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Journal of Physics A Mathematical and General

Quantum stochastic Feynman rules and extended Wigner statistics

John Edward Gough

Published Feb 27, 1998• 1 citation
Publisher

Abstract

We introduce a Feynman diagram calculus for quantum stochastic fields and show that from the weak coupling limit of a Bose quantum field reservoir in a Gaussian state one obtains a quantum stochastic field process which has extended Wigner statistics. That is, we calculate explicitly the moments using the diagram rules and show that only moments corresponding to non-crossing diagrams are non-trivial. We further give the physical mechanism underlying this fact; namely the combination of momentum conservation (arising from the correct treatment of the full responsive interaction) and the mass-shell energy conservation for the virtual (stochastic) reservoir quanta. In our analysis we introduce a volume cut-off initially for the system particle to discretize its energy spectrum and then take the weak coupling limit. The individual noise fields associated with energy state transitions were Gaussian however, when the volume cut-off is then removed and we recover the noise correlations. Thus for this problem the removal of the cut-off and the weak coupling limit are in fact interchangeable.

DOI
10.1088/0305-4470/31/8/013

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We introduce a Feynman diagram calculus for quantum stochastic fields and show that from the weak coupling limit of a Bose quantum field reservoir in a Gaussian state one obtains a quantum stochastic field process which has extended Wigner statistics. That is, we calculate explicitly the moments using the diagram rules and show that only moments corresponding to non-crossing diagrams are non-trivial. We further give the physical mechanism underlying this fact; namely the combination of momentum conservation (arising from the correct treatment of the full responsive interaction) and the mass-shell energy conservation for the virtual (stochastic) reservoir quanta. In our analysis we introduce a volume cut-off initially for the system particle to discretize its energy spectrum and then take the weak coupling limit. The individual noise fields associated with energy state transitions were Gaussian however, when the volume cut-off is then removed and we recover the noise correlations. Thus for this problem the removal of the cut-off and the weak coupling limit are in fact interchangeable.
The problem of obtaining the quantum stochastic limit for a system interacting with a reservoir is investigated when both are considered as quantum fields. The pre-limit fields are as encountered in standard quantum field theory and in particular we do not resort to the simplifying responseless interaction for the reservoir field. In the separation of time scales between the unperturbed (slow) and interaction (fast) dynamics we find a new quantum stochastic noise emerging which describes multi-particle scatterings and has extended free statistics. We give explicit diagrammatic rules to compute the 2n-point correlators of the noise.
We show that in a model of a particle interacting with a quantum field, the field operators rescaled according to the prescriptions of the stochastic limit, obey q -commutational relations with q depending on time. After the stochastic limit, due to the nonlinearity, the particle and field degrees of freedom become entangled even at a kinematical level in the sense that the field and the atomic variables no longer commute but give rise to a new algebra with new commutation relations replacing the boson ones. This allows to give a simple proof of the fact that the non-crossing half-planar diagrams give the dominating contribution in a weak-coupling regime and to calculate explicitly the correlations associated with the new algebra.
Within the framework of the first-order smoothing approximation and the long-time, Markovian approximation, kinetic equations are derived for the stochastic Wigner equation (the exact equation of evolution of the phase-space Wigner distribution function) and the stochastic Liouville equation (correspondence limit approximation) associated with the quantized motion of a particle described by a stochastic Schrödinger equation. In the limit of weak fluctuations and long times, the transport equation for the average probability density of the particle in momentum space which was reported recently by Papanicolaou is recovered. Also, on the basis of the Novikov functional formalism, it is established that several of the approximate kinetic equations derived in this paper are identical to the exact statistical equations in the special case that the potential field is a δ-correlated (in time), homogeneous, wide-sense stationary, Gaussian process.
In standard nonrelativistic quantum mechanics the expectation of the energy is a conserved quantity. It is possible to extend the dynamical law associated with the evolution of a quantum state consistently to include a nonlinear stochastic component, while respecting the conservation law. According to the dynamics thus obtained, referred to as the energy-based stochastic Schrodinger equation, an arbitrary initial state collapses spontaneously to one of the energy eigenstates, thus describing the phenomenon of quantum state reduction. In this paper, two such models are investigated: one that achieves state reduction in infinite time and the other in finite time. The properties of the associated energy expectation process and the energy variance process are worked out in detail. By use of a novel application of a nonlinear filtering method, closed-form solutions-algebraic in character and involving no integration-are obtained of both these models. In each case, the solution is expressed in terms of a random variable representing the terminal energy of the system and an independent noise process. With these solutions at hand it is possible to simulate explicitly the dynamics of the quantum states of complicated physical systems
We build an exact framework to evaluate heat, energy, and particle transport between Gaussian reservoirs mediated by a quadratic quantum system. By combining full counting statistics with newly developed non-Markovian master equation approaches, we introduce an effective master equation whose solution can be used to generate arbitrary moments of the heat statistics for any number of reservoirs. This theory applies equally to fermionic and bosonic systems, holds at arbitrarily strong coupling, and resolves out-of-equilibrium transient dynamics determined by the system's initial state. In the steady-state, weak-coupling limit, we recover results analogous to those of the well-known Landauer-Büttiker formalism. We conclude our discussion by demonstrating an application of the method to a prototypical fermionic system. Our results uncover a regime of transient negative heat conductance contingent upon the initial system preparation, providing a clear signature of non-trivial out-of-equilibrium dynamics.
In standard nonrelativistic quantum mechanics the expectation of the energy is a conserved quantity. It is possible to extend the dynamical law associated with the evolution of a quantum state consistently to include a nonlinear stochastic component, while respecting the conservation law. According to the dynamics thus obtained, referred to as the energy-based stochastic Schrödinger equation, an arbitrary initial state collapses spontaneously to one of the energy eigenstates, thus describing the phenomenon of quantum state reduction. In this paper, two such models are investigated: one that achieves state reduction in infinite time and the other in finite time. The properties of the associated energy expectation process and the energy variance process are worked out in detail. By use of a novel application of a nonlinear filtering method, closed-form solutions—algebraic in character and involving no integration—are obtained of both these models. In each case, the solution is expressed in terms of a random variable representing the terminal energy of the system and an independent noise process. With these solutions at hand it is possible to simulate explicitly the dynamics of the quantum states of complicated physical systems.