Experimental evidence that a photon can spend a negative amount of time in an atom cloud
Daniela Angulo Kyle Thompson Vida-Michelle Nixon Andy Jiao Howard M. Wiseman Aephraim M. Steinberg


A New First-Order Meta-Learning Algorithm
 with Convergence Guarantees
    
  El Mahdi Chayti

  Machine Learning and Optimization Laboratory (MLO), EPFL


  Martin Jaggi

  Machine Learning and Optimization Laboratory (MLO), EPFL

    September 9, 2024
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§ INTRODUCTION


One of the foundational scenarios in the field of light-matter interaction is the propagation of light through a dielectric medium. Although this has been extensively studied and, in many ways, thoroughly understood (see, for example, <cit.>), it has also sparked controversy in specific areas such as the definition of the speed of a propagating electromagnetic signal <cit.> and the mechanisms governing energy transport in dispersive media <cit.>. It is also fundamental to nonlinear optics <cit.> and to technologies such as quantum memories <cit.>. Examining energy transport from a quantum perspective gives rise to another question: How does an individual photon spend its time while propagating through the medium? In the case of light far from resonance, a commonly repeated explanation relies on the uncertainty principle <cit.>. Even when atoms are driven by light with a very large detuning δ, goes the story, they can still experience excitation for a brief period, so long as this is of order 1/δ. This extra time spent as a non-propagating excitation would be associated with the group delay (viz. the familiar treatment of slow light in terms of polaritons<cit.>). While this 1/δ dependence captures the behavior of the index of refraction far from resonance, one would probably not expect this association to hold for small detunings, where the group delay is well-known to become negative.

















 
We define the average time that the atoms spend in the excited state, or atomic excitation time (τ_0), as the time integral of the average number of atoms in the excited state:


    τ_0 ≡∫⟨N̂_e(t)⟩ dt ,

where N̂_e is the operator representing the number of excited atoms; i.e., ⟨N̂_e(t)⟩ is the single-atom excitation probability multiplied by the total number of atoms. This expectation value, ⟨N̂_e(t) ⟩, can be measured using a separate “probe” beam coupled to the atoms through a Kerr nonlinear interaction based on atomic saturation: the average phase shift on the probe beam is proportional to ⟨N̂_e(t) ⟩. 
While this expectation value is straightforward to calculate from semiclassical theory – for a single incident photon on resonance, τ_e takes the value τ_0 ≡( 1- e^-  OD) / Γ, i.e., the atomic lifetime multiplied by the probability of the photon being scattered – in this work, ,


When atoms are illuminated by a pulse of resonant light—which we will refer to as the `signal' beam—they become polarized and have some probability of being found in the excited state at any given time. For a single photon input, the number of excited atoms is represented by the operator N̂_e, i.e., ⟨N̂_e ⟩(t) is the single-atom excitation probability multiplied by the total number of atoms. 
The expectation value ⟨N̂_e ⟩(t) is always between zero and one, and can be thought of as a probability. It can be measured using a separate `probe' beam coupled to the atoms through a Kerr nonlinear interaction based on atomic saturation: the instantaneous phase shift on the probe beam is proportional to ⟨N̂_e ⟩(t). 

We define the average time that the atoms spend in the excited state (τ_0), or average atomic excitation time, as the time integral of the expectation value of the number of atoms in the excited state, τ_0 ≡∫⟨N̂_e ⟩(t) dt. While this quantity is straightforward to calculate from semiclassical theory as τ_0 ≡ P_S / Γ—i.e., the atomic lifetime multiplied by the probability of the photon being scattered— in this work, we are interested in a different (intrinsically quantum) question: if a single resonant photon incident on a medium is transmitted, how much time do the atoms spend in the excited state? One might imagine that, unlike scattered photons (mostly absorbed within the first optical decay length), transmitted photons interact with the entire sample and hence cause more integrated excitation; alternatively, one might think that excited atoms are those that lead to loss via spontaneous emission, so that post-selecting on only transmitted photons would result in little or no atomic excitation.



Our group previously carried out a measurement of this excitation time, <cit.> and found that in a particular parameter regime, a transmitted photon spent nearly as much time as an atomic excitation as the average incident photon. The implication that a large fraction of the excited atoms re-emitted in the forward direction was attributed to coherent forward emission originating from the π phase-flip that a broadband pulse's envelope picks up when propagating through an optically thick medium <cit.>. However, such a semiclassical picture cannot make a quantitative prediction as to the phase shift imprinted on our probe when the incident photon is transmitted, because there is no semiclassical equivalent of that postselection; the probe phase can depend on the intensity of the signal beam but not on whether a particular photon is transmitted or not. 





Prompted by that first experiment, our group therefore employed quantum trajectory theory <cit.> and the weak-value formalism <cit.> to calculate<cit.> the time that a resonant photon spends as an atomic excitation while being transmitted through a medium. Those calculations predicted that this time is always equal to the group delay of the transmitted photon, even when that quantity is negative, and that this time depends only on the optical depth of the medium and the spectral content of the pulse. Operationally, this means that the nonlinear phase shift written on our probe by a transmitted photon should change sign when we move into the negative-delay regime (in our earlier work, the bandwidth and optical depth were such that the net delay was positive).

In the experiment reported here, we test this remarkable theoretical prediction for a wide range of parameters, including regimes in which the group delay is negative.




§ DESCRIPTION OF THE MEASUREMENT


The Kerr interaction used to probe N̂_e involves two beams of light: a) the `signal' beam, pulsed and resonant with the atomic transition, responsible for causing the excitation time; and b) the probe beam, continuous wave (CW) and off-resonant, which measures the degree of excitation caused by the signal (see Figure <ref>). While the signal pulse propagates through the medium, it weakly saturates the atoms, and
the probe picks up a phase shift proportional to this saturation.

As previously stated, our focus is on the effect of a transmitted photon. That is, we post-select on the event of transmission. Since our Kerr nonlinearity is very weak and we are interested in the behavior of a system, signal photon, defined by the preparation state and later post-selection, it follows that the quantity we measure can be calculated using the weak-value formalism <cit.>, suitably generalized <cit.>. 
Specifically, the atomic excitation time caused by a transmitted photon, τ_T, is given by the time integral of the real part of the weak value of N̂_e <cit.>
, with the initial state specified by the preparation of the input photon and the final state (post-selection) being that of the photon having traversed the medium without scattering. Note that, unlike expectation values, weak values are not restricted to be within the eigenvalue spectrum of the operator being measured, and can even be complex <cit.>.

One might expect that, in order to study the behaviour of an individual transmitted photon, we would need to work with single-photon signal pulses at the input. 

However, in our experiment, we leverage the simplicity and rapid rates of coherent-state pulses, combined with the post-selection technique first employed in <cit.>. In these experiments using coherent states as inputs, our group exploited the fact that upon single-photon detection, the inferred average number of photons present in a medium increases by one; one can thus discern the effect of a single photon on a probe by calculating the difference between that probe's phase in cases with and without a detection event. (This fact was recently proven rigorously, and extended, in<cit.>.) 



This ensures that our measurement of τ_T using coherent states should match the results of one being performed with single photons. In our experiment, subtracting the probe phase in cases without a detector click ϕ_NC(t) from cases with a click ϕ_C(t) allows us to determine the phase shift imparted by a transmitted photon, denoted as ϕ_T(t)=ϕ_C(t)-ϕ_NC(t), whose integral is proportional to τ_T. 


The atomic sample is a cold cloud of ^85Rb at a temperature of 60-70 μK, prepared in a magneto-optical trap formed using two beams of light and a magnetic field gradient of 10 G/cm. The `trap' beam is -25 MHz detuned from the cycling transition of the D2 line (F=3 to F'=4), while the `repump' beam is resonant with F=2 to F'=3. The probe and signal counter-propagate, overlapped with each other, and are focused to a waist of 25 μm inside the atomic cloud of around 1 mm of length. Both the signal and probe address the same atomic transition F=3 to F'=4 of the D2 line, with a lifetime of τ_ sp≈26 ns. The signal is a Gaussian pulse centered on resonance, while the CW probe is detuned by Δ≈-20 MHz (see Figure <ref>). Additionally, we introduce a sideband at +100 MHz relative to the carrier (probe), with the carrier power ranging from 6-9 nW (about 15% of resonant saturation intensity) and the sideband being three times stronger. This sideband serves to establish a beat-note interferometer and acts as the phase reference, remaining unaffected by the presence of atoms. We found that setting the probe carrier as far from resonance as the experimental capabilities allowed was the optimal approach for conducting the experiment (see the Supplementary information).


After the interaction inside the atoms has taken place, each beam is collected at opposite sides of the setup. The probe is coupled into a multi-mode fiber and sent to an avalanche photodiode (APD) operating in the linear gain region. The APD's output is subjected to IQ demodulation to extract the probe's phase and amplitude. The signal beam is optically attenuated to ensure that the subsequent power coupled into a single-mode fiber and transmitted to a single-photon counting module (SPCM) prevents multiphoton counts. Furthermore, to eliminate the possibility of spurious correlations driven by the electronic pulse from the SPCM (TTL), we introduce an optical delay of approximately 480 ns before the SPCM using a 100 m-long optical fiber. We use signal pulses with an incident mean photon number of approximately |α|^2≈ 100. The attenuation after the atoms was adjusted to maintain an SPCM firing probability of ∼ 0.2 during a time window of 576 ns, which we refer to as a `shot' and contains a single signal pulse. The choice of a 576 ns duration for each shot provides sufficient spacing between the pulses, ensuring that the dynamics of each pulse remain independent of the others. 


    If you take weak-values of photon numbers in networks populated by coherent states and post-select in single photons, that difference is gauranteed to give you the correct weak-value for the single photon. You really are learning about the history of that one extra-photon. 
    This experiment should give the same result as one done with a single-photon source
    it is possible to learn about the history of that extra photon despite the background of other photons in the coherent state.

Each measurement cycle begins with a 3-4 ms period dedicated to trapping and cooling the atoms, during which the magnetic field is active for the first 2 ms. Subsequently, the trap beam is turned off, and to prevent atoms from falling to a dark ground state, the repump beam remains active during data acquisition, which begins immediately. Samples are captured every 16 ns. 

Each trace of data starts with two frequency scans of the probe: one with atoms present and the other without atoms. These scans, lasting 300 µs each, serve to calibrate the resonant optical depth (OD) of the medium, which matches the one experienced by the signal. We then send a sequence of 1500 signal pulses, one every 576 ns, totalling 864 µs of data acquisition, for a duty cycle of ∼25%. During this time (see Figure <ref>), we record both the phase and amplitude of the probe, in addition to monitoring the TTL signal from the SPCM.

We anticipated that the phase shift imparted by a transmitted photon would be on the same order of magnitude as the phase shift imparted by the average photon (without post-selection) ϕ_0(t), whose peak value ranges from 10 to 20 μrad. This experimental value is consistent with an estimate given by the inverse of the number of atomic cross-sections addressed by our beams, which is approximately 10^4. Detecting such a minute effect presents significant challenges. Our phase noise is ∼120 mrad, which is four times greater than the quantum noise expected from 5 nW of probe power. This excess noise stems from the noise equivalent power of our detector, which is 100 fW/√(Hz), and a measurement bandwidth of 25 MHz. As a result, our measurement required tens of millions of atom cycles for a single set of parameters, translating to about ten hours of data collection.



§ OBSERVATIONS AND ANALYSIS

We investigated resonant optical depths (OD) of around 2 and 4 (the legends in Figure <ref> contain the exact values), for three pulses with root mean square (rms) durations of 10, 18, and 27 ns. We also took data for a 36 ns rms duration, but the highest OD we could use was 3 due to the increasing presence of background light becoming appreciable for more narrowband pulses (see also the Supplementary information). These choices were made to effectively explore the parameter space while adhering to the technical constraints of our apparatus. 

After synchronizing the measurement of the probe's phase with the corresponding signal detection or lack thereof, accounting for the aforementioned optical delay, we average the phase shift for the shots where the SPCM fired (ϕ_C(t)) and where it did not (ϕ_NC(t)) and then subtract these two averages. A shot is tagged as `click' if the SPCM fired anywhere in the 576 ns time window. This procedure is repeated for all atom cycles, resulting in a trace of 36 datapoints with ϕ_T(t) and the corresponding covariance matrix. The results are depicted in Figure <ref>. The blue dots and green squares represent the data for ϕ_T(t) and ϕ_0(t), respectively, with error bars for ϕ_T(t) indicating the standard error of the mean. The orange solid line and the dashed pink line correspond to the theoretical results for the post-selected and average cross-phase shifts, respectively. These predictions were calculated using the theory in reference <cit.>. These calculations only require the rms duration of the pulse and the average resonant OD seen by the signal during the data runs as inputs.

The theory curves, originally in arbitrary units and time axis, are height-scaled and time-shifted to facilitate comparison with the experimental results (See the Methods section for a full description of the time shifting). The scaling factor, applied to both theory results for ϕ_0(t) and ϕ_T(t), is chosen such that the peak of the theory for ϕ_0(t) matches the peak of the experimental ϕ_0(t). The theory presented in Figure <ref> has been adjusted accordingly, using a different scaling factor for each set of parameters while using the same time-shift for all of them. We observe that, for most parameters, the relative size of ϕ_T(t) compared to ϕ_0(t) measured in our experiment is qualitatively consistent with the theoretical predictions. However, for the case of 18 ns and OD∼2 (see Figure <ref>c), it is evident that the amplitude of the feature exceeds the predicted value.

All of our data is subjected to low-pass filtering at 25 MHz to reduce noise in the phase measurement. This filtering smooths the sharp features, especially noticeable in the 10 ns data, mildly altering the original shape. Despite the fact that the theory curves are not filtered, we still observe good qualitative agreement between the experimental data and the theoretical curves in the horizontal direction. This indicates that, within the error bars, the peaks and troughs of the data and theory coincide in their timing.



We proceed to integrate the traces for ϕ_T(t) and ϕ_0(t) to determine the ratio of the atomic excitation times due to a transmitted photon (τ_T) and an average photon (τ_0) (see Figure <ref>). 

Within each 576 ns shot, the effect of the signal pulse is confined to the central 100–150 ns and the rest of the data is used to determine the background. We use the theory trace |ϕ_T(t)| to select a range of integration which keeps the statistical error bars low without introducing a significant systematic error by discarding too much of the dynamics – to ensure consistency, we implement this by using the arbitrary cutoff at the point where the

tails of the curve reach 30% of the peak value. 

The black solid vertical lines in Fig <ref> depict the boundaries of the integration intervals. 


Figure <ref> shows the ratio of τ_T/τ_0 for four different pulse durations and two ODs distinguished by their respective markers. Except for 36 ns, where only one data point was taken. The theoretical results for the integrals over the regions shown in Figure <ref> are represented as hollowed squares, color-coded according to their corresponding OD. Error bars were found using the covariance matrix to account for correlations in ϕ_T(t) as outlined in the Supplementary information. For completeness, we also include solid lines to show the theoretical predictions integrating over all time.
These results demonstrate a robust overall agreement between theory and experiment. However, there are specific cases that merit discussion. For instance, at 27 ns, although Figures <ref>e and <ref>f show good agreement for ϕ_T(t),τ_T exhibits a significantly more negative value than expected, and neither of the two data points aligns with their respective theoretical predictions. In contrast, the data obtained at 18 ns and OD∼2 is consistent with the theoretical value, despite a noticeable discrepancy in ϕ_T(t), as shown in Figure <ref>c. 


Figure <ref> shows the ratio of τ_T/τ_0 for four different pulse durations and two ODs distinguished by their respective markers (except for 36 ns, where only one data point was taken). These results demonstrate a robust overall agreement between theory and experiment. However, there are specific cases that merit discussion. For instance, at 27 ns, although Figures <ref>e and <ref>f show good agreement for ϕ_T(t),τ_T exhibits a significantly more negative value than expected, and neither of the two data points aligns with their respective theoretical predictions. In contrast, the data obtained at 18 ns and OD∼2 is consistent with the theoretical value, despite a noticeable discrepancy in ϕ_T(t), as shown in Figure <ref>c. 



The main sources of discrepancies between the theory predictions and experimental results are spurious correlations between the probe phase and the signal transmission. In other words, if the frequency of the probe is changing rapidly enough with respect to the repetition rate of the pulses, and in a way that is correlated with the intensity of the signal, one can mistake that fluctuation for the effect of a transmitted photon. Given the sensitivity of this measurement, we conducted several systematic checks for these types of correlations, as detailed in the Supplementary information. 



§ DISCUSSION

The measurement conducted by Sinclair et al. <cit.> was limited to one set of parameters due to several sources of noise. To explore a broader range of parameters beyond this previous measurement, we made significant progress in mitigating sources of spurious correlations, such as addressing the presence of ground loops. Additionally, we determined that detuning the probe beam further from the atomic resonance—specifically three times further than in the previous experiment—provides a cleaner background. Sinclair's experiment, using a pulse of 10 ns rms duration and an OD of 4, yielded a ratio of τ_T/τ_0=0.77± 0.16. In our study, we obtained a ratio of τ_T/τ_0=0.54± 0.28 for the same parameters, which is in agreement with the theoretical value of τ_T/τ_0=0.45 from<cit.>. It is notable that our approach to obtain τ_T/τ_0 also differs from Sinclair's in that we integrated over a region with multiple data points, rather than relying solely on the peak value. However, this resulted in greater statistical error in our measurement; despite us collecting the same amount of data, ϕ_0(t) is a much narrower feature than the integration window. This decision to integrate over a broad time window was informed by insights from theoretical work, knowing that the shape of ϕ_T(t) can significantly differ from ϕ_0(t). Despite the larger error bar, our result is more reliable as it considers more data that provides valuable information.

The results depicted in Figure <ref> undeniably raise the question: what does it mean for an atom to be excited for a negative time, or, equivalently, for a photon to spend a negative amount of time as an atomic excitation? 
The theory work in <cit.> shows that the atomic excitation time due to a transmitted photon is equal to the group delay given a single photon input. Additionally, it provides an illustrative explanation of how this time can be negative, discussing a similar system in which an anomalous dwell time arises from quantum interference in a cavity.


In our experiment, we use a coherent state input, relying on the proof in <cit.> to extract the effect of a single photon, and devise a probe to measure the time defined in Equation <ref>. The theory in <cit.> predicts that such a measurement should be equal to the group delay  Here, we present an intuitive argument that helps explain this fact for the particular measurement we perform.

It turns out that it is also possible to leave aside calculations of the weak value of “excitation time” observables entirely, and make a direct prediction for the correlation between the probe phase and the number of transmitted signal photons (which is what we measure experimentally), at least in the limit of single-mode signal and probe beams.

The state of the signal and probe after the interaction has taken place is given by 

    |ψ⟩=∑ |c_mn|e^iφ_mn|m⟩_p|n⟩_s.

Here, the subscripts p and s denote the probe and signal modes, respectively.

By definition, ϕ_T represents the change in the optical phase of the transmitted probe ϕ_PT per transmitted signal photon n. Using equation <ref>, it can be expressed heuristically as 

    ϕ_T≡∂ϕ_PT/∂ n=∂ (∂φ_mn/∂ m)/∂ n.
 

where we have used the fact that the optical phase of the probe is related to the variation of the phase φ_mn of the state with photon number in the probe mode, which we treat as a partial derivative.


The symmetry of second derivatives allows to interchange the order of partial derivatives, yielding

    ϕ_T=∂ (∂φ_mn/∂ n)/∂ m=∂ϕ_ST/∂ m.

This demonstrates that a phase shift is also induced on the transmitted signal, denoted as ϕ_ST, and its change with respect to the number of transmitted probe photons m corresponds to ϕ_T. The origin of this phase shift on the signal is the AC Stark shift of the atomic levels caused by the far-detuned probe. 

   Assuming that the probe undergoes negligible absorption, the previous statements can be summarized by 

    ϕ_T=∂ϕ_PT/∂ n=∂ϕ_ST/∂ m
 


The latter term in Equation <ref> can be further expressed, using the chain rule, as 

    ∂ϕ_ST/∂ m=∂ϕ_ST/∂ω_0∂ω_0/∂ m=-τ_g∂ω_0/∂ m.

Here, τ_g is the group delay for narrowband light and corresponds to the derivative of the spectral phase acquired by the signal beam with respect to its frequency. 
The previous results can be summarized by

    ϕ_T=∂ϕ_PT/∂ n=-τ_g∂ω_0/∂ m.

This indicates that the phase shift on the probe induced by a transmitted photon of the signal ϕ_T, as measured by our experiment, is proportional to the group delay of the signal τ_g with a scaling factor ∂ω_0/∂ m, which corresponds to the AC Stark shift induced by the probe (refer to the Supplementary information for further details of this calculation). Equation <ref> shares similarities with the expressions for anomalous drag found in <cit.>, suggesting that the group delay governs other effects in regimes where its meaning may be less apparent.

While it is widely known that the group delay can take on negative values, associated with the peak of the transmitted portion of a pulse appearing at times which may indicate superluminal or negative group velocities, it is commonly argued that this quantity does not correspond to the time anything actually “spends” in the medium, but merely to the happenstance of when interference is predominantly constructive.


Our observations, however, show that the group delay is a physically meaningful quantity. It not only provides the location of the centre of a transmitted pulse but also correctly describes the magnitude – and sign! – of the effect transmitted photons have on other systems they interact with. 





§ METHODS




 §.§ phase shift calibration

Before each data run, we conduct a calibration procedure to determine the phase shift imparted by an average input photon (ϕ_0(t)), specific to the chosen set of parameters (OD and pulse duration). This value serves to assess the overlap between the probe and signal inside the atoms and ensures  consistency in experimental conditions from day to day. The gaussian pulses corresponding to the signal are generated by sending the corresponding waveform to an acousto-optical modulator that pulses the light. We take multiple data points for the peak phase shift of the probe as a function of the number of photons in the signal pulse (while keeping the power below 2000 photons where saturation occurs). Subsequently, we fit the data to a straight line and extract the slope, which corresponds to the peak value of ϕ_0(t). 



 §.§ Data analysis and error bars

The data analysis begins by synchronizing the phase of the probe with the corresponding signal detection or lack thereof, while considering the prior delay of 400 ns introduced to prevent unwanted correlations. For each atom cycle, we obtain a trace for ϕ_T(t) by averaging the phase shift ϕ(t) for the shots where the SPCM fired (ϕ_C(t)) and where it did not (ϕ_NC(t)) and subtracting these two. Additionally, we compute the outer product of ϕ_T(t) to construct the covariance matrix. We repeat this procedure for all the atom cycles, resulting in a trace with ϕ_T(t) and the corresponding covariance matrix, denoted as M. This matrix M contains the information about correlations between any pair of elements of ϕ_T(t). The error bars depicted in <ref> correspond to the square root of the diagonal elements of M, representing the standard deviation of the mean. To integrate the traces from <ref> and obtain values for τ_T/τ_0 shown in <ref>, we use the trapezoidal rule. This integration can be seen as a multivariate function denoted as f(x_i,...,x_n), where i runs over the indices of the interval of integration as specified in <ref> in the main text. We calculate the Jacobian matrix defined as 𝐉_𝐢=∂ f/∂ x_i and find the error bar of the value of the integral as σ^2=𝐉^𝐓 𝐌_𝐑 𝐉, where 𝐉^𝐓 is the transpose of the Jacobian matrix and 𝐌_𝐑 is the covariance matrix including only the information related to the points in consideration by the interval of integration.




 §.§ Data and theory comparison


The theory curves are output on a time axis determined by the center of the input pulse. However, determining the center of the signal pulse in the experiment presented challenges for us. To align the time axes, we used ϕ_0(t) for the most narrowband set of parameters, as for this case, ϕ_0(t) closely follows and resembles the shape of the Gaussian input pulse. In our experiment, this occurs at 36 ns (OD ∼ 3). We obtained the experimental and theoretical values of ϕ_0(t) and fitted them to Gaussian profiles. Subsequently, we calculated the time difference between the centers of the fitted Gaussians, which was found to be 252 ns. To match the time axis of the theory, we adjusted the experimental values of both ϕ_0(t) and ϕ_T(t) by subtracting this time difference from the time variable. Consistently applying this adjustment across all data points, we created the plots shown in Figure <ref>.

The time alignment procedure solely relies on the average phase shifts and assumes a Gaussian lineshape for ϕ_0(t) across all pulse lengths, with negligible dispersion from elements other than the atomic medium. The assumption of Gaussianity in ϕ_0(t) breaks down when the pulse length goes well below τ_ sp, which includes, in this case, the data points taken at 10 ns; the shape of ϕ_0(t) is closer to an error function for the first half, followed by an exponential decay characterized by τ_ sp <cit.>.


