# Time Evolution of the Expectation Value of *X*, and also its Fluctuation, for Superconducting Josephson Junctions

Schematic diagram of the time evolution of both the expectation value
<f>(*t*) and the fluctuation <[Df]^2>(*t*)
of the phase operator f for different states:

**(a) **vacuum,

**(b) **number,

**(c) **coherent,

**(d)** and **(e) **squeezed.

Here dashed lines represent <f>(*t*),
while solid lines represent the envelopes <f>
± (<[Df]^2>(t))^0.5,
which procide the upper and lower bounds for the fluctuating quantity f(*t*).
(a) For the vacuum state |0>, <f>(*t*)
= 0 and <[Df]^2>(t) = 2(*EJ*/*EC*)^0.5.
(b) For a number state |*n*>, <f>(*t*)
= 0 and <[Df]^2>(t) = 2*n* +1. (c) For
a coherent state |a*>, *<f>(*t*)
= 2*Re*(aexp(-*i*w*t*))
= 2|a|cosw*t*,
which means that a is real, and <[Df]^2>(t)
= 2. (d) For a squeezed state |aexp(-*i*w*t*),
x(*t*)>, where the squeezing factor x(*t*)
satisfies

x(*t*) = *r*exp(-2*i*w*t*),
<f>(*t*) = 2|a|cosw*t*,

which means that a is real and its fluctuation is

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

(e) A squeezed state as in (d). Now the expectation value of f
is <f>(*t*) = 2|a|sinw*t*,
which means that a is purely imaginary, and
the fluctuation <[Df]^2>(*t*) has the
same time-dependence as in (d). Notice that the squeezing effect now appears
at the times when <f>(*t*) reaches its
maxima while in (d) the squeezing effect is present at the times when <f>(*t*)
is close to zero.

Image Source: X. Hu, *Quantum Fluctuations In Condensed Matter Systems*,
UM Ph.D. thesis 1997, Page 104.
** **X. Hu and F. Nori, UM Preprint, 1995.