Single interface effects dominate in exciton-condensate/normal-barrier/exciton-condensate (EC/N/EC) structures of long-barrier

We study theoretically the exciton-condensate/normal-barrier/exciton-condensate (EC/N/EC) structures in bilayers with a tunable relative phase ϕ0 between the two exciton condensates (ECs). It is a setup inspired initially by the superconducting Josephson junction but with a special ingredient added for bilayer systems, namely, the interlayer tunneling. Our results shows that in a EC/N/EC structure of long-barrier, the single Andreev reflection at one EC/N interface dominates—as opposed to the same structure of short-barrier, in which multiple Andreev reflections can be accommodated (similar to the superconducting Josephson junctions). The single interface effect turns the other EC inert and the system can no longer be understood as a Josephson junction. The supercurrent, however, still occurs at the N/EC interface since the current conservation is still fulfilled with the assistance of the interlayer tunneling in barriers. This exotic mechanism gives rise to only a half portion from a fractional soliton of a doubled topological charge 2Q = ϕ0/π (for the same relative phase ϕ0), as opposed to a full portion fraction soliton of charge Q = ϕ0/2π in the structures of short-barriers. We predict the current phase relation for the EC/N/EC structures of long-barriers which can be tested experimentally.

Intriguing physics can happen when employing two excitonic condensates to sandwich a normal-barrier, forming an EC/N/EC structure ( figure 1(a)). The two condensates are designed to hold a constant relative phase f 0 that can be generated by externally applying an spatially localized in-plane magnetic fields [47], or a vertical electric bias pulse with a controllable temporal width to reach the designated phase [48]. When the enclosing normal-barrier is short, this EC/N/EC structure resembles a superconducting Josephson junction [47,[49][50][51]

Theoretical method
We use the developed lattice model [52,[65][66][67] and tailor it to describe specifically our EC/N/EC structures. Consider that the electrons can only be in either top or bottom layer, the wave function of the eigenstate can be generally expressed as: where X labels the lattice site in the x-direction. The behavior in y-direction is assumed to be translationally invariant. Notice that in our notation, u(X), v(X) are real numbers and the phase information is contained in f (X). The creation operator c X  ( ) † represents the creation of an electron at position X in the top (bottom) layer. The vacuum state 0ñ | indicates the state of no electron in either layer. By performing an SU(2) to O(3) mapping, the exciton-condensate system is transformed into a classical spin-dynamics problem with the pseudospin defined as: m X sin cos , sin sin , cos .
Here u X 2 arccos q = [ ( )] and f=f(X). In this representation, the in-plane magnetization (m x , m y ) and the out-of-plane magnetization m z , correspond to the excitonic coherence and the population imbalance, respectively. The excitonic system is now readily described by the Landau-Lifshitz-Gilbert (LLG) equation Figure 1. (a) Schematic illustration of the setup of an exciton-condensation/normal-barrier/exciton-condensation (EC/N/EC) structure in a bilayer. The constant relative phase can be prepared between exciton-condensate region 1 (EC1) and region 2 (EC2) through adding a voltage pulse of a controllable temporal width to reach the designated phase. (b) Detailed geometry of the EC/N/EC structure in our calculation. The total length of the structure in the conducting direction x is L. The barrier takes a length of d J and the rest is equally distributed to EC1 and EC2. The other dimension y is assumed to be translationally invariant. In the pseudospin picture, the given relative phase f 0 can be viewed as the angle difference between the pseudospin Zeeman fields h  (red arrows) in EC1 and in EC2.
[ [67][68][69]: is the pseudospin magnetic field and α is the Gilbert damping parameter. We can see that the dynamics of m  is ultimately determined by the energy functional E m  [ ], which in the meanfield theory [65] is given by: Here L x and L y are the length and the width of each unit cell. The first term is the capacitive penalty. The positive anisotropy parameter β characterize the energy cost to introduce charge imbalance between layers. The second term is the exchange correlation in which the superfluid density ρ s aligns the phase of the excitonic coherence on each site with its neighbors. The third term is the interlayer tunneling energy where n is the density and Δ t is the interlayer tunneling strength. Note that this term is essentially the in-plane Zeeman energy, r n m h ) breaks the U(1) symmetry and aligns the pseudospins in the direction of f ex . This f ex can be controlled externally and is the key to tuning the phase of the excitonic phase. Using the above energy functional, we perform numerical calculations of the LLG equation for each discretized position X. Notice that the most important spatial variable f (X) is actually the average of phase over a lattice site. That means that the rapid spatial phase fluctuation, e.g. merons, have been averaged out. That will in effect, cause a reduction of ρ s .
Next we explain how the EC/N/EC structures are described under the above scheme. For easy analytic presentation, we will from now on use the continuous varying x instead of the discrete X. As illustrated in figure 1, the system with length L is composed of two exciton condensates (EC1 and EC2), each with a length of (L−d J )/2, and a barrier length d J in between. In EC2 in particular, an external phase f ex =f 0 is introduced by applying an interlayer voltage pulse. For convenience, we define the left edge of EC2 to be the origin of x so the barrier is now located at −d J <x<0 and EC1 at L d in the y-direction the behavior is assumed to be translationally invariant. With this notation, the external phase in equation (4) is then: where Θ(x) is the Heaviside step function. Meanwhile, superfluid density ρ s in the barrier is set to zero: ) These configurations will enter the energy functional and affect the pseudospin evolution described by LLG equation. On the other hand, if only the lowest-energy steady-state is of interest, it can also be approached by minimizing the energy functional with respect to the phase f. In the condensate regime, we have m z ∼0 and the phase derivative of the energy function yields the modified-sine-Gordon equation (MSGE): Here λ is the Josephson length defined as , the MSGE gives rise to a fractional one-soliton solution that exists in an EC/N/EC structure with d J =0 [52]: Finally, notice that the Josephson length λ characterizes the size of the ordinary one-soliton and it also serves as the length scale of the structure in the discussion.

Results and discussions
In this section, we demonstrate the effect of the barrier on EC/N/EC structures by solving for the static solution to the discretized LLG equation. The parameters used in the equation are: Since the excitonic superfluid loses its coherence after traveling in the barrier over a distance ξ, the correlation length also corresponds to the pixel size in the lattice model [70]. The Josephson length λ, on the other hand, is roughly 250l for the Δ t chosen above. Note that the choice of Δ t is not crucial. Physics of different parameters, including Δ t , can be accessed by scaling the system with λ. Finally, the supercurrents calculated later are usually in comparison with a more universal scale: J e 2 nA m s 0 0 0 1  r l m º~-. In the following, we discuss the EC/N/EC structures of long-barrier (d J <ξ) including both cases of short (L<λ) and long (L>λ) total structure length. Figure 2(a) shows the typical phase profile for an EC/N/EC structure of long barrier but with short structure length (specifically, d J =0.08λ and L=0.8λ). We first notice that the phase profiles show non-differentiable rises in the vicinity of the N-EC2 interface then the profiles saturate smoothly to the maxima. Although our approach cannot uniquely identify excitonic Andreev reflection, these rises do correspond to the sudden increase of supercurrent at the interface and are also consistent with the known effect of excitonic Andreev reflection [18-20, 64, 74]. In the following we provide our picture on the EC/N/EC structures of long-barrier.
First we recall that in superconducting Josephson junctions, the Josephson effect can be understood as caused by multiple Andreev reflections happening to-and-forth between the two normal-superconductor interfaces [75]. The Andreev reflection also happens in bilayer exciton-condensate, but in a different form [19]: at the N-EC interface, an electron in one layer will reflect into the other layer and drives an electron-hole pair (exciton) to flow in the exciton-condensation region. In an EC/N/EC structure of short-barrier (EJJ) as illustrated in figure 3(a), the Andreev reflections between the two N-EC interfaces form Andreev bound state, similar to their counterpart in the superconducting Josephson junction. In an EC/N/EC structure of longbarrier, the excitonic supercurrent happens even if the Andreev bound state cannot be formed ( figure 3(b)).  What happens is that each right-going electron in the bottom layer is Andreev-reflected at the N-EC2 interface. The reflected electron in the top layer then moves leftward until it comes back to bottom layer through singleparticle tunneling. It is important to note that the total charge is conserved through out the process-the supercurrent thus produced is physical.
With the physical picture in mind, we are now ready to discuss the detailed structure of the phase profile. The maximum values of the phase profiles are much less than the assigned f 0 , and even more interestingly, they do not depend monotonically on f 0 . Such behavior can be understood by simplifying the modified-sine-Gordon equation. Since f=f 0 , the f in sin )can be dropped in equation (6) so the solution should be well described by a simple quadratic function as: where we have implicitly employed the boundary conditions. The sin 0 f dependence in the solution above gives rise to the non-monotonic behavior when ranging f 0 from 0 to π.
The corresponding CPR is: In figure 4(b) we show the CPR again from both the LLG calculation (red circle) and the sin 2 0 f ( )fitting derived from the above picture (blue dash line). The two again reach great agreements.
With the success of describing the CPR by our naive picture in both short (equation (10) and figure 2(b)) and long structure (equation (12) and figure 4(b)) regimes, the distinct behaviors in the two regimes lead us to wonder if a transition happens from the short to the long structure, or a crossover. To answer the question, we monitor how CPR evolves with increasing structure length L, as plotted in figure 5(a). Starting from a short structure (L=0.8λ) as a sin 0 f curve, the CPR skews gradually until it finally turns into a sin 2 0 f ( )curve. The evolution is a crossover. It is also interesting to note that this sin 2 0 f ( )to sin 0 f ( )crossover reminds us of the dirty-to-clean crossover in superconductor [76]. Their similar behaviors may not be a coincidence. In EC/N/EC structures, the total transmission through interlayer tunneling increases when increasing the structure length L. This can be analog to the increase in transmission coefficient D of the superconducting Josephson junction in the clean-to-dirty crossover(see figure 3 in [76]).
Next we summarize the crossover behavior for different f 0 , in particular that of f 0 π/2 and of π/2<f 0 <π. Here we pick f 0 =π/2 and f 0 =π for easy comparison. In the main content of figure 5 (b) we plot J s versus L for the two designated f 0 . For f 0 =π/2, the curve ascends progressively as L increases from zero, and saturates when L∼3λ. For f 0 =π, however, the supercurrent is essentially negligible until L hits 3λ, then it rapidly rises to saturation. The two saturation values can be read off and compared with those for d J =0 (inset of figure 5(b)). The maximum J s /J 0 are roughly 1.3 and 2 for f 0 =π/2 and f 0 =π, respectively in the EC/N/EC structures of long-barriers, while they are 1.7 and 1.3 for those of short-barriers; the saturation values are always greater in the structures of long-barrier than those of short-barrier. This is also a manifestation of the tunnelingassisted supercurrent.
At the end of the results section, we would like to comment briefly on our LLG approach. It can essentially be understood as the Ginzburg-Landau theory. As shown in the method section, this approach reduces the complicate electron-wave functions to the local value of order parameter. It is therefore very efficient in obtaining the self-consistent result, which is non-trivial in the presence of interlayer tunneling. We also note that if the disorder becomes important or the information of the actual electron-wave function is required, we will have to step back to the microscopic description of scattering [74] or the non-equilibrium Green's function method [19]. However, we are confident that for the current quality of bilayer and when the interlayer tunneling still plays an important role, the CPR we predicted is fairly reliable. Next we comment on the experimental realization of EC/N/EC structures. Among all proposed systems for excitonic condensation, GaAs-based quantum Hall bilayer remains to be the most developed one. The Josephson length in this system typically ranges from 4.5 to 45 μm, while the system size can easily reach hundreds of microns. This system is ready for what we have proposed in this manuscript. Aside from the GaAs-based bilayers, graphene, especially the double bilayer graphene [36, 38-40, 42, 44-46] is another potential candidate for realizing EC/N/EC structures. Graphene is relatively clean and small (typically around 10μm), and is most suitable for studying the quantum coherence. Moreover, the graphene bilayer, or other novel two-dimensional materials each carries exotic properties including chirality and spin-orbit coupling, etc, that can be inherited by the excitons therein. This is a land yet to be explored. We expect that the exotic nature will bring in exciting twists to the physics in EC/N/EC structures.

Summary
In this paper, we investigate the exciton-condensate/normal-barrier/exciton-condensate structures (EC/N/ EC) of long barrier by solving the pseudospin LLG equation. In a structure as such, the supercurrent is created by Andreev reflection at one of the N/EC interfaces and fulfills current conservation with the assistance of the coherent interlayer tunneling. The corresponding phase profile exhibits a half portion of a fractional soliton with doubled topological charge 2Q=f 0 / π. When varying structure length, we find that the CPRs skews gradually from a sin 0 f curve to a sin 2 0 f ( )as increasing the length from short-to long structure limit. In the calculation, the crossover happens at L∼3λ. We believe that the finding summarized here on the EC/N/EC structures of long barrier, together with our previous discussion [52] on the same structure of short-barrier can shed light on the realization of excitonic-based structures and fractional solitons-which might lead to new types of quantum qubits.