##
<*X*>(*t*) and its Fluctuations for Coherent and Squeezed
States

Schematic diagram of the time evolution of the expectation value <*X*>,
and the fluctuation (<(D*X*)^2>)^0.5
of the dimensionless quadrature *X* (of an electric field, for instance)
in a **coherent state** (a) and a **squeezed state** (b). Here dashed
lines represent <*X*>, while solid lines represent the envelopes
<*X*> ± (<(D*X*)^2>)^0.5,
which are the upper and lower bounds of the fluctuating quantity *X*.
**(a)** A single-mode photon

coherent state |a>, where <*X*>
= 2*R*[*e*(aexp(-*i*w*t*))]
= 2|a|sinw*t*,
which means that a is purely imaginary; and
<(D*X*)^2> = 2.

**(b)** A single-mode photon squeezed state |aexp(-*i*w*t*),
x(*t*)>, where the squeezing factor x(*t*)
satisfies x(*t*) = *r*exp(-2*i*w*t*).
Here, <*X*> = w|a|cosw*t*,
which means that a is real, and its fluctuation
is

<(D*X*)^2> = 2[(exp(-2*r*)cos^2w*t*
+ exp(2*r*)sin^2w*t*)].

Notice that the squeezing effect appreas at the times when <*X*>
reaches its maxima.

Image Source: X. Hu, *Quantum Fluctuations In Condensed Matter Systems*,
UM Ph.D. Thesis 1997, Page 6.