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arXiv:2104.04927 · Low Temperature Physics

Spontaneous decay of artificial atoms in a multi-qubit system

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

Published Oct 1, 2021• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/10.0006063

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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 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 $\mathrm{\ensuremath{\Gamma}}$.
In the present paper, we have proposed the experimentally achievable method for the characterization of the collective states of qubits in a linear chain. We study a temporal dynamics of absorption of a single-photon pulse by three interacting qubits embedded in a one-dimensional open waveguide. Numerical simulations were performed for a Gaussian-shaped pulse with different frequency detunings and interaction parameters between qubits. The dynamic behavior of the excitation probability for each qubit is investigated. It was shown that the maximum probability amplitudes of excitation of qubits are reached when the frequency of external excitation coincides with the frequency of excitation of the corresponding eigenstate of the system. In this case, the magnitude of the probability amplitude of each qubit in the chain unambiguously correlates with the contribution of this qubit to the corresponding collective state of the system, and the decay of these amplitudes is determined by the resonance width arising from the interaction of the qubit with the photon field of the waveguide. Therefore, we show that the pulsed harmonic probe can be used for the characterization of the energies, widths, and the wave functions of the collective states in a one dimensional qubit chain.
We theoretically investigate quantum Rabi oscillations in a system consisting of a two-level atom (qubit) strongly coupled to a one-dimensional open waveguide. In contrast to conventional cavity quantum electrodynamics, the qubit interacts with a continuum of propagating modes, which gives rise to fundamentally different dynamical behavior. Within the rotating-wave approximation, we express the multimode Jaynes-Cummings Hamiltonian in terms of collective bosonic operators and show that the system possesses two integrals of motion, enabling an exact diagonalization of the Hamiltonian in the single-excitation subspace. We show that for a strongly interacting qubit-photon system, the dynamics are captured by a reduced two-level model. In this framework, each level is defined as the product of the atomic excited state and a specific field mode, which takes the form of a multiphoton Fock-like state. In this picture, the Rabi oscillations represent a collective phenomenon corresponding to oscillations between multiphoton states differing by a single photon. We then extend our analysis to multiphoton processes in which the initial field is a coherent state with a continuous spectrum. In this case, the Rabi frequency is shown to be sensitive to the spectral profile of the function that generates the coherent state.
For a system of two spatially separated qubits (two-level atoms) coupled to a one-dimensional waveguide we have described the time evolution of singly or doubly excited states of the atomic subsystem. When the interatomic distance lll takes special (``resonant'' or ``antiresonant'') values, the singly excited system of resonant atoms can form metastable (dark) states. If lll slightly deviates from one of the special values or the atomic frequencies do not coincide, the dark states slowly decay and we have calculated the decay rate. Also, we have found that the doubly excited state of two resonant atoms located at the special positions does not completely decay but, with a finite probability, can evolve (with the emission of a single photon) to one of the metastable singly excited states. Metastable states of pairs of qubits may find applications (e.g., as memory elements) in information processing or as detectors sensitive to external perturbations.
Waveguide quantum electrodynamics studies photon-mediated interactions of quantum emitters in a one-dimensional radiation channel. Although signatures of such interactions have been observed previously in a variety of physical systems, observation of coherent cooperative dynamics has been obscured by radiative decay of atoms into the waveguide. Employing transmon qubits as artificial atoms coupled to a microwave coplanar waveguide, here we observe dynamical oscillations in an open system where a designated probe qubit interacts with an entangled dark state of an array of qubits which effectively traps radiation as an atomic cavity. The qubit-cavity system is shown to achieve a large cooperativity of C=172\mathcal{C}=172C=172 due to collective enhancement of photon-mediated interactions, entering the strong coupling regime. The quantum coherence of the dark state cavity is also explored through its nonlinear response at the single-excitation level. With realistic refinements, this system is suitable for studying the many-body dynamics of large (N>10N>10N>10) quantum spin chains, synthesizing highly non-classical radiation fields on demand, and implementing universal quantum logic operations with high fidelity on information encoded within decoherence-free subspaces.
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 ${N}^{\ensuremath{-}3}$.