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Fortschritte der Physik

Wave packet motion and decoherence in an anharmonic potential

Péter Földi, M. G. Benedict, Attila Czirják, Bence Molnár

Published Feb 26, 2003• 6 citations
Publisher

Abstract

Abstract The time evolution of wave packets in the Morse potential is investigated under the influence of the environment consisting of harmonic oscillators. We obtain a Markovian master equation that incorporates the fact that due to the nonequidistant spectrum, the relaxation rate of each energy eigenstate is different. The nonunitary time evolution is visualized by the aid of the Wigner function related to the reduced density operator of the Morse system. The time scale of decoherence is much shorter than that of dissipation, and gives rise to states which are mixtures of localized states along the phase space orbit of the corresponding classical particle. This behaviour is to a large extent independent of the coupling strength, the temperature of the environment and also of the initial state.

DOI
10.1002/prop.200310014

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Abstract The time evolution of wave packets in the Morse potential is investigated under the influence of the environment consisting of harmonic oscillators. We obtain a Markovian master equation that incorporates the fact that due to the nonequidistant spectrum, the relaxation rate of each energy eigenstate is different. The nonunitary time evolution is visualized by the aid of the Wigner function related to the reduced density operator of the Morse system. The time scale of decoherence is much shorter than that of dissipation, and gives rise to states which are mixtures of localized states along the phase space orbit of the corresponding classical particle. This behaviour is to a large extent independent of the coupling strength, the temperature of the environment and also of the initial state.
The time evolution of wave packets in the Morse potential is investigated under the influence of the environment consisting of harmonic oscillators. These oscillators represent photon or phonon modes and are assumed to be in thermal equilibrium. Our model explicitly incorporates the fact that in the case of a nonequidistant spectrum the rates of the environment induced transitions are different for each transition. The nonunitary time evolution is visualized by the aid of the corresponding Wigner function. The time scale of decoherence is much shorter than that of dissipation, and gives rise to states that are mixtures of localized states along the phase-space orbit of the corresponding classical particle. This behavior is to a large extent independent of the coupling strength, the temperature of the environment, and also the initial state.
The effect of the environment on a quantum system is studied on an exactly solvable model: a harmonic oscillator interacting with a one-dimensional massless scalar field. We show that in an open quantum system, dissipation can cause decorrelation on a time scale significantly shorter than the relaxation time which characterizes the approach of the system to thermodynamic equilibrium. In particular, we demonstrate that the density matrix decays rapidly toward a mixture of ``approximate eigenstates'' of the ``pointer observable,'' which commutes with the system-environment interaction Hamiltonian. This observable can be regarded as continuously, if inaccurately, monitored by the scalar field environment. Both because in a harmonic oscillator the state of the system rotates in the phase space and because the effective environment ``measurement'' is weak, the system, on the short ``collision'' time scale (1/\ensuremath{\Gamma}), maintains a coherence in this pointer observable on time scales of order [\ensuremath{\gamma}/\ensuremath{\Omega}ln(\ensuremath{\Gamma}/\ensuremath{\Omega})]1/2{)]}^{1/2})]1/2 and on longer time scales settles into a mixture of coherent states with a dispersion approximately consistent with the vacuum state. The master equation satisfied by the exact solution differs from the other master equations derived both for the high-temperature limit and for T=0. We discuss these differences and study the transition region between the high- and low-temperature regimes. We also consider the behavior of the system in the short-time ``transient'' regime. For T=0, we find that, in the long-time limit, the system behaves as if it were subject to ``1/f noise.'' The generality of our model is considered and its predictions are compared with previous treatments of related problems. Some of the possible applications of the results to experimentally realizable situations are outlined. The significance of the environment-induced reduction of the wave packet for cosmological models is also briefly considered.
We have studied the quantum dissipative problem of a Gaussian wave packet under the influence of a harmonic potential. A phenomenological approach to dissipation is adopted in the light of the well-known model in which the environment is composed of a bath of noninteracting harmonic oscillators. As one of the effects of the coupling to the bath is the evolution of an initially pure wave packet into a statistical mixture, we estimate the characteristic time elapsed for this to occur for different regimes of temperature, damping, and also different initial states.
We find an exact analytical solution of the reduced density matrix from the Feynman-Vernon influence functional theory for a wave packet influenced by an environment containing a few discrete modes. We obtain two intrinsic energy scales relating to the time scales of the system and the environment. Different relationship between these two scales alters the overall form of the solution of the system. We also introduce a decoherence measure for a single wave packet which is defined as the ratio of Schrödinger uncertainty over the delocalization extension of the wave packet and characterizes the time-evolution behavior of the off-diagonal reduced density matrix element. We utilize the exact solution and the docherence measure to study the wave packet decoherence dynamics. We further demonstrate how the dynamical diffusion of the wave packet leads to non-Markovian decoherence in such a microscopic environment.
The dynamics of an initially excital Gaussian wave packet (harmonic oscillator coherent state, HOCS) in a general one-dimensional anharmonic system is studied analytically with the aid of a second quantization formalism for times shorter than a classical oscillation period. The initial state population probability PQ(t), its classical counterpart PC(t), and the overlap probability PQC(t) between a time evolved wave packet and a classically moving Gaussian with stationary shape are calculated in order to analyze the spreading of the wave packet. It was found that, in general, PQ(t) and PC(t) agree very well if the potential energy function has no strongly changing low order derivatives. This agreement is held even if the wave packet spreads drastically within the time scale considered as can be demonstrated by analyzing PQC(t). The analytical expressions for PQ(t), PC(t), and PQC(t) are applied to a dissociative third-order oscillator and a strongly anharmonic fourth-order oscillator. While for a harmonic system, the initial decay time of PQ(t) and PC(t) (which are identical in this case) are only dependent on the energy expectation value 〈H〉 with respect to the initial HOCS, differences up to an order of magnitude occur for HOCS with the same energy in the model systems.
Environment-induced destruction of quantum coherence is investigated in a simple model where the system is a harmonic oscillator, the environment is a collection of harmonic oscillators, and the interaction between them is linear in the position coordinate of the system. We study decoherence for initial states consisting of coherent superpositions of two Gaussian wave packets in either position or momentum. A new measure of the effectiveness of decoherence appropriate to the model and choice of initial conditions is proposed. By studying the dependence of the decoherence rate on the location of the initial peaks of the Wigner function, we clarify the sense in which position is a preferred observable even though position eigenstates are not the pointer states of this model. We analyze decoherence in the low-temperature regime and show that the usual "high-temperature" approximation is remarkably accurate in its domain of applicability. We also examine the relationship between the decoherence process and the frequency distribution of the environment oscillators (in particular, we focus attention on a specific "supra-Ohmic" environment). Implications of our results for the quantum to classical transition in various contexts are briefly explored.