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Lobachevskii Journal of Mathematics

Quantum dissipative dynamics a particle in a double-well potential

Chang Sub Kim, D. S. Pashin, Сатанин Аркадий Михайлович

Published Sep 1, 2017• 2 citations
Publisher

Abstract

The results of analytical and numerical investigation of quantum dissipative dynamics a particle moving in a potential well with two equilibrium states is presented. A classical analogue of this problem is the capture into the resonance a particle in a double-well potential, when two states of equilibrium in the phase space are separated by a separatrix. The probability of finding the system in one of states of equilibrium for the quantum particle, depending on initial conditions and noise, is discussed.

DOI
10.1134/s1995080217050201

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The results of analytical and numerical investigation of quantum dissipative dynamics a particle moving in a potential well with two equilibrium states is presented. A classical analogue of this problem is the capture into the resonance a particle in a double-well potential, when two states of equilibrium in the phase space are separated by a separatrix. The probability of finding the system in one of states of equilibrium for the quantum particle, depending on initial conditions and noise, is discussed.
We investigate the capture of a quantum particle in free space by a moving trapping potential. The capture is investigated for various initial conditions. We examine the dependence of the probability of capture on the shape and depth of the trapping potential, initial speed and mass of the particle, and other parameters. We take the trapping potential to move with an initial speed v 0 and to decelerate with constant acceleration a c until the well stops moving. We study the motion during the time the well is moving and after the well has stopped. We compare the probability of capture to the probability that the particle is in a stationary state (while the well is moving) and the probability the particle is in a bound state (after the well's motion has stopped). Our work could be of interest to instructors and students in upper-year undergraduate quantum mechanics courses.
Non-equilibrium thermodynamics can provide strong advantages when compared to more standard equilibrium situations. Here, we present a general framework to study its application to concrete problems, which is valid also beyond the assumption of a Gaussian dynamics. We consider two different problems: 1) the dynamics of a levitated nanoparticle undergoing the transition from an harmonic to a double-well potential; 2) the transfer of a quantum state across a double-well potential through classical and quantum protocols. In both cases, we assume that the system undergoes to decoherence and thermalisation. In case 1), we construct a numerical approach to the problem and study the non-equilibrium thermodynamics of the system. In case 2), we introduce a new figure of merit to quantify the efficiency of a state-transfer protocol and apply it to quantum and classical versions of such protocols.
We describe the dynamics of a bound state of an attractive δ-well under displacement of the potential. Exact analytical results are presented for the suddenly moved potential. Since this is a quantum system, only a fraction of the initially confined wave function remains confined to the moving potential. However, it is shown that besides the probability to remain confined to the moving barrier and the probability to remain in the initial position, there is also a certain probability for the particle to move at double speed. A quasi-classical interpretation for this effect is suggested. The temporal and spectral dynamics of each one of the scenarios is investigated.
Abstract: In this work, we study the dissipative dynamics of a double-well Bose-Einstein condensate (BEC) out-coupled to reservoir at each side of its trap. The sub-system comprises of a simple Bose-Hubbard model, where the interplay of atom-tunneling current and inter-particle interaction are the main quantum features. The contact with two separate heat baths causes dissipation and drives the system into a non-equilibrium state. The system is well described by the Generalized Quantum Heisenberg-Langevin equation. We considered two Markovian dissipative BEC systems based on (i) the mean-field model (MF), where the internal noise has been averaged out and (ii) the noise-correlated model (FDT). Physical quantities, such as population imbalance, coherence and entanglement of the system, are computed for the models. The two-mode BEC phases, such as the quantum tunneling state and the macroscopic quantum-trapping state, evolved into complicated dynamics by controlling the non-linear interaction and dissipation strengths. We found that many important quantum features produced by the noise-correlated FDT model are not captured by the mean-field model. Keywords: Double-well BEC, Dissipation, Noise, Markovian, Non-Markovian, Fixed points. PACS: 03.75 Lm, 03.65 Yz, 03.75 Gg, 05.
We theoretically study the phase dynamics in Josephson junctions, which maps onto the oscillatory motion of a pointlike particle in the washboard potential. Under appropriate driving and damping conditions, the Josephson phase undergoes intriguing bistable dynamics near a saddle point in the quasienergy landscape. The bifurcation mechanism plays a critical role in superconducting quantum circuits with relevance to nondemolition measurements such as high-fidelity readout of qubit states. We address the question "what is the probability of capture into either basin of attraction?" and answer it concerning both classical and quantum dynamics. Consequently, we derive the Arnold probability and numerically analyze its implementation of the controlled dynamical switching between two steady states under the various nonequilibrium conditions.
Quantum dynamics of wave packet in one-dimensional system with distributed potential and dissipation is investigated in context of Schrodinger-Langevin-Kostin equation. Quadratic potential and double well potential bounded by external impenetrable walls are represented as examples. Numerical simulations are carried out for initial Gaussian packet at different values of dissipation parameters. For the quantum oscillator with dissipation the asymptotic dynamical limit in time is established. The mean values of coordinate and velocity tend to zero but uncertainty relation tends to minimal value 0.5 (in atomic units). Oscillations and resonances at external time-dependent action were studied. Dynamical properties of system with double well potential are investigated at different frequencies of external action. Fourier spectra of mean coordinate are characterized by sharply expressed singular components.