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arXiv:2007.14814 · Physical Review A

Waveguide band-gap N -qubit array with a tunable transparency resonance

Ya. S. Greenberg, Alexander A. Shtygashev, Aleksey G. Moiseev

Published Feb 8, 2021• 21 citations
arXivHTMLPDF

Abstract

We study a single-photon transmission through a one-dimensional $N$-qubit chain. The qubits are supposed to be identical with equal distance between neighbors. We express the transfer matrix of the $N$-qubit chain in terms of Chebyshev polynomials, which allows us to obtain simple expressions for the transmission and reflection amplitudes for arbitrarily large $N$. If the distance between neighbor qubits is equal to a half wavelength, then the transmission spectrum exhibits a flat band gap structure with very steep walls. We show that, for odd $N$, the tuning of the excitation frequency of a central qubit gives rise to the appearance within a band gap of a narrow resonance with a full transmission. The position of the resonance and its width can be monitored with the frequency of a central qubit. We show that the formation of the band gap and the transmission resonance result from the overlapping the widths of individual qubits, which is due to the strong coupling between qubits and waveguide photons.

DOI
10.1103/physreva.103.023508

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We study a single-photon transmission through a one-dimensional NNN-qubit chain. The qubits are supposed to be identical with equal distance between neighbors. We express the transfer matrix of the NNN-qubit chain in terms of Chebyshev polynomials, which allows us to obtain simple expressions for the transmission and reflection amplitudes for arbitrarily large NNN. If the distance between neighbor qubits is equal to a half wavelength, then the transmission spectrum exhibits a flat band gap structure with very steep walls. We show that, for odd NNN, the tuning of the excitation frequency of a central qubit gives rise to the appearance within a band gap of a narrow resonance with a full transmission. The position of the resonance and its width can be monitored with the frequency of a central qubit. We show that the formation of the band gap and the transmission resonance result from the overlapping the widths of individual qubits, which is due to the strong coupling between qubits and waveguide photons.
We investigate the propagation of microwave photons in a one-dimensional open waveguide interacting with a number of artificial atoms (qubits). Within the formalism of projection operators and a non-Hermitian Hamiltonian approach we develop a one-photon approximation scheme for the calculation of the transmission and reflection factors of the microwave signal in a waveguide which contains an arbitrary number NNN of noninteracting qubits. We considered in detail the resonances and photon-mediated entanglement for two and three qubits in a chain. We showed that in the non-Markovian case the resonance widths, which define the decay rates of the entangled state, can be much smaller than the decay width of an individual qubit. It is also shown that for identical qubits in the long-wavelength limit a coherent superradiant state is formed with the width being equal to the sum of the widths of spontaneous transitions of NNN individual qubits. The results obtained in the paper are of general nature and can be applied to any type of qubits. The specific properties of the qubit are only encoded in the two parameters: the qubit energy $\mathrm{\ensuremath{\Omega}}$ and the rate of spontaneous emission $\mathrm{\ensuremath{\Gamma}}$.
We demonstrate the emergence of transparent behavior in a chain of periodically spaced nonidentical quantum emitters coupled to a waveguide, in the special case when the nearest neighbor separation is a half-integral multiple of the resonant wavelength, i.e., kLkLkL is an integral multiple of $\ensuremath{\pi}$, with kkk being the wave number and LLL the spatial periodicity. When equal but opposite frequency detunings are assigned in pairs to a system of even number of atoms, perfect transmission ensues. When the chain size is odd, a similar assignment leads to the disappearance of collective effects as the odd atom determines the spectral behavior. We also manifest the robustness of these features against dissipative effects and show how the spectral behavior hinges significantly on the relative detunings between the atoms as compared to the decay rate. A key distinction from the phenomenon of electromagnetically induced transparency is that in the waveguide case, the presence of an intrinsic waveguide-mediated phase coupling between the atoms strongly affects the transport properties. Furthermore, while reciprocity in single-photon transport does not generally hold due to the phase coupling, we observe an interesting exception for $kL=n\ensuremath{\pi}$ at which the waveguide demonstrates reciprocal behavior with regard to both the transmission and reflection coefficients.
We study the localization and scattering of a single photon in a waveguide coupled to qubit arrays with quasiperiodic spacings. As the quasiperiodic strength increases, localized subradiant states with extremely long lifetime appear around the resonant frequency and form a continuum band. In stark contrast to the fully disordered waveguide QED where all states are localized, we analytically find that the fraction of localized states is up to (3−5)/2(3-\sqrt{5})/2(3−5​)/2 when the modulation frequency is (1+5)/2(1+\sqrt{5})/2(1+5​)/2. The localized and delocalized states can be related to excitation in flat and curved inverse energy bands under the approximation of large-period modulation. When the quasiperiodic strength is weak, an extended subradiant state can support the transmission of a photon. However, as the quasiperiodic strength increases, localized subradiant states can completely block the transmission of a single photon in resonance with the subradiant states, and enhance the overall reflection. At a fixed quasiperiodic strength, we also find mobility edge in transmission spectrum, below and above which the transmission is either turned on and off as system size increases. Our work give new insights into the localization in non-Hermitian systems.
Abstract We study the dynamics of a single-photon pulse traveling through a linear qubit chain coupled to continuum modes in a one-dimensional (1D) photonic waveguide. We derive a time-dependent dynamical theory for qubits’ amplitudes and for transmitted and reflected spectra. We show that the requirement for the photon-qubit coupling to exist only for positive frequencies can significantly change the dynamics of the system. First, it leads to the additional photon-mediated dipole-dipole interaction between qubits which results in the violation of the phase coherence between them. Second, the spectral lines of transmitted and reflected spectra crucially depend on the shape of the incident pulse and on the initial distance between the pulse center and the first qubit in the chain. We apply our theory to one-qubit and two-qubit systems. For these cases we obtain the explicit expressions for the qubits’ amplitudes and for the photon radiation spectra as time tends to infinity. Specific calculations are performed for superconducting qubits operating in GHz frequency range. For the incident Gaussian wave packet we calculate the line shapes of transmitted and reflected photons.
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.
We study the dynamics of a single-photon pulse travelling through a linear qubit chain coupled to continuum modes in a one-dimensional (1D) photonic waveguide. We derive a time-dependent dynamical theory for qubit amplitudes and for transmitted and reflected spectra. We show that the requirement for the photon-qubit coupling to exist only for positive frequencies can significantly change the dynamics of the system. First, it leads to an additional photon-mediated dipole-dipole interaction between qubits which results in the violation of the phase coherence between them. Second, the spectral lines of transmitted and reflected spectra crucially depend on the shape of the incident pulse and the initial distance between the pulse center and the first qubit in the chain. We apply our theory to one-qubit and two-qubit systems. For these two cases, we obtain the explicit expressions for the qubits' amplitudes and the photon radiation spectra as time tends to infinity. For the incident Gaussian wave packet we calculate the line shapes of transmitted and reflected photons.