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arXiv:2107.01203 · Physical Review Letters

Quantum Mollow Quadruplet in Nonlinear Cavity QED

Thomas Allcock, Wolfgang Langbein, Egor A. Muljarov

Published Mar 24, 2022• 5 citations
arXivHTMLPDF

Abstract

We develop an exact analytical approach to the optical response of a two-level system coupled to a microcavity for arbitrary excitation strengths. The response is determined in terms of the complex amplitudes of transitions between the rungs of the Jaynes-Cummings ladder, explicitly isolating nonlinearities of different orders. Increasing the pulse area of the excitation field, we demonstrate the formation of a quantum Mollow quadruplet (QMQ), quantizing the semiclassical Mollow triplet into a coherent superposition of a large number of transitions between rungs of the ladder, with inner and outer doublets of the QMQ formed by densely lying inner and outer quantum transitions between the split rungs. Remarkably, a closed-form analytic approximation for the QMQ of any order of nonlinearity is found in the high-field low-damping limit.

DOI
10.1103/physrevlett.128.123602

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We develop an exact analytical approach to the optical response of a two-level system coupled to a microcavity for arbitrary excitation strengths. The response is determined in terms of the complex amplitudes of transitions between the rungs of the Jaynes-Cummings ladder, explicitly isolating nonlinearities of different orders. Increasing the pulse area of the excitation field, we demonstrate the formation of a quantum Mollow quadruplet (QMQ), quantizing the semiclassical Mollow triplet into a coherent superposition of a large number of transitions between rungs of the ladder, with inner and outer doublets of the QMQ formed by densely lying inner and outer quantum transitions between the split rungs. Remarkably, a closed-form analytic approximation for the QMQ of any order of nonlinearity is found in the high-field low-damping limit.
We develop an exact analytical approach to four-wave mixing and higher-order optical nonlinearities of a quantum dot-microcavity system. This approach allows us \nto analyse the non-linear response in terms of coherent transitions between rungs \nof the Jaynes-Cummings ladder, finding their complex amplitudes and frequencies. \nIncreasing the pulse area of the excitation field, a transition from quantum strong \ncoupling to coherent classical regime is observed, which manifests itself in a formation of the famous Mollow triplet, with the observation in the three-level model \nof a quantum dot, including a biexciton, to predict two new effects: One is the \nexistence of a Mollow-like quintuplet, the other is a complete elimination of the \nfour-wave-mixing optical non-linearity under certain parameter constraints. We further develop a simple and accurate approximation in the low-damping limit of the \nJaynes-Cummings model of the quantum dot-microcavity system, for nonlinear response of any order in a closed analytic form, predicting the positions and widths \nof the spectral lines. For higher damping, we demonstrate a strong relaxation down \nthe ladder, shown by both numerical and analytic calculation.
The effect of phonons on a nonlinear optical response of a quantum dot-cavity system in quantum strong coupling regime can be accounted for by a fully analytical treatment, provided that the exciton-phonon dynamics is much faster than the exciton-cavity dynamics. Modern experiments involving semiconductor quantum dots embedded in optical microcavities typically meet this criterion. We find that, for a relatively small exciton-cavity coupling, the effect of phonons is concentrated mainly in the polaron shift of the exciton frequency and reduction of exciton-cavity coupling by the Huang-Rhys factor. We have generalized this result to an arbitrary optical nonlinearity and demonstrated a good agreement with the exact solution in a wide range of temperatures. This generalization provides access to higher rungs of the Jaynes-Cummings ladder, where exact numerical approaches are impractical or even impossible. At larger coupling strengths and low temperatures, our approximation is also in good agreement with the exact solution, which makes it a very useful tool for addressing the phonon contribution to the coherent dynamics of a nonlinear optical system. We demonstrate our results for third-order optical polarization with varying observation time and delay time between excitation pulses in the form of two-dimensional spectra. These spectra provide useful information about the coherent coupling between the exciton and the cavity modes. The presented analytical approach is also compared with another useful approximation, having the form of a matrix product, which is a special case of the asymptotically exact solution in the limit of a short phonon memory time.
Light and matter can be unified under the strong coupling regime, creating superpositions of both, called dressed states or polaritons. After initially being demonstrated in bulk semiconductors and atomic systems ,strong coupling phenomena have been realized in solid state optical microcavities. They form an essential ingredient in the exciting physics spanning from many-body quantum coherence phenomena, like Bose-Einstein condensation and superfluidity, to cavity quantum electrodynamics (cQED). A widely used approach within cQED is the Jaynes-Cummings (JC) model that describes the interaction of a single fermionic two-level system with a single bosonic photon mode. For a photon number larger than one, known as quantum strong coupling (QSC), a significant anharmonicity is predicted for the ladder-like spectrum of dressed states. For optical transitions in semiconductor nanostructures, first signatures of the quantum strong coupling were recently published. In our latest report we applied advanced coherent nonlinear spectroscopy to explore a strongly coupled exciton-cavity system. Specifically, we measured and simulated its four-wave mixing (FWM) response, granting direct access to the first two rungs of the JC ladder. This paper summarizes the main results of Ref. 15 and adds FWM experiments obtained on a micropillar cavity in which a doublet of quantum dot (QD) excitons interacts with the cavity mode in the limit of weak to strong coupling.
A counterpart of the Mollow triplet (luminescence line shape of a two-level system under coherent excitation) is obtained for the case of incoherent excitation in a cavity. The system acquires coherence through the strong-coupling between the cavity and the emitter. Analytical expressions, in excellent agreement with numerical results, pinpoint analogies and differences between the conventional resonance fluorescence spectrum and its cavity QED analogue under incoherent excitation. Most notably, the satellites broaden and split sublinearly with increasing incoherent pumping.
In the strong coupling regime of cavity QED where the characteristic damping rates are orders of magnitude below the coherent coupling, resolved multi-photon resonances can be observed for strong enough driving strength. In a certain frequency range, the external coherent drive field hits multi-photon resonances which correspond to very high order transitions that cannot be excited at the given power of the driving field. We found that in this range the cavity field evolves into a robust semiclassical solution which is the mixture of the vacuum and an excited quasi-classical state. The system is reminiscent of optical bistability, however, here a single atom produces a telegraph signal for the outgoing intensity by switching between two states with large photon number difference. We calculate this behavior also for the Tavis-Cummings model representing three qubits in an existing circuit QED experimental setting.
Collapse and revivals in the Jaynes-Cummings model are studied within the context of the Mollow transformation for a single-mode coherent-state cavity field. The Mollow-transformed Jaynes-Cummings Hamiltonian has two atom-field terms, one corresponding to a classical field and the other to a quantized field driving the atomic transition. As such it maps onto the complementary problem of an atom in an optical cavity, whose field is initially in the vacuum state, driven by an external classical field. Both problems have the same atomic-state dynamics. It is shown that the revivals can be associated with two distinct properties of the quantized field of the transformed Hamiltonian. Revivals occurring at even (odd) multiples of the revival time are correlated with field states that are close to the initial field state and for which the average energy in the field is a minimum (maximum). Using semiclassical dressed states, we are able to map the problem for the field states onto one involving two uncoupled oscillators driven by off-resonant fields, provided effects related to dispersion are neglected. Analytic expressions for the photon number distribution of the transformed cavity field, including dispersion, are obtained for the even revival times.