Spectral theory of Liouvillians for dissipative phase transitions

Minganti, Fabrizio; Biella, Alberto; Bartolo, N.; Ciuti, Cristiano

doi:10.1103/physreva.98.042118

Cited by 323 publications

(396 citation statements)

References 80 publications

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“…3-(e)], we notice that the eigenvalue with largest real part is purely real, signaling the disappearance of the long-lived oscillation of the DTC-phase. Moreover, we can notice also that this eigenvalue goes to zero in the thermodynamic limit: this behavior can be associated to the closing of the Liouvillian gap in the vicinity of a critical point, and hence supports the evidence for a first-order dissipative phase transition [67], as already indicated by the results in Fig. 1.…”

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confidence: 87%

Dissipative time crystal in an asymmetric nonlinear photonic dimer

2020

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We investigate the behavior of two coupled non-linear photonic cavities, in presence of inhomogeneous coherent driving and local dissipations. By solving numerically the quantum master equation, either by diagonalizing the Liouvillian superoperator or by using the approximated truncated Wigner approach, we extrapolate the properties of the system in a thermodynamic limit of large photon occupation. When the mean field Gross-Pitaevskii equation predicts a unique parametrically unstable steady-state solution, the open quantum many-body system presents highly non-classical properties and its dynamics exhibits the long lived Josephson-like oscillations typical of dissipative time crystals, as indicated by the presence of purely imaginary eigenvalues in the spectrum of the Liouvillian superoperator in the thermodynamic limit.Open many-body quantum systems [1-3] have become a major field of study over the last decade. The open nature is common to a vast class of modern experimental platforms in quantum science and technology, such as photonic systems [4], ultracold atoms [5-9], optomechanical systems [10][11][12][13] or superconducting circuits [14][15][16], for which driving and losses are omnipresent. Open quantum systems also display emergent physics, in particular dissipative phase transitions and topological phases [40][41][42][43][44][45].Several studies have highlighted the possibility for a continuous-wave driven-dissipative quantum system to reach a non-stationary state in the long time limit in which undamped oscillations arise spontaneously [46][47][48][49][50][51][52]. This phenomenon has been dubbed as boundary or dissipative time crystal (DTC), in analogy with the time crystals in some Hamiltonian systems [53]. Formally, DTCs are associated with the occurrence of multiple eigenvalues of the Liouvillian with vanishing real and finite imaginary part [54][55][56]. The experimental feasibility of DTC has been confirmed by their observation in phosphorousdoped silicon [57]. The research for further platforms showing this phenomenon is very active and important to understand the mechanisms behind the spontaneous breaking of the time-translation symmetry in open quantum many-body systems.One of the main difficulties in the realization of DTCs in real system is related to the fragility of this phase to external perturbations which affect the symmetric structure of the model. Indeed, in most of the cases considered so far, the engineering of the DTCs relies on the exploitation of certain symmetries (either manifest [48,52] or emergent [50]) in the Hamiltonian or in the dissipation mechanism, which can be hard to maintain in real driven-dissipative systems out of equilibrium.In this letter, we show that a DTC can arise in a simple system of two coupled photonic cavities, whose equation of motion does not preserve any symmetry but the timetranslation invariance. In a broad region of the parameter space, the dynamics of this system presents limit cycles associated to parametric instabilities [58], which can be regar...

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confidence: 87%

Dissipative time crystal in an asymmetric nonlinear photonic dimer

2020

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“…Eq. (12) shows that for the estimation of ω 0 , the presence of dissipation restores the shot-noise scaling, similar to what happens in a lossy interferometric protocols. In the case in which the parameter to be estimated is the spin frequency Ω, the presence of dissipation replaces the quartic time-scaling obtained in the Hamiltonian case (8) by a quadratic one.…”

Section: Analysis Of Resourcessupporting

confidence: 62%

Critical Quantum Metrology with a Finite-Component Quantum Phase Transition

et al. 2020

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Physical systems close to a quantum phase transition exhibit a divergent susceptibility, suggesting that an arbitrarily-high precision may be achieved by exploiting quantum critical systems as probes to estimate a physical parameter. However, such an improvement in sensitivity is counterbalanced by the closing of the energy gap, which implies a critical slowing down and an inevitable growth of the protocol duration. Here, we design different metrological protocols that make use of the superradiant phase transition of the quantum Rabi model, a finite-component system composed of a single two-level atom interacting with a single bosonic mode. We show that, in spite of the critical slowing down, critical quantum optical systems can lead to a quantum-enhanced time-scaling of the quantum Fisher information, and so of the measurement sensitivity.In a system close to a critical point, small variations of physical parameters may lead to dramatic changes in the equilibrium state properties. The possibility of exploiting this sensitivity for metrological purposes is well known, and it has already been applied in classical devices, e.g. in superconducting transition-edge sensor [1]. Besides, the development of quantum metrology has extensively shown that quantum states can outperform their classical counterparts for sensing tasks [2]. Therefore, a question naturally arises: what sensitivity can be achieved using interacting systems close to a quantum-critical point? In the last few years, this question has attracted growing interest and it has been addressed by different methods [3][4][5][6][7][8][9]. These studies may be roughly divided in two classes.The first approach, which we will call the "dynamical" paradigm [5,7], focus on the time evolution induced by a Hamiltonian close to a critical point. In this approach, one prepares a probe system in a suitably chosen state, lets it evolve according to the critical Hamiltonian, and finally measures it. This bear close similarity to the standard interferometric paradigm of quantum metrology [2]. On the other hand, the "static" approach [3, 6] is based on the equilibrium properties of the system. It consists in preparing and measuring the system ground state in the unitary case, or the system steady-state when open quantum systems are considered. In proximity of the phase transition the susceptibility of the equilibrium state diverges, and so it does the achievable measurement precision. Unfortunately, the time required to prepare the equilibrium state diverges as well, both in the unitary [10] and in the driven-dissipative case [11,12], a behavior called critical slowing down. Only very re-cently, it has been demonstrated that for a large class of spin models these two approaches are formally equivalent [9], and that they both make it possible to achieve the optimal scaling limit of precision with respect to system size and to measurement time. These results were obtained considering spin systems that undergo quantum phase transitions in the thermodynamic limit, where the numb...

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“…This could be particularly useful in extended systems, e.g. to determine a dissipative phase transition in which the eigenvalue λ α with smallest non-zero real part becomes purely imaginary [40][41][42]. We note that in general it is difficult to compute the spectrum of a Liouvillian, whereas the Green's function may be found via many body techniques(see, e.g.…”

Section: Open System Casementioning

confidence: 99%

Spectral functions and negative density of states of a driven-dissipative nonlinear quantum resonator

2019

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Keywords: open quantum systems, quantum van der pol oscillator, driven and dissipative quantum systems AbstractWe study the spectral properties of Markovian driven-dissipative quantum systems, focusing on the nonlinear quantum van der Pol oscillator as a paradigmatic example. We discuss a generalized Lehmann representation, in which single-particle Greenʼs functions are expressed in terms of the eigenstates and eigenvalues of the Liouvillian. Applying it to the quantum van der Pol oscillator, we find a wealth of phenomena that are not apparent in the steady-state density matrix alone. Unlike the steady state, the photonic spectral function has a strong dependence on interaction strength. Further, we find that the interplay of interaction and non-equilibrium effects can result in a surprising 'negative density of states', associated with a negative temperature, even in absence of steady state population inversion.

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Spectral theory of Liouvillians for dissipative phase transitions

Cited by 323 publications

References 80 publications

Dissipative time crystal in an asymmetric nonlinear photonic dimer

Dissipative time crystal in an asymmetric nonlinear photonic dimer

Critical Quantum Metrology with a Finite-Component Quantum Phase Transition

Spectral functions and negative density of states of a driven-dissipative nonlinear quantum resonator

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