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Coherent control of the waveforms of recoillessc-ray photonsFarit Vagizov1,2,3, Vladimir Antonov1,4,5, Y. V. Radeonychev1,4,5, R. N. Shakhmuratov2,3 & Olga Kocharovskaya1

The concepts and ideas of coherent, nonlinear and quantum opticshave been extended to photon energies in the range of 10–100 kilo-electronvolts, corresponding to soft c-ray radiation (the term usedwhen the radiation is produced in nuclear transitions) or, equiva-lently, hard X-ray radiation (the term used when the radiation isproduced by electron motion). The recent experimental achievementsin this energy range include the demonstration of parametric down-conversion in the Langevin regime1, electromagnetically inducedtransparency in a cavity2, the collective Lamb shift3, vacuum-assistedgeneration of atomic coherences4 and single-photon revival in nuc-lear absorbing multilayer structures5. Also, realization of single-photon coherent storage6 and stimulated Raman adiabatic passage7

were recently proposed in this regime. More related work is discussedin a recent review8. However, the number of tools for the coherentmanipulation of interactions between c-ray photons and nuclearensembles remains limited. Here we suggest and implement an effi-cient method to control the waveforms of c-ray photons coherently.In particular, we demonstrate the conversion of individual recoil-less c-ray photons into a coherent, ultrashort pulse train and into adouble pulse. Our method is based on the resonant interaction ofc-ray photons with an ensemble of nuclei with a resonant transitionfrequency that is periodically modulated in time. The frequency mod-ulation, which is achieved by a uniform vibration of the resonantabsorber, owing to the Doppler effect, renders resonant absorptionand dispersion both time dependent, allowing us to shape the wave-forms of the incident c-ray photons. We expect that this techniquewill lead to advances in the emerging fields of coherent and quan-tum c-ray photon optics, providing a basis for the realization ofc-ray-photon/nuclear-ensemble interfaces and quantum interfer-ence effects at nuclear c-ray transitions.

Quantum optics deals with the interaction of photons with the quan-tum transitions of matter, providing the basis for the new, fast-growingfields of quantum cryptography, quantum communication and quan-tum information (see, for example, ref. 9 and references therein). So far,experiments in these fields have been implemented with either micro-wave or optical photons interacting with atomic transitions, and typ-ically require cryogenic temperatures.

The c-ray photons (for which we use the shorthand ‘c-photons’) inthe 10–100-keV energy range have important potential advantages overthe microwave and optical photons for applications in quantum cryp-tography, communication and information, for a number of reasons.First, with such photons there is nearly 100% detector efficiency andlow noise (almost no false detections). Second, there is extremely high(potentially sub-angstrom and, at the moment, nanometre10) spatialresolution. This removes what is now a fundamental limitation on thecreation of nanoscale photonic circuits set by the diffraction limit, whichis ,1mm for optical photons. Third, they have a high penetration depthin many materials that are opaque to optical photons. Fourth, with suchphotons the capacity of information channels is potentially much higher.

Nuclear transitions in solids in this energy range have a number ofadvantages compared with electronic optical transitions. They may pre-sent perfect two-level systems with very narrow linewidths (for example1.1 MHz for the 14.4-keV transition in 57Fe) owing to the Mossbauereffect (even at room temperature) and deep shielding from the envir-onment. The quality factor for the 14.4-keV transition is Q 5 3 3 1012,as per definition: Q is the ratio of resonance energy to bandwidth.Coherent multiple scattering of a c-photon in an optically thick, resonantabsorber enables potential applications for single c-photon/nuclear-ensemble interfaces owing to the collective enhancement of the photon–nucleus interaction. We note that in the optical range such single-photon/atomic-ensemble interfaces have been developed recently as a powerfulalternative to cavity-enhanced single-photon/single-atom interfaces11.Moreover, natural sources of single c-photons exist in the form of low-activity radioactive sources, for which discrete emissions well separatedin time (compared with the natural lifetime of the level) ensure the single-photon nature of the radiation. Also, the cascade scheme of the decay,characteristic of some radioactive Mossbauer sources (for example 57Co;Fig. 1a), ensures heralding of the emission of the second c-photon bydetection of the first c-photon in the cascade, which ‘starts the clock’for measuring the waveform of the second photon12.

The methods for coherently controlling temporal waveforms of singlephotons (well developed in optics owing to the use of high-finesse cav-ities and bright, coherent sources of radiation) are limited in c-optics,although the emerging state-of-the-art facilities, such as self-seededhard X-ray free-electron lasers13 and nearly 100% reflecting mirrors14,will provide an impetus for the development of such techniques.

We propose an efficient method to shape c-photons in an opticallythick, uniformly vibrating, resonant, recoilless absorber, and perform a

1Department of Physics and Astronomy and Institute for Quantum Science and Engineering, Texas A&M University, College Station, Texas 77843-4242, USA. 2Kazan Federal University, Kazan 420008,Russia. 3Kazan Physical-Technical Institute of the Russian Academyof Sciences, Kazan 420029,Russia. 4Institute of AppliedPhysics of the Russian Academyof Sciences, Nizhny Novgorod 603950,Russia.5N.I. Lobachevsky State University, Nizhny Novgorod 603950, Russia.

57Fe

57Co

|c⟩ 12 ns

|d⟩ 272 d

Electron capture

14.4 keV

122 keV

57Feb

a

|a⟩

|b⟩ 141 ns |2⟩

|1⟩

Figure 1 | Energy scheme of emission and resonant absorption of a14.4-keV photon. a, The 57Co nuclide in the state | dæ decays with half-life ofT1/2 5 272 d, producing an 57Fe nucleus in the state | cæ, which decays with alifetime of Tc 5 12 ns to the state | bæ, emitting a 122-keV c-photon. The state| bæ decays with lifetime of Tr 5 141 ns to the ground state, | aæ, emitting a14.4-keV photon. b, The 14.4-keV transition between the ground state, | 1æ, andthe first excited state, | 2æ, of the absorbing 57Fe nucleus with the linewidth1.13 MHz.

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proof-of-principle experiment with a 57Co source of single c-photonsand a 57Fe absorber (Fig. 1).

The basic idea is that the vibration of an absorber leads to periodicmodulation of the resonant 1j i« 2j inuclear transition frequency (Fig. 1b)with respect to the frequency of the incident photons owing to theDoppler effect. As a result, the quasi-monochromatic incident radi-ation is transformed during its propagation into a spectral comb at thepoint where it exits the absorber. The relative amplitudes and phases ofthe produced spectral components are defined by the vibration ampli-tude and frequency, the detuning of the central frequency of the sourcefrom the resonant frequency of the absorber, the source and absorberlinewidths, and the absorber optical depth. By changing these para-meters, we can adjust the amplitudes and phases of the output spectralcomponents and produce waveforms different from the incident c-photons. In particular, the creation of a constant phase difference betweenequidistant neighbouring spectral components of comparable amplitudewould result in a sequence of pronounced, bandwidth-limited pulses.

The discussed method to produce ultrashort c-ray pulses essentiallyconstitutes a c-ray implementation of a general approach based on theultrafast variation of the parameters of a resonant quantum transition(see ref. 15 and references therein). Whereas in the case of atomic tran-sitions the modulation of the transition frequency was produced by thequasi-static Stark effect in a strong laser field, in the case of nuclear tran-sitions it is due to the Doppler effect. This resonant parametric technique,unlike the traditional techniques of nonlinear optics, does not require ahigh intensity of incoming radiation and, hence, can be implementedeven in the single-photon regime.

The frequency modulation of recoilless emission by mechanical vibra-tion has been intensively studied since the late 1970s in applications tospectroscopic measurements and the calibration of mechanical dis-placements (see refs 16, 17 and references therein). Recently, there wasa suggestion to produce ultrashort c-pulses using a vibrating source,emitting frequency-modulated radiation, with a far-off-resonance absorbercompensating the phase mismatch of the incident c-radiation spectralcomponents18. Although sound in principle, that proposal is difficultto implement experimentally, because weak non-resonant interactionrequires an absorber so thick that off-resonance losses are undesirablylarge. Use of a resonant absorber resolves that problem. Also, in prac-tice it is easier to realize uniform oscillations of a thin, resonant absor-ber than of a radiative source.

The scheme of our experimental set-up is given in Fig. 2. Obser-vation of a single-photon waveform, shaped by the vibrating, resonantabsorber, is based on the time-delayed coincidence counts of the sequen-tially emitted 122-keV and 14.4-keV c-photons (Fig. 1a). Registering a122-keV photon defines the initial moment, t0, of the formation of the14.4-keV state and the beginning of photon emission (equation (2); allnumbered equations are in Methods). We measure the delay, t 5 t 2 t0,in the detection of the 14.4-keV photon at the exit of the absorber witha 5-ns bin width, repeating this procedure many times. As a result, weobtain the coincidence count rate, as a function of delay, N(t), corres-ponding to the single-photon waveform Y(t)j j2 (refs 5, 12, 16–21).These experiments are essentially inspired by the recent demonstra-tion of a new technique for optical single-photon shaping22.

Transmission through the absorber vibrating with frequency V resultsin the reshaping of the exponential waveform of the incident photon.The output shape depends on the vibration phase, q0~Vt0, at time t0,but because emission of 122-keV photon is a stochastic process, q0

is random. To reveal the coherent effect of the absorber vibration onthe 14.4-keV-photon waveform, we collect only the coincidence countsof 122-keV photons and those 14.4-keV photons whose 122-keV pho-tons were detected within short intervals, dt, around the times t(n)

0 ~(q0z2pn)=V, where n 5 0, 1, 2, … (Fig. 2), corresponding to the samevibration phase, q0.

This technique allowed us to demonstrate the temporal compres-sion of an individualc-photon into a decaying train of ultrashort pulsesthat were an order of magnitude shorter than the decay time of the

excited nuclear state (Fig. 3b). In the following, we show experimentalresults that were obtained under conditions leading to pulses with max-imum peak height for the given absorber thickness.

The physical origin of the waveform transformation is revealed byspectral analysis. The spectrum (Fig. 3a, inset) of the quasi-monochromaticincoming photon (Fig. 3b, red dashed curve) in the laboratory referenceframe is ‘seen’ by the vibrating nuclei of the absorber as a comb of equi-distant spectral components separated by the vibration frequency, V(Fig. 3a and equations (4) and (5)). Under the chosen modulation index,p 5 1.8 (see Methods), there are seven major spectral components. Itcan be seen that the upper-frequency sidebands are phase-matchedwith a phase difference of p between each neighbouring pair (brownsquares), whereas three lower-frequency sidebands are phase matchedwith a phase difference of 0 (green triangles). Taking into account theequivalence of 2pn-shifted phases of the ‘22’ and ‘12’ sidebands(Fig. 3a), we can match five major spectral components, eliminatingeither the ‘21’ or the ‘11’ sideband by tuning it to the absorber res-onance (by properly choosing the constant velocity of the source withrespect to the absorber). This leads to the waveforms shown by the redline in Fig. 3b (elimination of ‘21’) and blue line in the inset of Fig. 3b(elimination of ‘11’). In terms of the shift of the produced pulse trainrelative to the front edge of the photon, these waveforms differ fromeach other by half the vibration cycle, p/V. Owing to constructiveinterference of the generated sidebands, the peak values of the countrate rise above the exponential waveform of the incident photon. Evenhigher peak heights will be achieved if, instead of suppression of thephase-mismatched sideband through resonant absorption, its phase isshifted by p through resonant dispersion. Such a phase shift requiresdetuning from the exact resonance and a several-fold greater resonantoptical depth; without strong off-resonance photoelectric absorption,it can be realized in a stainless-steel film highly enriched with 57Fe.

We also demonstrate the possibility of a single photon splitting intotwo pulses (Fig. 3c), for the first time realizing a time-bin qubit in thec-ray frequency range23. The time interval between the pulses and the ratioof their amplitudes can be controlled by the vibration phase (Methods).

A direct count rate of the whole flow of the 14.4-keV photons versustime starting from t(n)

0 (without binding to the 122-keV photons) cor-responds to averaging the 14.4-keV-photon count rate, N(t 2 t0), overthe time t0 (equation (12)). In this case, the random flow of c-photons

G D1

z

R sin( t)

E

z′

Vem

Stop

Gate-in

122 keV14.4 keV

D2

Start

TAC

PHATransducer

t0 t0 + dt t

A

Ω

Figure 2 | Experimental set-up for c-photon waveform control. The emitter(E) is a radioactive 57Co:Rh foil. It can be moved by the Mossbauer transducerwith tunable constant velocity, Vem, relative to the absorber (A), which is25-mm-thick stainless-steel foil with a natural abundance (,2%) of 57Fe(optical depth, TM 5 5.18). It is glued on the polyvinylidene fluoridepiezo-transducer that transforms the sinusoidal signal from theradio-frequency generator (G) into the uniform vibration (R, vibrationamplitude). The time-to-amplitude converter (TAC), operating in thecoincidence mode, receives the ‘start’ pulse, produced by detector D1 onregistration of a 122-keV photon, and the ‘stop’ pulse, produced by detector D2on registration of a 14.4-keV photon transmitted by the vibrating absorber. Thedelay, t, between the stop and start pulses is measured by the pulse-heightanalyser (PHA). At times t(n)

0 , matching the chosen vibration phase,Vt(n)

0 ~q0z2pn, the generator also produces ‘gate-in’ signals, allowing TACand PHA to measure the t provided that the 122-keV photon is received onlywithin the short interval between t(n)

0 and t(n)0 zdt.

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transforms into a train of nanosecond pulses (Fig. 4). The pulse repe-tition period is equal to 2p/V, and the pulse duration is defined by theinverse of the product of the number of phase-matched components

(proportional to the modulation index) and the modulation frequency.An increase in the vibration frequency at a fixed vibration amplituderesults in a proportional shortening of both the pulse duration and therepetition period, owing to an increase in the separation between thesidebands (compare Fig. 4a with Fig. 4b and Fig. 4c with Fig. 4d). Anincrease in the vibration amplitude at a fixed vibration frequency resultsin a pulse shortening without change in the repetition period, owing toan increase in the number of sidebands (compare Fig. 4a with Fig. 4cand Fig. 4b with Fig. 4d). However, the short, intense pulses are accom-panied by the additional weak pulses appearing as a result of the phase-mismatched components.

The maximum potentially available vibration frequency is 1 GHzwith rather small modulation index, p , 2. A much higher modulationindex (p . 10) may be achieved with moderate modulation frequen-cies (10–100 MHz), but in this case additional care should be taken inphase-matching the produced sidebands (using, for example, a chain ofproperly tuned resonantly absorbing foils or external phase synchron-ization by the diffraction gratings). Thus, the shortest pulses, whichpotentially could be produced by that technique, are limited to 100 ps(unless other methods providing higher modulation frequency andhigher modulation index of the nuclear transition are used).

The described set-up is at present the only known tabletop source ofultrashort c-ray pulses. The parameters of the pulses, including theshape and number of pulses, the repetition rate (potentially variable inthe megahertz–gigahertz range) and the duration (potentially 100 ns–100 ps), can be widely controlled. This allows for a variety of time-resolvedexperiments (including dynamic X-ray diffraction) directly in local labs,although such a tabletop set-up cannot compete, in terms of pulse dura-tion and photon flux, with modern synchrotrons24. However, it has twounique features leading to applications that are unavailable with otherexisting sources of ultrashort X-ray pulses.

First, the produced pulses have exclusive spectral–temporal prop-erties. At the moment, they are the only nearly transform-limited ultra-short pulses that can be produced in this frequency range. They can beused to generate frequency combs with a fixed frequency interval andphase-matched spectral components, providing a unique combinationof wide spectral coverage and narrow spectral width for each component.

100200300400500

194 ns 98 ns

18 ns

a b

dc

21 ns 32 ns

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100 200 300 400 100 200 300 400

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1.0

0.8

1.4

1.2

1.0

0.8

Cou

nt r

ate

(a.u

.)

Time (ns)

0 0

Figure 4 | Pulses of Mossbauer radiation. Count rate of the 14.4-keV photonsversus time in the case where detector D1 is removed and the ‘gate-in’ signal,matched with the zero phase of vibration, triggers TAC and PHA tomeasure time elapsed until detector D2 produces a ‘stop’ signal on registrationof 14.4-keV photon. Repeating this procedure many times results in a countrate of the 14.4-keV photons versus time corresponding to the intensity of14.4-keV radiation. The constant velocity, Vem, of the emitter towards theabsorber tunes the ‘21’ sideband to the absorber resonance (Fig. 3a).a, V/2p5 5.16 MHz, Vem 5 0.44 mm s21, R 5 0.25 A, (p 5 1.8);b, V/2p5 10.2 MHz, Vem 5 0.87 mm s21, R 5 0.25 A (p 5 1.8);c, V/2p5 5.16 MHz, Vem 5 0.44 mm s21, R 5 0.27 A (p 5 2);d, V/2p5 10.2 MHz, Vem 5 0.87 mm s21, R 5 0.27 A (p 5 2). Theexperimental data are blue dots and the error bars are estimated from photoncounting statistics as above. The red solid line is plotted according toequations (13) and (14).

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+2–2

+3–3

+1–1

N(τ

)/N

(0)

τ (ns)

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0.20.40.60.8

Figure 3 | Shaping of c-photon. a, Spectrum of the incident 14.4-keV-photonwaveform in the reference frame of the vibrating absorber and in the laboratoryreference frame (inset), calculated according to equation (5) (blue solid andred dashed lines are spectral amplitudes and phases, respectively). Centralphases of the lower-frequency, principle-frequency and upper-frequencycomponents are labelled by triangles, circle and squares, respectively. Thevibration frequency, phase and amplitude (modulation index) are respectivelyV/2p5 10.2 MHz, q0~0 and R 5 0.25 A (p 5 1.8). a.u., arbitrary units. b, Thenormalized coincidence count rate of 14.4-keV photons, N(t)/N(0), passedthrough the vibrating absorber (c-photon waveform). The ‘21’ sideband istuned to the absorber spectral line by moving the emitter towards the absorberwith velocity Vem 5 0.88 mm s21. The experimental data are blue dots, thebackground is subtracted and the error bars are estimated from photoncounting statistics as

ffiffiffiffiffiffiffiffiffiN(t)

p=N(0). Red solid and dashed curves are plotted

according to equation (11) and equation (3), respectively. Upper inset,sinusoidal vibration of the absorber and the acquisition interval, Vdt 5p/2,limited by the equipment used. Lower inset, the single-photon waveform in thecase where the ‘11’ sideband is tuned to the absorber resonance. c, The14.4-keV-photon waveform in the case where V/2p5 2.6 MHz, q0~{0:3p,R 5 0.9 A (p 5 6.58), the emitter is moved towards the absorber with velocityVem 5 1.13 mm s21 (tuning the ‘25’ sideband to the absorber resonance),and the acquisition interval is p/8 (upper inset). Lower inset, the 14.4-keV-photon waveform under the same conditions, except that Vem 5 21.13 mm s21

(tuning the ‘15’ sideband to the absorber resonance).

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Such pulses are promising for the realization of frequency comb spec-troscopy (broadband, high-spectral-resolution spectroscopy) of nucleiin solids in the 10–100-keV energy range as well as time-resolved Mos-sbauer spectroscopy. They can also be used for efficient preparation ofa coherent superposition of nuclear states, and for realization of quan-tum interference effects such as modulation-induced transparency25

and electromagnetically induced transparency26,27. The last effect hasnumerous applications in the optical frequency range, such as theimplementation of resonantly enhanced nonlinearities (including sin-gle-photon nonlinearities), slow and stored light, lasing without inver-sion, and so on (see ref. 28 and references therein). Similar applicationsmay be anticipated in the c-ray range.

Second, at the moment our tabletop set-up is the only source ofsingle-photon, ultrashort pulses with efficiently controllable waveforms.The splitting of the single photon into two pulses constitutes the firstrealization of a time-bin qubit in this range of frequencies. The pro-duction of single-photon, ultrashort-pulse trains with pulse durationmuch shorter than the natural lifetime of the emitting nuclear level andwith controllable waveforms provides a unique opportunity for therealization of quantum memories29,30 and other nuclear-ensemble/c-photon interfaces, opening the way to applications in quantum com-munication and information.

METHODS SUMMARYThe transformation of a recoilless, spontaneously emitted c-photon on its pro-pagation through a uniformly vibrating, resonant Mossbauer absorber is describedsemiclassically by numerically and analytically solving the Maxwell–Bloch equa-tions with a harmonically modulated nuclear transition frequency in the rotating-wave and linear approximations. Modification of the shape of the c-photon at theexit of the absorber is studied as a function of the parameters of the system, such asthe frequency and amplitude of modulation, the detuning of the central frequencyof the source from the resonance frequency of the absorber, and the vibration phaseat the time of the c-photon arriving at the absorber entrance.

We numerically plot waveforms for different parameter values to demonstratehow to produce a variety of waveforms from a single c-photon (including a decay-ing train of ultrashort pulses, a double pulse and a single triangular pulse). Theregime of formation of a train of c-ray pulses with pulse-to-pulse coherence and aduration an order of magnitude less than the natural relaxation time of the res-onant nuclear transition is described analytically within a simplified model.

Online Content Any additional Methods, Extended Data display items and SourceData are available in the online version of the paper; references unique to thesesections appear only in the online paper.

Received 25 August 2013; accepted 13 January 2014.

Published online 16 March 2014.

1. Shwartz, S. et al. X-ray parametric down-conversion in the Langevin regime. Phys.Rev. Lett. 109, 013602 (2012).

2. Rohlsberger, R., Wille, H.-C., Schlage, K. & Sahoo, B. Electromagnetically inducedtransparency with resonant nuclei in a cavity. Nature 482, 199–203 (2012).

3. Rohlsberger, R., Schlage,K., Sahoo,B., Couet, S.& Ruffer, R.Collective Lambshift insingle-photon superradiance. Science 328, 1248–1251 (2010).

4. Heeg, K. P. et al. Vacuum-assisted generation and control of atomic coherences atx-ray energies. Phys. Rev. Lett. 111, 073601 (2013).

5. Shakhmuratov, R., Vagizov, F. & Kocharovskaya, O. Single c-photon revival andradiation burst in a sandwich absorber. Phys. Rev. A 87, 013807 (2013).

6. Liao, W.-T., Palffy, A. & Keitel, C. H. Coherent storage and phase modulation ofsingle hard-X-ray photons using nuclear excitons. Phys. Rev. Lett. 109, 197403(2012).

7. Liao, W.-T., Palffy, A. & Keitel, C. H. Nuclear coherent population transfer with X-raylaser pulses. Phys. Lett. B 705, 134–138 (2011).

8. Adams, B. W. et al. X-ray quantum optics. J. Mod. Opt. 60, 2–21 (2013).9. Ma, X.-S. et al. Quantum teleportation over 143 kilometres using active feed-

forward. Nature 489, 269–273 (2012).10. Doring, F. et al. Sub-5 nm hard X-ray point focusing by a combined Kirkpatrick-

Baez mirror and multilayer zone plate. Opt. Express 21, 19311–19323 (2013).11. Hammerer, K., Sørensen, A. S. & Polzik, E. S. Quantum interface between light and

atomic ensembles. Rev. Mod. Phys. 82, 1041–1093 (2010).12. Hoy,G.R., Hamill, D.W.&Wintersteiner, P.P. inMossbauer Effect Methodology Vol. 6

(ed. Gruverman, I. J.) 109–121 (Plenum, 1970).13. Amann, J. et al. Demonstration of self-seeding in a hard-X-ray free-electron laser.

Nature Photon. 6, 693–698 (2012).14. Shvyd’ko, Yu, Stoupin, S., Blank, V. & Terentyev, S. Near-100% Bragg reflectivity of

X-rays. Nature Photon. 5, 539–542 (2011).15. Antonov, V. A., Radeonychev, Y. V. & Kocharovskaya, O. Formation of a single

attosecond pulse via interaction of resonant radiation with a strongly perturbedatomic transition. Phys. Rev. Lett. 110, 213903 (2013).

16. Helisto, P., Tittonen, I. & Katila, T. Enhanced transient effects due to saturatedabsorption of recoilless c-radiation. Phys. Rev. B 34, 3458–3461 (1986).

17. Smirnov, G. V. & Potzel, W. Perturbation of nuclear excitons by ultrasound.Hyperfine Interact. 123/124, 633–663 (1999).

18. Kuznetsova, E., Kolesov, R. & Kocharovskaya, O. Compression of c-ray photons intoultrashort pulses. Phys. Rev. A 68, 043825 (2003).

19. Smirnov,G. V.General properties of nuclear resonant scattering.Hyperfine Interact.123/124, 31–77 (1999).

20. Kagan, Yu Theory of coherent phenomena and fundamentals in nuclear resonantscattering. Hyperfine Interact. 123/124, 83–126 (1999).

21. Hannon, J. P. & Trammell, G. T. Coherent gamma-ray optics. Hyperfine Interact.123/124, 127–274 (1999).

22. Kolchin, P., Belthangady, C., Du, S., Yin, G. Y. & Harris, S. E. Electro-optic modulationof single photons. Phys. Rev. Lett. 101, 103601 (2008).

23. Pittman, T. It’s a good time for time-bin qubits. Physics 6, 110 (2013).24. Rohlsberger, R. Nuclear Condensed Matter Physics using Synchrotron Radiation

(Springer Tracts Mod. Phys. 208, Springer, 2005).25. Radeonychev, Y. V., Tokman, M. D., Litvak, A. G. & Kocharovskaya, O. Acoustically

induced transparency in optically dense resonance medium. Phys. Rev. Lett. 96,093602 (2006).

26. Kocharovskaya, O. A. & Khanin, Ya. I. Population trapping and coherent bleachingof a three-level medium by a periodic train of ultrashort pulses. Sov. Phys. JETP 63,945–949 (1986).

27. Harris, S. E. Electromagnetically induced transparency. Phys. Today 50, 36–42(1997).

28. Fleischhauer, M., Imamoglu, A. & Marangos, J. P. Electromagnetically inducedtransparency: optics in coherent media. Rev. Mod. Phys. 77, 633–673 (2005).

29. Lvovsky, A. I., Sanders,B. C.& Tittel,W.Optical quantum memory. NaturePhoton. 3,706–714 (2009).

30. Simon, C. et al. Quantum memories. Eur. Phys. J. D 58, 1–22 (2010).

Acknowledgements We acknowledge the support from the US NSF (grant no.PHY-1307346), the RFBR (grants nos 13-02-00831 and 12-02-00263) and TheMinistry of Education and Science of the Russian Federation (contract no.11.G34.31.0011). V.A. acknowledges support from the Dynasty Foundation.

Author Contributions F.V. developed the experimental methods, designed theexperiment and derived all the experimental results. V.A., Y.V.R. and R.N.S. developedthe theoretical description. V.A. and F.V. determined the optimal parameters for theexperiments and provided the theoretical fit to experimental data. Y.V.R. and F.V.suggested the technique for observing the single-photon waveforms. R.N.S. obtainedanalytical solutions for some limiting cases. O.K. suggested the idea, coordinated theefforts and wrote the paper. All authors discussed the results and edited themanuscript.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper. Correspondenceand requests for materials should be addressed to O.K. (kochar@physics.tamu.edu).

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METHODSThe transformation of a recoilless c-photon in a uniformly (piston-like) vibratingMossbauer absorber can be considered as follows. The resonant absorber vibratesalong the direction of propagation of the photon with amplitude R and frequencyV/2p (Fig. 2). The vibration set-up was designed to provide uniform vibration ofthe absorber as a solid body. In particular, a small absorber thickness, L= 2pVs/V,where Vs is the speed of sound in the absorber, ensures that the internal vibrationalmodes have frequencies far beyond V and are not excited by the absorber vibration.Correspondingly, the coordinate z9 in the reference frame of the vibrating absorberis related to the coordinate z in the laboratory reference frame by the relation

z~z’zR sin (Vt) ð1Þ

The electric field of an individual 14.4-keV c-photon emitted by the radioactivesource, Er(z, t), in the laboratory reference frame is represented by a classical quasi-monochromatic wave3,4,8,17,19

Er(z,t)~E0h(t{z=c)e{(ivrzCr=2)(t{z=c)ziQ0 ð2Þ

where h(x) is the Heaviside step function, t 5 t 2 t0 is the time passed from detec-tion of the preceding 122-keV photon at the moment t0 (Fig. 1a), Cr 5 1/Tr (Cr/2p isthe linewidth of the source and Tr is the natural lifetime of the state jbæ), vr/2p is thecarrier frequency of the emitted photon, c is the speed of light in vacuum and Q0 isthe random phase. The propagation distance of the photon from the emitter both tothe absorber and to the detector satisfies the condition Dz= cdt, where dt is themeasurement error. Therefore, the count rate of the emitted photons in front of theabsorber, calculated from equation (2), has the form (red dashed lines in Fig. 3b, c):

Nr(t)! jEr(t)j2~E20h(t)e{Crt ð3Þ

In the reference frame co-moving with the absorber, the electric field in equation (2)takes the form

Er(z’,t)~E0h(t{z’=c)e{(ivrzCr=2)(t{z’=c)ziQ0

| J0(p)zX?n~1

Jn(p)ein(Vtzq0)

zX?n~1

e{inpJn(p)e{in(Vtzq0)

! ð4Þ

where q0 5 Vt0, Jn(p) is the Bessel function of the first kind of order n (n is theinteger number), p 5 2pR/l is the modulation index and l 5 2pc/vr is the photonwavelength. Equation (4) results from substitution of equation (1) into equation (2)in the nonrelativistic approximation, assuming that Cr=vr and taking into

account eip sin a~X?

n~{?Jn(p)eina as well as J{n~e{ipnJn. Thus, the incident

electric field ‘seen’ by the vibrating nuclei is frequency modulated and correspondsto a superposition of exponentially decaying spectral components with the carrierfrequencies vr 6 nV. The amplitudes and phases of the sidebands are determinedby the modulation index, p, through the magnitudes and signs of Bessel functionsof corresponding orders, Jn(p). Thus, ‘6n’ sidebands, located symmetrically withrespect to the carrier frequency, vr, have the same amplitudes, and their phasesdiffer by np.

The Fourier transform, Er(z’,t)~eiz’vr=cÐ?{?

~E’r(v)e{ivt dv, of the field ofthe incident c-photon (equation (4)) in the case of a short, as compared withthe photon length, propagation distance, Dz9= cTr, yields a representation inthe form of the spectral comb

~E’r(v)~X?

n~{?

~E’n(v)

~E’n(v)~E0

2pJn(p)ei(Q0znq0)

i(vr{nV{v)zC r=2

ð5Þ

For p 5 1.8, which maximizes J+1(p), this structure is shown in Fig. 3a, where theblue line represents the squared spectral amplitudes and the red line represents thespectral phases, respectively. With an increase in modulation index, a larger num-ber of sidebands is produced.

Evolution of the electric field of the photon, E(z9, t), during its propagationthrough the absorber (0ƒz’ƒL), is described by the wave equation

L2E

Lz’2{

1

c’2L2ELt2

~2de

c’2LELt

z4p

ec’2L2PLt2

ð6Þ

where c’~c=ffiffiep

, e<1 is the background dielectric permittivity of the absorber, de

takes into account the non-resonant losses due primarily to the photoelectric

absorption, and P is the resonant nuclear polarization, related to the coherence,r21, of the resonant nuclear transition

P~faNd12r21 ð7Þ

via the probability of recoilless absorption, fa, the concentration of the resonantnuclei, N, and the dipole moment, d12, of the resonant transition 1j i« 2j i (Fig. 1b).Because the photons are depolarized and the dipole moments are randomlyoriented, we can neglect the vector character of E, P and d12. The density matrixequation for the coherence of the resonant absorber transition has the form

dr21

dtz(ivazca)r21~

iB

n12d21E ð8Þ

where va/2p is the frequency of the resonant transition, ca/2p is the half-linewidthof its spectral line and n12~r11{r22<1 is the population difference between theenergy levels j1æ and j2æ.

We seek a solution of equations (6)–(8) in the form F(z’,t)~eiz’vr=c

Ð?{?

~F’(z’,v)e{ivt dv, where F stands for E or P. In the rotating-waveapproximation (jv – vaj=jv 1 vaj), from equations (7) and (8) we find that

~P’(z’,v)~n12faNjd12j2

B(va{v{ica)~E’(z’,v) ð9Þ

Fourier transforming equation (6), implementing the slowly-varying-envelope approx-imation (jL~F’(z’,v)=Lz’j=j~F’(z’,v)jv=c, j~F’(z’,v)=~F’(z’,vr)j=vr=jvr{vj) andusing equation (9) results in the following equation for the field:

L~E’(z’,v)

Lz’z dez

TM=2L1zi(va{v)=ca

� �~E’(z’,v)~0 ð10Þ

where TM~4pfavaNn12jd12j2L=caBcffiffiep

is the Mossbauer optical thickness. Thesolution of equation (10), satisfying the input boundary condition ~E’(z’~0,v)

~~E’r(v) (equation (5)), has the form of a spectral comb:

~E’(z’,v)~e{dez’X?

n~{?

~E’n(v) exp {TM=2L

1zi(va{v)=caz’

� �

This gives us the time dependence of the output field emerging from the absorberboth in the vibrating reference frame, Eout(z’,t)~eiz’vr=c

Ð?{?

~E’(z’~L,v)e{ivt dv,

and in the laboratory frame of reference, Eout(z,t)~ei(z{R sin (Vt))vr=cÐ?{?

~E’(z’~L,v)e{ivt dv. Finally, the count rate, Nout(t)! jEout(z, t)j2, of c-photonsshaped by the vibrating resonant absorber has the form

Nout(t)!e{Te E20

X?n~{?

ð?{?

Jn(p)einq0

i(vr{nV{v)zC r=2

�����| exp {

TM=21zi(va{v)=ca

� �e{ivt dv

����2

ð11Þ

where Te ; 2deL is the non-resonant absorber optical thickness. It is worthwhile topoint out here that, because jEout(z, t)j2 5 jEout(z9, t)j2, the waveform of a photon,Nout(t)! jEout(z, t)j2, in the laboratory reference frame is the same as in the ref-erence frame of the vibrating absorber.

In general, the output c-radiation constitutes an amplitude- and frequency-modulated signal that has a form determined by the number of produced spectralcomponents (that is, the modulation index), as well as their relative amplitudesand phases. If the p , 2.4 (p 5 2.4 is the first zero of J0(p)), then all Bessel functionsJn(p) of non-negative order are positive and the lower-frequency sidebands of theincident c-photons in the vibrating reference frame (the first sum in equation (4))are phase-matched with the phase difference between each neighbouring pairequal to q0, whereas the upper-frequency sidebands (the second sum in equa-tion (4)) are phase-matched with the phase difference between each neighbouringpair equal to q0zp. This is illustrated in Fig. 3a for p 5 1.8 (which is the position ofthe global maximum of J1(p)), corresponding to production of three upper andthree lower sidebands, and q0 5 0. The phases of three upper-frequency sidebandsand three lower-frequency sidebands are marked by brown squares and greentriangles, respectively (see the right-hand vertical axis for the values of thesephases). An elimination of either the ‘11’ sideband or the ‘21’ sideband fromthe incident spectrum (in the vibrating frame of reference, (equation (4))) by itstuning to the absorber resonance via the Doppler shift, caused by a constantvelocity of the source relative to the absorber, leads to phase-matching of fiveout of the six remaining spectral components and results in the formation of pro-nounced ultrashort c-ray pulses (see Figs 3b and 4 and Extended Data Figs 1 and 5).The theoretical curves on these figures are plotted in accordance with the numer-ical solutions of equation (11), (13) and (14), as well as the analytical formula inequation (15). To fit the experimental data in Figs 3 and 4, the calculated time

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dependencies were averaged over the acquisition intervals Vdt 5 p/2 (Fig. 3b) andVdt 5p/8 (Fig. 3c), determined by the experimental set-up. As is seen fromExtended Data Fig. 1, the value of modulation index chosen for our experimentmaximizes both the amplitude and the contrast of the produced pulses (red solidline). This happens because p 5 1.8 corresponds to the global maximum of Besselfunction J1(p), which determines the amplitude of the ‘21’ sideband (in thevibrating reference frame), suppressed by the resonant absorber. Strengtheningthe spectrally selective impact of the absorber on the incident c-photon in thevibrating reference frame leads to deeper modulation of the output photon in thelaboratory frame of reference. As a result, reaching the peak of J1(p) correspondsto forming the most pronounced pulses. Reduction of the modulation index top 5 0.8 leads to a decrease in the contrast of the produced pulses because of thenarrowing of the generated spectrum (blue dashed line in Extended Data Fig. 1).With an increase in modulation index, larger numbers of sidebands are produced,but because tuning the single absorber resonance allows for the elimination ofonly one ‘wrong phase’ spectral component (either ‘11’ or ‘21’), it becomesimpossible to phase match all of them using a single absorber line. Thus, anincrease in modulation index above the optimal value (an example correspondingto p 5 2.8 is shown in Extended Data Fig. 1 by the green dashed line) leads to areduction in the pulse contrast and generation of additional spikes because of thegrowing influence of the phase-mismatched spectral components.

The waveform of an output c-photon (equation (11)) strongly depends ondetuning the central frequency of the source, vr, from the resonance frequencyof the absorber, va. The detuning, vr 2 va, determines which component of theincident spectrum (in the vibrating reference frame) is affected by the absorberresonance (attenuated and/or phase-shifted owing to the resonant dispersion),and how the output spectrum is related to the incident spectrum. The timedependences of a c-photon count rate, plotted for three different values of detun-ing, vr 2 va 5 0.5V, vr 2 va 5 V and vr 2 va 5 2V, corresponding to tuningthe absorber resonance into the middle between the ‘0’ and ‘21’ sidebands, to the‘21’ sideband and to the ‘22 sideband, respectively, are plotted in Extended DataFig. 2. As is seen from that figure, only tuning the ‘21’ sideband to the absorberresonance (as in our experiment; see Figs 3a, b and 4) leads to formation of theultrashort c-radiation pulses owing to elimination of the ‘wrong-phase’ spectralcomponent (red solid line). Tuning the ‘22’ sideband to the resonance leads to theless pronounced amplitude modulation of the output c-photon, caused by elim-ination of the ‘right-phase’ component (green dashed line). In turn, tuning theabsorber resonance between the sidebands leads to the weakest amplitude modu-lation (shown by blue dashed line) because of the limited influence of the absorberresonance on the spectrum of the incident c-photon.

Although using higher values of the modulation index does not allow for anincrease in the peak amplitude or a reduction in pulse duration, it does allow forefficient single-photon shaping (Fig. 3c), demonstrating spitting of the singlec-photon into two pulses. The time interval between the pulses and their relativeamplitudes can be controlled by variation of the vibration phase, q0, as shown inExtended Data Fig. 3.

As is seen in Extended Data Fig. 3, a change in the vibration phase at themoment of formation of the excited state jbæ of the radiating nucleus (Fig. 1a)leads to a shift in the modulation pattern, produced as a result of the absorber’svibration, with respect to the front edge of the photon. Accordingly, it leads to atime shift of the second pulse (generated inside the absorber) in the produced pulsepair relative to the first pulse (which is the Brillouin precursor). In the case ofFig. 3c and Extended Data Fig. 3, the vibration period, 2p/V 5 385 ns, consid-erably exceeds the lifetime of the incident photon Tr 5 141 ns, and a change in thevibration phase therefore strongly affects the output photon waveform. If the oscil-lation cycle is shorter than the incident photon lifetime, as in the case of Fig. 3a, b,where 2p/V 5 98 ns, the vibration phase has a minor role (see Fig. 3b and its inset).

The output photon waveform also depends on the frequency of vibration, V,which determines the timescale of the beatings in the photon detection probability(or the intensity of the output radiation) produced as a result of the absorber’svibration. This dependence is illustrated in Extended Data Fig. 4, where the outputcount rate is plotted for the parameters of the experiment on a double-pulse for-mation (Fig. 3c) and three different values of the vibration frequency: V/2p5 1.3 MHz(green dashed line), 2.6 MHz (the same as in Fig. 3c (red solid line)) and 5.2 MHz(blue dashed line). As is seen from Extended Data Fig. 4, an increase in the vib-ration frequency corresponds to a shrinking of the modulation structure in time,whereas a decrease in the frequency of vibration leads to a stretching of the mod-ulation pattern in time.

It follows from Extended Data Figs 3 and 4 that the parameter values that werechosen for the experiment on double-pulse (time-bin qubit) formation from anincident c-photon (Fig. 3c), that is, V/2p5 2.6 MHz and q0~{0:3p are optimalfor the generation of the two well-separated pulses of nearly the same amplitude.

So far, we have considered the single-photon waveforms described by equa-tion (11). Averaging the output photon count rate, Nout(t) 5 Nout(t 2 t0), over themoment t0 gives the time-dependence of the full flow of c-photons, or an intensityof the continuous c-radiation, which can be produced by a high-activity source:

Iout(t)! limT??

1T

ðt

{TNout(t{t0) dt0 ð12Þ

The substitution of equation (11) into equation (12) results in

Iout(t)!e{TeX?

n,m~{?

Jn(p)Jm(p)

|Bnm cosf(n{m)(Vtzq0)zwnmgð13Þ

where Bnm 5 Bmn and wnm 5 wmn are real parameters defined by the relation

Bnmeiwnm

~1

2p

ð?{?

1

v2z(C r=2)2

| exp {TM

2ca

caziDvnz

ca

ca{iDvm

� �� �dv

Dvn~va{vrznV{v

ð14Þ

Under the approximation of complete suppression of a single ‘11’ or ‘21’ spectralcomponent of the incident photon (in the vibrating frame of reference), and in theabsence of the influence of the resonant absorber on all the other sidebands,equation (13) takes a very simple form:

Iout(t)!

1zJ21 (p){2J1(p) cos½Vt{p sin (Vt)�

{1 sideband is eliminated

1zJ21 (p)z2J1(p) cos½Vtzp sin (Vt)�

z1 sideband is eliminated

8>>><>>>:

ð15Þ

Noting that cos(Vt 1p) 5 2cos(Vt) and sin(Vt 1p) 5 2sin(Vt), it is clearlyseen from equation (15) that tuning the absorber line to the opposite sidebandis equivalent to a change in the phase of vibration by p. If t0, the moment offormation of the excited state jbæ of the radiating nucleus, is undefined, this isequivalent to shifting the starting observation time, which can be chosen arbitrar-ily, and does not change the intensity envelope. However, in the case of a single-photon waveform, both t0 and q0~Vt0 are fixed by the instant of formation of theexcited nuclear state (emitting the photon). Therefore, the photons ‘born’ indifferent vibration phases can have qualitatively different waveforms, as shownin Fig. 3c, its inset and Extended Data Fig. 3. Equation (15) also shows that the peakamplitude of the produced pulses is given by Imax

out !(1zjJ+1(p)j)2. Thus, thepulses indeed reach their maximum intensity when the corresponding Besselfunctions, jJ+1(p)j, take their peak value, jJ+1(p)jmax < 0.58, that is, at p 5 1.8.

Although equation (15) does not account for the finite spectral width of both theabsorber resonance and the incident c-radiation (that is, the width of each side-band of the incident spectrum in the absorber reference frame), in the case of highvibration frequency (relative to the source (and absorber) linewidth) it is in goodqualitative as well as quantitative agreement with the experimental results (ExtendedData Fig. 5).

As demonstrated in this work, the harmonic vibrations (with frequencies inthe range of 1–10 MHz and modulation index ranging from 1 to 7) of the singlestainless-steel absorber (with a natural abundance of 57Fe and an optical thicknessof about 5) makes it possible to form ultrashort pulses with a duration an order ofmagnitude shorter than the lifetime of the upper nuclear level (141 ns) as well as toproduce a variety of waveforms from a single c-photon (including a decaying trainof ultrashort pulses, a double pulse and a single triangular pulse) by changing thevibration frequency, the modulation index, the vibration phase at the moment ofthe photon’s arrival, and the detuning of the incident photon from the absorberresonance.

The range of possibilities for c-photon shaping and ultrashort pulse productionbased on modulation of the nuclear transition frequency may be essentiallyextended by using both single and multiple absorbers of larger optical thicknesses(enriched with 57Fe), as well as absorbers with a split transition line; differentMossbauer isotopes (in particular 67Zn, which possesses a 93.3-keV recoillesstransition with a 12.2-kHz linewidth); higher modulation frequencies and modu-lation indices; anharmonic (and even aperiodic) vibrations; and other techniques(such as the Zeeman effect) for modulation of the nuclear transition frequency.These possibilities will be addressed in our future studies.

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Extended Data Figure 1 | Time dependences of a c-photon detectionprobability for different values of modulation index. The values ofmodulation index are p 5 0.8 (blue dashed line), p 5 1.8 (red solid line) and

p 5 2.8 (green dashed line). All the other parameters are the same as in ourexperiment (Fig. 3).

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Extended Data Figure 2 | Time dependences of a c-photon detectionprobability for different detunings of the central frequency of the source, vr,from the resonance frequency of the absorber, va. The values of detuning are

vr 2 va 5 0.5V (blue dashed line), vr 2 va 5 V (red solid line) andvr 2 va 5 2V (green dashed line). All the other parameters are the same as inour experiment (Fig. 3).

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Extended Data Figure 3 | Variation of the waveform of a 14.4-keV c-photonwith a change in the vibration phase, q0, at the moment of detection of thepreceding 122-keV c-photon. The parameter values are the same as in Fig. 3c,corresponding to double-pulse formation, except for the initial phases of

vibration, which are as follows: q0~{0:3p, the same as in Fig. 3c (red solidline), q0~0:2p (blue dashed line), q0~0:7p (the same as in the inset of Fig. 3c(green dashed line)) andq0~1:2p (cyan dashed line).

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Extended Data Figure 4 | Waveforms of 14.4-keV c-photons produced atdifferent frequencies of vibration, V. The parameter values are the same as inFig. 3c, corresponding to double-pulse (time-bin qubit) formation, except for

the vibration frequencies, which are as follows: V/2p5 1.3 MHz (green dashedline), V/2p5 2.6 MHz (the same as in Fig. 3c (red solid line)) andV/2p5 5.2 MHz (blue dashed line).

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Cou

nt r

ate,

arb

. uni

ts

0.8

1.0

1.2

1.4

1.6

Time, ns

100 200 300 400 500100 200 300 400 500

Extended Data Figure 5 | Count rate of 14.4-keV photons versus time forthe case of Fig. 4. The blue dots centred at the confidence intervals correspond

to the experimental data, and the red and green solid curves are plottedaccording to equations (13) and (14) and, respectively, equation (15).

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