The imaging of physical systems is a key component of experimental science. The techniques used vary as widely as the length scales they are used to image, from electron microscopy [1] and in vivo imaging of neurons, [2] to astronomy [3] and cosmology [4]. Imaging of dilute samples of ultra-cold neutral atoms is a particular area of ongoing and intensive development [5]. The interaction of ultra-cold neutral atoms with light presents two primary detection strategies: absorption and scattering, or an optical phase shift of the interacting light. Each individual imaging technique is best suited for a specific system, typically related to constraints on the number of photons scattered or limits imposed by high or low optical depth samples [6].

This Letter focuses on single-shot imaging of low optical depth samples of a ${}^{87}\mathrm{Rb}$ quantum gas in long baseline atom interferometry. Detection in this regime is important as the current state-of-the-art atom interferometers are likely limited by atom detection noise [7]. The standard approach to detection of optically thin atom clouds is via fluorescence imaging (FI) measurements [8]. This technique has long been considered ideal, due to its zero background nature. In practice, rejection of classical noise, such as background light and high photon collection efficiency, is difficult to achieve.

Frequency modulation imaging (FMI) has been suggested as a technique for measuring cold atoms as far back as 1983 [9]. More recently, FMI has been analyzed as a “nondestructive” probe of ultra-cold atoms [10–12] and has been used to create a “self-locked” magneto-optical trap [13]. FMI is a heterodyne detection scheme based on the addition of frequency sidebands, ${\omega }_{\pm }$, to a narrow linewidth laser at frequency, ${\omega }_0$. The system operates in the limit where only a single sideband or carrier interacts with an atomic resonance. A signal is acquired by measuring the resulting interference between carrier and sidebands using standard RF techniques. The complete details of the interaction of an atomic resonance with such a probe laser are given in Bjorklund et al. [9]. The initial applications of FMI had significant difficulties achieving high signal-to-noise ratio (SNR) due to residual amplitude modulation [14]. Modern fiber-coupled electro-optic modulators (EOMs) have vanishingly small residual amplitude modulation, justifying a reassessment of this technique for imaging in modern atomic physics experiments.

Here, FMI is used to acquire high bandwidth, large SNR measurements of low optical depth clouds of atoms. FMI offers many advantages over other imaging techniques in this context, including zero background, strong rejection of classical noise, near 100% collection efficiency, and linear scaling with imaging laser power [9]. A theoretical comparison between ideal FI and FMI detection in the current system is explored and shows a sizable advantage for FMI. The utility of FMI is demonstrated by detection of a $2\times {10}^6$ ultra-cold atom cloud after 530 and 750 ms time-of-flight (TOF) with SNR of $900:1$. This SNR is comparable to the theoretical limit of FMI under the experimental conditions. Finally, the technique is demonstrated to be readily applicable to ultra-high sensitivity atom interferometry.

The demonstration of FMI on ultra-cold atoms is undertaken on an apparatus designed for ultra-high resolution gravimetry with Bose–Einstein condensates (BECs) [15]. This device produces large condensates of up to $5\times {10}^6$ atoms and allows 2.75 m of free fall below the condensate chamber. Four imaging regions are available along the fall, which corresponds to 0–25, 220, 530, and 750 ms of free expansion.

Briefly, a standard experimental cycle is as follows. Hot ${}^{87}\mathrm{Rb}$ atoms are introduced into a ${10}^{-7}\mathrm{Torr}$ vapor cell via an electronic dispenser. The hot atoms are cooled in a two-dimensional (2D) magneto-optical trap (2D MOT) and transferred through a high impedance line to an ultra-high vacuum (UHV) ($1\times {10}^{-11}\mathrm{Torr}$) chamber with a blue detuned push beam. Once in the UHV chamber, the atoms are collected and cooled in a three-dimensional (3D) MOT. A standard compression and polarization gradient cooling sequence is then applied. The cold ensemble of atoms is then optically pumped into the $F=1$ manifold, and the $|F=1,m_f=-1〉$ state is loaded into a magnetic quadrupole trap with a vertical gradient of 150 G/cm. Forced evaporation is performed by systematically removing the hottest atoms with resonant microwave radiation. By reducing the magnetic trap to 25 G/cm, the atoms are finally loaded into an optical trap formed by a pair of 1070 nm 25 W broadband fiber lasers intersecting at 22.5° and beam waists of 300 μm. A second stage of evaporation is performed by lowering the intensity of the optical trap and simultaneously reducing the magnetic trap gradient to 0 G/cm. After a total time of 13 s, a pure BEC of up to $5\times {10}^6$ atoms is produced in an essentially spherical optical trap. The condensate has an initial diameter of approximately 50 μm with a thermal temperature estimated to be significantly below the 50 nK asymptotic kinetic energy measured for expansion times up to 750 ms. By adjusting the final intensity of the optical trap, the temperature of the atomic ensemble can be varied from a pure ultra-cold thermal source to a pure BEC. Following production of the desired cloud, the optical trap is extinguished suddenly ($\sim 10\mathrm{\mu s}$), and the atoms begin to fall under gravity.

The falling gas expands due to either its kinetic temperature if it is a thermal cloud, or to the mean-field interaction between atoms in the condensate [16]. The on-resonance optical depth changes from 50 at 25 ms through to 0.05 at 750 ms for a single falling cloud with a 50 nK kinetic temperature. For an atom interferometer, in which the cloud is split into many nonoverlapping states, the final optical depth per cloud can be lower than 0.01. Across these four orders of magnitude in optical depth, no single imaging technique is ideal. Here, absorption imaging is used to characterize the large optical depth samples, while FMI is introduced as a higher SNR alternative to fluorescence imaging for the lower optical depth samples.

Absorption imaging is the standard benchmark for imaging BECs [17]. Here, it is used to quantitatively characterize the cold clouds for drop times between 0 and 530 ms. A simple absorption imaging optics setup is shown in Fig. 1(a). As the atomic ensemble passes through an imaging region, it is repumped into the $F=2$ ground state via the ${\mathrm{D}}_2$, $F=1\to F^{\prime }=2$, transition resulting in no measurable $F=1$ population. Immediately following the repump pulse, the atoms are imaged with a 100 μs pulse from a $1/e^2$ probe beam of diameter 7.5 mm resonant with the ${\mathrm{D}}_2$, $F=2\to F^{\prime }=3$, transition of ${}^{87}\mathrm{Rb}$. The atoms absorb a fraction of the beam resulting in a “shadow” proportional to the column density. This shadow is then imaged with a $2f$ imaging system onto a charge coupled device (CCD). A second image with no atoms present is taken as a reference, and the OD is calculated via $I_f=I_0{e}^{-\mathrm{OD}}$, where $I_0$ is the background light intensity, and $I_f$ is the intensity of light after passing through the atoms.

 

Fig. 1. (a) Absorption imaging flow diagram. The shadow of the atomic sample is imaged on a CCD with a one-to-one magnification single lens imaging system. (b) Frequency modulation imaging flow diagram.

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For drop times of 530 and 750 ms, not only is the optical depth low, but the interaction time between the imaging light and atomic sample is limited by the average velocity (5.2 and 7.4 m/s) of the sample and the optical access of the imaging windows (2.5 cm vertical). The restricted interaction time is detrimental to imaging techniques, such as fluorescence imaging (FI), where the signal is fundamentally limited by integration time. FMI, although proportional to interaction time, is not limited by it. The photon shot-noise limited SNR for FI and FMI are given in Eqs. (1) and (2), respectively:

Here, $N$ is the total atom number, $\sigma$ is the photodiode collection efficiency, $\gamma$ is the spontaneous emission rate, $t$ is the atom light interaction time, $B$ is the bandwidth of the measurement, $P_{\text{tot}}$ is the total laser power, ${\eta }_{\mathrm{PD}}$ is the photodiode efficiency, $\delta$ is the sideband absorption ratio ($0=100\%$ transmission, $1=0\%$ transmission), $M$ is the modulation index, $h$ is Planck’s constant, and $\nu$ is the carrier frequency of the light.

FMI is based on the heterodyne detection of a sideband and, as such, has a SNR proportional to the total laser power as seen in Eq. (2). A theoretical comparison between FI and FMI was done for a $2\times {10}^6$ atom BEC with a temperature of 50 nK after 750 ms of free expansion and is shown in Fig. 2(a).

 

Fig. 2. Theoretical comparison between photon shot-noise limited FI and FMI techniques for a $2\times {10}^6$ atom BEC after 750 ms free expansion. (a) FI (dashed) SNR for varying atom–light interaction times, assuming $3\times {10}^7\mathrm{photons}/\mathrm{s}$ scattering rate, 10% collection efficiency, and a Hamamatsu S3884 avalanche photodiode. FMI (solid) SNR for varying total laser power and atom-light interaction time (0.001 to 100 ms), assuming a Newport 1601FS-AC photodiode. (b) Comparison under real system limitations due to the need for vertical spatial resolution and laser power.

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This comparison was made under optimal conditions, assuming photon shot-noise limited imaging for both techniques, and shows a dramatic advantage to FMI for imaging power $>10\mathrm{\mu W}$ and interaction times $>100\mathrm{\mu s}$. For the apparatus to be used for precision measurement, it is necessary to have spatial resolution of $\sim 200\mathrm{\mu m}$ to resolve the spatially separated final interferometer states. This further limits the interaction time ($\sim 500\mathrm{\mu s}$) and the number of atoms interacting with the imaging beam at any instant. The photon shot-noise limited comparison of the two imaging techniques in the actual optics configuration is shown in Fig. 2(b). For the current experimental parameters of a $2\times {10}^6$ atom, a BEC with a temperature of 50 nK, 750 ms of free expansion, $\sim 100\mathrm{\mu W}$ imaging power, and an interaction time of $\sim 500\mathrm{\mu s}$ an order-of-magnitude increase in SNR is expected for FMI over FI.

The optical and RF setup for FMI is shown in Fig. 1(b). An imaging beam ($\sim 100\mathrm{\mu W}$), 40 MHz detuned from the ${\mathrm{D}}_2$, $F=2\to F^{\prime }=3$, transition is coupled into a single mode, polarization maintaining 20 GHz EOSpace fiber EOM and delivered to the sensor head. The 40 MHz detuning is chosen due to a minimum in the photodiode (PD) electronic noise. The EOM is driven via a Rigol DG4152 two-channel phase-locked function generator such that the $-1$ sideband is resonant with the desired atomic transition. The second channel is used as a variable phase RF local oscillator. After being out-coupled, the light is passed through an iris to match the beam width to the atomic cloud width. The vertical resolution is achieved by focusing the imaging beam with a cylindrical lens. As the atoms fall through the light sheet, the $-1$ sideband is absorbed, while the carrier and $+1$ sideband are phase shifted. The imaging light is collected and focused onto a Newport 1 GHz alternating current (AC) PD. The AC signal from the PD is then mixed with a phase-shifted output from the function generator via a Mini-Circuits ZAD-8. The output of the mixer is then passed through a Stanford Research Systems SR560 amplifier and 3 kHz low-pass filter, before being recorded by a National Instruments PXI 6143 digitizer. The modulation index and frequency of the EOM, laser power, and mixing phase are all adjusted to maximize the output signal.

The standard SNR analysis for absorption imaging is as follows: initially, two images are taken, one without atoms present and another with atoms, and converted to OD [Figs. 3(a) and 3(c)]. The two images are integrated along a single axis [Figs. 3(a) and 3(c) traces]. A cumulative second integration is performed [Figs. 3(b) and 3(d)] showing total camera counts. The signal is given by the final value of the second integration of the atom image. The noise is found via the variance of the second integration of the background image. Figure 3(c) corresponds to a typical absorption image of a $2\times {10}^6$ atom BEC after 530 ms of free expansion. At this fall time, absorption measured with a CCD camera gives an integrated SNR of $135:1$.

 

Fig. 3. Typical background and absorption image of a $2\times {10}^6$ atom BEC after 530 ms of free expansion. (a)–(d) Typical absorption imaging SNR analysis. The average SNR at this expansion time is $135:1$.

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The SNR analysis for FMI is similar to that of the absorption image, with the exception that the PD acts as the first integrator, leaving the profiles seen in Figs. 4(a) and 4(c) as the initial background and measurement. The FMI signals of a $2\times {10}^6$ atom BEC after 530 and 750 ms of free expansion are shown overlaid in Fig. 4. These traces show that the FMI has a SNR of $900:1$ and $880:1$, for 530 and 750 ms of expansion, respectively. For the $2\times {10}^6$ atoms used in this measurement, the fundamental measurement limit is the atom shot-noise, $\sqrt{N}$, placing the FMI system within a factor of 2 of the fundamental limit.

 

Fig. 4. FMI SNR analysis of a $2\times {10}^6$ atom BEC for 530 (solid) and 750 ms (dashed) of free expansion. The average SNRs from the FMI signals at 530 and 750 ms free expansion are $900:1$ and $880:1$, respectively.

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As a demonstration of the application of this imaging technique, a BEC is used as an atomic source for a high-precision sensor based on Bragg atom interferometry [15]. A $2\times {10}^6$ atom $\mathrm{F}=1$ spinor condensate of ${}^{87}\mathrm{Rb}$ is released into free fall for 750 ms and probed with a Mach–Zehnder atom interferometer based on Bragg transitions. The Bragg interferometer simultaneously addresses the three magnetic states, $|m_f=1,0,-1〉$, producing three separate interferometers with differing magnetic responses. A Stern–Gerlach magnetic pulse separates these states in the vertical direction after completion of the atom interferometer sequence. A repumping pulse, resonant with the ${\mathrm{D}}_2$, $F=1\to F^{\prime }=2$, transition of ${}^{87}\mathrm{Rb}$, is applied immediately prior to the atom clouds falling through the FMI light sheet. The resulting FMI signal from a single realization of the experiment showing both output ports for each $m_f$ state (six total peaks) is seen in Fig. 5(a). The interferometric fringes are scanned by changing the optical phase of the final Mach–Zehnder beam splitter for each consecutive run. A typical fringe set is shown in Fig. 5(b). By appropriately combining the phase shifts of the interferometric fringes, a measurement of the acceleration due to gravity can be made with an asymptotic precision of $\mathrm{\Delta }g/g=2.1\times {10}^{-9}$.

 

Fig. 5. (a) Typical FMI image of a spinor interferometer after 750 ms of free fall. (b) Interference fringes produced by a 60 ms total time Mach–Zehnder interferometer with a spinor BEC source.

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From simple theoretical considerations, FMI is suggested as an alternative to FI for low optical depth atomic clouds. For a single BEC, expanded over 530 ms, a raw SNR of $900:1$ is measured, less than a factor of two from the atom shot-noise limit. The system is implemented on a high-precision spinor BEC-based atom interferometer obtaining an imaging SNR of approximately 100 on the output states and an estimated contribution to the interferometer noise of $\sim 10\mathrm{mrad}$. Due to FMI’s high SNR with limited interaction time, it is feasible to apply this imaging technique to measure the spatial fringes associated with single-shot phase estimation in a long baseline atom interferometer [18].