# Squeezed coherent state

In physics, a squeezed coherent state is any state of the quantum mechanical Hilbert space such that the uncertainty principle is saturated. That is, the product of the corresponding two operators takes on its minimum value:

Husimi distribution of the squeezed coherent state

${\displaystyle \Delta x\Delta p={\frac {\hbar }{2}}}$

The simplest such state is the ground state ${\displaystyle |0\rangle }$ of the quantum harmonic oscillator. The next simple class of states that satisfies this identity are the family of coherent states ${\displaystyle |\alpha \rangle }$.

Often, the term squeezed state is used for any such state with ${\displaystyle \Delta x\neq \Delta p}$ in "natural oscillator units". The idea behind this is that the circle denoting a coherent state in a quadrature diagram (see below) has been "squeezed" to an ellipse of the same area. [1] [2] [3] [4] [5]

## Mathematical definition

The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with ${\displaystyle \hbar =1}$)

${\displaystyle \psi (x)=C\,\exp \left(-{\frac {(x-x_{0})^{2}}{2w_{0}^{2}}}+ip_{0}x\right)}$

where ${\displaystyle C,x_{0},w_{0},p_{0}}$ are constants (a normalization constant, the center of the wavepacket, its width, and the expectation value of its momentum). The new feature relative to a coherent state is the free value of the width ${\displaystyle w_{0}}$, which is the reason why the state is called "squeezed".

The squeezed state above is an eigenstate of a linear operator

${\displaystyle {\hat {x}}+i{\hat {p}}w_{0}^{2}}$

and the corresponding eigenvalue equals ${\displaystyle x_{0}+ip_{0}w_{0}^{2}}$. 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 Heisenberg's 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 ${\displaystyle \pi /2}$.

 Figure 1: Measured quantum noise of the electric field of different squeezed states depends on 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.[6] Figure 2: Oscillating wave packets of the five states. 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 for a squeezed state is no longer independent of the phase of the light wave. 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 an 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 distribution Wigner quasi-probability distribution. 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.

 Figure 4: Experimental photon number distributions for an amplitude-squeezed state, a coherent state, and a phase squeezed state reconstructed from measurements of the quantum statistics. Bars refer to theory, dots to experimental values. [6] (source: link 1) 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 is wider. The opposite is true for the phase-squeezed light, which displays a large intensity (photon number) noise but a narrow phase distribution. Nevertheless the statistics of amplitude squeezed light was not observed directly with photon number resolving detector due to experimental difficulty.[7]

Figure 4: Reconstructed and theoretical photon number distributions for a squeezed-vacuum state. A pure squeezed vacuum state would have no contribution from odd-photon-number states. The non-zero contribution in the above figure is because the detected state is not a pure state - losses in the setup convert the pure squeezed vacuum into a mixed state. [6] (source: link 1)

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 (see optical parametric oscillator). This is achieved by a simple process of four-wave mixing with a ${\displaystyle \chi ^{(3)}}$ crystal; similarly traveling wave phase-sensitive amplifiers generate spatially multimode quadrature-squeezed states of light when the ${\displaystyle \chi ^{(2)}}$ crystal is pumped in absence of any signal. Sub-Poissonian current sources driving semiconductor laser diodes have led to amplitude squeezed light.[8] Squeezed states have also been realized via motional states of an ion in a trap, phonon states in crystal lattices, or atom ensembles.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states. Current state of the art in noise suppression, for laser radiation using squeezed light, amounts to 12.7 dB.[9]

## 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 readout 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, and generally, particle production in curved backgrounds and Bogoliubov transformation).

Recently, the use of squeezed states for quantum information processing in the continuous variables (CV) regime has been increasing rapidly.[10] Continuous variable quantum optics uses squeezing of light as an essential resource to realize CV protocols for quantum communication, unconditional quantum teleportation[11] and one-way quantum computing.[12] This is in contrast to quantum information processing with single photons or photon pairs as qubits. CV quantum information processing relies heavily on the fact that squeezing is intimately related to quantum entanglement, as the quadratures of a squeezed state exhibit sub-shot-noise quantum correlations.