Saturday, 30 December 2017

open quantum systems - Is hermicity of the reduced density matrix preserved here?


I am following along Breuer and Petruccione's book . I would like to know if the property $\rho^{\dagger} = \rho$ is preserved for evolution that is described by the Born Approximation.


For a Hilbert space $\mathscr{H}_s \otimes \mathscr{H}_{b}$ describing a system of interest and some reservoir/bath, we consider a time-independent Hamiltonian of the form $$ H = H_s \otimes \mathbb{I}_b + \mathbb{I}_s \otimes H_b + g H_{int} $$ where $g$ is some small coupling and $H_{int}$ describes some interaction between the system and the bath.


The full density matrix of the combined system and bath $\sigma(t)$ evolves via the von Neumann equation $$ \frac{d\sigma_I(t)}{dt} = - i [ V(t) , \sigma_{I}(t) ] $$ which is written in the interaction-picture, where $$ \sigma_{I}(t) := e^{+ i H_s t} \otimes e^{+ i H_b t} \sigma(t) e^{- i H_s t} \otimes e^{- i H_b t} \ \ \ \ \text{and} \ \ \ \ V(t) := e^{+ i H_s t} \otimes e^{+ i H_b t} H_{int} e^{- i H_s t} \otimes e^{- i H_b t} $$


If we define the reduced density matrix describing the system as the following partial trace $$ \rho(t) := \mathrm{Tr}_{b}[ \sigma(t) ] $$ and then $\rho_{I}(t) := e^{+ i H_s t} \rho(t) e^{- i H_s t}$, the Born Approximation says that $$ \frac{d\rho_I(t)}{dt} \simeq - g^2 \int_0^t ds\ \mathrm{Tr}_b\bigg( \big[ V(t), [V(s), \rho_I(s) \otimes \varrho_{B}] \big] \bigg) $$ where $\varrho_b$ is the initial state of the bath at $t=0$, where $\sigma(0) = \rho(0) \otimes \varrho_b$.


My Question: Suppose that $\rho(0)^{\dagger}= \rho(0)$ so that the initial density matrix is Hermitian. How can you use the above equation to show that $\rho(t)^{\dagger}= \rho(t)$ for $t>0$?



Is this possible? In the literature, I sometimes come across some statements that Lindblad equations preserve Hermicity (Lindblad equations being the above equation, after taking the Markov Approximation and then the secular/rotating-wave approximation).


Is it possible to prove this from this equation of motion? Or do we need additional assumptions? Or can this be more generally proven for $\rho$ without specifying to a particular evolution equation?




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...