Gaussian trajectory approach to dissipative phase transitions: The case of quadratically driven photonic lattices

Verstraelen, Wouter; Rota, Riccardo; Savona, Vincenzo; Wouters, Michiel

doi:10.1103/physrevresearch.2.022037

Cited by 33 publications

(20 citation statements)

References 69 publications

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“…This is an exciting perspective, particularly concerning the statistical convergence of the density matrix reconstruction with the quantum trajectory method (one needs to average over an infinite number of quantum trajectories to obtain the exact evolution of the master equation [27,123,124]). Another future outlook is to extend these ideas to approximated time evolutions of open quantum systems, such as those stemming from truncated Wigner [49,50,62,125] and Gaussian ansatzes [53,106]. In these cases, the expectation values along a trajectory allow to approximately reconstruct the expectation values of all the operators.…”

Section: Discussionmentioning

confidence: 99%

“…The single cavity problem (L = 1) is the standard Kerr resonator, and has been analytically solved for its steady state [100][101][102], while lattice-like models have been investigated through a variety of other methods [29,32,49,50,[52][53][54][103][104][105][106], including the exact diagonalization (ED) of the full Liouvillian [23,52]. The DDBH is known to be characterized by a dissipative phase transition in the thermodynamic limit of infinite cavities.…”

Section: The Time-independent Drivendissipative Bose-hubbard Modelmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Discussionmentioning

confidence: 99%

Section: The Time-independent Drivendissipative Bose-hubbard Modelmentioning

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

View full text Add to dashboard Cite

show abstract

“…This is an exciting perspective, particularly concerning the statistical convergence of the density matrix reconstruction with the quantum trajectory method (one needs to average over an infinite number of quantum trajectories to obtain the exact evolution of the master equation [27,119,120]). Another future outlook is to extend these ideas to approximated time evolutions of open quantum systems, such as those stemming from truncated Wigner [49,50,61,121] and Gaussian ansatzes [53,103]. In these cases, the expectation values along a trajectory allow to approximately reconstruct the expectation values of all the operators.…”

Section: Discussionmentioning

confidence: 99%

“…The single cavity problem (L = 1) is the standard Kerr resonator, and has been analytically solved for its steady state [97][98][99], while lattice-like models have been investigated through a variety of other methods [29,32,49,50,[52][53][54][100][101][102][103], none other than the exact diagonalization (ED) of the full Liouvillian [23,52]. The DDBH is known to be characterized by a dissipative phase transition in the thermodynamic limit of infinite cavities.…”

Section: The Time-independent Drivendissipative Bose-hubbard Modelmentioning

confidence: 99%

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

Minganti,

Huybrechts

2021

Preprint

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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. 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.

show abstract

“…A notable feature is that it treats integrable and nonintegrable problems on a similar footing, including those in higher dimensions. It also offers opportunities for developing links to a diverse body of phase space approaches which have attracted attention in recent years [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. In previous work [18], we showed that the stochastic approach to quantum spins could be significantly improved by a two-patch parameterization of the Bloch sphere, in conjunction with a higher-order numerical integration scheme.…”

Section: Introductionmentioning

confidence: 99%

Time-evolving Weiss fields in the stochastic approach to quantum spins

2021

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We investigate nonequilibrium quantum spin systems via an exact mapping to stochastic differential equations. This description is invariant under a shift in the mean of the Gaussian noise. We show that one can extend the simulation time for real-time dynamics in one and two dimensions by a judicious choice of this shift. This can be updated dynamically in order to reduce the impact of stochastic fluctuations. We discuss the connection to drift gauges in the gauge-P literature.

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Gaussian trajectory approach to dissipative phase transitions: The case of quadratically driven photonic lattices

Cited by 33 publications

References 69 publications

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

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

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

Time-evolving Weiss fields in the stochastic approach to quantum spins

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