Quantum chaotic system as a model of decohering environment

As a model of decohering environment, we show that quantum chaotic system behave equivalently as many-body system. An approximate formula for time-evolution of the reduced density matrix of a system interacting with a quantum chaotic environment is derived. This theoretical formulation is substantiated by numerical study of decoherence of two qubits interacting with a quantum chaotic environment modeled by chaotic kicked top. Like many-body model of environment, quantum chaotic system is efficient decoherer, and it can generate entanglement between the two qubits which have no direct interaction. Interaction of a quantum system with environment creates correlations between the states of the system and of the environment. These correlations destroy the superposition of the system states a phenomenon known as decoherence [1]. This phenomenon is believed to be responsible for quantum to classical transition. Decoherence is also a major obstacle for designing quantum computational and informational protocol [2]. Therefore, a deeper understanding of the phenomenon is required to address the fundamental question like quantum-classical transition and, to develop quantum computational protocol. In general, the environment is modeled by many-body system, e.g. infinitely many harmonic oscillators in thermal equilibrium (Feynman-Vernon or Caldeira-Leggett model) [3], spin-boson model [4], chaotic spin-chain [5], etc. In another approach, random matrix model of the environment is used [6,7]. Random matrix theory has well known connections with quantum chaotic systems. Hence, some studies have concentrated on the possibility of having quantum dissipation and decoherence due to the interaction with chaotic degrees of freedom [8]. Recently, as a model of decohering environment, single particle quantum chaotic system has been considered [9]. This paper shows that the kicked rotator, a well studied model of chaotic system, can reproduce the decohering effects of a manybody environment. In comparison to complex many-body system, this simple deterministic system is very convenient for numerical as well as analytical studies of decoherence. Hence, single particle quantum chaotic system warrants a (a)Present address : Department of Physics, National University of Singapore, 117542, Singapore special attention as a model of decohering environment. In this Letter, we establish direct equivalence of single particle quantum chaotic environment and CaldeiraLeggett type model of many-body environment by providing a rigorous but straightforward treatment. We keep our results vis-a-vis a recent study in which decoherence in a quantum system is investigated under the influence of a collection of harmonic oscillators environment [10]. Our derivation first assumes weak interaction between the system and the environment. Using the interaction strength as a small parameter, we perform perturbative theory calculation. By exponentiating the perturbative expansion, we get an approximate formula for the non-perturbative strong interaction effect of the environment on the system. The approximate formula is then justified by numerical evidences. In numerics, we study decoherence of two noninteracting qubits which are individually interacting with a common quantum chaotic environment. We use chaotic kicked top, a very well studied model of quantum chaotic system [11], as the environment. Most general form of the Hamiltonian of a system S, interacting with an environment E, is H = HS+HE+HSE, where HS and HE are the Hamiltonians of the system and the environment, respectively, andHSE is the systemenvironment coupling Hamiltonian. We assume throughout this Letter that the decoherence time is much smaller than the system characteristic time. Hence we can neglect any dynamics of the isolated system and can discard HS . We consider kicked quantum chaotic system as a model of the environment, so the general form of the timedependent system-environment Hamiltonian is : H(t) =