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arXiv:quant-ph/0108006 · Europhysics Letters (EPL)

Giant Lamb shift of atoms near lossy multimode optical micro-waveguides

Péter Horák, P. Domokos, Helmut Ritsch

Published Feb 1, 2003• 17 citations
arXivHTMLPDF

Abstract

Atoms coupled to optical fields strongly confined in two spatial dimensions, as in solid-state microstructures, can experience large Lamb shifts due to a spectrally strongly asymmetric mode density. We use the generic example of a quasi–one-dimensional waveguide structure driven close to cutoff frequency of a new transverse branch of propagating modes. We analytically find strong shifts of the atomic resonance frequency due to the modified vacuum, which can be an order of magnitude larger than the atomic linewidth. At the same time one gets significantly enhanced scattering of the guided light by the atom, which could be used as a tool to investigate these effects or to build non-destructive single-atom detectors.

DOI
10.1209/epl/i2003-00151-4

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Atoms coupled to optical fields strongly confined in two spatial dimensions, as in solid-state microstructures, can experience large Lamb shifts due to a spectrally strongly asymmetric mode density. We use the generic example of a quasi–one-dimensional waveguide structure driven close to cutoff frequency of a new transverse branch of propagating modes. We analytically find strong shifts of the atomic resonance frequency due to the modified vacuum, which can be an order of magnitude larger than the atomic linewidth. At the same time one gets significantly enhanced scattering of the guided light by the atom, which could be used as a tool to investigate these effects or to build non-destructive single-atom detectors.
We report experimental and theoretical results on the extremely large Lamb shift in a multimode circuit quantum electrodynamics (QED) system in the deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength is comparable to or larger than the qubit and resonator frequencies. The system comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar waveguide resonator ([Formula: see text] CPWR) that are coupled inductively through a shared edge that contains a Josephson junction to achieve the DSC regime. Spectroscopy is performed around the frequency of the fundamental mode of the CPWR, and the spectrum is fitted by the single-mode quantum Rabi Hamiltonian to obtain the system parameters. Since the qubit is also coupled to a large number of higher modes in the resonator, the single-mode fitting does not provide the bare qubit energy but a value that incorporates the renormalization from all the other modes. We derive theoretical formulas for the Lamb shift in the multimode resonator system. As shown in previous studies, there is a cut-off frequency [Formula: see text] for the coupling between the FQ and the modes in the CPWR, where the coupling grows as [Formula: see text] for [Formula: see text] and decreases as [Formula: see text] for [Formula: see text]. Here [Formula: see text] is the frequency of the nth mode. The cut-off effect occurs because the qubit acts as an obstacle for the current in the resonator, which suppresses the current of the modes above [Formula: see text] at the location of the qubit and results in a reduced coupling strength. Using our observed spectrum and theoretical formulas, we estimate that the Lamb shift from the fundamental mode is 82.3% and the total Lamb shift from all the modes is 96.5%. This result illustrates that the coupling to the large number of modes in a CPWR yields an extremely large Lamb shift but does not suppress the qubit energy to zero, which would happen in the absence of a high-frequency cut-off.
Quantum vacuum mode structure can be changed due to length scale fluctuation of the cross section of a metallic waveguide. Such a structure change in vacuum modes (particularly in cutoff vacuum modes) would lead to dramatic enhancement or inhibition of spontaneous emission decay of atoms and, if the waveguide is filled with a dilute atomic vapor consisting of quantum-coherent atoms of a four-level tripod-configuration system, an optical wave propagating inside the waveguide can be coherently manipulated by tunable constructive and destructive quantum interference between two control transitions (driven by two control fields) in a quite unusual way (e.g., the optical response, in which a three-level dark state is involved, is sensitive to the waveguide dimension variations at certain positions of resonance of the atomic spontaneous emission decay rate). Therefore, an intriguing effect that can be employed to designs of new photonic and quantum optical devices could be achieved based on the present mechanisms of quantum-vacuum manipulation and quantum coherence control.
We present a quantum theory of a coherently pumped two-level atom in a photonic band gap (PBG), coupled to both a multimode waveguide channel and a high-quality microcavity embedded within a photonic crystal. One mode is engineered to exhibit a sharp cutoff within the PBG, leading to a large discontinuity in the local photon density of states near the atom, and the cavity field mode is resonant with the central component of the Mollow spectrum of atomic resonance fluorescence. Another mode of the waveguide channel is used to propagate the pump beam. We derive analytical expressions for the optical amplitude, intensity, second-order correlation functions, and conjugate quadrature variances for the light emitted by the atom into the microcavity. The quantum degree of second-order coherence in the cavity field reveals enhanced, inversionless, nearly coherent light generation when the photon density of states jump between the Mollow spectral components is large. The cavity field characteristics are highly distinct from that of a corresponding high-QQQ cavity in ordinary vacuum. In the case of a vanishing photon density of states on the lower Mollow sideband and no dipolar dephasing, the emitted photon statistics is Poissonian, and the cavity field exhibits quadrature coherence.
We study the cooperative optical coupling between regularly spaced atoms in a one-dimensional waveguide using decompositions to subradiant and super-radiant collective excitation eigenmodes, direct numerical solutions, and analytical transfer-matrix methods. We illustrate how the spectrum of transmitted light through the waveguide, including the emergence of narrow Fano resonances, can be understood by the resonance features of the eigenmodes. We describe a method based on super-radiant and subradiant modes to engineer the optical response of the waveguide and to store light. The stopping of light is obtained by transferring an atomic excitation to a subradiant collective mode with the zero radiative resonance linewidth by controlling the level shift of an atom in the waveguide. Moreover, we obtain an exact analytic solution for the transmitted light through the waveguide for the case of a regular lattice of atoms and provide a simple description of how the light transmission may present large resonance shifts when the lattice spacing is close, but not exactly equal, to half of the wavelength of the light. Experimental imperfections such as fluctuations of the positions of the atoms and loss of light from the waveguide are easily quantified in the numerical simulations, which produce the natural result that the optical response of the atomic array tends toward the response of a gas with random atomic positions.