Dynamical beats of short pulses in waveguide QED

We study weak pulse propagation through near-resonant collective media. The timescales of the pulse and the lifetime of the media are comparable. The transmitted pulse develops temporal oscillations, known as dynamical beats. We use a collection of ${}^{133}\mathrm{Cs}$ atoms randomly captured by a nanofiber-based optical lattice. An effective macroscopic model derived from a microscopic many-atom input-output theory reproduces the observations. The results deepen our understanding of single-photon pulse propagation, crucial for many waveguide QED-based quantum information protocols.

Figure 1

(a) The atomic array is trapped in the evanescent field of an optical lattice near the nanofiber surface. The lattice constant $d$ is half the wavelength of the trapping light. The temporal shape of a Gaussian input pulse (green) gets modified after propagating through an array of trapped atoms (red). (b) Schematic of the apparatus includes dichroic mirrors (DMs), volume Bragg gratings (VBGs), and half wave plates $(\lambda /2)$ with the pulses coming from the left detecting them with a single-photon counter module (SPCM) on the right. The pulses are produced by acousto- and electro-optical modulators (AOM, EOM) controlled by an electronic pulse from an arbitrary wave generator (AWG) that triggers the time correlator for single-photon counters (TCSPCs) for further data processing.

Figure 2

Time dependence of the transmitted intensity (red) for a 10 ns FWHM pulse propagating in a medium of OD = 11.6, normalized by the peak intensity of the pulse without atoms (gray), in logarithmic scale. The solid green line shows the results for the multimode model with an ideal single Lorenzian absorption as input. The solid purple line shows the results of the model using a broadened absorption modeled by a sum of Lorentzians. The inset shows the measured absorption (gray dots) and the result of the sum of Lorenzians (solid gray line). Note the logarithmic vertical scale. In all plots, ${\mathrm{\Gamma }}^{\prime }/2\pi =5.2$ MHz.

Figure 3

Left: Evolution of the first zero with on-resonance excitation (valley) of the transmission as a function of the OD for two different pulse widths. Blue circles correspond to a FWHM of 10 ns and the red squares for a FWHM of 13 ns. The corresponding blue and red continuous lines indicate the prediction of the full theory. Right: Extracted decay rate (green) from the fall of the second peak when the input is a 10 ns pulse (bars), 13 ns (x's). The dashed green line is $1+\text{OD}\text{/}2$, the expected value from the theory. The error bars are the same for the two sets of data, but only the horizontal or the vertical are plotted for clarity.

Figure 4

Comparison of the measured transmittance (solid black in logarithmic scale) with the modeled one (dotted blue). The model consists of a sum of frequency-shifted Lorentzian spectra (dashed gray) log-normally distributed.