# Time dependence of density matrix

I was given an exercise to calculate the time dependence of 2x2 density matrix of a qubit. I have the density matrix at time $t=0$ and Hamiltonian $H=AI+BY$, where I is the identity matrix, Y is Pauli matrix and A, B are constants.

I started with describing density matrix in the form $\rho= (1/2)I+a(t)X+b(t)Y+c(t)Z=\begin{pmatrix}1/2+c(t) & a(t)-ib(t)\\a(t)+ib(t) &1/2-c(t)\end{pmatrix}$, so I can write four equations for each element in the differential form $i\hbar \dot{\rho_i}=H\rho$ or, as the Hamiltonian is independent of time, $\rho_i(t)=exp(-i(t-t_0)H/\hbar)\rho_i(t_0)$. I am not sure if it is correct way of doing it? I am stuck here, as when I try to calculate an element of density matrix I have coefficient plus real number on the left side of the equation and matix exponential on the right, which is a matrix as well.

In the case of a time independent Hamilotnian, the right expression for the time evolved density matrix is $\rho(t) \, = \, U(t,t0) \, \rho(t0) \, U^{\dagger}(t,t0))$ with $U(t,t0) \, = \, \exp[ - i (t-t0)H/\hbar]$. So in principle you just have to write an anzats for $\rho(t0)$ and then compute that product of matrices.
Since $I$ and $Y$ commute, you can compute $\exp[ - i (t-t0)(A I + B Y)/\hbar]$ as the product $\exp[ - i (t-t0)A I/\hbar] \, \exp[ - i (t-t0)B Y/\hbar]$.
• I think you can just drop $t0$, it is usually irrelevant. By the way, the right expression for Schrodinger's equation (in Schodinger's picture) for the density matrix is $\dot{\rho} \, = \, -\frac{i}{\hbar} [H,\rho]$. Commented Jan 22, 2018 at 15:31
In case of density matrix, one can start by solving the Heisenberg equation given by $\frac{d\rho(t)}{dt}=-i[\rho,H]+\frac{\partial \rho}{\partial t}$,where $[]$ is the commutator between $\rho$ and $H$. For general time dependence case, the equation can be solved numerically using Runge-Kutta Method. For special cases of time-dependence the Unitary operator $U(t)$ can be expressed as $U(t)=\exp[-iHt]$ and the evolution of the density matrix can be obtained by $\rho(t)=U(t)\rho(0)U^{\dagger}$, where $\rho(0)$ is the density matrix at $t=0$.