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arXiv:1705.09033 · Physical Review A

Time-dependent spectrum of a single photon and its positive-operator-valued measure

S. J. van Enk

Published Sep 19, 2017• 16 citations
arXivHTMLPDF

Abstract

Suppose we measure the time-dependent spectrum of a single photon. That is, we first send the photon through a set of frequency filters (which we assume to have different filter frequencies but the same finite bandwidth $\mathrm{\ensuremath{\Gamma}}$) and then record at what time (with some finite precision $\mathrm{\ensuremath{\Delta}}t$ and some finite efficiency $\ensuremath{\eta}$) and after passing what filter the photon is detected. What is the positive-operator-valued measure (POVM), the most general description of a quantum measurement, corresponding to such a measurement? We show how to construct the POVM in various cases, with special interest in the case $\mathrm{\ensuremath{\Gamma}}\mathrm{\ensuremath{\Delta}}t\ensuremath{\ll}1$ (time-frequency uncertainty still holds, even in that limit). One application of the formalism is to heralding single photons. We also find a Hong-Ou-Mandel type of interference effect with two photons entering a frequency filter.

DOI
10.1103/physreva.96.033834

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Suppose we measure the time-dependent spectrum of a single photon. That is, we first send the photon through a set of frequency filters (which we assume to have different filter frequencies but the same finite bandwidth \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ) and then record at what time (with some finite precision \ensuremathΔt\mathrm{\ensuremath{\Delta}}t\ensuremathΔt and some finite efficiency \ensuremathη\ensuremath{\eta}\ensuremathη) and after passing what filter the photon is detected. What is the positive-operator-valued measure (POVM), the most general description of a quantum measurement, corresponding to such a measurement? We show how to construct the POVM in various cases, with special interest in the case \ensuremathΓ\ensuremathΔt\ensuremath≪1\mathrm{\ensuremath{\Gamma}}\mathrm{\ensuremath{\Delta}}t\ensuremath{\ll}1\ensuremathΓ\ensuremathΔt\ensuremath≪1 (time-frequency uncertainty still holds, even in that limit). One application of the formalism is to heralding single photons. We also find a Hong-Ou-Mandel type of interference effect with two photons entering a frequency filter.
The time-frequency degree of freedom of the electromagnetic field is the final frontier for single-photon measurements. The temporal and spectral distribution a measurement retrodicts (that is, the state it projects onto) is determined by the detector's intrinsic resonance structure. In this paper, we construct ideal and more realistic positive operator-valued measures (POVMs) that project onto arbitrary single-photon wave packets with high efficiency and low noise. We discuss applications to super-resolved measurements and quantum communication. In doing so we will give a fully quantum description of the entire photodetection process, give prescriptions for (in principle) performing single-shot Heisenberg-limited time-frequency measurements of single photons, and discuss fundamental limits and trade-offs inherent to single-photon detection.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}∥1R​e−iHtψ∥22​≤Ct−3 have been derived, where ψψψ denotes the initial state, HHH the Hamiltonian, RRR a positive constant, and CCC is explicitly known. As intermediate step, bounds on the derivatives of the SSS-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3n=1,2,3n=1,2,3, and the constants Cn,KC_{n,K}Cn,K​ explicitly known.
Quantum mechanics cements the intimate relationship between the nature of light and its detection. Historically, quantum theories of photodetection have generally fallen into two categories: the first tries to determine what quantum field observable is measured when photoelectrons are detected, laying the theoretical groundwork for photodetection being possible. The second type are phenomenological theories, which take great care to model the details of specific photodetectors. In this dissertation, we fill in the gap between these two models in the modern literature on photodetection by constructing a fully quantum-mechanical and sufficiently realistic model that includes all stages of the photodetection process. We accomplish this within the framework of quantum information theory using the language of positive operator valued measures (POVMs). \n\nA POVM provides the most general description of a quantum measurement. In the context of single-photon detection, the photodetector POVM provides a complete characterization of the single-photon detector (SPD) from which all figures of merit can be calculated. Each element of the POVM is comprised of a weighted sum over single-photon state projectors. To construct SPD POVMs, we identify the states projected onto by a measurement and the associated weights. We first accomplish this for a simple time-dependent system, laying out how to efficiently project onto arbitrary single-photon states. Then, we describe the three stages of a realistic SPD: transmission, amplification, and measurement.\n\nIn the transmission stage of photodetection a photon enters a network of discrete energy levels, its energy propagating through that network and finally exiting into another output continuum of modes, a process entirely described by a single complex function. In the amplification stage of photodetection, a single excitation is amplified into a macroscopic signal via a nonlinear amplification mechanism. In the final stage, an inefficient ``classical'' measurement is made on the macroscopic signal. By combining these three stages we form a chain of inference from the final ``click'' to the input photon and construct a POVM projecting onto the input Hilbert space. Lastly, we discuss limits and tradeoffs that arise at each stage and implications for photodetection applications.\n\nThis dissertation contains previously published and unpublished material.
We examine the precision limits of Hong-Ou-Mandel (HOM) timing measurements, as well as precision limits applying to generalized two-photon measurements. As a special case we consider the use of two-photon measurements using photons with variable bandwidths and frequency correlations. When the photon bandwidths are not equal, maximizing the measurement precision involves a tradeoff between high interference visibility and strong frequency anticorrelations, with the optimal precision occurring when the photons share nonmaximal frequency anticorrelations. We show that a generalized measurement has precision limits that are qualitatively similar to those of the HOM measurement whenever the generalized measurement is insensitive to the net delay of both photons. By examining the performance of states with more general frequency distributions, our analysis allows for engineering of the joint spectral amplitude for use in realistic situations, in which both photons may not have ideal spectral properties.