Spatiotemporal dynamics of quantum jumps with Rydberg atoms

We study the nonequilibrium dynamics of quantum jumps in a one-dimensional chain of atoms. Each atom is driven on a strong transition to a short-lived state and on a weak transition to a metastable state. We choose the metastable state to be a Rydberg state so that when an atom jumps to the Rydberg state, it inhibits or enhances jumps in the neighboring atoms. This leads to rich spatiotemporal dynamics that are visible in the fluorescence of the strong transition.

Figure 1

(a) An atom has a ground state $|g\rangle$, short-lived excited state $|e\rangle$, and metastable state $|r\rangle$, which is chosen to be a Rydberg state. One observes the spontaneous emission from $|e\rangle$. (b) The $|g\rangle \to |r\rangle$ transition is originally on resonance (${\Delta }_r=0$), but when one atom is in $|r\rangle$, the other atom is off resonance. (c) The $|g\rangle \to |r\rangle$ transition is originally off resonance (${\Delta }_r=V$), but when one atom is in $|r\rangle$, the other atom is on resonance. (d) When ${\Delta }_r=0$, $|rr\rangle$ is weakly coupled to the other states. Note that (b) and (d) are equivalent.

Figure 2

Quantum trajectory simulations of a chain of $N=8$ atoms with periodic boundary conditions. The Rydberg population of each atom is plotted vs time, using the color scheme on the right. White color means that the atom is bright and not in the Rydberg state. Black color means that the atom is dark and in the Rydberg state. (a) ${\Omega }_e=0.2{\gamma }_e$, ${\Omega }_r=0.005{\gamma }_e$, ${\Delta }_r=0$, $V=0.1{\gamma }_e$. (b) ${\Omega }_e=0.2{\gamma }_e$, ${\Omega }_r=0.005{\gamma }_e$, ${\Delta }_r=V=0.1{\gamma }_e$. (c) ${\Omega }_e={\Omega }_r=0.1{\gamma }_e$, ${\Delta }_r=0$, $V=0.4{\gamma }_e$.

Figure 3

Quantum trajectory simulations of $N=2$ atoms. The Rydberg population of each atom is plotted vs time, using the color scheme on the right. The parameters are the same as in Fig. 2: (a) ${\Omega }_e=0.2{\gamma }_e$, ${\Omega }_r=0.005{\gamma }_e$, ${\Delta }_r=0$, $V=0.1{\gamma }_e$. (b) ${\Omega }_e=0.2{\gamma }_e$, ${\Omega }_r=0.005{\gamma }_e$, ${\Delta }_r=V=0.1{\gamma }_e$. (c) ${\Omega }_e={\Omega }_r=0.1{\gamma }_e$, ${\Delta }_r=0$, $V=0.4{\gamma }_e$.

Figure 4

Ratio of ${\Gamma }^{BD\to DD}$ to ${\Gamma }^{BD\to BB}$ for $N=2$ atoms with ${\Omega }_e=0.2{\gamma }_e$, ${\Omega }_r=0.005{\gamma }_e$, $V=0.1{\gamma }_e$.

Figure 5

Dynamics of dark regions in a chain of $N=8$ atoms with ${\Omega }_e=0.2{\gamma }_e$, ${\Omega }_r=0.005{\gamma }_e$, $V=0.1{\gamma }_e$. The rates of expansion (black squares), contraction (red circles), and merging (blue triangles) were determined from quantum trajectory simulations. The simulation for each value of ${\Delta }_r$ was run for a time of ${10}^6/{\gamma }_e$, and the rates were calculated by sampling at a rate of ${\gamma }_e$ and defining an atom to be dark if $\langle R_i\rangle >0.98$. The scatter of data points with low rates is due to statistical uncertainty. Analytical predictions are shown for the rates of expansion (black, solid line), contraction (red, dashed line), and merging (blue, dash-dotted line).

Figure 6

Jump rates of $N=2$ atoms with ${\Omega }_r={\Omega }_e=0.1{\gamma }_e$, ${\Delta }_r=0$. ${\Gamma }^{DD\to BB}$: analytical result (black, solid line) and numerical data (black circles). ${\Gamma }^{BB\to DD}$: analytical upper bound (blue, dashed line) and numerical data (blue triangles).

Figure 7

Probability that the atom has not emitted by time $t$, given that it emitted at time 0. Parameters are ${\Omega }_e=0.2{\gamma }_e$, ${\Omega }_r=0.005{\gamma }_e$, ${\Delta }_e={\Delta }_r={\gamma }_r=0$.