Squeezed coherent state

In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position ${\displaystyle x}$ and momentum ${\displaystyle p}$ of a particle, and the (dimension-less) electric field in the amplitude ${\displaystyle X}$ (phase 0) and in the mode ${\displaystyle Y}$ (phase 90°) of a light wave (the wave's quadratures). The product of the standard deviations of two such operators obeys the uncertainty principle:

Trivial examples, which are in fact not squeezed, are the ground state ${\displaystyle |0\rangle }$ of the quantum harmonic oscillator and the family of coherent states ${\displaystyle |\alpha \rangle }$. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with ${\displaystyle \Delta x_{g}=\Delta p_{g}}$ in "natural oscillator units" and ${\displaystyle \Delta X_{g}=\Delta Y_{g}=1/2}$.

The term squeezed state is actually used for states with a standard deviation below that of the ground state for one of the operators or for a linear combination of the two. The idea behind this is that the circle denoting the uncertainty of a coherent state in the quadrature phase space (see right) has been "squeezed" to an ellipse of the same area. Note that a squeezed state does not need to saturate the uncertainty principle.

Squeezed states of light were first produced in the mid 1980s. At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e. ${\displaystyle \Delta ^{2}X\approx \Delta ^{2}X_{g}/2}$. As of 2017, a squeeze factor of 31 (15 dB) has been directly observed.

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

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

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.