Crossover quantum-classical

Gillam M J 1987 Quantum-classical crossover of the transition rate in the damped double well J. Phys. C Solid State Phys. 20 3621... 

Gilian, M. j., Quantum-Classical Crossover of the Transition Rate in the Damped Double Well, J. Phys. C. Solid State Phys. 1987, 20, 3621-3641. 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

Fig. 5. The low temperature crossover diagram of a one-dimensional CDW. t and K are proportional to the temperature and the strength of quantum fluctuations, respectively. The amount of disorder corresponds to a reduced temperature tu 0.1. In the classical and quantum disordered region, respectively, essentially the t = 0 behavior is seen. The straight dashed line separating them corresponds to At 1, i.e., K 1, where At is the de Broglie wave length. In the quantum critical region, the correlation length is given by At- Pinning (localization) occurs only for t = 0, K

Of special interest is the case where the barrier is parabolic, as in Eq. (1.5). Here, it is possible to examine the crossover between the classical and quantum regimes in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the transmission coefficient of the parabolic barrier [Landau and Lifshitz, 1981]... 

Figure 3. In the short-memory limit (rac —> oo), the quantum time-dependent diffusion coefficient DP - 00 piotte(j as a function of yt for several different bath temperatures on both sides of Tc (full lines) yrth = 0.25(T — 2TC, classical regime) yt — 0.5 (T = Tc, crossover) yrth = 1 (T = Tc/2, quantum regime) yt = +oo (T = 0). The corresponding curves for the classical diffusion coefficient Dcl( ) are plotted in dotted lines in the same figure.

Zaitsev-Zotov, S.V. 1993. Classical-to-quantum crossover in charge-density wave creep at low temperatures. Phys Rev Lett 71 605. 

We shall in addition assume that the rate of quantum tunneling through the potential barriers is negligible compared with the rate obtained from the classical treatment. This is an assumption that generally always breaks down when the temperature becomes low enough. However, the typical crossover temperature below which the rate becomes dominated by quantum tunneling is very low (dependent on the barrier thickness and the masses of the tunneling particles). 

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