The Spin-Boson Hamiltonian

The applet on this page shows the numerical integration of the so-called “spin-boson Hamiltonian”

H =  ---σx + ˉhλσz(a† + a) + ˉhω0a†a

which, like the Jaynes-Cummings model, features a quantized harmonic oscillator coupled to a two-level system. The parameters in (1) are referred to, in the text fields below, as the “spin frequency” (Ω), “oscillator frequency” (ω0) and “coupling” (λ). This Hamiltonian appears in many physical models, especially in solid-state physics.

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Unlike the JCM, the Hamiltonian (1) is not analytically integrable, so the applet attempts to integrate it numerically. This requires one to truncate the (in principle) infinite-dimensional space of states of the oscillator, using a finite basis of states ∣0〉,∣1〉...∣n    〉
           max , where the states are labeled by the number of energy quanta. It seems that, in principle, the error could be made arbitrarily small just by making the cutoff nmax sufficiently large; in particular, since the initial state is a coherent state with an average number of quanta 〈n〉 = ∣α ∣2   , and a standard deviation Δn = α, one might think that it is enough to set nmax equal to 〈n〉 plus 5 or 6 standard deviations, and this is the default choice.

However, it seems that, at least for some values of the parameters, the eigenvalues of the truncated version of the Hamiltonian (1) are very sensitive to the truncation parameter nmax, and this causes the trajectories calculated for different values of nmax to diverge after some time T. We have made it possible to vary nmax in the applet so the user can explore this phenomenon.

The eventual breakdown of predictability for the truncated quantum system may be related to the well-known fact that its classical counterpart (with a classical rather that a quantized field) is chaotic in some parameter regions. There is a vast amount of literature on this subject, so we’ll just cite our own work: G. A. Finney and J. Gea-Banacloche, Phys. Rev. E 54, 1449 (1996).

In other regimes, various analytical approximations are possible. For the resonant (Ω = ω0) case, with small coupling, an approximation was developed in: G. A. Finney and J. Gea-Banacloche, Phys. Rev. A 50, 2040 (1994), which shows the existence of almost-pure-state trajectories similar to the ones found in the Jaynes-Cummings model. One may use this applet to reproduce many of the results in that paper, only noting that the roles of x and z are reversed in the Hamiltonian (1); to convert the Finney-Gea-Banacloche coordinates to the ones used here, subtract π∕2 from θ and add π to φ. (For instance, one can reproduce Fig. 4(a) in that paper by choosing Ω = ω0 = 1, λ = 0.05, α = √32--, θ = 0.0495π, and φ = π.)

Another parameter region that we explored recently was the “fast oscillator” regime: see E. K. Irish et al., Phys. Rev. B 72, 195410 (2005).

Since the time evolution in this system, in general, displays many different time scales, we have also made it possible to change the step size for the plots (T_step above) if the user wants to increase their time resolution.

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