1
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Saha AK, Ray DS, Deb B. Phase diffusion and fluctuations in a dissipative Bose-Josephson junction. Phys Rev E 2023; 107:034141. [PMID: 37073026 DOI: 10.1103/physreve.107.034141] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2022] [Accepted: 03/16/2023] [Indexed: 04/20/2023]
Abstract
We analyze the phase diffusion, quantum fluctuations and their spectral features of a one-dimensional Bose-Josephson junction (BJJ) nonlinearly coupled to a bosonic heat bath. The phase diffusion is considered by taking into account of random modulations of the BJJ modes causing a phase loss of initial coherence between the ground and excited states, whereby the frequency modulation is incorporated in the system-reservoir Hamiltonian by an interaction term linear in bath operators but nonlinear in system (BJJ) operators. We examine the dependence of the phase diffusion coefficient on the on-site interaction and temperature in the zero- and π-phase modes and demonstrate its phase transition-like behavior between the Josephson oscillation and the macroscopic quantum self-trapping (MQST) regimes in the π-phase mode. Based on the thermal canonical Wigner distribution, which is the equilibrium solution of the associated quantum Langevin equation for phase, coherence factor is calculated to study phase diffusion for the zero- and π-phase modes. We investigate the quantum fluctuations of the relative phase and population imbalance in terms of fluctuation spectra which capture an interesting shift in Josephson frequency induced by frequency fluctuation due to nonlinear system-reservoir coupling, as well as the on-site interaction-induced splitting in the weak dissipative regime.
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Affiliation(s)
- Abhik Kumar Saha
- School of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
| | - Deb Shankar Ray
- School of Chemical Sciences, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
| | - Bimalendu Deb
- School of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
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2
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Meyerov I, Kozinov E, Liniov A, Volokitin V, Yusipov I, Ivanchenko M, Denisov S. Transforming Lindblad Equations into Systems of Real-Valued Linear Equations: Performance Optimization and Parallelization of an Algorithm. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1133. [PMID: 33286901 PMCID: PMC7597275 DOI: 10.3390/e22101133] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/20/2020] [Revised: 09/26/2020] [Accepted: 09/30/2020] [Indexed: 11/30/2022]
Abstract
With their constantly increasing peak performance and memory capacity, modern supercomputers offer new perspectives on numerical studies of open many-body quantum systems. These systems are often modeled by using Markovian quantum master equations describing the evolution of the system density operators. In this paper, we address master equations of the Lindblad form, which are a popular theoretical tools in quantum optics, cavity quantum electrodynamics, and optomechanics. By using the generalized Gell-Mann matrices as a basis, any Lindblad equation can be transformed into a system of ordinary differential equations with real coefficients. Recently, we presented an implementation of the transformation with the computational complexity, scaling as O(N5logN) for dense Lindbaldians and O(N3logN) for sparse ones. However, infeasible memory costs remains a serious obstacle on the way to large models. Here, we present a parallel cluster-based implementation of the algorithm and demonstrate that it allows us to integrate a sparse Lindbladian model of the dimension N=2000 and a dense random Lindbladian model of the dimension N=200 by using 25 nodes with 64 GB RAM per node.
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Affiliation(s)
- Iosif Meyerov
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia; (I.M.); (E.K.); (A.L.); (V.V.); (I.Y.)
| | - Evgeny Kozinov
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia; (I.M.); (E.K.); (A.L.); (V.V.); (I.Y.)
| | - Alexey Liniov
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia; (I.M.); (E.K.); (A.L.); (V.V.); (I.Y.)
| | - Valentin Volokitin
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia; (I.M.); (E.K.); (A.L.); (V.V.); (I.Y.)
| | - Igor Yusipov
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia; (I.M.); (E.K.); (A.L.); (V.V.); (I.Y.)
- Department of Applied Mathematics, Lobachevsky University, 603950 Nizhni Novgorod, Russia
| | - Mikhail Ivanchenko
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia; (I.M.); (E.K.); (A.L.); (V.V.); (I.Y.)
- Department of Applied Mathematics, Lobachevsky University, 603950 Nizhni Novgorod, Russia
| | - Sergey Denisov
- Mathematical Center, Lobachevsky University, 603950 Nizhni Novgorod, Russia; (I.M.); (E.K.); (A.L.); (V.V.); (I.Y.)
- Department of Computer Science, Oslo Metropolitan University, N-0130 Oslo, Norway
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3
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Sinha S, Sinha S. Chaos and Quantum Scars in Bose-Josephson Junction Coupled to a Bosonic Mode. PHYSICAL REVIEW LETTERS 2020; 125:134101. [PMID: 33034498 DOI: 10.1103/physrevlett.125.134101] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2020] [Accepted: 08/31/2020] [Indexed: 06/11/2023]
Abstract
We consider a model describing Bose-Josephson junction (BJJ) coupled to a single bosonic mode exhibiting quantum phase transition (QPT). Onset of chaos above QPT is observed from semiclassical dynamics as well from spectral statistics. Based on entanglement entropy, we analyze the ergodic behavior of eigenstates with increasing energy density which also reveals the influence of dynamical steady state known as π-mode on it. We identify the imprint of unstable π-oscillation as many body quantum scar (MBQS), which leads to the deviation from ergodicity and quantify the degree of scarring. Persistence of phase coherence in nonequilibrium dynamics of such initial state corresponding to the π-mode is an observable signature of MBQS which has relevance in experiments on BJJ.
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Affiliation(s)
- Sudip Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia 741246, India
| | - S Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia 741246, India
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4
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Sawyer RF. Quantum Break in High Intensity Gravitational Wave Interactions. PHYSICAL REVIEW LETTERS 2020; 124:101301. [PMID: 32216441 DOI: 10.1103/physrevlett.124.101301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2019] [Revised: 12/11/2019] [Accepted: 02/10/2020] [Indexed: 06/10/2023]
Abstract
The lowest order amplitudes for [graviton+graviton → photon +photon] lead to cross sections of order G^{2}, where G is the gravitational constant. These are too small to be of any interest. However, in dense clouds of pure gravitons there are collective effects utilizing these same amplitudes that under the right circumstances can lead to copious production of photons.
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Affiliation(s)
- R F Sawyer
- Department of Physics, University of California Santa Barbara, Santa Barbara, California 93106, USA
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5
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Liniov A, Meyerov I, Kozinov E, Volokitin V, Yusipov I, Ivanchenko M, Denisov S. Unfolding a quantum master equation into a system of real-valued equations: Computationally effective expansion over the basis of SU(N) generators. Phys Rev E 2019; 100:053305. [PMID: 31869931 DOI: 10.1103/physreve.100.053305] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2019] [Indexed: 11/07/2022]
Abstract
Dynamics of an open N-state quantum system is often modeled with a Markovian master equation describing the evolution of the system density operator. By using generators of SU(N) group as a basis, the density operator can be transformed into a real-valued "coherence-vector." A generator of the dissipative evolution, so-called "Lindbladian," can be expanded over the same basis and recast in the form of a real matrix. Together, these expansions result is a nonhomogeneous system of N^{2}-1 real-valued linear ordinary differential equations. Now one can, e.g., implement standard high-performance algorithms to integrate the system of equations forward in time while being sure in exact preservation of the trace (norm) and Hermiticity of the density operator. However, when performed in a straightforward way, the expansion turns to be an operation of the time complexity O(N^{10}). The complexity can be reduced when the number of dissipative operators is independent of N, which is often the case for physically meaningful models. Here we present an algorithm to transform quantum master equation into a system of real-valued differential equations and propagate it forward in time. By using a specific scalable model, we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with N=10^{3} states on a single node of a computer cluster.
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Affiliation(s)
- A Liniov
- Software Engineering Department, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia
| | - I Meyerov
- Mathematical Software and Supercomputing Technologies Department, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia
| | - E Kozinov
- Mathematical Software and Supercomputing Technologies Department, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia
| | - V Volokitin
- Mathematical Software and Supercomputing Technologies Department, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia
| | - I Yusipov
- Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia
| | - M Ivanchenko
- Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia
| | - S Denisov
- Department of Computer Science, OsloMet - Oslo Metropolitan University, N-0130 Oslo, Norway
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6
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Yusipov II, Ivanchenko MV. Quantum Neimark-Sacker bifurcation. Sci Rep 2019; 9:17932. [PMID: 31784568 PMCID: PMC6884488 DOI: 10.1038/s41598-019-53526-2] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2019] [Accepted: 11/01/2019] [Indexed: 12/02/2022] Open
Abstract
Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on “quantumtorus” and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.
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Affiliation(s)
- I I Yusipov
- Department of Applied Mathematics, Lobachevsky University, Nizhny Novgorod, Russia
| | - M V Ivanchenko
- Department of Applied Mathematics, Lobachevsky University, Nizhny Novgorod, Russia.
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7
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Sinha S, Sinha S. Dissipative Bose-Josephson junction coupled to bosonic baths. Phys Rev E 2019; 100:032115. [PMID: 31640056 DOI: 10.1103/physreve.100.032115] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2019] [Indexed: 11/07/2022]
Abstract
We investigate the effect of dissipation in a Bose-Josephson junction (BJJ) coupled to baths of bosons at its two sites. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within a Holstein-Primakoff approximation, which exhibits vanishing of the energy gap and enhanced quantum fluctuations at the critical point. We study the dynamics of BJJ using a time-dependent variational method and analyze stability of different types of steady states. As a special case we study in detail the phase space dynamics of BJJ coupled to a single mode, which reveals diffusive and incoherent behavior with increasing coupling to the bath mode. The dynamical steady states corresponding to the π oscillation and self-trapped state become unstable when their oscillation frequencies are in resonance with the bath modes. We study the Josephson dynamics in the presence of an Ohmic bath with Gaussian noise to incorporate the thermal fluctuations and obtain the Josephson oscillation frequency and damping analytically. We also observe the transition to the symmetry-broken state for strong coupling as well as decay of π oscillation and a self-trapped state to the ground state due to dissipation. Variation of the phase fluctuation with temperature of the bath shows similar behavior as observed in experiment. Finally we discuss the experimental setup to study the observable effects of dissipation in BJJ.
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Affiliation(s)
- Sudip Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia-741246, India
| | - S Sinha
- Indian Institute of Science Education and Research-Kolkata, Mohanpur, Nadia-741246, India
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8
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How to probe the microscopic onset of irreversibility with ultracold atoms. Sci Rep 2019; 9:14169. [PMID: 31578363 PMCID: PMC6775121 DOI: 10.1038/s41598-019-50608-z] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2019] [Accepted: 09/10/2019] [Indexed: 11/11/2022] Open
Abstract
The microscopic onset of irreversibility is finally becoming an experimental subject. Recent experiments on microscopic open and even isolated systems have measured statistical properties associated with entropy production, and hysteresis-like phenomena have been seen in cold atom systems with dissipation (i.e. effectively open systems coupled to macroscopic reservoirs). Here we show how experiments on isolated systems of ultracold atoms can show dramatic irreversibility like cooking an egg. In our proposed experiments, a slow forward-and-back parameter sweep will sometimes fail to return the system close to its initial state. This probabilistic hysteresis is due to the same non-adiabatic spreading and ergodic mixing in phase space that explains macroscopic irreversibility, but realized without dynamical chaos; moreover this fundamental mechanism quantitatively determines the probability of return to the initial state as a function of tunable parameters in the proposed experiments. Matching the predicted curve of return probability will be a conclusive experimental demonstration of the microscopic onset of irreversibility.
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9
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Yusipov II, Vershinina OS, Denisov S, Kuznetsov SP, Ivanchenko MV. Quantum Lyapunov exponents beyond continuous measurements. CHAOS (WOODBURY, N.Y.) 2019; 29:063130. [PMID: 31266336 DOI: 10.1063/1.5094324] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2019] [Accepted: 06/10/2019] [Indexed: 06/09/2023]
Abstract
Quantum systems, when interacting with their environments, may exhibit nonequilibrium states that are tempting to be interpreted as quantum analogs of chaotic attractors. However, different from the Hamiltonian case, the toolbox for quantifying dissipative quantum chaos remains limited. In particular, quantum generalizations of Lyapunov exponents, the main quantifiers of classical chaos, are established only within the framework of continuous measurements. We propose an alternative generalization based on the unraveling of quantum master equation into an ensemble of "quantum trajectories," by using the so-called Monte Carlo wave-function method. We illustrate the idea with a periodically modulated open quantum dimer and demonstrate that the transition to quantum chaos matches the period-doubling route to chaos in the corresponding mean-field system.
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Affiliation(s)
- I I Yusipov
- Department of Applied Mathematics, Lobachevsky University, Nizhny Novgorod 603950, Russia
| | - O S Vershinina
- Department of Applied Mathematics, Lobachevsky University, Nizhny Novgorod 603950, Russia
| | - S Denisov
- Department of Computer Science, Oslo Metropolitan University, Oslo N-0130, Norway
| | - S P Kuznetsov
- Kotelnikovs Institute of Radio-Engineering and Electronics of RAS, Saratov 410019, Russia
| | - M V Ivanchenko
- Department of Applied Mathematics, Lobachevsky University, Nizhny Novgorod 603950, Russia
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10
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Polo J, Ahufinger V, Hekking FWJ, Minguzzi A. Damping of Josephson Oscillations in Strongly Correlated One-Dimensional Atomic Gases. PHYSICAL REVIEW LETTERS 2018; 121:090404. [PMID: 30230871 DOI: 10.1103/physrevlett.121.090404] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2017] [Indexed: 06/08/2023]
Abstract
We study Josephson oscillations of two strongly correlated one-dimensional bosonic clouds separated by a localized barrier. Using a quantum-Langevin approach and the exact Tonks-Girardeau solution in the impenetrable-boson limit, we determine the dynamical evolution of the particle-number imbalance, displaying an effective damping of the Josephson oscillations which depends on barrier height, interaction strength, and temperature. We show that the damping originates from the quantum and thermal fluctuations intrinsically present in the strongly correlated gas. Because of the density-phase duality of the model, the same results apply to particle-current oscillations in a one-dimensional ring where a weak barrier couples different angular momentum states.
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Affiliation(s)
- J Polo
- Univ. Grenoble Alpes, CNRS, LPMMC, F-38000 Grenoble, France
| | - V Ahufinger
- Departament de Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
| | - F W J Hekking
- Univ. Grenoble Alpes, CNRS, LPMMC, F-38000 Grenoble, France
| | - A Minguzzi
- Univ. Grenoble Alpes, CNRS, LPMMC, F-38000 Grenoble, France
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11
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Wang RRW, Xing B, Carlo GG, Poletti D. Period doubling in period-one steady states. Phys Rev E 2018; 97:020202. [PMID: 29548194 DOI: 10.1103/physreve.97.020202] [Citation(s) in RCA: 34] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2017] [Indexed: 11/07/2022]
Abstract
Nonlinear classical dissipative systems present a rich phenomenology in their "route to chaos," including period doubling, i.e., the system evolves with a period which is twice that of the driving. However, typically the attractor of a periodically driven quantum open system evolves with a period which exactly matches that of the driving. Here, we analyze a periodically driven many-body open quantum system whose classical correspondent presents period doubling. We show that by studying the dynamical correlations, it is possible to show the occurrence of period doubling in the quantum (period-one) steady state. We also discuss that such systems are natural candidates for clean and intrinsically robust Floquet time crystals.
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Affiliation(s)
- Reuben R W Wang
- Engineering Product Development Pillar, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372, Singapore
| | - Bo Xing
- Engineering Product Development Pillar, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372, Singapore
| | - Gabriel G Carlo
- Departamento de Física, CNEA, Libertador 8250, (C1429BNP) Buenos Aires, Argentina
| | - Dario Poletti
- Engineering Product Development Pillar, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372, Singapore
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12
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Volokitin V, Liniov A, Meyerov I, Hartmann M, Ivanchenko M, Hänggi P, Denisov S. Computation of the asymptotic states of modulated open quantum systems with a numerically exact realization of the quantum trajectory method. Phys Rev E 2018; 96:053313. [PMID: 29347681 DOI: 10.1103/physreve.96.053313] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/12/2017] [Indexed: 11/07/2022]
Abstract
Quantum systems out of equilibrium are presently a subject of active research, both in theoretical and experimental domains. In this work, we consider time-periodically modulated quantum systems that are in contact with a stationary environment. Within the framework of a quantum master equation, the asymptotic states of such systems are described by time-periodic density operators. Resolution of these operators constitutes a nontrivial computational task. Approaches based on spectral and iterative methods are restricted to systems with the dimension of the hosting Hilbert space dimH=N≲300, while the direct long-time numerical integration of the master equation becomes increasingly problematic for N≳400, especially when the coupling to the environment is weak. To go beyond this limit, we use the quantum trajectory method, which unravels the master equation for the density operator into a set of stochastic processes for wave functions. The asymptotic density matrix is calculated by performing a statistical sampling over the ensemble of quantum trajectories, preceded by a long transient propagation. We follow the ideology of event-driven programming and construct a new algorithmic realization of the method. The algorithm is computationally efficient, allowing for long "leaps" forward in time. It is also numerically exact, in the sense that, being given the list of uniformly distributed (on the unit interval) random numbers, {η_{1},η_{2},...,η_{n}}, one could propagate a quantum trajectory (with η_{i}'s as norm thresholds) in a numerically exact way. By using a scalable N-particle quantum model, we demonstrate that the algorithm allows us to resolve the asymptotic density operator of the model system with N=2000 states on a regular-size computer cluster, thus reaching the scale on which numerical studies of modulated Hamiltonian systems are currently performed.
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Affiliation(s)
- V Volokitin
- Mathematical Software and Supercomputing Technologies Department, Lobachevsky State University of Nizhny Novgorod, Russia
| | - A Liniov
- Mathematical Software and Supercomputing Technologies Department, Lobachevsky State University of Nizhny Novgorod, Russia
| | - I Meyerov
- Mathematical Software and Supercomputing Technologies Department, Lobachevsky State University of Nizhny Novgorod, Russia
| | - M Hartmann
- Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany
| | - M Ivanchenko
- Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod, Russia
| | - P Hänggi
- Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany
| | - S Denisov
- Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany.,Department of Applied Mathematics, Lobachevsky State University of Nizhny Novgorod, Russia
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13
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Two-Time Correlation Functions in Dissipative and Interacting Bose–Hubbard Chains. CONDENSED MATTER 2018. [DOI: 10.3390/condmat3010002] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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14
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Parametric excitation and squeezing in a many-body spinor condensate. Nat Commun 2016; 7:11233. [PMID: 27044675 PMCID: PMC4822055 DOI: 10.1038/ncomms11233] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2015] [Accepted: 03/03/2016] [Indexed: 11/24/2022] Open
Abstract
Atomic spins are usually manipulated using radio frequency or microwave fields to excite Rabi oscillations between different spin states. These are single-particle quantum control techniques that perform ideally with individual particles or non-interacting ensembles. In many-body systems, inter-particle interactions are unavoidable; however, interactions can be used to realize new control schemes unique to interacting systems. Here we demonstrate a many-body control scheme to coherently excite and control the quantum spin states of an atomic Bose gas that realizes parametric excitation of many-body collective spin states by time varying the relative strength of the Zeeman and spin-dependent collisional interaction energies at multiples of the natural frequency of the system. Although parametric excitation of a classical system is ineffective from the ground state, we show that in our experiment, parametric excitation from the quantum ground state leads to the generation of quantum squeezed states. Ultracold atomic gases with collisional interactions allow the exploration of quantum many-body physics. Here, the authors vary over time the contributions to the interaction energy to coherently control the spin state of an atomic Bose gas, thereby implementing a truly many-body control scheme.
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15
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Hoang TM, Gerving CS, Land BJ, Anquez M, Hamley CD, Chapman MS. Dynamic stabilization of a quantum many-body spin system. PHYSICAL REVIEW LETTERS 2013; 111:090403. [PMID: 24033006 DOI: 10.1103/physrevlett.111.090403] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2013] [Indexed: 06/02/2023]
Abstract
We demonstrate dynamic stabilization of a strongly interacting quantum spin system realized in a spin-1 atomic Bose-Einstein condensate. The spinor Bose-Einstein condensate is initialized to an unstable fixed point of the spin-nematic phase space, where subsequent free evolution gives rise to squeezing and quantum spin mixing. To stabilize the system, periodic microwave pulses are applied that rotate the spin-nematic many-body fluctuations and limit their growth. The stability diagram for the range of pulse periods and phase shifts that stabilize the dynamics is measured and compares well with a stability analysis.
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Affiliation(s)
- T M Hoang
- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA
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16
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Lee TE, Gopalakrishnan S, Lukin MD. Unconventional magnetism via optical pumping of interacting spin systems. PHYSICAL REVIEW LETTERS 2013; 110:257204. [PMID: 23829757 DOI: 10.1103/physrevlett.110.257204] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2013] [Indexed: 06/02/2023]
Abstract
We consider strongly interacting systems of effective spins, subject to dissipative spin-flip processes associated with optical pumping. We predict the existence of novel magnetic phases in the steady state of this system, which emerge due to the competition between coherent and dissipative processes. Specifically, for strongly anisotropic spin-spin interactions, we find ferromagnetic, antiferromagnetic, spin-density-wave, and staggered-XY steady states, which are separated by nonequilibrium phase transitions meeting at a Lifshitz point. These transitions are accompanied by quantum correlations, resulting in spin squeezing. Experimental implementations in ultracold atoms and trapped ions are discussed.
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Affiliation(s)
- Tony E Lee
- ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
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17
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Khripkov C, Cohen D, Vardi A. Coherence dynamics of kicked Bose-Hubbard dimers: interferometric signatures of chaos. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:012910. [PMID: 23410407 DOI: 10.1103/physreve.87.012910] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2012] [Revised: 11/12/2012] [Indexed: 06/01/2023]
Abstract
We study the coherence dynamics of a kicked two-mode Bose-Hubbard model starting with an arbitrary coherent spin preparation. For preparations in the chaotic regions of phase space we find a generic behavior with Floquet participation numbers that scale as the entire N-particle Hilbert space, leading to a rapid loss of single-particle coherence. However, the chaotic behavior is not uniform throughout the chaotic sea and unique statistics is found for preparations at the vicinity of hyperbolic points that are embedded in it. This is contrasted with the low log(N) participation that is responsible for the revivals in the vicinity of isolated hyperbolic instabilities.
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Affiliation(s)
- Christine Khripkov
- Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
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18
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Boukobza E, Moore MG, Cohen D, Vardi A. Nonlinear phase dynamics in a driven bosonic Josephson junction. PHYSICAL REVIEW LETTERS 2010; 104:240402. [PMID: 20867282 DOI: 10.1103/physrevlett.104.240402] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2010] [Indexed: 05/29/2023]
Abstract
We study the collective dynamics of a driven two-mode Bose-Hubbard model in the Josephson interaction regime. The classical phase space is mixed, with chaotic and regular components, which determine the dynamical nature of the fringe visibility. For a weak off-resonant drive, where the chaotic component is small, the many-body dynamics corresponds to that of a Kapitza pendulum, with the relative phase φ between the condensates playing the role of the pendulum angle. Using a master equation approach we show that the modulation of the intersite potential barrier stabilizes the φ=π "inverted pendulum" coherent state, and protects the fringe visibility.
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Affiliation(s)
- Erez Boukobza
- Department of Chemistry, Ben-Gurion University of the Negev, Post Office Box 653, Beer-Sheva 84105, Israel
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19
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Sacchetti A. Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential. PHYSICAL REVIEW LETTERS 2009; 103:194101. [PMID: 20365923 DOI: 10.1103/physrevlett.103.194101] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2009] [Indexed: 05/29/2023]
Abstract
Here we consider stationary states for nonlinear Schrödinger equations in any spatial dimension n with symmetric double well potentials. These states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a supercritical pitchfork bifurcation, and a subcritical pitchfork bifurcation with two asymmetric branches occurring as the result of saddle-node bifurcations. We show that in the semiclassical limit, or for a large barrier between the two wells, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value; in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is a universal constant in the sense that it does not depend on the shape of the double well potential and on the dimension n.
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Affiliation(s)
- Andrea Sacchetti
- Faculty of Sciences, University of Modena e Reggio Emilia, Via Campi 213/B, I-41100 Modena, Italy
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20
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Zheng Q, Wang WG, Qin P, Wang P, Zhang X, Ren Z. Decay of Loschmidt echo in a Bose-Einstein condensate at a dynamical phase transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:016214. [PMID: 19658799 DOI: 10.1103/physreve.80.016214] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/13/2009] [Indexed: 05/28/2023]
Abstract
We study the quantum Loschmidt echo (LE) in a Bose-Einstein condensate (BEC) in a double-well potential. The BEC may undergo a dynamical phase transition between two phases: a tunneling phase and a self-trapping phase. For sufficiently weak perturbation, the LE has Gaussian decay in both phases. While, for relatively strong perturbation, the LE has a Gaussian decay in the self-trapping phase and has a stretched exponential decay in the tunneling phase. This qualitative difference in the decaying law of the LE in the two phases provides a characterization of the dynamical phase transition of the BEC. The semiclassical theory is used to explain the numerically observed behaviors of the LE decay.
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Affiliation(s)
- Qiang Zheng
- Department of Physics, Nanjing University, Nanjing 210093, China
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21
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Smith-Mannschott K, Chuchem M, Hiller M, Kottos T, Cohen D. Occupation statistics of a Bose-Einstein condensate for a driven Landau-Zener crossing. PHYSICAL REVIEW LETTERS 2009; 102:230401. [PMID: 19658908 DOI: 10.1103/physrevlett.102.230401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2009] [Indexed: 05/28/2023]
Abstract
We consider an atomic Bose-Einstein condensate loaded in a biased double-well trap with tunneling rate K and interatomic interaction U. The Bose-Einstein condensate is prepared such that all N atoms are in the left well. We drive the system by sweeping the potential difference E between the two wells. Depending on the interaction u=NU/K and the sweep rate E, we distinguish three dynamical regimes: adiabatic, diabatic, and sudden and consider the occupation statistics of the final state. The analysis goes beyond mean-field theory and is complemented by a semiclassical picture.
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22
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Boukobza E, Chuchem M, Cohen D, Vardi A. Phase-diffusion dynamics in weakly coupled bose-einstein condensates. PHYSICAL REVIEW LETTERS 2009; 102:180403. [PMID: 19518847 DOI: 10.1103/physrevlett.102.180403] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2008] [Indexed: 05/27/2023]
Abstract
We study the phase sensitivity of collisional phase diffusion between weakly coupled Bose-Einstein condensates, using a semiclassical picture of the two-mode Bose-Hubbard model. When weak coupling is allowed, zero relative phase locking is attained in the Josephson-Fock transition regime, whereas a pi relative phase is only locked in Rabi-Josephson point. Our analytic semiclassical estimates agree well with the numerical results.
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Affiliation(s)
- Erez Boukobza
- Department of Chemistry, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
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23
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Lee C. Universality and anomalous mean-field breakdown of symmetry-breaking transitions in a coupled two-component Bose-Einstein condensate. PHYSICAL REVIEW LETTERS 2009; 102:070401. [PMID: 19257648 DOI: 10.1103/physrevlett.102.070401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2008] [Indexed: 05/27/2023]
Abstract
We study both mean-field and full quantum dynamics of symmetry-breaking transitions (SBTs) in a coupled two-component Bose-Einstein condensate. By controlling s-wave scattering lengths and the coupling strength, it is possible to stimulate SBTs between normal and spontaneously polarized ground states. In static transitions, the probability maxima of full quantum ground states correspond to the mean-field ground states. In dynamical transitions, due to the vanishing of excitation gaps, the mean-field dynamics shows universal scalings obeying the Kibble-Zurek mechanism. Both mean-field and full quantum defect modes appear as damped oscillations, but they appear at different critical points and undergo different oscillation regimes. The anomalous breakdown of mean-field dynamics induced by SBTs depends on the approaching direction.
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Affiliation(s)
- Chaohong Lee
- Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia.
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24
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Witthaut D, Trimborn F, Wimberger S. Dissipation induced coherence of a two-mode Bose-Einstein condensate. PHYSICAL REVIEW LETTERS 2008; 101:200402. [PMID: 19113318 DOI: 10.1103/physrevlett.101.200402] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2008] [Indexed: 05/27/2023]
Abstract
We discuss the dynamics of a Bose-Einstein condensate in a double-well trap subject to phase noise and particle loss. The phase coherence of a weakly interacting condensate as well as the response to an external driving show a pronounced stochastic resonance effect: Both quantities become maximal for a finite value of the dissipation rate matching the intrinsic time scales of the system. Even stronger effects are observed when dissipation acts in concurrence with strong interparticle interactions, restoring the purity of the condensate almost completely and increasing the phase coherence significantly.
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Affiliation(s)
- D Witthaut
- QUANTOP, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark.
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25
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Graefe EM, Korsch HJ, Niederle AE. Mean-field dynamics of a non-Hermitian Bose-Hubbard dimer. PHYSICAL REVIEW LETTERS 2008; 101:150408. [PMID: 18999581 DOI: 10.1103/physrevlett.101.150408] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/11/2008] [Revised: 08/29/2008] [Indexed: 05/27/2023]
Abstract
We investigate an N-particle Bose-Hubbard dimer with an additional effective decay term in one of the sites. A mean-field approximation for this non-Hermitian many-particle system is derived, based on a coherent state approximation. The resulting nonlinear, non-Hermitian two-level dynamics, in particular, the fixed point structures showing characteristic modifications of the self-trapping transition, are analyzed. The mean-field dynamics is found to be in reasonable agreement with the full many-particle evolution.
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Affiliation(s)
- E M Graefe
- FB Physik, TU Kaiserslautern, D-67653 Kaiserslautern, Germany
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26
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Khodorkovsky Y, Kurizki G, Vardi A. Bosonic amplification of noise-induced suppression of phase diffusion. PHYSICAL REVIEW LETTERS 2008; 100:220403. [PMID: 18643407 DOI: 10.1103/physrevlett.100.220403] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2008] [Revised: 04/17/2008] [Indexed: 05/26/2023]
Abstract
We study the effect of noise-induced dephasing on collisional phase diffusion in the two-site Bose-Hubbard model. Dephasing of the quasimomentum modes may slow down phase diffusion in the quantum Zeno limit. Remarkably, the degree of suppression is enhanced by a bosonic factor of order N/logN as the particle number N increases.
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Affiliation(s)
- Y Khodorkovsky
- Department of Chemistry, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva 84105, Israel
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27
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Wang WG, Liu J, Li B. Stability of Fock states in a two-component Bose-Einstein condensate with a regular classical counterpart. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:056218. [PMID: 18643153 DOI: 10.1103/physreve.77.056218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/27/2007] [Revised: 04/10/2008] [Indexed: 05/26/2023]
Abstract
We study the stability of a two-component Bose-Einstein condensate (BEC) in the parameter regime in which its classical counterpart has regular motion. The stability is characterized by the fidelity for both the same and different initial states. We study as initial states the Fock states with definite numbers of atoms in each component of the BEC. It is found that for some initial times the two Fock states with all the atoms in the same component of the BEC are more stable than Fock states with atoms distributed in the two components. An experimental scheme is discussed, in which the fidelity can be measured in a direct way.
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Affiliation(s)
- Wen-Ge Wang
- Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
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28
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Lee C. Adiabatic Mach-Zehnder interferometry on a quantized Bose-Josephson junction. PHYSICAL REVIEW LETTERS 2006; 97:150402. [PMID: 17155301 DOI: 10.1103/physrevlett.97.150402] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2006] [Indexed: 05/12/2023]
Abstract
We propose a scheme to achieve Mach-Zehnder interferometry using a quantized Bose-Josephson junction with a negative charging energy. The quantum adiabatic evolution through a dynamical bifurcation is used to accomplish the beam splitting and recombination. The negative charging energy ensures the existence of a path-entangled state which enhances the phase measurement precision to the Heisenberg limit. A feasible detection procedure is also presented. The scheme should be realizable with current technology.
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Affiliation(s)
- Chaohong Lee
- Nonlinear Physics Centre and ARC Centre of Excellence for Quantum-Atom Optics, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia
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29
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Chong G, Hai W, Xie Q. Controlling chaos in a weakly coupled array of Bose-Einstein condensates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:016202. [PMID: 15697693 DOI: 10.1103/physreve.71.016202] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2004] [Indexed: 05/24/2023]
Abstract
The spatial structure of a Bose-Einstein condensate loaded into an optical lattice potential is investigated and the spatially chaotic distributions of the condensates are revealed under the tight-binding approximation. Adding a laser pulse on a proper site of the lattice and treating it as a control signal, control of the chaos in the system is carried out by using the Ott-Grebogi-Yorker scheme. For an appropriate laser pulse, we can suppress the chaos and push the system onto a stable manifold of a target orbit. After the control, a regular distribution, which may be expected in experiments or practical applications, of the condensates in the coordinate space is obtained.
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Affiliation(s)
- Guishu Chong
- Department of Physics, Hunan Normal University, Changsha 410081, China
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30
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Zhang C, Liu J, Raizen MG, Niu Q. Transition to instability in a kicked bose-einstein condensate. PHYSICAL REVIEW LETTERS 2004; 92:054101. [PMID: 14995306 DOI: 10.1103/physrevlett.92.054101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2003] [Indexed: 05/24/2023]
Abstract
A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor. For weak interactions between atoms, periodic motion (antiresonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.
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Affiliation(s)
- Chuanwei Zhang
- Department of Physics, and Center for Nonlinear Dynamics, The University of Texas, Austin, Texas 78712-1081, USA
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31
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Liu J, Wu B, Niu Q. Nonlinear evolution of quantum states in the adiabatic regime. PHYSICAL REVIEW LETTERS 2003; 90:170404. [PMID: 12786058 DOI: 10.1103/physrevlett.90.170404] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2002] [Indexed: 05/24/2023]
Abstract
We investigate adiabatic evolution of quantum states as governed by the nonlinear Schrödinger equation and provide examples of applications with a nonlinear tunneling model for Bose-Einstein condensates. Our analysis not only spells out conditions for adiabatic evolution of eigenstates but also characterizes the motion of noneigenstates which cannot be obtained from the former in the absence of the superposition principle. We find that Aharonov-Anandan phases play the role of classical canonical actions and are conserved in the adiabatic evolution of noneigenstates.
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Affiliation(s)
- Jie Liu
- Department of Physics, The University of Texas, Austin, Texas 78712, USA
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32
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Kohler S, Sols F. Oscillatory decay of a two-component bose-einstein condensate. PHYSICAL REVIEW LETTERS 2002; 89:060403. [PMID: 12190569 DOI: 10.1103/physrevlett.89.060403] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2001] [Indexed: 05/23/2023]
Abstract
We study the decay of a two-component Bose-Einstein condensate with negative effective interaction energy. With a decreasing atom number due to losses, the atom-atom interaction becomes less important and the system undergoes a transition from a bistable Josephson regime to the monostable Rabi regime, displaying oscillations in phase and number. We study the equations of motion and derive an analytical expression for the oscillation amplitude. A quantum trajectory simulation reveals that the classical description fails for low loss rates, as expected from analytical considerations. Observation of the proposed effect will provide evidence for negative effective interaction.
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Affiliation(s)
- Sigmund Kohler
- Departamento de Física Teórica de la Materia Condensada and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, Spain
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33
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Berman GP, Smerzi A, Bishop AR. Quantum instability of a Bose-Einstein condensate with attractive interaction. PHYSICAL REVIEW LETTERS 2002; 88:120402. [PMID: 11909432 DOI: 10.1103/physrevlett.88.120402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2001] [Indexed: 05/23/2023]
Abstract
We study the quantum and the mean-field Gross-Pitaevskii (GP) dynamics of a Bose-Einstein condensate gas confined in a toroidal trap. According to GP, if the interatomic interaction is attractive, the rotational states of the system can be dynamically stable or unstable depending on the strength of the mean-field energy. The full quantum analysis, however, reveals that the condensate is always unstable. Quantum fluctuations are particularly important close to the GP stability borderline, even for systems with a relatively large number of condensate atoms.
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Affiliation(s)
- G P Berman
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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34
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Zhou C, Xie D, Chen R, Yan G, Guo H, Tyng V, Kellman ME. Quantum calculation of highly excited vibrational energy levels of CS2(X) on a new empirical potential energy surface and semiclassical analysis of 1:2 Fermi resonance. SPECTROCHIMICA ACTA. PART A, MOLECULAR AND BIOMOLECULAR SPECTROSCOPY 2002; 58:727-746. [PMID: 11991492 DOI: 10.1016/s1386-1425(01)00666-7] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
We report a refined potential energy function for the ground electronic state of CS2 based on a least-squares fitting to several low-lying experimental vibrational frequencies. Energy levels up to 20,000 cm(-1) have been obtained on this empirical potential using the Lanczos algorithm and potential optimized discrete variable representation. Among them, 329 levels below 10,000 cm(-1) are assigned with approximate normal mode quantum numbers (n1, n(0)2, n3), based on expectation values of one-dimensional (1D) reference Hamiltonians. An effective Hamiltonian is extracted from these assigned levels. The agreement with experimental data, including those of several isotopically substituted species, is excellent. In addition, some Fermi and anharmonic resonances are analyzed. The nearest neighbor level spacing and delta3 distributions indicated that the vibrational spectrum of CS2 is largely regular in the energy range up to 20,000 cm(-1). Semiclassical phase space analysis, including bifurcation analysis of the spectroscopic Hamiltonian, is used to interpret subtle anomalies signaled by expectation values used in normal mode assignments. The meaning of Fermi resonance is clarified by contrasting the semiclassical analysis of CS2 and CO2.
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Affiliation(s)
- Cheng Zhou
- Department of Chemistry, University of New Mexico, Albuquerque 87111, USA
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