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arXiv:1801.08065 · Physical Review A

Perturbation approach for computing frequency- and time-resolved photon correlation functions

David I. H. Holdaway, Valentina Notararigo, Alexandra Olaya-Castro

Published Dec 19, 2018• 21 citations
arXivHTMLPDF

Abstract

We propose an alternative formulation of the sensor method presented in Phys. Rev. Lett. 109, 183601 (2012) for the calculation of frequency-filtered and time-resolved photon correlations. Our approach is based on an algebraic expansion of the joint steady state of quantum emitter and sensors with respect to the emitter-sensor coupling parameter $\ensuremath{\epsilon}$. This allows us to express photon correlations in terms of the open quantum dynamics of the emitting system only and ensures that computation of correlations are independent on the choice of a small value of $\ensuremath{\epsilon}$. Moreover, using time-dependent perturbation theory, we are able to express the frequency- and time-resolved second-order photon correlation as the addition of three components, each of which gives insight into the physical processes dominating the correlation at different time scales. We consider a bioinspired vibronic dimer model to illustrate the agreement between the original formulation and our approach.

DOI
10.1103/physreva.98.063828

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We propose an alternative formulation of the sensor method presented in Phys. Rev. Lett. 109, 183601 (2012) for the calculation of frequency-filtered and time-resolved photon correlations. Our approach is based on an algebraic expansion of the joint steady state of quantum emitter and sensors with respect to the emitter-sensor coupling parameter \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. This allows us to express photon correlations in terms of the open quantum dynamics of the emitting system only and ensures that computation of correlations are independent on the choice of a small value of \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. Moreover, using time-dependent perturbation theory, we are able to express the frequency- and time-resolved second-order photon correlation as the addition of three components, each of which gives insight into the physical processes dominating the correlation at different time scales. We consider a bioinspired vibronic dimer model to illustrate the agreement between the original formulation and our approach.
A theory of correlations between N photons of given frequencies and detected at given time delays is presented. These correlation functions are usually too cumbersome to be computed explicitly. We show that they are obtained exactly through intensity correlations between two-level sensors in the limit of their vanishing coupling to the system. This allows the computation of correlation functions hitherto unreachable. The uncertainties in time and frequency of the detection, which are necessary variables to describe the system, are intrinsic to the theory. We illustrate the power of our formalism with the example of the Jaynes-Cummings model, by showing how higher order photon correlations can bring new insights into the dynamics of open quantum systems.
We develop a formalism based on a time-dependent wave-function ansatz to study correlations of photons emitted from a collection of two-level quantum emitters. We show how to simulate the system dynamics and evaluate the intensity of the scattered photons and the second-order correlation function g(2)g^{(2)}g(2) in terms of the amplitudes of the different components of the wave function. Our approach is efficient for considering systems that contain up to two excitations. To demonstrate this we first consider the example of spectral filtering of photons emitted from a single quantum emitter. We show how our formalism can be used to study spectral filtering of the two-photon component of the emitted light from a single quantum emitter for various kinds of filters. Furthermore, as a general application of our formalism, we show how it can be used to study photon-photon correlations in an optically dense ensemble of two-level quantum emitters. In particular we lay out the details of simulating correlated photon transport in such ensembles reported recently by S. Mahmoodian {\it et.al.} [Phys. Rev. Lett. {\bf 121}, 143601 (2018)]. Compared to other existing techniques, the advantage of our formalism is that it is applicable to any generic spectral filter and quantum many-body systems involving a large number of quantum emitters while requiring only a modest computational resource.
The study of quantum phenomena in biology has received significant attention in the last decade. One of the problems of most interest is the understanding of quantum effects during the first steps of photosynthesis. Ultrafast two-dimensional electronic spectroscopy has revealed that pigment-protein complexes responsible for light- harvesting and charge separation in photosynthetic organisms can support quantum coherent dynamics in the excited state, for up to few hundreds of femtoseconds. The leading hypothesis on the mechanisms supporting this coherent behaviour is quan- tum interactions between electronic and some specific vibrational motions in the excited state. This hypothesis, however, awaits unambiguous confirmation. Among the most powerful techniques to investigate the quantum behaviour of an emitter is the study of quantum optical properties of the light it emits. This thesis de- velops theoretical studies showing that frequency-filtered and time-resolved photon counting statistics of the light emitted by a prototype photosynthetic unit can give important insight into the quantum coherent nature and the mechanisms underlying excited state dynamics in single photosynthetic complexes. By developing a pertur- bative and efficient approach to the computation of frequency- and time- resolved photon correlation functions, we show that such correlations have the potential to give unambiguous signatures of coherence contributions to the steady state emis- sion. For a light-harvesting unit emitting in free space, the signatures of excited state coherence manifest themselves as non-trivial antibunching. This feature can- not be probed by measuring unfiltered photon correlations. We then consider the situation in which a prototype energy transfer unit is embedded in an optical cavity such that its emission rate is enhanced. In this case we observed a rich behaviour of the frequency-filtered, second-order photon correlations that allows a clear distinc- tion of coherence contributions, and their variation, depending of the electronic and vibrational interactions in the system of interest.
Using second-order perturbation theory in the light-matter interaction, we derive an analytical approximation for the vibronic populations of a diatomic system excited by ultrabroadband frequency-entangled photons and evaluate the population dynamics for different degrees of entanglement between photon pairs. Our analytical approach makes the same predictions as previously derived via numerical solutions of the complete Schrödinger equation [H. Oka, ], with the added advantage of providing clear physical insights into the vibronic selectivity as a function of the degree of photon correlations while requiring significantly reduced computational effort. Specifically, our analytical expression for the probability of vibronic excitation includes a factor which predicts the enhancement of vibrational selectivity as a function of the degree correlation between the entangled photon pairs, the targeted vibrational energy level, and the vibrational molecular structure encoded in the Franck-Condon factors. Our results illustrate the importance of going beyond the usual approximations in second-order perturbation theory to capture the relevance of the vibrational structure of the molecular system of interest in order to gain a deeper understanding of the possible quantum-enhancement provided by the interaction between quantum light and matter.
The development of spectroscopic techniques able to detect and verify quantum coherence is a goal of increasing importance given the rapid progress of new quantum technologies, the advances in the field of quantum thermodynamics, and the emergence of new questions in chemistry and biology regarding the possible relevance of quantum coherence in biochemical processes. Ideally, these tools should be able to detect and verify the presence of quantum coherence in both the transient dynamics and the steady state of driven-dissipative systems, such as light-harvesting complexes driven by thermal photons in natural conditions. This requirement poses a challenge for standard laser spectroscopy methods. Here, we propose photon correlation measurements as a new tool to analyze quantum dynamics in molecular aggregates in driven-dissipative situations. We show that the photon correlation statistics of the light emitted in several models of molecular aggregates can signal the presence of coherent dynamics. Deviations from the counting statistics of independent emitters constitute a direct fingerprint of quantum coherence in the steady state. Furthermore, the analysis of frequency resolved photon correlations can signal the presence of coherent dynamics even in the absence of steady state coherence, providing direct spectroscopic access to the much sought-after site energies in molecular aggregates.
Multiphoton correlations from quantum emitters coupled to vibrational environments lie beyond the reach of standard tools such as the quantum regression theorem. Here, we introduce a Markovian framework for computing frequency-resolved N-photon correlation functions that overcomes this limitation. Applying our approach to a driven semiconductor quantum dot provides a tractable description of phonon effects on fluorescence beyond the single-photon spectrum. Our method accurately captures the emergence of the phonon sideband, missed by conventional quantum regression theorem treatments, and reveals rich phonon-induced structure in the filtered two-photon spectrum. Strikingly, we find that photons emitted via the phonon sideband inherit second-order coherence properties of the Mollow triplet.