Roles of energy eigenstates and eigenvalues in equilibration of isolated quantum systems

We show that eigenenergies and energy eigenstates play different roles in the equilibration process of an isolated quantum system. Their roles are revealed numerically by exchanging the eigenenergies between an integrable model and a nonintegrable model. We find that the structure of eigenenergies of a nonintegrable model characterized by nondegeneracy ensures that quantum revival occurs rarely whereas the energy eigenstates of a nonintegrable model suppress the fluctuations for the equilibrated quantum state. Our study is aided with a quantum entropy that describes how randomly a wave function is distributed in quantum phase space. We also demonstrate with this quantum entropy the validity of Berry's conjecture for energy eigenstates. This implies that the energy eigenstates of a nonintegrable model appear indeed random.

Figure 1

Ripple billiard. The two curved boundaries are given by $x=\pm b\mp acos(\pi y/b)$, respectively.

Figure 2

Nearest-neighbor level spacing statistics for square billiard $a/b=0$ in (a) and ripple billiard $a/b=0.2$ in (b) with even-even modes.

Figure 3

The 100th eigenfunction of a one-dimensional harmonic oscillator in the quantum phase space. The red circle is the corresponding classical trajectory. $j_x$ and $j_k$ are indices labeling Planck cells.

Figure 4

Time evolution for the quantum entropy in four situations: (a) integrable eigenstates and integrable eigenenergies; (b) integrable eigenstates and nonintegrable eigenenergies; (c) nonintegrable eigenstates and integrable eigenenergies; (d) nonintegrable eigenstates and nonintegrable eigenenergies. The scale of the ripple billiard is $a=1.1,b=5.5$ while the length of side for the square billiard is $b$. Initial Gaussian wave packet parameters: $\sigma =1,k_y=0;k_x=a+b$ for ripple billiard and $k_x=b$ for square billiard. $T$ is the time when the envelope of the initial wave packet with group velocity $V_g=2k_x$ return to the center of the billiard after reflecting once from one boundary along $x$ direction. $L$ is the length unit; $1/L$ is the unit for $k$.

Figure 5

(a1) the 1000th eigenstate in the position space; (b1) the corresponding wave function ${\psi }_B$ constructed according to Berry's conjecture in the position space; (c1) the 857th eigenstate (which is a scar state) in the position space. (a2), (b2), (c2) Their respective representation in the momentum space. (a3), (b3), (c3) Their representations in the quantum phase space with the position and momentum along the $y$ direction fixed at $j_y=5,j_{k_y}=0$. For the billiard, $a=0.55$ and $b=5.5$. $L$ is the unit of length.

Figure 6

(a1)–(a5) Quantum pure state entropy $S_w$ of eigenfunctions of the ripple billiards and their corresponding ${\psi }_B$ constructed according to Berry's conjecture. The $x$ axis is the eigenenergy level. Blue lines represent the results of the ripple billiard; red lines represent the results for ${\psi }_B$. Yellow lines are obtained from the blue dots by averaging the nearest 30 eigenstates (the standard microcanonical ensemble average). (a6) Average entropy fluctuation around its microcanonical ensemble average for different $a/b$.