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Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas

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Science  09 Jan 2015:
Vol. 347, Issue 6218, pp. 167-170
DOI: 10.1126/science.1258676

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.

Science, this issue p. 167; see also p. 127

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 Embedded Image.

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 Embedded Image, 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, Embedded Image, where Embedded Image is the critical temperature and Embedded Image is the (static) correlation-length critical exponent. Importantly, the corresponding relaxation time Embedded Image, needed to establish a diverging Embedded Image, also diverges: Embedded Image, where Embedded Image 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 Embedded Image is reduced toward Embedded Image at a finite rate, beyond some point in time the correlation length can no longer adiabatically follow its diverging equilibrium value. Consequently, at time Embedded Image, the transition occurs without Embedded Image 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).

Fig. 1 Domain formation during spontaneous symmetry breaking in a homogeneous Bose gas.

(A) Red points depict thermal atoms and blue areas coherent domains, in which the Embedded Image gauge symmetry is spontaneously broken. The arrows indicate the independently chosen condensate phase at different points in space, and dashed lines delineate domains over which the phase is approximately constant. The average size Embedded Image of the domains formed at the critical point depends on the cooling rate. Further cooling can increase the population of each domain before the domain boundaries evolve. (B) Phase inhomogeneities in a deeply degenerate gas are revealed in TOF expansion as density inhomogeneities. Shown are three realizations of cooling the gas in 1 s from Embedded ImagenK, through Embedded ImagenK, to Embedded Image nK. Each realization of the experiment results in a different pattern, and averaging over many images results in a smooth, featureless distribution. (C) Preparing a Embedded ImagenK gas more slowly (over 5 s) results in an essentially pure BEC with a spatially uniform phase.

The main quantitative prediction of the KZ theory is that, under some generic assumptions (5), the average domain size Embedded Image follows a universal scaling law. The crucial KZ hypothesis is that in the nonadiabatic regime close to Embedded Image, the correlations remain essentially frozen. Then, for a smooth temperature quench, the theory predictsEmbedded Image (1)with the KZ exponentEmbedded Image (2)where Embedded Image is the quench time defined so that close to the transition Embedded Image, and Embedded Image and Embedded Image 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 (911), atomic Bose-Einstein condensates (BECs) (1218), multiferroics (19), and trapped ions (2022). 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 4He. For this class, mean-field (MF) theory predicts Embedded Image and Embedded Image, giving Embedded Image, whereas a beyond-MF dynamical critical theory, the so-called F model (2), gives Embedded Image and Embedded Image, so Embedded Image. We prepared a homogeneous Bose gas by loading 3 × 105 87Rb atoms into a cylindrical optical-box trap (23) of length L ≈ 26 μm along the horizontal x axis and radius R ≈ 17 μm. Initially, Embedded Image nK, corresponding to Embedded Image. We then evaporatively cooled the gas by lowering the trap depth, crossed Embedded Image nK with 2 × 105 atoms, and had Embedded Image atoms at Embedded Image nK (Embedded Image). In our system, Embedded Image is expected to be set by the thermal wavelength at the critical point, Embedded Imagem, and Embedded Image by the elastic scattering time Embedded Image (13, 24); for our parameters, a classical estimate gives Embedded Image 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 << Tc in Embedded Image s). In our finite-sized box, we can also produce essentially pure and fully coherent (single-domain) BECs, by cooling the gas slowly (over Embedded Image 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

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

Fig. 2 Two-point correlation functions in equilibrium and quenched gases.

(A) Homodyne interferometric scheme. The first Bragg-diffraction pulse (Embedded Image) creates a superposition of a stationary cloud and its copy moving with a center-of-mass velocity Embedded Image. After a time Embedded Image, a second pulse is applied. In the region where the two copies of the cloud displaced by Embedded Image overlap, the final density of the diffracted atoms depends on the relative phase of the overlapping domains; Embedded Image is deduced from the diffracted fraction Embedded Image (see text). (B) Correlation function Embedded Image measured in equilibrium (blue) and after a quench (red) for, respectively, two different Embedded Image values and two different quench times. (Inset) 1D calculation of Embedded Image for a fragmented BEC containing Embedded Image (red) and Embedded Image (light red) domains of random sizes and phases. The solid lines correspond to Embedded Image.

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

We now turn to a quantitative study of Embedded Image 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 Embedded Image. Specifically, for Embedded Image, which in our case holds for both MF theory and the F model, the freeze-out time of Embedded Image for Embedded Image is expected to be (4) Embedded Image (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 Embedded Image is still a subject of theoretical work (29). Practically, a crucial question is when one should measure Embedded Image 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 Embedded Image and Embedded Image.

Fig. 3 KZ scaling and freeze-out hypothesis.

(A) Quench protocols. The self-similar QP1 trajectories are shown in blue for total cooling time Embedded Image s (upper panel) and Embedded Image s (lower panel). We use polynomial fits to the data (solid lines) to deduce Embedded Image and Embedded Image. QP2 is shown in the lower panel by the orange points, with the kink at Embedded Image. (B) Coherence length Embedded Image as a function of Embedded Image. Blue points correspond to QP1. The shaded blue area shows power-law fits with Embedded Image to the data with Embedded Image s. The horizontal dotted line indicates our instrumental resolution. (C) Coherence length Embedded Image measured following QP2, as a function of Embedded Image, for Embedded Image s (orange), Embedded Image s (green), and Embedded Image s (purple). The shaded areas correspond to the essentially constant Embedded Image (and its uncertainty) in the freeze-out period Embedded Image. (For Embedded Image the system never unfreezes.) The (average) Embedded Image values within these plateaus are shown in their respective colors as diamonds in (B).

In the first quench protocol (QP1), we followed cooling trajectories such as shown in Fig. 3A and varied only the total cooling time Embedded Image. We restricted Embedded Image to values between Embedded Image and Embedded Image s, for which we observed that the cooling curves were self-similar (as seen in Fig. 3A). We always crossed Embedded Image nK at Embedded Image (vertical dashed line) and always had the same atom number (within Embedded Image) 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 Embedded Image 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 Embedded Image is simply proportional to Embedded Image. (For Embedded Image s, the evaporation is less efficient and the cooling trajectories are no longer self-similar.)

In Fig. 3B we plot Embedded Image versus Embedded Image, measured using QP1 (blue points). For Embedded Image s, we observed a slow power-law growth of Embedded Image, in good agreement with the expected KZ scaling (blue shaded area). However, for longer Embedded Image, this scaling breaks down and Embedded Image 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 Embedded Image and the end of cooling is Embedded Image, whereas the KZ freeze-out time is Embedded Image, so for slow enough quenches, Embedded Image inevitably exceeds Embedded Image. Hence, although it may be impossible to adiabatically cross Embedded Image, 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, Embedded Image s, we posit that for Embedded Image, we have Embedded Image and hence, from Eq. 5, more generally Embedded Image.

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 Embedded Image, but then at a variable “kink” time Embedded Image, 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 Embedded Image we can complete the cooling and measure Embedded Image before the system has time to unfreeze.

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

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

Fig. 4 Critical exponents of the interacting BEC transition.

Orange circles and diamonds show Embedded Image values obtained using QP2, as in Fig. 3C; the diamonds show the same three data points as in Fig. 3B. Blue circles show the same QP1 data, with Embedded Image s as in Fig. 3B. We obtained Embedded Image (solid line), in agreement with the F-model prediction Embedded Image, corresponding to Embedded Image and Embedded Image, and excluding the mean-field value Embedded Image.

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 4He. Whereas Embedded Image has been measured in both liquid helium [see (33)] and atomic gases (34), the dynamical exponent Embedded Image has presented a challenge to experiments [see (2, 35)]. Using the well-established Embedded Image and Eq. 2, we obtain Embedded Image. In contrast, MF theory does not provide a self-consistent interpretation of our results, as fixing Embedded Image yields an inconsistent Embedded Image. Interestingly, if we instead fix Embedded Image, which holds at both the MF and F-model levels, from Eq. 2 we obtain a slightly more precise Embedded Image and also recover Embedded Image.

It would be interesting to study the effect of tunable interactions in a homogeneous atomic gas on the value of Embedded Image. According to the Ginzburg criterion, near the critical point, MF breaks down for Embedded Image, where Embedded Image 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 Embedded Image if on approach to Embedded Image the KZ freeze-out occurs before MF breaks down, and the F value of Embedded Image if the reverse is true. In our experiments, Embedded Imagem and the freeze-out values of Embedded Image 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

  1. See the supplementary materials on Science Online.
  2. Near Embedded Image, 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 Embedded Image is uniform across the sample.
  3. The unfreezing and healing of the system reconcile the small Embedded Image and the fact that a fully coherent homogeneous BEC (as in Fig. 1C) can be produced in a relatively modest cooling time, Embedded Image. In harmonically trapped Bose gases, Embedded Image is observed (5, 15), which makes the physics significantly different. In that case, cooling with Embedded Image is sufficiently slow to directly produce essentially fully coherent BECs.
  4. Scaling Embedded Image to Embedded Image and Embedded Image to Embedded Image, for each data series separately, affects Embedded Image by <0.01.
  5. 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.
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