Tracking Evaporative Cooling of a Mesoscopic Atomic Quantum Gas in Real Time

The fluctuations in thermodynamic and transport properties in many-body systems gain importance as the number of constituent particles is reduced. Ultracold atomic gases provide a clean setting for the study of mesoscopic systems; however, the detection of temporal fluctuations is hindered by the typically destructive detection, precluding repeated precise measurements on the same sample. Here, we overcome this hindrance by utilizing the enhanced light-matter coupling in an optical cavity to perform a minimally invasive continuous measurement and track the time evolution of the atom number in a quasi-two-dimensional atomic gas during evaporation from a tilted trapping potential. We demonstrate sufficient measurement precision to detect atom-number fluctuations well below the level set by Poissonian statistics. Furthermore, we characterize the nonlinearity of the evaporation process and the inherent fluctuations of the transport of atoms out of the trapping volume through two-time correlations of the atom number. Our results establish coupled atom-cavity systems as a novel test bed for observing thermodynamics and transport phenomena in mesoscopic cold atomic gases and, generally, pave the way for measuring multitime correlation functions of ultracold quantum gases.

Popular Summary

Much of the rich variety of physical phenomena in our environment arises from the interplay of randomness and correlated nonlinear dynamics. Examples include the growth of cell complexes, turbulence in liquids, and the spreading of wildfires. Despite their abundance in nature, these phenomena are notoriously hard to describe. In the quantum world, dedicated experiments in well-controlled environments are often the only viable way to foster our understanding. Our work presents the evaporative cooling of an ultracold quantum gas as another instance where stochastic noise meets nonlinear dynamics.

We monitor the number of atoms in our gas in real time by employing a novel, sensitive detection scheme that strongly couples the quantum gas to an optical resonator. With this technique, the quantum gas can be measured more efficiently than in standard detection schemes, and the sensitivity of this measurement becomes high enough to detect stochastic fluctuations in the number of atoms—a direct consequence of the fact that our gas is composed of atoms leaving the gas one at a time.

Our work allows us to establish continuous nondestructive measurements of stochastic nonlinear cold-atom gases as a new paradigm of exploring quantum systems. The presented method is also directly applicable to studying a wider range of transport phenomena in neutral-atom quantum simulators and opens the path to the nondestructive continuous detection of quantum systems taken out of equilibrium.

Figure 1

Experimental setup and measurement imprecision. (a) Schematic of the tilted evaporation of atoms (red) with temperature $T$ in the trap potential (blue) with dynamically lowered depth $U$. Atoms with energies exceeding the trap depth are spilled. (b) The atoms dispersively couple to the cavity, resulting in an atom-number-dependent shift ${\mathrm{\Delta }}_N$ of the transmission line shape (blue solid traces) compared to an empty cavity (blue dashed line). Tracking the side of fringe (marked by a red dot and line) provides a dynamical measurement of atom number. (c) The cavity resonance is tracked using a feedback loop involving the cavity-coupled atomic cloud (dark red), located at the intensity maximum of a probe beam in the cavity (light red). The transmitted intensity of probe light is detected using a heterodyne receiver with local oscillator (LO) and a radio-frequency power detector (Det). This power is kept constant using a feedback loop (PID) adjusting the frequency of probe and LO through an acousto-optic modulator (AO). The atom number is derived from an in-loop measurement of the voltage used to adjust a voltage controlled oscillator driving the AO. (d) A single unfiltered trace of equivalent atom number $N$ versus time with the filter procedure indicated (gray shaded areas). (e) The Allan deviation $\mathrm{\Delta }N$ is dominated by photonic shot noise for small integration times $\tau$ and by dynamics of the atom number for large $\tau$. A low-pass filter of the traces with optimal integration time $\tilde{\tau }$ minimizes the noise associated with both effects. The solid lines represent guides to the eye. Larger photon number $n$ [orange, $n=1.9(1)$; blue, $n=3.2(1)$; red, $n=6.0(1)$; green, $n=9.7(1)$] results in reduced shot noise but also faster loss of the atoms and, therefore, increased imprecision at longer integration times. The colored ticks mark the noise level set by Poissonian statistics for our lowest measured mean atom number for each $n$. The light symbols show a reduced Allan deviation, where the average dynamics is subtracted from each trace before calculating $\mathrm{\Delta }N$.

Figure 2

Two-time correlations of atom numbers from real-time trajectories. (a) Traces of equivalent atom number $N$ versus observation time $t$, calculated from the number-dependent cavity shift, for varying intracavity photon number $n$ (indicated in the upper right corner). The bright curves represent individual runs of the experiment (approximately 150 per panel). Selected traces are shown in slightly darker color to illustrate their stochastic character. The dark curves are the mean over all traces. The traces are filtered using their corresponding optimal integration time; see Fig. 1. The vertical lines mark the two times $t_2=25\mathrm{ms}$ ($t_2=275\mathrm{ms}$) used to produce the scatter plots shown above the time traces. They illustrate the standardized atom number $N_{2,s}$ measured at the earlier (later) $t_2$ versus $N_{1,s}$ measured at $t_1=5\mathrm{ms}$. The diagonal (dashed) line and the slope extracted from a linear ordinary least-squares fit to the data (dash-dotted line) are indicated in all scatter plots. The corresponding measurement noise is indicated by the ellipse in the center.

Figure 3

Nonlinearity of the evaporation process. (a) Schematic illustrating the nonlinearity measure. Superlinearity (sublinearity) is equivalent to a slope $a$ of the scatter of $N_2$ versus $N_1$ that is larger (smaller) than the remaining fraction of atoms $p$. For a linear process, $a=p$. (b) Subtracted slope $a_s=a-p$ for different initial times $t_1$ (dark to bright color, indicated in the legend) used to quantify the nonlinearity of the evaporation process versus remaining fraction of atoms $p$. The shaded region shows the standard deviation calculated by bootstrapping for the smallest and largest initial times.

Figure 4

Mesoscopic fluctuations of the evaporation process. (a) Unexplained variance ${\sigma }_u^2$ versus remaining fraction of atoms $p$ for intracavity photon number $n=1.9(1)$ at initial time $t_1=10\mathrm{ms}$ and (b) at $t_1=180\mathrm{ms}$, with the standard deviation calculated by bootstrapping indicated by the shaded region. The solid gray lines show the model prediction including Poissonian stochastic noise and measurement noise; the dashed lines show the prediction without the contribution of Poissonian stochastic noise. (c) Unexplained variance ${\sigma }_u^2$ for $p\approx 1$ ($t_2=t_1+\tau$, darker points) and $p=0.60(2)$ [$t_2=t_1+95(5)\mathrm{ms}$, lighter points] for different integration times $\tau$. For $p\approx 1$, the unexplained variance corresponds to twice the reduced Allan variance (solid orange line). The bright orange line denotes the full Allan variance without subtracting the mean dynamics [see Fig. 1]. The gray tick indicates the integration time $\tau =4\mathrm{ms}$ that minimizes ${\sigma }_u^2$ and is used to produce the curves shown in (a) and (b). (d) Scaling of unexplained variance for fixed lost fraction $p=0.60(2)$ with initial atom number $N_1$. The constant measurement noise is subtracted from the data for each $n$, to compare different $n$ (color coded as in Fig. 2). The gray solid line denotes the linear Poissonian expectation ${\tilde{\sigma }}_u^2(N_1)=p(1-p)N_1$. The error bars denote one standard deviation.

Figure 5

Experimental sequence. Upper: heterodyne magnitude $S_{\mathrm{het}}$ with atoms present (not present) in orange (blue). The heterodyne magnitude signal is not recorded for times $t_s$ between 30 and 170 ms. Middle: Corresponding VCO frequency $f$. Bottom: Current in our atomic chip wires (red). Evaporation is induced by ramping up the current, which lowers the trap depth dynamically (see Fig. 7), and halted at 320 ms by ramping the current back down quickly. During evaporation (indicated by gray shading in all panels), the heterodyne magnitude is kept constant by a side-of-fringe lock. The atom-induced cavity shift is extracted from an in-loop measurement of the control voltage fed back to the VCO. At the end of an experimental run, a swept atom-number measurement is performed, where ${\delta }_{pc}$ is varied by sweeping $f$ down and up across cavity resonance. In these sweep measurements, the atom number is extracted from the shift of the cavity profile visible in the heterodyne magnitude signal and is used just for calibration purposes.

Figure 6

Temperature after evaporation. Temperature of the gas measured in time of flight after the final swept atom-number measurement for different intracavity photon numbers. The gray line is a linear fit guiding the eye.

Figure 7

Potential depth during forced evaporation. The trap depth of the combined potential of far off-resonant optical lattice trap, magnetic field, and probe-induced potential as the magnetic gradient is ramped up from zero to its maximal value. The inset shows the correction factor required to correct $g^2$ for the shift of the cloud relative to the maximal probe intensity. Different colors correspond to different intracavity photon numbers, with the color code the same as in Fig. 2.

Figure 8

Theoretical modeling of evaporation. (a) Simulation of atom-number dynamics during forced evaporation for the trap ramp shown in Fig. 7. We show 200 traces with initial atom numbers $N_0$ and temperatures $T_0$ sampled from Gaussian distributions in light color, for varying intracavity photon number $n$ as indicated in the top right corner. The atom-number samples have a mean $⟨N_0⟩$ and a standard deviation of $2{⟨N_0⟩}^{1/2}$. The temperature distribution is centered about the initial temperature $T_0$ and has a standard deviation of $\mathrm{\Delta }{T}_0=0.6\mu \mathrm{K}$. We include a three-body loss contribution scaled by $c_3=0.2$. The dark line shows the measured atom-number dynamics including shift correction. (b) Ratio of trap depth to temperature, $\eta =U/k_{B}T$, for the same ensemble as in (a). The initially broad distribution is rapidly narrowed down and then remains nearly constant during further evaporation. (c) Subtracted slope $a_s$ for $n=1.9$ for different temperature spreads (indicated in the top right corner) in the absence of three-body loss ($c_3=0$). We observe the same trends as observed in our experiment, with a fast rise for small $p$ and a decay with increasing initial time $t_1$ (indicated in the legend). The shaded region indicates the standard deviation extracted from a bootstrap analysis of the calculated samples. (d) Subtracted slope $a_s$ for $n=3.2$ and $\mathrm{\Delta }{T}_0=0.6\mu \mathrm{K}$ for increasing contribution of three-body loss characterized by the scaling factor $c_3$ (indicated in the top right corner). Three-body loss leads to a negative contribution to $a_s$, which suppresses the effect of initial temperature variation.