# Require help understanding Glauber Coherent States of the Quantum Harmonic Oscillator

I have been doing some reading into Glauber Coherent States and I am struggling to get a grasp on how they are composed, ie. how to determine when they exist. I kind of get the idea (qualitatively speaking) let me try to explain what I think;

- They are composed of superpositions of many of the quantum harmonic oscillator (single particle) states, where the amplitudes of each of these states is such that it gives a pendulum like 'hump' (thinking of the graph of probability density) that swings back and forth within the potential, the coherent state exists only when this pendulum like behaviour is observed. Is this correct?

I just don't know how to define them mathematically and I'm getting lost in the text, I'll persist with reading so I may get the answer eventually but if someone could someone help me to understand it clearly I would be grateful. I have found a few texts so far but they seem to be going over my head, I am currently drudging through "Optical Coherence & Quantum Optics" (Mandel & Wolf) specifically chapter 11.

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tom.stoer

You can construct coherent states starting with

$$(a-z)|z\rangle = 0$$

where a is the annihilation operator and z is an arbitrary complex number.

Solving this equation tells you that |z> is

$$|z\rangle = e^{-|z|^2/2}\sum_{n=0}^\infty \frac{z^n}{\sqrt{n!}}|n\rangle = e^{-|z|^2/2} e^{a^\dagger z}|0\rangle$$

The time evolution can be calculated using

$$H = \omega a^\dagger a$$

(where I omitted the 1/2 b/v it's trivial) and

$$|z,t\rangle = e^{-iHt}|z,0\rangle = |z(t)\rangle$$

with

$$z(t) = e^{-i\omega t}\,z(0)$$

So time evolution takes a coherent state with z=z(0) inot a new coherent state z(t) with constant |z|