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arXiv:2310.00432 · APL Quantum

How much time does a photon spend as an atomic excitation before being transmitted through a cloud of atoms?

Kyle Thompson, Kehui Li, Daniela Angulo, Vida-Michelle Nixon, Josiah Sinclair, Amal Vijayalekshmi Sivakumar, Howard M. Wiseman, Aephraim M. Steinberg

Published Sep 1, 2025• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0288743

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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 scattered vs. transmitted? In particular, if it is not scattered, meaning that it emerges in the direction it entered, does it cause atoms to spend any time in the excited state, and if so, how much? In this thesis, I report an experimental measurement of the time a transmitted photon causes atoms to spend in the excited state. The experiment is carried out in an ultra-cold gas of Rubidium 85 atoms and involves simultaneously recording changes to the excited state occupation number induced by incident ("signal") photons and whether those photons are transmitted. The time atoms spend in the excited state, which is defined in terms of the time integral of the average excited state occupation number, is measured weakly using a separate, off-resonant “probe” laser to monitor the index of refraction of the sample—that is, I measure the nonlinear phase shift written by a signal pulse on this probe beam—and direct detection is used to isolate the effect of a single transmitted photon. 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), I report 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. This observation of “excitation without loss” is attributed to coherent forward emission, which arises, for instance, due to the sign-flip the instantaneous Rabi frequency (pulse envelope) picks up when a broadband pulse propagates through an optically thick medium with highly frequency-dependent absorption [1]. The results of this thesis 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.

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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 $\textit{not}$ 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.

arXiv:2010.02200•Oct 5, 2020•0 citations
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 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 scattered vs. transmitted? In particular, if it is not scattered, meaning that it emerges in the direction it entered, does it cause atoms to spend any time in the excited state, and if so, how much? In this thesis, I report an experimental measurement of the time a transmitted photon causes atoms to spend in the excited state. The experiment is carried out in an ultra-cold gas of Rubidium 85 atoms and involves simultaneously recording changes to the excited state occupation number induced by incident ("signal") photons and whether those photons are transmitted. The time atoms spend in the excited state, which is defined in terms of the time integral of the average excited state occupation number, is measured weakly using a separate, off-resonant “probe” laser to monitor the index of refraction of the sample—that is, I measure the nonlinear phase shift written by a signal pulse on this probe beam—and direct detection is used to isolate the effect of a single transmitted photon. 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), I report 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. This observation of “excitation without loss” is attributed to coherent forward emission, which arises, for instance, due to the sign-flip the instantaneous Rabi frequency (pulse envelope) picks up when a broadband pulse propagates through an optically thick medium with highly frequency-dependent absorption [1]. The results of this thesis 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.
Recent weak-measurement experiments performed by the University of Toronto reported an apparent negative transmission delay associated with photons propagating through a cold rubidium atomic cloud, raising fundamental questions concerning the physical interpretation of negative interaction time. In this work, we develop a second-quantized synchronization framework to describe resonant photon transmission through a collective atomic ensemble. Both the incident photon and the collective atomic excitation are treated as coupled quantum resonators interacting through an effective synchronization Hamiltonian with finite lifetime broadening. An analytical expression for the transmission amplitude and its phase response is derived, demonstrating that the apparent transmission delay naturally becomes negative on the resonance wings owing to coherent interference between direct transmission and resonant absorption–reemission pathways. In contrast, the microscopic excitation time, defined by the time integral of the atomic excitation-number operator, is rigorously proven to remain nonnegative. Numerical simulations using representative parameters for the Toronto cold-rubidium experiment reproduce the experimentally observed order of magnitude of the negative weak delay without introducing empirical fitting parameters. The calculations further show how the apparent delay depends systematically on frequency detuning and the effective collective coupling strength, providing experimentally testable predictions beyond the existing weak-value interpretation. The present synchronization framework therefore resolves the apparent paradox of negative transmission time by demonstrating that the observed temporal advance originates from phase synchronization and quantum interference rather than from a negative microscopic interaction time.
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.
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.