# Quantum coherence and decoherence

In Quantum Mechanics coherent states are defined as eigenstates to some annihilation operator. Afaik this notion is due to Roy Glauber.

Now, I just read that if you have a spin-state for example, then the coherence of the spin-state is measured by the off-diagonal elements of the density matrix. This meant that there is no coherence if the off-diagonal entries are zero, because then I have just spin-up or spin-down.

Are these two notions related to each other, cause I currently don't see it.

If anything is unclear, please let me know.

• Your last sentence is not clear: "This meant that there is no coherence, if the off-diagonal entries are zero, cause then I have just spin-up or spin-down and some coherence otherwise." Do you mean that there is no coherent spin state? Or, do you mean that if the off-diagonal entries are zero we don't have a coherent state? Which one? – Sofia Feb 6 '15 at 15:20
• @Sofia the latter one. – Xin Wang Feb 6 '15 at 15:25
• If the off-diagonal entries are zero, of course you have a mixture od states, not a coherent superposition, but what means to you the phrase "because I have just spin up and spin down"? Anyway, see my answer. – Sofia Feb 6 '15 at 15:33

Coherent state is one thing and decoherence is something else.

The coherent state has the form

$(\text I) \ |\alpha \rangle = e^{-\alpha ^2/2} \sum _n \frac {\alpha ^n}{\sqrt {n!}} |n \rangle.$

where $|n \rangle$ is a Fock state of n identical particles. This state is a coherent superposition of Fock states, and its density matrix has diagonal and non-diagonal elements.

Decoherence means that the density matrix is diagonal, i.e. you have a mixture, not a pure state in the form of quantum superposition.

• but how is this notion of decoherence related to the coherence we know from classical mechanics, where there has to be a constant phase difference between two objects or are all these ideas different from each other? – Xin Wang Feb 9 '15 at 18:43
• @XinWang Yes, the notion of decoherence means that the coherence is destroyed. Here coherence is quite associated to the notion that you mention, a constant phase difference between two things. In quantum mechanics we say that we have a coherent superposition of two states, $\psi_1$ and $\psi_2$ when the global state is $\Psi = A(\psi_1 + Be^{i\phi}\psi_2)$, where A is some constant and B is a real and positive number. The difference in phase between $\psi_1$ and $\psi_2$ in this superposition is $\phi$. This is a coherent superposition, even if $\phi$ is not constant in time. (I continue) – Sofia Feb 9 '15 at 19:20
• @XinWang For keeping the coherence it is sufficient that each system we create and obeys function $\Psi$, the difference in phase vary in time according to the same function $\phi (t)$. If these rules are respected, the density matrix of $\Psi$ has off-diagonal elements. Well, the decoherence erases the off-diagonals. We can't write anymore $\Psi$ as $A(\psi_1 + Be^{i\phi}\psi_2)$. What we get is a system which is either in the state $\psi_1$ or in the state $\psi_2$ with probabilities $|A|^2$, respectively $|A|^2 \ B^2$. Finally, as you see the coherent state has nothing to do with all this. – Sofia Feb 9 '15 at 19:39