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Tracing the history of a photon: excitation times and group delays

Daniela Angulo, Kyle Thompson, Vida-Michelle Nixon, Andy Jiao, Aephraim M. Steinberg

Published Jan 1, 2024• 0 citations
Publisher

Abstract

Do photons decelerate in a medium due to spending time as atomic excitations? We measured the atomic excitation time for transmitted photons and found it to be the group delay, even when negative.
DOI
10.1364/cleo_at.2024.jw2a.104

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When a single photon propagates through a cloud of two-level atoms, part of its energy is temporarily stored as a collective atomic excitation. This raises a fundamental question: is it possible to meaningfully quantify how long the photon spends in the form of an atomic excitation? If so, does this duration depend on whether the photon is ultimately transmitted through the cloud or scattered by the atoms? These questions motivated recent experimental work by some of us [Sinclair et al., PRX Quantum 3, 010314 (2022)], but until now there has been no theoretical prediction for these conditional durations. We answer these questions by extending the concept of a dwell time from quantum tunneling to light–matter interactions, introducing an experimentally measurable “atomic excitation time.” Using the weak-value formalism combined with quantum trajectory theory, we derive analytical expressions for this time conditioned on the photon being transmitted through the cloud or scattered into a side mode. We show that, for transmitted photons, this time is equal to the (spectrally averaged) group delay, even in regimes where the group delay is negative. For scattered photons, the time is given by the sum of an ensemble-averaged group delay and an always-positive Wigner time delay, well known in elastic scattering theory. Finally, we present a toy model that explains how such negative dwell times can arise as a result of quantum interference. This work provides new insight into the complex and surprising histories of photons traveling through absorptive media.
When a resonant photon traverses a sample of absorbing atoms, how much time do atoms spend in the excited state? Does the answer depend on whether the photon is ultimately absorbed or transmitted? In particular, if it is not\textit{not}not absorbed, does it cause atoms to spend any time in the excited state, and if so, how much? In an experiment with ultra-cold Rubidium atoms, we simultaneously record whether atoms are excited by incident ("signal") photons and whether those photons are transmitted. We measure the time spent by atoms in the excited state by using a separate, off-resonant "probe" laser to monitor the index of refraction of the sample - that is, we measure the nonlinear phase shift written by a signal pulse on this probe beam - and use direct detection to isolate the effect of single transmitted photons. For short pulses (10 ns, to be compared to the 26 ns atomic lifetime) and an optically thick medium (peak OD = 4, leading to 60% absorption given our broad bandwidth), we find that the average time atoms spend in the excited state due to one transmitted photon is not zero, but rather (77 ±\pm± 16)% of the time the average incident photon causes them to spend in the excited state. We attribute this observation of "excitation without loss" to coherent forward emission, which can arise when the instantaneous Rabi frequency (pulse envelope) picks up a phase flip - this happens naturally when a broadband pulse propagates through an optically thick medium with frequency-dependent absorption [1]. These results unambiguously reveal the complex history of photons as they propagate through an absorbing medium and illustrate the power of utilizing post-selection to experimentally investigate the past behaviour of observed quantum systems.
When a photon traverses a cloud of atoms without scattering, how much time does it spend as an atomic excitation? To address this question, we used the cross-Kerr effect to weakly probe the degree of atomic excitation caused by a transmitted resonant "signal" photon by measuring the phase shift induced on a (separate, off-resonant) "probe" beam, postselected on cases when the signal photon is transmitted. The time integral of this observed phase shift, properly normalized and averaged over many runs in which the photon is detected after transmission, is the excitation time of interest, in a weak-valued sense. We measured mean atomic excitation times ranging from (-0.82±0.31)τ_{0} for the most narrow band pulse to (0.54±0.28)τ_{0} for the most broadband signal pulse, where τ_{0} is the non-post-selected excitation time, given by the scattering probability multiplied by the atomic lifetime τ_{sp}. Across a range of pulse durations and optical depths, our results are consistent with the recent theoretical prediction that the weak value of the atomic excitation time caused by a transmitted photon equals the group delay experienced by the light. Our experimental results show that negative times are not limited to describing the shift in the peak of a reshaped wave packet (as resulting from group delay), but they also provide the appropriate description of the weak-valued time the photon spends as an atomic excitation.
When a resonant photon traverses a sample of absorbing atoms, how much time do atoms spend in the excited state?Does the answer depend on whether the photon is ultimately absorbed or transmitted?In particular, if it is not absorbed, does it cause atoms to spend any time in the excited state and if so, how much?To answer these questions, we perform an experiment with ultracold rubidium atoms in which we simultaneously record whether atoms are excited by incident ("signal") photons and whether those photons are transmitted.We measure the time spent by atoms in the excited state by using a separate off-resonant "probe" laser to monitor the index of refraction of the sample-that is, we measure the nonlinear phase shift written by a signal pulse on this probe beam-and use direct detection to isolate the effect of single transmitted photons.For short pulses (10 ns, to be compared to the 26-ns atomic lifetime) and an optically thick medium (peak optical depth equals 4, leading to 60% absorption given our broad bandwidth), we find that the average time atoms spend in the excited state due to one transmitted photon is not zero but, rather, (77 ± 16)% of the time the average incident photon causes them to spend in the excited state.We attribute this observation of "excitation without loss" to coherent forward emission, which can arise when the instantaneous Rabi frequency (pulse envelope) picks up a 180 • phase flip-this happens naturally when a broadband pulse propagates through an optically thick medium with frequency-dependent absorption.These results unambiguously reveal for the first time the complex history of photons as they propagate through an absorbing medium and illustrate the power of utilizing postselection to experimentally investigate the past behavior of observed quantum systems.