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

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


$$ B = \sum_n \kappa_n\sqrt{\frac{\hbar}{2m_n\omega_n}}(b_n+b_n^\dagger)$$

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 $\hbar=1$)

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

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

And correlation function can be easily analytically calculated.

My question is, if we define more general spectral density


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

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

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 ?


1 Answer 1


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): $$ J(\omega)=\eta_s \omega^s\omega_{ph}^{1-s}e^{-\omega/\lambda}, $$ 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.


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