The Caldeira-leggett (CL) model is sometimes used to describe quantum mechanical effects in a system (like a biomolecule) immersed in a solvent. The Hamiltonian of the CL model consists of an arbitrary system coupled to an infinite number of quantum harmonic oscillators.

$$H = \frac{\hat{p}^2}{2m} + V(\hat{x}) + \sum_n \frac{\hat{p}^2_n}{2m_n}+ \frac{1}{2}m_n \omega_n ^2 \hat{x}^2 - \hat{x} \sum_n \kappa_n \hat{x}_n$$

Here, $\hat{p}, \hat{x}, \hat{p_n}, \hat{x_n}$ are, respectively, system momentum and position operator and nth bath oscillator momentum and position operator.

My question is regarding the validity of this model for a system immersed in a solvent. The actual physical environment consists of several solvent molecules at some temperature T, colliding with the system at some rate. I do not understand how it is possible to replace this with a picture consisting of quantum harmonic oscillators.

Now, the paper that I linked earlier as well mentions the following-

Consider a system that consists of a single particle of mass M described by one degree of freedom and coupled to a large environment, which can be represented by a bath of harmonic oscillators. This is equivalent to representing some arbitrary environment in terms of its normal modes.

I don't understand this explanation. Is it always possible to replace any bath with a set of harmonic oscillators?

One plausible explanation that I could think of was if the bath consisted of identical noninteracting bosonic particles in a box, let us say. Then, we know that we can express the state of the thermal bath in momentum basis, in terms of the number of particles occupying a particular energy eigenstate. This would result in the bath to be a collection of uncoupled harmonic oscillators.

The problem is that the treatment above assumes the particles in the box to be noninteracting. Also, if the particles were fermionic instead of bosonic, each eigenstate would represent a qubit rather than a harmonic oscillator. Would this require us to treat the environment as infinite qubits instead?

So, my question is, how does one go from a bath consisting of molecules in a solvent colliding with the system to a picture with uncoupled quantum harmonic oscillators interacting with the system?

Any reference where this is discussed will also be very helpful.


1 Answer 1


Note that this is not the only model for an environment, see for example this review where collisional models are considered (which might be closer to this classical picture of molecules colliding).

That being said the most common approach to modelling an environment is using harmonic oscillators because: it's convenient; some environments are actually composed of non-interacting harmonic oscillators (the electromagnetic field, phonons in a crystal etc).

In the end the only thing that matters from the perspective of the system are the proprieties that your environment has, such as markovianity, additivity, time dependences, and this model consisting of harmonic oscillators is very well understood allowing for a precise modelling of an environment.

So, is your environment composed of harmonic oscillators? Maybe, maybe not, but in the end it doesn't matter that much from the perspective of the system, since there are only so many possible CPTP maps it can evolve with.


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