Recent advances in silicon nanofabrication have allowed the manipulation of spin qubits that are extremely isolated from noise sources, being therefore the semiconductor equivalent of single atoms in vacuum. We investigate the possibility of directly coupling an electron spin qubit to a superconducting resonator magnetic vacuum field. By using resonators modified to increase the vacuum magnetic field at the qubit location, and isotopically purified 28Si substrates, it is possible to achieve coupling rates faster than the single spin dephasing. This opens up new avenues for circuit-quantum electrodynamics with spins, and provides a pathway for dispersive read-out of spin qubits via superconducting resonators.
Natural atoms in vacuum are the cleanest and most reproducible quantum systems, but they pose limitations to the way they can be made to interact with their environment. In the cavity-Quantum Electrodynamics (cavity-QED) scheme, atoms interact with photons in a high-finesse cavity, but the strength and duration of the interaction is limited by the electric dipole of the atoms and the dwell time in the cavity.1 Ten years ago, the progress in nanofabrication and in coherent control of nanoscale electrical circuits opened a new avenue in this field, known as circuit-QED.2–4 Large artificial atoms are fabricated with superconducting thin films and Josephson junctions, and coupled to the quantized electromagnetic modes of a high-Q on-chip superconducting resonator. The dipole moment can be made almost arbitrarily large, and the dwell time is infinite. Unlike natural atoms, it is very easy to tune in-situ the energy spectrum and various other properties of artificial atoms. This architecture has brought about some of the most exquisite demonstrations of control over individual and multiple quantum systems, including quantum logic gates5 and quantum teleportation.6 Because of their large size, and the presence of amorphous materials and interfaces in their vicinity, the superconducting qubits used in circuit-QED are not the most long-lived quantum systems. Their lifetime has steadily improved over the years, reaching up to 10 μs in 3D cavities,7 but still does not match that of true atomic systems. The “ultimate setup” in this field would be to combine the purity of atoms in vacuum with the convenience and tuneability of circuits in solids.
The term “semiconductor vacuum”8 has been adopted to describe the exceptional properties of isotopically purified 28Si. Ultra-high purity samples are being produced for the purpose of redefining the kilogram in the SI units,9 but they are also used as hosts for the most coherent quantum systems demonstrated so far in solid state. A substitutional group V donor atom in Si (such as P, As, Sb or Bi) behaves to a good approximation like hydrogen in vacuum, with an energy spectrum renormalized by the effective mass and dielectric constant of the host material.10 The absence of nuclear spins and paramagnetic states in 28Si implies that the electron and nuclear spins of a donor atom behave almost as if they really were held in a magnetic vacuum. Indeed, extraordinary coherence times have been measured in bulk samples for both the electron (T2e = 10 s11) and the nucleus (T2n = 3 hours12) of 31P donor atoms in 28Si. Moreover, the weakness of spin-orbit coupling in P donors13 makes the donor electron insensitive to electric field fluctuations, tremendously reducing the impact of charge noise so common in nanostructures.
In this paper we investigate the possibility of using the spin of a 31P donor atom in 28Si to realize the ultimate circuit-QED setup – coupling a single atom in solid state to a single photon in a microwave circuit. In contrast to recent proposals in which the electron is coupled to the resonator electric field via different spin-orbit interaction mechanisms,14–18 here we consider the case where the coupling is directly provided via the resonator magnetic field.
II. SPIN-RESONATOR COUPLING
The interaction between an electron spin-1/2 and a photonic mode confined inside a resonator is described by the Jaynes-Cummings Hamiltonian:
where εz is the electron Zeeman energy, ν0 the photon frequency and g the coupling constant.
We consider an electron bound to a 31P dopant under an applied constant magnetic field B0. In this case the electron Zeeman energy is
The spin-photon coupling rate g/h is assumed to be equal to half the Rabi frequency of an electron under the resonator magnetic vacuum field, which has amplitude Bvac and direction perpendicular to B0:
In order to calculate g, we therefore have to calculate the strength of the resonator magnetic vacuum field. We consider a resonator in which the central line of the coplanar waveguide (CPW) is capacitively interrupted at two points separated by a distance l (Fig. 1(a)). The resonator transmits signals whose frequencies are integer multiples of the fundamental mode, which is the one whose half-wavelength is equal to the resonator length, λ0/2 = l. This condition implies the frequency of the fundamental mode to be:
where εeff is the effective dielectric constant of the CPW and c the speed of light. The magnetic field profile of this mode is maximum in the center of the resonator, which is where the 31P donor has to be placed (Fig. 1(a)).
To avoid losses, the waveguide layer is made of a superconducting material, e.g. Nb. This layer sits on a few-nanometers-thick layer of SiO2 followed by a 28Si substrate. For the present purpose, it is perfectly acceptable to use an isotopically enriched epilayer of ∼1 μm thickness, grown on top of natural silicon.19 In this limit where the substrate is much thicker than any other layer,
The value of Bvac depends on the amplitude of the zero-point current in the resonator, Ivac. The latter can be calculated by assuming the energy of the vacuum field to be stored in the resonator equivalent lumped inductance L:
The equivalent inductance of the resonator, for the fundamental mode, is known to be:21,22
where Z0 is the characteristic impedance of the transmission line, which depends on the line width w and gap s (Fig. 1(a)). Impedance matching to the outside circuitry requires Z0 = 50 Ω. From Eqs. (4) and (5), we find the resonator vacuum current:
For conventional CPW resonators used in circuit-QED experiments, the central line is wide enough so that one can consider the vacuum magnetic field a few nanometers underneath to be proportional to the vacuum current density Ivac/w,23
where μ0 is the vacuum permeability. For w = 20 μm and B0 = 200 mT (ν0 = 5.6 GHz), Bvac ≈ 2 nT, yielding spin-photon coupling rates g/h = γeBvac/4 ≈ 14 Hz.
In order to increase the resonator vacuum field at the donor location, therefore increasing the coupling strength, we propose to shrink the central line width as to increase the vacuum current. A similar procedure has been used to couple resonator magnetic fields to flux qubits,24 achieving coupling rates as high as to reach the ultrastrong coupling regime.25 Here we assume w = 30 nm, compatible electron beam lithography techniques. In order to avoid losses, the characteristic impedance of the constricted region has to be kept Z0 = 50 Ω which implies a transmission line gap width s = 70 nm.26 Such a small gap, if constant along all the resonator length, would result in very high electric fields Evac = Vvac/s, which can greatly deteriorate the resonator Q-factor by driving dissipative dynamics of charge fluctuators in and around the gap. Note however that our region of interest is only at the center of the resonator length, where the magnetic vacuum field has an antinode and the electric field a node. In this region, electric losses are negligible. We therefore choose the s = 70 nm gap to be localized in a constriction where the donor is to be implanted, and s = 10 μm everywhere else (Fig. 1(a)).
Considering Nb film thickness tNb = 20 nm and a donor implanted 20 nm below the oxide-Nb interface, Bvac at the donor location is approximately given by the Ampere's law,
where r ≈ 30 nm is the distance between the donor and the center of the central line (Fig. 1(b)). Since the zero-point current is given by Eq. (6), the spin-resonator coupling, g/h = γeBvac/4, is also found to depend linearly on the frequency,
In order to achieve the strong coupling regime, g has to be higher than the qubit dephasing, γ*, and photon decay,
The other requirement, g > κ, translates into Q > 2 × 106. Even though such high-Q resonators are feasible,30,31 the presence of magnetic fields (B0 ≈ 200 mT) is likely to introduce extra losses through the creation of vortices in the superconducting film. We find therefore the peculiar situation where it is the cavity decay κ instead of the qubit dephasing γ* that poses the greatest hurdle to achieving the strong coupling regime.
In addition to the microwave engineering aspects, this architecture also requires ensuring that there is one and only one electron bound to the 31P donor. For a donor near (e.g. ≃ 20 nm under) a Si/SiO2 interface, fixed charge in the SiO2 and at the Si/SiO2 interface above the donor can lift its electrochemical potential μD and lead to donor ionization.32 To circumvent this problem we consider the addition of an electron reservoir in the vicinity of the donor. The reservoir is induced with the help of an aluminum ‘top-gate', held at voltage Vtop (beneath the Nb ground plane in Fig. 1(b)), which attracts electrons from a heavily doped n+ source region (Fig. 1(a)), held at voltage Vs. The electron reservoir is induced when Vtop − Vs is larger than some threshold (typically around 0.6 V), but both voltages can float with respect to ground. Here, V = 0 ground is the potential of the resonator ground planes and center conductor. Therefore, it is possible to choose Vs such that the reservoir Fermi level EF is higher than μD, and ensure that the donor is neutral. We note that the donor-reservoir distance ≈70 nm is larger than the typical distances ≈25 nm used in donor-qubit devices.33 However this is not an issue, because the reservoir's only role here is to ensure donor charge neutrality – we do not seek to produce fast spin-dependent tunneling events between donor and reservoir to achieve spin readout.34
In Figure 1(b), we plot the conduction band energy Ec – computed with TCAD35 – along a slice of the device, having set the reservoir Fermi level EF as the zero-energy reference. We set the Nb ground planes and center conductor at ground (V = 0) and choose Vtop = +1.63 V, with Vs = −0.37 V. The electrochemical potential
III. SPIN CONTROL AND READ-OUT
In order to avoid spin-to-photon conversion while performing quantum gate operations on the electron spin, it is convenient to detune the spin Larmor frequency from the resonator mode. We therefore assume Δ ≫ g, where Δ = hν0 − εz.
In this so-called dispersive regime, the diagonalized Hamiltonian is approximately:38
The corresponding eigenstates are approximately the same as the uncoupled Hamiltonian, with a small deviation proportional to (g/Δ)2.39 In order to have eigenstates with 99% fraction of uncoupled modes, therefore protecting the qubits from decaying into photons, we will assume from now on:
Importantly, Eq. (10) implies that the cavity resonance depends on the spin state. Therefore, the measurement of the cavity transmission with a weak microwave signal allows for the quantum non-demolition readout of the spin state. Conversely the only spin readout method demonstrated so far with donor spins34,40 causes the physical loss of the electron upon readout.
The spin-dependent cavity resonance shift can be measured through the resonator phase-shift, whose spin-state dependent values are
Note that we have not assumed any operation frequency when deriving the limit for the Q-factor. Indeed, Eqs. (9) and (11) impose that the spin-dependent cavity shift is
The operation frequency is also important in determining the enhancement of the spin decay rate due to its coupling to resonator photons with finite lifetime. This enhancement is simply given by the photon fraction of the Hamiltonian (Eq. (10)) eigenstates times the photon decay rate:38
which is equal to ν0/(100Q) for our choice of detuning Δ = 10g (Eq. (11)). Such a dependence is plotted in Fig. 2(a). For instance, at an operating frequency ν0 = 5.6 GHz (corresponding to B0 = 200 mT), a quality factor as low as Q = 1.5 × 104 yields an increase in spin decay equal to 3.5 kHz (black square in Fig. 2(a)), which is of the same order as the intrinsic dephasing γ* of the isolated electron spin.
Another source of dephasing comes from thermal fluctuations of the photon number in the resonator. Indeed, the terms in Eq. (10) can be rearranged as to highlight that the spin resonance depends on the photon number, εz → εz + (g2/Δ)(2a†a + 1). This implies that the spin resonance linewidth, and therefore the qubit dephasing rate, increases with thermal photon occupation. The photon number in the fundamental mode43 is given by the Bose-Einstein distribution,
Such a dependence is plotted in Fig. 2(b) for a range of temperatures and operating frequencies. The enhanced spin dephasing remains on the order of its uncoupled dephasing rate for temperatures up to liquid helium (4.2 K), for all ranges of operating frequencies.
Note that the spin-dependent cavity shift does not depend on the photon number inside the resonator, and therefore the effectiveness of the readout method should not depend on temperature (until the superconducting resonator starts to degrade). Moreover, the relaxation rate
The electron spin state can be rotated by applying to the resonator a microwave pulse with the same frequency as the AC Stark-shifted spin Larmor frequency. Note that high input powers have to be used, since the drive is out-of-resonance with the resonator and therefore it is mainly reflected at the input port. The maximum Rabi-frequency of the electron spin is given by the critical current density in the center line before superconductivity is lost. The critical current of niobium films, 1 − 10 × 106 A/cm2,28,29 is enough to drive the spin at 1 − 10 MHz rates, there orders of magnitude faster than its dephasing rate.
IV. CAVITY-MEDIATED MACROSCOPIC ENTANGLEMENT
One of the greatest advantages of coupling qubits to CPW resonators is that the latter can be used as a bus to entangle qubits placed at different points along the resonator, and therefore separated by macroscopic distances. This is also done in the dispersive regime, with the coupling provided by virtual photons.44 The qubit-qubit coupling strength is then given by:38
For our chosen set of parameters (Eqs. (9) and (11)), this coupling is proportional to the resonator frequency, g2q/h ≈ 5.3 × 10−8ν0. In order to have a macroscopic coupling rate higher than the intrinsic qubit dephasing rate, g2q > γ*, operating frequencies ν0 > 22.5 GHz (B0 > 800 mT) are therefore required. Note however that such high frequencies would also increase the qubit decay rate induced by photon losses (see Eq. (12) and Fig. 2). One therefore has to carefully choose the set of parameters that maximizes the ratio g2q/Γ, where Γ is the total single qubit linewidth. Let us neglect the qubit dephasing due to thermal photons,
We first set the qubit-photon detuning to our previous assumption, Δ = 10g, and plot the g2q/Γ dependence on ν0 and Q in Fig. 3(a). As expected, the ratio increases with magnetic field and Q-factor, being equal to one for Q = 106 and B0 = 1 T. Even though such a high field is below the critical one that breaks up superconductivity of Nb films,45 the high losses introduced by proliferation of vortices are likely to lower the resonator Q-factor by a significant amount. In the following we attempt to lower the need for high B0 by investigating the g2q/Γ dependence on spin-photon detuning Δ. We assume Q = 106 and plot, in Fig. 3(b), the dependence of Eq. (15) on ν0 and Δ. For a fixed B0-field, we see that g2q/Γ increases with Δ, which is expected since g2q decreases linearly with Δ whereas Γ decreases quadratically. After a maximum detuning, however, the ratio g2q/Γ starts decreasing again. This happens whenever γQ < γ* and therefore the intrinsic spin dephasing rate γ* is the main loss channel. At this point, g2q decreases with Δ whereas Γ is unaffected. We find an optimal operating point at Δ = 5g and B0 = 650 mT (black square), at which g2q/Γ = 1. We note however that operating at such small detuning decreases the entanglement fidelity, since the spin eigenstantes of the Hamiltonian in Eq. (10) contain 4% (g2/Δ = 0.04) of photon fraction.
V. CONCLUSIONS AND PERSPECTIVES
The architecture presented here takes full advantage of the exquisite isolation from the environment of a single electron spin bound to donor atoms in isotopically purified 28Si. Even though reaching the strong-coupling regime will be probably limited by the resonator Q-factor, coherent control and non-demolition readout of the qubit state can be performed via the resonator with no significant increase in the qubit dephasing, even for resonator Q-factors as low as Q = 104 and liquid helium temperatures T = 4.2 K.
The low spin-photon coupling rate makes however strong coupling of macroscopically separated qubits via virtual resonator photons extremely hard to achieve, also mainly due to expected low resonator Q-factors under high magnetic fields. A solution to this problem would be to introduce vortex pinning structures that limit their movement and therefore dissipation, increasing the resonator Q-factor.46,47 In this case it is desirable to have pinning centers whose size is on the order of the coherence length of Nb (around 40 nm48) separated by a distance comparable to the London penetration depth (also around 40 nm49). Therefore an array of nanoscale holes would be the optimum vortex trapping structure.50 Note that here we propose to use Nb films whose thickness is smaller than the London penetration depth, and therefore cannot sustain a complete flux exclusion, resulting in lower diamagnetic energy which then leads to a higher critical field.45 On the other hand the film thickness is also smaller than the Nb coherence length, which implies that the transition temperature will be slightly smaller.29
Instead of relying on high Q-factors, one could look for resonator geometries that provide higher spin-photon coupling rates. An example is to introduce an artificial spin-orbit coupling17,18 as to couple the spin state to the resonator electric field.
It is important to notice that the coupling rates derived in this paper rely on shrinking the resonator central line to a few tens of nanometers. Such a constriction, on the order of the Nb coherence length, will most certainly behave as a weak link and therefore determine a nanobridge-like Josephson junction.51,52 Even though this increases the local inductance at the constriction,53 this is not associated with an increase of the magnetic vacuum field, since the junction inductance is purely kinetic and therefore not associated with any magnetic field. This is the reason why we ignored such an effect in this paper.
Finally, we note that the present proposal can apply also to electron spins in isotopically purified 12C, such as Nitrogen-Vacancy centers in diamond, which also can show intrinsic spin dephasing rates in the kHz range.54
We thank A. Laucht, J. T. Muhonen, J. P. Dehollain, R. Kalra and A. Blais for helpful discussions. This research was funded by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027) and the US Army Research Office (W911NF-13-1-0024). H. H. acknowledges financial support by DFG (Grant No. SFB 631, C3).