Open quantum systems and random matrix theory

A simple model for open quantum systems is analyzed with random matrix theory. The system is coupled to the continuum in a minimal way. In this paper the effect on the level statistics of opening the system is seen. In particular the ${\Delta }_3\left(L\right)$ statistic, the width distribution and the level spacing are examined as a function of the strength of this coupling. The emergence of a super-radiant transition is observed. The level spacing and ${\Delta }_3\left(L\right)$ statistics exhibit the signatures of missed levels or intruder levels as the super-radiant state is formed.

Figure 1

The evolution of the energy levels with $\kappa$. Here the 300 unfolded energies of one particular matrix have been evaluated. $E_n$ vs $\kappa$ for levels 130 to 195 of the unfolded spectrum have been plotted. The curves, or “trajectories,” around $\kappa =1$ vary from matrix to matrix.

Figure 2

The evolution of the energy and entropy as $\kappa$ increase for an $N=50$ system. The blue (gray) lines correspond to the basis which the original Hamiltonian $H^0$ is written in. The black lines correspond to the basis in which $H^0$ is diagonal. In both cases, the super-radiant state is obvious. Notice that one blue (gray) line drops down to low entropy as $\kappa$ increases. One can think of Fig. 1 as a bird's eye view of this plot, where just the energy and $\kappa$ values can be seen. Note that here the GOE average value of the entropy is $S=3.2$, which is the level most of the blue (gray) lines stay at. Conversely the states in the energy basis grow in complexity. Notice how some of the black lines rise to a higher entropy as $\kappa$ increases.

Figure 3

The density of an ensemble of 2000 matrices with $N=100$ and $\kappa =0,0.4,1.2$, and 2.8. The level density is close to the semicircle of the GOE even for $\kappa =0.4$. The deviations are consistent with Fig. 1 where levels migrate to the center of the energy range.

Figure 4

When $\kappa$ is turned on, the levels acquire a width, $\Gamma$. Here the set of $300\Gamma$ for each spectrum in an ensemble of 200 was sorted. For example, ${\Gamma }_3$ is the third largest width. The 9 lines in this plot are the ensemble average of ${\Gamma }_i$ vs $\kappa$, with $i=2\cdots 10$. The insert includes the largest width, ${\Gamma }_1$, which eventually becomes linear in $\kappa$.

Figure 5

This is a log-log plot of the average of the lines in Fig. 4.

Figure 6

The ensemble result for the distribution of reduced widths $p(\gamma /\overline{\gamma })$. The full set of 2.5 million widths, which are a superposition of 250 000 matrices with $N=100$ and $\kappa =1.5$, is in red (gray). The mean of this set is $\overline{\gamma }=0.15$. In black are the results for the smallest 100 widths of 88 646 matrices with $N=102$. The mean of this subset is $\overline{\gamma }=0.0549$. The blue (gray) line is a plot of the function $P\left(x\right)=\frac{1}{\sqrt{2\pi x}}exp(-\frac{x}{2})$.

Figure 7

The ensemble average of ${\Delta }_3\left(L\right)$ vs $\kappa$. The 23 lines are for the 23 values of $L$, with $5\le L\le 50$ in steps of 2. There are 200 spectra in the ensemble, with $N=300$.

Figure 8

The ensemble average of ${\Delta }_3(L,\kappa )$ vs $\kappa$ for $20\le L\le 50$, but now each line is divided by ${\Delta }_3(L,0)$. The lower $L$ values are not as sensitive to $\kappa$ as the window of $L$ levels is too narrow to probe long-range correlations, so it is best to not include the range $L<20$. The legend refers to the three bold lines in the plot. The lines for various $L$ become closer as $L$ increases.

Figure 9

The results of tests to determine the fraction of levels missed in incomplete spectra vs $\kappa$ for open but complete GOE spectra. $N$ is 300 in all cases.