squeezed

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



x


{\displaystyle x}

and momentum



p


{\displaystyle p}

of a particle, and the (dimension-less) electric field in the amplitude



X


{\displaystyle X}

(phase 0) and in the mode



Y


{\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:




Δ
x
Δ
p
≥


ℏ
2





{\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}\;}

and




Δ
X
Δ
Y
≥


1
4




{\displaystyle \;\Delta X\Delta Y\geq {\frac {1}{4}}}

, respectively.

Trivial examples, which are in fact not squeezed, are the ground state




|

0
⟩


{\displaystyle |0\rangle }

of the quantum harmonic oscillator and the family of coherent states




|

α
⟩


{\displaystyle |\alpha \rangle }

. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with



Δ

x

g


=
Δ

p

g




{\displaystyle \Delta x_{g}=\Delta p_{g}}

in "natural oscillator units" and



Δ

X

g


=
Δ

Y

g


=
1

/

2


{\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.




Δ

2


X
≈

Δ

2



X

g



/

2


{\displaystyle \Delta ^{2}X\approx \Delta ^{2}X_{g}/2}

. As of 2017, squeeze factors larger than 10 (10 dB) have been directly observed.

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