Antoine Royer

Measurement of quantum states and the Wigner function

AbstractIn quantum mechanics, the state of an individual particle (or system) is unobservable, i.e., it cannot be determined experimentally, even in principle. However, the notion of “measuring a state” is meaningful if it refers to anensemble of similarly prepared particles, i.e., the question may be addressed: Is it possible to determine experimentally the state operator (density matrix) into which a given preparation procedure puts particles. After reviewing the previous work on this problem, we give simple procedures, in the line of Lambs operational interpretation of quantum mechanics, for measuring a translational state operator (whether pure or mixed), via its Wigner function. These procedures closely parallel methods that might be used in classical mechanics to determine a true phase space probability distribution; thus, the Wigner function simulates such a distribution not only formally, but operationally also. There is no way to determine what the wave function (or state vector) of a system is—if arbitrarily given, there is no way to “measure” its wave function. Clearly, such a measurement would have to result in afunction of several variables, not in a relatively small set ofnumbers .... In order to verify the [quantum] theory in its generality, at least a succession of two measurements are needed. There is in general no way to determine the original state of the system, but having produced a definite state by a first measurement, the probabilities of the outcomes of a second measurement are then given by the theory.E. P. Wigner(1)

Combining projection superoperators and cumulant expansions in open quantum dynamics with initial correlations and fluctuating Hamiltonians and environments

Abstract The evolution of a small system a interacting with a bath b has been described by two different kinds of master equations for its reduced density matrix ρa(t): (i) Nakajima–Zwanzig ‘memory’ equations resulting from the use of projection superoperators; (ii) Time-local equations based on cumulant expansions. It is pointed out that their solution ρa(t) may be expressed in the ‘hybrid’ form (≻ signifies time-ordering) ρ a (t)= B (t,τ)ρ a (τ)+ ∫ τ t ds B (t,s) C (s,τ), B (t,t′)=e ≻ ∫ t′ t ds L (s,t′) where L (s,t′) is a cumulant expansion independent of initial correlations, while C (s,τ) , defined in terms of projectors, is the initial correlation term appearing in the ‘memory’ equation. Thus, the convolution represents the effect of initial correlations on ρa(t). We analyse the physical meanings of weak coupling approximations to the ‘memory’ and ‘time-local’ equations, elucidating why the latter are more accurate in general. We allow time-dependent Hamiltonians and non-stationary bath states.

Ehrenfest's Theorem Reinterpreted and Extended with Wigner's Function

For a wave packet evolving quantum mechanically, the rates of change of the expectations and uncertainties of the position and momentum are exactly the same as if Wigners function instantaneously obeyed a classical Liouville equation (whatever the Hamiltonian). This extension of Ehrenfests theorem should be useful for dealing with the evolution and manipulation of quantum states.