Decay of Loschmidt Echo Enhanced by Quantum Criticality

Abstract

We study the transition of a quantum system S from a pure state to a mixed one, which is induced by the quantum criticality of the surrounding system E coupled to it. To characterize this transition quantitatively, we carefully examine the behavior of the Loschmidt echo (LE) of E modelled as an Ising model in a transverse field, which behaves as a measuring apparatus in quantum measurement. It is found that the quantum critical behavior of E strongly affects its capability of enhancing the decay of LE: near the critical value of the transverse field entailing the happening of quantum phase transition, the off-diagonal elements of the reduced density matrix describing S vanish sharply. Introduction: Nowadays quantum -classical transitions described by a reduction from a pure state to a mixture [1, 2] renew interests in many areas of physics, mainly due to the importance of quantum measurement and decoherence problem in quantum computing. To study this transition, some exactly-solvable models were proposed for a system coupled to the macroscopic [3,4,5] or classical [6,7] surrounding systems. Relevantly, in association with the quantum -classical transition in quantum chaos, the concept of Loschmidt echo (LE) from NMR experiments was introduced to describe the hypersensitivity of the time evolution to the perturbations experienced by the surrounding system [8,9]. In this letter, by a concrete example, we will show how quantum phase transition (QPT) [10] of the surrounding system can also sensitively affect the decay of its own LE, which means a dynamic reduction of its coupled system from pure state to a mixed one. Here, we note that a QPT effect has been explored for the Dicke model at the transition from quasi-integrable to quantum chaotic phases [11].

Decoherence of a quantum system coupled to an XY spin chain: Role of the initial state of the spin chain

“…The effects of the effective Hamiltonian on the decoherence has been studied in many papers [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Especially, dramatic manifestation of the decoherence has often been found in the vicinity of the quantum critical point of the effective Hamiltonian.…”

“…[9][10][11], it has been shown that the coherence factor F (t) decay more dramatically and rapidly in the vicinity of the quantum critical point λ i =λ e =1.0 [9][10][11] for small N (N≈10 2 ). In this paper we will study the decoherence with different initial states.…”

“…Especially, dramatic manifestation of the decoherence has often been found in the vicinity of the quantum critical point of the effective Hamiltonian. Hence much work have been focused on the critical properties of the decoherence [7,9,10,[15][16][17][18][19].…”

Loschmidt echo and Berry phase of a quantum system coupled to anXYspin chain: Proximity to a quantum phase transition

“…Recently, Sengupta et al [14] have studied time evolution of the Ising order correlations under a time-dependent transverse field and shown that the order parameter is best enhanced in the vicinity of the quantum critical point. Quan et al [15] have studied transition dynamics of a quantum two-level system from a pure state to a mixed one induced by the quantum criticality of the surrounding many-body system. They have shown that the decaying behavior of the LE is best enhanced by the QPT of the surrounding system.…”

Instabilities of the Dicke model with field interactions

“…It is known that quantum critical phenomena exhibits deep relations to the Berry phase (BP) [19,20,21,22] and the fidelity [23,24,25,26,27,28,29,30]. The BP has been extensively studied by the geometric time evolution of a quantum system, providing means to detect the quantum effects and critical behavior, such as quantum jumps and collapse [31,32,33].…”