1

I am rather new to the field of open quantum systems and I have a seemingly basic question for which I somehow cannot find a complete answer.

Consider a closed system which we divide into two subsystems A and B which have empty intersection and which jointly cover the closed system. To acquire an effective description of subsystem A (resp. B), one traces over the degrees of freedom of subsystem B (resp. A). My question is as follows: is there a relation between the eigenenergies of subsystem A found after tracing over B and the eigenenergies of subsystem B found after tracing over A?

A relatively simple case would be to consider a closed system which we divide into two identical subsystems. In this case, both subsystems have the same number of eigenenergies and it seems reasonable to suppose that there some simple relation between them. However, I have not found such a relation in the (admittedly limited) literature I have read.

Any help with this question would be much appreciated.

BioPhysicist
  • 56,248
WLV
  • 63
  • 5

1 Answers1

2

If the two systems can interact with each other, you cannot talk about the Hamiltonian of system A or B alone, so system A or B alone don't have eigenenergies. Furthermore, after you trace out the degrees of freedom of system B, the state of system A will generally not be described by a pure state (a ket vector). Rather, the state is described by the reduced density matrix. The time evolution of the reduced density matrix is in general very hard to calculate. One common approach is the master equation, which describes the evolution of the subsystem on average. If you were to continously observe an open quantum system in the lab however, the evolution involves quantum jumps, which can be described via the method of quantum trajectories.

curio
  • 1,027
  • One can still talk about energies of open systems, which (can) arise by taking a closed system and tracing over the degrees of freedom of a subsystem. This should give an effective Hamiltonian for the open system that we acquire after performing the trace. – WLV Mar 11 '20 at 16:37
  • Which source are you using? Effective Hamiltonians are a matter of definition. When im talking about the Hamiltonian, I mean the generator of time translations. As I said, for open quantum systems the time evolution is not longer governed by the Schrödinger equation hence there is no Hamiltonian. If you are talking about stationary states (those that don't evolve in time), they are the steady state solutions of the master equation. However, quantum fluctuations (jumps) can pull a quantum system out of those steady states in certain cases. – curio Mar 11 '20 at 16:46
  • In the context of the master equation or quantum trajectories, the effective Hamiltonian isn't even hermitian. – curio Mar 11 '20 at 16:47
  • The effective Hamiltonian becoming non-hermitian is precisely what I am interested in. I suspect there to be a simple relation between the imaginary components of the eigenvalues of the effective Hamiltonians of subsystems A and B, as particles leaking away from one subsystem leak into the other. I guess the sum of the imaginary components of the effective Hamiltonians for A and B should sum to zero, but I am not sure. – WLV Mar 12 '20 at 02:05
  • The definition of effective Hamiltonian is really up to you, so please provide it in the question. If it is non-hermitian it might not be diagonalizable. Also, if you have two identical subsystems it would be weird if they dont have the same effective Hamiltonian so the imaginary parts must be identical and cant sum to zero. – curio Mar 12 '20 at 11:11
  • Good point. I am in fact not entirely sure how to define the effective Hamiltonian, and this is part of my confusion. Could you perhaps recommend a source where these things are properly explained? I will try to formulate my question more precisely.

    Regarding the two identical subsystems, if they together constitute a closed system whose time evolution is governed by a Hermitian Hamiltonian, then whatever particles leak out of the one subsystem necessarily leak into the other subsystem. However these are just the words and I am not sure how to phrase this more precisely/quantitatively.

     – WLV Mar 12 '20 at 12:38
  • Gernot Schallers lectures are a good reference for master equations. Open quantum systems are often discussed in quantum optics lectures. The point about the particles leaking like this is true, technically speaking we can replace the number operator of the whole system by its eigenvalue $N$. You might be able to draw some conclusions from that but you need to be more precise about what you want. – curio Mar 12 '20 at 13:21
  • Thank you for the recommendation. – WLV Mar 12 '20 at 14:48