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# Coherent state

In quantum mechanics a coherent state is a quantum state which describes coherent fields, fields where all of the particles involved share some measurable property that allows them to be thought of as being "in phase", or in other circumstances, one larger object.

In classical optics light is thought of as waves radiating from a source, and coherent light is thought of as the light from many such sources that are in phase. For instance a light bulb gives off light that is actually the result of the light being emitted at all the points along the filament, and such light is not coherent because the process is highly random. In the laser however, the light is emitted "all at once" from a carefully controlled mixture of atoms, the process is not random and the resulting light is highly ordered, or coherent.

In the quantum model we understand light to be somewhat more complex than a simple wave, including, among others, the properties traditionally measured in classical optics. However the quantum model describes all of nature using this description, with everything from light to solid objects sharing similar behaviours. Under the quantum model then, the concept of coherence takes on a broader and more general meaning that can be applied to any set of particles in the proper state.

In the wave-particle duality reminiscent in all quantum states, the coherent state stands on one extreme, namely, it behaves as much as possible as a wave, whereas the Fock state (like a photon) stands on the other end, namely the particle behaviour. In fact neither the particle or wave are "real" in this model, but they are useful fictions for describing the overall behaviour of systems, and in this case any system can display properties that make it easier to consider as wave-like or particle-like.

In quantum optics the coherent state is the quantum state of light emitted by an ideal laser. In condensed matter physics it describes fields with coherence such as Bose-Einstein condensates.

In a Fock space, the coherent state is the eigenstate of the annihilation operators:

$a(\phi)|\psi\rangle=\langle\phi|\psi\rangle|\psi\rangle$

if |ψ> is a coherent state, with a(φ) the operator which annihilate the particle which wavefunction is |φ>.

In the occupation number formalism (see Fock space and quantum harmonic oscillator), the coherent state becomes the eigenstate of the annihilation operator of a particle in its ground state, written just a. Let us call |α> this coherent state, so that by definition it formally reads:

$a|\alpha\rangle=\alpha|\alpha\rangle$

Since a is not hermitian, α can be complex. The coherent state was introduced by Schrödinger as the quantum state of the harmonic oscillator which minimizes the uncertainty relation with uncertainty equally distributed in both position P and impulsion X (if the uncertainty is not equally distributed, the state is a squeezed coherent state). From the generalized uncertainty relation, it is shown that such a state |α> must obey the equation

$(P-\langle P\rangle)|\alpha\rangle=i(X-\langle X\rangle)|\alpha\rangle$

which, if written back in terms of a and a, becomes:

$a|{\alpha}\rangle={1\over\sqrt{2\hbar}}(\langle X\rangle+i\langle P\rangle)|{\alpha}\rangle$

so that it is clear that the eigenvalues not only can be complex but actually span the whole complex space. Also the physical meaning of the coherent state is clear: it is the most classical state allowed by quantum physics, since it has lowest uncertainty in its conjuguate variables, and it can be located in the complex plane as the position in the phase space of the quantum oscillator, with position on real axis and momentum on imaginary axis. It is clear from this that a coherent state is a state of well defined phase.

The coherent state does not display all the nice mathematical features of a Fock state, for instance two different coherent states are not orthogonal:

$\langle\beta|\alpha\rangle=e^{-{1\over2}(|\beta|^2+|\alpha|^2-2\beta^*\alpha)}\neq\delta(\alpha-\beta)$

so that if the oscillator is in the quantum state |α> it is also with nonzero probability in the other quantum state |β> (but this is the more improbable the farther apart the states in the phase space). However any state can be decomposed on the set of coherent states which obey a closure relation. They hence form an overcomplete basis, which allows a diagonal decomposition of any state, which is the basis of Glauber P representation .

Another difficulty is that a has no eigenket (and a has no eigenbra). The formal following equality is the closest substitute and turns out to be very useful for technical computations:

$a^{\dagger}|\alpha\rangle=\left({\partial\over\partial\alpha}+{\alpha^*\over 2}\right)|\alpha\rangle$

This can be easily obtained, as can virtually all results involving coherent states, using the representation of the coherent state on the basis of Fock states:

$|\alpha\rangle=e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle$

Phase and particle number are conjugate variables and since the phase is well defined for a coherent state, the number of particle is poorly defined in return. This is confirmed by computing the average <aa>=|α|2 and the variance Var(aa)=|α|2. They are equal, as is characteristic of a Poissonian distribution, which is indeed the statistics of a coherent state.

In quantum field theory and string theory, a generalization of coherent states to the case of infinitely many degrees of freedom is used to define a vacuum state with a different vacuum expectation value from the original vacuum.