# Off-diagonal elements of density matrix, measurement of coherence?

I have a ensemble of systems and each system is made of a single one-dimensional quantum harmonic oscillator. Suppose all systems in the ensemble are in the following quantum state

$$|\Psi\rangle = \dfrac{1}{\sqrt{2}}\left(|\psi_0\rangle e^{-iE_0 t/\hbar} + |\psi_1\rangle e^{-iE_1t/\hbar}\right)$$

where $|\psi_0\rangle$ and $|\psi_1\rangle$ are eigenstates of the Hamiltonian, which correspond to the ground state and the first excited state respectively. $E_0$ and $E_1$ are the eigenvalues associated with $|\psi_0\rangle$ and $|\psi_1\rangle$, repectively. Now if we use $|\psi_0\rangle$ and $|\psi_1\rangle$ as a set of orthonormal basis functions, the density matrix of this ensemble can be calculated to be

$$\hat{\rho} = |\Psi\rangle \langle\Psi|=\dfrac{1}{2} \left[ \begin{array}{cc} 1&e^{i(E_1-E_0)t/\hbar}\\ e^{-i(E_1-E_0)t/\hbar}&1\\ \end{array} \right]$$

Now my question is: Does the off-diagonal density matrix means that there is coherence between $|\psi_0\rangle$ and $|\psi_1\rangle$? If there is, can we measure this coherence via some experimental approach?

Imagine we'd be able to clone an ensemble discribed by $\rho$ before making measurements on it. This magic technique would allow us to obtain probabilities for measurement results without destroying the original ensemble. In the first run, we would determine the probability for $|\psi_0\rangle$ and $|\psi_1\rangle$ which should be $\langle \psi_0 | \rho | \psi_0 \rangle$ and $\langle\psi_1|\rho|\psi_1\rangle$ respectively. For the next experiments we change the measuring device to measure another observable. Lets assume that the set of orthonormal eigenvectors of this second observable contains an element of the form $$|\sigma\rangle := \alpha |\psi_0\rangle + \beta|\psi_1\rangle.$$ If there is no coherency in the ensemble, we would expect the probability for $|\sigma\rangle$ beeing $$|\alpha|^2 \langle\psi_0|\rho|\psi_0\rangle + |\beta|^2 \langle \psi_1 | \rho| \psi_1 \rangle.$$ But if our experiments show a different probability, one would likely say: "bazinga, there is is some quantum mystery (aka. coherency) going on". Quantum theory says, that the probability for $|\sigma\rangle$ is $$\langle \sigma |\rho |\sigma \rangle = |\alpha|^2 \langle\psi_0|\rho|\psi_0\rangle + |\beta|^2 \langle \psi_1 | \rho| \psi_1 \rangle + 2\Re\big( \alpha^*\beta \langle \psi_0|\rho|\psi_1\rangle\big),$$ which differs from the above iff the off-diagonal element $\langle\psi_0|\rho|\psi_1\rangle$ does not vanish. Summing up we see decoherency if the density matrix has non vanishing off-diagonal elements.
There is another point, I'd like to mention in this context. Lets say our density matrix has all vanishing off-diagonal elements. Does this mean, that there is no coherency in the ensemble? I think the right answer to this question is "no, but...". Lets take a closer look. For a density matrix $$\rho = \sum_k p_k |\varphi_k\rangle\langle\varphi_k|,$$ where $p_k$ is the probability for a system beeing in the state $|\varphi_k\rangle$, vanishing off-diagonal elements only mean that $$0 = \sum_k p_k \langle \psi_0 | \varphi_k \rangle \langle \varphi_k | \psi_1 \rangle,$$ i.e. the elements $p_k \langle \psi_0 | \varphi_k \rangle \langle \varphi_k | \psi_1 \rangle$ sum up to zero. To surely say that there is no coherency, we need all elements $p_k \langle \psi_0 | \varphi_k \rangle \langle \varphi_k | \psi_1 \rangle$ being zero, but this is not what follows from vanishing off-diagonal elements. The key point is, that we have no chance to distinguish between different ensembles which are described by the same density matrix. So for our experiments, an ensemble where all $p_k \langle \psi_0 | \varphi_k \rangle \langle \varphi_k | \psi_1 \rangle$ sum up to zero behaves exactly like an ensembly where all $p_k \langle \psi_0 | \varphi_k \rangle \langle \varphi_k | \psi_1 \rangle$ vanish, i.e. an example without coherent states. Summarizing we can say, that an ensemble described by a density matrix with vanishing off-diagonal elements behaves like if there would be no coherent states. There may still be coherent states in our ensemble, but we have no chance to detect them.