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arXiv:0706.3633 · Physical Review A

Phase diffusion in quantum dissipative systems

Subhashish Banerjee, R. Srikanth

Published Dec 13, 2007• 23 citations
arXivHTMLPDF

Abstract

We study the dynamics of the quantum phase distribution associated with the reduced density matrix of a system for a number of situations of practical importance, as the system evolves under the influence of its environment, interacting via a quantum nondemolition type of coupling, such that there is decoherence without dissipation, as well as when it interacts via a dissipative interaction, resulting in decoherence as well as dissipation. The system is taken to be either a two-level atom (or, equivalently, a spin-$1∕2$ system) or a harmonic oscillator, and the environment is modeled as a bath of harmonic oscillators, starting out in a squeezed thermal state. The impact of the different environmental parameters on the dynamics of the quantum phase distribution for the system starting out in various initial states is explicitly brought out. An interesting feature that emerges from our work is that the relationship between squeezing and temperature effects depends on the type of system-bath interaction. In the case of a quantum nondemolition type of interaction, squeezing and temperature work in tandem, producing a diffusive effect on the phase distribution. In contrast, in the case of a dissipative interaction, the influence of temperature can be counteracted by squeezing, which manifests as a resistance to randomization of phase. We make use of the phase distributions to bring out a notion of complementarity in atomic systems. We also study the variance of the phase using phase distributions conditioned on particular initial states of the system.

DOI
10.1103/physreva.76.062109

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We study the dynamics of the quantum phase distribution associated with the reduced density matrix of a system for a number of situations of practical importance, as the system evolves under the influence of its environment, interacting via a quantum nondemolition type of coupling, such that there is decoherence without dissipation, as well as when it interacts via a dissipative interaction, resulting in decoherence as well as dissipation. The system is taken to be either a two-level atom (or, equivalently, a spin-1∕21∕21∕2 system) or a harmonic oscillator, and the environment is modeled as a bath of harmonic oscillators, starting out in a squeezed thermal state. The impact of the different environmental parameters on the dynamics of the quantum phase distribution for the system starting out in various initial states is explicitly brought out. An interesting feature that emerges from our work is that the relationship between squeezing and temperature effects depends on the type of system-bath interaction. In the case of a quantum nondemolition type of interaction, squeezing and temperature work in tandem, producing a diffusive effect on the phase distribution. In contrast, in the case of a dissipative interaction, the influence of temperature can be counteracted by squeezing, which manifests as a resistance to randomization of phase. We make use of the phase distributions to bring out a notion of complementarity in atomic systems. We also study the variance of the phase using phase distributions conditioned on particular initial states of the system.
We quantitatively analyze the dynamics of the quantum phase distribution associated with the reduced density matrix of a system, as the system evolves under the influence of its environment with an energy-preserving quantum nondemolition (QND) type of coupling. We take the system to be either an oscillator (harmonic or anharmonic) or a two-level atom (or equivalently, a spin-1/2 system), and model the environment as a bath of harmonic oscillators, initially in a general squeezed thermal state. The impact of the different environmental parameters is explicitly brought out as the system starts out in various initial states. The results are applicable to a variety of physical systems now studied experimentally with QND measurements.
We study a generic open quantum system where the coupling between the system and its environment is of an energy-preserving quantum nondemolition (QND) type. We obtain the general master equation for the evolution of such a system under the influence of a squeezed thermal bath of harmonic oscillators. From the master equation it can be seen explicitly that the process involves decoherence or dephasing without any dissipation of energy. We work out the decoherence-causing term in the high- and zero-temperature limits and check that they match with known results for the case of a thermal bath. The decay of the coherence is quantified as well by the dynamics of the linear entropy of the system under various environmental conditions. We make a comparison of the quantum statistical properties between QND and dissipative types of evolution using a two-level atomic system and a harmonic oscillator.
We study open quantum systems where the coupling between the system and its environment is of a quantum nondmolition (QND) type. Such a system undergoes decoherence without dissipation of energy. We obtain the master equation for the evolution of such a system under the influence of a squeezed thermal bath. From the master equation it can be seen explicitly that the process involves decoherence without any dissipation. We work out the decoherence causing term in the high and zero temperature limits and show that they match with known results in the different temperature regimes for the case of a thermal bath. We also make a comparison between the quantum statistical properties of QND and non-QND (i.e., involving decoherence as well as dissipation) types of evolution.
We consider a model of environment-induced dissipationless decoherence of a quantum system where the system is coupled to the bath degrees of freedom via the system Hamiltonian itself. We solve exactly for the reduced density operator of the system for an arbitrary spectral density of the thermal bath and also write down an exact master equation in the Lindblad form. We compare and contrast the above results with those obtained by considering the system frequencies to be randomly modulated as in stochastic models. We observe that a coupling to the bath as above necessarily induces a Kerr-like coherent contribution in the reduced dynamics of the system. This Kerr-like term is a reflection of the quantum nature of the bath and cannot be obtained from stochastic models. For the special case of a harmonic oscillator we consider the influence of decoherence and its relation with phase diffusion. Our numerical results exhibit oscillations in the evolution of system variables which overall is a signature of the quantum nature of the environment.
We derive and analyze the Born-Markov master equation for a quantum harmonic oscillator interacting with a bath of independent two-level systems. This hitherto virtually unexplored model plays a fundamental role as one of the four ``canonical'' system-environment models for decoherence and dissipation. To investigate the influence of further couplings of the environmental spins to a dissipative bath, we also derive the master equation for a harmonic oscillator interacting with a single spin coupled to a bosonic bath. Our models are experimentally motivated by quantum-electromechanical systems and micron-scale ion traps. Decoherence and dissipation rates are found to exhibit temperature dependencies significantly different from those in quantum Brownian motion. In particular, the systematic dissipation rate for the central oscillator decreases with increasing temperature and goes to zero at zero temperature, but there also exists a temperature-independent momentum-diffusion (heating) rate.
In this letter, we introduce a novel method for investigating dissipation (gain) and thermalization in an open quantum system. In this method, the quantum system is coupled linearly with a copy of itself or with another system described by a finite number of bosonic operators. The time-dependent coupling functions play a fundamental role in this scheme. To demonstrate the efficiency and significance of the method, we apply it to some ubiquitous open quantum systems. Firstly, we investigate a quantum oscillator in the presence of a thermal bath at the inverse temperature βββ, obtaining the reduced density matrix, the Husimi distribution function, and the quantum heat distribution function accurately. The results are consistent with existing literature by appropriate choices for the time-dependent coupling function. To illustrate the generalizability of this method to systems interacting with multiple thermal baths, we study the interaction of a quantum oscillator with two thermal baths at different temperatures and obtain compatible results. Subsequently, we analyze a two-level atom with energy or phase dissipation and derive the spontaneous emission and the pure dephasing processes consistently using the new method. Finally, we investigate the Markovianity in a dissipative two-level system.