Loop-count versus gate-time trade-off and robustness margins for the dispersive geometric quantum phase gate with Lambda-type atoms in cavity QED

Dong, Zhang, C. L. Zou, X. B. Zou and Guo (Phys. Lett. A 379, 2291 (2015)) proposed a two-qubit conditional pi phase gate in which two Lambda-type atoms are trapped in a driven optical cavity: in the large-detuning (dispersive) limit the excited atomic state is eliminated, the driven cavity follows state-dependent closed loops in phase space, and after a whole number of closed loops the cavity returns to vacuum while the atoms acquire a purely geometric conditional phase. The paper's decoherence analysis includes cavity decay only, concluding that the fidelity of the phase gate is robust to cavity decay and that a high-fidelity gate can be implemented with current experimental technology, and exhibits two high-fidelity operating points. Two quantitative questions it leaves undetermined define the open problem: (i) which operating point (the loop-count parameter m, loop index n, and hence the drive amplitude and detuning, which the gate conditions fix once m and n are chosen) is jointly optimal once all realistic decoherence channels compete, since increasing m suppresses the cavity-displacement amplitude and hence cavity-decay error, but proportionally lengthens the gate toward other decoherence times; and (ii) what experimental tolerance margins on the detuning ratio g_c/omega_a and on the symmetry of the two Lambda couplings keep residual spontaneous emission negligible. No published work resolves either question.

Background

Cavity quantum electrodynamics (cavity QED) studies single atoms strongly coupled to a confined light mode in a resonator, and is a leading platform for quantum information processing because it coherently interfaces stationary atomic qubits with photons. A two-qubit quantum phase gate applies a conditional phase to one two-qubit basis state (a pi phase makes it a controlled-Z gate), which together with single-qubit rotations yields universal quantum computation. Early resonant cavity-QED gates transiently populated excited atomic states and thereby suffered spontaneous-emission errors, the motivation for dispersive constructions.

In a Lambda-type atom, two long-lived ground states |g> and |s> are both optically coupled to one excited state |e>; in the dispersive (large-detuning) regime, where the atom-cavity detuning omega_a greatly exceeds the single-photon coupling strength g_c, |e> is only virtually populated and can be removed by adiabatic elimination (a perturbative reduction of the Hilbert space that keeps only the low-lying states). Dong et al. (2015) applied this to two identical symmetric Lambda atoms in a cavity driven by a classical field of amplitude F, obtaining the effective Hamiltonian

$$H = \Delta\, a^{\dagger}a + \Omega\, a^{\dagger}a\,(\sigma_{z,1}+\sigma_{z,2}) + F\,(a^{\dagger}+a),$$

where $a^{\dagger}$ and $a$ are cavity photon creation and annihilation operators, $\Delta = \omega_c - \omega_0$ is the detuning between cavity mode and drive, $\sigma_{z,i}$ distinguishes the two ground states of atom $i$, and the dispersive shift is $\Omega = g_c^2/\omega_a$. Because the cavity resonance seen by the field depends on the atomic state, each two-atom basis state displaces the cavity field along a circle in optical phase space. When $\Delta = 2n\Omega$ (integer $n \neq 0, \pm 1$) and the interaction time is $T = m\pi/\Omega$ (positive integer $m$), each trajectory completes a whole number of closed loops ($m(n-1)$, $mn$ and $m(n+1)$ for the four two-atom basis states, which carry dispersive shifts $(2n-2)\Omega$, $2n\Omega$ and $(2n+2)\Omega$): the cavity returns to vacuum (so no atom-field entanglement survives), and the atoms keep only geometric phases, that is, phases determined by the closed path traced in phase space rather than by the elapsed time. The conditional phase is $\phi = \varphi_{gg} + \varphi_{ss} - 2\varphi_{gs}$, accumulated by the four two-atom basis states, and the drive amplitude $F = \Omega\sqrt{n(n^2-1)/m}$ enforces $\phi = \pi$. This construction belongs to the family of 'unconventional' geometric phase gates in cavity QED developed by Zheng and by Liu et al., and is the cavity analogue of geometric phase gates used with trapped ions.

The open issue sits in the decoherence analysis. Including only cavity photon loss at rate $\kappa$ through a Lindblad master equation (the standard open-system evolution equation for a density matrix), the authors found numerically that larger $m$ shrinks the phase-space loop radius, weakens transient atom-cavity entanglement, and suppresses fidelity loss from cavity decay, quoting two operating points for $(g_c,\kappa,\gamma)/2\pi = (16,1.4,3)$ MHz with $\omega_a = 10 g_c$: $(m,n) = (400,30)$ with gate time $125\,\mu$s and fidelity above 0.97, and $(m,n) = (3200,100)$ with gate time near $1$ ms and fidelity above 0.99, where $\gamma$ denotes the atomic spontaneous-emission rate. But the loop radius shrinks only because the gate gets longer: cavity-decay error falls with $m$ while the growing duration $T = m\pi/\Omega$ exposes the gate to every other noise source (residual spontaneous emission, ground-state dephasing, drive and cavity frequency noise), and millisecond gate times are already comparable to typical atomic ground-state coherence in optical cavity QED. The neglect of spontaneous emission is, moreover, argued only for the ideal symmetric Lambda atom with large detuning: unequal couplings of the two transitions spoil the destructive interference that keeps the excited state dark, and the paper quantifies neither how small $g_c/\omega_a$ nor how symmetric the couplings must be.

Problem Statement

For the dispersive geometric quantum phase gate of Dong et al. (2015), realized with two symmetric Lambda-type atoms trapped in a classically driven optical cavity, with effective Hamiltonian $H = \Delta a^{\dagger}a + \Omega a^{\dagger}a(\sigma_{z,1}+\sigma_{z,2}) + F(a^{\dagger}+a)$, $\Omega = g_c^2/\omega_a$, gate conditions $\Delta = 2n\Omega$, $T = m\pi/\Omega$ and conditional geometric phase $\phi = \pi$ (with $\phi = \varphi_{gg} + \varphi_{ss} - 2\varphi_{gs}$, where $\varphi_{jk}$ are the geometric phases of the four two-atom basis states) at drive amplitude $F = \Omega\sqrt{n(n^2-1)/m}$, with $m$ a positive integer that fixes the gate duration $T = m\pi/\Omega$ and scales the number of closed phase-space loops (the four two-atom basis states close $m(n-1)$, $mn$ and $m(n+1)$ loops), and $n$ an integer loop index with $n \neq 0, \pm 1$, determine the gate's realistic performance limits and optimal operation, specifically:

1. Joint optimization under combined decoherence. Extend the paper's cavity-decay-only analysis to a master equation that simultaneously includes cavity decay at rate $\kappa$, atomic spontaneous emission at rate $\gamma$ with the residual excited-state occupation induced by finite $g_c/\omega_a$ and by unequal couplings of the two Lambda transitions, and ground-state dephasing at a rate representative of current optical cavity-QED experiments (the dependence of the results on this dephasing rate may be reported parametrically over that realistic range). Over the operating parameters $(m, n)$ (equivalently the loop counts, the loop radius and the gate duration) and for realistic cavity-QED parameter sets such as the paper's $(g_c,\kappa,\gamma)/2\pi = (16,1.4,3)$ MHz with $\omega_a = 10 g_c$, report the gate fidelity under a single, explicitly stated metric (for example, the average over the four computational-basis inputs of the overlap of the evolved two-qubit state with the ideal output state the gate would produce without decoherence), establish whether an interior optimum of that fidelity exists, where it lies, and what maximum fidelity and gate time it affords; in particular, whether an operating point with fidelity comparable to the paper's quoted values survives once the competing channels are included, given gate durations from tens of microseconds to milliseconds, treating the paper's quoted 0.97 and 0.99 as qualitative unless the same fidelity convention is reproduced.

2. Robustness margins for the spontaneous-emission neglect. Derive quantitative tolerance bounds, under an explicitly stated quantitative criterion for negligibility: how small $g_c/\omega_a$ must be, and how close to equal the two Lambda couplings must be made, for the spontaneous-emission contribution to the gate error to remain negligible against the dominant decoherence channel at high-fidelity operating points. The suppression relies on a dark-state interference that coupling asymmetry breaks, so the accuracy of the adiabatically eliminated description near the derived margins is itself a substantive concern.

The overall objective is a complete realistic-decoherence performance map of this gate, culminating in an assessment of whether, and under what experimental tolerances, it can operate at high fidelity in realistic devices.

References

1. Dong Dong, Yan-Lei Zhang, Chang-Ling Zou, Xu-Bo Zou, Guang-Can Guo. Quantum phase gate through the dispersive atom-field interaction with atoms trapped in optical cavity QED. Physics Letters A 379(38), 2291-2294 (2015). DOI: 10.1016/j.physleta.2015.07.020. https://doi.org/10.1016/j.physleta.2015.07.020
2. Nilakantha Meher, S. Sivakumar. A review on quantum information processing in cavities. EPJ Plus 137, 985 (2022). DOI: 10.1140/epjp/s13360-022-03172-x. arXiv:2204.01322. https://doi.org/10.1140/epjp/s13360-022-03172-x
3. Kai Muller, Kimmo Luoma, Walter T. Strunz. Geometric phase gates in dissipative quantum dynamics. Physical Review A 102, 032611 (2020). DOI: 10.1103/PhysRevA.102.032611. arXiv:1907.08033. https://doi.org/10.1103/PhysRevA.102.032611
4. M. Palmero, S. Martinez-Garaot, D. Leibfried, J. J. Wineland, J. G. Muga. Fast phase gates with trapped ions. Physical Review A 95, 022328 (2017). DOI: 10.1103/PhysRevA.95.022328. https://doi.org/10.1103/PhysRevA.95.022328
5. Shi-Biao Zheng. Unconventional geometric quantum phase gates with a cavity QED system. arXiv:1202.5387 (2012). https://arxiv.org/abs/1202.5387
6. Tong Liu, Xiao-Zhi Cao, Qi-Ping Su, Shao-Jie Xiong, Chui-Ping Yang. Multi-target-qubit unconventional geometric phase gate in a multi-cavity system. Scientific Reports 6, 21562 (2016). DOI: 10.1038/srep21562. arXiv:1506.06044. https://doi.org/10.1038/srep21562
7. Andreas Reiserer, Gerhard Rempe. Cavity-based quantum networks with single atoms and optical photons. Reviews of Modern Physics 87, 1379 (2015). DOI: 10.1103/RevModPhys.87.1379. https://doi.org/10.1103/RevModPhys.87.1379
8. M. Akbari, S. N. Andrianov, A. A. Kalachev. Quantum logic gates based on off-resonant cavity-assisted interaction between three-level atoms and single photons. Laser Physics 27, 075202 (2017). DOI: 10.1088/1555-6611/aa7547. https://doi.org/10.1088/1555-6611/aa7547