Universality of spectra for interacting quantum chaotic systems

We analyze a model quantum dynamical system subjected to periodic interaction with an environment, which can describe quantum measurements. Under the condition of strong classical chaos and strong decoherence due to large coupling with the measurement device, the spectra of the evolution operator exhibit an universal behavior. A generic spectrum consists of a single eigenvalue equal to unity, which corresponds to the invariant state of the system, while all other eigenvalues are contained in a disk in the complex plane. Its radius depends on the number of the Kraus measurement operators and determines the speed with which an arbitrary initial state converges to the unique invariant state. These spectral properties are characteristic of an ensemble of random quantum maps, which in turn can be described by an ensemble of real random Ginibre matrices. This will be proven in the limit of large dimension.

Figure 1

Schematic representation of (a) an isolated quantum system $S$ characterized by a Hamiltonian $H$ and a unitary evolution operator $U=\text{exp}\left(-\mathbf{i}H\right)$; (b) open quantum system $S$. The influence of an environment $E$ can be described by an anti-Hermitian part of the Hamiltonian, $-\mathbf{i}\mu W{W}^{†}$; and (c) interacting systems $S$ and $E$, in which the global evolution is unitary, and the nonunitarity of the evolution of $S_A$ is due to the partial trace over the subsystem $S_B$. Panels $\left({\text{a}}^{\prime }\right)–\left({\text{c}}^{\prime }\right)$ show exemplary spectra of the corresponding evolution operators, which belong to the unit disk on the complex plane $z=x+iy$.

Figure 2

Under the assumption of strong chaos and large decoherence a deterministic dynamics of (a) an interacting quantum system can be described by (b) an ensemble of random operations (completely positive, trace preserving maps). These, in turn, can be mimicked by (c) random matrices of the real Ginibre ensemble, (which do not imply CP and TP properties).

Figure 3

Sketch of the classical dynamical system acting on the torus: (b) reversible asymmetric $\left(K=4\right)$ baker map $B_K$, (c) irreversible sloppy baker map $B_{K,\Delta }$ in which in each step the upper part of the phase space is shifted down by $\Delta /2$, (d) double sloppy baker map $B_{K,\Delta ,\Delta }$ in which both parts of the phase space are shifted vertically by $\Delta /4$.

Figure 4

Exemplary spectra of the evolution operator for the sloppy baker map for several values of the parameters of the model. The dimension of the Hilbert space $N=64$, parameter $M=2$, and the shift width $\Delta =1/4$ are kept fixed. A generic spectrum for $K=4$, $L=16$ is shown on panel (b). The subplots (a) and (c) are obtained for the cases of a weak classical chaos for $K=64$, $L=1$ and $K=L=32$, respectively, while the last case (d) shows the spectrum for the double sloppy map $B_{K,\Delta ,\Delta }$ for $K=64$ and $L=64$.

Figure 5

Spectra of superoperators corresponding to typical random maps generated according to (a) the ensemble ${\Phi }_{\mathsf{E}}$ and (b) ensemble ${\Phi }_{\mathsf{P}}$ (for definitions see Sec. 2). All maps act on quantum states of size $N=64$, while the parameter of the model reads $M=2$. Note that in all cases the spectrum is contained in the disk of radius $R=1/\sqrt{2}$ apart of the leading eigenvalue marked by “$\times$.”

Figure 6

Modulus of the subleading eigenvalue $R=\left|z_2\right|$ as a function of the dimension $M$ of the environment for random quantum operation ${\Phi }_{\mathsf{E}}$ for $N=4(\times )$ and for the sloppy baker map for $N=64$, $K=4$, $L=16$, $M=2$, $\Delta =1/4$ (◼). Solid line shows the fit according to Eq. (12).

Figure 7

Radial density of complex eigenvalues of spectra of superoperators corresponding to a deterministic model of sloppy baker map $(\times )$, projective random operations (○), and the spectra of real random matrices of the Ginibre ensemble (◼). The size of each matrix is $N^2={32}^2$, the number of Kraus operators is $M=16$ so the density is shown as a function of the rescaled radius $r_M≔r\sqrt{M}=4r$. The tail of the distribution beyond the point $r_M=1$ (equivalent to $r=1/4$) does not violate therefore the quantum analogue of the Frobenius-Perron theorem.

Figure 8

Rescaled ratio of the real eigenvalues $\eta$ of the superoperator for random operations with $M=N^2$ (◻), $M=N$ (○), and real Ginibre matrices (△) as a function of the matrix size $N$. Solid horizontal line at $\sqrt{2/\pi }$ represents the asymptotic value of the normalized ratio implied by Eq. (29).

Figure 9

Numerical data for the density $P_1^{\mathbb{C}}\left(y_M\right)$ of complex eigenvalues along the rescaled imaginary axis obtained for an ensemble of random operations ${\Phi }_{\mathsf{E}}$ of dimension $N=8$ and with parameter $M=N^2=64$ (●) are compared with lower and upper bounds [Eq. (33)] obtained for small $\left|y\right|$ from the real Ginibre ensemble and represented by thick lines.