Variational study of a two-level system coupled to a harmonic oscillator in an ultrastrong-coupling regime

The nonclassical behavior of a two-level system coupled to a harmonic oscillator is investigated in the ultrastrong coupling regime. We revisit the variational solution of the ground state and find that the existing solutions do not account accurately for nonclassical effects such as squeezing. We suggest a trial wave function and demonstrate that it has an excellent accuracy for the quantum correlation effects as well as for the energy.

Figure 1

Profile of $H_{\mathrm{cl}}^{-}(x,0)$: (a) for $\Omega =10$ and (b) for $\Omega ={10}^5$. Different curves correspond to $g=0.1,1,2,3$ (from bottom to top). The inset in (b) zooms in on the region around $x=0$.

Figure 2

Comparison of variational solutions with the exact solution. The open (blue) square represents the DCS wave function, while the filled (red) circle indicates the DSS wave function. (a, e) Error in the energy with respect to the exact energy. (b, f) Error in the tangle. (c, g) Error in the momentum variance. (d, h) Momentum variance of the ground state; the solid (black) line indicates the exact solution.

Figure 3

Optimized variational parameters for the double squeezed state [Eq. (9)] for $\Omega =10$ (a–c) and for $\Omega ={10}^5$ (d–f). The variational parameters for the double coherent state [Eq. (10)] are qualitatively similar (not shown) except for $r_1=r_2=0$. (a, d) Relative weight $t$ in the superposition. (b, e) Displacement parameters ${\alpha }_1$ and ${\alpha }_2$. (c, f) Squeezing parameters $r_1$ and $r_2$.