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arXiv:2205.00174 · Physical Review A

Single-photon scattering on a qubit: Space-time structure of the scattered field

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

Published Jan 24, 2023• 5 citations
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Abstract

We study the space-time structure of the scattered field induced by the scattering of a narrow single-photon Gaussian pulse on a qubit embedded in a one-dimensional open waveguide. For weak excitation power we obtain explicit analytical expressions for the space and time dependence of reflected and transmitted fields which are, in general, different from plane traveling waves. The scattered field consists of two parts: a damping part which represents spontaneous decay of the excited qubit and a coherent, lossless part. We show that for a large distance $x$ from the qubit and at times $t$ long after the scattering event our theory provides the result which is well known from stationary photon transport. However, the approach to the stationary limit is very slow. The scattered field decreases as the inverse powers of $x$ and $t$ as both the distance from the qubit and the time after the interaction increase.

DOI
10.1103/physreva.107.013519

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We study the space-time structure of the scattered field induced by the scattering of a narrow single-photon Gaussian pulse on a qubit embedded in a one-dimensional open waveguide. For weak excitation power we obtain explicit analytical expressions for the space and time dependence of reflected and transmitted fields which are, in general, different from plane traveling waves. The scattered field consists of two parts: a damping part which represents spontaneous decay of the excited qubit and a coherent, lossless part. We show that for a large distance xxx from the qubit and at times ttt long after the scattering event our theory provides the result which is well known from stationary photon transport. However, the approach to the stationary limit is very slow. The scattered field decreases as the inverse powers of xxx and ttt as both the distance from the qubit and the time after the interaction increase.
In this paper, we study the spatiotemporal distribution of the photon electric field produced by the scattering of a single photon narrow pulse from a system of two identical qubits coupled to continuum modes in a one-dimensional (1D) open waveguide. We derive the time-dependent dynamical equations for qubits' and photon amplitudes which allow the calculation of the photon backward and forward scattering fields in the whole space: before qubits, between qubits, and behind the qubits. The scattered field consists of several contributions that describe a free field of incoming photon, a spontaneous exponential decay of excited qubits, a slowly decaying part that dies out as the inverse powers of ttt, and a lossless part that represents a steady state solution as t→∞t\rightarrow\inftyt→∞. For our system, we find the transmittance and reflectance fields as both time and distance from the qubits tend to infinity. We show that as the time after the event of scattering tends to infinity, the steady state photon the field is being formed in the whole one-dimensional space. If the distance ddd between qubits is equal to the integer of the wavelength λλλ, the field energy exhibits temporal beatings between the qubit frequency ΩΩΩ and the photon frequency ωSω_SωS​ with the period T=2π/(ωS−Ω)T=2π/(ω_S-Ω)T=2π/(ωS​−Ω).
We present a time-dependent quantum calculations of the first order and second order photon correlation functions for the scattering of a single-photon pulse on a two-level atom (qubit) embedded in a one-dimensional open waveguide. Within Markov approximation we find the analytic expression for the quantum operator of positive frequency electric field. We restricted Hilbert space of initial states by the states with one and two excitations and show that the photon probability amplitudes are given by the off-diagonal matrix elements of the electric field operator between these states. For two-excitation initial state where the atom is excited and there exists a single photon in a waveguide we calculate the second order correlation function which describes the measurements by two detectors at two different space-time points. The second order correlation function exhibits the interference term showing that the measurements of two detectors are correlated. This interference is similar to that found in the Hanbury Brown and Twiss correlation experiment with two indistinguishable photons.
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.
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 present a detailed quantum theory and simulations of few-photon Fock state pulses interacting with a two-level system (TLS) in an open waveguide-QED system. For a rectangular pulse shape, we present an exact temporal scattering theory to derive analytical expressions for the TLS population, using one-photon and two-photon pulses, for both chiral and symmetric emitters. We also derive the stationary (long-time) and time-dependent spectra for one-photon excitation, and show how these differ at a fundamental level when considering TLS population effects. Numerically, we also present matrix product state (MPS) simulations, which allow us to compute more general photon correlation functions for arbitrary quantum pulses, and we use this approach to also show results for Gaussian quantum pulses and to confirm the accuracy of our analytical solutions. In the case of a chiral TLS, we show how a one-photon pulse, of any temporal shape, yields a transmitted long-time spectrum that is identical to the input pulse, despite significant TLS population effects. However, the dynamical spectra and spectral intensity show rich population effects, allowing for their detection in spectroscopic experiments. We also show the striking differences between one-photon and two-photon excitation, where the latter shows clear signatures of nonlinear saturation effects. In addition, we demonstrate how significant population TLS dynamics also occur for pulses that are relatively long compared to the radiative decay time (showing that a weak excitation approximation, which neglects finite TLS population effects, cannot be made) and investigate the population signatures, nonlinear features and dynamical behavior as a function of quantum pulse length.
We present a time-dependent quantum calculation of the scattering of a few-photon pulse on a single atom. The photon wave packet is assumed to propagate in a transversely strongly confined geometry, which ensures strong atom-light coupling and allows a quasi-one-dimensional treatment. The amplitude and phase of the transmitted, reflected, and transversely scattered part of the wave packet strongly depend on the pulse length (bandwidth) and energy. For a transverse mode size of the order of ${\ensuremath{\lambda}}^{2},$ we find nonlinear behavior for a few photons already, or even for a single photon. In a second step we study the collision of two such wave packets at the atomic site and find striking differences between the Fock state and coherent state wave packets of the same photon number.