Quantum trajectories for realistic photodetection: I. General formalism

Abstract

Quantum trajectories describe the stochastic evolution of an open quantum system conditioned on continuous monitoring of its output, such as by an ideal photodetector. In practice an experimenter has access to an output filtered through various electronic devices, rather than the microscopic states of the detector. This introduces several imperfections into the measurement process, of which only inefficiency has previously been incorporated into quantum trajectory theory. However, all electronic devices have finite bandwidths, and the consequent delay in conveying the output signal to the observer implies that the evolution of the conditional state of the quantum system must be non-Markovian. We present a general method of describing this evolution and apply it to avalanche photodiodes (APDs) and to photoreceivers. We include the effects of efficiency, dead time, bandwidth, electronic noise, and dark counts. The essential idea is to treat the quantum system and classical detector jointly, and to average over the latter to obtain the conditional quantum state. The significance of our theory is that quantum trajectories for realistic detection are necessary for sophisticated approaches to quantum feedback, and our approach could be applied in many areas of physics.

Chaos and Quantum Mechanics

“…Nevertheless, this fundamental notion of measurement can be easily extended 7 to devise schemes that extract information continuously. [27][28][29][30][31][32][33][34] The basic idea is to have the system of interest interact weakly with another (e.g., atom interacting with an electromagnetic field) and make projective measurements on the auxiliary system (e.g., photon counting). Because of the weak interaction, the state of the auxiliary system gathers very little information about the system of interest, and therefore this system, in turn, is only perturbed slightly by the measurement backaction.…”

“…Suppose we have a single quantum degree of freedom, position in this case, undergoing a weak, ideal continuous measurement. [27][28][29][30][31][32][33][34] Here "ideal" refers to no loss of information during the measurement, that is, a fine-grained evolution with no increase in entropy. Then, we have two coupled equations, one for the measurement record y(t),…”

Model for monitoring of a charge qubit using a radio-frequency quantum point contact including experimental imperfections

“…Our second main result is the extension of our idealized quantum trajectory (37) to consider conditioning the qubit state on a corrupted (filtered, more noisy) measurement signal available to a realistic observer. Our realistic quantum trajectory 21,22,23 equation is Eq. (42).…”

“…(B9)]. This step is performed via 21,23 ρ V,c (x) + dρ V,c (x) = P V,c (x) + dP V,c (x) × [ρ c (t) + dρ c (t)] .…”

Dynamical parameter estimation using realistic photodetection

“…To treat realistic detection we use the theory developed in Refs. [8][9][10]. This generalized the theory of quantum trajectories [11][12][13][14][15][16] by determining the state of the system conditioned upon the output of a detector that has dead time ͑ dd ͒, dark counts at rate ␥ dk , inefficiency , and finite bandwidth ␥ r .…”