Photon counting for Bragg spectroscopy of quantum gases

We demonstrate a photon-counting technique for detecting Bragg excitation of an ultracold gas of atoms. By measuring the response of the light field to the atoms, we derive a signal independent of traditional time-of-flight atom-imaging techniques. With heterodyne detection we achieve photon shot-noise limited detection of the amplification or depletion of one of the Bragg laser beams. Photon counting for Bragg spectroscopy will be useful for strongly interacting gases where atom-imaging detection fails. In addition, this technique provides the ability to resolve the evolution of excitations as a function of pulse duration.

Figure 1

Bragg spectrum of a ${}^{85}\mathrm{Rb}$ BEC of $5.5\times {10}^4$ atoms at k = 16 $\mu$m${}^{-1}$, measured in two different ways. The horizontal axis shows the frequency difference between the Bragg beams. The vertical scale shows the number of excitations, which is measured using traditional absorption imaging of ejected atoms (open circles), as well as with the photon-counting technique presented here (solid triangles). The error bars on black points represent the shot noise in the photon-counting measurements. The photon-counting measurements used three pulses of equal length. The first and the third pulses used only a single weak beam to make an average background measurement, with no Bragg excitation. During the second pulse, both Bragg beams illuminate the condensate to induce Bragg scattering and we subtract the averaged background to count the number of photons gained or lost in the weak beam due to the Bragg excitation.

Figure 2

Diagram of the Bragg beams (a) and the heterodyne setup for Bragg scattering via photon counting (b). The Bragg beam frequencies (${\omega }_{\text{laser}},{\omega }_{\text{laser}}+\omega$) are offset by $\omega$, and create excitations at momentum $\hslash k=2\hslash k_{\text{laser}}\sin (\theta /2)$, where $\theta$ is the angle between the Bragg beams and $k_{\text{laser}}=2\pi /780$ nm in our case. In (b) the three beams (labeled weak, LO, and strong) are derived from the same laser source and individually fiber coupled. The optics for the LO beam are chosen to give the same spatial mode as that of the weak beam. After being combined on a beam splitter, the weak and LO beams illuminate a photodiode and the beat signal is sent to a demodulating mixer. The two quadrature outputs of the demodulator, I and Q, are sent to an oscilloscope for the measurement. Also illustrated in the figure is a servo loop that functions to minimize the phase fluctuations of the rf input with respect to the “LO” drive of the mixer. The servo minimizes the demodulator output, I, by feeding back to a phase modulation input on the synthesizer that drives the “LO” port of the mixer. This synthesizer is phase locked to the synthesizer driving the weak beam acousto-optic modulator (AOM), illustrated with a double-headed arrow. Not shown in the schematic is the synthesizer driving the strong-beam AOM, which is also phase referenced to the synthesizer driving the weak-beam AOM.

Figure 3

Noise performance of heterodyne detection for an LO power of 250 $\mu$W. Vertical scale is normalized to the shot noise expected for the relevant pulse length and laser power. The legend shows the different weak beam powers used. Low-frequency drifts make our heterodyne scheme no longer shot-noise limited at long time scales. Inset has the same normalized vertical units, with the data shown at different weak beam intensities, corresponding to a constant ${10}^5$ photons.

Figure 4

Bragg excitations as a function of time, measured with photon counting. The number of Bragg excitations is given by $N_{\text{exc}}(\tau )={\int }_0^{\tau }[\mathrm{ṅ}(t)-{\mathrm{ṅ}}_{\text{av}}]\mathit{dt}$, where ${\mathrm{ṅ}}_{\text{av}}$ is the average rate of weak beam photons measured when no strong light is present. The Bragg pulse begins at 0 ms. The deviation from expected ${\tau }^2$ behavior (for small $\tau$) is discussed in the text. These measurements were performed on resonance, with a BEC of 4 00 000 ${}^{87}\mathrm{Rb}$ atoms.