The scattering of an incident wave, whether classical or quantum, upon a single particle is not an instantaneous process. In 1955, E. P. Wigner showed that the time-delay associated to the elastic scattering of a wave upon a scatterer is the derivative of the phase shift acquired by the incident wave with respect to its energy [1]. Later, F. T. Smith pointed out that this so-called Wigner delay is the lifetime of a resonant state excited during the scattering [2], therefore being largest at resonance. Since its derivation, the Wigner delay has been used as an important parameter in the problem of near-resonant scattering in dense media, as it governs the transport of energy [3–7].

Scattering processes are common in many areas of physics, and therefore many systems are candidates for the measurement of this delay. However, in most systems it is expected to be very short, explaining why experimental demonstrations are scarce and have required involved techniques. In 1976, a delay in the ${10}^{-20}\mathrm{s}$ range was measured in the elastic near-resonant scattering of protons on a target of carbon using interferences in the Bremsstrahlung radiation [8]. The advent of ultrashort pulse laser-based metrology made it possible to measure time delays associated to the scattering of light by condensed matter systems or by atomic vapors. Femtosecond (fs) laser techniques, for instance, allowed to measure delays of a few fs associated with the bouncing of light off a metallic surface in the vicinity of a plasmon resonance [9]. More recently, attosecond metrology led to the measurement of delays in the 10–100 attoseconds range in the photoemission from a surface [10] or from a vapor [11,12]. Yet, the Gedanken experiment imagined initially by Wigner has never been realized, i.e., the direct measurement of the time delay induced by a single scatterer upon an incident wave. Here, we do so by sending a Gaussian wavepacket of near-resonant light on an isolated individual atom and by scanning the frequency of the light across an atomic resonance.

The Wigner delay can be understood in a semiclassical model, where a two-level atom elastically scatters a classical light field. In the limit of weak intensities, the atom responds linearly to the incoming light field, and the associated dispersive behavior leads to a time delay of the re-emitted field that is maximum at resonance and is given by ${\tau }_W=d\varphi /d\omega$. Here, $\omega$ is the frequency of light, $\varphi (\omega )=\arctan (\mathrm{\Gamma }/2({\omega }_0-\omega ))$ is the phase of the atomic polarizability [13,14], ${\omega }_0=2\pi c/{\lambda }_0$ is the frequency of the atomic resonance, and $\mathrm{\Gamma }$ is the inverse lifetime of the excited state $|e〉$ (see Fig. 1). ${\tau }_W$ is thus given by

 

Fig. 1. (a) Implementation of the Wigner Gedanken experiment: we excite a single atom by a pulse of light and measure the arrival time of the scattered pulse on a photon counter; we use the large numerical aperture lenses to isolate a single atom with a tightly focused laser beam (not shown), and collect the scattered photons efficiently. (b) Atomic levels used in the experiment.

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To check this prediction experimentally, we use a single cold ${}^{87}\mathrm{Rb}$ atom that we initially isolate in an optical dipole trap with microscopic size [15]. The temperature of the atom in the trap is 70 μK, measured by a release-and-recapture method [16]. We first prepare the trapped atom in the hyperfine ground state level ($5S_{1/2}$, $F=2$). We then release it in free space and illuminate it with a series of weak Gaussian pulses of circularly polarized near-resonant laser light at ${\lambda }_0=780\mathrm{nm}$ (see Fig. 1). Using a large numerical aperture lens ($\text{N.A.}=0.5$), we collect the photons scattered at 90° with respect to the direction of excitation and detect them with a fiber-coupled avalanche photodiode (APD) operating in the single photon counting mode. The signal is sent to a counting card with a 256 ps resolution. The excitation light is produced from a continuous laser locked on the ($5S_{1/2}$, $F=2$) to ($5P_{3/2}$, $F^{\prime }=3$) transition. We chop this light into pulses with a Gaussian temporal shape using an acousto-optic modulator and a fast arbitrary waveform generator. We set the peak intensity $I$ of the pulse to $I/I_{\text{sat}}=0.1$ ($I_{\text{sat}}=1.6\mathrm{mW}/{\mathrm{cm}}^2$) to operate in the weak excitation limit. For the temporal width of the pulses, we choose $\mathrm{\Delta }t=66\mathrm{ns}$ sufficiently large to approach the limit $\mathrm{\Delta }t\gg 1/\mathrm{\Gamma }$ (here, $1/\mathrm{\Gamma }=26\mathrm{ns}$), and sufficiently small to determine with a good accuracy the temporal center of the scattered pulse, and thus the time delay ${\tau }_W$ [see Fig. 2(a)]. The acousto-optic modulator allows us to tune the center frequency ${\omega }_L$ of the laser light around the frequency ${\omega }_0$ of the closed transition between the states $|g〉=|5S_{1/2},F=2,M_F=2〉$ and $|e〉=|5P_{3/2},F^{\prime }=3,M_F^{\prime }=3〉$. The optical pumping of the atom in the Zeeman sublevel $|g〉$ is ensured by the first excitation pulses when the laser is on resonance with the atom. When the laser is tuned away from the resonance, the optical pumping is less efficient. However the Wigner delay is unaffected by the events when the atom does not cycle on the closed transition, as both the center frequency and the width of the resonance, which sets the value of the delay, are independent of the transition between the states $|5S_{1/2},F=2,M_F〉$ and $|5P_{3/2},F^{\prime }=3,M_F^{\prime }〉$. During the excitation, repumping light tuned to the ($5S_{1/2}$, $F=1$) to ($5P_{3/2}$, $F^{\prime }=2$) transition is also sent on the atom.

 

Fig. 2. (a) Intensity of the excitation pulse with rms width $\mathrm{\Delta }t=66\mathrm{ns}$, (b) histograms of the number of photons detected after scattering for $\delta =({\omega }_L-{\omega }_0)/\mathrm{\Gamma }=(0;-0.3;2)$ (time bins: 5.9 ns). Dotted lines are the measured data and solid lines are Gaussian fits. In (b), the rms widths of the scattered pulses are, respectively, 73, 73, 65 ns, in agreement with the solution of Eqs. (3).

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To maximize the number of collected photons scattered by the same atom, we use a time sequence where we interleave excitation pulses in free space with recapturing periods of 1.3 μs. After 50 such excitation-and-recapture periods, photon scattering has heated the atom and the probability that the atom escapes the trap has increased. We therefore apply a 1 ms period of three dimensional laser cooling with the dipole trap on. In this way, we keep the Doppler shift below 100 kHz. We repeat this pattern 120 times, corresponding to a total of 6000 pulses sent on the same atom before we start again with a newly prepared atom.

Figure 2(b) shows the temporal responses obtained on the photon counter for different values of the detuning $\delta =({\omega }_L-{\omega }_0)/\mathrm{\Gamma }$ of the excitation laser. Each temporal response results from an integration over 2000 individual atoms having experienced the sequence described above, and is fitted with a Gaussian to extract the arrival time of the scattered pulse. At resonance, the scattered pulse is maximally delayed (and most intense), as expected from Eqs. (1) and (2). Figure 3 summarizes the variation of the arrival time versus $\delta$. The comparison to far off-resonance measurements reveals a Wigner time delay ${\tau }_W$ as large as $42\pm 2\mathrm{ns}$ at resonance, not far from the $2/\mathrm{\Gamma }=52\mathrm{ns}$ predicted by Eq. (1).

 

Fig. 3. Time delay of the scattered light pulse versus the detuning $\delta$ of the excitation laser. Dashed line: prediction of Eq. (1). Solid line: solution of Eqs. (3) for a two-level atom excited by the pulse shown in Fig. 2(a), with a fitted chirp rate $\alpha /2\pi =5\pm 0.8\mathrm{MHz}/\mathrm{\mu s}$. The vertical error bars are from the fits of the scattered pulses. Horizontal error bars are the rms uncertainty in the laser frequency ($0.12\mathrm{\Gamma }$). The inset shows the number of scattered photons detected on the APD, and the solid line is the solution of Eqs. (3).

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The discrepancy is actually due to the conditions $\mathrm{\Delta }t\gg 1/\mathrm{\Gamma }$ and $I/I_{\text{sat}}\ll 1$ not being exactly fulfilled experimentally. In particular, the latter condition means that the scattering is not only elastic, as implicit from the model discussed above, but has also a small inelastic component. To take this into account, we solved the optical Bloch equations governing the evolution of the density matrix for a two-level atom, using the measured shape of the excitation pulse as a driving term. These equations read in the rotating-wave approximation [17]:

To characterize the scattering process fully we also analyze the number of collected photons as a function of the excitation detuning $\delta$. To do so, we integrate the temporal signal obtained on the APD. The data are again in good agreement with the solution of Eqs. (3), and exhibit a full width at half maximum of $1.2\mathrm{\Gamma }$ (see Fig. 3 inset), larger than the Lorentzian profile predicted by Eq. (2). This larger width is also consistent with the reduced time delay due to the conditions $\mathrm{\Delta }t\gg 1/\mathrm{\Gamma }$ and $I/I_{\text{sat}}\ll 1$ not being exactly fulfilled experimentally.

In conclusion, we have measured the time delay introduced by an individual atom in an essentially elastic scattering process. We have found delays as large as 42 ns, in good agreement with the theoretical limit predicted by the optical Bloch equations. In the future, it will be interesting to extend these measurements to the case of dense ensembles of cold interacting atoms to probe collective scattering [3].