1
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Álvarez P, Pittilini D, Miserocchi F, Raamamurthy S, Margiani G, Ameye O, Del Pino J, Zilberberg O, Eichler A. Biased Ising Model Using Two Coupled Kerr Parametric Oscillators with External Force. PHYSICAL REVIEW LETTERS 2024; 132:207401. [PMID: 38829099 DOI: 10.1103/physrevlett.132.207401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2023] [Accepted: 04/09/2024] [Indexed: 06/05/2024]
Abstract
Networks of coupled Kerr parametric oscillators (KPOs) are a leading physical platform for analog solving of complex optimization problems. These systems are colloquially known as "Ising machines." We experimentally and theoretically study such a network under the influence of an external force. The force breaks the collective phase-parity symmetry of the system and competes with the intrinsic coupling in ordering the network configuration, similar to how a magnetic field biases an interacting spin ensemble. Specifically, we demonstrate how the force can be used to control the system, and highlight the crucial role of the phase and symmetry of the force. Our Letter thereby provides a method to create Ising machines with arbitrary bias, extending even to exotic cases that are impossible to engineer in real spin systems.
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Affiliation(s)
- Pablo Álvarez
- Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - Davide Pittilini
- Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - Filippo Miserocchi
- Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | | | - Gabriel Margiani
- Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - Orjan Ameye
- Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
| | - Javier Del Pino
- Institute for Theoretical Physics, ETH Zürich, CH-8093 Zürich, Switzerland
| | - Oded Zilberberg
- Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
| | - Alexander Eichler
- Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland
- Quantum Center, ETH Zürich, CH-8093 Zürich, Switzerland
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2
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Downing CA, Vidiella-Barranco A. Parametrically driving a quantum oscillator into exceptionality. Sci Rep 2023; 13:11004. [PMID: 37419917 PMCID: PMC10329046 DOI: 10.1038/s41598-023-37964-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2023] [Accepted: 06/30/2023] [Indexed: 07/09/2023] Open
Abstract
The mathematical objects employed in physical theories do not always behave well. Einstein's theory of space and time allows for spacetime singularities and Van Hove singularities arise in condensed matter physics, while intensity, phase and polarization singularities pervade wave physics. Within dissipative systems governed by matrices, singularities occur at the exceptional points in parameter space whereby some eigenvalues and eigenvectors coalesce simultaneously. However, the nature of exceptional points arising in quantum systems described within an open quantum systems approach has been much less studied. Here we consider a quantum oscillator driven parametrically and subject to loss. This squeezed system exhibits an exceptional point in the dynamical equations describing its first and second moments, which acts as a borderland between two phases with distinctive physical consequences. In particular, we discuss how the populations, correlations, squeezed quadratures and optical spectra crucially depend on being above or below the exceptional point. We also remark upon the presence of a dissipative phase transition at a critical point, which is associated with the closing of the Liouvillian gap. Our results invite the experimental probing of quantum resonators under two-photon driving, and perhaps a reappraisal of exceptional and critical points within dissipative quantum systems more generally.
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Affiliation(s)
- C A Downing
- Department of Physics and Astronomy, University of Exeter, Exeter, EX4 4QL, UK.
| | - A Vidiella-Barranco
- Gleb Wataghin Institute of Physics, University of Campinas - UNICAMP, Campinas, SP, 13083-859, Brazil
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3
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Roberts D, Clerk AA. Competition between Two-Photon Driving, Dissipation, and Interactions in Bosonic Lattice Models: An Exact Solution. PHYSICAL REVIEW LETTERS 2023; 130:063601. [PMID: 36827562 DOI: 10.1103/physrevlett.130.063601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2022] [Accepted: 01/06/2023] [Indexed: 06/18/2023]
Abstract
We present an exact solution in arbitrary dimensions for the steady states of a class of quantum driven-dissipative bosonic models, where a set of modes is subject to arbitrary two-photon driving, single-photon loss, and a global Hubbard (or Kerr)-like interaction. Our solutions reveal a wealth of striking phenomena, including the emergence of dissipative phase transitions, nontrivial mode competition physics and symmetry breaking, and the stabilization of many-body SU(1,1) pair-coherent states. Our exact solutions enable the description of spatial correlations, and are fully valid in regimes where traditional mean-field and semiclassical approaches break down.
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Affiliation(s)
- David Roberts
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, 60637 Illinois, USA
- Department of Physics, University of Chicago, Chicago, 60637 Illinois, USA
| | - A A Clerk
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, 60637 Illinois, USA
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4
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Suzuki Y, Watabe S, Kawabata S, Masuda S. Measurement-based preparation of stable coherent states of a Kerr parametric oscillator. Sci Rep 2023; 13:1606. [PMID: 36709379 PMCID: PMC9884232 DOI: 10.1038/s41598-023-28682-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2022] [Accepted: 01/23/2023] [Indexed: 01/30/2023] Open
Abstract
Kerr parametric oscillators (KPOs) have attracted increasing attention in terms of their application to quantum information processing and quantum simulations. The state preparation and measurement of KPOs are typical requirements when used as qubits. The methods previously proposed for state preparations of KPOs utilize modulation of external fields such as a pump and drive fields. We study the stochastic state preparation of stable coherent states of a KPO with homodyne detection, which does not require modulation of external fields, and thus can reduce experimental efforts and exclude unwanted effects of possible imperfection in control of external fields. We quantitatively show that the detection data, if averaged over an optimal averaging time to decrease the effect of measurement noise, has a strong correlation with the state of the KPO, and therefore can be used to estimate the state (stochastic state preparation). We examine the success probability of the state estimation taking into account the measurement noise and bit flips. Moreover, the proper range of the averaging time to realize a high success probability is obtained by developing a binomial-coherent-state model, which describes the stochastic dynamics of the KPO under homodyne detection.
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Affiliation(s)
- Yuta Suzuki
- grid.143643.70000 0001 0660 6861Department of Physics, Faculty of Science Division I, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 Japan ,grid.208504.b0000 0001 2230 7538Research Center for Emerging Computing Technologies (RCECT), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1, Umezono, Tsukuba, Ibaraki 305-8568 Japan
| | - Shohei Watabe
- grid.143643.70000 0001 0660 6861Department of Physics, Faculty of Science Division I, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 Japan ,grid.419152.a0000 0001 0166 4675Department of Computer Science and Engineering, College of Engineering, Shibaura Institute of Technology, 3-7-5 Toyosu, Koto-ku, Tokyo 135-8548 Japan
| | - Shiro Kawabata
- grid.208504.b0000 0001 2230 7538Research Center for Emerging Computing Technologies (RCECT), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1, Umezono, Tsukuba, Ibaraki 305-8568 Japan ,grid.208504.b0000 0001 2230 7538NEC-AIST Quantum Technology Cooperative Research Laboratory, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568 Japan
| | - Shumpei Masuda
- grid.208504.b0000 0001 2230 7538Research Center for Emerging Computing Technologies (RCECT), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1, Umezono, Tsukuba, Ibaraki 305-8568 Japan ,grid.208504.b0000 0001 2230 7538NEC-AIST Quantum Technology Cooperative Research Laboratory, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568 Japan
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5
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Zhang Y, Barthel T. Criticality and Phase Classification for Quadratic Open Quantum Many-Body Systems. PHYSICAL REVIEW LETTERS 2022; 129:120401. [PMID: 36179179 DOI: 10.1103/physrevlett.129.120401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2022] [Accepted: 08/15/2022] [Indexed: 06/16/2023]
Abstract
We study the steady states of translation-invariant open quantum many-body systems governed by Lindblad master equations, where the Hamiltonian is quadratic in the ladder operators, and the Lindblad operators are either linear or quadratic and Hermitian. These systems are called quasifree and quadratic, respectively. We find that steady states of one-dimensional systems with finite-range interactions necessarily have exponentially decaying Green's functions. For the quasifree case without quadratic Lindblad operators, we show that fermionic systems with finite-range interactions are noncritical for any number of spatial dimensions and provide bounds on the correlation lengths. Quasifree bosonic systems can be critical in D>1 dimensions. Last, we address the question of phase transitions in quadratic systems and find that, without symmetry constraints beyond invariance under single-particle basis and particle-hole transformations, all gapped Liouvillians belong to the same phase.
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Affiliation(s)
- Yikang Zhang
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
| | - Thomas Barthel
- Department of Physics, Duke University, Durham, North Carolina 27708, USA
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6
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Mylnikov VY, Potashin SO, Sokolovskii GS, Averkiev NS. Dissipative Phase Transition in Systems with Two-Photon Drive and Nonlinear Dissipation near the Critical Point. NANOMATERIALS 2022; 12:nano12152543. [PMID: 35893511 PMCID: PMC9332203 DOI: 10.3390/nano12152543] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/14/2022] [Revised: 07/15/2022] [Accepted: 07/21/2022] [Indexed: 11/16/2022]
Abstract
In this paper, we examine dissipative phase transition (DPT) near the critical point for a system with two-photon driving and nonlinear dissipations. The proposed mean-field theory, which explicitly takes into account quantum fluctuations, allowed us to describe properly the evolutionary dynamics of the system and to demonstrate new effects in its steady-state. We show that the presence of quantum fluctuations leads to a power-law dependence of the anomalous average at the phase transition point, with which the critical exponent is associated. Also, we investigate the effect of the quantum fluctuations on the critical point renormalization and demonstrate the existence of a two-photon pump “threshold”. It is noteworthy that the obtained results are in a good agreement with the numerical simulations.
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Affiliation(s)
- Valentin Yu. Mylnikov
- Ioffe Institute, 194021 St. Petersburg, Russia; (S.O.P.); (N.S.A.)
- Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia
- Correspondence: (V.Y.M.); (G.S.S.)
| | | | - Grigorii S. Sokolovskii
- Ioffe Institute, 194021 St. Petersburg, Russia; (S.O.P.); (N.S.A.)
- Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia
- Correspondence: (V.Y.M.); (G.S.S.)
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7
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Bloch J, Cavalleri A, Galitski V, Hafezi M, Rubio A. Strongly correlated electron-photon systems. Nature 2022; 606:41-48. [PMID: 35614214 DOI: 10.1038/s41586-022-04726-w] [Citation(s) in RCA: 23] [Impact Index Per Article: 11.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2020] [Accepted: 12/02/2021] [Indexed: 11/09/2022]
Abstract
An important goal of modern condensed-matter physics involves the search for states of matter with emergent properties and desirable functionalities. Although the tools for material design remain relatively limited, notable advances have been recently achieved by controlling interactions at heterointerfaces, precise alignment of low-dimensional materials and the use of extreme pressures. Here we highlight a paradigm based on controlling light-matter interactions, which provides a way to manipulate and synthesize strongly correlated quantum matter. We consider the case in which both electron-electron and electron-photon interactions are strong and give rise to a variety of phenomena. Photon-mediated superconductivity, cavity fractional quantum Hall physics and optically driven topological phenomena in low dimensions are among the frontiers discussed in this Perspective, which highlights a field that we term here 'strongly correlated electron-photon science'.
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Affiliation(s)
- Jacqueline Bloch
- Centre de Nanosciences et de Nanotechnologies (C2N), Universite Paris Saclay - CNRS, Palaiseau, France
| | - Andrea Cavalleri
- Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany
| | - Victor Galitski
- Department of Physics, University of Maryland, College Park, MD, USA.
| | - Mohammad Hafezi
- Departments of Physics and ECE, University of Maryland, College Park, MD, USA
| | - Angel Rubio
- Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany.,Center for Computational Quantum Physics (CCQ), Flatiron Institute, New York, NY, USA
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8
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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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9
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Diessel OK, Diehl S, Chiocchetta A. Emergent Kardar-Parisi-Zhang Phase in Quadratically Driven Condensates. PHYSICAL REVIEW LETTERS 2022; 128:070401. [PMID: 35244410 DOI: 10.1103/physrevlett.128.070401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2021] [Accepted: 12/24/2021] [Indexed: 06/14/2023]
Abstract
In bosonic gases at thermal equilibrium, an external quadratic drive can induce a Bose-Einstein condensation described by the Ising transition, as a consequence of the explicitly broken U(1) phase rotation symmetry down to Z_{2}. However, in physical realizations such as exciton polaritons and nonlinear photonic lattices, thermal equilibrium is lost and the state is rather determined by a balance between losses and external drive. A fundamental question is then how nonequilibrium fluctuations affect this transition. Here, we show that in a two-dimensional driven-dissipative Bose system the Ising phase is suppressed and replaced by a nonequilibrium phase featuring Kardar-Parisi-Zhang (KPZ) physics. Its emergence is rooted in a U(1)-symmetry restoration mechanism enabled by the strong fluctuations in reduced dimensionality. Moreover, we show that the presence of the quadratic drive term enhances the visibility of the KPZ scaling, compared to two-dimensional U(1)-symmetric gases, where it has remained so far elusive.
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Affiliation(s)
- Oriana K Diessel
- Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
| | - Sebastian Diehl
- Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany
| | - Alessio Chiocchetta
- Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany
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10
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Yuan D, Wang HR, Wang Z, Deng DL. Solving the Liouvillian Gap with Artificial Neural Networks. PHYSICAL REVIEW LETTERS 2021; 126:160401. [PMID: 33961454 DOI: 10.1103/physrevlett.126.160401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2020] [Accepted: 03/30/2021] [Indexed: 06/12/2023]
Abstract
We propose a machine-learning inspired variational method to obtain the Liouvillian gap, which plays a crucial role in characterizing the relaxation time and dissipative phase transitions of open quantum systems. By using "spin bi-base mapping," we map the density matrix to a pure restricted-Boltzmann-machine (RBM) state and transform the Liouvillian superoperator to a rank-two non-Hermitian operator. The Liouvillian gap can be obtained by a variational real-time evolution algorithm under this non-Hermitian operator. We apply our method to the dissipative Heisenberg model in both one and two dimensions. For the isotropic case, we find that the Liouvillian gap can be analytically obtained and in one dimension even the whole Liouvillian spectrum can be exactly solved using the Bethe ansatz method. By comparing our numerical results with their analytical counterparts, we show that the Liouvillian gap could be accessed by the RBM approach efficiently to a desirable accuracy, regardless of the dimensionality and entanglement properties.
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Affiliation(s)
- Dong Yuan
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
- Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China
| | - He-Ran Wang
- Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China
- Institute for Advanced Study, Tsinghua University, Beijing 100084, People's Republic of China
| | - Zhong Wang
- Institute for Advanced Study, Tsinghua University, Beijing 100084, People's Republic of China
| | - Dong-Ling Deng
- Center for Quantum Information, IIIS, Tsinghua University, Beijing 100084, People's Republic of China
- Shanghai Qi Zhi Institute, 41st Floor, AI Tower, No. 701 Yunjin Road, Xuhui District, Shanghai 200232, China
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11
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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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12
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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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13
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Garbe L, Wade P, Minganti F, Shammah N, Felicetti S, Nori F. Dissipation-induced bistability in the two-photon Dicke model. Sci Rep 2020; 10:13408. [PMID: 32770061 PMCID: PMC7414202 DOI: 10.1038/s41598-020-69704-6] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2019] [Accepted: 07/13/2020] [Indexed: 11/09/2022] Open
Abstract
The Dicke model is a paradigmatic quantum-optical model describing the interaction of a collection of two-level systems with a single bosonic mode. Effective implementations of this model made it possible to observe the emergence of superradiance, i.e., cooperative phenomena arising from the collective nature of light-matter interactions. Via reservoir engineering and analogue quantum simulation techniques, current experimental platforms allow us not only to implement the Dicke model but also to design more exotic interactions, such as the two-photon Dicke model. In the Hamiltonian case, this model presents an interesting phase diagram characterized by two quantum criticalities: a superradiant phase transition and a spectral collapse, that is, the coalescence of discrete energy levels into a continuous band. Here, we investigate the effects of both qubit and photon dissipation on the phase transition and on the instability induced by the spectral collapse. Using a mean-field decoupling approximation, we analytically obtain the steady-state expectation values of the observables signaling a symmetry breaking, identifying a first-order phase transition from the normal to the superradiant phase. Our stability analysis unveils a very rich phase diagram, which features stable, bistable, and unstable phases depending on the dissipation rate.
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Affiliation(s)
- Louis Garbe
- Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, CNRS UMR 7162, Université de Paris, 75013, Paris, France.
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, 351-0198, Japan.
| | - Peregrine Wade
- Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, CNRS UMR 7162, Université de Paris, 75013, Paris, France
| | - Fabrizio Minganti
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, 351-0198, Japan
| | - Nathan Shammah
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, 351-0198, Japan
- Unitary Fund, 340 S Lemon Ave. 7770, Walnut, CA, 91789, USA
| | - Simone Felicetti
- Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049, Madrid, Spain
- Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche (IFN-CNR), Piazza Leonardo da Vinci 32, 20133, Milano, Italy
| | - Franco Nori
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama, 351-0198, Japan
- Physics Department, The University of Michigan, Ann Arbor, MI, 48109-1040, USA
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14
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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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Vicentini F, Biella A, Regnault N, Ciuti C. Variational Neural-Network Ansatz for Steady States in Open Quantum Systems. PHYSICAL REVIEW LETTERS 2019; 122:250503. [PMID: 31347877 DOI: 10.1103/physrevlett.122.250503] [Citation(s) in RCA: 36] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2019] [Indexed: 06/10/2023]
Abstract
We present a general variational approach to determine the steady state of open quantum lattice systems via a neural-network approach. The steady-state density matrix of the lattice system is constructed via a purified neural-network Ansatz in an extended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof of principle, we apply the method to the dissipative quantum transverse Ising model.
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Affiliation(s)
- Filippo Vicentini
- Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013, Paris, France
| | - Alberto Biella
- Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013, Paris, France
| | - Nicolas Regnault
- Laboratoire de Physique de l'Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Sorbonne Paris Cité, F-75005, Paris, France
| | - Cristiano Ciuti
- Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013, Paris, France
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Hartmann MJ, Carleo G. Neural-Network Approach to Dissipative Quantum Many-Body Dynamics. PHYSICAL REVIEW LETTERS 2019; 122:250502. [PMID: 31347862 DOI: 10.1103/physrevlett.122.250502] [Citation(s) in RCA: 44] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2019] [Indexed: 06/10/2023]
Abstract
In experimentally realistic situations, quantum systems are never perfectly isolated and the coupling to their environment needs to be taken into account. Often, the effect of the environment can be well approximated by a Markovian master equation. However, solving this master equation for quantum many-body systems becomes exceedingly hard due to the high dimension of the Hilbert space. Here we present an approach to the effective simulation of the dynamics of open quantum many-body systems based on machine-learning techniques. We represent the mixed many-body quantum states with neural networks in the form of restricted Boltzmann machines and derive a variational Monte Carlo algorithm for their time evolution and stationary states. We document the accuracy of the approach with numerical examples for a dissipative spin lattice system.
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Affiliation(s)
- Michael J Hartmann
- Institute of Photonics and Quantum Sciences, Heriot-Watt University Edinburgh EH14 4AS, United Kingdom
- Google Research, Erika-Mann-Str. 33, 80636 München, Germany
- Department of Physics, University of Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany
| | - Giuseppe Carleo
- Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA
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