Ladder operators and coherent states for the Jaynes-Cummings model in the rotating-wave approximation

Hussin, Véronique; Nieto, L. M.

doi:10.1063/1.2137718

Cited by 29 publications

(60 citation statements)

References 16 publications

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“…In this so called rotating-wave approximation the Hamiltonian (1) reduces to the JaynesCumming (JC) model, H JC = ω r a † a + ω q σ + σ − + g a † σ − + σ + a used widely in discussions of CQED physics. While appropriate in many relevant cases, recent implementations of circuit QED [23,24] achieved coupling strengths g where the counter-rotating terms begin to show significant deviations from the expectations of the JC model [1,[25][26][27][28][29][30][31]. This is the so called "ultra-strong coupling" regime of parameters g ∼ ω r .…”

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confidence: 99%

Phase Transition of Light in Cavity QED Lattices

et al. 2012

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Systems of strongly interacting atoms and photons, that can be realized wiring up individual cavity QED systems into lattices, are perceived as a new platform for quantum simulation. While sharing important properties with other systems of interacting quantum particles here we argue that the nature of light-matter interaction gives rise to unique features with no analogs in condensed matter or atomic physics setups. By discussing the physics of a lattice model of delocalized photons coupled locally with two-level systems through the elementary light-matter interaction described by the Rabi model, we argue that the inclusion of counter rotating terms, so far neglected, is crucial to stabilize finite-density quantum phases of correlated photons out of the vacuum, with no need for an artificially engineered chemical potential. We show that the competition between photon delocalization and Rabi non-linearity drives the system across a novel Z2 parity symmetry-breaking quantum criticality between two gapped phases which shares similarities with the Dicke transition of quantum optics and the Ising critical point of quantum magnetism. We discuss the phase diagram as well as the low-energy excitation spectrum and present analytic estimates for critical quantities. Introduction -Interaction between light and matter is one of the most basic processes in nature and represents a cornerstone in our understanding of a broad range of physical phenomena. In the study of strongly correlated systems and collective phenomena, light has traditionally assumed the role of a spectroscopic probe. The increasing level of control over light-matter interactions with atomic and solid-state systems [1-3] has brought forth a new class of quantum many body systems where light and matter play equally important roles in emergent phenomena: photon lattices [4][5][6][7][8][9][10][11][12][13][14][15]. The basic building block of such systems is the elementary Cavity QED (CQED) system formed by a two-level system (TLS) interacting with a single mode of an electromagnetic resonator. When CQED systems are coupled to form a lattice, the interplay between photon blockade [17][18][19] and inter-cavity photon tunnelling leads to phenomenology akin to those of Hubbard models of massive bosons as realized e.g. by ultracold atoms in optical lattices [20]. The possibility of quantum phase transitions of light between Mott-like insulating and superfluid phases has stimulated a great deal of discussion recently [4][5][6][7][8][9][11][12][13][14]. The excitement about these systems stems from their potential as dissipative quantum simulators that provide full access to individual sites through continuous weak measurements [16].

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Phase Transition of Light in Cavity QED Lattices

et al. 2012

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“…2 are the dressed states [13], which we label as |±, n defined in the Methods section, where n is the number of excitations in the cavity. The ground state for the JaynesCummings Hamiltonian is qualitatively different from the other dressed states, implying a non-trivial form of the raising operator [14]. This constitutes a further departure from usual Hubbard-like condensed-matter models, where the raising operator is not dependent on the number of excitations.…”

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confidence: 96%

Quantum phase transitions of light

et al. 2006

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Recently, condensed matter and atomic experiments have reached a length-scale and temperature regime where new quantum collective phenomena emerge. Finding such physics in systems of photons, however, is problematic, as photons typically do not interact with each other and can be created or destroyed at will. Here, we introduce a physical system of photons that exhibits strongly correlated dynamics on a meso-scale. By adding photons to a two-dimensional array of coupled optical cavities each containing a single two-level atom in the photon-blockade regime, we form dressed states, or polaritons, that are both long-lived and strongly interacting. Our zero temperature results predict that this photonic system will undergo a characteristic Mott insulator (excitations localised on each site) to superfluid (excitations delocalised across the lattice) quantum phase transition. Each cavity's impressive photon out-coupling potential may lead to actual devices based on these quantum manybody effects, as well as observable, tunable quantum simulators.The Jaynes-Cummings [1] model is arguably the most important model for understanding light-matter interactions. It describes the interaction of a single, quasiresonant optical cavity field with a two-level atom. The coupling between the atom and the photons leads to optical nonlinearities and an effective photon-photon repulsion. Perhaps the most extreme demonstration of this photonic repulsion is photon blockade, demonstrated recently by Birnbaum et al. [2], where photonic repulsion prevents more than one photon from being in the cavity at any one time. Photon blockade was initially theoretically described with a four-state system [3], with multiplication of the weak Kerr nonlinearity effected by placing a large number of atoms within each cavity. However, it was quickly realised that the photonic blockade mechanism does not persist in the limit of many atoms [4], rapidly degrading as the number of atoms per cavity is increased [5]. Later Rebic et al. showed that the nonlinear interaction afforded by placing a single two-level atom inside a cavity would suffice for realising photon blockade [6]. This observation was highly significant as it allowed the full weight of the Jaynes-Cummings model to be used to attack and understand this problem.To create an atom-photon system whose dynamics mirror those traditionally associated with strongly interacting condensed matter systems, we consider a twodimensional array of photonic bandgap cavities. Each cavity contains a single two-level atom, quasi-resonant with the cavity mode. Evanescent coupling between the cavities due to their proximity allows inter-cavity photon hopping. This configuration is depicted schematically in Fig. 1(a), where we have explicitly chosen three nearest neighbours per cavity (coordination number z = 3), for reasons explained below. Because we are considering small cavities, with volumes of order λ 3 where λ is the wavelength of the light, there will be strong atom-photon couplings that will dominate over the...

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“…(1) are the dressed states [13], which we define as j; i i for one photon with spin per cavity and j; 0 i i for two photons with spin and 0 per cavity.…”

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Quantum Magnetic Dynamics of Polarized Light in Arrays of Microcavities

2007

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We describe an optical system that allows for a direct experimental observation of the quantum magnetic correlated dynamics of polarized light. By adjusting the Zeeman and the Raman fields, we could realize a ferromagnetic phase, super-counter-fluidity phase, and antiferromagnetic phase of polarized light, that are of interest for studying spin-dependent photon-photon interactions. We also design an experimental protocol for the observation of these phases. Moreover, the technique of controlling photospin correlation may be used for building quantum information devices.

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Ladder operators and coherent states for the Jaynes-Cummings model in the rotating-wave approximation

Cited by 29 publications

References 16 publications

Phase Transition of Light in Cavity QED Lattices

Phase Transition of Light in Cavity QED Lattices

Quantum phase transitions of light

Quantum Magnetic Dynamics of Polarized Light in Arrays of Microcavities

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