Quantum master equations for a system interacting with a quantum gas in the low-density limit and for the semiclassical collision model

Abstract

A quantum system interacting with a diluted gas experiences the irreversible dynamics. The corresponding master equation can be derived within two different approaches: the fully quantum description in the low-density limit and the semiclassical collision model, where the motion of gas particles is classical whereas their internal degrees of freedom are quantum. The two approaches have been extensively studied in the literature but their predictions have not been compared. This is mainly due to a fact that the low-density limit is extensively studied for mathematical physics analysis, whereas the collision models have been essentially developed for quantum information tasks as a tractable description of the open quantum dynamics. Here we develop and for the first time compare both approaches for a spin system interacting with a gas of spin particles. Using some approximations, we explicitly find the corresponding master equations including the Lamb shifts and the dissipators. The low density limit in the Born approximation for fast particles is shown to be equivalent to the semiclassical collision model in the stroboscopic approximation. We reveal that the both approaches give exactly the same master equation if the gas temperature is high enough. This allows to interchangeably use complicated calculations in the low density limit and rather simple calculations in the collision model.

Quantum collision models: Open system dynamics from repeated interactions

“…65 We will consider unitary AA collisions, yet the property can be extended to non-unitary collisions (as those in Section 8.2). 66 At variance with Section 8.1, here we explicitly show the S-dependence of SA collision unitaries, which facilitates establishing the connection with the notation used for introducing composite CMs.…”

“…Ref. [66] showed that this is equivalent to the low-density, fast-particle limit of a fully quantum treatment.…”

Two-Qubit Entanglement Generation through Non-Hermitian Hamiltonians Induced by Repeated Measurements on an Ancilla

“…If the measurement repetition rate is much greater than the maximum Bohr frequency of H , then one can neglect the term in Equation ( 13 ). This means that the system evolution in between the measurement acts is infinitesimal, so that the stroboscopic time is quasi-continuous in full analogy with the quantum collision models [ 48 , 49 , 50 , 51 ]. As a result, we obtain and the resulting non-Hermitian Hamiltonian, effectively acting on S , under the Zeno experimental protocol for repeated measurements on the ancilla, may be taken as …”

“…Equation ( 19 ). Therefore, the physical dynamics of are governed by the non-Hermitian Hamiltonian , which reads as follows on a conventional eigenbasis of operator : The obtained Hamiltonian provides an adequate description within the stroboscopic approximation, which is justified if [ 48 , 49 , 50 , 51 ]. However, the quantity can be comparable with if the coupling strength .…”

Complete Integrability of Quantum and Classical Dynamical Systems

“…Note that there is also another direction in studying properties of quantum systems when one takes the large time limit together with small coupling or small density limit, see [15,16,17].…”