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Quantum Gibbs distribution from dynamical thermalization in classical nonlinear lattices (video abstract 4mins)

This paper investigates the old fundamental problem of emergence of statistical laws in purely dynamical systems. The story started in 1876 in Wein from the famous dispute between Boltzmann and Loschmidt on thermalization, entropy growth and time reversibility of Newton equations. We extend the research line of dynamical thermalization in classical nonlinear lattices considering the case of disordered linear mode frequencies corresponding to a generic situation. Surprisingly, our results show that in lattices with weak or moderate nonlinearity there is emergence of a quantum Gibbs distribution over energies of linear eigenmodes. We discuss the conditions under which such a quantum Gibbs replaces a usually expected energy equipartition over linear modes, predicted by the classical thermalization theory.

Schrodinger cat animated on a quantum computer

Time evolution of the Schrodinger cat: probability distribution W(x) at -3.14 < x < 3.14 is shown for different number of map iterations t, changing along y-axis from t=0 (top) to t=180 (bottom). Here for the double well map K=0.04, a=1.6 and hbar = 4 x 3.1415/N with N=32. Quantum computation is done with 6 qubits and and noisy gates of noise strength eps = 0.02, and 2090 gates per one map iteration. At t=0 initial coherent packet is located at x=-a.
From Ref. 131

Quantum fractal survivor

Quantum fractal eigenstate (Husimi function) with minimal decay rate in the open quantum kicked rotator with the chaos parameter K=7; probability is absorbed outside of box size N=59049, kick strength k=N/4.
From Ref. 97 (Physica D 131 (1999) 311)

Quantum chaos & quantum computers

Quantum computer melting induced by the coupling between qubits. Color represents the level of quantum eigenstate entropy Sq (blue - minimal Sq=0, red - maximal Sq=11) for 12 qubits. Vertical axis gives the coupling strength, horizontal one gives the energy of computer eigenstate.
From Ref. 107 (1999)

Dynamical turbulent flow on the Galton board with friction (video)

See the color animated turbulent flow from Ref. 120 (2001) at

Delocalization of two-particle ring
near the Fermi level of 2d Anderson model

Probability distributions f and f_d for TIP in 2d disordered lattice of size L=40, and interaction of radius R=12 and width dR = 1. Left column, one-particle probability f for W=8V: ground state at U=0 (top); ground state with binding energy E = -1.05 V at U=-2V (middle); coupled state with E = -0.19 V at U=-2V (bottom). Right column: f for coupled state, compare to bottom left, at W=12V and U=-2V with E = -0.19 V(top); inter-particle distance probability f_d related to the middle left case (middle); f_d related to the bottom left case (bottom). All data are shown for the same disorder realisation.
From Ref. 110 (2000)

Generalized Cooper problem in the vicinity of Anderson transition

Probability distributions for two interacting particles in the 3d Anderson model at the ground state. Probability is projected on (x,y)-plane: one particle probability f_p for Hubbard interaction U=-4 V (left column), interparticle distance probability f_pd for U=-4 V (centrum column), f_p for U=0 (right column); the disorder strength is W/W_c=1.1 (upper line), W/W_c=0.5 (middle line), W/W_c=0.3 (bottom line); W_c=16.5 V is the critical disorder for Anderson transition at half filling. All data are given for the same realisation of disorder for the system size L=16. Upper line corresponds to the insulating noninteracting phase while two others are in the metallic one at U=0.
From Ref. 109 (1999)

Poincare recurrences: quantum and classical fractals

Quantum Husimi function in the kicked rotator with absorption at time t=100 (top left); 5000 (middle left), 3x10^5 (bottom left); right top (t=100) and middle (t=5000) are classical probabilities, number of levels N=3^8 , kick strength k=N/4, chaos parameter K=7; right bottom shows Husimi function for N=3^6, t=10^7.
From Ref. 100 (Phys. Rev. Lett. 82 (1999) 524)

Quantum fractals in hydrogen atom

Probability distribution in action-phase plane (y,x) for classical (left) and quantum (right) atom in a microwave field after 50 microwave periods, initial level n=1200.
From Ref. 108 (1999)

Quantum ergodicity in rough billiards

Transition from localization to Shnirelman ergodicity on energy surface (n,l) for level number N =2250, l_max=95 and M=20; shown are the absolute amplitudes of one eigenstate: a) localization for D=20, b) Wigner ergodicity for D=80 and c) Shnirelman ergodicity for D=1000.
From Ref. 92 (Phys. Rev. Lett. 79 (1997) 1833)