# <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.