Coherent states

If you are not working on quantum optics, you might be in a similar situation as I was a couple of days ago regarding coherent states: You had encountered them in a homework exercise on the harmonic oscillator where you had to prove that they are eigenstates of the creation operator and have minimal uncertainty. And you know that they are important to quantum optics. At least this was my state of knowledge until very recently. Since then, I have read this review (with which I do not agree in all parts) and have spent some thoughts on the subject I would like to share my current understanding.

Let's suppose we have two hermitean operators A and B and want to find states such that the uncertainty is minimal. To this, let's briefly go through the derivation of the uncertainty relation: You assume any state and from it form new states and similarly for B where is the expectation value of A. For these two new states, you use the Cauchy-Schwarz inequality (which basically says in the form ), expand and find


Finally, we use that the absolute value of the imaginary part of a number is less or equal to that number and realise that as A and be are hermitean the imaginary part of the left expectation value is to arrive at which is the usual uncertainty relation.

If you want to rest for a minute think about the following puzzle: Consider a particle on a circle (or on the interval with periodic boundary conditions). Take the wave function and compute that for this state while . This seems to clash with what we just derived. Where is the flaw? (Hint: see a previous post about quantum mechanics)

Back to the main argument. We want to find a state which saturates the inequality. To have that we have to saturate the inequality in the two places where used inequalities: The Cauchy Schwarz and the abs less Im parts of the argument. Cauchy Schwarz is saturated (the scalar product is maximal for vectors of fixed length) if the two vectors are proportional to each other, that is if there is a complex number such that in our case . We can rearrange that to that is has to be an eigenvector of the operator . Furthermore, for the absolute value of a number to be equal to its imaginary part, the number has to be purely imaginary. In our case, this means has to be purely imaginary which can only be if is purely imaginary.

So we found that the uncertainty of operators A and B in a state is minimal if the state is an eigenvector of an operator with real .

So much for the general theory. Now, we can specialise to the usual case A=x and B=p and conclude that states of minimal uncertainty are eigenstates of for some real . Note that so far we have not talked about the harmonic oscillator at all. We have just picked two operators and asked for states in which they have minimum uncertainty. This was a question at the level of Hilbert space operators and we did not specify any sort of dynamics.

Thus, coherent states are not about the harmonic oscillator at all. It just happens that they are eigenstates of annihilation operators for some harmonic oscillator. Above any real does the job and this translates directly to the frequency of the oscillator: What people call "squeezed states" are just coherent states for a different that can in a similar way be related to the annihilation operators at different frequencies.

This so far is my current understanding. In the above mentioned review there is another generalisation which does involve dynamics which I do not yet fully understand. It somehow splits a Hamiltonian into sums of products of 'elementary' operators and then considers the Lie algebra generated by these elementary operators upon commutators. Then you exponentiate this algebra to a group and consider the orbit of the ground state of that Hamiltonian under the action of this group. The part I do not yet understand is how physical this is and how the different choices on the way (the set of elementary operators for example) influence the result.