Ultracold gases and multi-Josephson junctions as simulators of out-of-equilibrium phase transformations in superfluids and superconductors

The experiments on strongly nonequilibrium symmetry-breaking phase transformations in superfluids and superconductors revealed that the topological defects (e.g. vortices) are produced most efficiently in the systems of microscopic size or low dimensionality (D=1), while in the macroscopic two-dimensional (2D) and 3D samples the efficiency of their formation was substantially suppressed (by a few orders of magnitude) as compared to theoretical predictions. A reasonable explanation for this behaviour is based on the specific thermal correlations between the phases of Bose–Einstein condensates formed in the spatial subregions disconnected during the phase transformation. Such correlations were initially revealed in the multi-Josephson-junction loop experiment (Carmi R et al 2000 Phys. Rev. Lett. 84 4966) and were confirmed recently by the experiments with ultracold atoms in periodic potentials (Hadzibabic Z et al 2006 Nature 441 1118). We begin our theoretical consideration from a phase transformation in the simplest φ4-model of the real scalar field and show that, under the presence of the above-mentioned correlations, the final symmetry-broken states are described by the effective Ising model. Its behaviour changes dramatically in passing from finite to infinite size of the system and from the low (D=1) to higher (D⩾2) dimensionality, which is in qualitative agreement with the experimental results.


The concept of topological defects
Formation of topological defects by the strongly out-of-equilibrium symmetry-breaking phase transformations is the subject of interest both in condensed-matter and elementary-particle physics. This is because of a close similarity between the Lagrangian of Landau-Ginzburg theory, widely used to describe phase transitions in condensed matter (superconductors, superfluids, liquid crystals etc), and the Lagrangians of the modern field theories (such as the standard electroweak model or various kinds of Grand Unification Theories), which are also substantially based on the concept of spontaneous symmetry breaking. After the phase transformations in all the above-mentioned cases, stable topological defects of the order parameter can arise, depending on the symmetry group involved such as the monopoles, strings (vortices) and domain walls.
Qualitative prediction of the defect formation, due to the independent establishment of the symmetry-broken states in spatially separated subregions, was done by Bogoliubov [1] soon after the appearance of the first field models based on the concept of spontaneous symmetry breaking. Later, a detailed quantitative theory of this phenomenon was developed by Kibble [2] and Zurek [3], and it is usually called the Kibble-Zurek (KZ) mechanism.
The KZ scenario is based on simple causality arguments. Namely, if during a phase transformation the information about an order parameter can spread over the distance ξ eff , then phases of the order parameter should be established independently in the regions of characteristic size ξ eff . 2 As a result, after some relaxation following the phase transformation, stable defects of the order parameter can be formed at the typical separation ξ eff from each other. Therefore, their concentration, for example, in a three-dimensional (3D) system can be roughly estimated as n ≈ 1/ ξ d eff , where d = 3, 2 and 1 for the monopoles, strings (vortices) and domain walls, respectively, while the effective correlation length ξ eff depends on the particular substance under consideration. For example, in the symmetry breaking of Higgs fields by the cosmological phase transitions (i.e. generation of mass of the elementary particles) ξ eff is commonly taken to be ξ eff c /H PT [4], where c is the speed of light and H PT is the Hubble constant at the instant of phase transition. In the consideration of vortex generation by a superfluid phase transformation, the corresponding quantity is defined as ξ eff ≈ c 2 τ Q , where c 2 is the speed of the second sound (which is a characteristic rate of propagation of information about the phase of the order parameter), and τ Q is the so-called quench time (a characteristic time of the phase transformation), which can be determined as 1/τ Q = (1/T )(dT /dt)| T =T c . Similar definitions of ξ eff are used also for other condensed-matter systems (superconductors, liquid crystals etc).

Review of experimental data
Although the KZ mechanism was proposed initially in the context of elementary-particle models admitting the symmetry breaking, much work has been undertaken in the last 15 years to study the same phenomenon in laboratory experiments with condensed matter. These works were originated by Chuang et al [5], dealt with liquid crystals, and about a dozen of subsequent experiments were performed with various superfluid and superconducting systems. They are listed in table 1, which outlines the design 4 of each experiment, cites the first publications by each group and summarizes their results 3 .
Analysis of the table reveals a quite interesting tendency: the topological defects are formed most efficiently in the systems of small size or low dimensionality, for example, quasi-1D multi-Josephson-junction loops (MJJL) [13] and annular Josephson tunnel junctions [14], as well as in microscopic hot bubbles of 3 He produced by neutron irradiation [10,11]. On the other hand, the concentration of defects in the macroscopic systems of higher dimensionality, e.g. 2D superconductor films [8,9,12] and 3D volume samples of 4 He [6,7], was found to be considerably less than theoretical predictions 4 .
The aim of the present work is to show that the above-mentioned features are natural consequences of the corrections which should be introduced into the standard KZ mechanism in view of the recent experiments on thermal dynamics of Bose-Einstein condensates (BECs) in periodic potentials [13,15]. As will be seen in the next sections, the corresponding thermal corrections result in a quite universal behaviour of the efficiency of defect formation as a function of size and dimensionality of the system.

Initial assumptions and equations
Let us consider the simplest ϕ 4 -model of a real scalar field (the order parameter) whose admits the discrete Z 2 symmetry breaking. (See also discussion of the same model in [16].) As is known, two stable vacuum states of this field (which, for the sake of convenience, will be marked by oppositely directed arrows) are The structure of a domain wall (kink) between them, located at x = x 0 , is described as and the specific energy concentrated in this wall (per unit length or area, depending on the dimensionality) equals Let a domain structure formed after a strongly nonequilibrium phase transformation be approximated by a regular (square, cubic etc) grid with a cell size of about the effective correlation length ξ eff , whose definition in the KZ scenario was already discussed in section 1.1. The particular value of ξ eff is not of importance here, but we shall assume that it is sufficiently large in comparison with a characteristic thickness (∼1/µ) of the domain wall. As a result, the final pattern of vacuum states (equation (2)) after the phase transformation will look like a distribution of spins on the regular grid.
The key assumption of the KZ mechanism is that the final symmetry-broken states of the field ϕ in two neighbouring cells of the size ξ eff are essentially independent of each other. Then, the probability of formation of a domain wall between them is given by the ratio of the number of statistical configurations involving the domain wall to the total number of configurations, P KZ = 2/4 = 1/2; and the resulting concentration of the defects (domain walls) will be where D is the effective dimensionality of the system. However, formula (5) is not sufficiently accurate, because it is based entirely on classical field dynamics and does not take into account the specific coherent effects of quantum fields. Namely, as follows from the latest experimental studies of BECs formed in isolated potential wells, there are pronounced residual correlations between their phases, i.e. a kind of 'footprint' of the initial thermal state of the entire system.

Overview of the experiments exhibiting the residual thermal correlations
The residual thermal correlations were observed for the first time in the MJJL experiment [13], whose sketch is presented in figure 1. A thin quasi-1D winding strip was engraved at the boundary between two crystalline grains of YBa 2 Cu 3 O 7−δ high-temperature superconductor film, thereby forming a loop of 214 superconductor segments separated by the grainboundary Josephson junctions. This system experienced multiple heating-cooling cycles in the temperature range 77 K to ∼100 K, which covers both the critical temperature of superconducting phase transformation in the segments of the loop (T c = 90 K) and in the junctions between them (T cJ = 83-85 K).
There is evidently no order parameter in the entire loop as long as T > T c . Next, when the temperature drops below T c but remains above T cJ (i.e. T cJ < T < T c ), some value of the order parameter should be established in each segment, as is schematically shown by arrows in the right-hand part of figure 1. Since these segments are separated by nonconducting Josephson junctions, it is reasonable to assume that the phase jumps between them are random (i.e. uncorrelated to each other). Finally, when the temperature drops below T cJ , the entire loop becomes superconducting and, due to the above-mentioned jumps, a phase integral along the loop, in general, should be nonzero. As a result the electric current I circulating along the loop, and the corresponding magnetic flux , penetrating the loop, will be spontaneously generated. So, if the phase jumps in the intermediate state T cJ < T < T c were absolutely uncorrelated, then distribution of spontaneously trapped magnetic flux in the particular experimental setup [13] would be given by the normal (Gaussian) law with a characteristic width of 3.6φ 0 (where φ 0 is the magnetic-flux quantum). On the other hand, the actual experimental distribution was found to be over twice as wide; and this anomaly was explained by authors of the experiment assuming that the phase jumps in the intermediate state were not random but correlated to each other, so that the probability P(δ i ) of the phase difference δ i , in the ith junction, was given just by the Boltzmann law: where E J is the energy concentrated in the Josephson junction, and T c is the phase transformation temperature, measured in energy units. Therefore, MJJL experiment provided the first evidence that a system experiencing the symmetry-breaking phase transformation 'remembers' its initial thermal state even if its separate parts cannot communicate with each other immediately during phase transformation. Unfortunately, this experiment did not verify the particular form of the functional dependence (6), because T c and E J were fixed by the material properties of the superconductor.
This obstacle was overcome a few years later by studying the interference between BECs of ultracold atoms. In particular, in the experiment of [15] two independent condensate clouds were formed in the wells of an optical potential. Then, the potential was abruptly switched off, the clouds began to expand and interfere with each other, and the experimentalists measured the number of sharp phase jumps (dislocations) in the interference pattern as a function of the initial temperature of the condensates (see figure 4 of [15]). Although the error bars were quite large, it was clearly seen that the probability of defect formation qualitatively followed the same Boltzmann dependence (6), i.e. quickly decreased with a decrease in temperatures. 5 We will not discuss further the problem of thermal influences on the occurrence of phase defects, but shall assume that formula (6) is typical for all BECs formed by the symmetrybreaking phase transformations. Next, we shall try to answer the question: what will be the resulting effect in the systems of variable size and dimensionality? 7

Improvement of the classical estimates
Following from the above section, the probabilities of various field configurations after the phase transformation in our ϕ 4 lattice model should be calculated by taking into account the Boltzmann factors (6). As a result, instead of a random distribution of the domain phases (spins), we obtain a distribution exactly equivalent to the Ising model at some temperature T c , which formally coincides with a critical temperature T c of the initial ϕ 4 -model, while the energy of the elementary domain wall E plays the role of an effective spin-spin interaction constant. 6 All aspects of this formal correspondence are summarized in table 2. (The item 5 about the equivalence between the state formed after a strongly nonequilibrium phase transformation in ϕ 4 -model and a thermodynamically equilibrium state of the Ising model, implies a mathematical identity between statistical sums for distribution of the domains of the symmetry-broken phase in ϕ 4 -model, on one hand, and for the spin distribution in the Ising model, on the other hand.) Then, the probability of a domain wall formation can be calculated just as an average energy of the system with domain walls divided by the energy of the elementary domain wall E (i.e. at one boundary between two neighbouring cells) and by the total number of the sites DN D , where these domain walls can be located: Here, N is the number of cells along each side of the lattice, and is the usual statistical sum over all possible spin configurations of the Ising model, where ε i is the total energy of the ith configuration.
Particularly, for a 1D system with periodic boundary conditions, for a 2D system, and so on. Here, s k and s kl are the spin-like variables describing the symmetry-broken states in the kth and (kl)th cells, respectively. As is known [17], the Ising model for 1D as well as the finite-size higher-dimensional systems does not experience a phase transition to the ordered state at any value of the ratio E/T c . From the viewpoint of domain wall formation by strongly out-of-equilibrium phase transformation in the original ϕ 4 -model (table 2), this means that concentration of the defects will not differ considerably from the standard KZ estimate, because the probability of defect formation P at the scale of effective correlation length ξ eff will not deviate substantially from P KZ = 1/2 ∼ 1.
On the other hand, the Ising model for the sufficiently large (infinite-size) 2D and 3D systems does experience a phase transition to the ordered state at some value of E/T c ∼ 1. As a result, the concentration of domain walls in the corresponding ϕ 4 -model at large ratios E/T c should be suppressed dramatically due to formation of macroscopic regions with the same value of the order parameter, covering a great number of cells of the effective correlation length ξ eff . (To avoid misunderstanding, let us emphasize again that the different values of T c should be understood here as critical temperatures of various physical systems described by the ϕ 4 -model, and they formally correspond to a variable temperature of the fixed Ising system.)

Particular example
The general conclusions formulated above can be illustrated by the particular example in figure 2, which represents the refined concentration of domain walls n normalized to standard KZ value, n KZ , as a function of E/T c for the following three cases: 1. 1D infinite-size Ising model, which admits the exact solution (see, for example [18]); 2. 2D 6 × 6-cell Ising model with periodic boundary conditions, which simulates quite well the system of infinite size; and 3. 2D 6 × 6-cell Ising model with free boundaries (where no energy is concentrated), which is an example of a microscopic system.
(The particular size of 6 cells along each side of the lattice was taken arbitrarily, just as the value at which a numerical computation of the statistical sum (10) is not too cumbersome; for details, see the appendix.) As is seen in figure 2, at E/T c ∼ 1 the concentration of defects in the 1D system (dashed green curve) differs from the standard KZ value by less than a factor of two; in microscopic 2D system (dotted blue curve), by three times; while in the macroscopic 2D system (solid red curve) it is suppressed by order of magnitude. Such suppression becomes much stronger when the ratio E/T c increases: for example, at E/T c ∼ 2 the difference between each of the three values is over an order of magnitude.

Conclusions
Following from the above consideration, specific thermal correlations between the phases of BECs in separated spatial subregions revealed for the first time in the MJJL experiment and confirmed later by the experiments with ultracold atoms, can be a promising method for the explanation of the data presented in table 1. The effective Ising model discussed in sections 2.3 and 2.4 predicts a strong suppression of the defect formation in the macroscopic systems with dimensionality D 2, as it was observed by studying the strongly out-of-equilibrium phase transformations of superfluids and superconductors (see rows marked by the minus sign in the table). Unfortunately, results of the very simplified model (1) with a real field ϕ cannot be compared quantitatively with the works cited in table 1, because superfluids and superconductors possess more complex order parameters and Lagrangians than (1). So, consideration of the more realistic models with complex order parameters (possessing U (1) symmetry group) is necessary.
Although a general treatment of the complex case is not easy, it can be expected that the above-mentioned properties will remain valid. For example, a strongly out-of-equilibrium phase transformation in the ϕ 4 -model, with a complex field ϕ in 2D, should be approximately reduced to the XY-model (where spins are in the plane of the lattice). The presence of the wellknown Berezinskii-Kosterlitz-Thouless phase transition in this model suggests that conclusions derived for the Ising model may be still valid.
Next, it is important to emphasize that the refined concentration of defects, obtained in the previous section, n = f (E/T c ) n KZ (ξ eff (τ Q )) possesses exactly the same dependence on the quench time τ Q as in the classical KZ scenario. So, this dependence, often measured in the experiments, cannot serve by itself for a discrimination between the mechanisms; the absolute values of the defect concentration are always necessary.
Finally, let us mention that the ideas described in the present work can also be applied to solving the problem of excessive concentration of topological defects predicted after the cosmological phase transitions of Higgs fields [19,20].  Next, the relative concentration of domain walls is obtained by substitution of (A.3) into (7), and the final result takes the form: . (A.4) In the particular cases plotted in figure 2, we used D = 2 and N = 6.