# Squeezed coherent state

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:

[itex]\Delta x \Delta p = \frac{\hbar}2[itex]

The simplest such state is the ground state [itex]|0\rangle[itex] of the quantum harmonic oscillator. The next simple class of states that satisfies this identity ist the family of coherent states [itex]|\alpha\rangle[itex].

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## Mathematical definition

The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with [itex]\hbar=1[itex])

[itex]\psi(x) = C\,\exp\left(-\frac{(x-x_0)^2}{2 w_0^2} + i p_0 x\right)[itex]

where [itex]C,x_0,w_0,p_0[itex] 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 [itex]w_0[itex], which is the reason why the state is called "squeezed".

The squeezed state above is an eigenstate of a linear operator

[itex] \hat x + i\hat p w_0^2[itex]

and the corresponding eigenvalue equals [itex]x_0+ip_0 w_0^2[itex]. 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 amplitude-squeezed and phase-squeezed 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 [itex]\pi/2[itex].

 Missing imageNoise_squeezed_states.jpg Figure 1: Measured quantum noise of the electric field of different squeezed states in dependence of the phase of the light field. For the first two states a 3π-interval is shown, for the last three states, belonging to a different set of measurements it is a 4π-interval. (source: link 1 and ref. 3) Missing imageWave_packet_squeezed_states.jpg Figure 2: Oscillating wave packets of the five states. Missing imageWigner_function_squeezed_states.jpg Figure 3: Wigner functions of the five states. The ripples are due to experimental inaccuracies.

From the top:

• Vacuum state
• Squeezed vacuum state
• Phase-squeezed state
• arbitrary squeezed state
• Amplitude-squeezed 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 amplitude-squeezed 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 phase-squeezed 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 imagePhoton_numbers_squeezed_coherent_states_subpoisson.jpg Figure 4: Measured photon number distributions for an amplitude-squeezed state, a coherent state, and a phase squeezed state. Bars refer to theory, dots to experimental values. (source: link 1 and ref. 2) Missing imagePhase_distribution_squeezed_coherent_states_subpoisson.jpg Figure 5: Pegg-Barnett phase distribution of the three states.

For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in sub-poissonian light, whereas its phase distribution ist wider. The opposite ist true for the phase-squeezed light, which displays a large intensity (photon number) noise but a narrow phase distribution.

Missing image
Photon_numbers_squeezed_vacuum.jpg
Figure 4: Measured photon number distributions for a squeezed-vacuum state. (source: link 1 and ref. 3)

For the squeezed vacuum state the photon number distribution displays odd-even-oscillations. This can be explained by the mathematical form of the squeezing operator, that resembles the operator for two-photon 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 non-linear 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 phase-squeezed light can improve the phase read out of interferometric measurements (see for example gravitational waves). Amplitude-squeezed 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).

## references

1. Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), [ISBN 0198501773]
2. D.F. Walls and G.J. Milburn, Quantum Optics, Springer Berlin 1994
3. G. Breitenbach, S. Schiller, and J. Mlynek, "Measurement of the quantum states of squeezed light", Nature, 387, 471 (1997) (http://www.exphy.uni-duesseldorf.de/Publikationen/1997/N387/471z.htm)

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