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Spontaneus Emission LETTER https://doi.org/10.1038/s41586-018-0348-z Spontaneous emission of matter waves from a tunable open quantum system Ludwig Krinner1, Michael Stewart1, Arturo Pazmiño1, Joonhyuk Kwon1 & Dominik Schneble1* The decay of an excited atom undergoing spontaneous photon with the prospect of engineering systems with optical long-range emission into the fluctuating quantum-electrodynamic vacuum interactions27. is an emblematic example of the dynamics of an open quantum Here, we realize an atom–optical analogue4–6 of emission in a system. Recent experiments have demonstrated that the gapped one-dimensional photonic-bandgap material, where the singularity photon dispersion in periodic structures, which prevents photons in the mode density near the edge of the continuum leads to particu- in certain frequency ranges from propagating, can give rise to larly strong deviations from Markovian behaviour. In our system of unusual spontaneous-decay behaviour, including the formation of matter-wave emitters, the free tunability of the excitation energy and dissipative bound states1–3. So far, these effects have been restricted decay strength allows for a systematic exploration of the emergence to the optical domain. Here we demonstrate similar behaviour in of non-Markovian dynamics, including partial reversibility and the a system of artificial emitters, realized using ultracold atoms in an formation of a matter-wave bound state that can be directly detected. optical lattice, which decay by emitting matter-wave, rather than Importantly, the close spacing of emitters gives rise to collectively optical, radiation into free space. By controlling vacuum coupling enhanced dynamics beyond the Weisskopf–Wigner model. and the excitation energy, we directly observe exponential and The experimental configuration is shown in Fig. 1a. Using a deep partly reversible non-Markovian dynamics and detect a tunable three-dimensional optical lattice with state selectivity along one axis, bound state that contains evanescent matter waves. Our system we prepare a sparse array of atoms confined to sites that are embedded provides a flexible platform for simulating open-system quantum in a system of isolated tubes acting as one-dimensional waveguides electrodynamics and for studying dissipative many-body physics (see Methods for details). An atom’s internal state (|r〉, red) is coher- with ultracold atoms4–6. ently coupled to a second, unconfined internal state (|b〉, blue) using an The Weisskopf–Wigner model of spontaneous emission7,8, a cen- oscillatory magnetic field. Each site thus acts as a two-level matter-wave tral concept in quantum optics9, describes how an excited atom can emitter, with harmonic-oscillator ground-state occupational levels |g〉 decay to its ground state owing to coupling to zero-point oscillations (empty) and |e〉 (full), supporting both the emission (for |e〉 → |g〉) and of the electromagnetic vacuum. It simultaneously represents one of the absorption (for |g〉 → |e〉) of a |b〉 atom. The excitation energy of the first open quantum systems discussed in the literature and an area the emitter, which is given by the detuning, Δ, of the coherent coupling of research that has recently seen a resurgence of intense theoretical from the atomic resonance, is converted into kinetic energy for atomic efforts10–13. In its usual Markovian formulation, the model makes the motion along the axis of the waveguide. assumption that the decay proceeds on a much slower timescale than One of the main features of each matter-wave emitter is its ability to the optical period, which leads to a memoryless, exponential decay undergo spontaneous decay, as described by the Weisskopf–Wigner of the excited-state amplitude and to an associated Lamb shift of the model. Assuming no lattice potential, the driven atom performs simple transition frequency. For free-space emission, the Markovian approx- Rabi oscillations between two internal states |r and |b . These oscilla- 〉 〉 † imation is generally fulfilled to high accuracy. tions are described by the Hamiltonian Hħˆ =/(2Ω )eitδ rbˆˆ +.hc., where On the other hand, modifications to the mode density of the vacuum Ω denotes the strength and δ the detuning of the coupling from the bare can change the features of spontaneous decay. This was recognized first atomic resonance, ħ is the Planck constant and ‘h.c.’ denotes the in the 1940s14 and again decades later15, during the development of Hermitian conjugate. The tight confinement of just one of the states 16–18 cavity quantum electrodynamics , where the decay can be altered (here, |r〉) strongly couples the atom’s internal and motional degrees of to the extreme point of coherent vacuum Rabi oscillations when the freedom, producing a zero-point energy shift of εˉ00=/ħħωΩ2 , spectrum is restricted to a single mode. Between these two limits lies where ω0 is the harmonic-oscillator frequency in the potential, as well 2 2 the regime of a vacuum with a bounded continuous spectrum, in as a kinetic-energy shift of εk = ħ k /(2m) for the motion of the free which a strong modification of spontaneous decay behaviour occurs |b〉 state at ħk momentum. As a consequence, the detuning and close to the boundary. An example is photonic crystals (also called strength of the coupling are shifted to Δ =+δε()ˉ −/ε ħ and Ωk = 19,20 kk0 photonic-bandgap materials) , where a periodic spatial modula- Ωγk, respectively, with γk = 〈k|ψe〉 denoting the overlap of the tion of the refractive index gives rise to a gapped dispersion relation. external wavefunctions. Integration over all possible momenta k then Δ † For emission close to a bandgap, the Markovian approximation can 6 ˆ = itk ˆ +.. =Ω yields Hħ∑k ggke ∣⟩⟨∣ebk hc, with gk k/2; that is, the stand- no longer be applied, and novel features appear, including oscillatory ard Weisskopf–Wigner Hamiltonian describing spontaneous emission decay dynamics for energies above the band edge and the formation into a vacuum of modes (k, εk) (see Fig. 1b). In contrast to optical 21 of atom–photon bound states below it . Over the past two decades, emission in free space, the dispersion relation εk is quadratic, as in a experiments on spontaneous emission in photonic-bandgap materi- photonic crystal (see Fig. 1c). In such crystals, the emission energy als, including the microwave domain, have observed some of these relative to the edge of the continuum may be adjusted through the 22,23 effects, specifically modified spontaneous emission rates and Lamb crystal’s band structure; in our system, the excitation energy, 24 1 shifts , as well as spectral signatures for non-exponential decay . Very ħΔ ≡ ħΔk=0, itself is tunable, including the case Δ < 0. Importantly, recently, experiments have probed the long-predicted atom–photon the tunability also includes the vacuum coupling, gk, which is set by Ω. 25,26 bound state , using both transmon qubits coupled to corrugated A common scenario considered in the Wigner–Weisskopf model is 3 2 microwave guides and atoms in photonic-crystal waveguides , emission deep into the continuum, such that the decay dynamics is 1 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, USA. *e-mail: [email protected] 26 JULY 2018 | VOL 559 | NATURE | 589 © 2018 Springer Nature Limited. All rights reserved. RESEARCH LETTER a a 1.0 b 1.0 n Populatio Population 0.6 0.6 2 0 24 10 20 30 Evolution time (ms) Δ/(2π) (kHz) 0 c k/k r 2 1 –2 r 0 0.5 b k/k (i) Bare states (ii) Rotating frame (iii) State-selective optical lattice –2 0 Ω H |b〉 k k 6.8 0210 0 30 GHz Δ /(2 ) (kHz) |F = 2, mF = 0〉 G Δ π 87Rb d 2 e 2 Ω 5 S1/2 0 r 0 ) (kHz) k/k –1 0 0 π |r 〉 /(2 0 L –2 G –2 |F = 1, mF = –1〉 1.5 3.7 1.53.7 c k/kr kπ/a Ω/(2π) (kHz) Ω /(2π) (kHz) 0011 Fig. 2 | Markovian regime. a, Time evolution of the lattice population for Hk |e〉 Δ Ω = 2π × 0.74(5) kHz and Δ = 2π × 1.9(3) kHz (red symbols). Each point |bk〉 is the average of at least three measurements and error bars show the standard error of the mean (s.e.m.). The red line is a phenomenological *:( ) exponential decay curve with a fitted rate of 2π × 94(3) Hz and an offset of 0.503(4). The light-grey lines represent the Markovian approximation |g 0 0 (dashed, Γ = 2π × 72(12) Hz) and the exact analytical solution for an 〉 6 PBG isolated emitter (solid). b, Lattice population as a function of Δ for t = 0.4 ms and Ω = 2π × 1.5(1) kHz. The solid line is the Markovian expectation with the overall decay width, Γ, scaled by 0.61(1). c, Detected Fig. 1 | Realization of matter-wave emitters. a, Experimental momentum distribution of |b〉 atoms versus Δ for parameters as in b. The configuration. An occupied site of an optical lattice embedded in dashed line is the single-particle dispersion; data for small positive and for a single-mode matter waveguide acts as an elementary emitter of a negative detunings are outside the Markovian regime. d, Raw time-of- single atom; adjacent empty lattice sites act as absorbers. The bottom flight data for extracting the energy shift at Δ = 2π × 6.0(3) kHz. Colour illustration shows the momentum distribution in the waveguide after scale as in c. e, Measured shifts δΔˉ =−Δ in the regime Ω/Δ < 1 for release and free expansion, where ħk = ħ(2π/λ ) (with a wavelength of L kˉ r z Ωt̴ =.124 and averaged over Δ = 2π × {1, 2, 4, 6} kHz. The data are λ = 790.1 nm) is the recoil momentum.
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