Emergence of Unique Steady Edge States in Trapped Ultracold Atom Systems

Abstract

We show that, for a one - dimensional open quantum system of ultracold atoms trapped in an array of harmonic potentials that is weakly coupled to a background Bose - Einstein Condensate (BEC), a unique steady state emerges at either of the two edges of the array due to the combined effects of excitation via lasers of these ultracold atoms and decay back to their initial energy levels via emission of excitations into the BEC, acting as an excitation reservoir. We then solve, both numerically and analytically, for the steady states of the master equation that describes the dynamics of this open quantum system, and show that these steady states occur at the edges of the array of harmonic potentials trapping these atoms. Using the open quantum system's master equation to evolve it numerically over time, we demonstrate that these steady states at the edge of the system will emerge regardless of the number of atoms trapped in each of the harmonic potentials in the array, establishing both their existence and uniqueness, and demonstrating that this driven trapped ultracold atom system coupled to a BEC is a topological material whose topological invariant is characterized by its master equation.

Figures (9)

• Figure 1: Schematic diagram of a portion of the harmonic oscillator array used to trap the ultracold atoms in this open quantum system. The atoms are initially in the ground state energy level $\epsilon_g$ of the harmonic oscillators centered at $x_l$ and $x_{l+1}$ (green lines), and are excited by Rabi lasers (pink lines) with Rabi frequencies $\Omega_l$ and $\Omega_{l+1}$ to the excited energy level (dashed orange line) located below the higher energy level $\epsilon_e$ of the harmonic potential centered at $x_{l+1/2}$ (solid orange line). Emission of an excitation with energy $E_\mathbf{k}$ (dashed sky blue line) into the background BEC will cause the excited atoms to return to the ground state energy level in either of the two neighboring harmonic oscillators centered at $x_l$ and $x_{l+1}$.
• Figure 2: Time evolution of the expectation values of the particle number $n_j(t)$ in the harmonic traps located at $x_j = -2x_0$$n_{1}(0)=N_1$, $x_j = x_0$$n_{2}(0)=N_2$ and $x_j = 2x_0$$n_{3}(0)=N_3$, with the values of $N_1, N_2, N_3$ indicated for each plot, such that $N_2 >N_1$ and $N_2 > N_3$.
• Figure 3: Time evolution of the expectation values of the particle number $n_j(t)$ in the harmonic traps located at $x_j = -3x_a$$n_{1}(0)=N_1$, $x_j = -x_a$$n_{2}(0)=N_2$, $x_j = x_a$, $n_{3}(0)=N_3$ and $x_j = 3x_0$$n_{4}(0)=N_4$, with the values of $N_1, N_2, N_3, N_4$ indicated for each plot, such that $N_1 = N_4$, $N_2 >N_1$ and $N_3 > N_4$.
• Figure 4: Time evolution of the expectation values of the particle number $n_j(t)$ in the harmonic traps located at $x_j = -2x_0$$n_{1}(0)=N_1$, $x_j = x_0$$n_{2}(0)=N_2$ and $x_j = 2x_0$$n_{3}(0)=N_3$, with the values of $N_1, N_2, N_3$ indicated for each plot, such that $N_2 = 100$, $N_1$ decreases from $N_1 = 600$ to $N_1 = 100$ and $N_3$ increases from $N_3 = 300$ to $N_3 = 800$.
• Figure 5: Time evolution of the expectation values of the particle number $n_j(t)$ in the harmonic traps located at $x_j = -3x_0$$n_{1}(0)=N_1$, $x_j = -x_0$$n_{2}(0)=N_2$, $x_j = x_0$$n_{3}(0)=N_3$ and $x_j = 3x_0$$n_{4}(0)=N_3$, with the values of $N_1, N_2, N_3, N_4$ indicated for each plot, such that $N_2 = N_3 = 50$, $N_1$ decreases from $N_1 = 600$ to $N_1 = 45$ and $N_4$ increases from $N_3 = 300$ to $N_3 = 855$.