Definition of geometric phase for mixed states and test in experiment

Question

In quantum mechanics, an open system interacting with the environment is described by mixed state, which is represented by the density operator $\rho(t)$ acting on a finite dimensional Hilbert space $\mathscr{H}_n$. By the spectral theorem, $\rho(t)$ can be decomposed as $\rho(t)=\sum_{k=1}^n\omega_k(t)|\phi_k(t)\rangle\langle\phi_k(t)| $, where $\omega_k(t)$ are the eigenvalues, $|\phi_k(t)\rangle$ are the eigenvectors and $\langle\phi_k(t)|$ are the corresponding vectors in the dual space of $\mathscr{H}_n$. For simplicity, here only non-degenerate case is considered. I am now trying to assign a geometric phase for the mixed state represented by $\rho(t)=\sum_{k=1}^n\omega_k(t)|\phi_k(t)\rangle\langle\phi_k(t)| $. The geometric phase is defined as $$\sum_{k=1}^n\int_0^t \omega_k(t')\frac{d\,\gamma_k(t')}{d\,t'}d\,t'$$, where $\gamma_k(t)=\arg\langle\phi_k(0)|\phi_k(t)\rangle+i\int_0^t\langle\phi_k(t')|\dot{\phi}_k(t')\rangle d\,t'$ are the geometric phases for eigenvectors $|\phi_k(t)\rangle$. Here we assume that $|\phi_k(t)\rangle$ are not orthogonal to $|\phi_k(0)\rangle$ for any time t and $\gamma_k(t)$ are smooth functions of time t. Note that $\gamma_k(t)$ are put in differentiation of time t. There is no ambiguity caused by multiple values of $\gamma_k(t)$. Thus the above geometric phase is well-defined. Question: How to test in experiment the geometric phase for mixed states defined above?

Comments

I'm a bit puzzled by the question. Isn't one of the key features of density matrix that the relative phase information is erased? In your case, before calculating $\alpha_k(t)$, how do you measure $|\phi_k(t)\rangle$? I mean, if you redefine $|\phi_k(t)\rangle\to e^{i\beta_k(t)}|\phi_k(t)\rangle$ , the density matrix $\rho(t)$ stays the same, and this means $\alpha_k(t)$ cannot be uniquely defined if all you have is a $\rho(t)$.(note this ambiguity doesn't go away by just differentiating $\alpha_k(t)$ since $\beta_k$ is time-dependent)

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You are right, @Yiyang. When I said the relative phase, the relative phase should be dependent of a certain "purification" and is surely not gauge invariant. Actually I want to know how to measure the geometric phase stated in the question after edition. If the relative phase of a certain "purification" can be measured like pure state case, then the geometric phase can be tested after the parallel transportation condition for each eigenvector is applied.

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@Guang-LeDu, "purification" means the same thing as before edition?

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@Yiyang. The well-known purification of $\rho(t)$ is of the form $|\psi(t)\rangle=\sum_{k=1}^n\sqrt{\omega_k(t)}|\phi_k(t)\rangle\otimes|a_k\rangle$. The relative phase of $|\psi(t)\rangle$ is given by $\arg\langle\psi(0)|\psi(t)\rangle=\arg(\sum_{k=1}^n\sqrt{\omega_k(0)\omega_k(t)}\langle\phi_k(0)|\phi_k(t)\rangle)$. Before edition, I wanted to know if it is possible to construct a pure state related to $\rho(t)$, like the well-known purification, giving the relative phase I mentioned. In this sense, the "purification". It may be a possible way to give an experimental scheme to test the geometric phase defined here.