## Breaking the symmetry in an atomic gas

Cooling a physical system through a phase transition typically makes it less symmetrical. If the cooling is done very slowly, this symmetry change is uniform throughout the system. For a faster cooling process, the system breaks up into domains: The faster the cooling, the smaller the domains. Navon *et al.* studied this process in an ultracold gas of Rb atoms near its transition to a condensed state (see the Perspective by Ferrari). The authors found that the size of the domains froze in time in the vicinity of the transition and that it depended on the cooling speed, as predicted by theory.

## Abstract

Kibble-Zurek theory models the dynamics of spontaneous symmetry breaking, which plays an important role in a wide variety of physical contexts, ranging from cosmology to superconductors. We explored these dynamics in a homogeneous system by thermally quenching an atomic gas with short-range interactions through the Bose-Einstein phase transition. Using homodyne matter-wave interferometry to measure first-order correlation functions, we verified the central quantitative prediction of the Kibble-Zurek theory, namely the homogeneous-system power-law scaling of the coherence length with the quench rate. Moreover, we directly confirmed its underlying hypothesis, the freezing of the correlation length near the transition. Our measurements agree with a beyond-mean-field theory and support the expectation that the dynamical critical exponent for this universality class is .

Continuous symmetry-breaking phase transitions are ubiquitous, from the cooling of the early universe to the λ transition of superfluid helium. Near a second-order transition, critical long-range fluctuations have a diverging correlation length , and details of the short-range physics are largely unimportant. Consequently, all systems can be classified into a small number of universality classes, according to their generic features such as symmetries, dimensionality, and range of interactions (*1*). Close to the critical point, many physical quantities exhibit power-law behaviors governed by critical exponents characteristic of a universality class. Specifically, for a classical phase transition, , where is the critical temperature and is the (static) correlation-length critical exponent. Importantly, the corresponding relaxation time , needed to establish a diverging , also diverges: , where is the dynamical critical exponent (*2*). An elegant framework for understanding the implications of this critical slowing down for the dynamics of symmetry breaking is provided by the Kibble-Zurek (KZ) theory (*3*, *4*).

Qualitatively, as is reduced toward at a finite rate, beyond some point in time the correlation length can no longer adiabatically follow its diverging equilibrium value. Consequently, at time , the transition occurs without ever having reached the size of the whole system. This results in the formation of finite-sized domains that display independent choices of the symmetry-breaking order parameter (Fig. 1A). [At the domain boundaries, rare long-lived topological defects can also form (*5*), their nature and density depending on the specific physical system.] Such domain formation was discussed in a cosmological context and linked to relativistic causality (*3*), whereas the connection to laboratory systems, critical slowing down, and universality classes was made in (*4*).

The main quantitative prediction of the KZ theory is that, under some generic assumptions (*5*), the average domain size follows a universal scaling law. The crucial KZ hypothesis is that in the nonadiabatic regime close to , the correlations remain essentially frozen. Then, for a smooth temperature quench, the theory predicts
(1)with the KZ exponent
(2)where is the quench time defined so that close to the transition , and and are a system-specific microscopic length scale and time scale, respectively.

Signatures of KZ physics have been observed in a wide range of systems, including liquid crystals (*6*), liquid helium (*7*, *8*), superconductors (*9*–*11*), atomic Bose-Einstein condensates (BECs) (*12*–*18*), multiferroics (*19*), and trapped ions (*20*–*22*). However, despite this intense activity, a direct quantitative comparison with Eqs. 1 and 2 has remained elusive; some common complications include system inhomogeneity, modified statistics of low-probability defects, and uncertainties over the nature of the transition being crossed [for a recent review, see (*5*)]. In this work, we studied the dynamics of spontaneous symmetry breaking in a homogeneous atomic Bose gas, which is in the same universality class as the three-dimensional (3D) superfluid ^{4}He. For this class, mean-field (MF) theory predicts and , giving , whereas a beyond-MF dynamical critical theory, the so-called F model (*2*), gives and , so . We prepared a homogeneous Bose gas by loading 3 × 10^{5} ^{87}Rb atoms into a cylindrical optical-box trap (*23*) of length *L* ≈ 26 μm along the horizontal *x* axis and radius *R* ≈ 17 μm. Initially, nK, corresponding to . We then evaporatively cooled the gas by lowering the trap depth, crossed nK with 2 × 10^{5} atoms, and had atoms at nK (). In our system, is expected to be set by the thermal wavelength at the critical point, m, and by the elastic scattering time (*13*, *24*); for our parameters, a classical estimate gives ms.

Qualitatively, random phase inhomogeneities in rapidly quenched clouds are revealed in time-of-flight (TOF) expansion as density inhomogeneities (*14*, *25*), such as shown in Fig. 1B (here the gas was cooled to *T* << *T*_{c} in s). In our finite-sized box, we can also produce essentially pure and fully coherent (single-domain) BECs, by cooling the gas slowly (over s). In TOF, such a BEC develops the characteristic diamond shape (*26*) seen in Fig. 1C.

To quantitatively study the coherence of our clouds, we probed the first-order two-point correlation function

(3)where is the Bose field. Our method (Fig. 2A) is inspired by (*27*). We use a short (0.1 ms) Bragg-diffraction light pulse to create a small copy of the cloud (containing of the atoms) moving along the *x* axis with recoil velocity mm/s (*26*). A second identical pulse is applied a time later, when the two copies are shifted by and for still partially overlap. This results in interference of the two displaced copies of the cloud in the overlap region of length . After the second Bragg pulse, the fraction of diffracted atoms (for ) is (*28*)
(4)where is the correlation function corresponding to periodic boundary conditions and normalized so that , and is the area of each Bragg pulse (in our case ). Allowing the recoiling atoms to fully separate from the main cloud (in 140 ms of TOF) and counting and , we directly measured , with a spatial resolution of m. Our resolution was limited by the duration of the Bragg pulses and the (inverse) recoil momentum; we experimentally assessed it by measuring in a thermal cloud with a thermal wavelength <0.5 μm.

In Fig. 2B we show examples of functions measured in equilibrium (blue) and after a quench (red). In an essentially pure equilibrium BEC (prepared slowly, as for Fig. 1C), and is simply given by the triangular function (dark blue solid line). In equilibrium at , we see a fast initial decay of , reflecting the significant thermal fraction. However, importantly, the coherence still spans the whole system, with the slope of the long-ranged part of giving the condensed fraction (light blue line is a guide to the eye). By comparison, the functions for quenched clouds clearly have no equilibrium interpretation. Here , corresponding to a phase space density >25, and yet coherence extends over only a small fraction of . These data are fitted well by (red lines), which provides a simple and robust way to extract the coherence length. This exponential form is further supported by a 1D calculation shown in the inset of Fig. 2B. Here we generated a wave function with a fixed number of domains , randomly positioning the domain walls and assigning each domain a random phase. Averaging over many realizations, we obtained that is fitted very well by an exponential with . (In our 3D experiments, the total number of domains was and was effectively averaged over 1D distributions.)

We now turn to a quantitative study of for different quench protocols (Fig. 3). For the KZ scaling law of Eqs. 1 and 2 to hold, a crucial assumption is that the correlation length is essentially frozen in time near . Specifically, for , which in our case holds for both MF theory and the F model, the freeze-out time of for is expected to be (*4*)
(5)where *f* is a dimensionless number of order unity. Although intuitively appealing, this assumption is in principle only approximative, and the dynamics of the system coarsening (i.e., merging of the domains) at times is still a subject of theoretical work (*29*). Practically, a crucial question is when one should measure in order to verify the universal KZ scaling. We resolved these issues by using two different quench protocols outlined in Fig. 3A, which allow us both to observe the KZ scaling and to directly verify the freeze-out hypothesis, without an a priori knowledge of the exact values of and .

In the first quench protocol (QP1), we followed cooling trajectories such as shown in Fig. 3A and varied only the total cooling time . We restricted to values between and s, for which we observed that the cooling curves were self-similar (as seen in Fig. 3A). We always crossed nK at (vertical dashed line) and always had the same atom number (within ) at the end of cooling. The self-similarity of the measured cooling trajectories and the essentially constant evaporation efficiency indicate that for this range of values, the system is always sufficiently thermalized, the temperature (as determined from the thermal wings in TOF) is well defined during the quench (*30*), and to a good approximation is simply proportional to . (For s, the evaporation is less efficient and the cooling trajectories are no longer self-similar.)

In Fig. 3B we plot versus , measured using QP1 (blue points). For s, we observed a slow power-law growth of , in good agreement with the expected KZ scaling (blue shaded area). However, for longer , this scaling breaks down and grows faster, quickly approaching the system size. Importantly, this breakdown can also be fully understood within the KZ framework. We note that the time between crossing and the end of cooling is , whereas the KZ freeze-out time is , so for slow enough quenches, inevitably exceeds . Hence, although it may be impossible to adiabatically cross , in practice the system can unfreeze and heal significantly before it is observed (*31*). From the point where the KZ scaling breaks down in Fig. 3B, s, we posit that for , we have and hence, from Eq. 5, more generally .

To verify this picture, we employed a second quench protocol (QP2), which involved two cooling steps, as shown by the orange points in the bottom panel of Fig. 3A. We initially followed the QP1 trajectory for a given , but then at a variable “kink” time , we accelerated the cooling; the last part of the trajectory always corresponds to the final portion of our fastest, 0.2-s cooling trajectory. This way, even for we can complete the cooling and measure before the system has time to unfreeze.

In Fig. 3C, the orange points show the QP2 measurements of for s and various values of the kink position . These data reveal two notable facts. First, for a broad range of values, is indeed constant (within errors), and the width of this plateau agrees with our estimate s for s, indicated by the horizontal arrow. Second, the value of within the plateau region falls in line with the KZ scaling law in Fig. 3B. We also show analogous QP2 measurements for s (green) and s (purple); in these cases, , so is longer than , the system never unfreezes, and thus the acceleration of the cooling has no effect on . These results provide direct support for the KZ freeze-out hypothesis.

To accurately determine the KZ exponent , we made extensive measurements following QP2, extracting from the plateaued regions of width , as in Fig. 3C. In Fig. 4, we combine these data with the QP1 measurements for s, and plot versus . The plotted values of and their uncertainties include the small systematic variation of the derivative of our cooling trajectory between and . We finally obtain , which strongly favors the F-model prediction over the MF value (*32*).

Having observed excellent agreement with the KZ theory, we discuss the implications of our measurements for the critical exponents of the interacting BEC phase transition, which is in the same universality class as the λ transition of ^{4}He. Whereas has been measured in both liquid helium [see (*33*)] and atomic gases (*34*), the dynamical exponent has presented a challenge to experiments [see (*2*, *35*)]. Using the well-established and Eq. 2, we obtain . In contrast, MF theory does not provide a self-consistent interpretation of our results, as fixing yields an inconsistent . Interestingly, if we instead fix , which holds at both the MF and F-model levels, from Eq. 2 we obtain a slightly more precise and also recover .

It would be interesting to study the effect of tunable interactions in a homogeneous atomic gas on the value of . According to the Ginzburg criterion, near the critical point, MF breaks down for , where is the s-wave scattering length. It is tempting to combine the (dynamical) KZ and (equilibrium) Ginzburg arguments and speculate that one should observe the MF value of if on approach to the KZ freeze-out occurs before MF breaks down, and the F value of if the reverse is true. In our experiments, m and the freeze-out values of are systematically higher. However, with use of a Feshbach resonance, the opposite regime should be within reach. Another interesting study could focus on the dynamics of domain coarsening. Finally, our methods could potentially be extended to studies of higher-order correlation functions and the full statistics of the domain sizes.

## Supplementary Materials

## References and Notes

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See the supplementary materials on
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- ↵ Near , the mean free path for classical elastic collisions is about four times larger than the size of the box. Hence, although the evaporation takes place at the box walls, we can also safely assume that is uniform across the sample.
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The unfreezing and healing of the system reconcile the small and the fact that a fully coherent homogeneous BEC (as in Fig. 1C) can be produced in a relatively modest cooling time, . In harmonically trapped Bose gases, is observed (
*5*,*15*), which makes the physics significantly different. In that case, cooling with is sufficiently slow to directly produce essentially fully coherent BECs. - ↵ Scaling to and to , for each data series separately, affects by <0.01.
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**Acknowledgments:**We thank M. Robert-de-Saint-Vincent for experimental assistance; R. Fletcher for comments on the manuscript; and N. Cooper, J. Dalibard, G. Ferrari, B. Phillips, and W. Zwerger for insightful discussions. This work was supported by AFOSR, ARO, DARPA OLE, and EPSRC (grant no. EP/K003615/1). N.N. acknowledges support from Trinity College, Cambridge, and R.P.S. from the Royal Society.