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International Journal of Modern Physics B

QUANTUM BROWNIAN AND THE CONSTRAINED PATH INTEGRAL

Sutee Boonchui, Virulh Sa‐Yakanit, P. PALOTAIDAMKERNG

Published Oct 20, 2008• 0 citations
Publisher

Abstract

We show that the environment affects a quantum system in the form of the constrained trajectory. Our result allows one to describe the quantum system in terms of stochastic state vector rather than quantum history. Moreover we can alternatively reduce the time evolution operator. Then the trajectory of system is constrained.
DOI
10.1142/s0217979208050516

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A quantum trajectory represents a single history of the state vector of an open quantum system. The stochastic nature of its evolution can be viewed as resulting from the quantum probabilities for the outcomes of measurements performed on the bath. We derive stochastic Schrödinger equations for various trajectory models, starting from the full Schrödinger equation for the system and the bath combined. The state reduction of the bath at a measurement also projects the state of the system on a pure state, thereby destroying the entanglement. The master equation for the system's density matrix is obtained as a result of the discussion, and it is not taken as its starting point. The general concepts are illustrated in the specific case of an atom coupled to the radiation field.
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Starting from the standard Lindblad form of a dissipative master equation, we identify a stochastic parameter in the quantum time evolution. This lacks dynamic evolution but its value is affected by the dissipative part of the master equation. An unravelling of the master equation in terms of quantum trajectories then shows this parameter performing the erratic variation characteristic of a stochastic variable. When the two evolutions separate, the stochastic Markov process can be extracted and solved independently and imposed on the parameter of the quantum evolution. This is the traditional form of treatment of stochastic variables in quantum mechanics. The relations to earlier works are established, and the approach is illustrated by simple model examples.
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When suitably generalized and interpreted, the path-integral offers an alternative to the more familiar quantal formalism based on state-vectors, selfadjoint operators, and external observers. Mathematically one generalizes the path-integral-as-propagator to a {\it quantal measure} μμμ on the space ΩΩΩ of all ``conceivable worlds'', and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such ``histories-based'' schemes new, and more ``realistic'' possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of μμμ in its sets of measure zero: such sets are to be ``precluded''. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of ``quantum logic'', would accommodate the contradictions by dualizing ΩΩΩ to a space of ``co-events'' and effectively identifying reality with an element of this dual space.