Doubt about a time convolution master equation

In summary, the two equations for the time convolution master equation are not equivalent, with the second equation being more general and applicable to a wider range of systems.
  • #1
Ark236
26
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TL;DR Summary
The same equation in 2 different books is different
I study of interaction between a system with a reservoir considering a weak coupling between them. I consider a bosonic bath, the initial state are separable and the operator of interaction between the system and bath is linear in the displacements of the oscillators.
.

In the book "Quantum Effect in Biology, Mohseni" , show that the time convolution master equation is:

$$\dfrac{d}{dt}\rho_{s,I}(t)=-\int_{0}^{t}d\tau Tr_{B}[\cal{L}_{SB,I}(t)\cal{L}_{sB,I}(\tau)\rho_{B}(0)]\rho_{S,I}(\tau)$$
where $$\cal{L}_{SB,I}(t) \hat{A}=\dfrac{1}{\hbar}[H_{I}(t),\hat{A}]$$ and $$H_I$$ is the hamiltonian system in the interaction picture.

but in "The Theory of Open Quantum Systems, Breuer" show that the time convolution master equation is:

$$\dfrac{d}{dt}\rho_{s,I}(t)=-\int_{0}^{t}d\tau Tr_{B}[\cal{L}_{SB,I}(t)\cal{L}_{sB,I}(\tau)\rho_{S,I}(\tau)\otimes \rho_{B}(0)]$$

I think that bot equation are not equivalent?
 
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  • #2
No, the two equations are not equivalent. The equation from "The Theory of Open Quantum Systems, Breuer" is more general and can be used for a wider range of systems than the equation from "Quantum Effects in Biology, Mohseni". The difference between the two equations is that the second equation includes an additional term, which is the tensor product of the system state at time $\tau$ and the bath state at time 0. This additional term is necessary to account for the full dynamics of the system-bath interaction.
 

FAQ: Doubt about a time convolution master equation

1. What is a time convolution master equation?

A time convolution master equation is a mathematical equation used in the field of quantum mechanics to describe the time evolution of a quantum system. It takes into account the effects of both internal and external forces on the system, and is often used to study the dynamics of open quantum systems.

2. How does a time convolution master equation differ from a regular master equation?

A regular master equation only considers the effects of internal forces on a quantum system, while a time convolution master equation also takes into account external forces. This allows for a more accurate description of the system's dynamics, particularly in cases where the system is interacting with its environment.

3. What are the limitations of a time convolution master equation?

One limitation of a time convolution master equation is that it assumes the system is in a steady state, meaning that its properties do not change over time. Additionally, it may not accurately describe systems with strong interactions or highly non-linear dynamics.

4. How is a time convolution master equation derived?

A time convolution master equation is typically derived using the quantum Liouville equation, which describes the time evolution of the density matrix of a quantum system. By applying certain approximations and assumptions, the time convolution master equation can be obtained.

5. What are some applications of a time convolution master equation?

A time convolution master equation has many applications in the field of quantum mechanics, including in the study of quantum optics, quantum computing, and quantum information processing. It is also used in various fields of physics, such as in the study of quantum phase transitions and quantum thermodynamics.

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