Decoherence and dissipation of a quantum harmonic oscillator coupled to two-level systems

We derive and analyze the Born-Markov master equation for a quantum harmonic oscillator interacting with a bath of independent two-level systems. This hitherto virtually unexplored model plays a fundamental role as one of the four “canonical” system-environment models for decoherence and dissipation. To investigate the influence of further couplings of the environmental spins to a dissipative bath, we also derive the master equation for a harmonic oscillator interacting with a single spin coupled to a bosonic bath. Our models are experimentally motivated by quantum-electromechanical systems and micron-scale ion traps. Decoherence and dissipation rates are found to exhibit temperature dependencies significantly different from those in quantum Brownian motion. In particular, the systematic dissipation rate for the central oscillator decreases with increasing temperature and goes to zero at zero temperature, but there also exists a temperature-independent momentum-diffusion (heating) rate.

Figure 1

(a) Oscillator-spin model. (b) Oscillator-spin model with each spin coupled to an additional bosonic bath. (c) Limiting case of (b) in which the oscillator interacts with a single environmental spin coupled to a bosonic bath.

Figure 2

(Color online) Dissipation and normal-diffusion coefficients (a) $\gamma$ and (b) $D$ for the oscillator-spin model (solid line) and for quantum Brownian motion (circles), assuming the ohmic spectral density (25). The vertical axis is normalized in units of the zero-temperature values $\gamma \left(T=0\right)$ and $D_{\text{QBM}}\left(T=0\right)$, respectively. The horizontal (temperature) axis is displayed in units of ${\Omega }_0/k_B$. We use $D_0=D_{\text{QBM}}\left(T=0\right)$.

Figure 3

(Color online) Comparison of the adiabatic approximation (crosses) and the full master equation (circles) for the simple model discussed in Sec. 3. We show the mean occupation number $⟨N⟩$ of the central harmonic oscillator as a function of time for three different temperatures [a larger value of $\overline{n}$, see Eq. (35), corresponds to higher temperature]. The time axis is in units of inverse $\Delta$ (with $\hslash \equiv 1$). We use $\Delta =1$, ${\Omega }_0=1$, $g=1$, and (a) $\gamma =10$, (b) $\gamma =100$.