$${\mathscr{PT}}$$ -symmetry from Lindblad dynamics in a linearized optomechanical system

Ávila, B. Jaramillo; Ventura-Velázquez, C.; León‐Montiel, Roberto de J.; Joglekar, Yogesh N.; Rodríguez-Lara, B. M.

doi:10.1038/s41598-020-58582-7

Cited by 31 publications

(15 citation statements)

References 37 publications

(52 reference statements)

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“…Most models [35][36][37][38][39] start with a non-Hermitian Hamiltonian (containing complex frequencies) that is applicable only in the classical limit where the number of energy quanta is much larger than one. For dissipative open quantum systems, one can write down a Lindblad equation from which one can get a non-Hermitian Hamiltonian by ignoring quantum jumps [62,66,84,85,88,121,122]. This approach has been generalized to gain-loss systems, with and without neglecting the quantum jumps, by writing down Lindblad terms for gain and loss phenomenologically [44,86,87,116,123].…”

Section: Discussionmentioning

confidence: 99%

“…The dynamics generated by a non-Hermitian Hamiltonian is not unitary, and such dynamics can result from an open quantum system coupled to one or more environments (baths). Non-Hermitian Hamiltonians arising out of phenomenological Lindblad equations have been theoretically explored in several recent works (for example, [84][85][86][87][88]). However, a complete microscopic theory of an open quantum system showing the emergence of a non-Hermitian Hamiltonian with a PT transition starting from a fully Hermitian Hamiltonian description of a system coupled to multiple baths has not yet been explored.…”

Section: Introductionmentioning

confidence: 99%

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Emergent PT symmetry in a double-quantum-dot circuit QED setup

Purkayastha

Kulkarni

Joglekar

2020

Phys. Rev. Research

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Open classical and quantum systems with effective parity-time (PT) symmetry, over the past five years, have shown tremendous promise for advances in lasers, sensing, and nonreciprocal devices. And yet, how such effective PT-symmetric non-Hermitian models emerge out of Hermitian quantum mechanics is not well understood. Here, starting from a fully Hermitian microscopic Hamiltonian description, we show that a non-Hermitian Hamiltonian emerges naturally in a double-quantum-dot (DQD) circuit-QED setup, which can be controllably tuned to the PT-symmetric point. This effective Hamiltonian governs the dynamics of two coupled circuit-QED cavities with a voltage-biased DQD in one of them. Our analysis also reveals the effect of quantum fluctuations on the PT-symmetric system. The PT transition is, then, observed both in the dynamics of cavity observables as well as via an input-output experiment. As a simple application of the PT transition in this setup, we show that loss-induced enhancement of amplification and lasing can be observed in the coupled cavities. By comparing our results with two conventional local Lindblad equations, we demonstrate the utility and limitations of the latter. Our results pave the way for an on-chip realization of a potentially scalable non-Hermitian system with a gain medium in the quantum regime, as well as its potential applications for quantum technology.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Emergent PT symmetry in a double-quantum-dot circuit QED setup

Purkayastha

Kulkarni

Joglekar

2020

Phys. Rev. Research

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“…But majority of works phenomenologically assume a non-Hermitian Hamiltonian and quantum and thermal fluctuations are ignored. Only recently some works have gone beyond such descriptions, deriving non-Hermitian dynamics from more microscopic quantum theories (for example, [70,72,[81][82][83][84][85][86][87][88]). On the other hand, classifying all kinds of non-Hermitian Hamiltonians and their relations to topology has remained an extremely active direction of research (for example, [76,[89][90][91][92][93][94]).…”

Section: Introductionmentioning

confidence: 99%

Lyapunov equation in open quantum systems and non-Hermitian physics

Purkayastha

2022

Preprint

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The continuous-time differential Lyapunov equation is widely used in linear optimal control theory, a branch of mathematics and engineering. In quantum physics, it is known to appear in Markovian descriptions of linear (quadratic Hamiltonian, linear equations of motion) open quantum systems, typically from quantum master equations. Despite this, the Lyapunov equation is seldom considered a fundamental formalism for linear open quantum systems. In this work we aim to change that. We establish the Lyapunov equation as a fundamental and efficient formalism for linear open quantum systems that can go beyond the limitations of various standard quantum master equation descriptions, while remaining of much less complexity than general exact formalisms. This also provides valuable insights for non-Hermitian quantum physics. In particular, we derive the Lyapunov equation for the most general number conserving linear system in a lattice of arbitrary dimension and geometry, connected to an arbitrary number of baths at different temperatures and chemical potentials. Three slightly different forms of the Lyapunov equation are derived via an equation of motion approach, by making increasing levels of controlled approximations, without reference to any quantum master equation. Then we discuss their relation with quantum master equations, positivity, accuracy and additivity issues, the possibility of describing dark states, general perturbative solutions in terms of single-particle eigenvectors and eigenvalues of the system, and quantum regression formulas. Our derivation gives a clear understanding of the origin of the non-Hermitian Hamiltonian describing the dynamics and separates it from the effects of quantum and thermal fluctuations. Many of these results would have been hard to obtain via standard quantum master equation approaches.

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“…With respect to an NHH, the Liouvillian also accounts for the presence of quantum jumps. The extension of EPs of NHHs to those based on Liouvillians [57] has shown that quantum jumps can play a crucial role in the properties of EPs [44,[58][59][60][61][62][63]. Furthermore, the evolution of a density matrix of an open quantum system is described by a completelypositive and trace-preserving (CPTP) linear map.…”

Section: Introductionmentioning

confidence: 99%

Generating high-order quantum exceptional points in synthetic dimensions

Arkhipov,

Minganti,

Miranowicz

et al. 2021

Preprint

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Recently, there has been intense research in proposing and developing various methods for constructing high-order exceptional points (EPs) in dissipative systems. These EPs can possess a number of intriguing properties related to, e.g., chiral transport and enhanced sensitivity. Proposals to realize high-order EPs have been based on the use of non-Hermitian Hamiltonians (NHHs) of composite systems, i.e., the operators describing the evolution of coupled post-selected systems or coupled intense light fields. In both cases, quantum jumps play no role. Here, by considering the full quantum dynamics of a quadratic Liouvillian superoperator, we introduce a simple and effective method for engineering NHHs with high-order quantum EPs, derived from evolution matrices of system operators moments. That is, by quantizing higher-order moments of system operators, e.g., of a quadratic two-mode system, the resulting evolution matrices can be interpreted as the new NHHs describing, e.g., networks of coupled resonators. Notably, such a mapping allows to correctly reproduce the results of the Liouvillian dynamics, including quantum jumps. By applying this mapping, we demonstrate that quantum EPs of any order can be engineered in dissipative systems and can, thus, be probed by the coherence and spectral functions. As an example, we consider a U (1)-symmetric quadratic Liouvillian describing an optical cavity with incoherent mode coupling, which can also possess anti-PT -symmetry. Compared to their PT -symmetric counterparts, such anti-PT -symmetric systems could be easier to scale and, thus, can serve as a promising platform for engineering quantum systems with high-order EPs.

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$${\mathscr{PT}}$$ -symmetry from Lindblad dynamics in a linearized optomechanical system

Cited by 31 publications

References 37 publications

Emergent PT symmetry in a double-quantum-dot circuit QED setup

Emergent PT symmetry in a double-quantum-dot circuit QED setup

Lyapunov equation in open quantum systems and non-Hermitian physics

Generating high-order quantum exceptional points in synthetic dimensions

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