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Observation of Ground-State Quantum Beats in Atomic Spontaneous Emission

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Observation of Ground-State Quantum Beats in Atomic Spontaneous Emission week ending PRL 105, 123602 (2010) PHYSICAL REVIEW LETTERS 17 SEPTEMBER 2010 Observation of Ground-State Quantum Beats in Atomic Spontaneous Emission D. G. Norris,1 L. A. Orozco,1 P. Barberis-Blostein,1,2 and H. J. Carmichael1,3 1Joint Quantum Institute, Department of Physics, University of Maryland and National Institute of Standards and Technology, College Park, Maryland 20742-4111, USA 2Instituto de Investigacio´n en Matema´ticas Aplicadas y en Sistemas, Universidad Nacional Auto´noma de Me´xico, Me´xico, DF 01000, Me´xico 3Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand (Received 4 February 2010; published 15 September 2010) We report ground-state quantum beats in spontaneous emission from a continuously driven atomic ensemble. Beats are visible only in an intensity autocorrelation and evidence spontaneously generated coherence in radiative decay. Our measurement realizes a quantum eraser where a first photon detection prepares a superposition and a second erases the ‘‘which path’’ information in the intermediate state. DOI: 10.1103/PhysRevLett.105.123602 PACS numbers: 42.50.Pq, 03.65.Yz, 32.60.+i Quantum beats are oscillations in the radiation intensity driven variation, the scheme of Zajonc [15] for generating of an ensemble of excited atoms due to interfering emis- ground-state quantum beats on the principle of the quan- sion pathways. They must be counted amongst the earliest tum eraser [16]. predictions of quantum mechanics [1]. So-called ‘‘Type I’’ Beginning with an outline of the scheme, we consider atoms exhibit beats at the separation frequency of two first one idealized atom then a realistic atomic ensemble. excited states which, prepared in a superposition, decay Consider an atom with Zeeman structure in its ground and to a common ground state. The preparation may be excited states interacting with degenerate, orthogonally achieved in a number of ways, e.g., through pulsed optical polarized cavity modes, H and V; a weak magnetic field excitation [2,3] or cascade emission [4]. In all cases the sets the quantization axis in the V direction, and mode V is coherence lasts for the excited state lifetime, unlike weakly and continuously driven [Fig. 1]. The atom is ground-state coherence, which can last long enough to prepared in state jg0i from which it is excited to je0i by be interrogated later and, for this reason, is favored by the V mode. It may return to the ground state emitting a , the field of quantum information. þ,orÀ photon, or any linear combination conserving We consider an unusual situation where ground-state angular momentum. In the given geometry, only þ or À coherence gives rise to a long-lived quantum beat in light couples to the H mode, with the helicity undeter- spontaneous emission. As repeatedly noted [1,5,6], QED mined; thus, if the emitted photon escapes the cavity before predicts no beat in the decay of a ‘‘Type II’’ atom to being reabsorbed, its detection places the atom in the 0 nondegenerate ground states, since they are orthogonal. superposition jc i¼jgÀ1iþjgþ1i [light grey (red) ar- Nevertheless, while beats may be absent from the mean rows in Fig. 1(a)]. The atom is now in the ground state intensity, they can still lie hidden in the fluctuations. In an elegant experiment in the 1950s, Forrester et al. [7] showed this for the (classical) beating of light from a pair of incoherent sources. We proceed in similar spirit; we re- cover a long-lived quantum beat from fluctuations. In contrast to recent experiments [8,9], which aim for deterministic quantum control, the ground-state coherence in our experiment is both prepared and read out by sponta- neous emission. Moreover, our measured beat is different from that seen by Schubert et al. [10], where an interfer- ence occurs in absorption rather than emission. Creation of coherence through spontaneous emission, so-called spontaneously generated coherence, has been discussed in the theoretical literature [11–13] and indirect experimental evidence reported for spontaneous creation FIG. 1 (color online). (a) -excitation of an F ¼ 3 to F0 ¼ 4 of electron spin coherence in charged GaAs quantum dots transition with scattering of a first, light grey (red), and second, [14]. We detect only spontaneous emission, and therefore dark grey (blue) photon, into the H mode; (b) schematic of the make a direct and unambiguous observation of spontane- apparatus, HWP: half-wave plate, PBS: polarizing beam splitter, ously generated coherence. We realize, in a continuously BS: beam splitter, APD: avalanche photodiode. 0031-9007=10=105(12)=123602(4) 123602-1 Ó 2010 The American Physical Society week ending PRL 105, 123602 (2010) PHYSICAL REVIEW LETTERS 17 SEPTEMBER 2010 with angular momentum perpendicular to the magnetic and jaji omitted from the product. This base state evolves field, and thus performs Larmor precession. When subse- under the coherent drive and coupling to the H mode, the quently reexcited by the driven V mode, state coming and going of atoms as they transit the cavity, and 0 iðÞ ÀiðÞ spontaneous emission. From it, at regular sample times, we jc i¼e jeÀ1iþe jeþ1i (1) initiate the collapsed state jc 0i¼b^jc i, where b^ annihilates is reached, with phase ÆðÞ gained through its precession an H-mode photon; thus, the ground-state coherence is in the ground state. From here the atom can decay back to prepared as an entangled state of N atoms, which then jg0i, emitting a second H-mode photon. The probability to evolves in parallel with jc i: do so depends on ðtÞ, giving rise to quantum beats. In summary, there are two paths for scattering a pair of XN0 jc 0i¼j0i jb0ijAi photons into the H mode [neglect for now the paths to i i i¼1 jgÆ2i in Fig. 1(a)]: jg0i!je0i!jgþ1i!jeþ1i!jg0i XN0 XN and jg0i!je0i!jgÀ1i!jeÀ1i!jg0i. The phase þj1i j 0 ij 0ij i þj 00ij i aj bi A ij bi A i ; (2) gained from the ground-state Zeeman shift (precession) i¼1 jÞi¼1 differs along the two paths, which interfere to produce j 0i¼j 0i j 00i¼2j 00i oscillations in the rate of delayed coincidences—i.e., in where bi ai and bi ai at the start of a sample, ð2Þð Þ j 0i j 00i j 0i 2j 00i the correlation function g . Note that after the first after which bi ( bi ) and ai ( ai ) differ due to their photon is detected ‘‘which path’’ information is available, correlation with zero (one) rather than one (two) H-mode since jgþ1i and jgÀ1i are distinguishable in principle. This photons; N0 N is the number of surviving entangled information is erased by the second detection. atoms, i.e., those remaining in the cavity. What we have presented is a single-atom idealization. It The source of the quantum beat is the atomic ground- neglects the presence of more than one atom in the cavity state coherence preserved in the vacuum of the cavity, i.e., (not admissible for an atomic beam), spontaneous emission the first term in Eq. (2). The system ground state—both to noncavity modes, the finite cavity decay rate, and the atoms and cavity—does not decay, and through excita- full complement of magnetic sublevels for the employed tion drives a sustained excited-state oscillation in the atom 0 F ¼ 3 to F ¼ 4 transition. These features are included mirroring Eq. (1): in a full quantum trajectory treatment, including a À ig ig Monte Carlo simulation of an atomic beam [17]. Two gm À1e gmiþ1e i je À1iþ je þ1i; (3) approximations are adopted: (i) the driven mode is treated mi þ ð þ 1ÞÁ mi À iðmi À 1ÞÁ i mi semiclassically (with absorption still taken into account), and (ii) reabsorption of H-mode photons is neglected. The with Á ¼ e À g, where g (e) are ground-state approximations are justified by our moderate dipole cou- (excited-state) Zeeman detunings, gmiÆ1 are Clebsch- pling strength. Gordon coefficients, and is the excited-state linewidth; Consider first an atom prepared in jg0i that has not yet m tracks the state reached by atom i through possible j i j 0i j 00i i suffered a spontaneous emission. Let ai , ai , and ai spontaneous emissions. From Eq. (3), oscillation at the denote unnormalized states expanded, respectively, over the ground-state frequency Æ is passed to the probability ¼ 0 ¼1 ¼ 0 Æ2 g mi , mi , and mi , submanifolds, as indi- amplitudes for emitting a second H-mode photon through cated by the black, light grey (red), and black plus dark grey j 00i j 0 ij 0i j 0i bi and aj bi in Eq. (2); note that bi is a source term (blue) levels of Fig. 1(a). These states correlate with the j 00i driving the equation of motion for bi [the dark grey scattering of zero, one, and two photons into the H mode. (blue) wavy lines plus excitation in Fig. 1(a)]. Only Should the atom undergo a spontaneous emission (to non- the amplitude and phase, but not the frequency of the cavity modes), the expansion manifolds, after the quantum oscillation, will be affected by a detuning of the drive jump, are unchanged for a emission but move one step to ! Æ 1 from the atom. the right or left for a emission (mi mi ). Keeping The probability for emitting a second H-mode photon track of these shifts, the system state is expanded as contains a term proportional to hb00jb00i, summed over all i i XN surviving entangled atoms. It accounts for the scattering of jc i¼j0ij iþj1i j 0ij i A ai A i a first and second photon by the same atom and shows the i¼1 quantum beat introduced above.
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