Semifluxons in Superconductivity and Cold Atomic Gases

Josephson junctions and junction arrays are well studied devices in superconductivity. With external magnetic fields one can modulate the phase in a long junction and create traveling, solitonic waves of magnetic flux, called fluxons. Today, it is also possible to device two different types of junctions: depending on the sign of the critical current density, they are called 0- or pi-junction. In turn, a 0-pi junction is formed by joining two of such junctions. As a result, one obtains a pinned Josephson vortex of fractional magnetic flux, at the 0-pi boundary. Here, we analyze this arrangement of superconducting junctions in the context of an atomic bosonic quantum gas, where two-state atoms in a double well trap are coupled in an analogous fashion. There, an all-optical 0-pi Josephson junction is created by the phase of a complex valued Rabi-frequency and we a derive a discrete four-mode model for this situation, which qualitatively resembles a semifluxon.


Introduction
During the past two decades, the field of cold atomic gases has come a long way starting from almost lossless trapping and cooling techniques [1] to reaching quantum degeneracy of Bosons and Fermions [2]. Many phenomena that are the hallmarks of condensed matter physics, whether in superfluid or superconducting materials [3], are revisited within this novel context. Due to the remarkable ease with which it is possible to isolate the key mechanisms from rogue processes, one can clearly identify phase transitions, for example, Bose-Einstein condensation, the Mott phase transition or the BEC-BCS crossover. Today however, degenerate gases are still at a disadvantage if we consider robustness, portability or the ability for a mass production compared to solid-state devices, which is the great achievement of the semiconductor industry. Strong attempts to miniaturize cold gas experiments [4,5] and to make them portable [6,7,8,9] are currently under way in many laboratories.
Due to the great importance and practical relevance of the Josephson effect in superconducting systems [10,11,12], it has also received immediate attention after the first realization of Bose-Einstein condensates [13,14,15,16,17,18,19,20,21,22,23,24,25]. In particular, the combination of optical lattices with ultracold gases [26,27] has boosted the possibilities to investigate junction arrays experimentally. Remarkably, even the absence of phase-coherence between neighboring sites can lead to interference as demonstrated in [28]. The possibility to study atomic Josephson vortices in the mean field description was raised first in connection with the sine-Gordon equation [29,30].
In the present article we will report on such a transfer of concepts from a superconducting device [31], i.e., in various realizations of Josephson junction arrays and their unusual state properties of traveling (fluxons) and pinned (semifluxons) magnetic flux quanta to an analogous set up for neutral bosonic atoms in a trap. In particular, we will investigate an alloptical 0-π Josephson junction that can be created with a jumping phase of an optical laser.
This article organized as follows: in Sec. 2, we will give a brief review of the current status of the superconductor physics of Josephson junctions. In particular, we will refer to the most relevant publications in this thriving field of fluxon and semifluxon physics; in Sec. 3, we will discuss a similar setup, which allows to find a pinned semifluxon in an atomic 0-π Josephson junction and we compare the results. Finally, we will discuss further open questions in a Conclusion.

Fluxons and Semifluxons in Superconductivity
The Josephson effect is a well established phenomenon in the solid state physics. A Josephson junction (JJ) consists of two weakly coupled superconducting condensates. JJs are usually fabricated artificially using low-or high-T c superconducting electrodes separated by a thin insulating (tunnel), normal metal, or some other (exotic) barrier. JJs can also be present intrinsically in a anisotropic layered high-T c superconductors such as Bi 2 Sr 2 Ca 1 Cu 2 O 8 [32,33].
The dc Josephson effect (flow of current through a Josephson junction without producing voltage drop, i.e. without dissipation) is expressed using the first Josephson relation, which in the simplest case has the form where I s is the supercurrent flowing through the junction, I c is the critical current, i.e. the maximum supercurrent which can pass through the JJ, and φ = θ 2 − θ 1 is the difference between the phases of the quantum mechanical macroscopic wave functions ψ 1,2 = √ n s e iθ 1,2 of the superconducting condensates in the electrodes. Recent advances in physics and technology allow to fabricate and study the so-called π-Josephson junctions -junctions which formally have negative critical current I c < 0. This can be achieved by using a ferromagnetic barrier, i.e. in Superconductor-Ferromagnet-Superconductor (SFS) [34,35,36,37,38] or Superconductor-Insulator-Ferromagnet-Superconductor (SIFS) [39,40] structures. One can also achieve the same effect using a barrier which effectively flips the spin of a tunneling electron, e.g. when the barrier is made of a ferromagnetic insulator [41], of a carbon nanotube [42] or of a quantum dot created by gating a semiconducting nanowire [43].
The change in the sign of a critical current has far going consequences. For example, analyze the Josephson energy (potential energy related to the supercurrent flow). In a conventional JJ with I c > 0 and has a minimum at φ = 0 + 2πn (the ground state), where E J = Φ 0 I c /2π is the Josephson energy. If I c < 0, we define E J = Φ 0 |I c |/2π > 0 and Obviously, the minimum of energy is reached for φ = π + 2πn. Thus, in the ground state (the JJ is not connected to a current source, no current flows through it) the phase drop across a conventional JJ with I c > 0 is φ = 0+2πn, while for a junction with I c < 0 it is φ = π +2πn. Therefore, one speaks about "0-JJs" and "π-JJs". Further, connecting the two superconducting electrodes of a π-JJ by a not very small inductor L (superconducting wire), the supercurrent ∝ π/L will start circulating in the loop. Note, that this supercurrent is spontaneous, i.e., it appears by itself, and has a direction randomly chosen between clockwise and counterclockwise [44]. The magnetic flux created by this supercurrent inside the loop is equal to Φ 0 /2, where Φ 0 = h/2e ≈ 2.07 · 10 −15 Wb is the magnetic flux quantum. Thus, the π-JJ works as a phase battery. This phase battery will work as described, supplying a supercurrent through the loop with inductor, provided the inductance L ≫ Φ 0 /I c . If the inductance L is not that large, the battery will be over loaded, providing a smaller phase drop and supporting smaller current. For very small inductance the battery will stop working completely.
Similar effects can be observed in a π dc superconducting quantum interference device (SQUID: one 0-JJ, one π-JJ and an inductor L connected in series and closed in a loop) or in 0-π Josephson junction. Let us focus on a latter case. Consider a long (along x) Josephson junction (LJJ) one half of which at x < 0 has the properties of a 0-JJ (critical current density j c > 0) and another half at x > 0 has the properties of a π-JJ (critical current density j c < 0). Long means that the length is much larger than the so called Josephson length λ J , which characterizes the size of a Josephson vortex; typically λ J ∼ 10 − 20 µm. What will be the ground state of such a 0-π LJJ? It turns out that if the junction is long enough (formally infinitely long), then far away from the 0-π boundary situated at x = 0, i.e., at x → ±∞ the phase φ will have the values 0 or ±π (we omit 2πn here), while in the vicinity of 0-π boundary the phase φ(x) smoothly changes from φ(−∞) = 0 to φ(+∞) = ±π, see Fig. 1a. The exact profile can be derived analytically [45,46,31]. Since the phase bends, the local magnetic magnetic field H ∝ dφ/dx will be localized in the vicinity of the 0-π boundary and carry the total flux equal to ±Φ 0 /2, see Fig. 1b. The sign depends on whether the phase bends from φ(−∞) = 0 to φ(+∞) = +π or to φ(+∞) = −π. Thus, such an object is called a semifluxon or an antisemifluxon. If one analyzes the Josephson supercurrent density flowing though the barrier j s (x) = j c (x) sin φ(x), one can see in Fig. 1c that the supercurrent has different directions on different sides from the 0-π boundary. Since we do not apply any external current, the flow of current should close in the top and bottom electrodes, i.e. the supercurrent circulates (counter)clockwise in the case of (anti)semifluxon. Thus, a semifluxon is a Josephson vortex of supercurrent. It is pinned at the 0-π boundary and has two degenerate ground states with the localized magnetic field carrying the flux ±Φ 0 /2.
Semifluxons in various types of JJ has been actively investigated during the last years. In fact, the first experiments became possible because of deeper understanding the symmetry of the superconducting order parameter in cuprate superconductors. This order parameter with the so-called d-wave symmetry is realized in anisotropic superconductors, such as YBa 2 Cu 3 O 7 or Nd 2−x Ce x CuO 4 . It allowed to fabricate 0-π grain boundary LJJs [47,48] and, later, more controllable d-wave/s-wave ramp zigzag JJs [49,50] and directly see and manipulate semifluxons using a SQUID microscope [51,52].
Semifluxons are very interesting non-linear objects: they can form a variety of ground states [53,54,55,56], may flip [51,47] emitting a fluxon [57,55,58], or be rearranged [59] by a bias current. Huge arrays of semifluxons were realized [51] and predicted to behave as tunable photonic crystals [60]. Semifluxons are also promising candidates for storage devices in the classical or quantum domain and can be used to build qubits [61] as they behave like macroscopic spin 1/2 particles. Now, an interesting question arises: Can one realize π or even 0-π JJs in an atomic BEC? In the latter case, the degenerate ground state corresponding to a semifluxon should have a non-trivial spatial phase profile and semifluxon physics can also be studied using BEC implementation.

Semifluxons in Bose-Einstein Condensates
Here, we will address this question and examine a configuration were the two-state atoms are trapped in a quasi-one dimensional cigar-shaped trap with an additional superimposed double well potential in the longitudinal direction. The spatial localization of the two-state atoms inside the double well potential leads to two internal atomic Josephson junctions that are driven via an optical, complex valued "0-π" laser field and they are motionally connected via tunneling.
First, we will describe details of the model and introduce the Hamiltonian of the system. Then, we will examine the classical limit of the field theory and study the ground state of the Gross-Pitaevskii equation. Finally, we will exploit the fact that the spatial wavefunctions are localized inside a deep double well and study a simple four-mode quantum model derived from a Wannier basis state representation.

0-π-junction in a Bose-Einstein condensate
To model the 0-π-Junction [29,30] in a BEC, we are guided by the the condensed matter physics setup depicted in Fig. 2. As in the single atom case, we replace the two superconductors by an atomic two-level BEC in a cigar shaped trap. The two states of the atom, i.e., the excited state |e and ground state |g couple via a position dependent Rabi frequency Ω 0 (x), which exhibits a phase jump at the origin of the x-axes In this quasi one-dimensional scenario, we will represent the two-state atoms by a spinorial bosonic quantum fieldΨ, which satisfies the commutator relation This field can be decomposed in any complete single particle basis |σ, l , which resolves the spatial extent of the field and the internal structure of the atoms, i.e., and we denote the corresponding discrete bosonic field amplitudes byâ σ,l . Here, σ characterizes the internal states by (e, g) and the external motion in the double well potential by a quantum label l. The dynamical evolution of the atomic field is governed by following HamiltonianĤ . In here, we use dimensionless units, in particular we have set = 1 and the mass of the atom m = 1. The energy consists of the single particle energy in a trap V (x), which is identical for both species, the electric dipole interaction of the two-state atom [62], as well as a generic collision energy proportional to the coupling constant g = g ee = g gg . To simplify the analysis, we have deliberately set the cross-component scattering length g eg = 0. No unaccounted loss channels are present. Therefore, we have number conservation as a symmetry. If we denote a generic state of the many-particle system in Fock-space by ψ n (t)|n = (n 0 , n 1 , . . .) , then one can obtain the dynamics of the system most generally from the Lagrangian formulation In this field theory, the canonical momentum is given by π n = δL δψn = iψ * n . From the Hamilton equationπ n = − δL δψn , one recovers the conventional Schrödinger equation in Fock space i∂ t |ψ =Ĥ|ψ .
The Lagrangian approach is obviously a central concept in the path integral formulation of quantum mechanics [63]. However, it is also of great utility in the approximate description of the dynamics if we connect it with concepts of classical mechanics as we will see in the following.

Spatially extended classical model: the Gross-Pitaevskii equation
The classical limit of the field equations [2] can be recovered quickly by approximating the state of the system by a coherent statê Within this approximation, we obtain from the Lagrangian of Eq. (10) the two component Gross- For a macroscopically occupied field this equation models the spatial evolution of the coupled Josephson junctions very well [17,18,19,20,21,22,23,24,30].

Discrete quantum model: two coupled Josephson junctions
To gain more insight into the quantum properties of the groundstate of the system [61], one can decompose the field into its principal components and disregard small corrections. A double well potential can be considered as the limiting case of a periodic lattice. While even and odd parity modes relate to delocalized Bloch-states in a periodic system, left and right localized modes i.e., φ 0 and right φ 1 , resemble the Wannier basis. These localized modes are depicted in Fig. 3. With respect to these basis states, we can approximate the field with four modesΨ The four bosonic amplitudes {ĝ 0 ,ê 0 ,ĝ 1 ,ê 1 } satisfy the usual commutation relations and we will disregard the small corrections of order δΨ σ If this approximate field is substituted in the Hamiltonian, Eq. (7), we can exploit the orthogonality of the wave functions and obtain the two-body matrix elements φ ijkl Out of the sixteen combinations for φ ijkl , we only retain the physically most relevant contributions and disregard others deliberately. This leads to the following model Hamiltonian for two coupled Josephson junctionŝ ) where all coupling constants are implicitly rescaled by the corresponding single particle or two-body matrix elements and the new parameter Λ measures the spatial hopping rate between the sites.
x Figure 4. Coupling pattern of two coupled Josephson junctions JJ 0 and JJ 1 with, Ω 1 = −Ω 0 . The classical phase θ refers to a particle current between the spatial sites, while the phases φ 0 and φ 1 are currents within the two-state atoms.

Fock-space representation of the four mode model
In principle, it is possible to solve the four-mode Schrödinger equation in Fock space by projecting it on the N-particle sector ψ n |n = (n e 0 , n g 0 , n e 1 , n g 1 ) .
The number constraint on the state remains valid throughout the time-evolution as number conservation is encoded into our Hamiltonian from the beginning in Eq. (8). This reduces the discrete d = 4 dimensional eigenvalue problem to an effective three-dimensional problem with nontrivial boundaries. We have illustrated the finite support of the amplitude field ψ n in Fig. 5. It is a (d − 1)-dimensional simplex embedded into a d-dimensional Fock space. The full analysis of this problem is an interesting problem in its own right and will be presented in a forthcoming publication.

The classical limit of the four mode model
It is not necessary to solve the four-mode problem in Fock-space to understand the principal features of the equilibrium configuration. Thus, we will again resort to the classical approximation and use the number-symmetry broken coherent state approximation for the quantum state |ψ = |α = (e 0 , g 0 , e 1 , g 1 ) ⊤ ,ê i |ψ = e i |ψ ,ĝ i |ψ = g i |ψ .
The dynamics is simply obtained from the Lagrangian if we introduce the classical Hamilton functions H(α, α * ) = α|Ĥ|α and number expectation values N (α, α * ) = α|N |α , as + Ω 0 (e * 0 g 0 + g * 0 e 0 ) + Ω 1 (e * 1 g 1 + g * 1 e 1 ) + g(|e 0 | 4 + |g 0 | 4 + |e 1 | 4 + |g 1 | 4 ), If the dynamical coordinate is α, then we find the canonical momentum as Consequently, variables and momenta obey the Poisson bracket {α j , π k } = δ jk . By construction, number conservation is satisfied dynamically as and the Hamilton equations of motion for the coordinates read In classical mechanics, we can deliberately choose new coordinates and momenta to accounts for symmetries of the Hamiltonian. If a new variable of a canonical transformation matches a conserved quantity, it follows that the conjugate variable becomes cyclic. In this spirit, we will introduce the following pairs of action-angle variables: (Φ, N ) measures the global phase and the total particle number N = n g 0 + n e 0 + n g 1 + n e 1 of the system, (θ, M) measures the relative phase between left and right sites and the population imbalance M = n e 0 +n g 0 −(n e 1 +n g 1 ) inbetween, (φ 0 , M 0 = n e 0 −n g 0 ), and (φ 1 , M 1 = n e 1 −n g 1 ) measure the relative internal phase of the atoms on each site and the corresponding population difference. This coupling scheme for the phases and population imbalances has been illustrated in Fig. 4. By inverting the population relations, one finds the individual occupation number n σ l per site as Finally, we can use these physical coordinates in a canonical transformation from complex amplitudes to real action-angle variables For later use it is also useful to introduce the auxiliary phases θ i , which are global phases of the subsystem on site i: By substituting field amplitudes into the Lagrangian of Eq. (21), one can again identify the variables and corresponding canonical momenta as With this new coordinates and momenta we obtain a Hamilton function For the parameters δ = 1, Ω 0 = 1, Ω 1 = −1, Λ = 0.1, g = 0.005, N = 100, we have numerically minimized the energy of Eq. (31) and find two minima E = E 0 , E 1 . Please note that the amplitudes of the solutions α are real valued. Due to the invariance of the solutions under global phase change, we can deliberately modify them and choose the global phase of the left site as the common ground θ 0 = 0. In Tab. 1, we have listed the energies, the real valued amplitudes and the phases calculated according Eqs. (28) and (29). Table 1. Two stationary solutions of the Hamilton equation, energies E 0 , E 1 , field amplitudes α = (e 0 , g 0 , e 1 , g 1 ) and phases. The global phase of the left site, i.e., θ 0 = 0 was chosen as the common ground.

Conclusion and Perspectives
In the present article we have briefly summarized the status of the fluxon and semifluxon physics in superconductivity. In particular, we were focusing on an effect that exists in a long 0-π geometry where two Josephson junctions were in the groundstate and one obtains a Josephson vortex of fractional magnetic flux, pinned at the 0-π boundary. Depending on the preparation procedure, this "classical" state of the superconducting device can occur in two different configurations: with the magnetic flux equal to +Φ 0 /2 or −Φ 0 /2. This arrangement of superconducting junctions has been analyzed and transfered to the context of an atomic bosonic quantum gas, where two-state atoms in a double well trap are coupled in a similar fashion. There, the optical 0-π junction is represented by the left/right localized internal atomic Josephson junctions and a jumping phase of the complex-valued Rabi-frequency. We have derived a simple four-mode model for this case and showed that in the "classical" approximation it qualitatively resembles the semifluxons seen in superconductivity.
In the superconducting case, one observes a smooth spatial behaviour of the phase across the 0-π boundary. In the atomic case, this will emerge also by a more realistic modeling for the Rabi-frequency and spatial motion. This is currently under investigation. Eventually, the fourmode, 0-π Josephson model will be also instrumental in examining the quantum properties of these macroscopic semifluxon states and we will explore the macroscopic tunneling between them [61]. This is also work in progress and will be reported in a forthcoming publication.