cavity mode and a radiation reservoir. Three methods are employed: a numerically exact variational approach with a multiple Davydov ansatz, the rotating-wave approximation (RWA), and the transformed RWA. Agreement between the variational results and the transformed RWA results is found in the regime of validity of the latter, where the RWA breaks down completely. We illustrate that the Lamb shift plays an essential role in modifying the vacuum Rabi splitting in the ultrastrong coupling regime, leading to off-resonant qubit-cavity coupling even though the cavity frequency equals the bare transition frequency of the qubit. Specifically, the emission spectrum exhibits one broad low-frequency peak and one narrow high-frequency peak in the presence of relatively weak cavity-qubit coupling. As the cavity-qubit coupling increases, the low-frequency peak narrows while the high-frequency peak broadens until they have similar widths.
cavity mode and a radiation reservoir. Three methods are employed: a numerically exact variational approach with a multiple Davydov ansatz, the rotating-wave approximation (RWA), and the transformed RWA. Agreement between the variational results and the transformed RWA results is found in the regime of validity of the latter, where the RWA breaks down completely. We illustrate that the Lamb shift plays an essential role in modifying the vacuum Rabi splitting in the ultrastrong coupling regime, leading to off-resonant qubit-cavity coupling even though the cavity frequency equals the bare transition frequency of the qubit. Specifically, the emission spectrum exhibits one broad low-frequency peak and one narrow high-frequency peak in the presence of relatively weak cavity-qubit coupling. As the cavity-qubit coupling increases, the low-frequency peak narrows while the high-frequency peak broadens until they have similar widths.
We investigate the spontaneous emission (SE) spectrum of a qubit in a lossy resonant cavity. We use neither the rotating-wave approximation nor the Markov approximation. For the weak-coupling case, the SE spectrum of the qubit is a single peak, with its location depending on the spectral density of the qubit environment. Then, the asymmetry (of the location and heights of the two peaks) of the two SE peaks (which are related to the vacuum Rabi splitting) changes as the qubit–cavity coupling increases. Explicitly, for a qubit in a low-frequency intrinsic bath, the height asymmetry of the splitting peaks is enhanced as the qubit–cavity coupling strength increases. However, for a qubit in an Ohmic bath, the height asymmetry of the spectral peaks is inverted compared to the low-frequency bath case. With further increasing the qubit–cavity coupling to the ultra-strong regime, the height asymmetry of the left and right peaks is slightly inverted, which is consistent with the corresponding case of a low-frequency bath. This inversion of the asymmetry arises from the competition between the Ohmic bath and the cavity bath. Therefore, after considering the anti-rotating terms, our results explicitly show how the height asymmetry in the SE spectrum peaks depends on the qubit–cavity coupling and the type of intrinsic noise experienced by the qubit.
We study the emission spectra of the dissipative Rabi and Jaynes-Cummings models in the non-Markovian and ultrastrong coupling regimes. We have derived a polaron-transformed Nakajima-Zwanzig master equation (PTNZE) to calculate the emission spectra, which eliminates the well-known limitations of the Markovian approximation and the standard second-order perturbation. Using the time-dependent variational approach with the multiple Davydov ansatz as a benchmark, the PTNZE is found to yield accurate emission spectra in certain ultrastrong coupling regimes where the standard second-order Nakajima-Zwanzig master equation breaks down. It is shown that the emission spectra of the dissipative Rabi and Jaynes-Cummings models are, in general, asymmetric under various initial conditions. Direct comparisons of spectra for the two models illustrate the essential role of the qubit-cavity counter-rotating term and the spectral features under different qubit-cavity coupling strengths and system initial conditions.
The effects of counter-rotating terms (CRTs) on Rabi splitting and the dynamic evolution of atomic population in the Jaynes—Cummings model are studied with a coherent-state approach. When the coupling strength increases, the Rabi splitting becomes of multi-Rabi frequencies for the initial state of an excited atom in a vacuum field, and the collapses and revivals gradually disappear, and then reappear with quite good periodicity. Without the rotating-wave approximation (RWA), the initial excited state contains many eigenstates rather than two eigenstates under the RWA, which results in the multi-peak emission spectrum. An analytical approximate solution for the strong coupling regime is obtained, which gives a new oscillation frequency and explains the recovery of collapses and revivals due to the equal energy spacing.
We measure the transmission spectrum of an optical cavity filled with two-level atoms. For small intensities, the coupling between atoms and a single mode of the cavity splits the spectrum into two normal modes: the vacuum Rabi peaks. We gradually decouple the system in two ways and study the evolution of the vacuum Rabi peaks. First we lower the atom-cavity coupling frequency through its intensity dependence, showing that the peaks shift, deform, exhibit frequency hysteresis, and eventually merge into a single peak that approaches the empty cavity resonance. Second we detune the atoms and the cavity mode and observe an avoided crossing. The normal modes of the coupled system transform into the two resonances of the uncoupled system. We map out the transition region and find good agreement with theoretical predictions.
In recent years, quantum Rabi model has aroused considerable interest because of its fundamental importance and potential applications in quantum technologies. For a conventional cavity-quantum-electrodynamic (cavity-QED) system involving the interaction between an atom and photons in a cavity, the atom-photon coupling frequency is much smaller than the transition frequency of the atom and the frequency of the cavity mode. This cavity-QED system is usually described by the Jaynes-Cummings model in which the rotating-wave approximation can be adopted by neglecting the counter-rotating coupling terms in the Hamiltonian of the system. However, by designing the unique structure of the superconducting circuit, the ultrastrong-coupling regime can be achieved in a circuit-QED system in which the counter-rotating coupling terms become as important as the rotating terms. Thus, the rotating-wave approximation cannot be used in the ultrastrongly coupled circuit-QED system. Owing to the ultrastrong coupling, this circuit-QED system is described by the standard quantum Rabi model when a superconducting qubit is coupled only to a single resonator mode. In this work, we experimentally study an ultrastrongly coupled circuit-QED system consisting of a four-junction superconducting flux qubit and a muti-mode coplanar-waveguide resonator. The transmission-spectrum measurement and numerical simulations show that the system is in the ultrastrong-coupling regime. By changing the photon number in the resonator, we observe the frequency shift of the flux qubit via the spectroscopic measurement. This frequency shift contains the contributions from not only the rotating-coupling terms but also the counter-rotating terms, which is in good agreement with the theory. The result indicates that this ultrastrongly-coupled quantum system can be used as a good platform to investigate the quantum Rabi model and has potential applications in various aspects of quantum technology, such as quantum simulation, ultrafast quantum gates, entangled-state preparation and protected qubits.
We present a full quantum theory for the dissipative dynamics of an optical cavity in the ultrastrong light-matter coupling regime, in which the vacuum Rabi frequency is a significant fraction of the active electronic transition frequency and the antiresonant terms of the light-matter coupling play an important role. In particular, our model can be applied to the case of intersubband transitions in doped semiconductor quantum wells embedded in a microcavity. The coupling of the intracavity photonic mode and of the electronic polarization to the external, frequency-dependent, dissipation baths is taken into account by means of quantum Langevin equations in the input-output formalism. In the case of a time-independent vacuum Rabi frequency, exact analytical expressions for the operators are obtained, which allows us to characterize the quantum dissipative response of the cavity to an arbitrary initial condition (vacuum, coherent field, thermal excitation). For a vacuum input in both the photonic and electronic polarization modes, the ground state of the cavity system is a two-mode squeezed vacuum state with a finite population in both photonic and electronic modes. These excitations are, however, virtual and cannot escape from the cavity: for a vacuum input, a vacuum output is found, without any trace of the intracavity squeezing. For a coherent photonic input the linear optical response spectra (reflectivity, absorption, transmission) have been studied, and signatures of the ultrastrong coupling have been identified in the asymmetric and peculiar anticrossing of the polaritonic eigenmodes. Finally, we have calculated the electroluminescence spectra in the case of an incoherent electronic input: the emission intensity in the ultrastrong coupling regime results in being significantly enhanced as compared to the case of an isolated quantum well without a surrounding cavity.