Relaxation breakdown and resonant tunneling in ultrastrong-coupling cavity QED

We study the open relaxation dynamics of an asymmetric dipole that is ultrastrongly coupled to a single electromagnetic cavity mode. By using a thermalizing master equation for the whole interacting system we derive a phase diagram of the Liouvillian gap. It emerges that the ultrastrong coupling inhibits the system's relaxation toward the equilibrium state due to an exponential suppression of the dipole tunneling rate. However, we find that polaronic multiphoton resonances restore fast relaxation by a cavity-mediated dipole resonant tunneling process. Aside from the numerical evidence, we develop a fully analytical description by diagonalizing the Rabi model through a generalized rotating-wave approximation, valid in the so-called polaron frame. The relaxation physics of such ultrastrong-coupling systems is then reduced to a multiphoton polaron version of the standard textbook dressed states picture. At the end we discuss an extension to a multiwell dipole that can set the basis of a cascaded resonant tunneling setup in the ultrastrong coupling regime.

Figure 1

(a) Cavity QED system. The cavity is modeled as an LC circuit, where the inductor magnetic flux $\mathrm{\Phi }$ takes the role of the dynamical variable of the electromagnetic field, usually given by the vector potential $\vec{A}$. The dipole inside the capacitor couples to the voltage drop $U=\dot{\mathrm{\Phi }}$ between the plates, separated by the distance $d$. (b) The dipole is described as a particle in a tilted double-well potential. The position $x$ represents the displacement between the two charges $q, -q$, such that the dipole moment is $qx$. When the central well of the potential is large enough the system is approximated by only the two lowest levels (two-level approximation; see Appendix pp1). (c) Open-system schematic view. The cavity QED system can be interpreted as an element of a dissipative circuit.

Figure 2

(a) Phase diagram of the Liouvillian gap $\lambda$ as a function of the light-matter coupling $g$ and the dipole asymmetry $\epsilon$. Parameters: $\gamma =\kappa /4=0.05{\omega }_c, {\omega }_d={\omega }_c$. (b) A cut of the phase diagram at $\epsilon =0$ as a function of the light-matter coupling $g$, in logscale. (c) A cut of the phase diagram at $g/{\omega }_c=3$ as a function of the dipole asymmetry $\epsilon$, in logscale.

Figure 3

(a) Spectrum of the Rabi model as a function of the light-matter coupling $g$ at fixed $\epsilon =0$. The solid lines are the result of full diagonalization, and the red and blue are meant only to match the color code in (c). The yellow dots are given by the analytic (F6) in Appendix pp6. (b) $cos{\theta }_n/2, sin{\theta }_n/2$ given by Eq. (F8) for each $n$ block as a function of the light-matter coupling $g$. (c) Scheme of the relaxation mechanism. The cavity relaxes jumping mainly between $++$ or $--$ dressed states, while for the dipole it is mainly between $+-$ states. The orange curly arrows represent the decay of the photon from an upper state to a lower one, while the blue curly arrows represent the decay of the dipole. In the USC limit the dipole does not relax anymore. Parameters: $\epsilon =0, {\omega }_c={\omega }_d$.

Figure 4

Infidelity time evolution. (a), (b) Weak and intermediate coupling regime, the infidelity is calculated starting with the initial density matrix ${\rho }_{0,d}$ (blue solid line) and ${\rho }_{0,\mathrm{ph}}$ (red solid line). The black dashed line highlights the bare decay scaling $s_{0}exp[-\gamma t]$, while the green dot-dashed line marks the scaling given by the Liouvillian gap $s_0^{\prime }exp\left[2\lambda t\right]$ ($s_0,s_0^{\prime }$ are arbitrary offsets). The light-matter coupling is given in the panels. (c) USC regime, the infidelity is calculated starting with the initial density matrix ${\rho }_{0,d}$ (weak blue, middle blue, deep blue solid lines) and ${\rho }_{0,\mathrm{ph}}$ (orange, red, dark red solid lines). For both initial state the couplings are $g/{\omega }_c=2.5,3.5,4.5$ going from the lighter to the darker color. The middle blue and deep blue lines representing the infidelity starting from ${\rho }_{0,d}$ are almost overlapping and not well distinguishable. Parameters: ${\omega }_c={\omega }_d, \epsilon =0, \gamma =\kappa /4=0.1{\omega }_c, T=0$.

Figure 5

Spectrum of the Rabi model as a function of the dipole asymmetry $\epsilon$ for various $g/{\omega }_c=0.1,1,2.5,3.5$ light-matter couplings. The solid lines are the result of exact diagonalization, while the yellow dots are given by the analytical formula in Eq. (17). For $k>1$ the yellow dots do not cover the lower lines because our approximation treats these eigenstates as a bare photon state for which the energy is trivially $n{\omega }_c$. To highlight the part of the spectrum where cavity and dipole are effectively coupled we do not put the yellow dots on these trivial eigenvalues. Parameters: ${\omega }_d={\omega }_c$.

Figure 6

(a) Schematic view of the resonant tunnel mechanism. In the USC regime, the dipole can switch well by exchanging $k$ photons with the cavity. (b) Rabi oscillation data collapse. Each curve is labeled by its resonant index $k$ and represents $\langle {\tilde{s}}_x\rangle =e^{k\gamma /2t}(\langle s_x\rangle +1/2)-1/2$. For each $k$ curve the time is normalized on its respective $k$-Rabi frequency, ${\mathrm{\Omega }}_{(k,k)}/\left(2\pi \right)$. In this way it is clear how our analytical description fits very well the full numerics. Parameters: ${\omega }_d={\omega }_c, g/{\omega }_c=3, \gamma =\kappa /4=0.002{\omega }_c$.

Figure 7

Current transmission $\left|\mathcal{T}\right(\omega \left)\right|$ as a function of the dipole asymmetry $\epsilon$ and the probe frequency $\omega$, for various $g/{\omega }_c=0.1,0.5,2,2.5$ light-matter couplings. Parameters: ${\omega }_d={\omega }_c, k_{b}T=0.2\hslash {\omega }_c, Q={\omega }_c/\gamma ={10}^2$.

Figure 8

Dipole radiation impedance $|Z_{\mathrm{rad}}\left(\omega \right)|$ as a function of the dipole asymmetry $\epsilon$ and the probe frequency $\omega$ for various $g/{\omega }_c=2,2.5$ light-matter couplings. Parameters: ${\omega }_d={\omega }_c, k_{b}T=0.5\hslash {\omega }_c$. In this plot we assumed a linewidth ${\gamma }_{{\mathcal{S}}_{\mathrm{dip}}}=0.05{\omega }_c$.

Figure 9

Total relaxation rate ${\mathrm{\Gamma }}_T^{\mathrm{tot}}\left(\omega \right)$ defined in Eq. (35) as a function of frequency $\omega$ for various $g/{\omega }_c=0.1,0.5,1,2$ coupling strengths (normalized over ${\mathrm{\Gamma }}_d={\omega }_d^{2}N/\gamma$). The blue line corresponds to $T=0$, and the red line is for $k_{b}T=2\hslash {\omega }_c$. Parameters: $\gamma /{\omega }_c=0.1$.