Localization behavior in a Hermitian and non-Hermitian Raman lattice

TL;DR

This work investigates localization phenomena in a one-dimensional quasi-periodic Raman lattice implemented with alkaline-earth-like atoms, focusing on how spin-dependent incommensurate potentials and non-Hermitian dissipation shape Anderson localization. The authors formulate a tight-binding model $H=H_0+\sum_{j,\sigma}(\delta_{\sigma}^{j}+i\gamma_{\sigma})n_{j,\sigma}$ with $\delta_{\sigma}^{j}=M_{z,\sigma}\cos(2\pi\beta j)$ and spin-dependent hopping, and discuss an experimental scheme using $^{173}$Yb to realize spin-selective incommensurate lattices via detuning control near the $^1S_0(F=5/2)\rightarrow{}^3P_1(F=7/2)$ transition. In the Hermitian regime, a fully spin-dependent lattice ($M_{z,\uparrow}/M_{z,\downarrow}=-1$) yields a critical phase with $0<\bar{\eta}<1$ and a phase boundary $M_z=2|t_0\pm t_{so}|$, while partial spin-dependence leads to mobility-edge-like coexistence of extended, critical, and localized states. Introducing non-Hermitian dissipation ($\gamma\neq0$) suppresses the critical phase, producing a mixed extended/localized regime, and eliminating the pure critical region by destroying generalized incommensurate zeros in the on-site potential. The study provides a practical pathway to explore localization phenomena and non-Hermitian effects with ultracold atoms, linking theory to potential experiments with alkaline-earth systems.

Abstract

We propose a flexible Raman lattice system for alkaline-earth-like atoms to theoretically investigate localization behaviors in a quasi-periodic lattice with controllable non-Hermiticity. Our analysis demonstrates that critical phases and mobility edges can arise by adjusting spin-dependence of the incommensurate potentials in the Hermitian regime. With non-Hermiticity introduced by spin-selective atom loss, our calculations reveal that critical localization behaviour in this system can be suppressed by dissipation. Our work provides insights into interplay between quasi-periodicity and non-Hermitian physics, offering a new perspective on localization phenomena.

Figures (11)

• Figure 1: Experimental implementation scheme of critical phase in optical Raman lattice with $^{173}$Yb atoms. (a) Experimental setup of the optical Raman lattice consists of a standard 1D optical Raman lattice $E_1$, $E_3$ with another spin-dependent incommensurate lattice $E_{21}$ and $E_{22}$. (b) Schematic energy diagram with relevant transitions. (c) The incommensurate lattice induces a spin-dependent offset $V_{2,\sigma}(x)$ to the standard optical Raman lattice potential. (d) For the implementation of a perfect spin-dependent incommensurate lattice, the laser detuning should be set to approximately 1.9 GHz blue-detuned from the $^1S_0(F=5/2)\rightarrow{}^3P_1(F=7/2)$ transition.
• Figure 2: Phase diagram of the optical Raman lattice system with additional incommensurate lattice. (a) The various phases, namely extended, critical, and localization phases, are distinguished based on the mean fractal dimension $\bar{\eta}$ when $M_{z,\uparrow}/M_{z,\downarrow}=-1$. $M_z$ and $t_{so}$ are presented in units of $t_0$. (b) Fractal dimension $\eta$ and energy of individual states for distinct values of $M_z$ with $t_{so}=0.3$ and $M_{z,\uparrow}/M_{z,\downarrow}=-1$. (c) Phase diagram of mean fractal dimension $\bar{\eta}$ as a function of $M_z$ and $t_{so}$ when $M_{z,\uparrow}/M_{z,\downarrow}=1$. Other parameters are identical to those in b. (d) Fractal dimension $\eta$ and energy of individual states for distinct values of $M_z$ with $t_{so}=0.3$ and $M_{z,\uparrow}/M_{z,\downarrow}=1$. All results are simulated with $L=1597$ and $\beta=987/1597$ in periodic boundary condition.
• Figure 3: Simulation of detecting critical phase with expansion dynamics. (a) Time evolution of the wave packet for the system located in the extended phase ($M_z=0.5$), critical phase ($M_z=2.0$), and localized phase ($M_z=3.5$). (b) Expansion dynamics of the wave packet, characterized by $W$, as a function of $M_z$, with $t_{so}=0.3$ and an initial width of $a=5$. (c) Time-averaged $\bar{W}$ as a function of $M_z$ with different initial width $a$. All results are simulated with $L=400$ and $\beta=\cos{52^\circ}$ under open boundary condition.
• Figure 4: Evolution of spin polarization for different phases. (a) Spin dynamics as a function of $M_z$, with $t_{so}=0.3$, and an initial distribution in the spin up state with uniform distribution. (b) Time-averaged spin polarization as a function of $M_z$ and $t_{\text{so}}$. (c, d) Corresponding reconstructed momentum distribution during the time evolution at extended phase ($M_z=0.5$) and critical phase ($M_z=2.0$). All simulations are conducted with $L=400$ and $\beta=\cos{52^\circ}$ under open boundary conditions.
• Figure 5: Evolution of density imbalance for different phases. (a, b) Time evolution of the density imbalance for spin-up (a) and spin-down (b) atoms in the critical phase ($M_z=2.0, t_{\text{so}}=0.3$) and the localized phase ($M_z=5.0, t_{\text{so}}=0.3$). (c) Time-averaged density imbalance as a function of $M_z$ and $t_{\text{so}}$. The markers indicate the parameter sets corresponding to the dynamics shown in (a) and (b). (d) Time-averaged spin polarization $\bar{I}_p$ and density imbalance $\bar{I}_d$ as a function of $M_z$ averaged over the time interval $t=0-200$. The expected phase transition position is marked by a vertical dashed line. All simulations are conducted with $L=400$ and $\beta=\cos{52^\circ}$ under open boundary conditions.