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Physical Review A

Quantum dynamics of Bose-Fermi mixtures via the stochastic-wave-function approach

Paweł Zin

Published Oct 4, 2018• 0 citations
Publisher

Abstract

We present a reformulation of the nonrelativistic many-body quantum dynamics of Bose-Fermi mixtures via stochastic wave functions. We show that, within a wide range of two-particle interactions and a certain class of greater-than-two-particle interactions, the quantum dynamics can be mapped exactly onto a set of single-particle wave functions, with their evolution governed by a stochastic process.

DOI
10.1103/physreva.98.043608

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We present a reformulation of the nonrelativistic many-body quantum dynamics of Bose-Fermi mixtures via stochastic wave functions. We show that, within a wide range of two-particle interactions and a certain class of greater-than-two-particle interactions, the quantum dynamics can be mapped exactly onto a set of single-particle wave functions, with their evolution governed by a stochastic process.
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We show that any non-relativistic quantum N-body dynamics problem with pairwise interactions can be exactly reformulated in terms of N well behaved 1-body stochastic density equations. Specifically, the time evolving N-body density matrix is written as an average of tensor products of 1-body densities each of which obeys a stochastic evolution equation. Such decompositions can be constructed for any mixture of fermions and bosons. The evolution equations for the 1-body densities preserve norm, Hermiticity and positivity.
Following the strategy of stochastic quantization, based on variational principles for processes taking values on a discrete configuration space, we give a detailed and self-contained description of the stochastic processes associated to a two-level quantum system, of interest for the physical description of spin-\textonehalf{} particles and Fermi fields. In particular we investigate the class of processes related to unperturbated quantum evolution and those simulating quantum-measurement procedures leading to mixture formation.
In this work we investigate the exact classical stochastic representations of many-body quantum dynamics.We focus on the representations in which the quantum states and the observables are linearly mapped onto classical quasiprobability distributions and functions in a certain (abstract) phase space.We demonstrate that when such representations have regular mathematical properties, they are reduced to the expansions of the density operator over a certain overcomplete operator basis.Our conclusions are supported by the fact that all the stochastic representations currently known in the literature (quantum mechanics in generalized phase space and, as it recently has been shown by us, the stochastic wave-function methods) have the mathematical structure of the above-mentioned type.We illustrate our considerations by presenting the recently derived operator mappings for the stochastic wave-function method.
We establish a conceptual framework for the identification and the characterization of induced interactions in binary mixtures and reveal their intricate relation to entanglement between the components or species of the mixture. Exploiting an expansion in terms of the strength of the entanglement among the two species enables us to deduce an effective single-species description. In this way, we naturally incorporate the mutual feedback of the species and obtain induced interactions for both species which are effectively present among the particles of same type. Importantly, our approach incorporates few-body and inhomogeneous systems extending the scope of induced interactions where two particles interact via a bosonic bath-type environment. Employing the example of a one-dimensional ultracold Bose-Fermi mixture, we obtain induced Bose-Bose and Fermi-Fermi interactions with short-range attraction and long-range repulsion. With this, we show how beyond species mean-field physics visible in the two-body correlation functions can be understood via the induced interactions.