Validity of mean-field theory in a dissipative critical system: Liouvillian gap, 
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-symmetric antigap, and permutational symmetry in the 
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 model

Huybrechts, Dolf; Minganti, Fabrizio; Nori, Franco; Wouters, Michiel; Shammah, Nathan

doi:10.1103/physrevb.101.214302

Cited by 57 publications

(33 citation statements)

References 152 publications

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“…In this paper we have mainly focused on the steady state ρ 0 , which is determined for all parameters by the zero eigenvector of L. In classical systems, PT -symmetry breaking is usually discussed in terms of a transition from purely oscillatory to exponentially damped or amplified dynamics, which is associated with the appearance of exceptional points in the eigenspectrum of the dynamical matrix. This has motivated similar studies of the spectra of Liouville operators, where the appearance of exceptional points [56,57] or additional symmetries in the complex eigenvalue structure [33,47,48] have been discussed. In Fig.…”

Section: Resultsmentioning

confidence: 96%

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2020

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The effect of PT-symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of PT symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected PTsymmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe PT-symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes. B Fully symmetric steady state 11 C Random jump operators 12 References 13

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Section: Resultsmentioning

confidence: 96%

“…While the systems studied in Refs. [33,47,48] satisfy Eq. (4) with a redefinition of P, none of the examples discussed below exhibits the symmetry considered in these references when d > 2.…”

Section: Pt -Symmetric Quantum Systemsmentioning

confidence: 99%

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“…[32] and analyzed for different spin models in Refs. [33,47,48]. In those systems, the eigenvalues of the Liouvillian exhibit a particular symmetry in terms of a cross-shaped pattern in the complex plane, which is broken at larger values of Γ .…”

Section: Pt -Symmetric Quantum Systemsmentioning

confidence: 99%

“…However, this property is rather specific and while the systems studied in Refs. [32,33,47,48] satisfy Eq. (4) with a redefinition of P, none of the examples discussed below exhibits the symmetry considered in these references when d > 2.…”

Section: Pt -Symmetric Quantum Systemsmentioning

confidence: 99%

Emergence of PT-symmetry breaking in open quantum systems

et al. 2020

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The effect of \mathcal{PT}𝒫𝒯-symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT}𝒫𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT}𝒫𝒯-symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT}𝒫𝒯-symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.

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“…[28] proposes a variational ansatz valid for two weakly-coupled resonators. Methods based on mean-field approximations determine efficiently the state of lattices of nonlinear resonators [29][30][31][32][33][34], but their predictions are ensured to be valid only for weak nonlinearities or for high-connectivity models [35]. Monte Carlo [36], tensor network [37][38][39], and neural network [40][41][42] algorithms proved efficient even for highly-entangled phases, but can be burdened by various factors, such as the exponential growth of a density matrix, entropy, or correlations.…”

Section: Introductionmentioning

confidence: 99%

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

Minganti

Huybrechts

2022

Quantum

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The characterization of open quantum systems is a central and recurring problem for the development of quantum technologies. For time-independent systems, an (often unique) steady state describes the average physics once all the transient processes have faded out, but interesting quantum properties can emerge at intermediate timescales. Given a Lindblad master equation, these properties are encoded in the spectrum of the Liouvillian whose diagonalization, however, is a challenge even for small-size quantum systems. Here, we propose a new method to efficiently provide the Liouvillian spectral decomposition. We call this method an Arnoldi-Lindblad time evolution, because it exploits the algebraic properties of the Liouvillian superoperator to efficiently construct a basis for the Arnoldi iteration problem. The advantage of our method is double: (i) It provides a faster-than-the-clock method to efficiently obtain the steady state, meaning that it produces the steady state through time evolution shorter than needed for the system to reach stationarity. (ii) It retrieves the low-lying spectral properties of the Liouvillian with a minimal overhead, allowing to determine both which quantum properties emerge and for how long they can be observed in a system. This method is general and model-independent, and lends itself to the study of large systems where the determination of the Liouvillian spectrum can be numerically demanding but the time evolution of the density matrix is still doable. Our results can be extended to time evolution with a time-dependent Liouvillian. In particular, our method works for Floquet (i.e., periodically driven) systems, where it allows not only to construct the Floquet map for the slow-decaying processes, but also to retrieve the stroboscopic steady state and the eigenspectrum of the Floquet map. Although the method can be applied to any Lindbladian evolution (spin, fermions, bosons, …), for the sake of simplicity we demonstrate the efficiency of our method on several examples of coupled bosonic resonators (as a particular example). Our method outperforms other diagonalization techniques and retrieves the Liouvillian low-lying spectrum even for system sizes for which it would be impossible to perform exact diagonalization.

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Validity of mean-field theory in a dissipative critical system: Liouvillian gap, PT -symmetric antigap, and permutational symmetry in the XYZ model

Cited by 57 publications

References 152 publications

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Emergence of PT-symmetry breaking in open quantum systems

Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems

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