Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations

This review describes the emerging field of waveguide quantum electrodynamics concerned with the interaction of photons propagating in a waveguide with localized quantum emitters. The collective emitter-photon interactions can lead to both enhanced and suppressed coupling compared to the case of independent emitters. Here the focus is on guided photons and ordered emitter arrays, manifesting superradiant and subradiant states, bound photon states, and quantum correlations with promising quantum information applications. Recent groundbreaking experiments performed with different quantum platforms, including cold atoms, superconducting qubits, semiconductor quantum dots, and quantum solid-state defects, are highlighted. The review also provides a comprehensive introduction to theoretical techniques to study the interactions and dynamics of these emitters and the photons in the waveguide.

Figure 1

Schematic of light-matter interactions in various atomic ensembles. (a) Light beam propagation and scattering in a disordered dilute atomic array in free space. (b) Interaction of an ordered atomic array with the guided photon mode. Subradiant and bright collective array excitations that are out of phase and in phase with the photon mode (blue, long-dashed wavy arrow) are shown. (c) Interaction of an ordered atomic array with a photon mode propagating through a waveguide in the transverse direction. As in (b), bright and dark collective excitations are shown.

Figure 2

(a) Comparison of a waveguide coupling efficiency $\beta$ and number of emitters $N$ for different platforms of waveguide quantum electrodynamics. (b)–(e) Schematics of different platforms. More information is given in Table 2. (b) From [207]. (c) From [83]. (d) From [287]. (e) From [57]. (f) From [93]. (g) From [145].

Figure 3

Schematics of potential future platforms for WQED. (a) Three clouds of Rydberg atoms in free space. From [302]. (b) Planar two-dimensional atomic array. From [260]. (c) Top panel: array of superconducting $LC$ resonators with alternating short and long spacings. From [154]. Bottom panel: schematics of the topologically nontrivial Su-Schrieffer-Heeger model and the edge state realized in this array. (d). Topologically nontrivial photonic state excited by a quantum dot and propagating between two photonic crystals. From [21].

Figure 4

Linewidth of the optical resonance of the atomic array with the period $a/{\lambda }_0\approx 0.7$ depending on the array filling fraction. Decrease of the linewidth with the filling fraction indicates a collective subradiant response. From [260].

Figure 5

(a) Schematics of two quantum emitters interacting via directional bound states in a Su-Schrieffer-Heeger lattice of coupled cavities. Vertical bars show the amplitudes of the photon wave function. (b) Measured directional coupling between two emitters in the topological Su-Schrieffer-Heeger superconducting qubit array. From [154]. (c),(d) Schematics of two emitters coupled via the photonic graphene bath and tuned to the Dirac point. Depending on whether emitters are in the same sublattice or a different sublattice, the coupling is either dissipative or long-range coherent. From [96].

Figure 6

Real and imaginary parts of the eigenmodes calculated for (a) an atomic array in free space, (b) an atomic array near the fiber waveguide, with collective emission into the free space neglected, and (c) the full model. The vertical dotted lines in (a) and (c) show the light cone boundaries ($K=\pm \omega /c$). The vertical solid blue lines in (b) and (c) show the guided mode wave vector $\pm \beta$. The black solid curves in (a) and (b) show the analytical results of Eqs. (11) and (12) for an infinite array. The eigenmodes have been calculated for $N=40$ atoms perpendicularly polarized to the fiber. The polariton wave vector has been extracted from the Fourier transform of the eigenmodes. The other parameters are given in the text.

Figure 7

Spectrum of light reflection from two superconducting qubits with the spacing $3\lambda ({\omega }_0)/4$ coupled to the waveguide depending on the driving. The driving strength is characterized by the Rabi frequencies ${\mathrm{\Omega }}_R$, shown in the graph; larger reflection corresponds to larger driving. From [310].

Figure 8

(a) Imaginary and (b) real parts of the complex eigenfrequencies of the array of $N=10$ atoms coupled to a waveguide depending on the period of the array $d$. The shaded areas in (b) show the edges of the polariton band gaps, calculated using a diagonalization of the Hamiltonian (15) for the vanishing nonradiative decay rate $\mathrm{\Gamma }$. Each value of $d/\lambda$ corresponds to $N=10$ eigenvalues.

Figure 9

Polariton dispersion in the atomic arrays with different periods. (a)–(c) Three different values of $d/d_{\text{Bragg}}$, with the values as indicated. Bragg and polariton band gaps are provided. (d) Dependence of the band gap positions on the period of the array. The calculation was performed for ${\gamma }_{1\mathrm{D}}/{\omega }_0=0.03$ and $\gamma =0$.

Figure 10

(a),(d) Transmission, (b),(e) reflection, and (c),(f) absorption spectra of the array of $N=10$ atoms coupled to a waveguide, depending on the period of the array $d$. The bottom panels are calculated for four specific periods, indicated by corresponding dashes in the upper panels. The thin white lines in (a)–(c) show the real parts of the polaritonic eigenfrequencies ${\omega }^{\nu }$. The calculation was performed for $\gamma =0.1{\gamma }_{1\mathrm{D}}$ and ${\gamma }_{1\mathrm{D}}/{\omega }_0={10}^{-3}$. The value of the parameter $d/{\lambda }_0$ used for the calculations in (d)–(f) is indicated for each curve.

Figure 11

Two destructively interfering pathways for the absorption of two photons by two different atoms.

Figure 12

Schematics of single- and two-photon scattering on a two-level atom under resonant excitation.

Figure 13

Dependence of the photon-photon correlation function $g^{(2)}(0)$ on the number of atoms $N$ and on the ratio of nonradiative and radiative damping rates $\gamma /{\gamma }_{1\mathrm{D}}$ calculated with Eq. (36). (a),(b) Reflection and transmission geometries, respectively, calculated for the light incident at the resonant frequency $ϵ={\omega }_0$ of the two-level atoms ($U\to \infty$).

Figure 14

Schematic phase diagram showing different possible types of two-polariton states in an array of atoms coupled to a waveguide, depending on their radiative decay rate and the ratio of the array length $Nd$ to the wavelength at the atom frequency. Wave functions for these states are shown in Fig. 15.

Figure 15

Examples of different double-excited states in the finite array of $N=51$ two-level atoms coupled to a nonchiral waveguide. Top row: real-space two-polariton wave functions $|{\psi }_{nm}{|}^2$. Second row: Fourier transforms $|\psi (k){|}^2$. (b)–(e) Bottom panels: wave vectors of two polaritons corresponding to the two-polariton wave functions in the upper panels (not in scale). The calculation was performed for the array periods (a),(b),(d) ${\omega }_{0}d/c=0.01$ and (c),(e) ${\omega }_{0}d/c=0.4$. The complex energies of the displayed states are $(ϵ-{\omega }_0)/{\gamma }_{1\mathrm{D}}=8.4-48.9i$, $-5.0-1.0i$,$-0.2-4\times {10}^{-6}i$, and $1.9-2\times {10}^{-6}i$ for (a)–(e), respectively.

Figure 16

(a)–(c) Schematics of the $|6S_{1/2}F=4⟩\to |6P_{3/2}F=5⟩$ transition in the $D_2$ line of a ${}^{133}\mathrm{Cs}$ atom. (b) Level structure of the $|F_g=4⟩\to |F_e=5⟩$ transition of ${}^{133}\mathrm{Cs}$ in a magnetic field. (c) Atoms trapped in the edge state $|F_g=4,g_M=4⟩$. (d) Spontaneous emission rates for various magnetic sublevels of the excited state $|6P_{3/2}{F}^{\prime }=5⟩$ of a ${}^{133}\mathrm{Cs}$ atom. Blue dashed, green dash-dotted, and red solid lines show the rates of emission into guided modes, radiation modes, and total decay rates as functions of the atom distance from the fiber center. Different lines of each plot correspond to different Zeeman sublevels of the excited state. The fiber radius was chosen as $a=200\mathrm{nm}$. The wavelength of the $D_2$ line of ${}^{133}\mathrm{Cs}$ is ${\lambda }_0=852\mathrm{nm}$. The refractive indexes of the fiber and the vacuum clad are $n_1=1.45$ and $n_2=1$, respectively. The decay rates are normalized to the free-space decay rate ${\mathrm{\Gamma }}_0$.

Figure 17

(a) Schematics of the electromagnetically induced transparency effect. The transition $|g⟩\to |e⟩$ is coupled to the guided photon mode, and the transition $|s⟩\to |e⟩$ is driven by a classical strong control field with a Rabi frequency ${\mathrm{\Omega }}_c$ that is external to the nanofiber. (b) Calculated transmission spectrum $|t_N{|}^2$ with the characteristic transparency window. The number of atoms is $N=5$, the lattice period is $d={\lambda }_p/4$ (where ${\lambda }_p$ is the wavelength of the probe photon and the control field ${\mathrm{\Omega }}_c=2\mathrm{\Gamma }$), and the guided decay rate ${\gamma }_{1\mathrm{D}}=0.5\gamma$. From [11]

Figure 18

(a) Evanescent field at a waveguide interface. (b) Schematics of the waveguide mode dispersion with different electric field polarization states. (c) Directional coupling of the emission for right and left circularly polarized atomic dipole transitions. (d) Polariton dispersion in a chiral waveguide depending on the coupling asymmetry $\xi$. The calculation was performed following Eq. (59) for a regular array of circularly polarized ${\mathit{d}}_{-}$ atoms and separated with the phase $\phi ={\omega }_{0}d/c=\pi /2$.

Figure 19

Illustration of the protocol proposed by [111] for generation of (a) a GHZ state and (b) a quantum state transfer with an array of superconducting qubits coupled to two waveguides. The bottom panel in (b), which shows the quantum circuit, is described in Appendix pp8.

Figure 20

The frequencies and radiative rates of polariton eigenstates of a finite chain of regularly spaced atoms chirally coupled through a waveguide mode with $\xi =2\times {10}^{-5}$, depending on the corresponding Bloch wave vector. The radiative decay rate, indicated by the color of the dots, is normalized to the decay rate of the individual atom ${\gamma }_{1\mathrm{D}}$. The calculation was performed for $N=400$ atoms and the anti-Bragg spacing $\phi ={\omega }_{0}d/c=\pi /2$. From [78].

Figure 21

Dependence of the photon-photon correlation function $g^{(2)}(0)$ in a chiral waveguide on the number of atoms $N$ and on the ratio of nonradiative and radiative damping rates $\gamma /{\gamma }^{\to }$. The black curve shows the threshold value $N^{*}(\gamma )$ given by Eq. (63) when $g_{N*}^{(2)}(0)=1$. The calculation was performed using Eq. (i7) for light incident at the resonant frequency $ϵ={\omega }_0$ at the two-level atoms ($U\to \infty$).

Figure 22

Experimentally measured (crosses) and calculated (lines) photon-photon correlation functions depending on the number of atoms in the array and optical density. Solid orange and dash-dotted green curves were calculated without and with the experimental uncertainty in the optical density. The parameters of the experiment correspond to $\gamma /{\gamma }^{\to }\approx 130$. From [240].

Figure 23

(a) Collective radiative decay rate of an atomic array depending on the ratio of the array period $a$ to the light wavelength at the atomic resonance ${\lambda }_0$. (b) Light reflection spectra depending on the ratio $a/{\lambda }_0$. The thick black line shows the collective resonance frequency ${\tilde{\omega }}_0$ [Eq. (65)] and the thin red line shows the analytical solution found using Eq. (66), calculated while neglecting the nonradiative decay rate $\mathrm{\Gamma }$ for the state with the zero in-plane wave vector.

Figure 24

(a) Experimental spontaneous emission kinetics of photons in a nanofiber. Time is normalized to the natural atom lifetime (${\tau }_0=1/{\mathrm{\Gamma }}_0=26.24\mathrm{ns}$). Inset: illustrated setup and waveguide-mediated coupling between distant atoms. (b) Dependence of the spontaneous decay rate on the average number of atoms and the corresponding OD for two distant atomic clouds. The blue circles, dotted dark green diamonds, and solid red square correspond to the right atomic cloud, left atomic cloud, and a combination of the two clouds, respectively. The red dashed line is the theoretical prediction. From [290].

Figure 25

Waveguide-coupled collective excitation of an atomic array. Top panel: DLCZ scheme ([69]). One single-flip excitation is created in a chain of $N$ atoms trapped near an optical nanofiber by an external optical pulse that is detuned from the atomic $|g⟩\to |e⟩$ transition. This process is heralded by the detection of a photon in the guided field-1 mode. Later, a read pulse resonant to the $|s⟩\to |e⟩$ atomic transition converts the excitation into a single photon in the guided field-2 mode. Lower panel: characterization of the collective excitation. (a) Normalized cross-correlation function $g_{12}$ between field 1 and field 2. (b) Conditional retrieval efficiency $q_c$ as a function of the probability $p_1$ for detecting a heralding photon in field 1. (c) Suppression of the two-photon component of field 2. From [57].

Figure 26

(a) Nanofiber with a diameter 400 nm overlapped with an ensemble of cold atoms of ${}^{133}\mathrm{Cs}$. The signal pulse is a guided mode field, and the control pulse is external to the nanofiber. (b) Transmitted pulses for different control powers. The reference is measured without atoms. (c) Storage and retrieval of the guided light with an exponentially rising profile with a full width at half maximum of 60 ns. In the absence of a control field, the blue circles and purple triangles give the transmitted pulse without and with atoms. The red squares correspond to the memory sequence, showing leakage and retrieval. The solid black line indicates the control timing. (d) EIT for the guided light. The control field is on resonance with the $|s⟩\to |e⟩$ transition, while the signal is detuned by $\mathrm{\Delta }$ from the $|g⟩\to |e⟩$ atomic resonance. From [100].

Figure 27

(a) Slow guided light. A light delay with respect to a reference pulse (indicated as “input”) is visible. The solid lines correspond to Gaussian fits of the experimental data at different powers, which are indicated near the curves. (b) Storage of light in a nanofiber-trapped ensemble of cold atoms. A pulse duration of $\tau =0.2\mu \mathrm{s}$ contains 0.8 photon on average.The storage time was chosen as $1\mu \mathrm{s}$. The data corresponding to a green area with a slanted pattern represent the reference recorded without atoms. The black solid line is the simulated time trace. The homogeneous magnetic field is $B_{\mathrm{off}}=15\mathrm{G}$. From [266].

Figure 28

(a) Optical micrograph of the eight-qubit metamaterial composed of superconducting transmon qubits capacitively coupled to a coplanar waveguide. Local flux-bias lines provide individual qubit frequency control in the range of 3–8 GHz. (b) Transmission spectra $|S_{21}|$ for different numbers $N$ of resonant qubits with the resonant frequency $f=7.898\mathrm{GHz}$ and low drive powers. With an increasing $N$, the emergence of subradiant states (visible as peaks in transmission) can be observed. The black dotted lines are fits to the expected transmission using a transfer matrix calculation. Black vertical arrows indicate the calculated frequencies of the two brightest subradiant states for $N=8$. From [43].

Figure 29

Bragg reflection from atoms coupled to a one-dimensional waveguide. (a) $N$ atoms trapped near a waveguide and exhibiting radiative decay rates ${\gamma }^{\to }={\gamma }^{\leftarrow }$ into the right- and left-propagating modes, respectively, and $\gamma \sim {\gamma }_0$ into all other modes. (b) Electric field distribution in the transverse plane of a nanofiber for a guided probe field with a quasilinear polarization (indicated by the arrow). Theoretical reflection spectra for a probe quasilinearly polarized (c) along the $y$ direction (symmetric decay rates) and (d) along $x$ direction (asymmetric decay rates). The spectra are given for different distances between the atoms, with values close to the commensurate case. $\mathrm{\Delta }\lambda$ stands for the trap detuning from the resonance, with $d={\lambda }_0/2+\mathrm{\Delta }\lambda /2$. The calculation parameters are $N=2000$, ${\gamma }_{1\mathrm{D}}/{\gamma }_0=0.007$, ${\gamma }^{\to }=2.8{\gamma }_{1\mathrm{D}}$, and ${\gamma }^{\to }/{\gamma }^{\leftarrow }=12$. From [56].

Figure 30

Effect of the chiral character of the waveguide on the Bragg reflection. (a) Measured reflection spectra for $x$ and $y$ quasilinear polarizations, with $\mathrm{\Delta }\lambda =0.2\mathrm{nm}$. (b) Theoretical simulations done with $N=2000$, ${\gamma }_{1\mathrm{D}}/{\gamma }_0=0.007$, ${\gamma }^{\to }/{\gamma }^{\leftarrow }=12$, and $f=0.3$.

Figure 31

Schematics of the separation in Eq. (d2) of the problem with left- and right-propagating photons into two problems with even and odd symmetry.

Figure 32

Illustration of the Bethe Ansatz (d10) for the two-photon states depending on the first and second coordinates $x$ and $y$. Region I corresponds to the incident state. In regions II and ${\mathrm{II}}^{\prime }$ either the first or second photon is scattered on the atom. In regions III and ${\mathrm{III}}^{\prime }$ both photons have scattered.

Figure 33

Illustration of the so-called string between two rapidities, describing the bound two-photon state (d17) in the Bethe Ansatz.

Figure 34

Integration contours for the rapidities $k_1$ and $k_2$ used to calculate the two-photon scattering problem in Eq. (d19).

Figure 35

(a) Diagrammatic representation of the self-energy correction for the excited atomic state. The straight line is the bare Green’s function of the atom, while the wavy line is the photon’s Green’s function. (b) Diagram representing the single-photon reflection from an atom. The bold straight line represents the dressed Green’s function of the atom in the excited state. (c) Diagrammatic representation of the two-photon scattering by an atom.

Figure 36

(a) Dyson-like equation for the dressed exciton Green’s function (thick straight line). The thin straight and wavy lines represent bare exciton and bare photon Green’s functions. (b) Diagram representing single-photon reflection. (c) Series corresponding to the two-photon scattering. The bold dots indicate the exciton-exciton interaction $U$.