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Suppose we have a system that is coupled to a bath with non zero temperature. Through some mechanism one is able to prepare the system in its ground state (i.e. zero temperature state) and let it evolve under time. Quantum mechanics tells us that if a state is in the eigenstate of its Hamiltonian, the system would evolve coherently with $e^{iHt}$. However, since the system (which is in its ground state) is not an isolated system it would eventually evolve into the same state as the finite temperature bath (i.e. thermalize).

My question is first is my interpretation correct? If so, what determines the thermalization dynamics (is it connected to decoherence?) and for how long would the ground state evolve under its Hamiltonian before it "switch" to become thermalized?

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Quantum mechanics tells us that if a state is in the eigenstate of its Hamiltonian, the system would evolve coherently with $e^{iHt}$.

Not really, because $H_0$ which the eigenstate refers to is not the full Hamiltonian describing the evolution of the system accurately when the system is in contact with a bath. $H_0$ would be accurate if the system was isolated, and then the state evolution would be trivial $\psi(t) = \psi(0) e^{-iHt/\hbar}$, where $H$ is some eigenvalue of $H_0$.

However, since the system (which is in its ground state) is not an isolated system it would eventually evolve into the same state as the finite temperature bath (i.e. thermalize).

Not for the quantum state $\psi$ (ket) - that is supposed to evolve in a complicated way, due to interaction with the bath.

Above, you're using "state" in the other meaning, as density matrix. The relation between a quantum state and "quantum statistical physics state" = density matrix is like the relation in classical statistical physics between a microstate $(q_1,...q_N,p_1..p_N)$ in the phase space and probability density $\rho(q_1,...q_N,p_1,...p_N)$ on the phase space. Density matrix is "kind of" like probability on the space of some basis states which play privileged role (Hamiltonian eigenstates), in that it describes "non-quantum" probabilities, describing likelihood of various quantum states due to lack of knowledge of the actual quantum state. However, it is different from just probability distribution function, as it contains also off-diagonal terms, sometimes called coherences.

So the correct statement would be that due to interaction with the bath assumed in thermodynamic equilibrium, which we describe by density matrix $\exp(-\beta H_{bath})/Z_{bath}$, we expect the system's density matrix, due to interaction, and unlimited heat capacity of the bath, to become similarly $\exp(-\beta H_0)/Z_0$. That is, we expect system with density matrix having only one diagonal entry to be 1 and others zero, describing a system in its ground state (a pure density matrix, a pure state) to evolve, after enough time, into a density matrix where all diagonal entries are non-zero, and their values are Boltzmannian probabilities of observing the corresponding energies (eigenvalues of $H_0$), implied by their energy and degeneracy.

My question is first is my interpretation correct? If so, what determines the thermalization dynamics (is it connected to decoherence?) and for how long would the ground state evolve under its Hamiltonian before it "switch" to become thermalized?

Evolution of the density matrix towards equilibrium can be described by the so-called quantum master equations, which are models that describe how probabilities (diagonal elements) and coherences (off-diagonal elements) evolve in time. How long this takes depends on parameters in these models. It's like with populations in chemical kinetics, there are rate constants, except in addition to probabilities, we have coherences as well.

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  • $\begingroup$ To make sure I'm understanding you correctly. Is your claim that for an open system, one cannot describe "thermalization" dynamics as Hamiltonian evolution (that describes evolution of a single state $| \psi \rangle$ but a density state evolution that evolves a state into a superposition of states described by the relative weights in a density matrix? $\endgroup$
    – ljc
    Commented May 30 at 19:36
  • $\begingroup$ @ljc No. One can in principle try to describe thermalization of a single system using Hamiltonian evolution of a ket, but 1) this is "too detailed", as it requires initial condition $\psi_0$ and a time-dependent Hamiltonian $H(t)$ describing action of environment on the system; 2) this will provide only evolution of a single system, resulting $\psi(t)$ will be a complicated function of time that may be outlier and need not show any probabilistic characteristic (distribution of energies) that we expect for a set of similar systems at given temperature. $\endgroup$
    – Ján Lalinský
    Commented May 31 at 1:38
  • $\begingroup$ It would be like thermalization of single point mass in classical theory, this can be described by Hamiltonian evolution of point in phase space, but this won't show necessarily the Boltzmann probability distribution. $\endgroup$
    – Ján Lalinský
    Commented May 31 at 1:39
  • $\begingroup$ The density matrix does not give a single ket, but is its statistical description; when the ensemble consists of states $\psi_n$ with probabilities $p_n$, and there is a basis made of states $\Phi_k$, density matrix for the ensemble is defined as $\rho_{ik} = \sum_n p_n \langle\Phi_i|\psi_n\rangle \langle\psi_n|\Phi_k\rangle$. If $\psi$ is not a basis state, then the matrix has off-diagonal terms. This thing we're describing this way is called statistical mix, not a superposition. This manifests also in the fact that in definition of $\rho$ we're not adding $c_k\Phi_k$'s to get one ket ... $\endgroup$
    – Ján Lalinský
    Commented May 31 at 1:59
  • $\begingroup$ ... but for any basis index pair $ik$, we're adding terms describing members of the ensemble $\psi_n$'s, each with its weight, to get the matrix of probabilities $\rho_{kk}$ and coherences $\rho_{ik},~ i\neq k$. $\endgroup$
    – Ján Lalinský
    Commented May 31 at 2:01

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