Qubit-oscillator system: An analytical treatment of the ultrastrong coupling regime

We examine a two-level system coupled to a quantum oscillator, typically representing experiments in cavity and circuit quantum electrodynamics. We show how such a system can be treated analytically in the ultrastrong coupling limit, where the ratio $g/\Omega$ between coupling strength and oscillator frequency approaches unity and goes beyond. In this regime the Jaynes-Cummings model is known to fail because counter-rotating terms have to be taken into account. By using Van Vleck perturbation theory to higher orders in the qubit tunneling matrix element $\Delta$ we are able to enlarge the regime of applicability of existing analytical treatments, including, in particular, also the finite-bias case. We present a detailed discussion on the energy spectrum of the system and on the dynamics of the qubit for an oscillator at low temperature. We consider the coupling strength $g$ to all orders, and the validity of our approach is even enhanced in the ultrastrong coupling regime. Looking at the Fourier spectrum of the population difference, we find that many frequencies contribute to the dynamics. They are gathered into groups whose spacing depends on the qubit-oscillator detuning. Furthermore, the dynamics is not governed anymore by a vacuum Rabi splitting which scales linearly with $g$, but by a nontrivial dressing of the tunneling matrix element, which can be used to suppress specific frequencies through a variation of the coupling.

Figure 1

Energy levels against detuning $\delta =\Delta -\Omega$ for $\varepsilon /\Omega =0$, $g/\Omega =0.1$. Our VVP solution is compared to the GRWA and the JCM. The latter two agree well with numerical calculations for the whole detuning range (not shown), while VVP yields only reliable results for negative detuning, $\Delta <\Omega$.

Figure 2

Energy levels against detuning $\delta =\Delta -\Omega$ for $\varepsilon /\Omega =0$, $g/\Omega =0.5$. The JCM fails already completely for such a coupling strength (not shown). We compare VVP and the GRWA against numerical calculations. Both agree well with the numerics for negative detuning and even at resonance. For stronger positive detuning they both fail and strongest deviations can be seen for the lower energy levels.

Figure 3

Energy levels against detuning $\delta =\Delta -\Omega$ for $\varepsilon /\Omega =0$, $g/\Omega =1.0$. We compare VVP, the adiabatic approximation, and GRWA against a numerical calculation. For a negative detuning all three approaches agree very well with the exact numerics. However, for zero and positive detuning deviations occur. In particular, the ground level and the first excited level are not described correctly by the adiabatic approximation and the GRWA for strong positive detuning, while VVP yields good results.

Figure 4

Energy levels against detuning. Same as in Fig. 3, but for a coupling strength of $g/\Omega =1.5$. Adiabatic approximation and GRWA fail for positive detuning, while VVP gives the first four energy levels correctly even up to a detuning of $\delta /\Omega =2.0$. It also yields good results beyond the resonant case for the higher energy levels.

Figure 5

Energy levels against coupling strength $g/\Omega$ for negative detuning $(\delta /\Omega =-0.5)$. Numerical results are compared with the adiabatic approximation, GRWA, and VVP. All three approaches show only slight deviations.

Figure 6

Energy levels against coupling strength at resonance $(\delta /\Omega =0)$. For small coupling strength, the adiabatic approximation and VVP show small deviations from the correct values (see especially the higher energy levels). The GRWA works well in this regime. For stronger coupling strength, all three approaches agree well with the numerical results.

Figure 7

Energy levels against coupling strength for positive detuning $(\delta /\Omega =0.5)$. For coupling strengths with $g/\Omega \gtrsim 0.75$, VVP exhibits the best agreement with numerical results, while for smaller coupling and higher energy levels, the GRWA should be used.

Figure 8

Sketch of the validity regime of VVP and GRWA for $\varepsilon =0$. The GRWA is perferable to VVP at weak coupling, in particular close to resonance and positive detuning. On the contrary, VVP works better at strong coupling strengths.

Figure 9

Energy levels against static bias $\varepsilon$ for $g/\Omega =1.0$ at resonance $\Delta /\Omega =1.0$. VVP is compared to a numerical diagonalization of the Hamiltonian.

Figure 10

Energy levels against coupling $g/\Omega$ for $\varepsilon /\Omega =1.0$ and $\Delta /\Omega =0.5$. The adiabatic approximation and VVP agree almost perfectly with numerical results. Slight deviations can be seen for the adiabatic approximation at $g/\Omega \to 0$.

Figure 11

Energy levels against coupling $g/\Omega$ for $\varepsilon /\Omega =1.0$ and $\Delta /\Omega =1.0$. VVP is still valid compared to numerical results, while the adiabatic approximation fails specifically for weak coupling strengths.

Figure 12

Energy levels against coupling $g/\Omega$ for $\varepsilon /\Omega =1.0$ and $\Delta /\Omega =1.5$. In this regime, also VVP shows deviations from the numerical results for $g/\Omega \lesssim 0.75$, especially for the higher energy levels. It agrees well for stronger coupling.

Figure 13

Sketch of the validity regime of VVP and of the adiabatic approximation for $\varepsilon =1.0$. For positive detuning and simultaneously weak coupling both approaches fail. For stronger coupling VVP yields an improvement to the adiabatic approximation.

Figure 14

Energy levels against coupling $g/\Omega$ for $\varepsilon /\Omega =3.0$ and $\Delta /\Omega =1.5$. The three lowest energy levels have no degenerate partner. Despite the high value of $\Delta$, VVP still gives reliable results, while the adiabatic approximation differs from the numerical values even for the low energy levels.

Figure 15

Population difference for zero static bias. Further parameters are $\Delta /\Omega =0.5$, $\hslash \beta \Omega =10$, and $g/\Omega =1.0$. The adiabatic approximation and VVP are compared to numerical results. The first one covers the long-scale dynamics, while VVP also returns the fast oscillations. With increasing time small differences between numerical results and VVP become more pronounced.

Figure 16

Fourier transform of the population difference in Fig. 15. The left panel shows the whole frequency range. The lowest frequency peaks originate from transitions between levels of a degenerate subspace and are determined through the dressed oscillation frequency ${\Omega }_j^0$. Numerical calculations and VVP predict groups of peaks located around $\nu /\Omega =0,1.0,2.0,3.0$. The first group at $\nu /\Omega =0$ is shown in the middle panel. One can identify frequencies ${\Omega }_0^0$ and ${\Omega }_2^0$, which fall together, and ${\Omega }_1^0$. The small peak comes from the frequency ${\Omega }_3^0$. This first group of peaks is also covered by the adiabatic approximation. The other groups come from transitions between different manifolds. The adiabatic approximation does not take them into account, while VVP does. A blowup of the peaks coming from transitions between neighboring manifolds is given in the right panel. In the left panel, the Jaynes-Cummings peaks are also shown, which, however, fail completely.

Figure 17

Population difference for zero static bias. Same parameters as in Fig. 15 but for a coupling strength of $g/\Omega =2.0$. Both the adiabatic approximation and VVP agree well with the numerics, but show slight dephasing on a longer time scale.

Figure 18

Fourier spectrum of the population difference in Fig. 17. In the left panel a large frequency range is covered. Peaks are located around $\nu /\Omega =0$, $1.0$, $2.0$, $3.0$, etc. Even the adiabatic approximation exhibits the higher frequencies. The top right panel shows the first group close to $\nu /\Omega =0$. The two main peaks come from ${\Omega }_3^0$ and ${\Omega }_4^0$ and higher degenerate manifolds. Frequencies from lower manifolds contribute to the peak at zero. The adiabatic approximation and VVP agree well with the numerics. The bottom right panel shows the second group of peaks around $\nu /\Omega =1.0$. This group is also predicted by the adiabatic approximation and VVP, but they do not fully return the detailed structure of the numerics. Interestingly, there is no peak exactly at $\nu /\Omega =1.0$, indicating no nearest-neighbor transition between the low degenerate levels.

Figure 19

Population difference and Fourier spectrum for a biased qubit ($\varepsilon /\Omega =\sqrt{0.5}$) at resonance with the oscillator $({\Delta }_{\mathrm{b}}=\Omega )$ in the ultrastrong coupling regime ($g/\Omega =1.0$). Concerning the time evolution VVP agrees well with numerical results. Only for long times weak dephasing occurs. The inset in the left panel shows the adiabatic approximation only. It exhibits death and revival of oscillations which are not confirmed by the numerics. For the Fourier spectrum, VVP covers the various frequency peaks, which are gathered into groups as for the unbiased case. The adiabatic approximation only returns the first group.

Figure 20

Population difference and Fourier spectrum for $\varepsilon /\Omega =1.5$, $\Delta /\Omega =0.5$, and $g/\Omega =1.0$. VVP is confirmed by numerical calculations, while results obtained from the adiabatic approximation deviate strongly. In Fourier space, we find pairs of frequency peaks coming from the two dressed oscillation frequencies ${\Omega }_j^1$ and ${\Omega }_j^2$. The spacing between those pairs is about $0.5\Omega$. The adiabatic approximation only returns one of those dressed frequencies in the first pair.