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arXiv:1504.00752 · Physical Review A

Non-Hermitian Hamiltonian approach to the microwave transmission through a one-dimensional qubit chain

Ya. S. Greenberg, Alexander A. Shtygashev

Published Dec 21, 2015• 36 citations
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Abstract

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 $N$ 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 $N$ 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 $\mathrm{\ensuremath{\Gamma}}$.

DOI
10.1103/physreva.92.063835

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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 \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ and the rate of spontaneous emission \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ.
In this work, using the non-Hermitian Hamiltonian method, the transmission of a single photon in a one-dimensional waveguide interacting with the cavity containing an arbitrary number of photons and the two-level artificial atom is studied with allowance for the relaxation of the latter. For transport factors, analytical expressions which explicitly take into account the qubit relaxation parameter have been obtained. The form of the transmission (reflection) coefficient when there is more than one photon in the cavity qualitatively differs from the single-photon cavity and contains the manifestation of the photon blockade effect. The qubit lifetime depends on the number of photons in the cavity.
We analyze a photon transport through a one-dimensional open waveguide side coupled to the NNN-photon microwave cavity with embedded an artificial two-level atom (qubit). The qubit state is probed by a weak signal at the fundamental frequency of the waveguide. Within the formalism of projection operators and a non-Hermitian Hamiltonian approach we develop a one-photon approximation scheme to obtain the photon wave function, which allows for the calculation of the probability amplitudes of the spontaneous transitions between the levels of two Rabi doublets in an NNN-photon cavity. We obtain analytic expressions for the transmission and reflection factors of the microwave signal through a waveguide which contains the information of the qubit parameters. We show that for a small number of cavity photons the Mollow spectrum consists of four spectral lines, which is a direct manifestation of the quantum nature of light. The results obtained in the paper are of a 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 energy $\mathrm{\ensuremath{\Omega}}$ of the qubit and its coupling $\ensuremath{\lambda}$ to the cavity photons.
We investigate the interaction between a single photon and a chain of NNN equally spaced two-level atoms inside a one-dimensional waveguide. By solving the eigenvectors of the Hamiltonian in the subspace with a single excitation, we address quantum interference effects in two physical processes. First, we analyze the scattering of a single photon when atoms are all initially at their ground states. We discover a set of transmission peaks determined by NNN. These transmission peaks are narrow and sensitive to the relative position between atoms. Second, we examine the spontaneous emission of a photon from an excited atom in the chain in vacuum. The characteristic atomic decay rates are found to be enhanced and reduced by increasing the number of atoms. In particular, the slowest decay rate scales approximately as ${N}^{\ensuremath{-}3}$.
The influence of nonradiative damping of qubits on the microwave transport of photons propagating in a onedimensional microstrip line has been considered. Expressions for the transmission and reflection coefficients for two qubits have been obtained within the non-Hermitian Hamiltonian formalism. The indirect interaction between qubits caused by nonradiative decay into a common channel is explicitly taken into account in these expressions. It has been shown that this interaction leads to results significantly different from known results.
We consider a one-dimensional chain of N equidistantly spaced noninteracting qubits embedded in an open waveguide. In the frame of single-excitation subspace, we systematically study the evolution of qubits' amplitudes if the only qubit in the chain was initially excited. We show that the temporal dynamics of qubits' amplitudes crucially depend on the value of kd, where k is the wave vector, d is a distance between neighbor qubits. If kd is equal to an integer multiple of π, then the qubits are excited to a stationary level which scales as N−1. We show that in this case, it is the dark states which prevent qubits from decaying to zero even though they do not contribute to the output spectrum of photon emission. For other values of kd the excitations of qubits have the form of damping oscillations, which represent the vacuum Rabi oscillations in a multi-qubit system. In this case, the output spectrum of photon radiation is defined by a subradiant state with the smallest width.
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.