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) ± (<[Du(±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 <[Du(±q)]^2> = 2.

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

(c) A single-mode phonon coherent state |aq>, where <u(±q)> = 2Re(aqexp(-iwqt)) = 2|aq|coswqt, which means that aq is real, and <[Du(±q)]^2> = 2.

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

    <[Du(±q)]^2> = 2[exp(-2r)cos^2wqt + exp(2r)sin^2wqt].

(e) A single-mode phonon squeezed state, as in (d). Now the expectation value of u is <u(±q)> = 2|aq|sinwqt, which means that aq is purely imaginary, and the fluctuation <[Du(±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).