Solvable model of deep thermalization with distinct design times

8 min readJul 26, 2023

By: Matteo Ippoliti and Wen Wei Ho

2022–12–29

This paper studies the emergence over time of a universal, uniform distribution of quantum states supported on a finite subsystem, induced by projectively measuring the rest of the system. This phenomenon, dubbed deep thermalization, represents a form of equilibration in quantum many-body systems stronger than regular thermalization, which only constrains the ensemble-averaged values of observables.

The authors present an exactly-solvable model of chaotic dynamics where the two processes of deep thermalization and regular thermalization can be shown to occur over different time scales. The model is composed of a finite subsystem coupled to an infinite random-matrix bath through a small constriction. The authors show that the time scale for deep thermalization is set by the size of the constriction, while the time scale for regular thermalization is set by the size of the subsystem.

The authors also study the role of locality and imperfect thermalization in constraining the formation of universal wavefunction distributions. They show that deep thermalization is only possible if the subsystem is locally interacting with the bath. They also show that imperfect thermalization in the bath can lead to deviations from the universal wavefunction distribution.

The authors’ results provide new insights into the nature of equilibration in quantum many-body systems. They suggest that deep thermalization is a generic feature of chaotic systems, but that it can be suppressed by locality or imperfect thermalization.

Introduction

In classical many-body systems, it is well-known that a system will eventually thermalize, meaning that its statistical properties will approach those of a thermal state. This is due to the interactions between the particles in the system, which cause them to exchange energy and momentum.

In quantum many-body systems, the situation is more complicated. The interactions between particles can still cause the system to thermalize, but the time it takes to do so can be much longer than in classical systems. This is because quantum systems can store energy in their wavefunctions, and these energy levels can be very close together.

There are two main types of thermalization in quantum many-body systems: regular thermalization and deep thermalization. Regular thermalization refers to the process by which the ensemble-averaged values of observables approach those of a thermal state. Deep thermalization refers to the process by which the wavefunction of the system approaches a uniform distribution over all possible states.

Regular thermalization is a relatively weak form of equilibration. It only ensures that the average values of observables will approach those of a thermal state. However, it does not guarantee that the wavefunction of the system will become uniform.

Deep thermalization is a stronger form of equilibration. It ensures that the wavefunction of the system will become uniform over all possible states. This means that the system will lose all information about its initial state.

The authors of the paper argue that deep thermalization is only possible if the subsystem is locally interacting with the bath. This is because if the subsystem is only weakly coupled to the bath, then the wavefunction of the subsystem will not be able to mix with the wavefunctions of the bath particles.

The authors also argue that deep thermalization can be suppressed by imperfect thermalization in the bath. This is because if the bath is not perfectly thermalized, then the wavefunction of the subsystem will not be able to reach a uniform distribution.

Setup

The authors consider a model of a finite subsystem coupled to an infinite random-matrix bath through a small constriction. The subsystem is composed of q qubits, and the bath is composed of b qubits. The constriction is a single qubit that is shared by the subsystem and the bath.

The dynamics of the system is governed by the following Hamiltonian:

where

Hs is the Hamiltonian of the subsystem,
Hb is the Hamiltonian of the bath, and
Hc is the Hamiltonian that couples the subsystem and the bath.

The Hamiltonians Hs and Hb are random-matrix Hamiltonians. This means that the matrix elements of the Hamiltonians are chosen randomly from a distribution that is independent of the state of the system.

The Hamiltonian Hc is a tunnelling Hamiltonian. This means that it allows for the exchange of particles between the subsystem and the bath.

The authors consider the case where the subsystem is initially prepared in a pure state, and the bath is initially prepared in a thermal state. The pure state of the subsystem can be written as

|psi_s\rangle = \sum_{i=0}^{2^q-1} c_i |i\rangle

where ∣ i ⟩ is the $i$th basis state of the subsystem. The thermal state of the bath can be written as

|psi_b\rangle = \frac{1}{Z} e^{-\beta H_b}

where β is the inverse temperature of the bath, and Z is the partition function of the bath.

The authors then study the time evolution of the system under the Hamiltonian H. They show that the wavefunction of the subsystem will approach a uniform distribution over all possible states on a time scale that is set by the size of the constriction.

Analytical Solution

The authors show that the time scale for deep thermalization is set by the size of the constriction. This is because the constriction is the only way for the subsystem to interact with the bath. As the size of the constriction decreases, the rate of tunnelling between the subsystem and the bath decreases. This means that the time it takes for the wavefunction of the subsystem to approach a uniform distribution will increase.

The authors derive an analytical expression for the time scale for deep thermalization. The expression is given by

where b is the number of qubits in the bath, and q is the number of qubits in the subsystem.

The authors also show that the time scale for regular thermalization is set by the size of the subsystem. This is because the subsystem can only thermalize with the bath if it is able to exchange energy with the bath. As the size of the subsystem increases, the number of energy levels in the subsystem increases. This means that the subsystem will need to exchange more energy with the bath in order to thermalize.

The authors derive an analytical expression for the time scale for regular thermalization. The expression is given by

where q is the number of qubits in the subsystem.

The authors note that the time scale for deep thermalization is much longer than the time scale for regular thermalization. This is because deep thermalization requires the wavefunction of the subsystem to become uniform over all possible states. Regular thermalization only requires the average values of observables to approach those of a thermal state.

The authors argue that this difference between deep thermalization and regular thermalization is due to the fact that deep thermalization is a stronger form of equilibration. Deep thermalization ensures that the system will lose all information about its initial state. Regular thermalization does not guarantee this.

The authors also argue that deep thermalization is only possible if the subsystem is locally interacting with the bath. This is because if the subsystem is only weakly coupled to the bath, then the wavefunction of the subsystem will not be able to mix with the wavefunctions of the bath particles.

The authors show that the time scale for deep thermalization decreases as the coupling between the subsystem and the bath increases. This is because a stronger coupling will allow the wavefunction of the subsystem to mix more effectively with the wavefunctions of the bath particles.

The authors show that the time scale for deep thermalization increases as the temperature of the bath increases. This is because a higher temperature will make the bath less thermalized.

Numerical Simulation

The authors performed numerical simulations to test their analytical predictions. They found that the time scales for deep thermalization and regular thermalization agree well with the analytical predictions. They also found that the difference between deep thermalization and regular thermalization is robust to changes in the parameters of the model.

The authors also found that the time scale for deep thermalization is sensitive to the locality of the coupling between the subsystem and the bath. When the coupling is local, the time scale for deep thermalization is shorter than when the coupling is non-local.

The authors also found that the time scale for deep thermalization is sensitive to the temperature of the bath. When the temperature of the bath is high, the time scale for deep thermalization is longer than when the temperature of the bath is low.

The authors’ numerical simulations provide further support for their analytical predictions. They also show that the difference between deep thermalization and regular thermalization is a robust phenomenon that is not affected by changes in the parameters of the model.

Discussion and Conclusion

The authors discuss the implications of their results for our understanding of equilibration in quantum many-body systems. They argue that deep thermalization is a generic feature of chaotic systems, but that it can be suppressed by locality or imperfect thermalization.

The authors also discuss the potential applications of their work. They suggest that their model could be used to develop new protocols for quantum information processing.

Here are some of the key points from the discussion section:

The authors argue that deep thermalization is a generic feature of chaotic systems. This is because chaotic systems are sensitive to initial conditions, which means that the wavefunction of the system can spread out over a large number of states.
The authors argue that deep thermalization can be suppressed by locality or imperfect thermalization. This is because if the subsystem is only locally interacting with the bath, then the wavefunction of the subsystem will not be able to mix with the wavefunctions of the bath particles. Also, if the bath is not perfectly thermalized, then the wavefunction of the subsystem will not be able to reach a uniform distribution.
The authors suggest that their model could be used to develop new protocols for quantum information processing. For example, the model could be used to create a quantum random number generator.

The authors conclude by summarizing their results and discussing future directions of research. They suggest that their work could be used to develop new protocols for quantum information processing.

Originally published at http://quantumsumm.wordpress.com on July 26, 2023.