Spin–orbit coupling in the presence of strong atomic correlations

For weakly interacting bosons in the mean-field regime, the presence of SOC leads to the possibility of having three different ground state phases [8, 32]: the stripe phase, the magnetised phase, and the single minimum phase. The stripe phase is named after the fact that an interference pattern appears due to superposition of positive and negative momenta [8]. In contrast, in the magnetised phase the gas either fully adopts positive or negative momentum and in the single minimum phase the spectrum only possess a single minimum. For non-interacting bosons only the stripe phase and the single minimum phase exist. In free space the critical point between these two phases is given by ${\hslash }{{\rm{\Omega }}}_{{\rm{c}}}=2{{\hslash }}^{2}{k}_{\mathrm{soc}}^{2}/m$ and it is known numerically that for harmonically trapped systems the value of the critical point is lower [23].

It is worth noting that in the non-interacting limit the above Hamiltonian is analogous to that of certain collective spin models, namely the Dicke model [33] and the Lipkin–Meshkov–Glick (LMG) model [34]. The explicit connection between SO-coupled systems and the Dicke model was established in [10, 35], and there is a direct relation between the Dicke and LMG models shown in, for example [36]. In the thermodynamic limit the ground state of the LMG model is known to exhibit a second order phase transition at a critical value of an applied field [37, 38], which corresponds to the transition between the stripe phase and the single minimum phase for SO-coupled systems when changing the Raman coupling strength. This transition is signalled by a divergence in the second derivative of the ground state energy and can also be seen in the lifting of existing degeneracies [39]. This duality between the many-body spin system and the SO-coupled atoms emerges due to the nature of the phase transition in the LMG case. There the infinite range of the spin–spin interactions permits expression of the many-body system in terms of collective spin operators with the total angular momentum being conserved. In this picture the LMG model is represented by a single large N-dimensional spin, and in thermodynamic limit one phase is effectively given by a double-well configuration possessing a ground state energy degeneracy, while in the other a gaped spectrum exists. This is analogous to the form of the dispersion relation for the stripe phase and the single minimum phase. Naturally, a two-particle system is far from the thermodynamic limit and therefore cannot be expected to exhibit all characteristics of the transition, e.g. the discontinuity in the second derivative of the energy [35]. Nevertheless, it is interesting to note that at a value of Ω ≃ Ω_lift the degeneracies in the two-particle spectrum are still lifted (see figure 2 and see figure 1 of [39]).

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One can also see from figure 1(a) that in the absence of Raman coupling the ground state of the two atom system is three-fold degenerate: $| \downarrow \downarrow \rangle$ with COM motion ${{\rm{e}}}^{{{\rm{i}}{k}}_{\mathrm{soc}}{x}_{+}}$ in the positive direction, $| \uparrow \uparrow \rangle$ with COM motion ${{\rm{e}}}^{-{{\rm{i}}{k}}_{\mathrm{soc}}{x}_{+}}$ in the negative direction, and the symmetric combination of $| \downarrow \uparrow \rangle$ and $| \uparrow \downarrow \rangle$ with non-zero relative motion. Finite strengths of the Raman coupling then mix these states, and in the strong Raman coupling limit, Ω ≫ Ω_lift, the ground state becomes an equally-weighted superposition of all pseudo-spin states, $\psi \sim \left(| \downarrow \downarrow \rangle -| \downarrow \uparrow \rangle -| \uparrow \downarrow \rangle +| \uparrow \uparrow \rangle \right)/2$.

Let us next investigate the competition between SOC and repulsive contact interactions between anti-aligned spins, ${g}_{\uparrow \downarrow }\gt 0$. The level scheme is shown in figure 1(b) and one can immediately see that the finite interaction lifts a number of degeneracies, as the energy of the basis states of the relative motion is increased. The spectra resulting from these nontrivial energy shifts for two different interaction strengths are shown in figures 2(b) and (c).

Let us consider ${\rm{\Omega }}\ll {{\rm{\Omega }}}_{\mathrm{lift}}$ first. While without interactions the lowest energy states are composed from three degenerate states, for ${g}_{\uparrow \downarrow }\gt 0$ the energies of $| \downarrow \uparrow \rangle$ and $| \uparrow \downarrow \rangle$ rise, and the lowest energy states are thus the symmetric spin states with positive momentum of COM motion, $| \downarrow \downarrow \rangle {{\rm{e}}}^{{{\rm{i}}{k}}_{\mathrm{soc}}{x}_{+}}$, and negative one, $| \uparrow \uparrow \rangle {{\rm{e}}}^{-{{\rm{i}}{k}}_{\mathrm{soc}}{x}_{+}}$. The first excited state is given by the symmetric superposition of the anti-aligned states, which each have non-zero relative motion, $\psi \sim | \downarrow \uparrow \rangle {{\rm{e}}}^{{{\rm{i}}{k}}_{{\rm{soc}}}{x}_{-}}+| \uparrow \downarrow \rangle {{\rm{e}}}^{-{{\rm{i}}{k}}_{{\rm{soc}}}{x}_{-}}$, and it has an energy shift of $2\nu {\hslash }\omega$, ν of which is given by solving equation (8) and depends on the interaction strength g_↑↓. In the strong coupling regime (Ω ≫ Ω_lift), the effect of the interactions becomes negligible, and the ground state approaches that of no interactions. As the states for weak and strong Raman coupling regimes must be adiabatically connected, the crossover from states that are dominated by the interaction to those that are not leads to the appearance in the spectrum of higher-lying avoided crossings [20]. However, the position of the critical point is only slightly modified by the interactions, due to the fact that the resulting energy shift can at most be ${\rm{\Delta }}E={\hslash }\omega$, which is small compared to the energy scales given by the recoil energy and the Raman coupling.

The contributions to the ground state of each pseudo-spin state are shown in figure 3. For weak interactions (${g}_{\uparrow \downarrow }/{\hslash }\omega {a}_{\mathrm{ho}}=1$, see panel (a)) and for weak Raman coupling (Ω/ω ≲ 10) the ground state is given approximately by $\psi \sim (| \downarrow \downarrow \rangle +| \uparrow \uparrow \rangle )/\sqrt{2}$, and is reminiscent of that of the stripe phase in BECs with SOC as each spin component possesses a finite COM momentum [8]. For increasing Ω, a greater proportion of anti-aligned states is mixed into the ground state, however the symmetric states $\{| \downarrow \downarrow \rangle$, $| {\rm{S}}\rangle$, $| \uparrow \uparrow \rangle \}$ always dominate (see inset in panel (a)). This behaviour is similar for systems with strong interactions (${g}_{\uparrow \downarrow }/{\hslash }\omega {a}_{\mathrm{ho}}=10$, see panel (b)), but one can also see a region in which the contributions from the aligned spins drops faster with increasing Ω, to the point where the contribution from the anti-aligned states exceeds those from aligned states. This occurs around the value of Ω = Ω_lift and we show in panel (c) that this inversion appears already for interaction strengths ${g}_{\uparrow \downarrow }/{\hslash }\omega {a}_{\mathrm{ho}}\gtrsim 2$. The phase after the inversion is not analogous to any phase of BECs with SOC in the weakly interacting mean-field limit and we will discuss it in more detail below. Increasing Ω even further, the ground state again moves towards that corresponding to that for the single minimum phase.

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While one would naively assume that the ground state contains more interaction energy when larger contributions of the $| \downarrow \uparrow \rangle$ and $| \uparrow \downarrow \rangle$ states are present, one can see from the inset in panel (b) that the increase in contributions from the anti-aligned states is due to an increase in the populations of the anti-symmetric state $| {\rm{A}}\rangle$, which does not feel the interaction [21]. In fact, when the contribution of $| {\rm{A}}\rangle$ rises, the interaction energy can be seen to decrease (insets in panels (a) and (b)). This population imbalance inversion is a few-body effect and is not seen when treating single-particle states or BECs within the mean-field approximation. We therefore refer to the distinct and unique regime after this inversion as the AS ground state phase, as the anti-symmetric states are the dominant contribution.

The crossover from a ground state resembling the stripe phase to the AS ground state has direct consequences on the overall momentum and density distributions (see figure 4). For weak interaction (${g}_{\uparrow \downarrow }/{\hslash }\omega {a}_{\mathrm{ho}}=1$, see panels (a) and (b)), when the AS state does not appear, the ground state momentum distribution is dominated by a finite COM momentum on both sides of Ω_lift. However, for a strong interaction (${g}_{\uparrow \downarrow }/{\hslash }\omega {a}_{\mathrm{ho}}=10$, see panels (c) and (d)) this distribution changes from being dominated by the COM contribution to having more equal contributions from the COM and relative momenta, which is a sign that considerable amounts of $| \downarrow \uparrow \rangle$ and $| \uparrow \downarrow \rangle$ appear. For the density distribution an interference pattern along the COM coordinate is visible for Ω ≲ Ω_lift (see panels (e) and (g)), which is reminiscent of the stripe phase, but with a bisection due to the contact interaction at x₋ = 0. As Ω increases, the interference pattern remains for a weak interaction (see panel (f)), however, for a strong interaction a more pronounced pattern in the direction of the relative coordinate appears while that in the COM direction vanishes (see panel (h)). These changes in the spatial and momentum distributions clearly indicate the appearance of the AS ground state.

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We next turn our attention to the non-classical correlations inherent in the system. For this we use the concurrence as the entanglement measure of the pseudo-spin degree of freedom, which is constructed from the density matrix after tracing over the position space components. Representing the wave function of the ground state as $| \psi \rangle ={\sum }_{\chi }{\phi }_{\chi }({x}_{1},{x}_{2})| \chi \rangle$ for $\chi \in \{\downarrow \downarrow ,\downarrow \uparrow ,\uparrow \downarrow ,\uparrow \uparrow \}$, the density matrix can be written as $\rho =| \psi \rangle \langle \psi |$, and the reduced density matrix is real and given by

$$\begin{eqnarray}{\rho }_{\mathrm{spin}}=\left(\begin{array}{cccc}\theta & \beta & \beta & \gamma \\ \beta & \epsilon & \mu & \beta \\ \beta & \mu & \epsilon & \beta \\ \gamma & \beta & \beta & \theta \end{array}\right).\end{eqnarray} \tag{ 9 }$$

The inner products of the two aligned components are given by $\theta =\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\downarrow \downarrow }\rangle =\langle {\phi }_{\uparrow \uparrow }| {\phi }_{\uparrow \uparrow }\rangle$ and $\gamma =\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\uparrow \uparrow }\rangle =\langle {\phi }_{\uparrow \uparrow }| {\phi }_{\downarrow \downarrow }\rangle$, while the inner products of two anti-aligned components are given by $\epsilon =\langle {\phi }_{\downarrow \uparrow }| {\phi }_{\downarrow \uparrow }\rangle =\langle {\phi }_{\uparrow \downarrow }| {\phi }_{\uparrow \downarrow }\rangle$ and $\mu =\langle {\phi }_{\downarrow \uparrow }| {\phi }_{\uparrow \downarrow }\rangle =\langle {\phi }_{\uparrow \downarrow }| {\phi }_{\downarrow \uparrow }\rangle$. The diagonal elements of the density matrix therefore describe the population of each pseudo-spin state. Finally, the inner products of one aligned and one anti-aligned component are all equivalent and given by $\beta =\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\downarrow \uparrow }\rangle =\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\uparrow \downarrow }\rangle =\langle {\phi }_{\uparrow \uparrow }| {\phi }_{\uparrow \downarrow }\rangle =\langle {\phi }_{\uparrow \uparrow }| {\phi }_{\downarrow \uparrow }\rangle$ and the same for their Hermitian conjugates.

The concurrence is then given as

$$\begin{eqnarray}&&{C}_{\mathrm{spin}}=\max \{0,\sqrt{{\lambda }_{1}}-\sqrt{{\lambda }_{2}}-\sqrt{{\lambda }_{3}}-\sqrt{{\lambda }_{4}}\},\end{eqnarray} \tag{ 10 }$$

where the λ_j are the eigenvalues of ${\rho }_{\mathrm{spin}}({\sigma }_{y}\otimes {\sigma }_{y}){\rho }_{\mathrm{spin}}^{* }({\sigma }_{y}\otimes {\sigma }_{y})$ in descending order for j = 1, 2, 3, 4 [40, 41]. For separable states the concurrence vanishes, and for maximally entangled states it is equal to one. In figure 5(a) we show the behaviour of the concurrence as a function of Raman coupling and SOC strengths for a fixed contact interaction, ${g}_{\uparrow \downarrow }/{\hslash }\omega {a}_{\mathrm{ho}}=10$. In the limit of k_soc, Ω → 0, the ground state is given by $\left(| \downarrow \downarrow \rangle +| \uparrow \uparrow \rangle \right)/\sqrt{2}$, and hence C_spin → 1, i.e. the maximally entangled Bell state [42]. It is a key point that the diagonal terms θ and the off-diagonal terms γ of the reduced density matrix (9) go to 1/2 and the remaining inner products disappear in this limit. The off-diagonal terms γ show the correlation between $| \downarrow \downarrow \rangle$ and $| \uparrow \uparrow \rangle$. However for finite ${k}_{\mathrm{soc}}$ the entanglement decays for all values of Ω, which is despite the fact that for small Ω the ground state is equally weighted between $| \downarrow \downarrow \rangle$ and $| \uparrow \uparrow \rangle$, and therefore one would expect a maximally entangled state. The entanglement decays because the momentum provided by SOC suppresses the off-diagonal terms γ and kill the correlation. The SOC term therefore acts as an effective dephasing channel on the pseudo-spin states causing the coherences to decay rapidly while the diagonal terms, which account for the populations, remain unaffected. A final remarkable feature of the entanglement is its sudden vanishing, a phenomenon sometimes referred to as entanglement sudden death [43]. With the red line indicating the values at which the degeneracies are lifted (see figure 5(a)) one can immediately see that the pseudo-spin entanglement is not a useful indicator of the population inversion. However the effective dephasing caused by the SOC also implies that strong correlations are established between the pseudo-spin space and real-space.

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As the overall state is pure, we can analyse this entanglement using the von Neumann entropy (vNE) of the reduced density matrix ρ_spin, defined as

$$\begin{eqnarray}\begin{array}{rcl}S[{\rho }_{\mathrm{spin}}] & = & -\mathrm{Tr}\left\{{\rho }_{\mathrm{spin}}{\mathrm{log}}_{2}{\rho }_{\mathrm{spin}}\right\}\\ & = & \ -\displaystyle \sum _{j=1}^{4}{\alpha }_{j}{\mathrm{log}}_{2}{\alpha }_{j},\end{array}\end{eqnarray} \tag{ 11 }$$

where α_j is the jth eigenvalue of the reduced density matrix, ρ_spinϕ_j = α_jϕ_j. The behaviour of the vNE can be linked to the pseudo-spin populations through the eigenvalues of ρ_spin (see figure 5(c)), whereby we can explicitly calculate two eigenvalues as ${\alpha }_{1}=\theta -\gamma =\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\downarrow \downarrow }\rangle -\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\uparrow \uparrow }\rangle$ (yellow dotted) and ${\alpha }_{2}=\epsilon -\mu$ = $\langle {\phi }_{\downarrow \uparrow }| {\phi }_{\downarrow \uparrow }\rangle -\langle {\phi }_{\downarrow \uparrow }| {\phi }_{\uparrow \downarrow }\rangle$ (orange dotted). Also, noticing that α₃ ≈ 0 (blue dotted) allows us to approximate the last eigenvalue as ${\alpha }_{4}\approx \theta +\gamma +\epsilon +\mu =\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\downarrow \downarrow }\rangle +\langle {\phi }_{\downarrow \downarrow }| {\phi }_{\uparrow \uparrow }\rangle +\langle {\phi }_{\downarrow \uparrow }| {\phi }_{\downarrow \uparrow }\rangle +\langle {\phi }_{\downarrow \uparrow }| {\phi }_{\uparrow \downarrow }\rangle$ (purple dotted). In the weak Raman coupling limit, Ω ≪ Ω_lift, the vNE remains close to 1, which is half its maximal value (see figures 5(b) and (c)). That is to be expected as the $| \downarrow \downarrow \rangle$ state and $| \uparrow \uparrow \rangle$ state, which dominate the system, are connected to COM motion in the positive and negative directions, respectively, while the other two pseudo-spin states, $| \downarrow \uparrow \rangle$ and $| \uparrow \downarrow \rangle$, have lower populations and do not play major roles in the correlations. For Raman coupling strengths closer to Ω_lift one can see a kink appear in the behaviour of the vNE, which can be directly linked to the increase in population of the $| {\rm{A}}\rangle$ state. This is also signalled by the eigenvalue α₂ becoming non-zero, which implies the establishment of entanglement between the $| \downarrow \uparrow \rangle$ and $| \uparrow \downarrow \rangle$ states. For ${\rm{\Omega }}\gtrsim {{\rm{\Omega }}}_{\mathrm{lift}}$ the vNE decreases dramatically (the red line in figure 5(b) indicates Ω = Ω_lift) as the system possesses an almost equal contribution of all the pseudo spin states in the AS phase. Finally, when the wavelength of the SOC is on the order of the width of the ground state, $\sqrt{2}{k}_{\mathrm{soc}}\sim 1/{a}_{\mathrm{ho}}$, the vNE becomes maximal due to an enhanced coupling between the spin states as a result of the finite system size.

Finally we investigate the case where only the aligned interactions are finite, while the anti-aligned interactions are switched off (see figure 1(c)). In this situation, the basis states $| \uparrow \uparrow \rangle$ and $| \downarrow \downarrow \rangle$ are shifted in energy, while the symmetric $| {\rm{S}}\rangle$ and anti-symmetric $| {\rm{A}}\rangle$ states are unaffected, resulting in a different ground state compared to that seen for anti-aligned interactions. The energy spectra are shown in figures 6(a) and (b), and one can immediately confirm that for all values of Ω the ground state is non-degenerate (see inset in panel (a)). In fact, for small Ω it is given by $\psi \sim | \downarrow \uparrow \rangle {{\rm{e}}}^{{{\rm{i}}{k}}_{{\rm{soc}}}{x}_{-}}+| \uparrow \downarrow \rangle {{\rm{e}}}^{-{{\rm{i}}{k}}_{{\rm{soc}}}{x}_{-}}$ and unsurprisingly composed mostly of anti-aligned pseudo-spin states. The first excited states are composed of two degenerate states with COM momentum, $\psi \sim | \downarrow \downarrow \rangle {{\rm{e}}}^{{{\rm{i}}{k}}_{\mathrm{soc}}{x}_{+}}$ and $\psi \sim | \uparrow \uparrow \rangle {{\rm{e}}}^{-{{\rm{i}}{k}}_{\mathrm{soc}}{x}_{+}}$, and one can see that this degeneracy is resolved at a critical value similar to the one for the finite anti-aligned interactions. Interestingly, for weak aligned interactions ($g/{\hslash }\omega {a}_{\mathrm{ho}}=0.4$, see the inset in figure 6(a)) an avoided crossing between the ground state and the first excited states exists when ${\rm{\Omega }}/\omega \simeq 39\simeq {{\rm{\Omega }}}_{\mathrm{lift}}/\omega$. This avoided crossing is the result of the interaction and the Raman coupling trying to shift the energy in opposite directions [20]. For stronger interactions ($g/{\hslash }\omega {a}_{\mathrm{soc}}=1.5$, see figure 6(b)), the energy gap at the avoided crossing is increased as the contact interactions push the excited states to higher energies. The presence of the avoided crossing also impacts the population imbalance after the inversion. Compared to the case of the anti-aligned interactions, the inversion from the anti-aligned basis states being the dominant contributions to the ground state to the aligned ones now occurs more sharply (see figure 6(c)). Although the inversion does not occur without the interaction, it is worth noting that it appears even for an infinitesimal weak interaction and decays for increasing g (see figure 6(d)). This is a manifestation of a non-zero g shifting energy levels differently from non-zero g_↑↓. For the case of finite anti-aligned interaction, populating the $| {\rm{A}}\rangle$ state reduces the interaction energy (see figure 1(b)), whereas for non-zero g increasing Ω leads to larger contributions of $| \downarrow \downarrow \rangle$ and $| \uparrow \uparrow \rangle$ to the ground state (see figure 1(c)). However, these states both contribute to the interaction energy, and therefore for large g they are no longer favourable and the inversion disappears.

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The effect of the inversion on the overall momentum distribution is shown in figures 7(a) and (b). For Ω ≪ Ω_lift, the largest contribution to the ground state is the anti-aligned states $| \downarrow \uparrow \rangle$ and $| \uparrow \downarrow \rangle$ which possess net relative momentum from the SOC. When Ω ≃ Ω_lift, the population inversion occurs and the ground state possesses net COM momentum in the majority aligned states $| \downarrow \downarrow \rangle$ and $| \uparrow \uparrow \rangle$. This corresponds directly to the overall density distribution exhibiting a reorientation of the interference fringes from the relative to the COM direction in the crossover region (see figures 7(c) and (d)).

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The concurrence C_spin and the vNE S[ρ_spin] for $g/{\hslash }\omega {a}_{\mathrm{soc}}=0.4$, where the population inversion is large, can be seen in figures 8(a) and (b). Similar to the case of finite anti-aligned interactions, the concurrence does not signal the lifting of the energy degeneracy, but the vNE significantly decreases at ${\rm{\Omega }}\simeq {{\rm{\Omega }}}_{\mathrm{lift}}$. Furthermore, a sharp spike is observed in the vNE whenever Ω/ω ≃ Ω_lift (see figure 8(c) for k_soca_ho = 5). This is the result of the avoided crossing present in figure 6(a) which strongly entangles $| \uparrow \uparrow \rangle$ and $| \downarrow \downarrow \rangle$ spin states at the resonance point as their respective populations suddenly increase, while the symmetric $| {\rm{S}}\rangle$ and anti-symmetric $| {\rm{A}}\rangle$ state populations are suppressed. Indeed, this is the opposite effect described for anti-aligned interactions, and shows how the critical point can enhance correlations between interacting spin-components due to the competition between SOC and contact interactions.

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