Kibble-Zurek scaling and its breakdown for spontaneous generation of Josephson
vortices in Bose-Einstein condensates
Atomic Bose-Einstein condensates confined to a dual-ring trap support Josephson vortices as topologically stable defects in the relative phase. We propose a test of the scaling laws for defect formation by quenching a Bose gas to degeneracy in this geometry. Stochastic Gross-Pitaevskii simulations reveal a power-law scaling of defect number with quench time for fast quenches, consistent with the Kibble-Zurek mechanism. Slow quenches show stronger quench-time dependence that is explained by the stability properties of Josephson vortices, revealing the boundary of the Kibble-Zurek regime. Interference of the two atomic fields enables clear long-time measurement of stable defects, and a direct test of the Kibble-Zurek mechanism in Bose-Einstein condensation.
A possible mechanism for the formation of domain structures in the early universe was proposed by Kibble Kibble (). He argued that the Universe cooled down after the hot Big Bang event and subsequently passed through a symmetry breaking phase transition at a critical temperature . Causally unconnected spatial domains settling into different vacua would lead to the formation of defects like domain walls, monopoles, strings, textures, etc Vilenkin94 (). Due to thermal fluctuations thwarting the emerging order, it was postulated that the number of defects eventually settled at the so-called Ginzburg temperature .
Later Zurek Zurek1 () put forward an alternative argument focusing on the nonequilibrium aspect of the phase transition. The density of the defects is determined at the critical temperature instead and its number is scaled with the quench rate. The scaling exponent depends on the critical exponents of the underlying phase transition. This scenario, known as the Kibble-Zurek mechanism (KZM) should equally apply to condensed matter phase transitions accessible to laboratory experiments Kibble07 (). The KZM proved to be robust and was verified by a number of recent experiments on annular Josephson tunnel junctions Monaco02 (); Monaco03 (); Monaco06 (); Monaco06prl () and theoretical research on Bose-Einstein condensates (BEC) Zurek09 (); Das (); Damski (); delCampo (). It also extends to quantum phase transitions Dziarmaga08 (); Sabbatini11 (); Chen11 ().
In the spirit of Kibble’s argument one might expect the KZM to fail in the limit of slow quenches where the time scale of other processes occurring in the system dominates over the quench time. Deviations from KZM predictions were observed in He experiments HeExpt () but the interpretation was controversial Rivers () and a manifestation of the Ginzburg temperature was ruled out in Ref. Bettencourt (). So far the transition between the regime of KZ scaling and its breakdown has not been studied systematically.
In this Letter we investigate the robustness of the KZ scaling in a system where departure from it can be understood in detail because the defects are easily quantified and are stable at the end of the quench. This avoids the difficulty of counting the decaying population of defects Weiler08 (); Damski () or their remnants Sikivie82 (). To this end, we study two linearly coupled quasi-1D atomic Bose gases in the ring configuration, as in Fig. 1. A quench through the Bose-Einstein condensation phase transition can generate Josephson vortices (JVs) confined between two BECs Kaurov05 (); Brand1 (). We show that the number of JVs obeys the KZ scaling law for fast quenches. On the contrary, for slow quenches, the predicted behavior deviates substantially and we observe a much stronger quench-time dependence than expected for critical phenomena in our simulations. This is due to decay processes occurring before the topological stability is established, in analogy to Kibble’s arguments.
The system under study can be realized by crossing a vertical Gaussian-Laguerre laser beam and two horizontal sheet beams Campbell () to form an optical dipole trap or with rf-dressing on an atom chip Fernholz (). Another way is trapping the atoms with two hyperfine states coupled via Raman transitions Sidorov () in a single ring trap Campbell (). Along the -axis the trapping potential can be treated as a double-well potential as shown in Fig. 1. Assuming tight confinement, the transverse motion can be eliminated. The resulting coupled Gross-Pitaevskii equations for the order parameter and in each ring assume the dimensionless form
where (), is the chemical potential, and the tunneling energy. Length, time, and energy are scaled by , , and , respectively, where is the atomic mass and is the transverse trapping frequency. Accordingly, the dimensionless non-linear interaction strength is related to the s-wave scattering length by .
Equation (1) supports topological and non-topological defects in the form of the JV and the dark soliton (DS), respectively, , where , and the scaling and has been applied. Both the DS with and and the JV with and for are localised excitations on the length scale above the vacuum where Kaurov05 (). The DS, where both components have identical profiles, is non-topological because it can continuously deform to the vacuum by a family of moving “grey” solitons with decreasing energy PSbook (). Although they may be present transiently during quenches through the phase transition, DSs will thus not survive the final stage of cooling. Furthermore, for , DSs are dynamically unstable with respect to decay into JVs, which have lower energy Qadir12 (). The stability properties of the JV, on the other hand, depend on the dimensionless parameter and may change during the quench. The JV bifurcates from the DS at as a time-reversal symmetry broken state (vortex and anti-vortex) with a characteristic phase winding of around a point located between the two rings (see Fig. 1), and only exists for smaller values of . From numerical simulations it is known that JVs can move with respect to the background BEC, although explicit solutions are unknown. For variational arguments indicate that the JV is energetically unstable Qadir12 (). For where the JV resembles the Sine-Gordon soliton Kaurov05 (); Brand1 (), the stationary solution is a metastable local energy minimum, since the energy increases with velocity. Thus, at sufficiently small , JVs are topologically stable, enabling experimental tests of the KZ scaling by counting the number of JVs at the end of quench in a dual-ring BEC. The defects would be immediately evident by the interference images of two expanding atomic fields. The situation is strikingly different from a single 1D BEC where the KZ scaling law was predicted to govern a transient population of eventually decaying DSs, which makes experimental detection more difficult Damski ().
where is the growth rate and is the thermal noise satisfying the fluctuation-dissipation relation , with being the temperature in units of . At the mean field equilibrium level the phase transition is described by the ground state of the energy , where we seek the minimum of the potential for . The symmetry imposes a common amplitude for the ground state fields. Taking , , and , the minimum of occurs at , , for . At the critical point the minimum is independent of and each field breaks symmetry.
The transition to the broken symmetry phase is simulated via Eq. (2) with time-dependent chemical potential
where is the quench time. The quench starts from a thermal gas with a chemical potential , and ceases in the Bose-condensed phase at . Due to inter-ring coupling, acts as the effective chemical potential; the precise location of the transition in a dynamical quench must be determined numerically. We evaluate the total number of JVs during the quench with . The net number is the difference between the number of clockwise and anti-clockwise vortices.
The KZ theory applied to the BEC phase transition gives the relaxation time and healing length close to the critical point as
where for our system . When exceeds , localized phase-domains start to grow in each ring. Typically a piece of the (anti-)vortex will fall within a -sized domain in which the phase is chosen randomly. Therefore, for small , in a ring with circumference , the total number of JVs is estimated to be
and thus obeys the power-law scaling with quench time. The number of JVs thus shows a stronger quench-time dependence than the winding number of a single-ring BEC, which was predicted to scale with Das ().
We consider a gas of Rb atoms with a transverse confining frequency of Hz. We numerically integrate Eq. (2) with , , and , which are realistic parameters with the set-up of Ref. Henderson (). The scaling in Eq. (6) is verified by averaging over 500 trajectories for and . The value of is chosen to be sufficiently large that the resulting defect number is independent of it. As shown in Fig. 2, the results for fast quenches compare favorably with the KZM prediction, yet the number of JVs deviates from the KZ scaling for slow quenches.
The stability of a JV depends on conditions that change during the quench. According to the KZM, two different regimes exist: For early and late times during the quench, relaxation is efficient and fluctuations in the Bose gas follow the changing chemical potential adiabatically. However, when the diverging relaxation time of Eq. (4) exceeds the time scale of the quench , fluctuations transiently freeze out and the system enters the impulse regime. This occurs when
giving the freeze-out timescale . At the following impulse-adiabatic transition the frozen fluctuations are imprinted onto the forming BEC. We thus expect that the stability properties of defects formed at this transition point determine their survival during the adiabatic phase of the quench.
In Fig. 3, the impulse-adiabatic transition can be clearly observed from the particle number, . A rapid increase of particle number takes place at , with . The value of appears to depend weakly on the details of the system, including the parameters and , consistent with the theoretical argument that is relevant for the quench dynamics. While the particle number is small in the impulse regime, it follows the dashed linear time-dependence in the adiabatic regime. Similar behavior was observed for a single ring BEC Das (). Therefore we can predict the chemical potential at the impulse-adiabatic transition as:
We denote the critical ratio of tunneling to chemical potential for JV stability by , where is the stabilizing chemical potential for given . As shown in Fig. 4(a), the defects are frozen in until for fast quenches ensuring the topological protection of JVs and hence the KZM signature. However, for slow quenches the impulse regime terminates earlier with , which causes the decay of the JVs in the shaded region in Fig. 4 until the topological stability of JV is established at . Although the critical for a moving JV at finite temperatures is unknown, we can estimate from the numerical simulations, at the point where KZ scaling breaks down. From Eq. (8) we obtain the criterion for obtaining stable JVs
The value is obtained from the data for , which suggests for , as shown by the vertical dashed line plotted in Fig. 2(b). This prediction agrees with the numerical data very well. The critical quench time depends on the growth rate through and we have also verified the prediction of Eq. (9) at different growth rates. In the slow quench regime, as seen in Fig. 2, the defect number falls off more rapidly with quench time than expected from the KZM. Since the variation is far from linear, we do not expect to enter a new regime of power-law scaling for slow quenches. Note that slow quenches show the same knee structure characterizing the impulse-adiabatic transition as the fast quenches that lead to KZ scaling (Fig. 3). We have also verified that continues to satisfy the KZ scaling for slow quenches (solid line in Fig. 2), when counted immediately after the impulse-adiabatic transition at . This supports our argument that the reduced defect number is due to thermal decay processes happening after the transition, and that defect formation is unaffected by thermal fluctuations during freeze-out. Moreover, by varying the circumference, we verify that the KZM departure is not due to the finite-size effects discussed in Ref. Weir (); Biroli (); zurek96 ().
For JVs, the slow quench regime is similar to Kibble’s idea Kibble (), where thermal fluctuations suffice to destroy the emerging order before the the system reaches the Ginzburg temperature. We observe a Ginzburg-like regime where thermal effects destroy the pattern of symmetry breaking inherited from criticality during the interval , shown in the shaded region of Fig. 4(b). This scenario is consistent with the evolution of the net number of vortices during a quench shown in Fig. 4(c) and (d). This measure is an indicator of the stability of individual JVs, unlike the total number that is affected by their pairwise annihilation (to which KZ scaling is immune).
The absence of any clear cut evidence of cosmological nature and the difficulty in observing the KZ scaling in condensed-matter experiments is usually not attributed to the failure of the mechanism but may be explained by the decay of defects in the post-quench era Sikivie82 (); Weiler08 (); Damski (). This is circumvented if the formed defects are topologically protected. The defects observed in the successful experiments of Refs. Monaco02 (); Monaco03 (); Monaco06 (); Monaco06prl () have this property, as do the JVs that are the subject of this Letter. While cosmological defects protected by topology may still survive in dark matter or dark energy, their detection is difficult Pospelov ().
Our work paves the way for a direct test of KZM in the Bose-Einstein condensation phase transition, by eliminating post-quench decay of defects Das (). The quench of could be supplanted by a controlled sweep of , providing another knob for varying the quench time.
Discussions with Brian Anderson and Ray Rivers are gratefully acknowledged. Simulations were performed on the Massey University CTCP computer cluster. SWS and SCG were supported by National Science Council, Taiwan (Grant No. 100-2112-M-018-001-MY3). OF and JB were supported by the Marsden Fund (Contract MAU0910). AB was supported by the Marsden Fund (Contract UOO162), and a Rutherford Discovery Fellowship of the Royal Society of New Zealand (Contract UOO004).
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