Quantum Spin Dimers from Chiral Dissipation in Cold-Atom Chains

We consider the nonequilibrium dynamics of a driven dissipative spin chain with chiral coupling to a one-dimensional (1D) bosonic bath, and its atomic implementation with a two-species mixture of cold quantum gases. The reservoir is represented by a spin-orbit coupled 1D quasicondensate of atoms in a magnetized phase, while the spins are identified with motional states of a separate species of atoms in an optical lattice. The chirality of reservoir excitations allows the spins to couple differently to left- and right-moving modes, which in our atomic setup can be tuned from bidirectional to purely unidirectional. Remarkably, this leads to a pure steady state in which pairs of neighboring spins form dimers that decouple from the remainder of the chain. Our results also apply to current experiments with two-level emitters coupled to photonic waveguides.

Figure 1

The 1D spin chain coupled to a 1D chiral bosonic reservoir. (a) Driven spins decay into right- and left-moving reservoir modes with rates ${\gamma }_R$ and ${\gamma }_L$. For ${\gamma }_R\ne {\gamma }_L$, quantum spin dimers (indicated by $|D⟩$) are formed as the unique pure steady state. (b-d) Implementation with a two-species mixture of cold atoms. (b) Spins are represented by the two lowest vibrational states of atoms $a$ on each site of a 1D optical lattice, which can “decay” due to collisions with a 1D SOC quasi-BEC, representing the bath. (c) SOC of atoms $b$ due to coupling of two internal states $|\uparrow ⟩$ and $|\downarrow ⟩$ via a Raman process [29]. (d) Dispersion relations $\hslash {\omega }_{k\beta }$ of the bath excitations in the plane wave phase. The red and blue arrows indicate excitations of atoms $b$ from the quasi-BEC (circle at $k_m$) to wave vectors $k_L$ and $k_R$, resonant with $\hslash \omega$.

Figure 2

Tunability of decay asymmetry into chiral left- and right-moving modes (a) ${\gamma }_L/{\gamma }_R$ as a function of $\hslash {\mathrm{\Omega }}_0/E_0$ for $g_{a\downarrow }=g_{a\uparrow }$ (solid line) and $g_{a\downarrow }=0$ (dashed line). The dash-dotted line shows the reservoir spin polarization ${\overline{\rho }}_{\downarrow }/{\overline{\rho }}_{\uparrow }$. (b) Density fluctuation coefficients $Q_{-}^{\lambda }(k)$ ($\lambda =\uparrow ,\downarrow$) in the lower branch for $\hslash {\mathrm{\Omega }}_0=0.2E_0$. The wave vectors for left- and right-moving excitations $k_s$ ($s=L,R$) are indicated, where $Q_{-}^{\lambda }(k_s)$ show their strong spin polarization. Other parameters are $\overline{\rho }=6.14k_0$, $m_a/m_b=2$, $g_{\uparrow \uparrow }=g_{\uparrow \downarrow }=g_{\downarrow \downarrow }=0.23E_0/k_0$, $g_{a\uparrow }=-0.37E_0/k_0$, $\hslash \omega =1.46E_0$, and ${\delta }_0=-0.004E_0$.

Figure 3

Dynamical formation of spin dimers as the unique steady state of the driven-dissipative spin chain. We plot the entropy $S_{j,j+1}(t)$ of all adjacent spin pairs (solid lines) and the purity $\mathcal{P}(t)$ of the total state (black dashed line) for the initial condition $|\mathrm{\Psi }(0)⟩={\otimes }_{j=1}^N|g{⟩}_j$. Results are shown for $\mathrm{\Omega }=0.5{\gamma }_R$ and (a) $N=10$, ${\gamma }_L=0$, (b) $N=10$, ${\gamma }_L=0.4{\gamma }_R$, (c) $N=9$, ${\gamma }_L=0$, (d) $N=9$, ${\gamma }_L=0.4{\gamma }_R$.

Figure 4

Robustness of the dimerized steady state against imperfections for $N=6$. (a) Pair purities ${\mathcal{P}}_{2j-1,2j}$ and total purity $\mathcal{P}$ as a function of $\epsilon$ (see text), for ${\gamma }_L=0.1$ (dashed line) and ${\gamma }_L=0.4{\gamma }_R$ (solid line). (b) ${\mathcal{P}}_{2j-1,2j}$ and $\mathcal{P}$ as a function of decay outside the 1D bath ${\gamma }^{\prime }$, for ${\gamma }_L=0$ (dashed line) and ${\gamma }_L=0.4{\gamma }_R$ (solid line). We fix $\mathrm{\Omega }=0.5{\gamma }_R$.