Phonon squeezing

T. Kobayashi I would like to make the comment that an interesting application of wavepacket control [1] is phonon squeezing in molecular systems and the creation of the Schrodinger cat state. It was theoretically predicted that there are several mechanisms that lead to squeezing of phonon states. 

The above-mentioned mechanism of squeezing the vibrational state prompted some controversial discussion in the literature [13-16]. The phenomenon is caused by the change of the frequency of the molecular vibration provided that the transition takes place in a fraction of time negligibly small as compared with the vibrational period. Recently we have shown that phonon squeezing, connected to the finite duration of the excitation pulse, occurs even in the absence of frequency change. This effect would be rather common in ultrashort laser pulse experiments [17-19],... 

Figure 1. Uncertainty of the quadrature AX of the phonons (squeezing occurs if AX becomes less than unity) after a resonant Franck-Condon transition induced by a chirped pulse of moderate duration (u 0.0437a)) as a function of the chirp parameter tv, which is in the units of the phonon frequency at. The electron-lattice constant is supposed to beg = 5. The markers a-d refer to Fig. 2.

In combination with DFT calculations, the time- and depth-dependent phonon frequency allows to estimate the effective diffusion rate of 2.3 cm2 s 1 and the electron-hole thermalization time of 260 fs for highly excited carriers. A recent experiment by the same group looked at the (101) and (112) diffractions in search of the coherent Eg phonons. They observed a periodic modulation at 1.3 THz, which was much slower than that expected for the Eg mode, and attributed the oscillation to the squeezed phonon states [9]. 

The -functions of phonon states after the electronic transition induced by differently ses. (a) The 2-function at the minimal AA"+, that is, maximal squeezing (the chirp parameter see marker a in Fig. 1). (b) The AX+ is in its next maximum, (c) An intermediate state maximum and minimum. As we consider lower and lower chirp parameters, the maxima and ome less prominent (4), approaching the number state with equal distribution along a circle. 

As we have seen, the role of phonons in two-dimensional tunneling can be elucidated by considering linearly and symmetrically coupled double-well potentials. In both cases the bending of the reaction path is caused by coupling to a vibration. The pure effect of the vibration-induced squeezing of the reactive channel (without bending) may be conventionally studied using the potential... 

In fact, it is squeezed displaced harmonic oscillator if one does not take into account the parameter 17 Introducing A does not invoke higher order nonlinearities with respect to phonon-1 coordinates. Thus (APj) = 0, like it should be for an harmonic oscillator, and the expressions for second momenta have the form ... 

A suitable choice of the variational wave functions for various electron-phonon two-level systems is a long-standing problem in solid state physics as well as in quantum optics. For two-level reflection symmetric systems with intralevel electron-phonon interaction the approach with a variational two-center squeezed coherent phonon wave function was found to yield the lowest ground state energy. The two-center wave function was constructed as a linear combination of the phonon wave functions related to both levels introducing new VP. 

An example of principal squeeze variances of compound mode (Si, A]) is given in Fig. 29 for three different asymmetric configurations. When yv = 0 (no damping) we have maximal squeezing and periodic dependence on z, whereas for yVl = 2.5 the oscillations are damped and X reaches some asymptotic value. Figure 29 also illustrates that nonzero mean number of chaotic phonons makes the squeezing less pronounced. 

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