# Time Evolution of the Expectation Value of *X*, and also its Fluctuation, for Phonons

Schematic diagram of the time evolution of the expectation value and
the fluctuation of the lattice amplitude operator *u*(±**q**)
in different states. Here dashed lines represent the average <*u*(±**q**)>(t),
while solid lines represent the envelopes <*u*(±**q**)>(t)
± (<[D*u*(±**q**)]^2>(t))^0.5
which provide the upper and lower bounds for the fluctuations in *u*(±**q**)(t).

**(a)** The phonon vacuum state |0>, where <*u*(±**q**)>
= 0 and <[D*u*(±**q**)]^2>
= 2.

**(b)** A phonon number state |*n*q, *n*-q>,
where <*u*(±**q**)> = 0 and
<[D*u*(±**q**)]^2>
= 2(*n*q* + n*-q) +
2.

**(c)** A single-mode phonon coherent state |aq*>*,
where <*u*(±**q**)> = 2*Re*(aqexp(-*i*wq*t*))
= 2|aq|coswq*t*,
which means that aq is
real, and <[D*u*(±**q**)]^2>
= 2.

**(d)** A single-mode phonon squeezed state |aqexp(-*i*wq*t*),
x(*t*)>, where the squeezing factor x(*t*)
satisfies x(*t*) = *r*exp(-2*i*wq*t*).
Here, <*u*(±**q**)> = 2|aq|coswq*t*,
which means that aq is
real, and its fluctuation is

<[D*u*(±**q**)]^2>
= 2[exp(-2*r*)cos^2wq*t*
+ exp(2*r*)sin^2wq*t*].

**(e)** A single-mode phonon squeezed state, as in (d). Now the expectation
value of *u* is <*u*(±**q**)>
= 2|aq|sinwq*t*,
which means that aq is
purely imaginary, and the fluctuation <[D*u*(±**q**)]^2>
has the same time-dependence as in (d). Notice that the squeezing effect
now appears at the times when <*u*(±**q**)>
reaches its maxima while in (d) the squeezing efffect is present at the
times when <*u*(±**q**)> is
close to zero.

Image Source: X. Hu, *Quantum Fluctuations In Condensed Matter Systems*,
UM Ph.D. Thesis 1997, Page 45.

X. Hu and F. Nori, *Physical Review B 53*,
2419 (1996).