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AIP conference proceedings

Formulation of Time-Resolved Counting Statistics Based on a Positive-Operator-Valued Measure

Wolfgang Belzig, Adam Bednorz, Massimo Macucci, Giovanni Basso

Published Jan 1, 2009• 0 citations
Publisher

Abstract

It is shown how to include the influence of detector into the description of time‐resolved counting statistics for a quantum point contact. Using properly constructed positive operator‐valued measure, one can predict correction to the noise and avoid the paradox of negative probability.
DOI
10.1063/1.3140498

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We show that the standard Keldysh formulation of full counting statistics breaks down, when trying to interpret it directly in terms of a probability for time-dependent currents, since it leads to negative probability density. We propose a positive operator valued measure, which effectively accounts for some detector dynamics, but reproduces the Keldysh result in the low-frequency limit. However, the detector dynamics adds a large white Gaussian detection noise, which should be visible in experiment. The original negative probability can be measured indirectly by two independent detectors.
We propose a derivation of the full counting statistics of electronic current based on a positive-operator-valued measure. Our approach justifies the Levitov-Lesovik formula in the long-time limit, but can be generalized to the detection of finite-frequency noise correlations. The combined action of the projection postulate and the quantum formula for current noise at high frequencies imply an additional white noise. Estimates for this additional noise are in accordance with known experiments. We propose an experimental test of our conjecture by a simultaneous measurement of high- and low-frequency noise.
The counting statistics give insight into the properties of quantum states of light and other quantum states of matter such as ultracold atoms or electrons. The theoretical description of photon counting was derived in the 1960s and was extended to massive particles more recently. Typically, the interaction between each particle and the detector is assumed to be limited to short time intervals, and the probability of counting particles in one interval is independent of the measurements in previous intervals. There has been some effort to describe particle counting as a continuous measurement, where the detector and the field to be counted interact continuously. However, the formalism based on continuous measurements does not provide a formula applicable to general time- and space-dependent fields. In our work, we derive a fully time- and space-dependent description of the counting process for linear quantum many-body systems, taking into account the back-action of the detector on the field. We apply our formalism to an expanding Bose-Einstein condensate of ultracold atoms, and show that it describes the process correctly, whereas the standard approach gives unphysical results in some limits. The example illustrates that, in certain situations, the back-action of the detector cannot be neglected and has to be included in the description.
What is the right statistics for the measurements of arrival times of a quantum particle? Although this question is very old, it is still open. The usual experiments are performed in the far-field regime and this question becomes unimportant, as a semiclassical analysis suffices. Nevertheless, the development in the detector technology will soon allow for near-field investigations; thus, a better understanding of arrival time measurements is needed. Since outcomes of quantum measurements are always described by positive-operator-valued measures (POVMs), various arrival time POVMs have been proposed. On the other hand, many physicists would agree that the arrival time statistics is given by the quantum flux. This urges the question whether a POVM exists, which agrees approximately with the quantum flux values on a reasonable set of wave functions. We answer this question negatively for a very natural set of wave functions, but we remark that the answer is very sensitive to the choice of the set and provide evidence for the existence of a POVM that agrees with the quantum flux on a more restrictive set.
We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.
A model for a one-dimensional quantum particle interacting with a point detector is developed and analyzed. The analysis leads to any simple, intuitively pleasing expression for the final density operator of the particle in terms of the freely propagating particle density operator, a projection operator representing the detector volume, and a probability density for the time at which the detector fires. This probability density is found explicitly to be the presence-time density given by ${\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}∕\ensuremath{\int}{\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}dt$ where ${\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)$ is the free particle wave function evaluated at the detector position (x=0)(x=0)(x=0). Moreover, it is shown that the resultant density operator transformation is a nonlinear transformation, not describable as a positive operator valued measure.