## Superconducting Quantum Optics

The coherence properties of superconducting circuits enable them to be developed as qubits in quantum information processing applications. **Astafiev et al.** (p. 840) now show that these macroscopic superconducting devices also behave as artificial atoms and can exhibit quantum optical effects. The ability to fabricate and integrate these superconducting devices in electronic circuitry may help toward developing a fully controlled quantum optics system on a chip.

## Abstract

An atom in open space can be detected by means of resonant absorption and reemission of electromagnetic waves, known as resonance fluorescence, which is a fundamental phenomenon of quantum optics. We report on the observation of scattering of propagating waves by a single artificial atom. The behavior of the artificial atom, a superconducting macroscopic two-level system, is in a quantitative agreement with the predictions of quantum optics for a pointlike scatterer interacting with the electromagnetic field in one-dimensional open space. The strong atom-field interaction as revealed in a high degree of extinction of propagating waves will allow applications of controllable artificial atoms in quantum optics and photonics.

A single atom interacting with electromagnetic modes of free space is a fundamental example of an open quantum system (Fig. 1A) (*1*). The interaction between the atom (or molecule, quantum dot, etc.) and a resonant electromagnetic field is particularly important for quantum electronics and quantum information processing. In three-dimensional (3D) space, however, although perfect coupling (with 100% extinction of transmitted power) is theoretically feasible (*2*), experimentally achieved extinction has not exceeded 12% (*3*–*7*) because of spatial mode mismatch between incident and scattered waves. This problem can be avoided by an efficient coupling of the atom to the continuum of electromagnetic modes confined in a 1D transmission line (Fig. 1B), as proposed in (*8*, *9*). Here, we demonstrate extinction of 94% on an artificial atom coupled to the open 1D transmission line. The situation with the atom interacting with freely propagating waves is qualitatively different from that of the atom interacting with a single-cavity mode; the latter has been used to demonstrate a series of cavity quantum electrodynamics (QED) phenomena (*10*–*18*). Moreover, in open space the atom directly reveals such phenomena known from quantum optics as anomalous dispersion and strongly nonlinear behavior in elastic (Rayleigh) scattering near the atomic resonance (*1*). Furthermore, spectrum of inelastically scattered radiation is observed and exhibits the resonance fluorescence triplet (the Mollow triplet) (*19*–*23*) under a strong drive.

Our artificial atom is a macroscopic superconducting loop, interrupted by Josephson junctions (Fig. 1B) [identical to a flux qubit (*24*)] and threaded by a bias flux Φ_{b} close to a half flux quantum Φ_{0}/2, and shares a segment with the transmission line (*25*), which results in a loop-line mutual inductance *M* mainly due to kinetic inductance of the shared segment (*26*). The two lowest eigenstates of the atom are naturally expressed via superpositions of two states with persistent current, *I*_{p}, flowing clockwise or counterclockwise. In energy eigenbasis, the lowest two levels and are described by the truncated Hamiltonian *H* = *ħ*ω_{a}σ* _{z}*/2, where is the atomic transition frequency and σ

*(*

_{i}*i*=

*x*,

*y*,

*z*) are the Pauli matrices. Here,

*ħ*ε = 2

*I*

_{p}δΦ (δΦ ≡ Φ

_{b}– Φ

_{0}/2) is the energy bias controlled by the bias flux, and

*ħ*ω

_{0}is the anticrossing energy between the two persistent current states. The excitation energies of the third and higher eigenstates are much larger than

*ħ*ω

_{ϕ}; therefore, they can be neglected in our analysis.

We considered a dipole interaction of the atom with a field of an electromagnetic 1D wave. In the semiclassical approach of quantum optics, the external field of the incident wave *I*_{0}(*x*,*t*) = *I*_{0}*e ^{ikx}*

^{–}

^{i}^{ω}

*(where ω is the frequency and*

^{t}*k*is the wavenumber) induces the atomic polarization. The atom with a characteristic loop size of ~10 μm (which is negligibly small as compared with the wavelength λ ~ 1 cm) placed at

*x*= 0 generates waves

*I*

_{sc}(

*x*,

*t*) =

*I*

_{sc}

*e*

^{ik}^{|}

^{x}^{|–}

^{i}^{ω}

*, propagating in both directions (forward and backward). The current oscillating in the loop under the external drive induces an effective magnetic flux ϕ (playing a role of atomic polarization). The net wave*

^{t}*I*(

*x*,

*t*) = (

*I*

_{0}

*e*+

^{ikx}*I*

_{sc}e^{ik}^{|}

^{x}^{|})

*e*

^{–}

^{i}^{ω}

*satisfies the 1D wave equation ∂*

^{t}*–*

_{xx}I*v*

^{–2}∂

*=*

_{tt}I*c*δ(

*x*) ∂

*ϕ, where the wave phase velocity is (*

_{tt}*l*and

*c*are inductance and capacitance per unit length, respectively), and the dispersion relation is ω =

*vk*.

At the degeneracy point (ε = 0), ω_{a} = ω_{0}, and the dipole interaction of the atom with the electromagnetic wave in the transmission line *H*_{int} = –ϕ_{p}Re[*I*_{0}(0,*t*)]σ* _{x}* is proportional to the dipole moment matrix element ϕ

_{p}=

*MI*

_{p}. In the rotating wave approximation, the standard form of the Hamiltonian of a two-level atom interacting with the nearly resonant external field is

*H*= –(

*ħ*δωσ

*+*

_{z}*ħ*Ωσ

*)/2. Here, δω = ω – ω*

_{x}_{0}is the detuning, and

*ħ*Ω = ϕ

_{p}

*I*

_{0}is the dipole interaction energy. The time-dependent atomic dipole moment can be presented for a negative frequency component as , and the boundary condition for the scattered wave generated because of the atomic polarization satisfies the equation 2, where σ

^{±}= (σ

*±*

_{x}*i*σ

*)/2. Assuming that the relaxation of the atom is caused solely by the quantum noise of the open line, we obtain the relaxation rate (where is the line impedance) (*

_{y}*27*) and find(1)This expression indicates that the atomic dissipation into the line reveals itself even in elastic scattering.

The atom coupled to the open line is described by the density matrix ρ, which satisfies the master equation . At zero temperature, the simplest form of the Lindblad operator describes energy relaxation (the first term) and the damping of the off-diagonal elements of the density matrix with the dephasing rate Γ_{2} = Γ_{1}/2 + Γ_{ϕ} (the second term), where Γ_{ϕ} is the pure dephasing rates. It is convenient to define reflection and transmission coefficients *r* and *t* according to *I*_{sc} = –*rI*_{0} and *I*_{0} + *I*_{sc} = *tI*_{0} and, therefore, *t* = 1 – *r*. From Eq. 1, we find the stationary solution(2)where the maximal reflection amplitude *r*_{0} = ηΓ_{1}/2Γ_{2} at δω = 0. Here, η presents dimensionless coupling efficiency to the line field, including nonradiative relaxation. The maximal possible power extinction (1 – |*t*|^{2}) can reach 100% when |*r*_{0}| = 1. It takes place for η = 1 and Γ_{2} = Γ_{1}/2, that is, in the absence of pure dephasing, Γ_{ϕ} = 0. In such a case, the wave scattered forward by the atom is canceled out because of destructive interference with the incident wave (*I*_{sc} = –*I*_{0}). Although Eq. 2 is obtained for the degeneracy point (ε = 0), it remains valid in the general case of ε ≠ 0 if the dipole interaction energy *ħ*Ω is multiplied by ω_{0}/ω_{a}.

The excitation energy of the atom was revealed by means of transmission spectroscopy (Fig. 1C). Owing to the broadband characteristics of the transmission line, we swept the frequency of the incident microwave in a wide range and monitored the transmission. As shown in the inset of Fig. 1C, the resonance is detected as a sharp dip in the power transmission coefficient |*t*|^{2}. At resonance, the power extinction reaches its maximal value of 94%, which suggests that the system is relatively well isolated from other degrees of freedom in the surrounding solid-state environment and behaves as a nearly isolated atom in open space, coupled only to the electromagnetic fields in the space. The resonance frequency ω_{a} is traced as a function of the flux bias δΦ. By fitting the data, we obtained ω_{0}/2π = 10.204 GHz at δΦ = 0 and the persistent current *I*_{p} = 195 nA.

The elastic response of the artificial atom shows typical anomalous dispersion. Figure 2A represents the reflection coefficient derived from the transmission according to *r* = 1 – *t* and obtained at δΦ = 0. Similarly to the case of a natural atom, we can define the polarizability α = α′ + *i*α′′ as and, therefore, α ∝ *ir*. In the vicinity of the resonance, Re(*r*) (∝α′′) is positive and reaches maximum at the resonance, whereas Im(*r*) (∝–α′) changes the sign from positive to negative.

With a weak driving field of Ω^{2}/(Γ_{1}Γ_{2}) << 1 (Fig. 2A, topmost curve), a peak in Re(*r*) {Re(*r*) = η*r*_{0}[1 + (δω/Γ_{2})^{2}]^{–1/2}} appears. Fitting by using Eq. 2 with η = 1 gives Γ_{1} = 6.9 × 10^{7} s^{–1} (Γ_{1}/2π = 11 MHz) and Γ_{2} = 4.5 × 10^{7} s^{–1} (Γ_{2}/2π = 7.2 MHz). From the expression for Γ_{1}, the mutual inductance between the atom and the transmission line is estimated to be *M* = 12 pH. Although our assumption of η = 1 has not been checked experimentally, it may be reasonable because (i) all the line current should effectively interact with the atom and (ii) the possible relaxation without emission measured for isolated atoms is weak, being typically less than 10^{6} s^{–1} (*28*). In a case of imperfect coupling (η < 1), the actual Γ_{1} could be slightly higher.

The nonlinearity of the atom manifests in the saturation of the atom excitation. With increasing the power of the incident microwave *W*_{0}, |*r*| monotonically decreases, and in the Smith chart (Fig. 2B) the shape of the trajectory changes from a large circle to a small ellipse. As a single two-level system, the atom is saturated at larger powers and can have large reflectance only for the weak driving case. Again, the nearly perfect agreement between the calculations and the measurements supports our model of a two-level atom coupled to a single 1D mode. Any artificial medium built of such “atoms” (*29*) will also have a strongly nonlinear susceptibility.

So far, we have investigated elastic Rayleigh scattering in which the incident and the scattered waves have the same frequency. However, the rest of the power is scattered inelastically and can be observed in the power spectrum. The spectrum was measured at the degeneracy point (δΦ = 0) under a resonant drive with the power corresponding to Ω/2π ≈ 57 MHz (Fig. 3A). It manifests the resonance fluorescence triplet, also known as the Mollow triplet (*19*–*23*). In the case of a strong driving field (Ω^{2} >> ), the expression for the inelastically scattered power simplifies to , which is independent of the incident power and can be rewritten as : The atom is half populated by the strong drive and spontaneously emits with rate Γ_{1}. Assuming η = 1, the spectral density measured in one of the two directions is expected to be
(3)where half-width of the central and side peaks are γ_{c} = Γ_{2} and γ_{s} = (Γ_{1} + Γ_{2})/2, respectively. The red curve in Fig. 3A is drawn by using Eq. 3 without any fitting parameters. The good agreement with the theory indicates the high collection efficiency of the emitted photons, which is due to the 1D confinement of the mode. The shift of the side peaks, ±Ω, from the main resonance depends on the driving power. The intensity plot in Fig. 3B shows how the resonance fluorescence emission depends on the driving power. The dashed white lines mark the calculated position of the side peaks as a function of the driving power, showing good agreement with the experiment.

The demonstrated resonance wave scattering from a macroscopic “artificial atom” in an open transmission line indicates that such superconducting quantum devices can be used as building blocks for controllable, quantum-coherent macroscopic artificial structures, in which a plethora of effects can be realized from quantum optics of atomic systems.

## Supporting Online Material

## References and Notes

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- This work was supported by the Core Research for Evolutional Science and Technology, Japan Science and Technology Agency and the Ministry of Education, Culture, Sports, Science and Technology kakenhi “Quantum Cybernetics.”