# Decoherence in the long time limit of density matrix element

For a state $|\Psi(t)\rangle = \sum_{k}c_k e^{-iE_kt/\hbar}|E_k\rangle$, the density matrix elements in the energy basis are

$\rho_{ab}(t) = c_a c^*_be^{-it(E_a -E_b)/\hbar}$

How is it that in the long time limit, this reduces to $\rho_{ab}(t) \approx |c_a|^2 \delta_{ab}$?

Is there some characteristic time scale here? Or has the density matrix been time averaged to get rid of the oscillatory terms (off diagonal coherences) ?

I'm studying the quantum harmonic oscillator, if that helps. Thanks!

EDIT: The Hamiltonian for the system described by $|\Psi(t)\rangle$ is just the standard harmonic oscillator hamiltonian. No interaction terms are present, so the problem is that of an isolated simple harmonic oscillator.

Demystifier
Gold Member
For a state $|\Psi(t)\rangle = \sum_{k}c_k e^{-iE_kt/\hbar}|E_k\rangle$, the density matrix elements in the energy basis are

$\rho_{ab}(t) = c_a c^*_be^{-it(E_a -E_b)/\hbar}$

How is it that in the long time limit, this reduces to $\rho_{ab}(t) \approx |c_a|^2 \delta_{ab}$?
It doesn't work that way. Your first density matrix is a pure state, and it cannot evolve to your second density matrix which is a mixed state. What you miss is a system which consists of two subsystems. Then the density matrix for the whole system is always a pure state (like your first density matrix), but the density matrix of a subsystem becomes mixed.

bhobba
It doesn't work that way. Your first density matrix is a pure state, and it cannot evolve to your second density matrix which is a mixed state. What you miss is a system which consists of two subsystems. Then the density matrix for the whole system is always a pure state (like your first density matrix), but the density matrix of a subsystem becomes mixed.

Thanks for the reply. I should've mentioned that I'm studying the dynamical aspects of thermalization. What I'm considering is a quantum system that is initially prepared in a pure state and then suddenly perturbed out of equilibrium, at say a time t=0. I'm trying to show that the density matrix (for t>0) becomes diagonal in the long time limit.

Like in classical stat. mech., if we take equilibrium vales of observables as asymptotic time averages of the actual time dependent fluctuating values (due to the oscillatory terms in the density matrix elements), then shouldn't the time averaged density matrix approximate the equilibrium density matrix?

That is, $\hspace{2mm} \bar{\rho} = \lim_{T\to\infty} \frac{1}{T} \int_0^T \rho(t) \, \mathrm{d}t$

Time averaging gets rid of the coherences, and therefore the equilibrium density matrix is diagonal. Are there any mistakes in this reasoning?

EDIT: This paper http://dx.doi.org/10.1063/1.3455998 (section II.A) does something along the same lines as I have above. What I don't understand is this; for the time-averaging argument above to be usable, the system must be allowed to equilibrate, and isn't this possible only if it is at least weakly coupled to the environment?

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Demystifier
Gold Member
Thanks for the reply. I should've mentioned that I'm studying the dynamical aspects of thermalization. What I'm considering is a quantum system that is initially prepared in a pure state and then suddenly perturbed out of equilibrium, at say a time t=0. I'm trying to show that the density matrix (for t>0) becomes diagonal in the long time limit.

Like in classical stat. mech., if we take equilibrium vales of observables as asymptotic time averages of the actual time dependent fluctuating values (due to the oscillatory terms in the density matrix elements), then shouldn't the time averaged density matrix approximate the equilibrium density matrix?

That is, $\hspace{2mm} \bar{\rho} = \lim_{T\to\infty} \frac{1}{T} \int_0^T \rho(t) \, \mathrm{d}t$

Time averaging gets rid of the coherences, and therefore the equilibrium density matrix is diagonal. Are there any mistakes in this reasoning?
The fact that pure density matrix of a closed quantum system cannot evolve into a mixed one has its classical analogue. It is called the Liouville theorem. According to this theorem, the phase volume associated with the classical probability density cannot change its volume. In particular, your classical equation above, strictly speaking, cannot be true.

Yet, in a certain sense your classical equation above may be true. It can be true when you don't consider the full fine-grained density, but only a macroscopic coarse-grained density. By coarse-graining you ignore some fine degrees of freedom, which makes the transition above possible. The Liouville theorem is not valid for the coarse-grained density.

Likewise, in quantum mechanics you ignore degrees of freedom associated with some unobserved subsystem, which leads to decoherence of the density matrix for the observed subsystem.

All this is not a direct answer to your question, but I hope it helps.

soviet1100
The fact that pure density matrix of a closed quantum system cannot evolve into a mixed one has its classical analogue. It is called the Liouville theorem. According to this theorem, the phase volume associated with the classical probability density cannot change its volume. In particular, your classical equation above, strictly speaking, cannot be true.

Yet, in a certain sense your classical equation above may be true. It can be true when you don't consider the full fine-grained density, but only a macroscopic coarse-grained density. By coarse-graining you ignore some fine degrees of freedom, which makes the transition above possible. The Liouville theorem is not valid for the coarse-grained density.

Likewise, in quantum mechanics you ignore degrees of freedom associated with some unobserved subsystem, which leads to decoherence of the density matrix for the observed subsystem.

All this is not a direct answer to your question, but I hope it helps.

Thanks. Is there a good reference from which I can study this in more detail? What I'm ultimately trying to do is calculate the equilibrium density matrix ($\bar{\rho}$) for an isolated harmonic oscillator and show that the distribution of high energy states is canonical, with some effective temperature.

Demystifier
Gold Member
Is there a good reference from which I can study this in more detail?
For the classical case, I like
https://www-physics.ucsd.edu/students/courses/spring2010/physics210a/LECTURES/210_COURSE.pdf
Chapter 3

For a simple explanation of both classical and quantum cases I highly recommend
https://www.amazon.com/dp/9812561315/?tag=pfamazon01-20
Secs. 8.1 and 8.2,
even if you are not interested in black holes, string theory and related exotics.

http://lanl.arxiv.org/abs/1103.4003

For a critique of relevance of ergodicity, I like