Rydberg crystallization detection by statistical means

We investigate an ensemble of atoms which can be excited into a Rydberg state. Using a disordered quantum Ising model, we perform a numerical simulation of the experimental procedure and calculate the probability distribution function $P\left(M\right)$ to create a certain number of Rydberg atoms $M$, as well as their pair-correlation function. Using the latter, we identify the critical interaction strength above which the system undergoes a phase transition to a Rydberg crystal. We then show that this phase transition can be detected using $P\left(M\right)$ alone.

Figure 1

Density distribution of Rydberg atoms; darker color indicates higher density. Left panel: 1D density as a function of interaction strength. The system boundaries are preferred. Strong interactions produce a single peak in the center. Parameters: $\Omega /\Delta =0.1$ and $N=M^{*}=6$ atoms (no cutoff, exact diagonalization). Right panel: 2D projection of a 3D system with $N=M^{*}=6$, $\Omega /\Delta =4$, and $C=10$ in a cube with open boundaries. There is a low density at the center of the volume, while it is high in the corners.

Figure 2

Histogram of the number of Rydberg atoms with Gaussian fit. The data points shown are the cumulative sum of all bins in the interval $[n-0.5,n+0.5]$. The features of the plot are explained in the text. Parameters: $\Omega /\Delta =0.1$, $C/\left(\Delta L^6\right)=4\times {10}^{-7}$, with $N=10=M^{*}$ atoms (which corresponds to no cutoff). Inset: Histogram for $\Omega /\Delta =0.64$, $C/\left(\Delta L^6\right)=5\times {10}^{-7}$, with $N=10=M^{*}$ atoms.

Figure 3

Average number of (a) Rydberg atoms $\mu$ and (b) variance $\sigma$ in dependence on $C$ (double-log plots). The straight lines in (a) are guides for the eye only. The parameters for all plots are $\Omega /\Delta =0.2$ with $N=10=M^{*}$ atoms.

Figure 4

(a) Correlation function with adjustment of the system length in such a way that main and secondary peaks coincide, $C/\left(\Delta L^6\right)=7.5\times {10}^{-6}$. (b) Correlation function as a function of distance. The aliasing can be seen in the shoulders of the second and third maximum, $C/\left(\Delta L^6\right)=4\times {10}^{-6}$. For both correlation functions, $N=30$ and $\Omega /\Delta =25$. The dashed vertical line indicates where to read off the blockade radius (used in later plots).

Figure 5

Schematic illustration of the effect shown in Fig. 4. The dashed line is part of the correlation function in the right interval shifted by the length of the interval. The sum of the dashed and solid curve in the left interval is a curve with main peaks that have secondary peaks on their shoulders. The secondary peak on the shoulder of the first peak is suppressed because of the blockade phenomenon.

Figure 6

$R_B$ as a function of $C$ (double-log plot). The data points clearly show a power law. The parameters are $\Omega /\Delta =0.1$ with $N=8=M^{*}$ atoms. The fit is discussed in the text. Inset: $R_B$ as a function of $\Omega$ (double-log plot). The parameters are $C/\left(\Delta L^6\right)={10}^{-5}$ with $N=7=M^{*}$ atoms.

Figure 7

Correlation function as a function of distance. The dashed line represents a fit explained in the text. Inset: The same plot as in the main plot shown on a logarithmic scale. The parameters of the plot are $\Omega /\Delta =1$, $C/\left(\Delta L^6\right)=2\times {10}^{-5}$, with $N=6=M^{*}$ atoms.

Figure 8

Correlation function of 1D samples measured in arbitrary units. The curves correspond to $C/\Delta L^6=4\times {10}^{-9}$ (dotted), $C/\Delta L^6=4\times {10}^{-7}$ (dashed), and $C/\Delta L^6=4\times {10}^{-6}$ (solid), where all other parameters remained the same: $\Omega /\Delta =0.4$ and $N=25$, with 300 basis states. The curves indicate that there is a critical value for $C/\Delta L^6$ between the two larger values given above.