Observation of the Efimovian expansion in scale-invariant Fermi gases

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Science  22 Jul 2016:
Vol. 353, Issue 6297, pp. 371-374
DOI: 10.1126/science.aaf0666

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


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 (113). 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 (1623), with recent work confirming the geometric scaling of the energy spectrum (2427).

For a harmonic trapped gas, the expansion dynamics offers great insight to the property of the gas (2833). 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 Embedded Image, 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 Embedded Image just increases as Embedded Image. 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, Embedded Image displays a sequence of plateaus, which means that at a set of discrete times Embedded Image the cloud expansion stops, despite the continuous decreasing of the trap frequency. The locations of the plateaus Embedded Image obey a geometric scaling behavior.

Fig. 1 The schematic of the Efimovian expansion.

(A and B) A scale-invariant ultracold gas is first held in a harmonic trap with frequency Embedded Image. Then, starting from Embedded Image, the trap frequency starts to decrease as Embedded Image, and the cloud expands. (C) The theoretical predication of the Efimovian expansion: The cloud size Embedded Image as a function of time t follows a log-periodic function and exhibits a series of plateaus. The locations of the plateaus obey a geometric scaling law.

To explain these dynamics, we first point out why Embedded Image is special. For simplicity, we first consider a three-dimensional (3D) isotropic trap Embedded Image. In the absence of a trapping potential, the system is invariant under a scale transformation Embedded Image, 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 Embedded Image, the time-dependent Schrödinger equation exhibits a space-time scaling symmetry under the transformation Embedded Image and Embedded Image.

Defining the cloud size as Embedded Image, the equation-of-motion for Embedded Image can be derived as Embedded Image, where Embedded Image 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 Embedded Image, we conclude that the cloud size Embedded Image obeys the differential equation (see supplementary text S1):

Embedded Image (1)

In the experiment, we start with a finite initial trap frequency Embedded Image before turning it down (Fig. 1B). The system is at equilibrium for Embedded Image, and at Embedded Image, Embedded Image and Embedded Image for Embedded Image. 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 Embedded Image (The constants Embedded Image, Embedded Image, and Embedded Image are determined by the boundary conditions), where Embedded Image and Embedded Image are two linear independent solutions (see supplementary text S1) of

Embedded Image (2)

By replacing Embedded Image with Embedded Image and t with r and regarding Embedded Image 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. Embedded Image is a special point for Eq. 2. For Embedded Image, there are two independent solutions of Eq. 2, Embedded Image and Embedded Image, where Embedded Image; Embedded Image then takes a log-periodic formEmbedded Image(3)where Embedded Image is determined by the boundary condition at Embedded Image. Equation 3 clearly reveals the discrete scaling symmetry—i.e., when Embedded Image, Embedded Image, and Embedded Image for all the m-th order derivatives. Therefore, at time Embedded Image, the first- and second-order time derivatives for Embedded Image become zero and the cloud expansion is strongly suppressed, that is to say, the expansion dynamics shows a series of plateaus around each Embedded Image. A similar conclusion can also be obtained from the hydrodynamics expansion equations (34, 35). Note that Embedded Image is tunable by the speed of the decrease of the trap frequency Embedded Image. When Embedded Image, Embedded Image simply follows a power law as Embedded Image for Embedded Image, where Embedded Image. 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 6Li fermions in the lowest two hyperfine states Embedded Image and Embedded Image. 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 Embedded Image 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 Embedded Image, the trap potential is lowered as

Embedded Image (4)

Because Embedded Image 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 Embedded Image obeys the same form as Eq. 3, exceptEmbedded Image (5)where Embedded Image is a factor related to the breathing mode frequency, Embedded Image for the noninteracting gas, and Embedded Image 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 Embedded Image is controlled by the decrease rate of the laser intensity, with the initial axial trap depth always fixed at Embedded Image, where Embedded Image is the full trap potential. Thus, different Embedded Image corresponds to different Embedded Image, where Embedded Image is the initial axial trap frequency. Finally, after certain expansion time texp 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 ttof = 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 Embedded Image, from which we obtain Embedded Image. Embedded Image is related to the in situ cloud size by a scale factor Embedded Image via Embedded Image; Embedded Image can be obtained from either hydrodynamic or ballistic expansion equation with the time-of-flight time ttof (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 Embedded Image only gives a quantitative correction to the results. Figure 2 shows the typical measurements of Embedded Image with different Embedded Image for both the noninteracting and the unitary Fermi gases. For instance, for Embedded Image, we decrease the trap frequency from Embedded ImageHz to Embedded ImageHz within 8 ms. Dots are the measured data, and the solid and the dashed lines are both theoretical curves based on Eq. 3, taking Embedded Image as a fitting parameter or using Embedded Image given by Eq. 5, respectively. Because Embedded Image is obtained by a Gaussian fit to the density profile, Embedded Image, and thus the theoretical expression for Embedded Image 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.

Fig. 2 Experimental observation of the Efimovian expansion.

The mean axial cloud size Embedded Image (with Embedded Image) versus the expansion time Embedded Image for (A) a noninteracting Fermi gas of 6Li measured at B = 528 G and (B) a unitary Fermi gas measured at B = 832 G. Dots are measured data. Black, blue, and green dots denote Embedded Image, Embedded Image, and Embedded Image for (A), and Embedded Image, Embedded Image, and Embedded Image for (B). The dashed lines are the theory curves based on Eq. 3 (with Embedded Image given by Eq. 5) without any free parameters, and the solid lines are the best fit using the function form of Eq. 3, with Embedded Image as a fitting parameter. Red dots in both figures denote the case with Embedded Image, and the shaded area is the regime where expansion does not show discrete scaling symmetry. The inset in (B) shows three successive density profiles (after the time-of-flight) when the time texp is located inside a plateau, as indicated by the arrows. Error bars, mean ± SD.

For smaller Embedded Image, 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 Embedded Image, where the mean square of the cloud size follows a linear expansion as expected.

For the critical value Embedded Image (red dots in Fig. 2), no plateaus are observed within finite expansion time. How the plateaus disappears as Embedded Image could not be measured here. This is because as Embedded Image, Embedded Image 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 Embedded Image, the faster the trapping frequency drops and the shorter the expansion time, the plateaus could not be observed even before reaching the critical value Embedded Image 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 Embedded Image is different from Embedded Image, 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 Embedded Image relates to Embedded Image via Eq. 5. In the experiment, Embedded Image is determined by the trap frequencies measured by the parametric resonance, and Embedded Image is extracted from the best fit of the expansion data in Fig. 2. The universal relation between Embedded Image and Embedded Image is plotted in Fig. 3A. Embedded Image can fit very well with a linear function Embedded Image, which gives the slope Embedded Image for the noninteracting case and Embedded Image for the unitary Fermi gas. These are in good agreement with Embedded Image for the noninteracting case and Embedded Image for the unitary case. The Efimovian expansion is also robust and insensitive to the temperature and atom number of the Fermi gas (Fig. 3B).

Fig. 3 Universality of the Efimovian expansion.

(A) Embedded Image obtained from fitting the expansion curves v.s. Embedded Image. The solid lines are the linear fitting curves, and the dashed lines are Embedded Image, with Embedded Image for the noninteracting fermions and Embedded Image for the unitary Fermi gas. (B) For a given Embedded Image and for the unitary Fermi gas, Embedded Image is obtained from fitting the expansion curves for different fermion numbers and temperatures as indicated (Embedded Image is the Fermi temperature). The solid line is the theory value for the unitary Fermi gas, and the arrow indicates the theory value for the noninteracting Fermi gas with the same Embedded Image. Embedded Image as a function of Embedded Image for the noninteracting (red dots) and the unitary Fermi gas (blue dots) with Embedded Image in (C) and Embedded Image in (D). Error bars, mean ± SD.

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

Finally, we study a time-reversed compression process. Consider an expansion process from Embedded Image to Embedded Image, where the trap frequency decreases from Embedded Image to Embedded Image. Now we consider an inverted process of increasing the trap frequency as Embedded Image, where the trap frequency increases from Embedded Image to Embedded Image when t changes from Embedded Image to Embedded Image. For the compression dynamics to really invert the expansion dynamics, Embedded Image has to be carefully chosen to satisfy Embedded Image. 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.

Fig. 4 Time-reversal symmetry of the Efimovian expansion.

Embedded Image for the expansion and its inverted compression process from Embedded Image to Embedded Image. Embedded Image. Black dots are the expansion process, with Embedded Image and the frequency changing from Embedded Image (at Embedded Image) to Embedded Image (at Embedded Image). Blue dots are the inverted compression process, with Embedded Image and the frequency changing from Embedded Image (at Embedded Image) to Embedded Image (at Embedded Image). Here, Embedded Image and the data are taken in the unitary regime. Error bars, mean ± SD.

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 (3639). 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 (4042).

Supplementary Materials

Materials and Methods

Supplementary Text

Figs. S1 to S3

Table S1

References (43, 44)

References and Notes

  1. 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.
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