Observation of Ground-State Quantum Beats in Atomic Spontaneous Emission

week ending PRL 105, 123602 (2010) PHYSICAL REVIEW LETTERS 17 SEPTEMBER 2010

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 Investigacioń en Matema´ticas Aplicadas y en Sistemas, Universidad Nacional Autońoma 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,or photon, or any linear combination conserving We consider an unusual situation where ground-state angular momentum. In the given geometry, only þ or 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 i¼jg1iþjgþ1i [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 spontaneous emission. Moreover, our measured beat is different from that seen by Schubert et al. [10], where an interference 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 iðÞ spontaneous emission. From it, at regular sample times, we jc i¼e je1iþe jeþ1i (1) initiate the collapsed state jc 0i¼b^jc i, where b^ annihilates is reached, with phase ðÞ 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 jg2i in Fig. 1(a)]: jg0i!je0i!jgþ1i!jeþ1i!jg0i XN0 XN and jg0i!je0i!jg1i!je1i!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 ji¼1 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 jg1i 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 ig ig Monte Carlo simulation of an atomic beam [17]. Two gm 1e gmiþ1e i je 1iþ je þ1i; (3) approximations are adopted: (i) the driven mode is treated mi þ ð þ 1Þ mi iðmi 1Þ i mi semiclassically (with absorption still taken into account), and (ii) reabsorption of H-mode photons is neglected. The with ¼ e g, where g (e) are ground-state approximations are justified by our moderate dipole cou- (excited-state) Zeeman detunings, gmi1 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 is passed to the probability ¼ 0 ¼1 ¼ 0 2 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 ! 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. There is also a term pffiffiffi XN 1 XN computed from the norm of ja0 ijb0iþja0ijb0 i, i j, þ 2j2i j 0ij 0 ij i þj 00ij i j i i j ai aj A ij ai A i ; which adds probability amplitudes for ‘‘a first photon ¼1 2 ¼1 i j i from atom i and a second from atom j,’’ and ‘‘a first photon where j0i, j1i, and j2i denote zero, one, and two photons in from atom j and a second from atom i.’’ If the scattered Eb Ea þ the H mode, N is the (time-varying) number of interacting fields were classical, i at time t and i at time t , one jEaEb þ EaEbj2 atoms, jAi¼ja1ija2i ...jaNi, jAii is the state jAi with jaii would have the intensity j i i j , where the inter- j i j i j i 2ReðEaEbEbEaÞ omitted from the product, and A ij is the state A with ai ference j j i i disallows assignment of the first

123602-2 week ending PRL 105, 123602 (2010) PHYSICAL REVIEW LETTERS 17 SEPTEMBER 2010 detection (superscript b) to the intensity of either source 4.9 MHz. The V mode is populated with 2–3 photons, on (similarly the second). An assignment may be made in average, when no atoms are present, and the atomic flux principle, however, since one and only one atom changes corresponds to N 0:2 effective maximally coupled its ground state when the photon is detected. The change atoms; most of the time there are no atoms well coupled could be seen if one looked, thus providing ‘‘which-path’’ to the mode [19]. The oscillation has low visibility and sits information. As we do not look, the two quantum paths, atop a raised Gaussian background whose correlation time ‘‘atom i then j’’ and ‘‘atom j then i,’’ interfere—after a is given by the transit time of an atom ( 2:5 s). The second detection both atoms have changed state, so the dominant beat for small N is that arising from the emission ‘‘which-path’’ information is erased. This interference also of two photons by the same atom [Fig. 4(c)]. Figure 2(b) creates a quantum beat. displays the measured correlation function with N larger A detailed description of the apparatus is given in [18]. by a factor of 10 but otherwise similar conditions. The beat The main elements appear in Fig. 1(b). We probe a dilute visibility is improved. The background is also removed, 85 beam of cold Rb atoms (speed 22 m=s) transiting a evidence that there is now an equal two-atom quantum beat 2.2 mm Fabry-Perot cavity with TEM00 mode waist [Fig. 4(d)]. The beat frequency is reduced to 4.7 MHz, an 56 m. The finesse is 11 000, and the cavity and atomic indirect effect, we believe, of increased absorption by the 6 1 decay rates, ð; Þ=2 ¼ð3:2; 6Þ10 s , are both additional atomic flux, which reduces the V-mode photon larger than the maximum dipole coupling, g=2 ¼ number—by a factor of 2—and thus also the light shifts. 0 0 1:5 MHz on the F ¼ 3, m ¼ 0 to F ¼ 4, m ¼ 0 transi- A detailed study of light shifts is planned for separate tion; we operate in the intermediate regime of cavity QED, 2 presentation. with single-atom cooperativity C1 ¼ g = ¼ 0:12 and 2 2 Figure 3 illustrates the change in the observed beat when saturation photon number n0 ¼ =3g ¼ 5:3. Splitting the polarization presented to the detector is not taken of the polarization modes by birefringence is less than orthogonal to the polarization of the drive but is allowed 200 kHz. Atoms enter the cavity optically pumped to the to rotate by a few degrees. The rotation is controlled by ¼ 3 ¼ 0 F , m ground state, from which the driven cavity changing the angle of the HWP placed between the cavity mode excites resonant transitions. and the PBS [Fig. 1(b)]. This mixes a small amount of drive A Glan-Thompson polarizer and zero-order half-wave light with the scattered light. With increasing fraction of plate (HWP) placed before the cavity linearly polarize drive light, the beat is eventually dominated by a homo- the drive with an extinction ratio that can reach better dyne term [Fig. 4(e)] arising from the correlation of a 5 105 than . A second HWP after the cavity aligns the photon scattered into the H mode with a photon from the polarization to a calcite Wollaston prism (PBS) to separate drive; thus, as in the two-atom case, interfering time orders the H from the V-mode. The former is divided between yield a quantum beat. This beat oscillates at half the two avalanche photodiodes at a second beam splitter. Each frequency and allows the correlation function to dip below detector output goes directly to a time stamp card, which one [e.g., Fig. 4(b)]. Generally, some drive light is coupled records a continuous stream of detection times with 164 ps into the H mode through a small birefringence of the cavity resolution. The typical duration of a time series is three mirrors. In Fig. 3, the evident asymmetry with respect to minutes or less. angle is possibly due to imperfect alignment of the mag- Figure 2(a) displays a measured correlation function at a netic field with respect to the light polarization. magnetic field of 5 G, yielding a beat frequency of Quantum trajectory simulations agree well with the

2.0 measurements. Figure 4(a) displays a computed correlation a function overlaying the data of Fig. 2(b). A mean velocity 1.8 22 ms1 ) of ﬁts the decay of coherence well. Other parame-

τ 1.6 (

(2) ters, such as the ﬁdelity of the optical pumping, atomic

g 1.4 1.2 1.0 2.0 1.8 b 1.6 τ ()

(2) 1.4 g 1.2 1.0 -4 -3 -2 -1 0 1 2 3 4 τ(µs) FIG. 3 (color online). Evolution of the quantum beat as V-mode light is mixed with the H mode. Upper and lower extremes FIG. 2 (color online). Intensity correlation function of the H correspond to approximately 6 times more detector counts from mode for 2–3 photons in the V mode with no atoms present and a the V mode than spontaneous emission. The parameters are 0.6 5 G magnetic ﬁeld: (a) N ¼ 0:2 and (b) N ¼ 2:0. photons in the V mode, a magnetic ﬁeld of 4 G, and N ¼ 1.

123602-3 week ending PRL 105, 123602 (2010) PHYSICAL REVIEW LETTERS 17 SEPTEMBER 2010

2.0 a 1.8 We have observed ground-state quantum beats in the 1.6 spontaneous emission from a continuously driven atomic τ ( ) 1.4 ensemble. Contrasting deterministic manipulations [8,9], (2) g 1.2 we demonstrate the spontaneous creation and readout of 1.0 ground-state coherence, where in the spirit of Forrester 0.8 2.0 et al. [7], we retrieve a hidden beat from the ﬂuctuations. b 1.8 Our theoretical treatment, which is in good agreement 1.6

τ with the measurements, decomposes the signal into a one- ( ) 1.4 (2) atom beat, a two-atom beat (interference of emission time g 1.2 order), and a homodyne beat due to interference with a 1.0 drive photon mixed through birefringenece. We plan to 0.2 c study a new class of quantum feedback in this system [21]. I 1 0.1 0.0 Work supported by NSF, CONACYT, Me´xico, and the 0.8 Marsden Fund of RSNZ. We thank I. Deutsch for stimu- I 2 0.4 d 0.0 lating discussions. 0.3 e I 3 0.0 -0.3 -3 -2 -1 0 1 2 3 τ(µ s) [1] G. Breit, Rev. Mod. Phys. 5, 91 (1933). 17 FIG. 4 (color online). Calculated intensity correlation function [2] E. B. Aleksandrov, Opt. Spectrosc. , 522 (1964) [3] J. N. Dodd, W. J. Sandle, and D. Zissermann, Proc. Phys. with the birefringence background set at 1% (a) and 10% (b) of 92 the H-mode photon number; (c)–(e) the three interference Soc. London , 497 (1967). [4] A. Aspect, J. Dalibard, P. Grangier, and G. Roger, Opt. terms contributing in (b); curve (a) is plotted against the data 49 of Fig. 2(b). Parameters are N ¼ 4, 4 photons in the V mode with Commun. , 429 (1984). no atoms present, and an atomic beam tilt of 1.3 degrees. [5] W. W. Chow, M. O. Scully, and J. O. Stoner Jr, Phys. Rev. A 11, 1380 (1975). [6] R. M. Herman, H. Grotch, R. Kornblith, and J. H. Eberly, beam tilt, and background from birefringence or elsewhere Phys. Rev. A 11, 1389 (1975). are more difﬁcult to accurately determine (no background [7] A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, 99 correction is made to the data). Note that with a back- Phys. Rev. , 1691 (1955). ground light amplitude from birefringence, the post- [8] T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, Science 0 317, 488 (2007). detection state is jc iþjc i, and carries noise from [9] B. Weber, H. P. Specht, T. Mu¨ller, J. Bochmann, M. the atomic beam and spontaneous emission (absorption on Mu¨cke, D. L. Moehring, and G. Rempe, Phys. Rev. Lett. the V mode). Plausible parameters are used for the plots of 102, 030501 (2009). Fig. 4, with the quality of the ﬁt primarily determined by [10] M. Schubert, I. Siemers, R. Blatt, W. Neuhauser, and P.E. the atomic beam density, which controls the relative size of Toschek, Phys. Rev. A 52, 2994 (1995). the one- and two-atom quantum beats, and the strength of [11] J. Javanainen, Europhys. Lett. 17, 407 (1992). the drive, which controls the level of spontaneous emis- [12] A. K. Patnaik and G. S. Agarwal, Phys. Rev. A 59, 3015 sion. For the parameters of Fig. 4(a), an atom passing near (1999). the cavity axis (within half a mode waist) typically under- [13] S. E. Economou, R. B. Liu, L. J. Sham, and D. G. Steel, 71 goes 10 spontaneous emissions to noncavity modes Phys. Rev. B , 195327 (2005). during its transit, yet the coherence, merely passed be- [14] M. V. Gurudev Dutt, J. Cheng, B. Li, X. Xu, X. Li, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, S. E. tween different m multiplets, is preserved. In contrast, i Economou, R. B. Liu, and L. J. Sham, Phys. Rev. Lett. spontaneous emission decoheres an excited-state beat. 94, 227403 (2005). Figure 4(b) displays an example of a correlation function [15] A. G. Zajonc, Phys. Lett. A 96, 61 (1983). with mixed drive light [Fig. 3], together with its breakdown [16] M. O. Scully and K. Dru¨hl, Phys. Rev. A 25, 2208 (1982). into three contributing quantum beats: one-atom interfer- [17] L. Horvath and H. J. Carmichael, Phys. Rev. A 76, 043821 ence (frame c), two-atom interference (frame d), and (2007). homodyne interference (frame e). The pieces lie in one- [18] D. G. Norris, E. J. Cahoon, and L. A. Orozco, Phys. Rev. A to-one correspondence with known terms in the intensity 80, 043830 (2009). correlation function for a source comprised of many scat- [19] H. J. Carmichael and B. C. Sanders, Phys. Rev. A 60, 2497 terers and a coherent background, e.g., Eq. (11) of [20], (1999). [20] H. J. Carmichael, P. Drummond, P. Meystre, and D. F. where the correlation function is the sum of a single-atom 11 ð2Þð Þ j ð1Þð Þj2 Walls, J. Phys. A , L121 (1978). term, gA , two-atom term, gA , and a homodyne [21] P. Barberis-Blostein, D. G. Norris, L. A. Orozco, and H. J. Re½ ð1Þð Þ 12 term, gA . Carmichael, New J. Phys. , 023002 (2010).

123602-4