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arXiv:2209.11193

On the static effective Lindbladian of the squeezed Kerr oscillator

Jayameenakshi Venkatraman, Xiao, Xu, Rodrigo G. Cortiñas, Michel Devoret

Published Sep 22, 2022• 0 citations
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Abstract

We derive the static effective Lindbladian beyond the rotating wave approximation (RWA) for a driven nonlinear oscillator coupled to a bath of harmonic oscillators. The associated dissipative effects may explain orders of magnitude differences between the predictions of the ordinary RWA model and results from recent superconducting circuits experiments on the Kerr-cat qubit. The higher-order dissipators found in our calculations have important consequences for quantum error-correction protocols and parametric processses.

DOI
10.48550/arxiv.2209.11193

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We derive the static effective Lindbladian beyond the rotating wave approximation (RWA) for a driven nonlinear oscillator coupled to a bath of harmonic oscillators. The associated dissipative effects may explain orders of magnitude differences between the predictions of the ordinary RWA model and results from recent superconducting circuits experiments on the Kerr-cat qubit. The higher-order dissipators found in our calculations have important consequences for quantum error-correction protocols and parametric processses.
Out-of-equilibrium driven nonlinear oscillators have attracted interest as a platform for tailoring Floquet Hamiltonian spectra and stabilizing quantum states for quantum information. However, it is usually ignored that the necessary parametric drive can amplify environmental fluctuations limiting applications. In this work, we capture this effect by computing the effective Lindbladian of a Kerr parametric oscillator beyond the rotating wave approximation. We find that the relevant dissipators depend strongly on the nonlinearity and the mean-photon number in the oscillator. In particular, we address recent measurements of the logical lifetime on a Schr\"odinger's-cat qubit stabilized by a parametric drive, thereby providing an explanation for the observed three-order-of-magnitude discrepancy between experimental data and an ordinary Lindblad treatment. Our systematic approach paves the path to treat dissipative effects in Floquet-engineered nonlinear quantum systems.
Parametric gates and processes engineered from the perspective of the static effective Hamiltonian of a driven system are central to quantum technology. However, the perturbative expansions used to derive static effective models may not be able to efficiently capture all the relevant physics of the original system. In this work, we investigate the conditions for the validity of the usual low-order static effective Hamiltonian used to describe a Kerr oscillator under a squeezing drive. This system is of fundamental and technological interest. In particular, it has been used to stabilize Schrödinger cat states, which have applications for quantum computing. We compare the states and energies of the effective static Hamiltonian with the exact Floquet states and quasi-energies of the driven system and determine the parameter regime where the two descriptions agree. Our work brings to light the physics that is left out by ordinary static effective treatments and that can be explored by state-of-the-art experiments.
In this thesis we investigate the interplay between dissipation and driving in nonlinear quantum systems for a special setup: a flux qubit read out by a DC-SQUID - a nonlinear quantum oscillator. The latter is embedded in a harmonic bath, thereby mediating dissipation to the qubit. Two different approaches are elaborated: First we consider a composite qubit-SQUID system and add the bath afterwards. We derive analytical expressions for its eigenstates beyond rotating wave approximation (RWA), by applying Van Vleck perturbation theory (VVPT) in the qubit-oscillator coupling. The second approach is an effective bath approach based on a mapping procedure, where SQUID and bath form an effective bath seen by the qubit. Here the qubit dynamics is obtained by applying standard procedures established for the spin-boson problem. This approach requires the knowledge of the steady-state response of the dissipative Duffing oscillator, which is studied within a resonant and an off-resonant approach: The first is applicable near and at an N-photon resonance using VVPT beyond a RWA. The second is based on the exact Floquet states of the nonlinear driven oscillator. The dissipative qubit dynamics is described analytically for weak system-bath coupling and agrees well for both approaches. We derive the effect of the nonlinearity on the qubit dynamics, on the Bloch-Siegert shift and on the vacuum Rabi splitting.
Abstract The quantum to classical transition is determined by the interplay of a trio of parameters: dissipation, nonlinearity, and macroscopicity. Why is nonlinearity needed to see quantum effects? And, is not an ordinary pendulum quite nonlinear already? In this manuscript, we discuss the parameter regime where the dynamics of a massive oscillator should be quantum mechanical in the presence of dissipation. We review the outstanding challenge of the dynamical generation of highly quantum mechanical cat states of a massive ‘pendulum’, known as Kerr-cats. We argue that state-of-the-art cold atom experiments may be in a position to reach such a nonlinear regime, which today singles out superconducting quantum circuits. A way to stabilize Schrödinger cat superpositions of a mechanical atomic oscillator via parametric squeezing and further protected by an unusual form of quantum interference is discussed. The encoding of a neutral atom Kerr-cat qubit is proposed.
By applying a microwave drive to a specially designed Josephson circuit, we have realized an elementary quantum optics model, the squeezed Kerr oscillator. This model displays, as the squeezing amplitude is increased, a cross-over from a single ground state regime to a doubly-degenerate ground state regime. In the latter case, the ground state manifold is spanned by Schrödinger-cat states, i.e. quantum superpositions of coherent states with opposite phases. For the first time, having resolved up to the tenth excited state in a spectroscopic experiment, we confirm that the proposed emergent static effective Hamiltonian correctly describes the system, despite its driven character. We also find that the lifetime of the coherent state components of the cat states increases in steps as a function of the squeezing amplitude. We interpret the staircase pattern as resulting from pairwise level kissing in the excited state spectrum. Considering the Kerr-cat qubit encoded in this ground state manifold, we achieve for the first time quantum nondemolition readout fidelities greater than 99%, and enhancement of the phase-flip lifetime by more than two orders of magnitude, while retaining universal quantum control. Our experiment illustrates the crucial role of parametric drive Hamiltonian engineering for hardware-efficient quantum computation.
We consider a simple model of a Josephson junction phase qubit coupled to a solid-state nanoelectromechanical resonator. This and many related qubit-resonator models are analogous to an atom in an electromagnetic cavity. When the systems are weakly coupled and nearly resonant, the dynamics is accurately described by the rotating-wave approximation (RWA) or the Jaynes-Cummings model of quantum optics. However, the desire to develop faster quantum-information-processing protocols necessitates approximate, yet analytic descriptions that are valid for more strongly coupled qubit-resonator systems. Here we present a simple theoretical technique, using a basis of dressed states, to perturbatively account for the leading-order corrections to the RWA. By comparison with exact numerical results, we demonstrate that the method is accurate for moderately strong coupling and provides a useful theoretical tool for describing fast quantum information processing. The method applies to any quantum two-level system linearly coupled to a harmonic oscillator or single-mode boson field.