# Super-ohmic bosonic bath correlation function

In quantum Brownian motion, bosonic/harmonic oscillator bath and interaction described by Hamiltonian

$$H_B = sum_{n}hbaromega_n(b_n^dagger b_n) \ H_I = -sigma_x otimes B$$

and

$$B = sum_n kappa_nsqrt{frac{hbar}{2m_nomega_n}}(b_n+b_n^dagger)$$

We have two correlation function to describe time evolution of system (See Breuer p.174)

$$D(tau) equiv iTr_R([B,B(-tau)]rho_B) \ D_1(tau) equiv Tr_R({B,B(-tau)}rho_B)$$

And using Spectral Density (approximating infinite mode) we get (setting $$hbar=1$$)

$$D(tau)=2int_0^infty domega , J(omega)sin(omegatau) \ D_1(tau)=2int_0^infty domega , J(omega)coth left(frac{hbaromega}{2k_BT} right)cos(omegatau)$$

In many references (including Breuer) ohmic spectral density with Drude-Lorentz cutoff is used
$$J(omega)=alpha^2omegafrac{lambda^2}{omega^2+lambda^2}$$

And correlation function can be easily analytically calculated.

My question is, if we define more general spectral density

$$J(omega)=alpha^2omega^somega_{ph}^{1-s}frac{lambda^2}{omega^2+lambda^2}$$

With $$S = 1$$ (Ohmic), $$S > 1$$ (Super-Ohmic), $$S < 1$$ (Sub-Ohmic). In Sub-ohmic region, working the integral for $$D(tau)$$ in Mathematica gives Hypergeometric function which is more complicated than Ohmic case but still a well behaved function. The problem arise when choosing spectral density as a Super-Ohmic

$$D(tau)=2alpha^2lambda^2omega_ph^{1-s}int_0^infty frac{omega^s sin(omegatau)}{omega^2+lambda^2}$$

For $$S > 1$$ the integral is divergent. From this result, is the boson bath "limited" to the ohmic and sub-ohmic spectral density(is there any microscopical model to suggest this) ? How the divergence of correlation function can be interpret in relation to super-ohmic ?

Physics Asked by Hezzuappu on December 31, 2020

Note that the term $$lambda^2/(omega^2+lambda^2)$$ in your spectral density is suitable only for the Ohmic case, i.e. it is the proper cutoff in scenarios where the spectral density is proportional to $$omega$$. A nice (classical) microscopic model displaying this behavior is the Lorentz-Drude model for electrical conduction. Resistors in (quantum) superconducting circuits also induce dissipation with an Ohmic dependency and corresponding cutoff [1].

If you want to study scenarios with Super-Ohmic or Sub-Ohmic spectral densities, then you need to introduce a different cutoff. In these situations the spectral density is usually written as [2] (Section 3.1.5): $$J(omega)=eta_s omega^somega_{ph}^{1-s}e^{-omega/lambda},$$ where $$eta_s$$ expresses the coupling strength of the interaction with the bath, while $$lambda$$ is the cutoff frequency. The exponential dependence on $$lambda$$ solves any divergence issue. The idea is that you aim to reproduce the "physical behavior" described by the dependence on $$omega^s$$, and then you insert a phenomenological "ad hoc" cutoff to remove any unphysical behavior at very high (or very low) frequencies. For more microscopic models have a look at the book by Weiss [2].

[1] U. Vool and M. Devoret. "Introduction to quantum electromagnetic circuits". International Journal of Circuit Theory and Applications 45, 897-934 (2017).

[2] U. Weiss, Quantum dissipative systems (3rd edition). World scientific, 2008.

Correct answer by Goffredo_Gretzky on December 31, 2020

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