1
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Daviet R, Zelle CP, Rosch A, Diehl S. Nonequilibrium Criticality at the Onset of Time-Crystalline Order. PHYSICAL REVIEW LETTERS 2024; 132:167102. [PMID: 38701486 DOI: 10.1103/physrevlett.132.167102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2024] [Accepted: 03/25/2024] [Indexed: 05/05/2024]
Abstract
We explore the phase transitions at the onset of time-crystalline order in O(N) models driven out of equilibrium. The spontaneous breaking of time translation symmetry and its Goldstone mode are captured by an effective description with O(N)×SO(2) symmetry, where the emergent external SO(2) results from a transmutation of the internal symmetry of time translations. Using the renormalization group and the ε=4-d expansion in a leading two-loop analysis, we identify a new nonequilibrium universality class. Strikingly, it controls the long-distance physics no matter how small the microscopic breaking of equilibrium conditions is. The O(N=2)×SO(2) symmetry group is realized for magnon condensation in pumped yttrium iron garnet films and in exciton-polariton systems with a polarization degree of freedom.
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Affiliation(s)
- Romain Daviet
- Institut für Theoretische Physik, Universität zu Köln, 50937 Cologne, Germany
| | - Carl Philipp Zelle
- Institut für Theoretische Physik, Universität zu Köln, 50937 Cologne, Germany
| | - Achim Rosch
- Institut für Theoretische Physik, Universität zu Köln, 50937 Cologne, Germany
| | - Sebastian Diehl
- Institut für Theoretische Physik, Universität zu Köln, 50937 Cologne, Germany
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2
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Begg SE, Hanai R. Quantum Criticality in Open Quantum Spin Chains with Nonreciprocity. PHYSICAL REVIEW LETTERS 2024; 132:120401. [PMID: 38579202 DOI: 10.1103/physrevlett.132.120401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2023] [Revised: 12/26/2023] [Accepted: 02/20/2024] [Indexed: 04/07/2024]
Abstract
We investigate the impact of nonreciprocity on universality and critical phenomena in open quantum interacting many-body systems. Nonreciprocal open quantum systems often have an exotic spectral sensitivity to boundary conditions, known as the Liouvillian skin effect (LSE). By considering an open quantum XXZ spin chain that exhibits LSE, we demonstrate the existence of a universal scaling regime that is not affected by the presence of the LSE. We resolve the critical exponents, which differ from those of free fermions, via tensor network methods and demonstrate that observables exhibit a universal scaling collapse, irrespective of the reciprocity. We find that the LSE only becomes relevant when a healing length scale ξ_{heal} at the system's edge (which is different from the localization length of the eigenstate of the Liouvillian) exceeds the system size, allowing edge properties to dominate the physics. We expect this result to be a generic feature of nonreciprocal models in the vicinity of a critical point. The driven-dissipative quantum criticality we observe has no classical analog and stems from the existence of multiple dark states.
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Affiliation(s)
- Samuel E Begg
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea
| | - Ryo Hanai
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea
- Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
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3
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Marino J. Universality Class of Ising Critical States with Long-Range Losses. PHYSICAL REVIEW LETTERS 2022; 129:050603. [PMID: 35960567 DOI: 10.1103/physrevlett.129.050603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2021] [Revised: 01/27/2022] [Accepted: 07/08/2022] [Indexed: 06/15/2023]
Abstract
We show that spatial resolved dissipation can act on d-dimensional spin systems in the Ising universality class by qualitatively modifying the nature of their critical points. We consider power-law decaying spin losses with a Lindbladian spectrum closing at small momenta as ∝q^{α}, with α a positive tunable exponent directly related to the power-law decay of the spatial profile of losses at long distances, 1/r^{(α+d)}. This yields a class of soft modes asymptotically decoupled from dissipation at small momenta, which are responsible for the emergence of a critical scaling regime ascribable to the nonunitary counterpart of the universality class of long-range interacting Ising models. For α<1 we find a nonequilibrium critical point ruled by a dynamical field theory described by a Langevin model with coexisting inertial (∼∂_{t}^{2}) and frictional (∼∂_{t}) kinetic coefficients, and driven by a gapless Markovian noise with variance ∝q^{α} at small momenta. This effective field theory is beyond the Halperin-Hohenberg description of dynamical criticality, and its critical exponents differ from their unitary long-range counterparts. Our Letter lays out perspectives for a revision of universality in driven open systems by employing dark states tailored by programmable dissipation.
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Affiliation(s)
- Jamir Marino
- Institut für Physik, Johannes Gutenberg Universität Mainz, D-55099 Mainz, Germany and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA
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4
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Singh VP, Weimer H. Driven-Dissipative Criticality within the Discrete Truncated Wigner Approximation. PHYSICAL REVIEW LETTERS 2022; 128:200602. [PMID: 35657854 DOI: 10.1103/physrevlett.128.200602] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2021] [Accepted: 04/14/2022] [Indexed: 06/15/2023]
Abstract
We present an approach to the numerical simulation of open quantum many-body systems based on the semiclassical framework of the discrete truncated Wigner approximation. We establish a quantum jump formalism to integrate the quantum master equation describing the dynamics of the system, which we find to be exact in both the noninteracting limit and the limit where the system is described by classical rate equations. We apply our method to simulation of the paradigmatic dissipative Ising model, where we are able to capture the critical fluctuations of the system beyond the level of mean-field theory.
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Affiliation(s)
- Vijay Pal Singh
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
- Zentrum für Optische Quantentechnologien and Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany
| | - Hendrik Weimer
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
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5
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Luo D, Chen Z, Carrasquilla J, Clark BK. Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation. PHYSICAL REVIEW LETTERS 2022; 128:090501. [PMID: 35302809 DOI: 10.1103/physrevlett.128.090501] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2020] [Revised: 05/19/2021] [Accepted: 01/05/2022] [Indexed: 06/14/2023]
Abstract
The theory of open quantum systems lays the foundation for a substantial part of modern research in quantum science and engineering. Rooted in the dimensionality of their extended Hilbert spaces, the high computational complexity of simulating open quantum systems calls for the development of strategies to approximate their dynamics. In this Letter, we present an approach for tackling open quantum system dynamics. Using an exact probabilistic formulation of quantum physics based on positive operator-valued measure, we compactly represent quantum states with autoregressive neural networks; such networks bring significant algorithmic flexibility due to efficient exact sampling and tractable density. We further introduce the concept of string states to partially restore the symmetry of the autoregressive neural network and improve the description of local correlations. Efficient algorithms have been developed to simulate the dynamics of the Liouvillian superoperator using a forward-backward trapezoid method and find the steady state via a variational formulation. Our approach is benchmarked on prototypical one-dimensional and two-dimensional systems, finding results which closely track the exact solution and achieve higher accuracy than alternative approaches based on using Markov chain Monte Carlo method to sample restricted Boltzmann machines. Our Letter provides general methods for understanding quantum dynamics in various contexts, as well as techniques for solving high-dimensional probabilistic differential equations in classical setups.
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Affiliation(s)
- Di Luo
- Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA
- IQUIST and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Illinois 61801, USA
| | - Zhuo Chen
- Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA
| | - Juan Carrasquilla
- Vector Institute for Artificial Intelligence, MaRS Centre, Toronto, Ontario, Canada
- Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1,Canada
| | - Bryan K Clark
- Department of Physics, University of Illinois at Urbana-Champaign, Illinois 61801, USA
- IQUIST and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Illinois 61801, USA
- NCSA Center for Artificial Intelligence Innovation, University of Illinois at Urbana-Champaign, Illinois 61801, USA
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6
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Lieu S, Belyansky R, Young JT, Lundgren R, Albert VV, Gorshkov AV. Symmetry Breaking and Error Correction in Open Quantum Systems. PHYSICAL REVIEW LETTERS 2020; 125:240405. [PMID: 33412027 DOI: 10.1103/physrevlett.125.240405] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2020] [Accepted: 11/10/2020] [Indexed: 06/12/2023]
Abstract
Symmetry-breaking transitions are a well-understood phenomenon of closed quantum systems in quantum optics, condensed matter, and high energy physics. However, symmetry breaking in open systems is less thoroughly understood, in part due to the richer steady-state and symmetry structure that such systems possess. For the prototypical open system-a Lindbladian-a unitary symmetry can be imposed in a "weak" or a "strong" way. We characterize the possible Z_{n} symmetry-breaking transitions for both cases. In the case of Z_{2}, a weak-symmetry-broken phase guarantees at most a classical bit steady-state structure, while a strong-symmetry-broken phase admits a partially protected steady-state qubit. Viewing photonic cat qubits through the lens of strong-symmetry breaking, we show how to dynamically recover the logical information after any gap-preserving strong-symmetric error; such recovery becomes perfect exponentially quickly in the number of photons. Our study forges a connection between driven-dissipative phase transitions and error correction.
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Affiliation(s)
- Simon Lieu
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Ron Belyansky
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Jeremy T Young
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Rex Lundgren
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Victor V Albert
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
- Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
- National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
| | - Alexey V Gorshkov
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
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7
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Crowley PJD, Martin I, Chandran A. Half-Integer Quantized Topological Response in Quasiperiodically Driven Quantum Systems. PHYSICAL REVIEW LETTERS 2020; 125:100601. [PMID: 32955337 DOI: 10.1103/physrevlett.125.100601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2019] [Revised: 12/19/2019] [Accepted: 08/04/2020] [Indexed: 06/11/2023]
Abstract
A spin strongly driven by two harmonic incommensurate drives can pump energy from one drive to the other at a quantized average rate, in close analogy with the quantum Hall effect. The pumping rate is a nonzero integer in the topological regime, while the trivial regime does not pump. The dynamical transition between the regimes is sharp in the zero-frequency limit and is characterized by a Dirac point in a synthetic band structure. We show that the pumping rate is half-integer quantized at the transition and present universal Kibble-Zurek scaling functions for energy transfer processes. Our results adapt ideas from quantum phase transitions, quantum information, and topological band theory to nonequilibrium dynamics, and identify qubit experiments to observe the universal linear and nonlinear response of a Dirac point in synthetic dimensions.
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Affiliation(s)
- P J D Crowley
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - I Martin
- Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
| | - A Chandran
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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8
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Lundgren R, Gorshkov AV, Maghrebi MF. Nature of the nonequilibrium phase transition in the non-Markovian driven Dicke model. PHYSICAL REVIEW. A 2020; 102:10.1103/PhysRevA.102.032218. [PMID: 34136732 PMCID: PMC8204515 DOI: 10.1103/physreva.102.032218] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The Dicke model famously exhibits a phase transition to a superradiant phase with a macroscopic population of photons and is realized in multiple settings in open quantum systems. In this paper, we study a variant of the Dicke model where the cavity mode is lossy due to the coupling to a Markovian environment while the atomic mode is coupled to a colored bath. We analytically investigate this model by inspecting its low-frequency behavior via the Schwinger-Keldysh field theory and carefully examine the nature of the corresponding superradiant phase transition. Integrating out the fast modes, we can identify a simple effective theory allowing us to derive analytical expressions for various critical exponents including the dynamical exponent. We find excellent agreement with previous numerical results when the non-Markovian bath is at zero temperature; however, contrary to these studies, our low-frequency approach reveals that the same exponents govern the critical behavior when the colored bath is at finite temperature unless the chemical potential is zero. Furthermore, we show that the superradiant phase transition is classical in nature, while it is genuinely nonequilibrium. We derive a fractional Langevin equation and conjecture the associated fractional Fokker-Planck equation that captures the system's long-time memory as well as its nonequilibrium behavior. Finally, we consider finite-size effects at the phase transition and identify the finite-size scaling exponents, unlocking a rich behavior in both statics and dynamics of the photonic and atomic observables.
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Affiliation(s)
- Rex Lundgren
- Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, College Park, Maryland 20742, USA
| | - Alexey V Gorshkov
- Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, National Institute of Standards and Technology and University of Maryland, College Park, Maryland 20742, USA
| | - Mohammad F Maghrebi
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
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9
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Geng Z, Peters KJH, Trichet AAP, Malmir K, Kolkowski R, Smith JM, Rodriguez SRK. Universal Scaling in the Dynamic Hysteresis, and Non-Markovian Dynamics, of a Tunable Optical Cavity. PHYSICAL REVIEW LETTERS 2020; 124:153603. [PMID: 32357047 DOI: 10.1103/physrevlett.124.153603] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2019] [Accepted: 03/31/2020] [Indexed: 06/11/2023]
Abstract
We investigate, experimentally and theoretically, the dynamics of a laser-driven cavity with noninstantaneous effective photon-photon interactions. Scanning the laser-cavity frequency detuning at different speeds across an optical bistability, we find a hysteresis area that is a nonmonotonic function of the speed. In the limit of fast scans comparable to the memory time of the interactions, we demonstrate that the hysteresis area decays following a universal power law with scaling exponent -1. We further demonstrate a regime of non-Markovian dynamics emerging from white noise. This regime is evidenced by peaked distributions of residence times in the metastable states of our system. Our results offer new perspectives for exploring the physics of scaling, universality, and metastability, in non-Markovian regimes using arrays of bistable optical cavities with low quality factors, driven by low laser powers, and at room temperature.
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Affiliation(s)
- Z Geng
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - K J H Peters
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - A A P Trichet
- Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
| | - K Malmir
- Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
| | - R Kolkowski
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
| | - J M Smith
- Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
| | - S R K Rodriguez
- Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
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10
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Landa H, Schiró M, Misguich G. Multistability of Driven-Dissipative Quantum Spins. PHYSICAL REVIEW LETTERS 2020; 124:043601. [PMID: 32058770 DOI: 10.1103/physrevlett.124.043601] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2019] [Indexed: 06/10/2023]
Abstract
We study the dynamics of lattice models of quantum spins one-half, driven by a coherent drive and subject to dissipation. Generically the mean-field limit of these models manifests multistable parameter regions of coexisting steady states with different magnetizations. We introduce an efficient scheme accounting for the corrections to mean field by correlations at leading order, and benchmark this scheme using high-precision numerics based on matrix-product operators in one- and two-dimensional lattices. Correlations are shown to wash the mean-field bistability in dimension one, leading to a unique steady state. In dimension two and higher, we find that multistability is again possible, provided the thermodynamic limit of an infinitely large lattice is taken first with respect to the longtime limit. Variation of the system parameters results in jumps between the different steady states, each showing a critical slowing down in the convergence of perturbations towards the steady state. Experiments with trapped ions can realize the model and possibly answer open questions in the nonequilibrium many-body dynamics of these quantum systems, beyond the system sizes accessible to present numerics.
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Affiliation(s)
- Haggai Landa
- Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, 91191 Gif-sur-Yvette, France
| | - Marco Schiró
- JEIP, USR 3573 CNRS, Collège de France, PSL Research University, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France
| | - Grégoire Misguich
- Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, 91191 Gif-sur-Yvette, France
- Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, Université de Cergy-Pontoise, 95302 Cergy-Pontoise, France
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11
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Young JT, Gorshkov AV, Foss-Feig M, Maghrebi MF. Nonequilibrium Fixed Points of Coupled Ising Models. PHYSICAL REVIEW. X 2020; 10:10.1103/physrevx.10.011039. [PMID: 33364075 PMCID: PMC7756198 DOI: 10.1103/physrevx.10.011039] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Driven-dissipative systems are expected to give rise to nonequilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their nonequilibrium origin. In this paper, we show that multicritical points in such systems lead to a rich and genuinely nonequilibrium behavior. Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high- to a low-density phase-reminiscent of a liquid-gas transition-and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) ℤ 2 symmetry. However, they coalesce at a multicritical point, giving rise to a nonequilibrium model of coupled Ising-like order parameters described by a ℤ 2 × ℤ 2 symmetry. Using a dynamical renormalization-group approach, we show that a pair of nonequilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition. As direct evidence of the nonequilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes "hotter" and "hotter" at longer and longer wavelengths. Finally, we argue that this nonequilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.
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Affiliation(s)
- Jeremy T. Young
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Alexey V. Gorshkov
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Michael Foss-Feig
- United States Army Research Laboratory, Adelphi, Maryland 20783, USA
| | - Mohammad F. Maghrebi
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
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12
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Nagy A, Savona V. Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems. PHYSICAL REVIEW LETTERS 2019; 122:250501. [PMID: 31347886 DOI: 10.1103/physrevlett.122.250501] [Citation(s) in RCA: 41] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2019] [Indexed: 06/10/2023]
Abstract
The possibility to simulate the properties of many-body open quantum systems with a large number of degrees of freedom (d.o.f.) is the premise to the solution of several outstanding problems in quantum science and quantum information. The challenge posed by this task lies in the complexity of the density matrix increasing exponentially with the system size. Here, we develop a variational method to efficiently simulate the nonequilibrium steady state of Markovian open quantum systems based on variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the quantum master equation. We test the effectiveness of the method by modeling the two-dimensional dissipative XYZ spin model on a lattice.
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Affiliation(s)
- Alexandra Nagy
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
| | - Vincenzo Savona
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Lausanne, Switzerland
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13
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Puel TO, Chesi S, Kirchner S, Ribeiro P. Mixed-Order Symmetry-Breaking Quantum Phase Transition Far from Equilibrium. PHYSICAL REVIEW LETTERS 2019; 122:235701. [PMID: 31298904 DOI: 10.1103/physrevlett.122.235701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2018] [Indexed: 06/10/2023]
Abstract
We study the current-carrying steady state of a transverse field Ising chain coupled to magnetic thermal reservoirs and obtain the nonequilibrium phase diagram as a function of the magnetization potential of the reservoirs. Upon increasing the magnetization bias we observe a discontinuous jump of the magnetic order parameter that coincides with a divergence of the correlation length. For steady states with a nonvanishing conductance, the entanglement entropy at zero temperature displays a bias dependent logarithmic correction that violates the area law and differs from the well-known equilibrium case. Our findings show that out-of-equilibrium conditions allow for novel critical phenomena not possible at equilibrium.
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Affiliation(s)
- T O Puel
- Beijing Computational Science Research Center, Beijing 100193, China
- CeFEMA, Instituto Superior Técnico, Universidade de Lisboa Av. Rovisco Pais, 1049-001 Lisboa, Portugal
- Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China
- Zhejiang Province Key Laboratory of Quantum Technology and Devices, Zhejiang University, Hangzhou 310027, China
| | - Stefano Chesi
- Beijing Computational Science Research Center, Beijing 100193, China
- Department of Physics, Beijing Normal University, Beijing 100875, China
| | - S Kirchner
- Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China
- Zhejiang Province Key Laboratory of Quantum Technology and Devices, Zhejiang University, Hangzhou 310027, China
| | - P Ribeiro
- Beijing Computational Science Research Center, Beijing 100193, China
- CeFEMA, Instituto Superior Técnico, Universidade de Lisboa Av. Rovisco Pais, 1049-001 Lisboa, Portugal
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14
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Rota R, Minganti F, Ciuti C, Savona V. Quantum Critical Regime in a Quadratically Driven Nonlinear Photonic Lattice. PHYSICAL REVIEW LETTERS 2019; 122:110405. [PMID: 30951358 DOI: 10.1103/physrevlett.122.110405] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2018] [Indexed: 06/09/2023]
Abstract
We study an array of coupled optical cavities in the presence of two-photon driving and dissipation. The system displays a critical behavior similar to that of a quantum Ising model at finite temperature. Using the corner-space renormalization method, we compute the steady-state properties of finite lattices of varying size, both in one and two dimensions. From a finite-size scaling of the average of the photon number parity, we highlight the emergence of a critical point in regimes of small dissipations, belonging to the quantum Ising universality class. For increasing photon loss rates, a departure from this universal behavior signals the onset of a quantum critical regime, where classical fluctuations induced by losses compete with long-range quantum correlations.
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Affiliation(s)
- Riccardo Rota
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Fabrizio Minganti
- Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot, CNRS-UMR 7162, 75013 Paris, France
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
| | - Cristiano Ciuti
- Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot, CNRS-UMR 7162, 75013 Paris, France
| | - Vincenzo Savona
- Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
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15
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Karl M, Cakir H, Halimeh JC, Oberthaler MK, Kastner M, Gasenzer T. Universal equilibrium scaling functions at short times after a quench. Phys Rev E 2017; 96:022110. [PMID: 28950605 DOI: 10.1103/physreve.96.022110] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/05/2017] [Indexed: 06/07/2023]
Abstract
By analyzing spin-spin correlation functions at relatively short distances, we show that equilibrium near-critical properties can be extracted at short times after quenches into the vicinity of a quantum critical point. The time scales after which equilibrium properties can be extracted are sufficiently short so that the proposed scheme should be viable for quantum simulators of spin models based on ultracold atoms or trapped ions. Our results, analytic as well as numeric, are for one-dimensional spin models, either integrable or nonintegrable, but we expect our conclusions to be valid in higher dimensions as well.
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Affiliation(s)
- Markus Karl
- Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Halil Cakir
- Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Jad C Halimeh
- Physics Department and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
| | - Markus K Oberthaler
- Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
| | - Michael Kastner
- National Institute for Theoretical Physics (NITheP), Stellenbosch 7600, South Africa
- Department of Physics, Institute of Theoretical Physics, University of Stellenbosch, Stellenbosch 7600, South Africa
| | - Thomas Gasenzer
- Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
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16
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Lorenzo S, Marino J, Plastina F, Palma GM, Apollaro TJG. Quantum Critical Scaling under Periodic Driving. Sci Rep 2017; 7:5672. [PMID: 28720852 PMCID: PMC5515946 DOI: 10.1038/s41598-017-06025-1] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/13/2017] [Accepted: 06/07/2017] [Indexed: 12/02/2022] Open
Abstract
Universality is key to the theory of phase transitions, stating that the equilibrium properties of observables near a phase transition can be classified according to few critical exponents. These exponents rule an universal scaling behaviour that witnesses the irrelevance of the model’s microscopic details at criticality. Here we discuss the persistence of such a scaling in a one-dimensional quantum Ising model under sinusoidal modulation in time of its transverse magnetic field. We show that scaling of various quantities (concurrence, entanglement entropy, magnetic and fidelity susceptibility) endures up to a stroboscopic time τbd, proportional to the size of the system. This behaviour is explained by noticing that the low-energy modes, responsible for the scaling properties, are resilient to the absorption of energy. Our results suggest that relevant features of the universality do hold also when the system is brought out-of-equilibrium by a periodic driving.
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Affiliation(s)
- Salvatore Lorenzo
- Dipartimento di Fisica e Chimica, Università degli Studi di Palermo, via Archirafi 36, I-90123, Palermo, Italy.,Quantum Technology Lab, Dipartimento di Fisica, Universita' degli Studi di Milano, 20133, Milano, Italy.,INFN, Sezione di Milano, I-20133, Milano, Italy
| | - Jamir Marino
- Institute of Theoretical Physics, University of Cologne, D-50937, Cologne, Germany
| | - Francesco Plastina
- Dip. Fisica, Università della Calabria, 87036, Arcavacata di Rende (CS), Italy.,INFN - Gruppo collegato di Cosenza, Cosenza, Italy
| | - G Massimo Palma
- Dipartimento di Fisica e Chimica, Università degli Studi di Palermo, via Archirafi 36, I-90123, Palermo, Italy.,NEST, Istituto Nanoscienze-CNR, Pisa, Italy
| | - Tony J G Apollaro
- Dipartimento di Fisica e Chimica, Università degli Studi di Palermo, via Archirafi 36, I-90123, Palermo, Italy. .,Quantum Technology Lab, Dipartimento di Fisica, Universita' degli Studi di Milano, 20133, Milano, Italy. .,NEST, Istituto Nanoscienze-CNR, Pisa, Italy. .,Centre for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and Physics, Queen's University Belfast, Belfast, BT7 1NN, United Kingdom.
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17
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Chiocchetta A, Gambassi A, Diehl S, Marino J. Dynamical Crossovers in Prethermal Critical States. PHYSICAL REVIEW LETTERS 2017; 118:135701. [PMID: 28409986 DOI: 10.1103/physrevlett.118.135701] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2016] [Indexed: 06/07/2023]
Abstract
We study the prethermal dynamics of an interacting quantum field theory with an N-component order parameter and O(N) symmetry, suddenly quenched in the vicinity of a dynamical critical point. Depending on the initial conditions, the evolution of the order parameter, and of the response and correlation functions, can exhibit a temporal crossover between universal dynamical scaling regimes governed, respectively, by a quantum and a classical prethermal fixed point, as well as a crossover from a Gaussian to a non-Gaussian prethermal dynamical scaling. Together with a recent experiment, this suggests that quenches may be used in order to explore the rich variety of dynamical critical points occurring in the nonequilibrium dynamics of a quantum many-body system. We illustrate this fact by using a combination of renormalization group techniques and a nonperturbative large-N limit.
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Affiliation(s)
- Alessio Chiocchetta
- Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany
- SISSA-International School for Advanced Studies and INFN, via Bonomea 265, I-34136 Trieste, Italy
| | - Andrea Gambassi
- SISSA-International School for Advanced Studies and INFN, via Bonomea 265, I-34136 Trieste, Italy
| | - Sebastian Diehl
- Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany
| | - Jamir Marino
- Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany
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18
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He L, Sieberer LM, Diehl S. Space-Time Vortex Driven Crossover and Vortex Turbulence Phase Transition in One-Dimensional Driven Open Condensates. PHYSICAL REVIEW LETTERS 2017; 118:085301. [PMID: 28282158 DOI: 10.1103/physrevlett.118.085301] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2016] [Indexed: 06/06/2023]
Abstract
We find a first-order transition driven by the strength of nonequilibrium conditions of one-dimensional driven open condensates. Associated with this transition is a new stable nonequilibrium phase, space-time vortex turbulence, whose vortex density and quasiparticle distribution show strongly nonthermal behavior. Below the transition, we identify a new time scale associated with noise-activated unbound space-time vortices, beyond which, the temporal coherence function changes from a Kardar-Parisi-Zhang-type subexponential to a disordered exponential decay. Experimental realization of the nonequilibrium vortex turbulent phase is facilitated in driven open condensates with a large diffusion rate.
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Affiliation(s)
- Liang He
- Institute for Theoretical Physics, Technical University Dresden, D-01062 Dresden, Germany
- Institute for Theoretical Physics, University of Cologne, D-50937 Cologne, Germany
| | - Lukas M Sieberer
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
- Department of Physics, University of California, Berkeley, California 94720, USA
| | - Sebastian Diehl
- Institute for Theoretical Physics, Technical University Dresden, D-01062 Dresden, Germany
- Institute for Theoretical Physics, University of Cologne, D-50937 Cologne, Germany
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19
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Supermode-density-wave-polariton condensation with a Bose-Einstein condensate in a multimode cavity. Nat Commun 2017; 8:14386. [PMID: 28211455 PMCID: PMC5321730 DOI: 10.1038/ncomms14386] [Citation(s) in RCA: 61] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2016] [Accepted: 12/22/2016] [Indexed: 11/16/2022] Open
Abstract
Phase transitions, where observable properties of a many-body system change discontinuously, can occur in both open and closed systems. By placing cold atoms in optical cavities and inducing strong coupling between light and excitations of the atoms, one can experimentally study phase transitions of open quantum systems. Here we observe and study a non-equilibrium phase transition, the condensation of supermode-density-wave polaritons. These polaritons are formed from a superposition of cavity photon eigenmodes (a supermode), coupled to atomic density waves of a quantum gas. As the cavity supports multiple photon spatial modes and because the light–matter coupling can be comparable to the energy splitting of these modes, the composition of the supermode polariton is changed by the light–matter coupling on condensation. By demonstrating the ability to observe and understand density-wave-polariton condensation in the few-mode-degenerate cavity regime, our results show the potential to study similar questions in fully multimode cavities. When a single mode optical cavity is coupled to a Bose-Einstein condensate, one usually observes a single mode of light when strongly pumped. Here the authors observe a supermode in the output of a multimode cavity and relate this to a signature of a nonequilibrium condensation phase transition.
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20
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Zhang G, Novais E, Baranger HU. Rescuing a Quantum Phase Transition with Quantum Noise. PHYSICAL REVIEW LETTERS 2017; 118:050402. [PMID: 28211744 DOI: 10.1103/physrevlett.118.050402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/04/2016] [Indexed: 06/06/2023]
Abstract
We show that placing a quantum system in contact with an environment can enhance non-Fermi-liquid correlations, rather than destroy quantum effects, as is typical. The system consists of two quantum dots in series with two leads; the highly resistive leads couple charge flow through the dots to the electromagnetic environment, the source of quantum noise. While the charge transport inhibits a quantum phase transition, the quantum noise reduces charge transport and restores the transition. We find a non-Fermi-liquid intermediate fixed point for all strengths of the noise. For strong noise, it is similar to the intermediate fixed point of the two-impurity Kondo model.
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Affiliation(s)
- Gu Zhang
- Department of Physics, Duke University, P.O. Box 90305, Durham, North Carolina 27708, USA
| | - E Novais
- Centro de Ciíncias Naturais e Humanas, Universidade Federal do ABC, Santo Andrè, São Paulo 09210-580, Brazil
| | - Harold U Baranger
- Department of Physics, Duke University, P.O. Box 90305, Durham, North Carolina 27708, USA
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21
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Foss-Feig M, Niroula P, Young JT, Hafezi M, Gorshkov AV, Wilson RM, Maghrebi MF. Emergent equilibrium in many-body optical bistability. PHYSICAL REVIEW. A 2017; 95:10.1103/PhysRevA.95.043826. [PMID: 31093586 PMCID: PMC6513354 DOI: 10.1103/physreva.95.043826] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Many-body systems constructed of quantum-optical building blocks can now be realized in experimental platforms ranging from exciton-polariton fluids to ultracold Rydberg gases, establishing a fascinating interface between traditional many-body physics and the driven-dissipative, nonequilibrium setting of cavity QED. At this interface, the standard techniques and intuitions of both fields are called into question, obscuring issues as fundamental as the role of fluctuations, dimensionality, and symmetry on the nature of collective behavior and phase transitions. Here, we study the driven-dissipative Bose-Hubbard model, a minimal description of numerous atomic, optical, and solid-state systems in which particle loss is countered by coherent driving. Despite being a lattice version of optical bistability, a foundational and patently nonequilibrium model of cavity QED, the steady state possesses an emergent equilibrium description in terms of a classical Ising model. We establish this picture by making new connections between traditional techniques from many-body physics (functional integrals) and quantum optics (the system-size expansion). To lowest order in a controlled expansion-organized around the experimentally relevant limit of weak interactions-the full quantum dynamics reduces to nonequilibrium Langevin equations, which support a phase transition described by model A of the Hohenberg-Halperin classification. Numerical simulations of the Langevin equations corroborate this picture, revealing that canonical behavior associated with the Ising model manifests readily in simple experimental observables.
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Affiliation(s)
- M Foss-Feig
- United States Army Research Laboratory, Adelphi, Maryland 20783, USA
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, Maryland 20742, USA
| | - P Niroula
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - J T Young
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
| | - M Hafezi
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Department of Electrical and Computer Engineering and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
| | - A V Gorshkov
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, Maryland 20742, USA
| | - R M Wilson
- Department of Physics, United States Naval Academy, Annapolis, Maryland 21402, USA
| | - M F Maghrebi
- Joint Quantum Institute, NIST and University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, Maryland 20742, USA
- Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
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22
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Overbeck VR, Maghrebi MF, Gorshkov AV, Weimer H. Multicritical behavior in dissipative Ising models. PHYSICAL REVIEW. A 2017; 95:10.1103/PhysRevA.95.042133. [PMID: 31093585 PMCID: PMC6513333 DOI: 10.1103/physreva.95.042133] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
Abstract
We analyze theoretically the many-body dynamics of a dissipative Ising model in a transverse field using a variational approach. We find that the steady-state phase diagram is substantially modified compared to its equilibrium counterpart, including the appearance of a multicritical point belonging to a different universality class. Building on our variational analysis, we establish a field-theoretical treatment corresponding to a dissipative variant of a Ginzburg-Landau theory, which allows us to compute the upper critical dimension of the system. Finally, we present a possible experimental realization of the dissipative Ising model using ultracold Rydberg gases.
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Affiliation(s)
- Vincent R. Overbeck
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
| | - Mohammad F. Maghrebi
- Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Alexey V. Gorshkov
- Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Hendrik Weimer
- Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
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23
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Sieberer LM, Buchhold M, Diehl S. Keldysh field theory for driven open quantum systems. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2016; 79:096001. [PMID: 27482736 DOI: 10.1088/0034-4885/79/9/096001] [Citation(s) in RCA: 73] [Impact Index Per Article: 9.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Recent experimental developments in diverse areas-ranging from cold atomic gases to light-driven semiconductors to microcavity arrays-move systems into the focus which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in equilibrium condensed matter physics. This concerns both their non-thermal stationary states and their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.
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Affiliation(s)
- L M Sieberer
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel
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24
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Marcuzzi M, Buchhold M, Diehl S, Lesanovsky I. Absorbing State Phase Transition with Competing Quantum and Classical Fluctuations. PHYSICAL REVIEW LETTERS 2016; 116:245701. [PMID: 27367395 DOI: 10.1103/physrevlett.116.245701] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/01/2016] [Indexed: 06/06/2023]
Abstract
Stochastic processes with absorbing states feature examples of nonequilibrium universal phenomena. While the classical regime has been thoroughly investigated in the past, relatively little is known about the behavior of these nonequilibrium systems in the presence of quantum fluctuations. Here, we theoretically address such a scenario in an open quantum spin model which, in its classical limit, undergoes a directed percolation phase transition. By mapping the problem to a nonequilibrium field theory, we show that the introduction of quantum fluctuations stemming from coherent, rather than statistical, spin flips alters the nature of the transition such that it becomes first order. In the intermediate regime, where classical and quantum dynamics compete on equal terms, we highlight the presence of a bicritical point with universal features different from the directed percolation class in a low dimension. We finally propose how this physics could be explored within gases of interacting atoms excited to Rydberg states.
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Affiliation(s)
- Matteo Marcuzzi
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - Michael Buchhold
- Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany
| | - Sebastian Diehl
- Institut für Theoretische Physik, Universität zu Köln, D-50937 Cologne, Germany
| | - Igor Lesanovsky
- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
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