Position-dependent spin-orbit coupling for ultracold atoms

We theoretically explore atomic Bose-Einstein condensates (BECs) subject to position-dependent spin-orbit coupling (SOC). This SOC can be produced by cyclically laser coupling four internal atomic ground (or metastable) states in an environment where the detuning from resonance depends on position. The resulting spin-orbit coupled BEC phase-separates into domains, each of which contain density modulations - stripes - aligned either along the x or y direction. In each domain, the stripe orientation is determined by the sign of the local detuning. When these stripes have mismatched spatial periods along domain boundaries, non-trivial topological spin textures form at the interface, including skyrmions-like spin vortices and anti-vortices. In contrast to vortices present in conventional rotating BECs, these spin-vortices are stable topological defects that are not present in the corresponding homogenous stripe-phase spin-orbit coupled BECs.

In this article, we focus on real atomic systems from which we simultaneously identify a pseudospin-1/2 system, and induce SOC with the desired spatial dependence. This must be achieved using terms naturally entering into the bare atomic Hamiltonian. Here it is shown that such a goal can be realized by first creating SOC by cyclically coupling together four ground (or metastable) atomic states via two-photon Raman transitions, and then by spatially varying the detuning from twophoton Raman resonance. We present an explicit construction for 87 Rb in which SOC and the desired spatial dependance coexist. We then explore spatial dependance of the spin-1/2 spin-orbit coupled Bose-Einstein condensates (SOBECs) resulting from this construction. The SOC leads to formation of domains of differently oriented stripe phases. When the stripe's projection onto the domain-boundaries are spatially mismatched (See Fig. 1), arrays of non-trivial topological structures such as vortices and anti-vortices form.
The paper is organized as follows: in Sec. 2 we present a simple physical picture elucidating implications of the position-dependent SOC; in Sec. 3 we formulate the light-atom interaction for the specific example of 87 Rb, and derive the associated position-dependent spin-orbit coupled Hamiltonian for ground-state atoms; and in Sec. 4 we use the Gross-Pitaevskii equation (GPE) to study the ground state structure of these inhomogeneous systems. Finally, the Section 5 summarises our findings.

Physical picture
Before delving into a detailed discussion of specific atomic systems, we first discuss the qualitative physics leading to the formation of topological defects in our system. Our focus is on spin-1/2 SOBECs containing mostly Rashba-type SOC contaminated by a small tunable contribution of Dresselhaus-type SOC; together, these are parametrized by a non-Abelian vector potential A, and are described by the single particle Hamiltonian , σ x,y,z are the Pauli operators, and I is the identity. Here, m is that atomic mass; k is the momentum; κ ≥ 0 both describes the Rashba SOC strength and defines the energy E κ = 2 κ 2 /2m; lastly, κ describes the Dresselhaus SOC strength. The eigenvalues of this Hamiltonian (shown in Fig. 1a) are where the second equation is valid to linear order in . For = 0, these energies depend only on |k|, so the ground state (minimum of lower energy band, E − (k)) is macroscopically degenerate on the ring |k/κ| = 1/2. Figure 1b plots the energy minimum of radial cuts through SOC dispersion relations as a function of the polar angle γ, where k = |k| (cos γe x + sin γe y ); the black line, independent of γ, indicates the degenerate ground states of the Rashba Hamiltonian. This massive degeneracy is lifted when = 0. In this case, the dispersion is two-fold degenerate with minima at k ± /κ = ∓(1 + )e y /2 for > 0 (red curve in Fig. 1b) and k ± /κ = ±(1 − )e x /2 for < 0 (blue curve in Fig. 1b). The corresponding eigenstates with minimum energy represent the states where the atomic spin points along k ± .
Under many realistic physical conditions, a SOBEC will Bose-condense into both of these minima simultaneously [18,23], and the spatial interference between these two states, differing in momentum by δk ≈ κ, will generate stripes in the atomic spin density with spatial period 2π/κ. These stripes are aligned parallel to e x for > 0 and parallel to e y for < 0.
Here we study physical systems where the magnitude of the Dresselhaus SOC κ varies linearly along a direction in the e x -e y plane defined by the unit vector e = cos θe x + sin θe y . In the half-plane with > 0 we expect horizontal stripes and in the half-plane with < 0 we expect vertical stripes (schematically shown in Fig. 1c), and we ask: how are these different patterns of stripes linked at the boundary line 0 = x cos θ + y sin θ delineating the two domains (grey line in Fig. 1c). This seemingly simple question is nontrivial because the horizontal stripes ( > 0) have period d + = |2π/κ sin θ| projected onto the delineating line, while vertical stripes ( < 0) have period d − = |2π/κ cos θ| along the delineating line (see Fig. 1c): when | cos θ| = | sin θ| stripes must terminate or originate at the domain boundary, leading to the formation of pinned topological defects.  Here 2δ is the frequency detuning of the atomic levels |2 and |4 from two-photon Raman resonance due to an inhomogeneous magnetic field. Each line or curve connecting the bare states depicts a two-photon Raman transition.

The electronic Hamiltonian and its eigenstates
Our inhomogeneous SOC may be created using any atom with four internal ground (or metastable) states {|1 , |2 , |3 , |4 } that can be coupled in the cyclic manner shown in Fig. 2a. In 87 Rb these might be the four ground hyperfine states [5] |f = 1, m F = 0, −1 and |f = 2, m F = 0, +1 illustrated in Fig. 2c. In that case the four states are Raman coupled with the position-dependent couplings characterised by the amplitude Ω j , recoil momentum k j+1 − k j and phase shift φ j . Here is the wave-vector of the j th Raman laser field, κ being its length. This coupling scheme can be realized using the combination of π and σ polarized laser fields laid out in Ref. [5]. The linear Zeeman shift, from a biasing magnetic field B 0 = B 0 e z , is rendered position-dependent by virtue of an additional magnetic field gradient B = B (r · e)e z , linearly varying in the e x − e y plane along the direction e = cos θe x + sin θe y . The combination B 0 + B provides a controllable detuning δ = δ 0 + δ (r · e) from Raman resonance to the states |2 and |4 , see Fig. 2c. Physically this can be realized by using atomic magnetic levels shown in Fig. 2c where one pair of states is field insensitive and the other pair share essentially the same E Z = g F µ B m F Zeeman shift, where µ B is the Bohr magneton and g F is the Landé g-factor.
The scheme of cyclically coupled four atomic internal states shown in Fig. 2 is formally equivalent to a four-site lattice with periodic boundary conditions, i.e., |5 ≡ |1 . In terms of the position-dependent states |j ≡ |j (r) = exp (−ik j · r) |j , the Hamiltonian describing the internal atomic degrees of freedom iŝ where the contribution due to the atom-light detuning has been represented in terms of the overall shift of the energy zero δI and an alternating detuning δ(−1) j in the second term. This corresponds exactly to the experimental situation illustrated in Fig. 2c, where the levels 2 and 4 experience the shift 2δ, whereas the levels 1 and 3 are not affected. Because we study the case where δ depends on position, one can not simply remove the overall energy shift δ, as it was done in the previous work [5]. However in the following the overall detuning δ will be incorporated into the trapping potential V (r) featured in Eq. (14), where the linear detuning leads to the shift of the harmonic potential minimum. The phases of the laser fields are taken such that j φ j = π, so that an atom acquires a π phase shift upon traversing the closed-loop |1 → |2 → |3 → |4 → |1 in state space. For zero detuning (δ = 0) and equal Rabi frequencies (Ω j = Ω) the eigenfunctions and corresponding eigenvalues are In the {|χ q } basis, the internal Hamiltonian iŝ It has two pairs of degenerate eigenstates |↑, ↓; ± labeled by the pseudospin index ↑, ↓ and by their energies ± √ δ 2 + 2Ω 2 . The eigenstates are given by | ↓, ± = a ∓ |χ 0 ± a ± |χ 2 , and | ↑, ± = a ∓ |χ 1 ± a ± |χ 3 , where

Adiabatic motion and spin-orbit coupling
We are interested in the situation where the separation energy 2 √ δ 2 + 2Ω 2 between the pairs of dressed states exceeds the kinetic energy of the atomic motion. In that case the atoms adiabatically move about within each manifold of two-fold degenerate internal states. Such adiabatic motion is affected by the matrix-valued geometric vector and scalar potentials A (±) and Φ (±) which result from the position-dependence of the atomic internal dressed states [6,12]. Here the upper (low) sign refers to the upper (lower) adiabatic manifold. The matrix elements of the gauge potentials are When the detuning is much smaller than the Rabi frequency, δ Ω, the lowest order in δ contribution to the gauge potentials A ≡ A (−) and Φ ≡ Φ (−) are linear and quadratic respectively, The effective scalar potential Φ, resulting from the adiabatic elimination of the excited states, is proportional to the unit matrix and hence provides only an additional stateindependent trapping potential. The matrix-valued vector potential can be equivalently understood as SOC with the spatially-dependence appearing via a position-dependence of the detuning δ ≡ δ (r). For zero detuning, the vector potential is proportional to σ x e x − σ y e y , so the SOC is cylindrically symmetric. For non-zero detuning the cylindrical symmetry is lost, leading to the formation of the stripe phases in the SOC BEC along e x or e y as was discussed in Sec. 2.

Equations of motion
Having now shown how to create inhomogeneous SOC, we shift our focus to its effects on ground state properties of BECs. At zero temperature, the mean-field energy functional of a spin-1/2 BEC with SOC is where Ψ = (ψ ↓ , ψ ↑ ) T is the spinor (vectorial) order parameter, V (r) = mω 2 r 2 /2 is the trapping potential and g is the nonlinear interaction strength. The synthetic vector and scalar gauge potentials A and Φ (Eqs. (10)-(11)) depend on the linearly varying detuning δ = δ (x cos θ + y sin θ) (15) introduced in Sec. 3.1. Here we assume that V (r) embodies all external potentials including that resulting from the spatially-dependent energy offset in Eq. (4). The spinor time-dependent GPE (TDGPE) can be derived via the Hartree variational principle i ∂ t ψ j = δE/δψ * j giving where ρ = |ψ ↓ | 2 + |ψ ↑ | 2 is the total density. Equation (16) governs the dynamics of the BECs with position-dependent SOC, at the mean-field level.

Ground-state phases of the SOBEC
In Sec. 2, we discussed the single particle properties we expect from our mixed Rashba-Dresselhaus spin-orbit coupled system and noted that when = 0 the spectrum is twofold degenerate at points k ± /κ = ±(1 + )e y /2 for > 0 and k ± /κ = ±(1 − )e x /2 for < 0. When a weak repulsive interaction is included, the bosons can condense either in: (1) a plane-wave phase (PW) in which one of k + or k − is macroscopically occupied; or in (2) a standing wave phase (SW, sometimes called a striped phase) in which the bosons condense into a coherent superposition of k + and k − . We focus on the case where the inter-and intra-spin interactions are identical, for which the ground state is in the SW phase [18,23]. The detuning δ vanishes along the separatrix x cos θ + y sin θ = 0 that segregates the system into two regions with δ > 0 and δ < 0. Since the wave vectors characterizing the two domains have differing projections onto the line where δ = 0, novel structures can form to heal the otherwise discontinuous pattern at one each side of the separatrix. For example, for tan θ = 1 and 2, we expect one-to one and two-to-one connections on the interface, respectively.
To determine the ground state of the SOBEC with position-dependent SOC, we minimize the energy functional Eq. (14) by propagating Eq. (16) with differing degrees of imaginary time, replacing i∂ t by (i − γ)∂ t in Eq. (16). Specifically, the groundstate was solved by using the imaginary time propagation involving the replacement i∂ t → ∂ t , as well as the damped GPE [24][25][26][27] implying i∂ t → (i − γ) ∂ t . Both simulations give the same ground-state solution. However, the damped GPE takes a much shorter computation time if the parameter γ is properly chosen. We confirm that we have obtained the ground state by the absence of any time-dependance when we evolve in real time. We considered a 2D 87 Rb BEC confined in a harmonic potential with frequency ω/2π = 100 Hz. For computational convenience, we adopt the dimensionless units where the frequency and length are scaled in units of the trap frequency ω/2π and the oscillator length /mω, respectively. We employ the Fourier pseudospectral method with N x = N y = 256 grid points.
We first consider the parameters of µ = 32 ω, δ = Ω/2, κ = 2, and θ = π/4. In this case cos θ = sin θ and the horizontal and vertical stripes are matched 1:1 at the boundary. The corresponding ground-state wavefunction is shown in Fig. 3. In this case, the interface lies along the line x + y = 0 and the stripes align along e y for x + y < 0 and along e x for x + y > 0. Clearly, the stripes in both domains  are connected one-to-one across the interface. Moreover, the orientation of the state along the Bloch sphere smoothly connects the two phases, as shown in Figs. 3b, c and d. The ground-state structure is consistent with the prediction of the noninteracting homogeneous system with our single particle arguments.
Next we investigate how the ground-state density and phase profiles vary over the interface region for the mismatched condition when two stripes in one domain are linked to a single stripe in the other. Specifically, we consider tan θ = 1/2 and δ = Ω √ 10/6, which gives a separatrix 2x + y = 0. Keeping the same values of µ, κ we used for θ = π/4 case, the ground state is shown in Fig. 4. We observe that, near the trap center, every two adjacent stripes in the region of 2x+y > 0 connect to one single stripe in the region of 2x + y < 0. Unlike the previous case with θ = π/4, where the density and phase profiles both vary smoothly across the interface, in the current case the phases profile does not vary smoothly across the interface. In the spin-projections shown in Fig. 4b,c and d we see that vortices form at the boundary. In this case, two stripes must merge-and also form a vortex-before a single stripe crosses the domain boundary. The existence of vortices in the ground state results from the merging of stripes. In conventional BECs, vortices are stabilized by the application of artificial magnetic fields. However in our spatially dependent SOC BEC, the ground state supports vortices at the interface between two distinct SW phases. Similar defect formation at the interface of two distinct ground state phases was studied for spin-1 Probability density Figure 5. The ground state of a spatially dependent SOC BEC with orthogonal stripe patterns. This simulation was performed with µ = 32 ω, δ = Ω/ √ 2, κ = 2 and θ = π/2. In this case, the boundary occurs at x = 0. Panels (a)-(d) plot the expectation values Ψ|I|Ψ , Ψ|σ z |Ψ , Ψ|σ x |Ψ , and Ψ|σ y |Ψ . A row of vortices forms to link the two completely incompatible SW patterns.
Finally we examine the case where θ = π/2 and δ = Ω/ √ 2: here the interface coincides with the x-axis. In this case the stripes for y < 0 are perpendicular to the interface while the stripes are parallel to the interface in the region of y > 0. The result is shown in Fig. 5. The orthogonal stripes result in the formation of a vortex chain on the interface that can be seen clearly in the spin-projections of Fig. 5c and d. Our results indicate that the unconventional BEC ground state contains chains of vortices and anti-vortices stabilized by the position-dependent SOC. Furthermore, the number of vortices is highly controllable by tuning the size of condensate, the spin-orbit coupling strength κ, and the orientation of the interface.

Concluding remarks
We have proposed a new technique for creating a position-dependent SOC for cold atomic BECs. This can be carried out combining a cyclic Raman coupling scheme [5] to induce SOC, and a magnetic field gradient [30,31] to impart a spatial dependance.
Subject to this combination, we find that a weakly interacting BEC separates into two domains with orthogonally oriented stripes. Depending on axes of the domain boundary-set by the spatial direction of the magnetic field gradient-the stripes from each domain can intersect the boundary with commensurate or incommensurate spatial periods. We show that when the stripe patterns intersect with different spatial periods, a chain of topological defects, including vortices and anti-vortices, form to link the mismatched stripe patterns. In contrast to vortices present in conventional rotating BECs, here the vortices are stable topological defects that are not present in the homogenous phase (here the SW phase). These vortices can form in an ordered chain when the relative periods at the domain wall are different, but commensurate, and they form a disordered chain when the relative periods are incommensurate.