Colloquium: Strongly interacting photons in one-dimensional continuum

Photon-photon scattering in vacuum is extremely weak. However, strong effective interactions between single photons can be realized by employing strong light-matter coupling. These interactions are a fundamental building block for quantum optics, bringing many-body physics to the photonic world and providing important resources for quantum photonic devices and for optical metrology. This Colloquium reviews the physics of strongly interacting photons in one-dimensional systems with no optical confinement along the propagation direction. It focuses on two recently demonstrated experimental realizations: superconducting qubits coupled to open transmission lines and interacting Rydberg atoms in a cold gas. Advancements in the theoretical understanding of these systems are presented in complementary formalisms and compared to experimental results. The experimental achievements are summarized alongside a description of the quantum optical effects and quantum devices emerging from them.

Figure 1

(a) A superconducting qubit (marked with an arrow) embedded in a one-dimensional transmission line waveguide. A cavity is formed by two capacitive gaps in the middle conductor. (b) Without the cavity. (c) Probe light focused through a dense atomic cloud, exciting a single Rydberg atom within the blockade sphere (dashed line).

Figure 2

Two configurations of emitter-photon coupling in an open waveguide: (a) side coupled and (b) direct coupled.

Figure 3

Linearization of the dispersion relation of the waveguide mode. The full dispersion relation ${\omega }_k$ is shown by the dashed curve. The linearized relations are denoted by the two solid lines around the wave vectors $\pm k_0$, corresponding to a frequency ${\omega }_0$.

Figure 4

The reflection coefficient (solid line) and transmission coefficient (dashed line) of a single photon propagating in a 1D continuum with a side-coupled lossless two-level emitter.

Figure 5

Reflection of single photons from a coherent-state input by one atom near a microtoroidal cavity. (a) Transmission ${\mathcal{T}}_0(t)$ and (b) reflection ${\mathcal{R}}_0(t)$ of the probe field after averaging over atom transit events with a selection criterion that the sum of the counts over a time interval of $4\mu \mathrm{s}$ is equal to or greater than a threshold count $C_{\mathrm{th}}$ shown in the legend of (a). (c), (d) The second-order (intensity) correlation functions $g_{T,R}^{(2)}(\tau )$ for the transmitted and reflected fields. The dip in $g_R^{(2)}(\tau )$ around $\tau =0$ indicates anticorrelation in the detection of photons, which is a signature of antibunching and indicates that the field is dominated by single photons. Solid lines are a theoretical calculation described in [10]. Adapted from [10].

Figure 6

Correlations between scattered photons in the lossless resonant case. The calculated two-photon scattering coefficients $|t_2{|}^2$ and $|r_2{|}^2$ are plotted vs the scaled separation $\mathrm{\Gamma }(x_1-x_2)/v_g$. (a) Bunching of transmitted photons indicated by the peak in $|t_2{|}^2$ at $x_1=x_2$. (b) Antibunching of reflected photons indicated by $|r_2{|}^2=0$ at $x_1=x_2$.

Figure 7

Second-order correlation function of the transmitted field at various emitter-photon coupling strengths for a coherent-state input with $\overline{n}\le 1$. The couplings are (left) $\mathrm{\Gamma }/\gamma =0.2$, (middle) $\mathrm{\Gamma }/\gamma =1.6$, and (right) $\mathrm{\Gamma }/\gamma =3$. Here $v_g{k}_0={\omega }_e=10\gamma$ and ${\mathrm{\Delta }}_k/\gamma =1$. Note the different vertical scales. The second-order correlation shows antibunching and bunching at different coupling strengths.

Figure 8

(a) $\mathrm{\Lambda }$-type and (b) ladder-type three-level emitters.

Figure 9

(a) Prefect reflection, (b), (c) electromagnetically induced transparency at the Raman resonance $\mathrm{\Delta }={\mathrm{\Delta }}_c$ for weak control field, and (d) Autler-Townes splitting for stronger control fields. The splitting between the Autler-Townes doublet is ${\mathrm{\Omega }}_c$. The parameters are ${\mathrm{\Delta }}_c/\mathrm{\Gamma }=-1/2$, $\gamma /\mathrm{\Gamma }=1/4$, and ${\gamma }_s/\mathrm{\Gamma }=1/40$.

Figure 10

One-photon transmission amplitudes $t$ vs the probe detuning $\delta {\omega }_p\equiv \mathrm{\Delta }$ for a superconducting three-level qubit coupled to a microwave waveguide. (a), (b) The experimentally measured real and imaginary parts of the transmission amplitudes for various control field amplitudes ${\mathrm{\Omega }}_c$, specified in (b). The curves show typical dispersion for the Autler-Townes splitting. $\mathrm{Re}(t)$ near $\delta {\omega }_p=0$ approaches unity with increasing ${\mathrm{\Omega }}_c$. (c), (d) Corresponding theoretical curves. Adapted from [1].

Figure 11

Spectroscopy of an artificial atom coupled to an open 1D transmission line. The color map shows the power transmission $T=|t{|}^2$ vs flux bias $\delta \mathrm{\Phi }$ and incident microwave frequency $\omega$. When the incident radiation is on resonance with the emitter, a dip in $T$ reveals a dark line. Inset: $|t{|}^2$ at $\delta \mathrm{\Phi }=0$ as a function of the probe detuning $\delta \omega \equiv \mathrm{\Delta }$ from the resonance frequency ${\omega }_e/2\pi =10.204\mathrm{GHz}$. The maximal power extinction of $1-T=94\%$ takes place on resonance ($\delta \omega =0$). Adapted from [12].

Figure 12

Second-order correlation function $g^{(2)}$ vs the time separation $\tau$ between the photons for a thermal state, a coherent state, and the scattered states generated by the artificial two-level emitter. (a) $g^{(2)}(\tau )$ of a thermal state and a coherent state. (b) $g^{(2)}(\tau )$ of the resonant transmitted microwaves for four different incident powers. (c) $g^{(2)}(\tau )$ of a resonant reflected field for two different incident powers. $g^{(2)}(0)$ does not reach zero because of well-understood experimental imperfections, including the bandwidth of the measurement system. (d) $g^{(2)}(\tau )$ of a resonant reflected field for different measurement bandwidths (BW), illustrating how decreasing bandwidth decreases the depth of the dip. The solid curves in (a)–(d) are theory curves. Adapted from [91].

Figure 13

Photon-mediated interaction between distant emitters in superconducting circuits. Transmon qubits ($\mathrm{length}\approx 300\mu \mathrm{m}$) acting as emitters are coupled to an open 1D transmission line. Adapted from [109].

Figure 14

(a), (b) Photon-mediated exchange interaction between two distant superconducting qubits in an open transmission line. Power spectral density (PSD) of the resonance fluorescence of two qubits in resonance at $d\sim 3\lambda /4$, driven at the indicated Rabi rates ${\mathrm{\Omega }}_R$. PSD falls with decreasing ${\mathrm{\Omega }}_R$. At drive rates much lower than the relaxation rate ${\mathrm{\Omega }}_R/2\pi \le 5\mathrm{MHz}$, the observed double-peak structure reveals the effective exchange interaction between the two qubits. Adapted from [199].

Figure 15

(a) Schematics of a typical Rydberg-polaritons setup: a cloud of ultracold alkali atoms is held between two confocal lenses. The outgoing probe photons are measured using single-photon detectors. (b) Normalized second-order correlation $g^{(2)}$ of the outgoing probe vs the time separation $\tau$ between the photons, showing the transition from antibunching in the dissipative regime ($\mathrm{\Delta }=0$) to bunching in the dispersive regime ($\mathrm{\Delta }=4.6\gamma$). The group delay in the medium was $0.25\mu \mathrm{s}$. Points are experimental data, lines are numerical simulations. (c) Conditional phase shift obtained from a tomographic reconstruction of the outgoing two-photon wave function. (d) Signature of the two-photon bound state: The interaction potential $U(r)$ is approximated by a well of width $2r_{\mathrm{B}}$ (bottom, dash-dotted line). The resulting bound state, observed in the shape of the measured $g^{(2)}$ (top, circles, where time is converted to distance via $x_2-x_1=\tau v_{\mathrm{EIT}}$), conforms to that calculated using Eq. (43) (top, solid line). The initial state $\psi (r,0)=1$ (bottom, dashed line) is a superposition of the bound state (bottom, thick line) and the manifold of scattering states (bottom, thin line). Adapted from [150], and [59].