Measuring the Time Atoms Spend in the Excited State Due to a Photon They Do Not Absorb

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, $(77\pm 16)\mathrm{\%}$ 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 ${180}^{\circ }$ 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.

Popular Summary

When light travels through matter, some photons are absorbed (exciting atoms, which later reemit this energy as fluorescence, in all directions) while others are transmitted. With modern techniques, it is possible to probe the effect of a single photon on the atoms as it propagates through. While it can be shown that the total time atoms spend in the excited state is equal to the number of absorbed photons times the atomic lifetime, there has not yet been any prediction of how this time is split between cases where the photon is transmitted and those where it is “scattered” by the medium. Here, we present the first experiment to address this fundamental question about how light and matter interact at the most basic level and find that even photons that are not absorbed by the atoms cause the latter to spend some time in the excited state; implying, in turn, that absorbed photons cause atoms to spend less than their natural lifetime in the excited state before reemitting. This work opens up a new experimental paradigm for studying the “history” of photons as they interact with matter and raises questions that call out for new theoretical calculations in quantum electrodynamics.

Figure 1

(a) The level scheme used in the experiment; ${\mathrm{\Omega }}_{p(s)}$ is the probe (signal) Rabi frequency, ${\mathrm{\Delta }}_p$ is the probe detuning, and $\mathrm{\Gamma }$ is the spontaneous decay rate of the excited state. (b) The measured optical depth and linear phase experienced by the probe as a function of ${\mathrm{\Delta }}_p$, with the signal turned off. (c) A simplified schematic of the experimental apparatus, consisting of signal and probe beams overlapped using 90:10 beam splitters (BSs), the rubidium MOT, photon detection, and phase measurement. (d) The experimental timing sequence used in the experiment. The green arrows indicate photon detection being reassociated with the correct shot. $I_s(t)$ is the average signal intensity measured with a fast APD. $\varphi (t)$ is the measured XPS during a shot. “Click?” indicates whether or not a photon is detected.

Figure 2

(a),(b) The probe cross-phase shift and (c)–(f) the probe cross-phase shift differences versus time, averaged over approximately 6 billion shots, in the presence of a 10-ns signal pulse. The probe XPS [$\varphi (t)$] is recorded for pulses containing $|\alpha {|}^2\approx 34$ photons incident at approximately $t=200$ ns for probe detunings of (a) $-5.6\mathrm{MHz}$ and (b) $+4.7\mathrm{MHz}$. The purple rectangle indicates the window of time wherein the XPS occurs. The shaded gold curve is a fit to a cubic polynomial (to model the background drift) plus a peaked function. The solid gold line shows only the cubic fit to background. Plots (c) and (d) show $\mathrm{\Delta }\varphi (t)$ for pulses containing $|\alpha {|}^2\approx 34$ photons and plots (e) and (f) show $\mathrm{\Delta }\varphi (t)$ for pulses containing $|\alpha {|}^2\approx 134$ photons. As before, the probe detuning is negative on the left [(c),(e)] and positive on the right [(d),(f)]. Here, the shaded gold region indicates a fit to a cubic polynomial plus a function with the shape of the peaks fitted to the relevant average XPS [e.g., for (c), the XPS in (a) is used, while the XPS used for (e) and (f) is not shown] but with an adjustable amplitude. As before, the solid gold line shows only the cubic background fit. The phase is sampled every 16 ns and a 25-MHz measurement bandwidth is used. The phase noise on a single sample is approximately 100–200 mrad and is determined by the sampling rate, the measurement bandwidth, and the probe power, which is set to approximately 5 nW in order to be much less than saturation (about 50 nW). The error bars shown are the standard error of the mean.

Figure 3

(a) A weighted average of $\mathrm{\Delta }\varphi (t)$ with $34$ and $134$ photons and with positive and negative probe detunings, in units of ${\varphi }_0$. The gold line represents our measurement of the effect of a transmitted photon, with the amplitude and shape taken from fits to $\varphi (t)$ and the effect of proportional noise subtracted off (see Sec. III of the Supplemental Material [26]). (b) The phase shift due to a transmitted photon (${\varphi }_T$) for $-5.6\mathrm{MHz}$ and $+4.7\mathrm{MHz}$ is shown. In the shaded region, the null results of checks for systematic effects are shown: ${\varphi }_T$ with the probe detuned far from resonance, with the MOT turned off, and with the signal detuned far from resonance [with the probe detuning set to be above ($+$) and below ($-$) resonance]. (c) Our best estimate of the relative phase shift due to a transmitted photon (${\varphi }_T/{\varphi }_0$), found by combining measurements from (b) at negative and positive detunings.

Figure 4

A comparison of our experimental results with the predictions of various semiclassical models. The predictions of a toy model (see Sec. IV of the Supplemental Material [26]) in which coherent forward emission occurs as a result of an interference effect arising when broadband small-area pulses propagate through an optically thick medium are shown both for signal pulses of 8-MHz bandwidth, as in our experiment (green dot-dashed line), and for narrow-band (1.5-MHz) ones (red dashed line). An “egalitarian” model in which photons affect the medium up until the point where they are scattered (or transmitted) is shown for the 8-MHz (orange dotted line) and 0.1-MHz (blue solid line) cases. Our experimental results are also shown (purple), with the peak on-resonant OD inferred from the optical depth seen by the probe (about 4), consistent with the amount of attenuation experienced by a resonant pulse with a bandwidth of 8 MHz. In the inset, ${\tau }_L/{\tau }_{\mathrm{sp}}$ is plotted for all four models, along with the inferred result of our measurement.