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arXiv:2409.10744 · Journal of Physics A Mathematical and Theoretical

Symmetries of Liouvillians of squeeze-driven parametric oscillators

FRANCESCO IACHELLO, Colin Coane, Jayameenakshi Venkatraman

Published Sep 13, 2024• 5 citations
arXivHTMLPDF

Abstract

Abstract We study the symmetries of the Liouville superoperator of one dimensional parametric oscillators, especially the so-called squeeze-driven Kerr oscillator, and discover a remarkable quasi-spin symmetry su(2) at integer values of the ratio η = ω / K of the detuning parameter ω to the Kerr coefficient K, which reflects the symmetry previously found for the Hamiltonian operator. We find that the Liouvillian of an su(2) representation | j , m j ⟩ has a characteristic double-ellipsoidal structure, and calculate the relaxation time T X for this structure. We then study the phase transitions of the Liouvillian which occur as a function of the parameters ξ = ε 2 / K and η = ω / K . Finally, we study the temperature dependence of the spectrum of eigenvalues of the Liouvillian. Our findings may have applications in the generation and stabilization of states of interest in quantum computing.

DOI
10.1088/1751-8121/ad7ae6

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Abstract We study the symmetries of the Liouville superoperator of one dimensional parametric oscillators, especially the so-called squeeze-driven Kerr oscillator, and discover a remarkable quasi-spin symmetry su(2) at integer values of the ratio η = ω / K of the detuning parameter ω to the Kerr coefficient K, which reflects the symmetry previously found for the Hamiltonian operator. We find that the Liouvillian of an su(2) representation | j , m j ⟩ has a characteristic double-ellipsoidal structure, and calculate the relaxation time T X for this structure. We then study the phase transitions of the Liouvillian which occur as a function of the parameters ξ = ε 2 / K and η = ω / K . Finally, we study the temperature dependence of the spectrum of eigenvalues of the Liouvillian. Our findings may have applications in the generation and stabilization of states of interest in quantum computing.
Abstract We study the symmetries of the static effective Hamiltonian of a driven superconducting nonlinear oscillator, the so-called squeeze-driven Kerr Hamiltonian, and discover a remarkable quasi-spin symmetry su(2) at integer values of the ratio η = Δ / K of the detuning parameter Δ to the Kerr coefficient K. We investigate the stability of this newly discovered symmetry to high-order perturbations arising from the static effective expansion of the driven Hamiltonian. Our finding may find applications in the generation and stabilization of states useful for quantum computing. Finally, we discuss other Hamiltonians with similar properties and within reach of current technologies.
We investigate the quantum properties of a nonlinear Kerr oscillator driven by a three-photon pump. We derive both exact and approximate analytical expressions for the ground state of this interacting model. The exact solution arises at an exact spectral degeneracy, while the approximate solution describes regimes of quasi-degeneracy of the energy spectrum. In both cases, the threefold (quasi)degenerate ground-state manifold consists of quantum superpositions of three macroscopically distinct states. These states differ qualitatively from conventional three-component Schrödinger's cat states due to the presence of squeezing with a distinctive parametric dependence. By varying the detuning between the oscillator and the three-photon pump, we show that the squeezing can be enhanced, suppressed, or even reversed, leading to a squeezing-to-anti-squeezing transition. We analyze the generation and stabilization of these superposition states, their robustness against perturbations and analytically quantify the leakage to excited states. Our analysis elucidates how the three-photon Kerr parametric oscillator can be used to encode a Kerr-cat qutrit protected against phase-flip errors.
In the present work we study the phase space picture of squeezing using the model of the driven generalized parametric oscillator. Exploiting the SU (1, 1)⊕ h (4) algebraic structure of the Wigner operator we find the exact solution of the time dependent Wigner equation as well as the variances of the operators q and p. Due to the fact that we have not any violation of the Liouville's theorem, this model is appropriate to study pure quantum effects (for instance tunneling time) with the classical trajectory molecular dynamics technique.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
One-dimensional quantum systems that undergo spontaneous symmetry-breaking, having a symmetric (non-degenerate) and a broken-symmetry (doubly-degenerate) phase, have been intensely studied in different branches of physics. In most cases, the spontaneously-broken symmetry is parity. However, it is possible to obtain similar phases in systems without parity symmetry, through an antiunitary symmetry that implies a two-fold symmetry either on momentum or coordinate in the system's classical limit. To illustrate this phenomenon, we use a Kerr parametric oscillator (KPO) with one- and two-photon drives that, despite the breaking of parity symmetry, may have doubly-degenerate levels. Different realizations of squeezed KPOs convey a great deal of attention, as effective Hamiltonians for driven superconducting circuits and the occurrence of degeneracy in such systems could be of practical interest in their application to obtain protected qubits in parity-breaking setups. In addition to this, the reported spectral features strongly indicate the existence of additional symmetries in the system.