Preparing Atomic Topological Quantum Matter by Adiabatic Nonunitary Dynamics

Motivated by the outstanding challenge of realizing low-temperature states of quantum matter in synthetic materials, we propose and study an experimentally feasible protocol for preparing topological states such as Chern insulators. By definition, such (nonsymmetry protected) topological phases cannot be attained without going through a phase transition in a closed system, largely preventing their preparation in coherent dynamics. To overcome this fundamental caveat, we propose to couple the target system to a conjugate system, so as to prepare a symmetry protected topological phase in an extended system by intermittently breaking the protecting symmetry. Finally, the decoupled conjugate system is discarded, thus projecting onto the desired topological state in the target system. By construction, this protocol may be immediately generalized to the class of invertible topological phases, characterized by the existence of an inverse topological order. We illustrate our findings with microscopic simulations on an experimentally realistic Chern insulator model of ultracold fermionic atoms in a driven spin-dependent hexagonal optical lattice.

Figure 1

Protocol for preparing topological states such as Chern insulators (CIs) from a trivial initial state in three steps (i)–(iii) (see text). Vertical arrows illustrate the distinction between CI and trivial states. As direct adiabatic preparation of a CI in the target system ($S$) is impossible, we intermittently couple $S$ to a conjugate system ($S^{*}$) so as to keep the total state topologically trivial at all times and avoid a topological quantum phase transition. Finally, $S^{*}$ is projected out to obtain the desired topological phase in $S$. Insets: data for the model in Eq. (2). The $S\text{−}{S}^{*}$ coupling $\lambda (\tau )={\lambda }_0\tau (1-\tau ){\sin }^2\pi \tau$ and the mass term $m(\tau )=m_0+(m_f-m_0)(10{\tau }^3-15{\tau }^4+6{\tau }^5)$. As long as ${\lambda }_0$ is finite, the gap $\mathrm{\Delta }(\tau )$ stays finite.

Figure 2

(a) A linearly shaking spin-dependent hexagonal lattice is subject to Raman lasers. The lattice is shaken along the ${\mathbf{e}}_F$ direction, where ${\mathbf{e}}_F=\cos {\phi }_F{\mathbf{e}}_x+\sin {\phi }_F{\mathbf{e}}_y$ is a unit vector on the $x\text{−}y$ plane. The spin $\uparrow$ and $\downarrow$ sectors—representing the system $S$ and the conjugate system $S^{*}$, respectively—lay on the same plane, but are artificially lifted for better visualization. Raman lasers are applied to activate local couplings (as indicated by the springs) between spin $\uparrow$ and $\downarrow$ atoms on the same lattice sites. (b) The spin-dependent optical potentials for spin $\uparrow$ and $\downarrow$ atoms within a unit cell of $\mathbf{A}$ and $\mathbf{B}$ sublattice sites. The solid (dashed) curve is the potential for spin $\uparrow$ ($\downarrow$) atoms. Raman lasers with a two-photon detuning $\delta$ flip spins locally. Note that the Raman coupling amplitude ($\pm {\mathrm{\Omega }}_R$) is designed to take a sublattice-dependent sign factor.

Figure 3

(a) The phase diagram of the system $S$ as a function of $\alpha$ and ${\phi }_F$ with $\delta =0$. The conjugate system $S^{*}$ has the opposite Chern number. (b) The phase diagram as a function of $\alpha$ and $\delta$ for ${\phi }_F=0.09\pi$. The (black) green dot indicates the initial (final) state of the ramping protocol. (c) The band gap $\mathrm{\Delta }(\tau )$ for different values of the Raman coupling. The ramping protocol for the detuning term is $\delta (\tau )={\delta }_i+({\delta }_f-{\delta }_i)f(\tau )$ with $f(\tau )=15{\tau }^7-70{\tau }^6+126{\tau }^5-105{\tau }^4+35{\tau }^3$ and $\tau =t/T$, being $T$ the ramp time. The coupling between $S$ and $S^{*}$ is $\mathrm{\Lambda }(\tau )={\mathrm{\Omega }}_R(\tau ){\sigma }_z$ with ${\mathrm{\Omega }}_R(\tau )={\mathrm{\Omega }}_0(1-\tau )\sin \pi \tau$. Here, $t_{\mathrm{NNN}}=1$ and $t_{\mathrm{NN}}=5$.

Figure 4

(a) Time evolution of the Chern number, the Hall conductivity, and the minimum of the purity for $T=80$. (b) The minimum of the purity gap $P(T)$ as a function of the ramping time $T$. Insets: the Chern number $C(T)$ and the Hall conductivity $\mathrm{\Sigma }(T)$ as a function of the ramping time $T$; here ${\phi }_F=0.09\pi$, $\alpha =3.5$, ${\delta }_i=2$, ${\delta }_f=0$, and ${\mathrm{\Omega }}_0=0.4$.