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:

${\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}\;}$ and ${\displaystyle \;\Delta X\Delta Y\geq {\frac {1}{4}}}$ , respectively.

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 ${\displaystyle \Delta X_{g}=\Delta Y_{g}=1/2}$. (In literature different normalizations for the quadrature amplitudes are used. Here we use the normalization for which the sum of the ground state variances of the quadrature amplitudes directly provide the zero point quantum number ${\displaystyle \Delta ^{2}X_{g}+\Delta ^{2}Y_{g}=1/2}$).

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.^[1][2][3] Note that a squeezed state does not need to saturate the uncertainty principle.

Squeezed states of light were first produced in the mid 1980s.^[4][5] 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, squeeze factors larger than 10 (10 dB) have been directly observed.^[6][7][8]