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Physical Review A

Quantum optics of lossy metasurfaces: Propagating the photon-moment matrix by the semiclassical Liouvillian

Wai Chun Wong, Kai Ming Lau, Hong Liang, Tsz Kit Yung, Bei Zeng, Jensen Li

Published Jul 11, 2022• 6 citations
Publisher

Abstract

In this paper we map the quantum optical scattering of a linear dissipative metasurface to an equivalent Liouvillian dynamics with respect to the effective Hamiltonian of the metasurface propagating in fictitious time. By using a lossy bianisotropic metasurface as a generic example, our formulation allows a complete specification of the generalized eigenspace of the Liouvillian superoperator starting from either a diagonalizable or defective classical scattering matrix of the metasurface. With a further factorization of the Liouvillian superoperator, we demonstrate the input-output relationship on the photon density matrix can be obtained by propagating the photon-moment matrix using a complete basis of the semiclassical Liouvillian superoperator. Our investigations will be useful for characterizing quantum scattering of generally bianisotropic metasurfaces, paving the way for investigating and designing passive parity-time symmetric and non-Hermitian metasurfaces for quantum information processing applications.

DOI
10.1103/physreva.106.013503

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In this paper we map the quantum optical scattering of a linear dissipative metasurface to an equivalent Liouvillian dynamics with respect to the effective Hamiltonian of the metasurface propagating in fictitious time. By using a lossy bianisotropic metasurface as a generic example, our formulation allows a complete specification of the generalized eigenspace of the Liouvillian superoperator starting from either a diagonalizable or defective classical scattering matrix of the metasurface. With a further factorization of the Liouvillian superoperator, we demonstrate the input-output relationship on the photon density matrix can be obtained by propagating the photon-moment matrix using a complete basis of the semiclassical Liouvillian superoperator. Our investigations will be useful for characterizing quantum scattering of generally bianisotropic metasurfaces, paving the way for investigating and designing passive parity-time symmetric and non-Hermitian metasurfaces for quantum information processing applications.
A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.
We present the point-coupling Hamiltonian as a model for broadband linear optical devices acting on propagating optical modes described as a continua of harmonic oscillators. We formally integrate the Heisenberg equations of motion for this Hamiltonian, calculate its quantum scattering matrix, and show that an application of the quantum scattering matrix on an input state is equivalent to applying the inverse of classical scattering matrix on the annihilation operators describing the optical modes. We show how to construct the point-coupling Hamiltonian corresponding to a general linear optical device described by a classical scattering matrix, and provide examples of Hamiltonians for some commonly used linear optical devices. Finally, in order to demonstrate the practical utility of the point-coupling Hamiltonian, we use it to rigorously formulate a matrix-product-state based simulation for time-delayed feedback systems wherein the feedback is provided by a linear optical device described by a scattering matrix as opposed to a hard boundary condition (e.g. a mirror with less than unity reflectivity).
Open quantum systems interacting with an environment exhibit dynamics described by the combination of dissipation and coherent Hamiltonian evolution. Taken together, these effects are captured by a Liouvillian superoperator. The degeneracies of the (generically non-Hermitian) Liouvillian are exceptional points, which are associated with critical dynamics as the system approaches steady state. We use a superconducting transmon circuit coupled to an engineered environment to observe two different types of Liouvillian exceptional points that arise either from the interplay of energy loss and decoherence or purely due to decoherence. By dynamically tuning the Liouvillian superoperators in real time we observe a non-Hermiticity-induced chiral state transfer. Our study motivates a new look at open quantum system dynamics from the vantage of Liouvillian exceptional points, enabling applications of non-Hermitian dynamics in the understanding and control of open quantum systems.
Over the past decade, classical optical systems with gain or loss, modeled by non-Hermitian parity-time symmetric Hamiltonians, have been deeply investigated. Yet, their applicability to the quantum domain with number-resolved photonic states is fundamentally voided by quantum-limited amplifier noise. Here, we show that second-quantized Hermitian Hamiltonians on the Fock space give rise to non-Hermitian effective Hamiltonians that generate the dynamics of corresponding creation and annihilation operators. Using this equivalence between PT-symmetry and symplectic Bogoliubov transformations, we create a quantum optical scheme comprising squeezing, phase-shifters, and beam-splitters for simulating arbitrary non-unitary processes by way of singular value decomposition. In contrast to the post-selection scheme for non-Hermitian quantum simulation, the success probability in this approach is independent of the system size or simulation time and can be efficiently Trotterised similar to a unitary transformation.
In this thesis, we study semiclassical phase-space methods for quantum evolution in Hermitian and non-Hermitian systems. We first present the dynamics of Gaussian wave packets under non-Hermitian Hamiltonians and interpret them as a classical dynamics for complex Hamiltonians. We use these to derive exact dynamics for wave packets in the quadratic Swanson oscillator. We show that in the case of unbroken PT-symmetry there can be periodic divergences in this system and relate this to the fact that any operator mapping the system to a Hermitian counterpart is unbounded. We apply the semiclassical wave-packet dynamics to two further anharmonic example systems: a PT-symmetric wave guide, a version of which we propose as a filtering device for optical beams, and a non-Hermitian single-band tight-binding model, for which we use classical equations of motion to model both narrowly and widely distributed initial states. We further develop an exact quantum propagator for non-Hermitian dynamics using lattices of wave packets, whose evolution is governed by the semiclassical equations of motion. We demonstrate that this accurately reproduces quantum dynamics compared to the split operator method. Finally we study a Hermitian two-mode many-particle model for bosonic atom-molecule conversion, for which the classical phase-space structure is an orbifold. We show that standard semiclassical tools may be applied to recover features such as the dynamics, spectrum and density of states of the many-particle system.
Abstract Open quantum systems that interact with a Markovian environment can be described by a Lindblad master equation. The generator of time-translation is given by a Liouvillian superoperator acting on the density matrix of the system. As the Fock space for a single bosonic mode is already infinite-dimensional, the diagonalization of the Liouvillian has to be done on the creation- and annihilation-superoperators, a process called ‘third quantization’. We propose a method to solve the Liouvillian for quadratic systems using a single symplectic transformation. We show that the non-Hermitian effective Hamiltonian of the system, next to incorporating the dynamics of the system, is a tool to analyze its symmetries. As an example, we use the effective Hamiltonian to formulate a -‘symmetry’ of an open system. We describe how the inclusion of source terms allows us to obtain the cumulant generating function for observables such as the photon current.