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arXiv:2111.10433 · Physical Review Research

Qubit decoherence and symmetry restoration through real-time instantons

Foster Thompson, Alex Kamenev

Published Apr 8, 2022• 12 citations
arXivHTMLPDF

Abstract

A parametrically driven quantum oscillator, stabilized by a nonlinear dissipation, exhibits a spontaneous breaking of the parity symmetry. It results in the quantum bistability, corresponding to a Bloch sphere of dark states. This makes such a driven-dissipative system an attractive candidate for a qubit. The parity symmetry breaking is exact both on the classical level and within the quantum mechanical perturbation theory. Here, we show that nonperturbative quantum effects lead to the symmetry restoration and result in an exponentially small but finite qubit decoherence rate. Technically, the symmetry restoration is due to real-time instanton trajectories of the Keldysh path integral, which represents the Lindbladian evolution of the driven-dissipative oscillator.

DOI
10.1103/physrevresearch.4.023020

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A parametrically driven quantum oscillator, stabilized by a nonlinear dissipation, exhibits a spontaneous breaking of the parity symmetry. It results in the quantum bistability, corresponding to a Bloch sphere of dark states. This makes such a driven-dissipative system an attractive candidate for a qubit. The parity symmetry breaking is exact both on the classical level and within the quantum mechanical perturbation theory. Here, we show that nonperturbative quantum effects lead to the symmetry restoration and result in an exponentially small but finite qubit decoherence rate. Technically, the symmetry restoration is due to real-time instanton trajectories of the Keldysh path integral, which represents the Lindbladian evolution of the driven-dissipative oscillator.
We study the quantum dynamics of a single bosonic cavity subject to two-photon driving and two-photon dissipation in the presence of finite detuning. Exploiting a hidden time-reversal symmetry, the Wigner representation and the WKB method, we introduce an effective phase-space potential for description of the steady state. It reveals two attracting points, which are metastable due to quantum fluctuations. By employing the Keldysh real-time path integral formalism, we compute the instanton trajectory governing the quantum activation process between these attractors and establish a fundamental connection with the Wigner representation. This relation unifies the steady-state phase-space description with dynamical quantum activation processes. We also derive an analytical expression for the decoherence rate of the system. Our work provides a coherent theoretical framework for analyzing quantum bistability, metastability, and decoherence in driven-dissipative nonlinear resonators, with direct implications for the design of bosonic qubits and quantum information processing.
Parametrically driven oscillators can emerge as a basis for the next generation of qubits. Classically, these systems exhibit two stable oscillatory states with opposite phases. Upon quantization, these states turn into a pair of closely spaced Floquet states, which can serve as the logical basis for a qubit. However, interaction with the environment induces phase-slip events which set a limit on qubit coherence. Such phase slips persist even at zero temperature due to a mechanism known as quantum activation \cite{QuantumActivation}. In contrast to conventional tunneling, the quantum activation is described by a {\em real-time} instanton trajectory in the complexified phase space of the system. In this work, we show that the phase-slip rate is exponentially sensitive to weak AC perturbations. The spectrum of the system's response -- captured by the so-called logarithmic susceptibility (LS) -- enables a direct observation of characteristic features of real-time instantons. Studying this spectrum suggests new means of efficient qubit control.
We consider the dynamics of the quantum Rabi model driven parametrically by a periodic modulation of a complex coupling. We show both analytically and numerically that instead of Rabi oscillations, this nonunitary coherent driving leads to a unidirectional instanton solution which mediates the rapid and deterministic one-way tunneling of any initial coherent state to the ground state, making the ground state a strong attractor in the quantum dynamics of the qubit. The timescale of this tunneling is shown to be inversely proportional to the effective resonant coupling, allowing for exceptionally fast, deterministic, and high-fidelity qubit reset through a purely coherent, PT-symmetric drive--without coupling to external dissipative baths, lossy resonators, or employing measurement-based feedback. Finally, we show how the drive can be engineered to place the strong attractor at any arbitrary point on the Bloch sphere.
Passive error correction schemes have so far been limited to static memories rather than dynamical qubits. We demonstrate that a driven-dissipative bosonic system, the Bose-Hubbard dimer (BHD), can encode a persistently oscillating qubit within a dynamical noiseless subsystem (DNS). The strong parity symmetry of the system leads to degenerate stationary states, whose mutual coherence breaks the time translation symmetry, giving rise to a time crystal in the thermodynamic limit. This time crystal undergoes a dissipative phase transition controlled by the driving force, with the DNS present above the transition point. This noiseless subsystem, and hence the qubit, is resilient to collective loss and at most quadratically affected when the strong symmetry condition is weakened. The decoherence from collective dephasing also cancels in the leading order, with the cancellation being exact at a specific drive strength, set by the photon-number difference between the two logical states. Tuning the drive extends this cancellation to unbalanced dephasing. These results establish dissipative time crystals as a medium for autonomously stabilized oscillating qubits.
Metastability in many-body quantum systems and its associated exponentially-long timescales have been the subject of considerable recent interest. Here, we focus on a class of driven-dissipative many-body open quantum systems described by a Lindbladian having hidden time-reversal symmetry (a form of quantum detailed balance). Examples include boundary-driven interacting spin chains, bosonic lattice models and driven-dissipative collective spin models. We suggest that for such systems, slow timescales in the vicinity of a dissipative first-order phase transition can be analytically predicted using a special purification of the non-equilibrium steady state. We show the accuracy of our conjecture through detailed studies of a dissipative transverse-field Ising model with collective and local decay, and a driven-dissipative nonlinear cavity model. Our results allow quantitative insights into metastability and slow dynamics for a range of systems, including cases where semiclassical or path-integral instanton approaches are intractable.
We follow the passage from complex amplitude bistability to phase bistability in the driven dissipative Jaynes-Cummings oscillator. Quasi -distribution functions in the steady state are employed, for varying qubit-cavity detuning and drive parameters, in order to track a first-order dissipative quantum phase transition up to the critical point marking a second-order transition and spontaneous symmetry breaking. We demonstrate the photon blockade breakdown in the dispersive regime, and find that the coexistence of cavity states in the regime of quantum bistability is accompanied by pronounced qubit-cavity entanglement. Focusing on the effect of quantum-activated switching for both coupled degrees of freedom, we move from a region of minimal entanglement in the dispersive regime, where we derive analytical perturbative results, to the threshold behaviour of spontaneous dressed-state polarization at resonance.