# Density matrix off-diagonal elements

auctor
The possible values of the diagonal elements of a density matrix are restricted by the condition $\mathrm{Tr}~\rho = 1$. Are there any restrictions on the possible values of off-diagonal elements, apart from the obvious $\mathrm{Re}~\rho_{nm} = \mathrm{Re}~\rho_{mn}$, $\mathrm{Im}~\rho_{nm} = - \mathrm{Im}~\rho_{mn}$? If the off-diagonals are written in the form $\left| \rho_{nm} \right| \exp{i \phi_{nm}}$, do the absolute value and the phase have a simple physical meaning?

## Answers and Replies

Einstein Mcfly
The possible values of the diagonal elements of a density matrix are restricted by the condition $\mathrm{Tr}~\rho = 1$. Are there any restrictions on the possible values of off-diagonal elements, apart from the obvious $\mathrm{Re}~\rho_{nm} = \mathrm{Re}~\rho_{mn}$, $\mathrm{Im}~\rho_{nm} = - \mathrm{Im}~\rho_{mn}$? If the off-diagonals are written in the form $\left| \rho_{nm} \right| \exp{i \phi_{nm}}$, do the absolute value and the phase have a simple physical meaning?

The magnitude of a coherence between two levels is bounded by the population in those levels:
$|\rho_{nm}|^2 \le \rho_{nn}\rho_{mm}\le 1$.

The phase differences determine the time dependent properties of the total system. These phases can be set by time and frequency shaped laser pulses in order to enhance or suppress properties of the system. This is the field of "optical quantum control". Take a look at, for example, Ben Fain's book "Irreveribilities in Quantum Mechanics" or better yet, David Tannor's EXCELLENT "Introduction to Quantum Mechanics: A Time-Dependent Perspective". There's also a review called "Optical control of molecular dynamics" edited by Stuart Rice. It's not that great of a source to learn from, but it's a nice collection of the ideas. Most of it will discuss the interference in the wave function formalism, but you'll get the basic picture.

1ndranil
For a system with time Independent hamiltonian , in the basis of energy eigen kets, every off diagonal element between two kets corresponding to two distinct energy eigen values will oscillate with Bohr Frequency.