Squeezed coherent state
From Academic Kids

In physics, a squeezed coherent state is every state in the Hilbert space of quantum mechanics that saturates the uncertainty principle that is the product of the corresponding two operators takes on its minimum value:
 <math>\Delta x \Delta p = \frac{\hbar}2<math>
The simplest such state is the ground state <math>0\rangle<math> of the quantum harmonic oscillator. The next simple class of states that satisfies this identity ist the family of coherent states <math>\alpha\rangle<math>.
Contents 
Mathematical definition
The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with <math>\hbar=1<math>)
 <math>\psi(x) = C\,\exp\left(\frac{(xx_0)^2}{2 w_0^2} + i p_0 x\right)<math>
where <math>C,x_0,w_0,p_0<math> are constants (a normalization constant, the center of the wavepacket, its width, and its average momentum). The new feature relative to a coherent state is the free value of the width <math>w_0<math>, which is the reason why the state is called "squeezed".
The squeezed state above is an eigenstate of a linear operator
 <math> \hat x + i\hat p w_0^2<math>
and the corresponding eigenvalue equals <math>x_0+ip_0 w_0^2<math>. In this sense, it is a generalization of the ground state as well as the coherent state.
Examples of squeezed coherent states
Depending on at which phase the state's quantum noise is reduced one can distinguish amplitudesqueezed and phasesqueezed states or general quadrature squeezed states. If no coherent excitation exists the state is called a squeezed vacuum. The figures below give a nice visual demonstration of the close connection between squeezed states and Heisenbergs uncertainty relation: Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of the complementary quadrature, that is the field at the phase shifted by <math>\pi/2<math>.
Missing image Noise_squeezed_states.jpg  Missing image Wave_packet_squeezed_states.jpg  Missing image Wigner_function_squeezed_states.jpg 
From the top:
 Vacuum state
 Squeezed vacuum state
 Phasesqueezed state
 arbitrary squeezed state
 Amplitudesqueezed state
As can be seen at once in contrast to the coherent state the quantum noise is not independent of the phase of the light wave anymore. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. The wave packet of a squeezed state is defined by the square of the wave function introduced in the last paragraph. They correspond to the probability distribution of the electric field strength of the light wave. The moving wave packets display an oscillatory motion combined with the widening and narrowing of their distribution: The "breathing" of the wave packet. For an amplitudesqueezed state, the most narrow distribution of the wave packet is reached at the field maximum, resulting in a amplitude that is defined more precisely than the one of a coherent state. For a phasesqueezed state the most narrow distribution is reached at field zero, resulting in an average phase value that is better defined than the one of a coherent state.
In phase space quantum mechanical uncertainties can be depicted by Wigner distributions. The intensity of the light wave, its coherent excitation is given by the displacement of the Wigner distribution from the origin. A change in the phase of the squeezed quadrature results in a rotation of the distribution.
Photon number distributions and phase distributions of squeezed states
The squeezing angle, that is the phase with minimum quantum noise, has a large influence on the photon number distribution of the light wave and its phase distribution as well.
Missing image Photon_numbers_squeezed_coherent_states_subpoisson.jpg  Missing image Phase_distribution_squeezed_coherent_states_subpoisson.jpg 
For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in subpoissonian light, whereas its phase distribution ist wider. The opposite ist true for the phasesqueezed light, which displays a large intensity (photon number) noise but a narrow phase distribution.
Photon_numbers_squeezed_vacuum.jpg
For the squeezed vacuum state the photon number distribution displays oddevenoscillations. This can be explained by the mathematical form of the squeezing operator, that resembles the operator for twophoton generation and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs.
Experimental realizations of squeezed coherent states
There has been a whole variety of successful demonstrations of squeezed states. The most prominent ones were experiments with light fields using lasers and nonlinear optics. But squeezed states have also been realized via motional states of an ion in a trap, phonon states in crystal lattices or atom ensembles. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states.
applications
Squeezed states of the light field can be used to enhance precision measurements. For example phasesqueezed light can improve the phase read out of interferometric measurements (see for example gravitational waves). Amplitudesqueezed light can improve the read out of very weak spectroscopic signals.
Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in quantum field theory, for example Unruh effect and Hawking radiation (generally: particle production in curved backgrounds).
See also
External links
 An introduction to quantum optics of the light field (http://gerdbreitenbach.de/gallery)
references
 Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), [ISBN 0198501773]
 D.F. Walls and G.J. Milburn, Quantum Optics, Springer Berlin 1994
 G. Breitenbach, S. Schiller, and J. Mlynek, "Measurement of the quantum states of squeezed light", Nature, 387, 471 (1997) (http://www.exphy.uniduesseldorf.de/Publikationen/1997/N387/471z.htm)