing multicomponent gases has also been accessed via the Yudson approach281 allowing the study of phenomena such a quantum Brownian motion or the dynamics of FFLO (Fulde-Ferrell-Larkin-Ovchinnikov) states.282,283 Similarly the quench dynamics of other models such as the Kondo and Anderson models are currently studied via the Yudson approach.284 They give access to such quantities as the time evolution of the Kondo resonance or of the charge or heat currents in voltage or temperature driven two lead quantum dot system.
These methods we discussed could be thought as being microscopic, starting from the exact eigenstates of the system. Recently these problems have been studied from a macroscopic perspective by combining integrability and ideas from hydrodynamics.285 Generalized hydrodynamics (GHD) provides a simple description of the nonequilibrium integrable models on long length scales and times. It has been utilized in studies of domain wall initial states in the Lieb-Linger and the emergence of light-cones in quenches of the XXZ model.286,287 This method allows the incorporation integrability breaking effects within the formalism, but is limited to “Euler scale” dynamics. It would be of great interest compare the results and expectations of GHD with the methods and results presented here to further understand the limitations of both the microscopic and macroscopic approaches.
VI. NONEQUILIBRIUM PROTOCOLS FOR ONE DIMENSIONAL BOSE GASES IN ATOMTRONIC CIRCUITS
L. Piroli, A. Trombettoni
A promising line of research in atomtronics is the realization of configurations where several waveguides in which ultracold atoms move are merged to form circuits.5 Among the challenges one has to face, an important one is the tailoring of the circuits in a way to reduce transverse instabilities during the dynamics of ultracold matter wavepackets.28 This would allow for the possibility of stable motion of the matter wavepackets across the whole circuit, including the passage through junctions and in the regions where the waveguides composing the circuit have to bend. Since transverse instabilites are suppressed in one-dimensional geometries, the lines of research of atomtronics and one-dimensional ultracold atoms have been developing tight connections in the last decade. On the one hand, the study of circuits made of one-dimensional waveguides open new directions of investigation for the community working on one-dimensional integrable systems, such as the study of junctions of one-dimensional waveguides: an example is given in,63 where a junction of three Tonks-Girardeau gases is studied, and connected to the literature of coupled/intersecting nanowires. On the other hand, the amount of available results in the field of one-dimensional integrable models provides an extremely useful basis for the characterization of ultracold matter wavepackets on such geoemtries, which has been at center of significant discussions in the Atomtronics@Benasque conference series.5
One-dimensional interacting bosons are well described by the integrable Lieb-Liniger model, which was extensively studied since its introduction in the sixties, also in connection with other one-dimensional integrable systems. Extensions and generalizations of the Lieb-Liniger model may apply to one-dimensional fermionic systems and mixtures, including Bose-Bose and Bose-Fermi mixtures. Therefore, the field of atomtronics circuits made of (possibly connected) one-dimensional ultracold systems is a natural arena to apply such a body of knowledge, and at the same time calls for new ideas and investigations using integrability techniques.
One-dimensional systems provide per se an exciting arena where, over the past decade, significant experimental technical advances have allowed for very precise studies of a series of nonequilibrium phenomena. At the same time, a number of powerful theoretical tools were developed to describe them. The study of one-dimensional systems plays a role as well in the field of atomtronics and in particular in atomtronics circuits, where matter-wave packets can be controlled and moved. When the transverse dimensions of the waveguides in which atoms move are small enough to create one-dimensional tightly confined traps and the energies involved are negligible with respect to the excitation energies of transverse degrees of freedom, then one enters the one-dimensional regime. Ultracold bosons are then effectively described by the Lieb-Liniger model,288–290 belonging to the family of integrable theories. In such one-dimensional regimes quantum fluctuations play a prominent role and a general issue is whether and for what applications such one-dimensional features hamper or at variance make it easier to realize atomtronics tasks.
Here we give an account of some interesting properties of the Lieb-Liniger model and how integrability gives access to the study of some of its local and global nonequilibrium properties. The following contribution focuses on the theoretical study of two of the more useful protocols to control the quantum dynamics of the Lieb-Liniger model: (i) integrable dynamics after a quench; (ii) Floquet engineering. They are relevant for atomtronics applications, both for the possibility to have quenches and time-periodic potentials as a tool to control the dynamics and induce desired dynamical regimes and for the remarkable progress in experimental techiniques enabling the possibility to vary interaction strengths, geometry of the trap and the time-dependence of the potentials acting on the atoms in one-dimensional ultracold systems.288–290 In the present contribution, L.P. wrote Sec. VI A, while A.T. wrote Sec. VI B.
A. Quench dynamics in the Lieb-Liniger model
In the early noughties, a series of cold-atomic experiments contributed to the emergence of a growing theoretical interest in the nonequilibrium dynamics of isolated quantum integrable systems.125,291 For instance, in the famous “quantum Newton's cradle” experiment,292 out-of-equilibrium arrays of trapped one-dimensional (1D) Bose gases were shown not to reach thermal equilibrium within the accessible time scales. This peculiar behavior was attributed to the approximate integrability of the system: indeed, in the idealized situation where longitudinal confining potentials are neglected, a 1D gas of N bosons with mass m and pointwise interactions can be described by the integrable Lieb-Liniger Hamiltonian.251 Denoting by L the length of the system, the Hamiltonian can be written as
where are bosonic creation and annihilation operators satisfying canonical commutation relations. Here, the interaction strength is related to the one dimensional scattering length through 293 and can be varied via Feshbach resonances294 to take either positive or negative values.
Given its relative simplicity and experimental relevance, in the past decade a large number of studies have focused on the nonequilibrium dynamics in the Lieb-Liniger gas, especially within simplified protocols such as the one of a quantum quench:295,296 in this setting one considers the ground state of some Hamiltonian (c0 denotes an internal parameter), which is suddenly changed at time t = 0 by an abrupt variation . These studies have played an important role for the development of a general theory of integrable systems out of equilibrium.297 In this section, we provide a review of a selected number of them, focusing exclusively on the simplest case of homogeneous settings (see Sec. VI C for recent further developments in the presence of confinement potentials and inhomogeneities).
1. The quench problem
Physical intuition suggests that after a quench an extended system should act as an infinite bath with respect to its own finite subsystems, and that local properties should relax to stationary values described by a thermal Gibbs ensemble. While for generic models this picture turns out to be correct,249,298,299 a quite different scenario emerges in the presence of integrability, due to the existence of an extensive number of local conservation laws which strongly constrain the dynamics. In this case, it was proposed in Ref. 300 that the correct post-quench stationary properties are captured by a generalized Gibbs ensemble (GGE), which is written in terms of all higher local conservation laws beyond the Hamiltonian.300–302 It was later discovered that quasi-local conservation laws must also be taken into account303–308 and the validity of the GGE is now widely accepted.
Despite the established conceptual picture, computations based on the GGE are hard, and more generally the characterization of the post-quench dynamics remains extremely challenging in practice. In order to explain the difficulties involved, it is useful to consider the time evolution of a physically relevant observable for the 1D Bose gas, namely so-called pair correlation function309
where is the particle density, with L the system size, while is the state of the system. Physically, g2 quantifies the probability that two particles occupy the same position. For a quantum quench, we have the formal expression (setting )
Here we denoted by , En the energy eigenstates and eigenvalues, respectively, while is the state of the system evolved at time t after the quench. For the Lieb-Liniger model the Bethe ansatz257 is a very efficient tool to obtain most of the ingredients appearing in Eq. (36), including the matrix elements of the local operator .310,311 However, due to the complicated form of the energy eigenfunctions, there appears to be no simple way to compute the overlaps for general initial states. Furthermore, Eq. (36) involves the evaluation of a double sum over all the eigenstates of the Hamiltonian, which is currently out of reach in most of the physically interesting situations.
Due to the above difficulties, initial studies in the Lieb-Liniger model were restricted to the limit of either vanishing312,313 or infinitely repulsive post-quench interactions,314–320 where the Hamiltonian can be mapped onto free fermions through a Jordan-Wigner transformation. While these works already made it possible to explore in some detail interesting phenomena such as local relaxation317 and “light-cone” spreading of correlation functions,312,317 it remained as an open problem to provide predictions in the case of finite values of the interactions.
2. The quench action
A conceptual and technical breakthrough came with the introduction, by Caux and Essler, of the so-called Quench Action method,254,321 which proved to be a powerful and versatile approach to the quench dynamics in integrable systems (other methods, that will not be discussed here, have also been developed, including a Yudson-representation approach, which is also suitable to study inhomogeneous initial states, see Refs. 260, 266 and the contribution of N. Andrei and C. Rylands).
It is well known that, in the thermodynamic limit, each eigenstate of an integrable system is associated with a distribution function , where λ are the quasi-momenta of the (stable) quasi-particle excitations.257 Based on physical arguments, it was proposed in Ref. 254 that this description could be exploited to replace the double sum in Eq. (36) with a functional integral over all distribution functions . This approach is particularly powerful to investigate the late-time limit, for which one can write (in the thermodynamic limit)254,321
where denotes an eigenstate corresponding to the distribution function . Here we introduced the “Quench Action” , which can be determined based on the knowledge of the overlaps . While, as we have already mentioned, it is not known how to obtain these in general, it turned out that they can be computed in several interesting cases.322–343
Given , the functional integral can be treated exactly by saddle-point evaluation, so that the r.h.s. of Eq. (37) can be replaced by , where . Crucially, the saddle-point distribution function determines all the post-quench local expectation values (which can be explicitly computed via exact Bethe ansatz formulas310,344–347) and thus represents an effective characterization of the late-times steady state.
The Quench Action approach was first applied in the Lieb-Liniger model for quenches from zero to positive values of the interactions, .348 It was found that the steady state displays quantitative different features from a thermal state, unequivocally proving the absence of thermalization. The same approach also allowed for the computation of the full time evolution of g2349 (see also350,351) unveiling a quite general power-law decay to stationary values for local observables, and for a detailed study of the statistics of the work performed by the quench.352–354
3. Quenches to the attractive regime
In the case of quenches to repulsive interactions, the late-time steady state appears to display features that are only quantitatively different from those observed at thermal equilibrium.348 In this respect, an even more interesting picture emerges for quenches to the attractive regime. These were investigated in Refs. 355 and 356 where the formalism of Refs. 254 and 348 was employed to study interaction quenches of the form .
The main results of these works are arguably the prediction of the dynamical formation of n-boson bound states with finite densities Dn, and the characterization of the corresponding distribution of quasi-momenta . Interestingly, it was shown that the value of n for which the density Dn is maximum decreases with the rescaled interaction . Although this result might appear counter-intuitive, there is in fact a simple physical interpretation: in the attractive regime, the bosons have a tendency to form multiparticle bound states. However, in the quench setup the total energy of the system is fixed by the initial state, while the energy of n-particle bound states increases, in absolute value, very rapidly with γ and n.355 Therefore, n-particle bound states cannot be formed for large values of γ, while they become accessible as γ decreases.
We note that the structure of the stationary state predicted in Refs. 355 and 356 is qualitatively very different from the super Tonks-Girardeau gas, which is obtained by quenching the one-dimensional Bose gas from infinitely repulsive to infinitely attractive interactions.315,357–362 Indeed, the latter features no bound state, even though it is more strongly correlated than the traditional Tonks-Girardeau gas, as has been observed experimentally.359 The findings of Refs. 355 and 356 are thus also interesting because they show that the physics emerging at late times after a quench depends qualitatively on the initial state of the system.
Importantly, the formation of bound states after the quench have consequences on the local correlation functions. For instance, the value of g2 at large times is always greater than 2, and increases with .356 This is displayed in Fig. 27, and is once again qualitatively different from the case of the super Tonks-Girardeau gas. We note that these results are consistent with subsequent numerical calculations reported in Ref. 363 and based on the method developed in Refs. 364 and 365.
B. Floquet Hamiltonian for the periodically tilted Lieb-Liniger Model
Another promising protocol for inducing and controlling interesting instances of quantum dynamics is provided by the Floquet engineering. In this scheme the original Hamiltonian—in this section the Lieb-Liniger model—is subject to a time-periodic driving V. The Floquet Hamiltonian control then the time dynamics of the system when observed at stroboscopic times, i.e., at times multiples of the period of V. The general goal is to design V in a way that the Floquet Hamiltonian is the one inducing the desired quantum dynamics.
In general, when a periodic driving acts on an integrable model, then the resulting Floquet Hamiltonian is nonintegrable. In this section, we consider the case of the Lieb-Liniger model subject to a potential periodic in time and linear in space, which we refer to as a periodic tilting.366 The Floquet Hamiltonian of the integrable Lieb-Liniger model for such linear potential with a periodic time–dependent strength is integrable and its quasi-energies can be determined using well known results for the undriven Lieb-Liniger model.
We pause here to comment about the relevance of the investigation of Floquet engineering, and periodic tilting in particular, starting from the Lieb-Liniger Hamiltonian for atomtronics applications and perspectives. Controlling matter-wave dynamics in waveguides and other atomtronics circuitry and components is in general an interesting perspective to be discussed and studied. A time-independent potential linear in space induces a motion in the atomtronics devices, and a time-dependent periodic tilt can be used to control the motion across, to and fro, a circuit. As discussed in the Introduction, to reduce trasnverse excitations it may be convenient to use and merge one-dimensional waveguides, and a natural question is what is the effect of a time-dependent periodic tilting in such one-dimensional systems.
We then consider the periodic tilting
with f(t) a periodic function with period T. The Lagrangian density of the system is
where denotes the hermitian conjugate of the first term and .
When the potential V is time-independent with f(t) constant, then it is well known that one can gauge away the potential linear in space by moving to the center-of-mass accelerating frame. Notice that this property is valid in any dimension and also for interacting systems, as long as the two-body interaction depends only on the relative distance (for a pedagogical presentation see, e.g., Ref. 367).
Let now come back to the case of f(t) periodic in time. Proceeding as one does for the single-particle and the two-particles cases,366,368 one can solve the Schrödinger equation of the many-body interacting model. To this aim, one introduces the following gauge transformation:
with the functions and to be suitably determined in order to gauge away the potential term V from the Lagrangian density when rewritten in terms of the field .
The functions ξ and θ are determined as it follows. We start by imposing
We now make the ansatz
finding the conditions