## Steps to ultracold gas expansion

Cold atomic gases are often studied while confined in parabolic traps, with the largest atomic density at the center of the trap. When the trap is made shallower, the gas radially expands as the energy cost for atoms that are farther from the trap center decreases. Deng *et al.* observed an interesting effect when they reduced the characteristic frequency of the parabolic trap so that it was at any moment inversely proportional to the elapsed time. Instead of expanding continuously, a strongly interacting Fermi gas held in such a trap stalled at certain time points. These time points formed a geometric progression, a consequence of scale invariance in the strongly interacting limit.

*Science*, this issue p. 371

## Abstract

Scale invariance plays an important role in unitary Fermi gases. Discrete scaling symmetry manifests itself in quantum few-body systems such as the Efimov effect. Here, we report on the theoretical prediction and experimental observation of a distinct type of expansion dynamics for scale-invariant quantum gases. When the frequency of the harmonic trap holding the gas decreases continuously as the inverse of time *t*, the expansion of the cloud size exhibits a sequence of plateaus. The locations of these plateaus obey a discrete geometric scaling law with a controllable scale factor, and the expansion dynamics is governed by a log-periodic function. This marked expansion shares the same scaling law and mathematical description as the Efimov effect.

Interaction between dilute ultracold atoms is described by the s-wave scattering length. For a spin-1/2 Fermi gas, when the scattering length diverges at a Feshbach resonance, there is no length scale other than the interparticle spacing in this many-body system, and therefore the system, known as the unitary Fermi gas, becomes scale invariant. The spatial scale invariance leads to universal thermodynamics and transport properties, as revealed by many experiments (*1*–*13*). On the other hand, in a boson system with an infinite scattering length, three-body bound states can form, where the extra length scale of the three-body parameter turns the continuous scaling symmetry into a discrete scaling symmetry and gives rise to an infinite number of three-body bound states whose energies obey a geometric scaling symmetry. This so-called Efimov effect (*14*, *15*) has been observed in cold atom experiments (*16*–*23*), with recent work confirming the geometric scaling of the energy spectrum (*24*–*27*).

For a harmonic trapped gas, the expansion dynamics offers great insight to the property of the gas (*28*–*33*). Here, we consider what happens to a scale-invariant quantum gas held in a harmonic trap when the trap is gradually opened up by decreasing the trap frequency ω as , where λ is a constant and *t* is time (Fig. 1, A and B). Naïvely, by dimensional analysis, one would expect that the cloud size just increases as . Here we show, both theoretically and experimentally, that when λ is smaller than a critical value, the expansion dynamics displays a discrete scaling symmetry in the time domain. As a function of *t*, displays a sequence of plateaus, which means that at a set of discrete times the cloud expansion stops, despite the continuous decreasing of the trap frequency. The locations of the plateaus obey a geometric scaling behavior.

To explain these dynamics, we first point out why is special. For simplicity, we first consider a three-dimensional (3D) isotropic trap . In the absence of a trapping potential, the system is invariant under a scale transformation , whereas in the presence of a static harmonic trap, the fixed harmonic length introduces an additional length scale that breaks this spatial scale invariance. Nevertheless, if ω changes as , the time-dependent Schrödinger equation exhibits a space-time scaling symmetry under the transformation and .

Defining the cloud size as , the equation-of-motion for can be derived as , where is the generator of a spatial scaling transformation. Using the fact that the system is scale invariant, and by taking higher-order time derivatives of , we conclude that the cloud size obeys the differential equation (see supplementary text S1):

(1)In the experiment, we start with a finite initial trap frequency before turning it down (Fig. 1B). The system is at equilibrium for , and at , and for . This sets a boundary condition for Eq. 1 that can turn its continuous scaling symmetry in the time domain into a discrete one.

The solution of Eq. 1 can be generally written in a form as (The constants , , and are determined by the boundary conditions), where and are two linear independent solutions (see supplementary text S1) of

(2)By replacing with and *t* with *r* and regarding as a real wave function and *r* as the hyper-radius, Eq. 2 becomes the zero-energy Schrödinger equation for the Efimov effect in the hyperspherical coordinate (*14*, *15*). This reveals a connection between this dynamical expansion and the Efimov problem. is a special point for Eq. 2. For , there are two independent solutions of Eq. 2, and , where ; then takes a log-periodic form(3)where is determined by the boundary condition at . Equation 3 clearly reveals the discrete scaling symmetry—i.e., when , , and for all the *m*-th order derivatives. Therefore, at time , the first- and second-order time derivatives for become zero and the cloud expansion is strongly suppressed, that is to say, the expansion dynamics shows a series of plateaus around each . A similar conclusion can also be obtained from the hydrodynamics expansion equations (*34*, *35*). Note that is tunable by the speed of the decrease of the trap frequency . When , simply follows a power law as for , where . A detailed comparison between this expansion and the Efimov effect is summarized in table S1. We will refer to this effect as the Efimovian expansion.

In our experiment, we use a balanced mixture of ^{6}Li fermions in the lowest two hyperfine states and . Fermionic atoms are loaded into a cross-dipole trap to perform evaporative cooling. The resulting potential has a cylindrical symmetry around the *z* axis, and the trap anisotropic frequency ratio is about 9. The above theoretical considerations hold for the isotropic case, but similar results can be obtained for an anisotropic trap (see supplementary text S2). Starting at the initial time , the trap potential is lowered as

Because the effect is more pronounced along the axial direction than in the transverse direction. Therefore, hereafter we focus on the cloud expansion along the axial direction. Theory shows (see supplementary text S2) that the axial cloud square size obeys the same form as Eq. 3, except (5)where is a factor related to the breathing mode frequency, for the noninteracting gas, and for the unitary Fermi gas along the axial direction. A Feshbach resonance is used to tune the interaction of the atoms either to the noninteracting regime with the magnetic field *B* = 528 G or to the unitary regime with *B* = 832 G. The trap frequency is lowered by decreasing the laser intensity, and is controlled by the decrease rate of the laser intensity, with the initial axial trap depth always fixed at , where is the full trap potential. Thus, different corresponds to different , where is the initial axial trap frequency. Finally, after certain expansion time *t*_{exp} with the trap, the trap is completely turned off and the cloud is probed by standard resonant absorption imaging techniques after a time-of-flight expansion time *t*_{tof} = 200 μs. Each data point is an average of five shots of the measurements at identical parameters.

The time-of-flight density profile along the axial direction is fitted by a Gaussian function as , from which we obtain . is related to the in situ cloud size by a scale factor via ; can be obtained from either hydrodynamic or ballistic expansion equation with the time-of-flight time *t*_{tof} (see supplementary text S5). Because the trap is quite anisotropic, the cloud expands slowly along the axial direction during a short time-of-flight, and the expansion factor only gives a quantitative correction to the results. Figure 2 shows the typical measurements of with different for both the noninteracting and the unitary Fermi gases. For instance, for , we decrease the trap frequency from Hz to Hz within 8 ms. Dots are the measured data, and the solid and the dashed lines are both theoretical curves based on Eq. 3, taking as a fitting parameter or using given by Eq. 5, respectively. Because is obtained by a Gaussian fit to the density profile, , and thus the theoretical expression for is simply a square root of Eq. 3. Figure 2 clearly shows the plateaus for the expansion dynamics and an excellent agreement between theory and experiment. Density profiles for three successive measurement times inside a plateau almost perfectly overlap with each other (Fig. 2B, inset), which confirms that the expansion stops at the plateau.

For smaller , the trap frequency decreases slower, the plateaus become denser, and the difference in height between two adjacent plateaus becomes smaller. The adiabatic limit is reached for , where the mean square of the cloud size follows a linear expansion as expected.

For the critical value (red dots in Fig. 2), no plateaus are observed within finite expansion time. How the plateaus disappears as could not be measured here. This is because as , decreases toward zero, the period increases exponentially, and therefore even the first plateau would appear after a very long expansion time. On the other hand, there is a lower limit for the trap frequency below which atoms cannot be trapped. Together with the fact that the larger the , the faster the trapping frequency drops and the shorter the expansion time, the plateaus could not be observed even before reaching the critical value within the finite expansion time. Nevertheless, for comparison, we have performed measurements where the trapping frequency decreases with similar average speeds in Fig. 2, but the time dependence of is different from , which breaks the aforementioned spatial-time scaling symmetry. The plateaus are indeed not observed in the expansion (fig. S2).

We now demonstrate that these dynamics are universal. First, we should verify that relates to via Eq. 5. In the experiment, is determined by the trap frequencies measured by the parametric resonance, and is extracted from the best fit of the expansion data in Fig. 2. The universal relation between and is plotted in Fig. 3A. can fit very well with a linear function , which gives the slope for the noninteracting case and for the unitary Fermi gas. These are in good agreement with for the noninteracting case and for the unitary case. The Efimovian expansion is also robust and insensitive to the temperature and atom number of the Fermi gas (Fig. 3B).

Second, we notice that the noninteracting and the unitary cases only differ in the relation between and , and once is given to be the same, the dynamics are exactly identical for these two different systems (Fig. 3, C and D). In other words, is a function of (or ) and is a universal function for all scale-invariant systems.

Finally, we study a time-reversed compression process. Consider an expansion process from to , where the trap frequency decreases from to . Now we consider an inverted process of increasing the trap frequency as , where the trap frequency increases from to when *t* changes from to . For the compression dynamics to really invert the expansion dynamics, has to be carefully chosen to satisfy . We perform such an experiment (Fig. 4) showing that the dynamical process with a carefully chosen boundary is time-reversal symmetric. The small asymmetry arises because the lowering of the trap during expansion (black dots) causes evaporative cooling, which decreases cloud sizes correspondingly.

Our results are universal for all scale-invariant quantum gases. Future experiments can test them with a Tonks gas in 1D and in a 2D quantum gas, where the deviation from the log-periodic behavior can be used to calibrate the scaling symmetry anomaly in 2D (*36*–*39*). In the 3D case, it will be interesting to investigate the scaling symmetry breaking when the system is tuned away from the scale-invariant points of zero and infinite s-wave scattering length. The study could also be generalized to observe a dynamic analogy of a recently proposed super-Efimov effect (*40*–*42*).

## Supplementary Materials

www.sciencemag.org/content/353/6297/371/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S3

Table S1

## References and Notes

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**Acknowledgments:**We thank P. Zhang for helpful discussions. This research is supported by the National Natural Science Foundation of China (NSFC) (grant nos. 11374101 and 91536112) and the Shu Guang project (14SG22) of Shanghai Municipal Education Commission and Shanghai Education Development Foundation. Z.S. and H.Z. are supported by MOST (grant no. 2016YFA0301604), Tsinghua University Initiative Scientific Research Program and NSFC grant no. 11325418. R.Q. is supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China under grant no. 15XNLF18 and no. 16XNLQ03.