Deterministic generation of highly squeezed GKP states in ultracold atoms

Fault-tolerant quantum information processing will require fast low error-rate operations on qubits.¹ This ensures large algorithms can be fully implemented without the exponential buildup of errors rendering the final result useless. However, current systems suffer from relatively high physical error rates.² This must, at least in the near term, be offset through the use of error-correcting codes.³ These rely upon a larger state space upon which a smaller subspace of information is encoded. This proceeds either through the representation of a single logical qubit with multiple physical qubits or through the use of a bosonic oscillator, which itself describes a larger state space.

The Gottesman–Kitaev–Preskill (GKP) states⁴ are the canonical example of this latter case. These states have been shown to have the highest error-correction performance compared to similar encodings,⁵ and they have been shown to have utility in cluster-state quantum computing.⁶ GKP states consist of periodic delta functions evenly spaced in position space. The two encoded qubit states are identical, except that one is displaced by half the spacing period. This nominally allows for perfect error correction up to a displacement by a quarter of the inter-delta spacing. Finite approximations to these ideal infinitely squeezed states provide consequently approximate error correction.

The difficulty in preparing these states lies in the challenge of generating adequate squeezing to offset the overhead of the encoding and reduce the logical error rate. Furthermore, the control of these states is very sensitive to noise, for more accurate approximations, due to the large amplitude in phase space.⁷ Therefore, these states must be considered not only in their error-correction ability but also in the difficulty with which they may be implemented and controlled. Nevertheless, approximate GKP states have been experimentally demonstrated in ions,⁸ transmon qubits,⁹ and photonic systems.¹⁰ 

Here, we present a proposal for the realization of GKP states in ultracold atoms, specifically, arrays of single atoms trapped in the individual sites of a deep optical lattice potential. Atoms are excellent candidates for qubits, and a reprogrammable tweezer-based quantum information processor with error correction has recently been demonstrated,¹¹ where quantum information is processed via atom-atom interactions mediated by selective excitation of atoms into high-lying Rydberg states.¹² Our protocol can be straightforwardly adapted to atoms in optical tweezers,^13–17 but we focus here on single atoms in optical lattices. This is due to their straightforward scalability and controllability, even at the single-atom level.^18–20 In addition, optical-lattice-based systems have found use in an array of high-precision quantum technologies, including atomic clocks,^21,22 quantum simulators,^23,24 and inertial sensors.^25–28 Recent work has also demonstrated the production and density matrix reconstruction of atoms in optical lattices.²⁹ 

Ideal GKP states—grids of infinitely narrow lines spanning the entirety of the phase space—are physically unrealizable, as they require infinite energy. For this reason, a large variety of approximations of GKP states have been proposed. A number of these possess attractive qualities, such as the representation of approximate GKP states as sums of translated coherent states, which naturally admits an experimental construction in ions.⁸ However, these approximations are equivalent up to a redefinition of parameters.²⁹ Therefore, we choose, without loss of generality, an approximate definition of GKP states as a superposition of evenly displaced, squeezed, vacuum states of squeezing parameter Δ weighted by a Gaussian envelope of standard deviation 1/σ, with $Δ=−lnσ2/(1−σ4)$ as in the original paper.⁴ In phase space, this leads to a grid of finite-sized peaks localized around the origin.

The finite GKP state is then characterized solely by the level of squeezing as ζ = −10 log₁₀(σ²) dB.

Here, we find optimal controls that can be used to generate GKP states with ζ = 10 dB of squeezing with atoms in optical lattices. This level of squeezing is generally recognized to be the point at which such states are useful from a quantum information perspective.^31,32 These states may also be concatenated with a surface code.³³ This provides enhanced error correction performance at the cost of additional overhead and complexity. As a result, fault-tolerant quantum computing can be achieved at as low as 8.1 dB of squeezing,³⁴ assuming zero losses in the system.

We assume the atom is trapped in a single site of a deep 1D lattice potential, which is straightforward to experimentally prepare.^19,38 We then define, as in Ref. 39, our Fock states |n⟩ to correspond to the bound vibrational levels within a given lattice site, i.e., n = 0, …, N. Thus, the total number N + 1 of Fock states bound to a single lattice site directly affects the maximum obtainable squeezing of our resulting GKP states. Figure 1 shows, for a given squeezing level, the required lattice depth and resulting Fock basis size (i.e., the number of vibrational states bound to a given lattice site). The purpose of our phase modulation is then to control transfers between these Fock states to generate and manipulate the desired GKP states.

In our simulations, we initialize our atomic state in a given initial state |ψ(t = 0)⟩ (here, |0⟩, considering the generation of GKP states from the ground state of the lattice potential). We then generate an initial guess (seed) for our shaking function u(t) where 0 ≤ t ≤ T and use the symmetrized split-step method⁴⁰ to evolve the state using the time-dependent Schrödinger equation with the Hamiltonian in Eq. (4) and the aforementioned u(t). This results in a final state |ψ(t = T)⟩ = |ψ_F⟩ where the time T is varied; higher values of T typically lead to higher fidelity protocols. The optimization protocol then adjusts u(t) in an attempt to maximize fidelity.

We define our state transfer fidelity as $F=|⟨ψD|ψF⟩|2$⁠, where |ψ_D,F⟩ are the desired state and final state after a given shaking protocol u(t), respectively. In Fig. 1 and this work in general, our target fidelity is $F≥0.99$⁠; extensions beyond this will require more bound states and a deeper lattice, posing increasing experimental challenges. It remains to be seen what fidelities are required for atomic GKP states to be useful in quantum information protocols; finite fidelity likely manifests similarly to loss in photonic systems.

To find our controls, we use the GRAPE method⁴¹ with BFGS built into the QEngine C++ library.⁴² Furthermore, bandwidth and amplitude limits for u(t) are implemented via regularization and soft boundary terms in our optimization functional. Practically, we used a smooth low pass filter centered at 0.5 MHz to ensure the experimental viability of the controls,⁴³ in addition to limiting the shaking amplitude to no more than half a lattice site.

The experimental proposal requires atoms to be loaded into the ground state of an optical lattice generated by two lasers with identical wavelengths but varying phases. Ground-state loading can be achieved by either loading from a Bose–Einstein condensate or sideband cooling^44,45 atoms loaded into the lattice directly after laser cooling. Phase modulation can be achieved by changing the phase of one lattice beam relative to the other, e.g., by controlling the RF tone sent to an acousto-optic modulator (AOM). In this way, we can implement the shaking protocols required for GKP state generation and manipulation.

State verification uses a recent proposal for direct measurement of the Wigner function of atoms in an optical lattice potential.³⁹ This method that was shown to work even for anharmonic traps requires one to take advantage of the differential light shift between two hyperfine ground states of the atoms (e.g., the |F, m_F⟩ = |2, −2⟩ and |1, −1⟩ hyperfine states of rubidium²⁰) due to their differing vector polarizabilities. Similar methods of differential lattice manipulation have been used, e.g., to perform quantum walk experiments,⁴⁶ but the drawback is the lattice wavelength must be between the D1 and D2 lines of the alkali atom used, limiting atom lifetime in the lattice.

It is important to note that another method for Wigner function determination via density matrix reconstruction has been demonstrated experimentally.⁴⁷ This method relies on time-of-flight data, and thus, the stringent limits on lattice wavelength are relaxed. Moving to a far-off-resonant trap (e.g., at the common lattice wavelength of 1064 nm) will require more lattice power to achieve a given depth (as depth goes as $I/Δ̃$⁠, where I is the intensity of the lattice light and $Δ̃$ is the detuning from resonance).⁴⁸ However, the associated atom lifetime will be longer, as the photon scattering rate $Γ∝I/Δ̃2$⁠. We will, however, consider a lattice between the D1 and D2 lines of an alkali atom as a worst-case scenario for our proposal.

As a first step, we need to determine the lattice depth that gives us the required squeezing. From Fig. 1, we see that, to obtain 10 dB of squeezing, we require 24 basis states and a lattice depth of U ≈ 1500E_r. Figure 2 shows a contour plot of the lattice depth and scattering-limited lifetimes for Rb and Cs as a function of laser power and wavelength, restricted between the D1 and D2 lines. From the figure, we see for U = 1500E_r, the best rubidium lifetime is between 5 and 10 ms; however, for cesium, this is about six times larger, assuming the laser power is below 100 mW (10% of what is plotted in Fig. 2), which is readily achievable in the laboratory. However, there is a minor trade-off due to the increased wavelength of the cesium lattice that causes, in the conversion from scaled to unscaled units, the cesium protocols to be roughly a factor of two longer than the rubidium protocols, so the overall gain of a cesium system is only about three times that of rubidium. Note that for lighter alkali atoms, the achievable lattice lifetime will be lower, as the D1/D2 splitting increases with atomic mass. As such, cesium, the heaviest easily trappable alkali (although there are efforts focusing on the heavier, radioactive francium⁴⁹), is the best alkali atom for these experiments.

The modulation times for the time-optimal (i.e., lowest T where $F≥0.99$⁠) protocols considered here are 141 µs (239 µs) for the $|0〉→GKP0$ transitions in Rb (Cs) and 158 µs (270 µs) for the $|0〉→GKP1$ transitions. The resulting rubidium Wigner functions are compared with the ideal in Fig. 3; Wigner results for cesium can be found in Fig. 4 in  Appendix A. We also present example controls and their spectra in Figs. 5 and 6 in  Appendix B. Our protocols reproduce the ideal states with good fidelity; in particular, the grid-like signature of the GKP state is readily apparent.

When considering the viability of our protocols with respect to atom lifetime in the lattice, one must also consider the time required for the Wigner measurement protocol; in Ref. 39, the protocol duration was on the order of hundreds of μs; the protocol time depends on the differential trap depth, and thus, with a sufficiently large variation in the power between the different circular polarizations of the lattice light, we expect this protocol time should be maintained even at the high lattice depths required for these experiments. As such, we find that, without accounting for measurement time, our modulation protocols take ≈3% of the atom lifetime in the lattice, assuming a very conservative Rb lattice lifetime of 5 ms. Cesium protocols require a fraction of a percent of the atom lifetime. Including an (again, conservative) 0.5 ms Wigner measurement time still keeps us within ≤15% of the atom lifetime in the lattice.

Finally, we would like to remark on the scalability of this method and its feasibility in real systems, given the intrinsic dynamics of the non-equilibrium GKP states considered here. Empirically, we find our protocols require the lattice depth to be stable to within 0.1% before the fidelity is reduced to 90%. If we assume our atoms are trapped in a three-dimensional lattice with sufficiently large lattice beam waists in each dimension, the lattice depth variation across an atom cloud can be minimized. Even for modest beam waists of 150 µm, one can populate $∼20$ lattice sites in each direction before the fidelity drops appreciably, allowing for 8000 atoms to be simultaneously interrogated so that high-fidelity GKP states can be created along one Cartesian dimension. In addition, state stabilization protocols can be applied along the orthogonal dimensions during state preparation; thus, the population remains in the ground state in these dimensions; such multidimensional optimization is considered in Ref. 50 and is straightforward to include in our system as well. Cross-coupling between trap axes can be accounted for if atoms are well-prepared in the absolute ground state of a deep 3D lattice trap designed such that the spacing between adjacent energy levels in the other dimensions is so high that excitation cross-axis coupling is unlikely. Finally, we expect that state stabilization can be applied (via lattice modulation) post-GKP production so that the intrinsically non-equilibrium GKP state maintains its fidelity when computations are not being performed, similar to what is shown in Ref. 50. Similarly, tunneling between adjacent lattice sites (which becomes a problem for finite lattice depths and state populations in higher bands) can be mitigated via applying a small tilt to the lattice, inhibiting tunneling via Wannier–Stark localization, although this tilt must be small to mitigate Bloch oscillations and Landau–Zener tunneling.⁵¹ Thus, we expect our protocols to be readily scalable to considerable atom numbers and feasible to implement with current experimental technology.

In this work, we demonstrate optimal control protocols that drive an atom from the ground state of an optical lattice potential into the $GKP0$ and $GKP1$ states. We additionally present an experimental protocol for the generation, manipulation, and measurement of these states in optical lattices that can be readily achieved with current experimental technology. Our protocols require large lattice depths U ≈ 1500E_R, which poses challenges for atom lifetime and laser power; however, we show that even in our worst-case scenario of rubidium atoms and lattice light between the D1 and D2 lines, the proposed protocols can be implemented within a few percent of the atom lifetime in the lattice with powers $<1W$⁠. We expect that as error correcting protocols improve, the required squeezing level for GKP state utility will also decrease, allowing useful atomic GKP states to be generated with lower power requirements. In addition, while this paper focuses on GKP state generation, extensions to gate optimization are straightforward⁵⁰ and will be the focus of future work. Such gate optimization protocols must crucially account for the evolution of the state during gate operations, which can be done similarly to the state stabilization discussed in Sec. III.

The efficient experimental generation of these states will also require finding modulation protocols that are robust, e.g., to variations in the lattice depth or noise in the modulation; this can be done via adaptation of the methods discussed in Ref. 52. There is likely also utility in exploring constrained basis methods⁵³ for these problems, as such methods have been adapted to efficiently find state transfer protocols in other modulated-lattice systems.⁵⁴ These will allow for controls that can mitigate off-resonant heating effects,⁵⁵ analogously to what was shown in Ref. 56 for phase-modulated shallow lattices. This will also reduce the beam waist (and thus power) requirements to achieve the required lattice depths. Cavity-enhanced lattices⁵⁷ can also lower power requirements.

Furthermore, while our proposal centers on atoms trapped in optical lattice potentials, similar protocols can also be realized in optical tweezer potentials. Arrays of tweezer potentials do not pack atoms as tightly as lattices can, but they have the advantage that each lattice site can be more easily independently controlled, allowing for flexible continuous-variable quantum information protocols (although lattice systems with high-resolution imaging and potential projection capabilities can allow for single-site manipulation²⁰). In addition, the proposed Wigner measurement protocol is applicable to tweezer systems.

Finally, for useful quantum information protocols to be realized, one must entangle atoms with one another. While entanglement protocols are a subject for future work, we would like to highlight that proposals for the implementation of $SWAP$ gates with atoms in optical lattices has been proposed in Ref. 58. Similar protocols could be used to generate cluster states of atoms,⁵⁹ leading to a new paradigm of continuous variable quantum information in atomic systems.

In the preparation of this work, the authors became aware of similar work by Grochowski et al.⁵⁰ Reference 50 is generally applicable to any anharmonic quantum system that admits similar driving terms to those considered here and considers a wider array of interesting quantum states, e.g., cat states.⁶⁰ Our work, in contrast, presents a feasible experimental proposal for the generation of atomic GKP states with 10 dB of squeezing.

This work was carried out using the computational facilities of the Advanced Computing Research Center, University of Bristol—http://www.bristol.ac.uk/acrc. H.C.P.K. was funded by the EPSRC CDT in quantum engineering (Grant No. EP/S023607/1). C.W. acknowledged funding from EPSRC (Grant No. EP/Y004728/1). The authors would like to thank G. Ferrini and A. Alberti for interesting discussions that motivated this work and A. Clark for valuable comments on the paper draft.

The authors have no conflicts to disclose.

Harry C. P. Kendell: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (supporting). Giacomo Ferranti: Conceptualization (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Project administration (supporting); Resources (supporting); Validation (supporting); Writing – review & editing (supporting). Carrie A. Weidner: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (lead); Supervision (lead); Validation (supporting); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Here, we present plots like those in Fig. 3 for cesium.

Here, we present our controls and their Fourier spectra for the transitions considered in this work.