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Observation of dark state polariton collapses and revivals

D. N. Matsukevich, T. Chaneliere, S. D. Jenkins, S.-Y. Lan, T.A.B. Kennedy, and A. KuzmichSchool of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430

(Dated: February 1, 2008)

By time-dependent variation of a control field, both coherent and single photon states of light arestored in, and retrieved from, a cold atomic gas. The efficiency of retrieval is studied as a function ofthe storage time in an applied magnetic field. A series of collapses and revivals is observed, in verygood agreement with theoretical predictions. The observations are interpreted in terms of the timeevolution of the collective excitation of atomic spin wave and light wave, known as the dark-statepolariton.

PACS numbers: 03.65.Ud,03.67.Mn,42.50.Dv

Atomic ensembles show significant promise as quantummemory elements in a quantum network [1, 2, 3, 4, 5].A “dark-state polariton” is a bosonic-like collective ex-citation of a signal light field and an atomic spin wave[6], whose relative amplitude is governed by a controllaser field. In the context of quantum memories, the darkstate polariton should enable adiabatic transfer of singlequanta between an atomic ensemble and the light field.Seminal “stopped-light” experiments that used laser lightexcitation [7, 8, 9] can be understood in terms of thedark-state polariton concept. In a recent work the stor-age and retrieval of single photons using an atomic en-semble based quantum memory was reported, and thestorage time was conjectured to be limited by inhomoge-neous broadening in a residual magnetic field [10].

We have recently predicted that dark-state polaritonswill undergo collapses and revivals in a uniform dc mag-netic field [11]. During storage, the dark-state polaritonconsists entirely of the collective spin wave excitation.According to the dark-state polariton concept, the re-trieved signal field should exhibit the collapse and re-vivals experienced by the spin wave. The revivals occurat integer multiples of one half the Larmor period, withdynamics that are sensitive to the relative orientation ofthe magnetic field and the light wavevector. The spinwave part of the dark-state polariton involves a partic-ular superposition of atomic hyperfine coherences (seeEq.(1) below), intimately related to the phenomenon ofelectromagnetically-induced transparency (EIT) [12, 13].Revivals of single atom coherences were observed in atominterferometery [14, 15]. Coupled exciton-polariton os-cillations in semiconductor microcavities have also beenreported [16, 17].

The remarkable protocol of Duan, Lukin, Cirac, andZoller (DLCZ) provides a measurement-based scheme forthe creation of atomic spin excitations [3]. In systemswhere EIT is operative, these excitations will in gen-eral contain a dark-state polariton component. The or-thogonal contribution may be regarded as a bright-statepolariton in that it couples dissipatively to the excitedatomic level in the presence of the control field [18].Observation of the retrieved signal field, however, picks

out the dark state polariton part, while the orthogo-nal component is converted into spontaneous emission[11]. A number of previous works reported generationand subsequent retrieval of DLCZ collective excitations[19, 20, 21, 22, 23, 24, 25, 26, 27]. Several of these stud-ies investigated the decoherence of these excitations incold atomic samples [19, 23, 24, 26, 27]. It has beensimilarly conjectured in these works that the decay ofthe coherence was due to spin precession in the ambientmagnetic field. While the observed decoherence timesare consistent with the residual magnetic fields believedto be present, the observation of revivals predicted inRef.[11] would be solid proof that Larmor precession isindeed the current limitation on the quantum memorylifetime. Moreover, controlled revivals could provide avaluable tool for quantum network architectures that in-volve collective atomic memories [3, 4, 5].

With this goal in mind, we report in this Letter obser-vations of collapses and revivals of dark-state polaritonsin agreement with the theoretical predictions [11]. Inour experiment, we employ two different sources for thesignal field, a coherent laser output and a conditionalsource of single photons [3]. The latter is achieved by us-ing a cold atomic cloud of 85Rb at Site A in the off-axisgeometry pioneered by Harris and coworkers [25]. An-other cold atomic cloud of 85Rb at Site B serves as theatomic quantum memory element, as shown in Fig. 1.Sites A and B are physically located in adjacent labora-tories connected by a 100 meter long single-mode opticalfiber. The fiber channel directs the signal field to the op-tically thick atomic ensemble prepared in level |b〉. Theinset in Fig. 1 indicates schematically the structure ofthe three atomic levels involved, |a〉, |b〉 and |c〉, where|a〉; |b〉 correspond to the 5S1/2, Fa = 3, Fb = 2 levelsof 85Rb, and |c〉 represents the 5P1/2, Fc = 3 level asso-ciated with the D1 line at 795 nm. The signal field isresonant with the |b〉 ↔ |c〉 transition and the controlfield with the |a〉 ↔ |c〉 transition.

When the signal field enters the atomic ensemble atSite B, its group velocity is strongly modified by thecontrol field. By switching off the control field withinabout 100 ns, the coupled excitation is converted into a

2

FIG. 1: A schematic diagram illustrates our experimentalsetup. A signal field from either a laser, or a DLCZ sourceof conditional single photons at Site A is carried by a single-mode fiber to an atomic ensemble at Site B, where it is reso-nant on the |b〉 ↔ |c〉 transition. The signal field is stored, fora duration Ts, and subsequently retrieved by time-dependentvariation of a control field resonant between levels |a〉 and |c〉.All the light fields responsible for trapping and cooling, aswell as the quadrupole magnetic field in the MOT, are shutoff during the period of the storage and retrieval process. Anexternally applied magnetic field created by three pairs ofHelmholtz coils (not shown) makes an angle θ with the signalfield wavevector. The inset shows the structure and the ini-tial populations of atomic levels involved. The signal field ismeasured by detectors D2 and D3, while detector D1 is usedin the conditional preparation of single photon states of thesignal field at Site A.

spin wave excitation with a dominant dark state polari-ton component, i.e., the signal field is “stored” [6, 7, 8, 9].An important condition to achieve this storage is a suf-ficiently large optical thickness of the atomic sample,which enables strong spatial compression of the incidentsignal field [5]. In our experiment the measured opticalthickness d ≃ 8. The subsequent evolution of a dark statepolariton in an external magnetic field is predicted to re-veal a series of collapses and revivals whose structure issensitive to the magnitude and orientation θ of the fieldrelative to the signal wavevector [11].

As we deal with an unpolarized atomic ensemble, wemust take into account the Zeeman degeneracy of theatomic levels. Choosing the same circular polarizationsfor both the probe and the control fields allows us to re-tain transparency [10]. For a σ+ polarized signal field, thedark state polariton annihilation operator for wavenum-

ber q is given by [11]

Ψ (q, t) =Ω(t)ak,+ −√

Npg∗∑

m RmSb ma m (q, t)

√

|Ω(t)|2 + Np |g|2 ∑

m |Rm|2(1)

where Ω(t) is the control field Rabi frequency, g the cou-pling coefficient for the signal transition, m is the mag-netic quantum number, Rm = CFb 1 Fc

m 1 m+1/CFa 1 Fc

m 1 m+1 is aratio of Clebsch-Gordan coefficients, N is the number ofatoms, p = 1/(2Fb+1), ak,+ is the field annihilation oper-ator for the mode of wavevector k = q+ω0/c and positivehelicity, ω0 is the Bohr frequency of the |b〉 ↔ |c〉 tran-

sition, Sb ma m(q, t) ≡ 1/

√Np

∑

µ σ(µ)b m, a m(t) exp(−i(qzµ −

∆(t − zµ/c))) is a collective spin wave operator, where

σ(µ)b m, a m(0) = |b, m〉µ〈a, m| is a hyperfine coherence op-

erator for atom µ = 1, ..N , zµ is the position of atomµ, and ∆ is the hyperfine splitting of the ground state.When Rm is finite for all m, the atomic configuration sup-ports EIT, but when one or more Rm is infinite, there isan unconnected lambda configuration, EIT is not possi-ble and dark state polaritons do not exist. Specifically,the excited state |c, m + 1〉 is not coupled by the controlfield to a state in the ground level |a〉. An atom in thestate |b, m〉 would absorb the signal field as if no controlfield were present.

The signal is stored in the form of spin wave ex-citations associated with the dark state polaritons ∼∑

m RmSb ma m(q) for some range of q’s. During the stor-

age phase, and in the presence of the magnetic field B,the atomic hyperfine coherences rotate according to thetransformation

Sbmam(q, t) =

Fb∑

m1=−Fb

Fa∑

m2=−Fa

D(b)†m1m(t)D(a)

m,m2(t)Sbm1

am2(q, 0),(2)

where D(s)m,m′(t) ≡ 〈s, m| exp(−igsΩ · Ft)|s, m′〉 is the ro-

tation matrix element for hyperfine level s, F is the to-tal angular momentum operator, Ω ≡ µBB/~, µB is theBohr magneton, ga and gb are the Lande g factors for lev-els |a〉 and |b〉 of 85Rb and, ignoring the nuclear magneticmoment, ga = −gb. This rotation dynamically changesthe dark state polariton population during storage.

The measured signal retrieved after a given storagetime Ts is determined by the remaining dark state polari-ton population. Stated differently, only the linear combi-nation of hyperfine coherences∼

∑

m RmSb ma m(q, Ts) con-

tributes to the retrieved signal. We calculate the numberof dark state polariton excitations as a function of Ts us-ing Eqs.(1) and (2), 〈N(Ts)〉 =

∑

q〈Ψ†(q, Ts)Ψ(q, Ts)〉,and find

〈N(Ts)〉〈N(0)〉

=

∣

∣

∣

∣

∣

∑

m1m2

Rm1Rm2

∑

m |Rm|2D(b)†

m2m1(Ts)D(a)

m1m2(Ts)

∣

∣

∣

∣

∣

2

.(3)

In Fig. 2, panels (f) through (j), we show the retrievalefficiency for various orientations of a magnetic field of

3

magnitude 0.47 G, corresponding to the Larmor periodof 4.6 µs. With the approximation ga = −gb it is clearthat the system undergoes a revival to the initial stateafter one Larmor period (2π/|gbΩ|), and thus the signalretrieval efficiency equals the zero storage time value. De-pending on the orientation of the magnetic field, a partialrevival at half the Larmor period is also observed. For amagnetic field oriented along the z axis (Fig. 2(f)), thepolariton dynamics is relatively simple. Each hyperfinecoherence Sb m

a m merely picks up a phase factor that oscil-lates at frequency 2m|gbΩ|, thus returning the system toits initial state at half the Larmor period. In this case,the partial revival is actually a full revival. On the otherhand, for θ = π/2 (Fig. 2(j)), a rotation through halfthe Larmor period causes the coherence transformationSb m

a m → −Sb −ma −m, and as a result, the retrieval efficiency

is reduced to (∑

m RmR−m/∑

m R2m)2. The substruc-

ture within a half Larmor period is associated with in-terference of different hyperfine coherences contributingto the dark-state polariton [11].

To test these predictions, we apply a uniform dc mag-netic field of magnitude 0.5 ± 0.05 G to the atomic en-semble using three pairs of Helmholtz coils. In our firstset of measurements, 150 ns long coherent laser pulsescontaining on average ≃ 5 photons serve as the signalfield. The outputs of the single-photon detectors D2 andD3 are fed into two “Stop” inputs of a time-interval an-alyzer which records the arrival times with a 2 ns timeresolution. The electronic pulses from the detectors aregated for the period [t0, t0 + Tg], with Tg = 240 ns, cen-tered on the time determined by the control laser pulseduring the retrieval stage. Counts recorded outside thegating period are therefore removed from the analysis.The recorded data allows us to determine the numberof photoelectric events N2 + N3, where Ni is the totalnumber of counts in the i-th channel(i = 1, 2, 3).

By measuring the retrieved field for different storagetimes and orientations of the magnetic field, we obtainthe collapse and revival signals shown in Fig. 2, (a)through (e). As expected, we observe revivals at in-teger multiples of the Larmor period. In addition, wesee partial revivals at odd multiples of half the Larmorperiod, except in the vicinity of θ = π/4. The mea-sured substructures within a single revival period are ingood agreement with the theory (cf., insets of Fig. 2,(e) and (j)). We attribute the overall damping of therevival signal in the experimental data to the magneticfield gradients. The evident decrease of this dampingwhile θ is varied from 0 to π/2 suggests that such gradi-ents are predominantly along the direction of the signalfield wavevector. Similarly, we attribute the additionalbroadening of the revival peaks at longer times to in-homogeneous magnetic fields, possibly ac fields, not in-cluded in the theoretical description. We are pursuingadditional investigations to determine the temporal andspatial characteristics of the residual magnetic fields.

0

0.05

0.1

(a) - 0

0

0.05

0.1

(b) - /8

0

0.05

0.1

(c) - /4

Re

trie

va

l e

ffic

ien

cy

0

0.05

0.1

(d) - 3 /8

0 5 100

0.05

0.1

(e) - /2

Storage time, s

0 2 4

0.005

0

0.5

1

(f)

0

0.5

1

(g)

0

0.5

1

(h)

0

0.5

1

(i)

0 5 100

0.5

1

Storage time, s

(j)

0 2 4

0.05

FIG. 2: Panels (a)-(e) show the ratio of the number of photo-electric detection events for the retrieved and incident signalfields for various orientations, θ = 0, π/8, π/4, 3π/8, π/2, ofthe applied magnetic field, and as a function of storage time.The incident signal field is a weak coherent laser pulse. In allcases the control pulse is switched off at Ts = 0. We observe aseries of collapses and revivals at multiples of the half Larmorperiod of 2.3 µs. The observed damping over several Larmorperiods is likely caused by residual magnetic field gradients.The inset in Panel (e) demonstrates the observed substruc-ture within the first Larmor period. Panels (f) through (j)are corresponding theoretical plots of the dark-state polari-ton number calculated using Eq.(2).

Theory predicts that both the collapse and the revivaltimes (TC and TR, respectively) scale inversely with themagnetic field [11]. In Fig. 3 the theoretical predictionTC ≈ 0.082TR (solid line) is compared with the experi-mentally measured values. We find very good agreementexcept for the lowest value of magnetic field B = 0.2 Gwhich may be explained by the presence of residual mag-

4

0

0.2

0.4

0.6

0 2 4 6

TC, ms

TR, ms

FIG. 3: Diamonds show the measured collapse time TC of thefirst revival at half the Larmor period as a function of themeasured revival time TR, for magnetic field values of 0.8,0.6, 0.4, and 0.2 G, respectively, and for fixed orientation θ =π/2. The line shows the corresponding theoretical predictionTC ≈ 0.082TR from Eq.(2).

netic field gradients.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

Coi

ncid

ence

s ra

te (

s-1)

Storage time, s

(a) - 0

0 1 2 3 4 50

0.2

0.4

0.6

0.8

Storage time, s

(b) - /2

FIG. 4: Squares show the measured rate of coincidence detec-tions between D1 and D2,3, Nsi = N12 + N13 as a functionof the storage time. Diamonds show the measured level ofrandom coincidences NR. The ratio of squares to diamondsgives gsi. Uncertainties are based on the statistics of the pho-toelectric counting events.

In the measurements presented above, classical, co-herent laser light was used as the signal field [29]. Wehave also investigated the revival dynamics of our atomicmemory with the signal field in a single photon state, asshown in Fig. 1. The procedure for production of a singlephoton state of the signal field at Site A is conditioned onthe detection of an idler photon by D1 (see Refs.[10, 26]for further details). If photoelectric detections in differ-ent channels 1, 2 or 1, 3 happen within the same gatingperiod, they contribute to the corresponding coincidencecounts between D1 and Dj, N1j , j = 2, 3. We evaluatethe α parameter of Grangier et al. [28], given by the ra-tio of various photoelectric detection probabilities whichare measured by the set of detectors D1,D2 and D3. Foran ideal single-photon state α = 0, whereas for coher-ent fields α = 1. We have experimentally determinedα = 0.51 ± 0.06, without any correction for background

or dark counts.The normalized intensity cross-correlation function

gsi ≡ (N12 + N13)/NR may be employed as a measureof non-classical signal-idler field correlations [29, 30], asdiscussed in detail in Ref.[19]. Here NR ≡ N1 ·(N2+N3)·Rrep is the level of random coincidences, where Rrep isthe repetition rate of the experimental protocol. The val-ues of gsi are obtained by the ratio of the upper and lowertraces in Fig. 4. The measurements presented there givevalues of gsi well in excess of two at the revival times,suggesting the dark-state polaritons have a non-classicalnature. One could further evaluate self-correlations forthe idler field gii, and for the signal field gss, and con-firm that the Cauchy-Schwarz inequality g2

si ≤ gssgii

is indeed violated [19, 29, 30]. We have measured, byadding a beamsplitter and an additional detector, thevalue gii = 1.42 ± 0.03. When the signal field is storedand retrieved 500 ns later, we find that both gss ≤ 2[10]. While the total number of recorded coincidencesbetween detectors D2 and D3 is not high enough to eval-uate gss for the revived polariton, it is also expected tobe less than two, leading to a substantial violation of theCauchy-Schwarz inequality.

In summary, we have demonstrated revivals of dark-state polaritons in a quantum memory element based ona cold atomic ensemble. The dynamical manipulationand control of collective matter-field excitations, at thelevel of single quanta, is encouraging for further develop-ments and applications in quantum information science.

We thank M. S. Chapman for discussions and E.T.Neumann for experimental assistance. This work wassupported by NASA, National Science Foundation, Officeof Naval Research, Research Corporation, Alfred P. SloanFoundation, and Cullen-Peck Chair.

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