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Resonance Fluorescence of a Single Artificial Atom

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Science  12 Feb 2010:
Vol. 327, Issue 5967, pp. 840-843
DOI: 10.1126/science.1181918

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% (37) 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 (1018). 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) (1923) under a strong drive.

Fig. 1

Resonance fluorescence: resonant wave scattering on a single atom. (A) Sketch of a natural atom in open space. The atom resonantly absorbs and reemits photons in a solid angle of 4π. (B) False-colored scanning-electron micrograph of an artificial atom coupled to a 1D transmission line. A loop with four Josephson junctions is inductively coupled to the line. The incident wave (blue arrow) is scattered only backward and forward (red arrows) and can be detected in either direction. The transmitted wave is indicated by a magenta arrow. (C) Spectroscopy of the artificial atom. Shown is the power transmission coefficient |t|2 versus flux bias δΦ and incident microwave frequency ω/2π. When the incident radiation is in resonance with the atom, a dip of |t|2 reveals a dark line. (Inset) Power transmission coefficient |t|2 at δΦ = 0 as a function of incident wave detuning δω/2π from the resonance frequency ω/2π = 10.204 GHz. The maximal power extinction of 94% takes place at the resonance (δΦ = 0).

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 Φb close to a half flux quantum Φ0/2, 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, Ip, flowing clockwise or counterclockwise. In energy eigenbasis, the lowest two levels Embedded Image and Embedded Image are described by the truncated Hamiltonian H = ħωaσz/2, where Embedded Image is the atomic transition frequency and σi (i = x,y,z) are the Pauli matrices. Here, ħε = 2IpδΦ (δΦ ≡ Φb – Φ0/2) is the energy bias controlled by the bias flux, and ħω0 is the anticrossing energy between the two persistent current states. The excitation energies of the third and higher eigenstates are much larger than ħωϕ; 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 I0(x,t) = I0eikxiωt (where ω is the frequency and 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 Isc(x,t) = Isceik|x|–iωt, 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 I(x,t) = (I0eikx + Isceik|x|)eiωt satisfies the 1D wave equation ∂xxIv–2ttI = cδ(x) ∂ttϕ, where the wave phase velocity is Embedded Image (l and c are inductance and capacitance per unit length, respectively), and the dispersion relation is ω = vk.

At the degeneracy point (ε = 0), ωa = ω0, and the dipole interaction of the atom with the electromagnetic wave in the transmission line Hint = –ϕpRe[I0(0,t)]σx is proportional to the dipole moment matrix element ϕp = MIp. 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 = –(ħδωσz + ħΩσx)/2. Here, δω = ω – ω0 is the detuning, and ħΩ = ϕpI0 is the dipole interaction energy. The time-dependent atomic dipole moment can be presented for a negative frequency component as Embedded Image, and the boundary condition for the scattered wave generated because of the atomic polarization satisfies the equation 2Embedded Image, where σ± = (σx ± iσy)/2. Assuming that the relaxation of the atom is caused solely by the quantum noise of the open line, we obtain the relaxation rate Embedded Image (where Embedded Image is the line impedance) (27) and findEmbedded Image(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 Embedded Image. At zero temperature, the simplest form of the Lindblad operator Embedded Image describes energy relaxation (the first term) and the damping of the off-diagonal elements of the density matrix with the dephasing rate Γ2 = Γ1/2 + Γϕ (the second term), where Γϕ is the pure dephasing rates. It is convenient to define reflection and transmission coefficients r and t according to Isc = –rI0 and I0 + Isc = tI0 and, therefore, t = 1 – r. From Eq. 1, we find the stationary solutionEmbedded Image(2)where the maximal reflection amplitude r0 = ηΓ1/2Γ2 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 |r0| = 1. It takes place for η = 1 and Γ2 = Γ1/2, that is, in the absence of pure dephasing, Γϕ = 0. In such a case, the wave scattered forward by the atom is canceled out because of destructive interference with the incident wave (Isc = –I0). 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 ħΩ is multiplied by ω0a.

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 ωa is traced as a function of the flux bias δΦ. By fitting the data, we obtained ω0/2π = 10.204 GHz at δΦ = 0 and the persistent current Ip = 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′ + iα′′ as Embedded Image 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.

Fig. 2

Elastic scattering of the incident microwave. The reflection coefficient r at δΦ = 0 (measured at different powers), being proportonal to the atomic polarizability, exhibits “anomalous dispersion.” (A) Real and imaginary parts of r as a function of the detuning frequency δω/2π from the resonance at ω0 = 10.204 GHz. The driving power W0 is varied from –132 dBm (largest r) to –84 dBm (smallest r) with an increment of 2 dB. (B) Smith charts of the microwave reflection. (Top) Experimentally obtained r is plotted in the coordinates of Re(r) and Im(r) for powers from –132 dBm to –102 dBm with a step of 2 dB. The color coding is the same as in (A). (Bottom) Calculation using Eq. 2 for the same signal powers as in the top panel.

With a weak driving field of Ω2/(Γ1Γ2) << 1 (Fig. 2A, topmost curve), a peak in Re(r) {Re(r) = ηr0[1 + (δω/Γ2)2]–1/2} appears. Fitting by using Eq. 2 with η = 1 gives Γ1 = 6.9 × 107 s–11/2π = 11 MHz) and Γ2 = 4.5 × 107 s–12/2π = 7.2 MHz). From the expression for Γ1, 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 106 s–1 (28). In a case of imperfect coupling (η < 1), the actual Γ1 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 W0, |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 Embedded Image 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 (1923). In the case of a strong driving field (Ω2 >> Embedded Image), the expression for the inelastically scattered power simplifies to Embedded Image, which is independent of the incident power and can be rewritten as Embedded Image: The atom is half populated by the strong drive and spontaneously emits with rate Γ1. Assuming η = 1, the spectral density measured in one of the two directions is expected to be Embedded Image(3)where half-width of the central and side peaks are γc = Γ2 and γs = (Γ1 + Γ2)/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.

Fig. 3

Resonance fluorescence triplet: spectrum of inelastically scattered radiation. (A) Linear frequency spectral density [S = 2πS(ω)] of emission power under a resonant drive with the Rabi frequency of Ω/2π = 57 MHz corresponding to the incident microwave power of W0 = –112 dBm or 6.3 × 10−15 W. Experimental data are shown by the open circles. The red solid curve is the emission calculated from Eq. 3 with no fitting parameters. A schematic of the triplet transitions in the dressed-state picture is presented in the inset: The atomic levels split by Ω because of strong driving, and transitions with frequencies ω0 – Ω, ω0, and ω0 + Ω (marked by colored arrows), give rise to three emission peaks. (B) Resonance fluorescence emission spectrum as a function of the driving power. The dashed white lines indicate the calculated position of the side peaks shifted by ±Ω/2π from the main resonance. The split peak was used for calibration of the field amplitude at the atom.

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

  • This author is on leave from Physical-Technical Institute, Tashkent 100012, Uzbekistan.

  • This author is on leave from Lebedev Physical Institute, Moscow 119991, Russia.

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. 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.”
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