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The Thomas–Reiche–Kuhn (TRK) sum rule is a fundamental consequence of the position–momentum commutation relation for an atomic electron, and it provides an important constraint on the transition matrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light–matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light–matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light–matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.

Plasmonic dimer cavities can induce extreme electric-field hot spots that allow one to access ultrastrong coupling regimes using Raman-type spectroscopy on single vibrating molecules. Using a generalized master equation, we study resonant Raman scattering in the strong coupling regime of cavity quantum electrodynamics, when also in the vibrational ultrastrong coupling regime, leading to ``phonon-dressed polaritons.'' The master equation rigorously includes spectral baths for the cavity and vibrational degrees of freedom, as well as a pure dephasing bath for the resonant two-level system, which play a significant role. Employing realistic parameters for gold dimer cavity modes, we investigate the emission spectra in several characteristic strong coupling regimes, leading to extremely rich spectral resonances due to an interplay of phonon-modified polariton states and bath-induced resonances. We also show explicitly the failure of the standard master equation in these quantum nonlinear regimes.

We introduce a protocol to transfer excitations between two noninteracting qubits via purely dissipative processes (i.e., in the Lindblad master equation there is no coherent interaction between the qubits). The fundamental ingredients are the presence of collective (i.e. nonlocal) dissipation and unbalanced local dissipation rates (the qubits dissipate at different rates). The resulting quantum trajectories show that the measurement backaction changes the system wave function and induces a passage of the excitation from one qubit to the other. While similar phenomena have been witnessed for a non-Markovian environment, here the dissipative quantum state transfer is induced by an update of the observer knowledge of the wave function in the presence of a Markovian (memoryless) environment -- this is a single quantum trajectory effect. Beyond single quantum trajectories and postselection, such an effect can be observed by histogramming the quantum jumps along several realizations at different times. By investigating the effect of the temperature in the presence of unbalanced local dissipation, we demonstrate that, if appropriately switched on and off, the collective dissipator can act as a Maxwell's demon. These effects are a generalized measure equivalent to the standard projective measure description of quantum teleportation and Maxwell's demon. They can be witnessed in state-of-the-art setups given the extreme experimental control in, e.g., superconducting qubits or Rydberg atoms., 11 pages, 5 figures

The interaction between quantized electromagnetic fields in cavities and natural or artificial atoms has played a crucial role in developing our understanding of light-matter interactions and quantum technologies. Recently, new regimes beyond the weak and strong light-matter coupling of cavity-QED have been explored in several settings, wherein the light-matter coupling rate becomes comparable to (ultrastrong coupling) or even exceeds (deep-strong coupling) the photon frequency. These ultrastrong coupling regimes can give rise to new physical effects and applications, and they challenge our understanding of cavity QED; fundamental issues like the proper definition of subsystems, their quantum measurements, the structure of light-matter ground states, and the analysis of time-dependent interactions are subject to gauge ambiguities that lead to even qualitatively distinct predictions. The resolution of these ambiguities is important for understanding and designing next-generation quantum devices that can operate in extreme coupling regimes. Here we discuss and provide solutions to these ambiguities by adopting an approach based on operational procedures involving measurements on the individual light and matter components of the interacting system., A.S., D.Z., and S.H. acknowledge RIKEN for its hospitality, D.Z. acknowledges the support by the Spanish Ministerio de Ciencia, Innovaciòn y Universidades within project MAT2017-88358-C3-1-R, the Aragòn Government project Q-MAD, EU-QUANTERA project SUMO and the Fundaciòn BBVA. F.N. is supported in part by: Nippon Telegraph and Telephone Corporation (NTT) Research, the Japan Science and Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP) program, the Moonshot R&D Grant No. JPMJMS2061, and the Centers of Research Excellence in Science and Technology (CREST) Grant No. JPMJCR1676], the Japan Society for the Promotion of Science (JSPS) [via the Grants-in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134 and the JSPS RFBR Grant No. JPJSBP120194828], the Army Research Office (ARO) (Grant No. W911NF-18-1-0358), the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06. S.H. acknowledges funding from the Canadian Foundation for Innovation, and the Natural Sciences and Engineering Research Council of Canada. S.S. acknowledges the Army Research Office (ARO) (Grant No. W911NF-19-1-0065).

The quantum Rabi model is a widespread description of the coupling between a two-level system and a quantized single mode of an electromagnetic resonator. Issues about this model's gauge invariance have been raised. These issues become evident when the light-matter interaction reaches the so-called ultrastrong coupling regime. Recently, a modified quantum Rabi model able to provide gauge-invariant physical results (e.g., energy levels, expectation values of observables, quantum probabilities) in any interaction regime was introduced [O. Di Stefano, A. Settineri, V. Macrì, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Nat. Phys. 15, 803 (2019)]. Here we provide an alternative derivation of this result, based on the implementation in two-state systems of the gauge principle, which is the principle from which all the fundamental interactions in quantum field theory are derived. The adopted procedure can be regarded as the two-site version of the general method used to implement the gauge principle in lattice gauge theories. Applying this method, we also obtain the gauge-invariant quantum Rabi model for asymmetric two-state systems, and the multimode gauge-invariant quantum Rabi model beyond the dipole approximation., F.N. is supported in part by: Nippon Telegraph and Telephone Corporation (NTT) Research, the Japan Science and Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP) program, the Moonshot R&D Grant No. JPMJMS2061, and the Centers of Research Excellence in Science and Technology (CREST) Grant No. JPMJCR1676], the Japan Society for the Promotion of Science (JSPS) [via the Grants-in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134 and the JSPS – RFBR Grant No. JPJSBP120194828], the Army Research Office (ARO) (Grant No. W911NF-18-1-0358), the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06. S.H. acknowledges funding from the Canadian Foundation for Innovation and the Natural Sciences and Engineering Research Council of Canada. S.S. acknowledges the Army Research Office (ARO) (Grant No. W911NF1910065).

Emission spectra in the ultrastrong coupling regime are traditionally computed in a gauge-dependent theory. We present gauge-invariant spectra which show significant disagreement with previous results obtained using the standard quantum Rabi model.

The Dicke model, which describes the dipolar coupling between $N$ two-level atoms and a quantized electromagnetic field, seemingly violates gauge invariance in the presence of ultrastrong light-matter coupling, a regime that is now experimentally accessible in many physical systems. Specifically, it has been shown that, while the two-level approximation can work well in the dipole gauge, the Coulomb gauge fails to provide the correct spectra in the ultrastrong coupling regime. Here we show that, taking into account the nonlocality of the atomic potential induced by the two-level approximation, gauge invariance is fully restored for arbitrary interaction strengths, even in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. Finally, we express the Hopfield model, a general description based on the quantization of a linear dielectric medium, in a manifestly gauge-invariant form, and show that the Dicke model in the dilute regime can be regarded as a particular case of the more general Hopfield model.

It has been shown elsewhere that two spatially separated atoms can jointly absorb one photon, whose frequency is equal to the sum of the transition frequencies of the two atoms. We describe this process in the presence of an ensemble of many two-level atoms and show that it can be used to generate spin squeezing and entanglement. This resonant collective process allows us to create a sizable squeezing already at the single-photon limit. It represents a way to generate many-body spin-spin interactions, yielding a two-axis twisting-like interaction among the spins, which is very efficient for the generation of spin squeezing. We perform explicit calculations for ensembles of magnetic molecules coupled to a superconducting coplanar cavities. This system represents an attractive on-chip architecture for the realization of improved sensing., D.Z. acknowledges RIKEN for its hospitality and the support by the Spanish Ministerio de Ciencia, Innovaciòn y Universidades within Project No. MAT2017-88358-C3-1-R, the Aragòn Government Project Q-MAD, EU-QUANTERA Project SUMO, and the Fundaciòn BBVA. F.N. is supported in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Japan Science and Technology Agency (JST) (via the Q-LEAP program, and the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (JSPS-RFBR Grant No. 17-52-50023, and JSPS-FWO Grant No. VS.059.18N), and the Grant No. FQXiIAF19-06 from the Foundational Questions Institute Fund (FQXi), a donor advised fund of the Silicon Valley Community Foundation. S.S. acknowledges the US Army Research Office (ARO) (Grant No. W911NF-19-1-0065).

We use a non-Lindbladian master equation of the Scully-Lamb laser model for the analysis of light propagation in a parity-time symmetric photonic system composed of coupled active and passive whispering-gallery microresonators. Performing the semiclassical approximation, we obtain a set of two nonlinear coupled differential equations describing the time evolution of intracavity fields. These coupled equations are able to explain the experimentally-observed light non-reciprocity [Peng {\em et al.}, Nature Physics {\bf 10}, 394 (2014), Chang {\em et al.}, Nature Photonics {\bf 8}, 524 (2014)]. We show that this effect arises from the interplay between gain saturation in the active microcavity, intercavity coupling, and losses in the cavities. Additionally, using this approach, we study the effect of the gain saturation on exceptional points, i.e., exotic degeneracies in non-Hermitian systems. Namely, we demonstrate that the inclusion of gain saturation leads to a modification of the exceptional points in the presence of intense intracavity fields. The Scully-Lamb master equation for systems of coupled optical structures, as proposed and applied here, constitutes a promising tool for the study of quantum optical effects in coupled systems with losses, gain, and gain saturation., Comment: 16 pages, 14 figures

To observe the dynamical Casimir effect (DCE) induced by a moving mirror is a long-standing challenge because the mirror velocity needs to approach the speed of light. Here, we present an experimentally feasible method for observing this mechanical DCE in an optomechanical system. It employs a detuned, parametric driving to squeeze a cavity mode, so that the mechanical mode, with a typical resonance frequency, can parametrically and resonantly couple to the squeezed cavity mode, thus leading to a resonantly amplified DCE in the squeezed frame. The DCE process can be interpreted as {\it mechanically-induced two-photon hyper-Raman scattering} in the laboratory frame. Specifically, {\it a photon pair} of the parametric driving absorbs a single phonon and then is scattered into an anti-Stokes sideband. We also find that the squeezing, which additionally induces and amplifies the DCE, can be extremely small. Our method requires neither an ultra-high mechanical-oscillation frequency (i.e., a mirror moving at nearly the speed of light) nor an ultrastrong single-photon optomechanical coupling and, thus, could be implemented in a wide range of physical systems., 19 pages, 17 figures

Ultrastrong coupling between light and matter has, in the past decade, transitioned from theoretical idea to experimental reality. It is a new regime of quantum light-matter interaction, going beyond weak and strong coupling to make the coupling strength comparable to the transition frequencies in the system. The achievement of weak and strong coupling has led to increased control of quantum systems and applications like lasers, quantum sensing, and quantum information processing. Here we review the theory of quantum systems with ultrastrong coupling, which includes entangled ground states with virtual excitations, new avenues for nonlinear optics, and connections to several important physical models. We also review the multitude of experimental setups, including superconducting circuits, organic molecules, semiconductor polaritons, and optomechanics, that now have achieved ultrastrong coupling. We then discuss the many potential applications that these achievements enable in physics and chemistry., Review article. 26 pages, 7+1 figures, 3 boxes, 1 table

Two close parallel mirrors attract due to a small force (Casimir effect) originating from the electromagnetic quantum vacuum uctuations of the electromagnetic field. These vacuum uctuations can also induce motional forces exerted upon one mirror when the other one moves. Here we consider an optomechanical system consisting of two vibrating mirrors coupled to an optical resonator. We find that motional forces can determine noticeable coupling rates between the two spatially separated vibrating mirrors. We show that, by tuning the two mechanical oscillators into resonance, energy is exchanged between them at the quantum level. This coherent motional coupling is enabled by the exchange of virtual photon pairs, originating from the dynamical Casimir effect. The process proposed here shows that the electromagnetic quantum vacuum is able to transfer mechanical energy somewhat like an ordinary uid. We show that this system can also operate as a mechanical parametric down-converter even at very weak excitations. These results demonstrate that vacuuminduced motional forces open up new possibilities for the development of optomechanical quantum technologies., Comment: 15 pages, 12 Figures

In quantum electrodynamics, the choice of gauge influences the form of light–matter interactions. However, gauge invariance implies that all physical results should be independent of this formal choice. The Rabi model, a widespread description for the dipolar coupling between a two-level atom and a quantized electromagnetic field, seemingly violates this principle in the presence of ultrastrong light–matter coupling, a regime that is now experimentally accessible in many physical systems. This failure is attributed to the finite-level truncation of the matter system, an approximation that enters the derivation of the Rabi model. Here, we identify the source of gauge violation and provide a general method for the derivation of light–matter Hamiltonians in truncated Hilbert spaces that produces gauge-invariant physical results, even for extreme light–matter interaction regimes. This is achieved by compensating the non-localities introduced in the construction of the effective Hamiltonians. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly in form from the standard one, but provides the same physical results obtained by using the dipole gauge. These results shed light on gauge invariance in the non-perturbative and extreme-interaction regimes, and solve long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models. The principle of gauge invariance in quantum electrodynamics may be violated by approximate models in the presence of strong light–matter interactions. A general approach solves gauge ambiguities and offers a way to construct gauge-invariant Hamiltonians.

We present scanning near-field images of surface plasmon modes around a single elliptical nanohole in 88 nm thick Au film. We find that rotating surface plasmon vortex modes carrying extrinsic orbital angular momentum can be induced under linearly polarized illumination. The vortex modes are obtained only when the incident polarization direction differs from one of the ellipse axes. Such a direct observation of the vortex modes is possible thanks to the ability of the SNOM technique to obtain information on both the amplitude and the phase of the near field. The presence of the vortex mode is determined by the rotational symmetry breaking of the system and it can be considered the counterpart of the photonic spin Hall effect. Finite element method calculations show that such a vorticity originates from the presence of nodal points where the phase of the field is undefined, leading to a circulation of the energy flow. The configuration producing vortex modes corresponds to a nonzero total topological charge (+1).

The coherent nonlinear process where a single photon simultaneously excites two or more two-level systems (qubits) in a single-mode resonator has recently been theoretically predicted. Here we explore the case where the two qubits are placed in different resonators in an array of two or three weakly coupled resonators. Investigating different setups and excitation schemes, we show that this process can still occur with a probability approaching one under specific conditions. The obtained results provide interesting insights into subtle causality issues underlying the simultaneous excitation processes of qubits placed in different resonators.

We show that the dynamical Casimir effect in an optomechanical system can be achieved under incoherent mechanical pumping. We adopt a fully quantum-mechanical approach for both the cavity field and the oscillating mirror. The dynamics is then evaluated using a recently-developed master equation approach in the dressed picture, including both zero and finite temperature photonic reservoirs. This analysis shows that the dynamical Casimir effect can be observed even when the mean value of the mechanical displacement is zero. This opens up new possibilities for the experimental observation of this effect. We also calculate cavity emission spectra both in the resonant and the dispersive regimes, providing useful information on the emission process., Comment: 10 pages, 6 figures

The interaction among the components of a hybrid quantum system is often neglected when considering the coupling of these components to an environment. However, if the interaction strength is large, this approximation leads to unphysical predictions, as has been shown for cavity-QED and optomechanical systems in the ultrastrong-coupling regime. To deal with these cases, master equations with dissipators retaining the interaction between these components have been derived for the quantum Rabi model and for the standard optomechanical Hamiltonian. In this article, we go beyond these previous derivations and present a general master equation approach for arbitrary hybrid quantum systems interacting with thermal reservoirs. Specifically, our approach can be applied to describe the dynamics of open hybrid systems with harmonic, quasi-harmonic, and anharmonic transitions. We apply our approach to study the influence of temperature on multiphoton vacuum Rabi oscillations in circuit QED. We also analyze the influence of temperature on the conversion of mechanical energy into photon pairs in an optomechanical system, which has been recently described at zero temperature. We compare our results with previous approaches, finding that these sometimes overestimate decoherence rates and understimate excited-state populations.

Photon antibunching in the light scattered by single quantum emitters is one of the hallmarks of quantum optics, providing an unequivocal demonstration of the quantized nature of the electromagnetic field. Antibunching can be intuitively understood by the need for a two-level system lying in its lower state after emitting a photon to be re-excited into the upper one before a second emission can take place. Here we show that such a picture breaks down in the ultrastrong light–matter coupling regime, when the coupling strength becomes comparable to the bare emitter frequency. Specializing to the cases of both a natural and an artificial atom, we thus show that a single emitter coupled to a photonic resonator can emit bunched light. The result presented herein is clear evidence of how the ultrastrong coupling regime is able to change the nature of individual atoms.

We propose a new method for frequency conversion of photons which is both versatile and deterministic. We show that a system with two resonators ultrastrongly coupled to a single qubit can be used to realize both single- and multiphoton frequency-conversion processes. The conversion can be exquisitely controlled by tuning the qubit frequency to bring the desired frequency-conversion transitions on or off resonance. Considering recent experimental advances in ultrastrong coupling for circuit QED and other systems, we believe that our scheme can be implemented using available technology., Comment: 16 pages, 10 figures

We show how analogues of a large number of well-known nonlinear-optics phenomena can be realized with one or more two-level atoms coupled to one or more resonator modes. Through higher-order processes, where virtual photons are created and annihilated, an effective deterministic coupling between two states of such a system can be created. In this way, analogues of three-wave mixing, four-wave mixing, higher-harmonic and -subharmonic generation (i.e., up- and downconversion), multiphoton absorption, parametric amplification, Raman and hyper-Raman scattering, the Kerr effect, and other nonlinear processes can be realized. The effective coupling becomes weaker the more intermediate transition steps are needed. However, given the recent experimental progress in ultrastrong light-matter coupling, especially in the field of circuit quantum electrodynamics, we estimate that many of these nonlinear-optics analogues can be realized with currently available technology., Comment: 23 pages, 10 figures, 2 tables

Cavity-QED systems have recently reached a regime where the light-matter interaction strength amounts to a non-negligible fraction of the resonance frequencies of the bare subsystems. In this regime, it is known that the usual normal-order correlation functions for the cavity-photon operators fail to describe both the rate and the statistics of emitted photons. Following Glauber's original approach, we derive a simple and general quantum theory of photodetection, valid for arbitrary light-matter interaction strengths. Our derivation uses Fermi's golden rule, together with an expansion of system operators in the eigenbasis of the interacting light-matter system, to arrive at the correct photodetection probabilities. We consider both narrow- and wide-band photodetectors. Our description is also valid for point-like detectors placed inside the optical cavity. As an application, we propose a gedanken experiment confirming the virtual nature of the bare excitations that enrich the ground state of the quantum Rabi model., Comment: 9 pages, 1 figure

We study the dynamical Casimir effect using a fully quantum-mechanical description of both the cavity field and the oscillating mirror. We do not linearize the dynamics, nor do we adopt any parametric or perturbative approximation. By numerically diagonalizing the full optomechanical Hamiltonian, we show that the resonant generation of photons from the vacuum is determined by a ladder of mirror-field {\em vacuum Rabi splittings}. We find that vacuum emission can originate from the free evolution of an initial pure mechanical excited state, in analogy with the spontaneous emission from excited atoms. By considering a coherent drive of the mirror, using a master-equation approach to take losses into account, we are able to study the dynamical Casimir effect for optomechanical coupling strengths ranging from weak to ultrastrong. We find that a resonant production of photons out of the vacuum can be observed even for mechanical frequencies lower than the cavity-mode frequency. Since high mechanical frequencies, which are hard to achieve experimentally, were thought to be imperative for realizing the dynamical Casimir effect, this result removes one of the major obstacles for the observation of this long-sought effect. We also find that the dynamical Casimir effect can create entanglement between the oscillating mirror and the radiation produced by its motion in the vacuum field, and that vacuum Casimir-Rabi oscillations can occur., Comment: 30 pages, 8 figures

Spontaneous parametric down-conversion is a well-known process in quantum nonlinear optics in which a photon incident on a nonlinear crystal spontaneously splits into two photons. Here we propose an analogous physical process where one excited atom directly transfers its excitation to a pair of spatially-separated atoms with probability approaching one. The interaction is mediated by the exchange of virtual rather than real photons. This nonlinear atomic process is coherent and reversible, so the pair of excited atoms can transfer the excitation back to the first one: the atomic analogue of sum-frequency generation of light. The parameters used to investigate this process correspond to experimentally-demonstrated values in ultrastrong circuit quantum electrodynamics. This approach can be extended to realize other nonlinear inter-atomic processes, such as four-atom mixing, and is an attractive architecture for the realization of quantum devices on a chip. We show that four-qubit mixing can efficiently implement quantum repetition codes and, thus, can be used for error-correction codes.

Light carries both spin and momentum. Spin-orbit interactions of light come into play at the subwavelength scale of nano-optics and nano-photonics, where they determine the behaviour of light. These phenomena, in which the spin affects and controls the spatial degrees of freedom of light, are attracting rapidly growing interest. Here we present results on the spin-momentum locking in the near field of metal nanostructures supporting localized surface resonances. These systems can confine light to very small dimensions below the diffraction limit, leading to a striking near-field enhancement. In contrast to the propagating evanescent waves of surface plasmon-polariton modes, the electromagnetic near-field of localized surface resonances does not exhibit a definite position-independent momentum or polarization. Our results can be useful to investigate the spin-orbit interactions of light for complex evanescent fields. Note that the spin of the incident light can control the rotation direction of the canonical momentum., 23 pages and 7 figures, ACS Photonics - open access

The following changes have been made to the original article: in the lower-right panel of Fig. 1, opoelectrics has been corrected to optoelectronics; in the Box 1 footnote, rotating wave-approximation has been corrected to rotating-wave approximation; in equation B1.3, an operator symbol has been added to the last term; and in the third paragraph of Box 2, |j→|n⟩ was changed to |j⟩→|n⟩. This has been corrected in the HTML and PDF versions of the article.

We theoretically study the carrier capture and distribution among the available energy levels of a symmetric semiconductor quantum dot under continuous-wave excitation resonant with the barrier energy levels. We find that at low temperature all the dot level-occupations but one, the second (dark) energy level,decrease monotonically with energy. The second energy level, displays a steady-state carrier density exceeding that of the lowest level more than a factor two (see the figure). This nonequilibrium effect does not origin from radiative recombination before relaxation to lower energy levels, but at the opposite, it is consequence of carrier trapping due to the symmetry-induced suppression of radiative recombination of the second (dark) energy level. While the observation of such effect by means of far-field spectroscopy is prevented, scanning near-field optical microscopy, able to detect the evanescent waves generated by dark-states occupations, can be adopted to inspect such nonequilibrium effects at the nanoscale. Exciton-level occupations of the dot (normalized at the maximum) calculated for two different temperatures. See Sect. 3 for a detailed description of the results here displayed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Polaritons in semiconductor microcavities are hybrid quasiparticles consisting of a superposition of cavity photons and two-dimensional excitons in embedded quantum wells (QWs), Owing to their exciton content, they have a strong mutual interaction which can be exploited for producing spontaneous parametric scattering of pump polaritons into signal-idler polariton pairs. This process is able to produce polariton states exhibiting quantum correlations, and can thus be exploited for potential applications as well as to study quantum optical processes within a complex many-body system. Here we present an overview of polariton parametric processes within a full quantum description based on the dynamics controlled truncation scheme. We also describe the results of an experiment that proves the existence of polariton pair correlations. This interpretation is confirmed by a microscopic theoretical analysis of the measured in terference visibility. Calculated time-resolved idler polariton number per mode (dashed) originating from parametrie emission; quantum and classical interference visibility when combining two idler bearns sharing a common signal.

We consider two separate atoms interacting with a single-mode optical or microwave resonator. When the frequency of the resonator field is twice the atomic transition frequency, we show that there exists a resonant coupling between one photon and two atoms, via intermediate virtual states connected by counterrotating processes. If the resonator is prepared in its one-photon state, the photon can be jointly absorbed by the two atoms in their ground state which will both reach their excited state with a probability close to one. Like ordinary quantum Rabi oscillations, this process is coherent and reversible, so that two atoms in their excited state will undergo a downward transition jointly emitting a single cavity photon. This joint absorption and emission process can also occur with three atoms. The parameters used to investigate this process correspond to experimentally demonstrated values in circuit quantum electrodynamics systems.

We propose a protocol for the deterministic preparation of entangled NOON mechanical states. The system is constituted by two identical, optically coupled optomechanical systems. The protocol consists of two steps. In the first, one of the two optical resonators is excited by a resonant external $\ensuremath{\pi}$-like Gaussian optical pulse. When the optical excitation coherently partly transfers to the second cavity, the second step starts. It consists of sending simultaneously two additional $\ensuremath{\pi}$-like Gaussian optical pulses, one at each optical resonator, with specific frequencies. In the optomechanical ultrastrong coupling regime, when the coupling strength becomes a significant fraction of the mechanical frequency, we show that NOON mechanical states with quite high Fock states can be deterministically obtained. The operating range of this protocol is carefully analyzed. Calculations have been carried out taking into account the presence of decoherence, thermal noise, and imperfect cooling.

We study the squeezing of output quadratures of an electro-magnetic field escaping from a resonator coupled to a general quantum system with arbitrary interaction strengths. The generalized theoretical analysis of output squeezing proposed here is valid for all the interaction regimes of cavity-quantum electrodynamics: from the weak to the strong, ultrastrong, and deep coupling regimes. For coupling rates comparable or larger then the cavity resonance frequency, the standard input–output theory for optical cavities fails to calculate the variance of output field-quadratures and predicts a non-negligible amount of output squeezing, even if the system is in its ground state. Here we show that, for arbitrary interaction strength and for general cavity-embedded quantum systems, no squeezing can be found in the output-field quadratures if the system is in its ground state. We also apply the proposed theoretical approach to study the output squeezing produced by: (i) an artificial two-level atom embedded in a coherently-excited cavity; and (ii) a cascade-type three-level system interacting with a cavity field mode. In the latter case the output squeezing arises from the virtual photons of the atom-cavity dressed states. This work extends the possibility of predicting and analyzing the results of continuous- variable optical quantum-state tomography when optical resonators interact very strongly with other quantum systems.

When an atom is strongly coupled to a cavity, the two systems can exchange a single photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED has been reached experimentally where the strength of the interaction between light and artificial atoms (qubits) becomes comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modify the concept of vacuum Rabi oscillations, enabling multiphoton exchanges between the qubit and the resonator. We find that experimental state-of-the-art circuit- QED systems can undergo two- and three-photon vacuum Rabi oscillations. These anomalous Rabi oscillations can be exploited for the realization of efficient Fock-state sources of light and complex entangled states of qubits., 25 pages,7 figures

The recent observation of the dynamical Casimir effect in a modulated superconducting waveguide, coronating thirty years of world-wide research, empowered the quantum technology community with a powerful tool to create entangled photons on-chip. In this work we show how, going beyond the single waveguide paradigm using a scalable array, it is possible to create multipartite nonclassical states, with the possibility to control the long-range quantum correlations of the emitted photons. In particular, our finite-temperature theory shows how maximally entangled $NOON$ states can be engineered in a realistic setup. The results here presented open the way to new kinds of quantum fluids of light, arising from modulated vacuum fluctuations in linear systems.

We present an extension of the previous descriptions based on the Dynamics Controlled Truncation Scheme of light-matter interaction beyond mean-field, including the microscopic description of the exciton-photon interaction. This enables the microscopic analysis of the influence of decoherence and noise on the polariton quantum correlations originating from nonlinear optical processes. We expand the operators involved in the dynamics in terms of exact eigenstates of the electron system, the photon and phonon operators and treat phonon-assisted transitions within the Markov approximation. In particular, we present quantum Heisenberg-Langevin equations describing light-induced excitations in semiconductor systems interacting with the phonon bath. This theoretical framework is applied to study the influence of dephasing and noise due to photoluminescence on polariton quantum correlations generated by parametric emission in microcavities. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We investigate the influence of environmental noise on polariton entangled pairs in the self-stimulated regime. By adopting a recent separability criterion, we show that self-stimulation may suppress the detrimental influence of noise on entanglement, but once effective, a noise-equipped classical model of parametric emission provides the same results of quantum theory with respect to the separability criterion. More generally we show that, in the macroscopic limit, it is not possible to observe violations of local realism with measurements of finite order n -particle correlations only. These results provide a prototypical case of the emergence of macroscopic local realism in the presence of strong entanglement even in the absence of decoherence. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We present an overview of our results concerning the influence of two-pair continuum coherences on the transient nonlinear optical response of semiconductor microcavities. We show that the interplay between these 4-particle coherences and the nonperturbative light-matter interaction produces highly desirable almost decoherence-free exciton-exciton collisions on the lower polariton branch. This effect gives rise to the very different nonlinear absorption rates on the two polariton branches observed in many experiments and make possible to reach a very high degree of amplification in samples with large Rabi splitting. Moreover, we show that the availability of almost decoherence free exciton-exciton collisions can be used for the realization of coherent trapping of polariton emission and amplification. This coherent manipulation and trapping of many-particle polariton states can be performed employing both ultrafast and continuous wave operation.

We present an overview of our results concerning the influence of two-pair continuum coherences onthe transient nonlinear optical response of semiconductor microcavities. We show that the interplaybetween these 4-particle coherences and the nonperturbative light-matter interaction produces highlydesirable almost decoherence-free exciton-exciton collisions on the lower polariton branch. This effectgives rise to the very different nonlinear absorption rates on the two polariton branches observed inmany experiments and make possible to reach a very-high degree of amplification in samples withlarge Rabi splitting.

We discuss the influence of many-body and correlation effects on the transient optical response of semiconductor microcavities. The complexity induced by the Coulomb interaction between electrons determines the non-instantaneous character of exciton–exciton collisions. We show that the exciton–photon coupling in semiconductor microcavities is able to alter the exciton dynamics during collisions strongly affecting the effective scattering rates. This influence determines the very different nonlinear absorption rates of the two polariton branches observed in many experiments. This analysis also clarifies the origin of the great enhancement of polariton parametric amplification observed when increasing the polariton splitting. It demonstrates that exciton–exciton collisions in semiconductor microcavities can be controlled and engineered to produce almost decoherence-free collisions for the realization of all-optical microscopic devices.

Near-field optical spectroscopy and microscopy give access to excitations that cannot be revealed in the far zone. In order to investigate these remarkable differences, we present a theoretical analysis of the local optical properties of semiconductor quantum wells including the effects of disorder arising from interface fluctuations. The far-field absorption spectrum is compared with spatially averaged absorption spectra calculated at different spatial resolutions. We find that summing up local optical spectra does not reproduce the global spectrum in contrast to findings at diffraction-limited resolutions.

We describe how, exploiting the current scanning near-field optical techniques, it is possible to map phase and modulus of the wavefunctions of a quantum dot system. In particular we propose an experimental setup based on an interferometric technique able to measure simultaneously modulus and phase of wavefunctions. We present numerical calculations simulating the measurements on dots arising from interface fluctuations of GaAs quantum wells.

We describe how ultrastrong interactions in optomechanical systems can be used to force the system ground state to evolve into an arbitrary quantum state of mechanical motion in a completely controlled and deterministic manner. If the target quantum state is a superposition of $N$ Fock states, it can be obtained by applying in single-step $N$ classical optical signals of different frequencies for a common time interval. This protocol can be applied to various strongly interacting quantum systems as trapped ions beyond the Lamb-Dicke regime and cavity QED into the ultrastrong coupling regime.

We present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowled...

Istituto Nazionale per la Fisica della Materia, Universita`di Messina, Salita Sperone 31, I-98166 Messina, Italy~Received 29 March 2000; revised manuscript received 4 May 2000!A theoretical approach for the simulation of scanning local optical spectroscopy in disordered quantumstructures is proposed. The local optical spectra at subwavelength resolution show the spectral lines from thelowest states of the exciton center-of-mass motion located at the individual potential minima as well as fromthe excited states that merge into quasicontinua near the potential barriers in agreement with the experimentalfindings. The simulations show the crucial impact that spatial resolution may have on the local spectra. Whenthe spatial resolution is lowered many lines disappear owing to destructive spatial interference of quantumstates.I. INTRODUCTION

We present a numeric calculation of the exciton-exciton correlation for a one-dimensional semiconductor model with long-range Coulomb interactions, taking into account the effect of the effect of the exciton continuum. we use the obtained correlation functions to calculate the time integrated intensity of the four wave mixing signal. Changing the central frequency of the exciting pulses we observe changes in the profile and in the rise time of the time integrated signal for negative delays in qualitative agreement with experimental results. Our results show theoretically how the exciton continuum affects the coherence of four-particle correlations.