Nonequilibrium and Nonperturbative Dynamics of Ultrastrong Coupling in Open Lines

The time and space resolved dynamics of a qubit with an Ohmic coupling to propagating 1D photons is studied, from weak coupling to the ultrastrong coupling regime. A nonperturbative study based on matrix product states shows the following results, (i) The ground state of the combined systems contains excitations of both the qubit and the surrounding bosonic field. (ii) An initially excited qubit equilibrates through spontaneous emission to a state, which under certain conditions is locally close to that ground state, both in the qubit and the field. (iii) The resonances of the combined qubit-photon system match those of the spontaneous emission process and also the predictions of the adiabatic renormalization [A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987)]. Finally, nonperturbative ab initio calculations show that this physics can be studied using a flux qubit galvanically coupled to a superconducting transmission line.

Figure 1

Ground state number of photons in each site of the bosonic lattice (3) without a qubit ($\alpha =0$, blue line below), with a flux qubit ($\alpha =1$, solid peaked line), and charge qubit ($\alpha =1$, dashed line). The qubit is located in the middle of the lattice. Note how the perturbation of line due to the flux qubit extends over the whole system.

Figure 2

Excitation probability, $P_z=\frac{1}{2}(⟨{\sigma }_z⟩+1)$, of an initially excited flux qubit in the open transmission line. (a) Full dynamics and (b) final average excitation at the white line $t{\omega }_{\mathrm{at}}=20$ (black solid line), with the master equation prediction (blue dashed) and the ground state value (crosses). Horizontal axes show both the value of the microscopic coupling $g$ and the parameter $\alpha$ for $L=121$ and ${\omega }_{\mathrm{at}}/{\omega }_0=1/3$. The qubit relaxes to the ground state Bloch vector, but for small $g$ and $\alpha$, the relaxation rate decreases as $g^2$ and $P_z$ is Gaussian. The vertical line marks the value $\alpha =1/2$.

Figure 3

(a) Normalized distribution of photons in frequency space vs coupling $g/{\omega }_0$ or $\alpha$, at $t=45/{\omega }_{\mathrm{at}}$ after spontaneous emission; RG prediction (6) in dashed line and $\alpha =1/2$ line in dash-dotted line. (b) Transmitted photons for a coherent wave packet with average number of photons 1, as a function of the photon frequency, $\omega$, and the coupling strength $g/{\omega }_0$ or $\alpha$.

Figure 4

Real space distribution of photons relative to the squeezed vacuum. (a) Spontaneous emission from a qubit at $x=0$, with $g=0.475{\omega }_0$. (b) Scattering of an incoming photon by a qubit at $x=0$, $g=0.7{\omega }_0$, $\omega =0.186{\omega }_{\mathrm{at}}$. In both cases $L=121$, ${\omega }_{\mathrm{at}}=1/3{\omega }_0$.

Figure 5

Susceptibility of the qubit along the $X$ direction (solid line) when spontaneously emitting a photon while subject to a perturbation $\epsilon {\sigma }_x/2$, with a fit to $a/{\omega }_{\mathrm{eff}}$ (dotted line).

Figure 6

(a) Lumped element circuit for a flux qubit ultrastrongly coupled to a microwave resonator by inserting it in the transmission line. (b) Effective coupling strengths for the respective circuits, together with a fit $0.84/{\beta }^{0.92}$ (dashed blue line) for the 4-JJ setup, consistent with Ref. 21. We assume an impedance $Z_0=100\Omega$ and a small junction ${\alpha }_q=0.7$.