Single spontaneous photon as a coherent

0 downloads 0 Views 738KB Size Report
Apr 3, 2011 - a transition to the ground state and emits a single photon. As- sociated with the ... The photon can be regarded as a beamsplitter for an atomic matter-wave and ... mirror the observation of the emitted photon direction implies the knowledge of the ... In the presence of the mirror the detection of a photon in a ...
LETTERS PUBLISHED ONLINE: 3 APRIL 2011 | DOI: 10.1038/NPHYS1961

Single spontaneous photon as a coherent beamsplitter for an atomic matter-wave Jiˇrí Tomkoviˇc1 *, Michael Schreiber2 , Joachim Welte1 , Martin Kiffner3 , Jörg Schmiedmayer4 and Markus K. Oberthaler1 In spontaneous emission an atom in an excited state undergoes a transition to the ground state and emits a single photon. Associated with the emission is a change of the atomic momentum due to photon recoil1 . Photon emission can be modified close to surfaces2,3 and in cavities4 . For an ion, localized in front of a mirror, coherence of the emitted resonance fluorescence has been reported5,6 . Previous experiments demonstrated that spontaneous emission destroys motional coherence7–9 . Here we report on motional coherence created by a single spontaneous emission event close to a mirror surface. The coherence in the free atomic motion is verified by atom interferometry10 . The photon can be regarded as a beamsplitter for an atomic matter-wave and consequently our experiment extends the original recoiling slit Gedanken experiment by Einstein11,12 to the case where the slit is in a robust coherent superposition of the two recoils associated with the two paths of the quanta. We consider an atom passing by a mirror which spontaneously emits a single photon (see Fig. 1a). As a result of the photon momentum the atom receives a corresponding recoil kick in the direction opposite to the photon emission. In the absence of the mirror the observation of the emitted photon direction implies the knowledge of the atomic momentum resulting from the photon– atom entanglement8 . In the presence of the mirror the detection of a photon in a certain direction does not necessarily reveal if it has reached the observer directly or via the mirror. For the special case of spontaneous emission perpendicular to the mirror surface the two emission paths are in principle indistinguishable for small atom–mirror distances d  c/Γ , where c is the speed of light and Γ the natural linewidth. This general limit is always fulfilled in our experiments. Thus the atom after this emission event is in a superposition of two motional states. This is also true for the more general case of tilted emission, as revealed in Fig. 1b for emission close to the mirror surface. One expects residual coherence for emission angles where the optical absorption cross-section of the atom and the mirror atom observed by a fictitious observer in the emission direction still overlap. This is visualized in Fig. 1b, where the corresponding cross-sections are indicated with the bars. The overlap as a function of emission direction is depicted on the sphere (blue no coherence, red full coherence). The result on the atomic motion is indicated for one special trajectory which starts with an atom moving parallel to the mirror surface and a single photon emission at an angle to the mirror normal. This case leads to an imperfect coherent superposition of two momentum states separated by less than two photon momenta hk ¯ 0 . The spatial distribution of the atoms at the

position of the detector is shown, where the colour corresponds to the degree of coherence. In Fig. 1c we contrast this to the case of a larger distance to the mirror, where the portion of coherent atomic momentum is strongly reduced. It is important to keep in mind that a single particle detector cannot distinguish between coherent superpositions and mixtures but only gives the probability distribution. Thus an interferometric measurement13 has to be applied to reveal the expected coherent structure (see Fig. 2). For that, the two momentum states of interest have to be overlapped and the coherence (that is, well-defined phase difference) is verified by observing an interference pattern as a function of a controlled phase shift applied to one of the momentum states. The two outermost momentum states are expected to show the highest coherence. Their recombination can be achieved by a subsequent Bragg scattering off an independent standing light wave (see Fig. 2b) with a suitable wavelength10,14 . The relative phase φB is straightforwardly changed by shifting the probing standing light wave. This is implemented by moving the retroreflecting mirror by distance L. The upper graph depicts the results obtained for large distances (>54 µm) of the atom to the mirror (that is, a free atom). In this case no interference is observed, and thus spontaneous emission induces a fully incoherent modification of the atomic motion. For a mean distance of 2.8 µm, clear interference fringes are observed, demonstrating that a single spontaneous emission event close to a mirror leads to a coherent superposition of outgoing momentum states. In the following we describe the essential parts of the experimental set-up shown in Fig. 2b, lower graph. Further details are provided in the Supplementary Information. As the effect depends critically on the distance between the atom and mirror, a well-collimated and localized beam of 40 Ar atoms in the metastable 1s5 state is used. To ensure the emission of only a single photon we induce a transition 1s5 → 2p4 (λE = 715 nm). From the excited state 2p4 , the atom predominantly decays to the metastable 1s3 state via spontaneous emission of a single photon (λSE = 795 nm) (branching ratio of 1s5 /1s3 = 1/30). The residual 1s5 are quenched to an undetectable ground state by an additional laser. Choosing the appropriate polarization of the excitation laser the atomic dipole moment is aligned in the mirror plane, leading to the momentum distribution after spontaneous emission shown in Fig. 2a. The interferometer is realized with a far-detuned standing light wave on a second mirror. Finally the momentum distribution is detected by a spatially resolved multi channel plate (MCP) ≈1 m behind the spontaneous emission, enabling different momenta to be distinguished.

1 Kirchhoff-Institut

für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany, 2 Ludwig-Maximilians-Universität, Schellingstr. 4, 80799 München, Germany, 3 Physik Department I, Technische Universität München, James-Franck-Straße, 85747 Garching, Germany, 4 Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria. *e-mail: [email protected].

NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics

© 2011 Macmillan Publishers Limited. All rights reserved.

1

NATURE PHYSICS DOI: 10.1038/NPHYS1961

LETTERS b

Atom detector

Prob a distr bility ibuti on

a p

c x

z y

Prob a distr bility ibuti on z x

x

y

z y

p

Figure 1 | Motional coherence generated by a single spontaneous emission event. a, The situation of interest is depicted—an atom in front of a mirror spontaneously emits a single photon. For emission perpendicular to the mirror surface an observer can in principle not distinguish if the photon has been reflected or not. Momentum conservation in the atom–photon system implies that the atom after the emission is in a coherent superposition of two different momentum states separated by twice the photon recoil. b, Indistinguishability is also given for more general emission directions. With the spatial extension of the atom corresponding to the optical absorption cross-section, indistinguishability can be estimated by the projected overlap of the atom and its mirror image. This overlap is represented by a colour code on a sphere for all emission directions (red: full coherence, blue: no coherence). Repeating the experiment—a single atom emits a single photon—leads to the indicated pattern at the atom detector. The colour code indicates the probability of generating a coherent superposition for the corresponding event (red: full coherence, blue: no coherence). c, In the case of large distances to the mirror, the coherent portion is drastically reduced, approaching the limit of vanishing coherence in free space.

a

b

Bragg mirror

L

Atom detector

5,500

4,000

Counts

Momentum

+¬hk0

Mirror position L (μm) 2.58 2.75 2.92 3.09 3.26

0

5,250

3,750

5,000

3,500

4,750

¬¬hk0 π 2π 0 Phase Mirror position L (μm) 2.58 2.75 2.92 3.09 3.26

¬2π

Atoms/pixel

Bragg mirror Entangling mirror

L

0

¬¬hk0

z x

Counts

Momentum

+¬hk0

y

2,800

3,400

2,600

3,200

2,400

3,000 ¬2π

Atoms/pixel

¬π

¬π

π 0 Phase



Figure 2 | Experimental confirmation of coherence induced by spontaneous emission. a, Experimental observation of the momentum distribution does not reveal the coherence. In both cases—close to and far from the mirror—the momentum distribution is the same (blue line). To compare the observed momentum distribution after spontaneous emission with theory (light grey) the data has been deconvoluted by the initial momentum distribution. The deviation results from a residual filtering of high spatial frequencies. b, The coherence is revealed if the spontaneous emission event is employed as the first beamsplitter of an atom interferometer. The recombination is accomplished by Bragg scattering from a standing light wave. The relative phase of the two paths can be changed by moving the ‘Bragg’ mirror as indicated. In the case of a mean distance of 54 µm between atoms and ‘entangling’ mirror (upper graph, error bars indicate Poisson noise) no interference signal is observed, confirming the free space limit. The inset depicts the position of the ‘entangling’ mirror with respect to the atomic beam. For a mean distance of 2.8 µm (lower graph) the two complementary outputs of the interferometer reveal an interference pattern with a maximal visibility of 5.9% ± 1.1%.

For systematic studies of the coherence we analyse the probability of finding a particle in a coherent superposition of momentum states as a function of atom–mirror distance d. This is done by analysing the final momentum distribution for different phases φB within the interferometer and fitting to each resolved momentum (≈1/8 of a photon momentum) an interference pattern given by N = N0 + NA cos(φB + φ0 ) 2

In Fig. 3 we plot the visibility V = NA /N0 (with N0 the constant atom number and NA the oscillatory part) revealing that the coherence vanishes within distances of a few micrometres from the mirror. For a basic understanding of the physics behind the experimental observation we use a simple semiclassical model. We follow the picture of an atom and its image by Morawitz15 and Milonni and Knight2 and assume a two-level system with ground state |g i and excited |ei. To deduce the indistinguishability between the atom and its mirror atom, that is, the photon emission

NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics © 2011 Macmillan Publishers Limited. All rights reserved.

NATURE PHYSICS DOI: 10.1038/NPHYS1961

LETTERS where

7

 |ψs i = rs ∗ eik0 uˆz + e−ik0 uˆz |ψ0 i

(1)

6

 |ψp i = −rp ∗ eik0 uˆz + e−ik0 uˆz |ψ0 i

(2)

The operators e±ik0 uˆz in equations (1) and (2) describe the transverse recoil momentum ±h¯ k0 u transferred to the atom by the spontaneously emitted photon. The Fresnel coefficient rs (rp ) accounts for the reflection of the transversal electric (transversal magnetic) mode at the mirror and |ψ0 i describes the motional state of the atom before spontaneous emission. The normalization is ensured by the normalization constant α. For a quantitative comparison with the experiment we assume that

Visibility (%)

5 4 3 2

Z |ψ0 i =

1 0

3

4 5 Mean distance (µm)

6

54

Figure 3 | Dependence of visibility on the mean atom–mirror distance. Measured data is depicted as blue points. The mean distance is calculated from the position of the ‘entangling’ mirror with respect to the centre of the atomic beam, as indicated in the insets. The error bars indicate a 95% confidential interval resulting from the fitting procedure to the interference pattern. The expectation from the simple cross-section overlap model is shown by the blue line. The quantum mechanical treatment is depicted as a green line. One finds good agreement between theory and experiment by including details such as initial spatial and momentum distribution, averaging over all distances, details of Bragg scattering and the final spatial resolution of the atom detector. The mean atom–mirror distance is adjusted by the position of the ‘entangling’ mirror with respect to the collimation slit of the atomic beam.

towards and away from the mirror, we attribute to the atom a size corresponding the optical absorption cross-section (σ = 3λ2 /2π). In the direction perpendicular to the mirror an observer cannot in principle distinguish between the atom and mirror atom and √ thus a coherent superposition of momentum states |ψi = 1/ 2 (|+ h¯ k0 i+|− hk ¯ 0 i) emerges with the photon momentum prec = h¯ k0 . For emission directions other √ than perpendicular, the probability P for generating |ψ 0 i = 1/ 2(| + h¯ k 0 i + | − h¯ k 0 i) can be estimated by the overlap region of an atom and mirror atom with the assigned effective size, as shown in Fig. 1b. This overlap depends on the distance between the atom and the mirror and on the observation angle (for details see Supplementary Information). To quantitatively compare with the experimental data, the finite resolution of momentum detection has to be taken into account, leading to an integration over different observation directions. Further averaging due to the finite extension of the atomic beam (width in transverse direction of 10 µm) and the initial momentum distribution results in a reduction of the visibility V . The prediction within this model is shown as the solid blue line in Fig. 3. The comprehensive quantum mechanical model (for details see Supplementary Information) takes into account the modified mode structure of the electromagnetic field due to the presence of the mirror16 . We derive a master equation for the internal degrees of freedom of the atom and its centre of mass motion perpendicular to the mirror surface. It is found that the quantum state of the atomic centre of mass motion after spontaneous emission can be written as 3 %ˆ gg (t = ∞) = α 8

Z1

 du |ψs ihψs | + u2 |ψp ihψp |

0

i

dp f (p,d)e h¯ pd eiφf (p) |pi

is a coherent wave packet. The quantity |f (p,d)|2 represents the initial momentum distribution of atoms and is inferred from an independent measurement of the momentum distribution. The description of the initial atomic state by a pure state is a sensible assumption because the width of the slit collimating the atoms is chosen to be close to the diffraction limit. The phase φf (p) determines the shape of the wavefunction in position space. The Bragg grating is modelled as a beamsplitter with a momentum dependent splitting ratio determined from experimental measurements. After free evolution of the atom we determine the probability to detect the atom within the given resolution of the detector. The result of this calculation is shown as the green line in Fig. 3, where only the phase φf (p) of the wavefunction in front of the first mirror cannot be fully reconstructed acting as a free parameter. The uncertainty of this phase explains the smaller visibility and the asymmetry between different diffraction orders (see Fig. 4). So far we have discussed the maximum coherence observed in the experiment. In Fig. 4 the momentum dependence of the coherence is shown for a mean distance of 3.3 µm from the mirror. This reveals that only the outermost parts of the momentum distribution are in a coherent superposition, which is consistent with the simple picture of an atom and mirror atom. It is important to note that Bragg scattering itself exhibits a momentum dependence (Bragg acceptance). For the chosen short interaction length the Bragg acceptance is indicated by the grey line in Fig. 4. As the observed coherence decays significantly within the Bragg acceptance we can experimentally confirm that only the most extreme emission events, that is, perpendicular to the mirror surface, lead to a significant generation of coherence. This angular dependence is similar for all investigated mirror distances as it is essentially given by the coherent momentum spread of the strongly confined initial atomic beam. Finally, we would like to point out the differences from other experiments where the connection between spontaneous emission and coherence has been investigated. For example, the experiment in ref. 8 shows that the spontaneous photon carries away information from the atom about its position, and therefore destroys the coherence when the two paths can be distinguished. Another experiment5 , on the other hand, provides direct proof for the coherence of photons emitted in the resonance fluorescence of a laser-driven ion in front of a mirror. The observed interference pattern can be regarded as indirect evidence for the motional coherence of the trapped ion, well within the Lamb–Dicke limit6 . A different example in the context of laser cooling is velocity selective coherent population trapping17 , where spontaneous emission populates motional dark states. Here the direction of the emitted photon is indistinguishable because it is emitted in the direction of a macroscopic classical field that drives the atom. The most salient

NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics

© 2011 Macmillan Publishers Limited. All rights reserved.

3

NATURE PHYSICS DOI: 10.1038/NPHYS1961

LETTERS

100

0.5 p

80 60 40 20 0

Prob dist ability ribu tion

Bragg acceptance

Coherent atoms per pixel

Received 6 September 2010; accepted 1 March 2011; published online 3 April 2011

¬¬hk0

0 Momentum

+¬hk0

Figure 4 | Observation of angular dependence of coherence. The schematics show an idealized case of coherent momenta for an atom at a fixed distance and an initial momentum parallel to the mirror (red area within the momentum distribution). Owing to the finite momentum distribution of the atomic beam, the narrow coherent momenta is smeared out in the experimental realization. The measured width of the coherent momenta (red points) is smaller than the angle-acceptance of the Bragg-crystal (grey line), revealing that mainly atoms with momenta of ±h¯ k0 are in a coherent superposition. The data is shown for a mean distance of 3.3 µm (in contrast to Fig. 2b (lower graph), where the atom is much closer to the mirror). Error bars are defined as in Fig. 3.

feature of our experiment is that a single spontaneous emission event in front of a mirror creates a coherent superposition in freely propagating atomic matter-waves, without any external coherent fields being involved. The emission directions of the spontaneous photon become indistinguishable because of the mirror. In work by Bertet et al.12 photons from transitions between internal states are emitted into a high finesse cavity. Their first experiment demonstrates the transition from indistinguishability when emission is into a large classical field to distinguishability and destruction of coherence between the internal atomic states when emission is into the vacuum state of the cavity. In their second experiment they show that, using the same photon for both beamsplitters in an internal state interferometer sequence, coherence can be obtained even in the empty cavity limit. In our experiment the photon leaves the apparatus. We observe coherence only when the photon cannot carry away ‘which path’ information. This implies that the generated coherence in motional states is robust and lasts. In this sense it is an extension of Einstein’s famous recoiling slit Gedanken experiment11 . The single photon is the ultimate lightweight beamsplitter, which can be in a robust coherent superposition of two motional states. In free space the momentum of the emitted photon allows the measurement of the path of the atom. This corresponds to a well-defined motional state of the beamsplitter (that is, no coherence). Close to the mirror the reflection renders some paths indistinguishable, realizing a coherent superposition of the beamsplitter. The large mass of the mirror ensures that even in principle the photon recoil cannot be seen. Thus, the atom is in a coherent superposition of the two paths. We measure this generated coherence by matter-wave interference.

4

References 1. Milonni, P. W. The Quantum Vacuum (Academic, 1994). 2. Milonni, P. W. & Knight, P. L. Spontaneous emission between mirrors. Opt. Commun. 9, 119–122 (1973). 3. Drexhage, K. H. Progress in Optics Vol. 12, 163–192, 192a, 193–232 (Elsevier, 1974). 4. Goy, P., Raimond, J., Gross, M. & Haroche, S. Observation of cavity-enhanced single-atom spontaneous emission. Phys. Rev. Lett. 50, 1903–1906 (1983). 5. Eschner, J., Raab, C., Schmidt-Kaler, F. & Blatt, R. Light interference from single atoms and their mirror images. Nature 413, 495–498 (2001). 6. Eschner, J. Sub-wavelength resolution of optical fields probed by single trapped ions: Interference, phase modulation, and which-way information. Eur. Phys. J. D 22, 341–345 (2003). 7. Pfau, T., Spälter, S., Kurtsiefer, C., Ekstrom, C. R. & Mlynek, J. Loss of spatial coherence by a single spontaneous emission. Phys. Rev. Lett. 73, 1223–1226 (1994). 8. Chapman, M. S. et al. Photon scattering from atoms in an atom interferometer: Coherence lost and regained. Phys. Rev. Lett. 75, 3783–3787 (1995). 9. Kokorowski, D. A., Cronin, A. D., Roberts, T. D. & Pritchard, D. E. From single- to multiple-photon decoherence in an atom interferometer. Phys. Rev. Lett. 86, 2191–2195 (2001). 10. Oberthaler, M. et al. Dynamical diffraction of atomic matter waves by crystals of light. Phys. Rev. A 60, 456–472 (1999). 11. Bohr, N. in Albert Einstein: Philosopher Scientist (ed. Schilpp, P.A.) (Library of Living Philosophers, Evanston, 1949); reprinted in Quantum Theory and Measurement (eds Wheeler, J. A. and Zurek, W. H.) 9–49 (Princeton Univ. Press, 1983). 12. Bertet, P. et al. A complementarity experiment with an interferometer at the quantum-classical boundary. Nature 411, 166–170 (2001). 13. Cronin, A. D., Schmiedmayer, J. & Pritchard, D. E. Optics and interferometry with atoms and molecules. Rev. Mod. Phys. 81, 1051–1129 (2009). 14. Martin, P., Oldaker, B., Miklich, A. & Pritchard, D. Bragg scattering of atoms from standing light-wave. Phys. Rev. Lett. 60, 515–518 (1988). 15. Morawitz, H. Self-coupling of a two-level system by a mirror. Phys. Rev. 187, 1792–1796 (1969). 16. Di Stefano, O., Savasta, S. & Girlanda, R. Three-dimensional electromagnetic field quantization in absorbing and dispersive bounded dielectrics. Phys. Rev. A 61, 023803 (2000). 17. Aspect, A., Arimondo, E., Kaiser, R., Vansteenkiste, N. & Cohen-Tannoudji, C. Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping. Phys. Rev. Lett. 61, 826–829 (1988).

Acknowledgements We wish to thank Florian Ritterbusch for assistance throughout the preparation of this manuscript. We gratefully acknowledge support from the Forschergruppe FOR760, Deutsche Forschungsgemeinschaft, the German-Israeli Foundation, the Heidelberg Center of Quantum Dynamics, Landesstiftung Baden-Württemberg, the ExtreMe Matter Institute and the European Commission Future and Emerging Technologies Open Scheme project MIDAS (Macroscopic Interference Devices for Atomic and Solid-State Systems). M.K. acknowledges financial support within the framework of the Emmy Noether project HA 5593/1-1 funded by the German Research Foundation (DFG). J.S. acknowledges financial support through the Wittgenstein Prize.

Author contributions J.S and M.K.O. conceived the conceptual idea, M.K.O. and J.T. designed the experiment, J.T., M.S. and J.W. performed the experiment, J.T., M.K. and M.K.O. analysed the data, J.T. and M.K.O. developed the semiclassical model, M.K. contributed the quantum mechanical model. All authors discussed the results and wrote the manuscript.

Additional information The authors declare no competing financial interests. Supplementary information accompanies this paper on www.nature.com/naturephysics. Reprints and permissions information is available online at http://npg.nature.com/reprintsandpermissions. Correspondence and requests for materials should be addressed to J.T.

NATURE PHYSICS | ADVANCE ONLINE PUBLICATION | www.nature.com/naturephysics © 2011 Macmillan Publishers Limited. All rights reserved.

SUPPLEMENTARY INFORMATION doi: 10.1038/nPHYS1961

Single spontaneous photon as a coherent beamsplitter for an atomic matter-wave Supplementary Information Experimental details Atomic beam preparation For the experimental implementation a well collimated beam of metastable 40 Ar atoms is used. The atomic beam is transversally cooled and collimated by a 2D-magneto-optical trap and furthermore collimated by a setup of 2 variable slits, leading to a beam width of 10 µm (FWHM) at the position of the first mirror, a spread in momentum distribution of 0.4 photon recoil (HWHM) and a longitudinal velocity of 31 m/s. The beam is polarized by applying a Stern-Gerlach magnet configuration. Spatially resolved atom detection Metastable argon atoms allows for spatially resolved single-particle detection using a multichannel plate (MCP) in combination with a resistive anode. By tilting the MCP with respect to the beam axis, the resolution is enhanced to 17 µm per Pixel. We evaluated the signal on 3 adjacent pixel. The distance between the experimental chamber and the MCP is 1.05 m so that a momentum transfer of less than a photon recoil can be resolved clearly. Electronic structure and detection of coherence The internal electronic structure of argon allows for a straight forward implementation of spontaneous emission of a single photon (see Fig. 1). Starting in the 1s5 (J = 2, mj = 0) level (Paschen- notation), a laser with a wavelength of 714.903 nm excites the atom to the 2p4 level. From there, it can either decay to the ground state – which is not detectable by the MCP, or to two metastable states. A fraction of 96.75 % of all metastable atoms decay by spontaneous emission of a single photon with a wavelength of 795.036 nm to the metastable 1s3 (J = 0) state for which the excitation laser is far detuned. Thus reexcitation is strongly suppressed. Although a small fraction (3.25 %) of atoms decay back to the 1s5 (J = 2) state, the interaction time of atoms with the excitation laser is chosen to be smaller than the cycle time of RabiOscillations of the cycle 1s5 → 1s3 → 1s5 to avoid spontaneous emission of more than one photon. An additional laser at 801.702 nm behind the interaction zone optically quenches the remaining 1s5 to the ground state. Thus all detected metastable atoms have undergone a single photon emission event.

nature physics | www.nature.com/naturephysics

1

1

supplementary information

doi: 10.1038/nphys1961

2p� (J=1) 2p� (J=2)

802nm

nm 715

795 nm 1s� (J=0)

1s� (J=2)

Figure 1: Relevant level scheme of

40

Ar

The Bragg-crystal is conventionally implemented by a standing light wave on a second mirror. The laser is red detuned by 10 GHz to the 1s3 ↔ 2p4 transition, the same transition where the spontaneous photon is emitted. The width of the standing light wave was chosen to 1.5 mm resulting in a sufficiently large angle acceptance of the Bragg-crystal allowing for compensating small shifts in the original momentum distribution of the atomic beam. The intensity of the Bragg-crystal was adjusted to ensure approximately a 50/50 beam splitting ratio. Mirror setup and adjustment The spontaneous emission takes place in front of a gold coated mirror. It is important to note that for a dielectric mirror the photon reflection is ’deep’ in the mirror surface. Thus the effective distance between atom and the reflection point is typically too large to make the effect observable. The second mirror is mounted on a 3-axis piezo mirror mount so it can be aligned parallel to the first mirror and translated along its normal. Thus the phase of the standing light wave can be shifted with respect to the first mirror and we can scan over a range of more than two interference fringes. Both mirrors are mounted together on a custom made translation/rotation stage, so they can be aligned parallel to the atomic beam. The translation stage transversal to the beam allows for implementing different distances between the beam and the mirror setup without changing the optical alignment. The intensity of the Bragg crystal and the angle between the atomic beam and the mirrors are adjusted in preliminary experiments by tilting the mirrors in the Bragg angle, shining in only the second standing light wave and probing for Bragg diffraction of about 50%. It was carefully checked that no stray light from the Bragg lattice hits the first mirror.

2

2

nature physics | www.nature.com/naturephysics

doi: 10.1038/nphys1961

supplementary information

Quantification of the intuitive picture In order to obtain a quantitative prediction within the intuitive picture, we choose the following assumptions. We assign to the atom a size corresponding to the optical absorption cross section σ = 3λ2 /2π. The overlap of the atom and the mirror atom in the projection to the observer is a measure for the indistinguishability and thus for the coherence (see Main text, Fig. 1b). This overlap depends on the distance d between atom and mirror and the observation angle ϕ to the mirror normal and can be written as:   2 P (d, ϕ) = arccos γ − γ 1 − γ 2 (1) π d tan ϕ . with γ =  σ/π

The detector integrates over all emission directions located on a spherical calotte. Furthermore the spatial extension of the atomic beam leads to an integration over different atom-mirror distances. Including experimental details as the initial momentum distribution of the atomic beam and the finite angle acceptance of Bragg scattering leads to the prediction for the visibility.

Quantum mechanical model The experiment measures the probability P to find an atom after spontaneous emission of a single photon and after diffraction by the Bragg grating within the interval z0 ± ∆z /2, where ∆z = 51.6µm (the z direction is perpendicular to the mirror plane). The probability P at z0 depends on the phase φB imprinted by the Bragg grating, P = P0 + A cos(φB + φ0 ),

(2)

where φ0 is an irrelevant constant phase. The calculation of the visibility V = A/P0 of the interference pattern consists of three steps that we illustrate in the following. First, we determine the quantum state of the atom after the spontaneous emission in front of the mirror. Second, we evaluate the state after the Bragg grating, and the third and final step concerns the detection of the atoms. Throughout the derivation, unit vectors v = v/|v| are labeled ˆ by a hat. by an arrow, and operators X Spontaneous emission. We model the atom by a two-level system with excited state |e, ground state |g, transition frequency ω0 and transition ˆ dipole matrix element d = e|d|g. The density operator ˆ describes the quantum state of the atomic system and contains internal as well as external 3

nature physics | www.nature.com/naturephysics

3

supplementary information

doi: 10.1038/nphys1961

degrees of freedom. At t = 0, the electromagnetic field is assumed to be in the vacuum state ˆF , and the density operator of the total system is ˆT = ˆ ⊗ ˆF . In electric-dipole coupling and rotating-wave approximation, the interaction between the atom and the radiation field is governed by the Hamiltonian ˆ int = −d · E ˆ + (ˆ H r)|eg| + h.c.,

(3)

ˆ + (r) is the positive frequency part of the electric field operator and where E ˆr is the position operator. In our model, an absorbing medium with a high reflectivity (z > 0) forms a planar interface at z = 0 with the vacuum (z < 0). For this geometry, the positive frequency part of the electric field operator is given by [1] ∞ + ˆ (r) = dω E ˆ + (r, ω) + noise, E (4) 0

where

  ˆ + (r, ω) = i eiK·R αK (ω) UK (z, ω)ˆ aK (ω), E A K 

(5)

A is a quantization surface and R = xex +yey is the projection of r = (x, y, z) onto the x − y plane. Similarly, K = kxex + kyey denotes the in-plane component of the wave vector k =(kx , ky , kz ), and for a given frequency ω, |kz | = κ ˆK are photon anniis determined by κ(ω) = ω 2 /c2 − K 2 . In Eq. (5), a hilation operators that obey the usual bosonic commutation relations and  αK (ω) = ω /4πε0 c2 κ(ω). The index  ∈ {s, p} labels the two orthogonal modes UsK (z, ω) = eiκzs + e−iκz rss, UpK (z, ω) = eiκz p+ + e−iκz rp p− ,

(6) (7)

where s labels the TE (transversal electric) and p the TM (transversal magnetic) mode characterized by the unit vectors  × ez , s = K

 p± = (Kez ∓ κK)c/ω.

(8)

In Eqs. (6) and (7), rs and rp are the Fresnel coefficients associated with the reflection of the TE and TM modes at the mirror, respectively. The noise term in Eq. (4) is related to the losses in the half-space z > 0 according to the fluctuation-dissipation theorem. In the following, we will neglect this noise term which is justified for a mirror with near-perfect reflectivity. With standard projection operator techniques [2], we derive a master equation for the reduced density operator ˆ of the atom including the center 4

4

nature physics | www.nature.com/naturephysics

supplementary information

doi: 10.1038/nphys1961

of mass motion. The momentum components in the mirror plane are traced out such that only the z coordinate of the atomic motion is retained. With ˆgg = g|ˆ |g and ˆee = e|ˆ |e, we find 3 ∂t ˆgg (t) = γ 8

1

du[rs∗ eik0 uˆz + e−ik0 uˆz ]

(9)

0

+

1 0

׈ ee (t)[rs e−ik0 uˆz + eik0 uˆz ] duu2 [rp∗ eik0 uˆz − e−ik0 uˆz ]  ׈ ee (t)[rp e−ik0 uˆz − eik0 uˆz ] ,

where γ is the full decay rate of the excited state |e in free space and we assumed that the dipole moment d is oriented parallel to the mirror. At t = 0, we suppose that the atom is prepared in the state ˆgg (0) = 0 and ˆee (0) = |ψ0 ψ0 |, where |ψ0  is the motional state of the atom before spontaneous emission that we specify below. The quantum state of the atom after ∞ spontaneous emission is given by ˆgg (∞) = 0 dt∂t ˆgg (t) and can  ∞be obtained from the right-hand side of Eq. (9) if ˆee is replaced by ˆ0 = 0 dtˆ ee (t). If the mirror were absent, the integral would be given by ˆ0 = [ˆ ee (t = 0)/γ]. The mirror gives rise to a position-dependent decay rate that varies on a length scale determined by the wavelength of the optical transition. In order to account for this effect, we take ˆ0 = (α/γ)|ψ0 ψ0 |, where α is fixed by the condition Tr[ˆ gg (∞)] = 1. Note that in the present case, the effect of the mirror on ˆ0 is small since the spatial extend of |ψ0  is larger than the wavelength of the transition. The quantum state of the atom after spontaneous emission can then be written as 3 ˆgg (∞) = α 8

1 0

�  du |ψs ψs | + u2 |ψp ψp | ,

(10)

where  rs∗ eik0 uˆz + e−ik0 uˆz |ψ0 ,  � |ψp  = −rp∗ eik0 uˆz + e−ik0 uˆz |ψ0 . |ψs  =



(11) (12)

The operators e±ik0 uˆz in Eqs. (11) and (12) describe the transverse recoil momentum ±k0 u transferred to the atom by the spontaneously emitted 5

nature physics | www.nature.com/naturephysics

5

supplementary information

doi: 10.1038/nphys1961

photon. It follows that |ψs  and |ψp  represent a coherent superposition of two wave packets that travel at different mean velocities. The state  i |ψ0  = dpf (p, d)e  pd eiφf (p) |p (13) describes the motional state of the atom before spontaneous emission and is modeled as a coherent wave packet, where f (p, d) > 0 and [f (p, d)]2 represents the initial momentum distribution of the atoms for a given atom-mirror distance d. This distribution is inferred from an independent measurement i and approximately given by an assymetric Lorentzian. The factor e  pd in Eq. (13) ensures that the distance between the center of the wavepacket and the mirror is given by d. On the other hand, the phase φf determines the unknown shape of the wave packet in position space and acts as a free parameter that we set to zero in the following. Note that a mixture of momentum states would not contain any information about the position, since a momentum state is completely delocalized in space. Therefore, the experimentally observed dependence of the interference signal on the atom-mirror distance can only be captured in a model where the atoms are described by a coherent wave packet. Furthermore, the description of the initial atomic state by a pure state, rather than a mixture of pure states, is a good approximation here since the width of the slit collimating the atoms is chosen to be close to the diffraction limit. Bragg grating. The Bragg grating can be described by a unitary operator Ubragg that transfers a momentum state |p with p > 0 into the superposition state Ubragg |p = cos ϕp |p + sin ϕp e−iφB |p − 2kL ,

(14)

where φB is a phase factor and kL is the wave number of the grating. cos ϕp and sin ϕp are the amplitudes of the transmitted and the diffracted components, respectively. These amplitudes are inferred from an independent measurement that characterizes the Bragg grating. Similarly, a momentum state |q with q < 0 becomes Ubragg |q = cos ϕq |q − sin ϕq eiφB |q + 2kL .

(15)

After the diffraction at the Bragg grating, the atomic wave packet undergoes a free time evolution with Ufree = exp[−iˆ p2z /(2m)T ] for T = 33.5ms. The quantum state at the detector is thus given by ˆ(detector) = (Ufree Ubragg )ˆ gg (∞)(Ufree Ubragg )† ,

(16)

6

6

nature physics | www.nature.com/naturephysics

supplementary information

doi: 10.1038/nphys1961

where ˆgg (∞) is defined in Eq. (10). In order to determine ˆ(detector), it suffices to consider the time evolution of the states |ψ  ( ∈ {s, p}) that are defined in Eqs. (11) and (12). The Bragg grating followed by free time evolution transfers |ψ  into a state comprised of four terms, |ψ˜  = Ufree Ubragg |ψ  = |T−  + e−iφB |D−  + |T+  + eiφB |D+ .

(17)

In Eqs. (11) and (12), the wave packet described by e−ik0 uˆz |ψ0  travels in the negative z direction due to the momentum kick by spontaneous emission. The Bragg grating splits this wave packet into a transmitted and a diffracted part denoted by |T−  and |D+ , respectively. Here the superscript ± indicates the propagation direction after the grating. In addition, the wave packet eik0 uˆz |ψ0  travelling in the positive z direction after spontaneous emission is split into a transmitted and a diffracted part denoted by |T+  and |D− , respectively. Detection. The detector measures the probability to find an atom in the region z0 ± ∆z /2, where ∆z = 51.6µm. Next we determine the probability P for the detection of an atom in state |ψ˜  and in an interval around z0 < 0. If |z0 | is sufficiently large, only the wave packets |T−  and |D−  overlap such that P = I + 2 ReA , (18) where Re denotes the real part and I =

z0 +∆  z /2

z0 −∆z /2

  dz |z|T− |2 + |z|D− |2 ,

A = e−iφB

z0 +∆  z /2

dz z|D− z|T− ∗ .

(19)

(20)

z0 −∆z /2

In order to obtain the full detection probability P , we have to take the average over the incoherent mixture of states according to Eq. (10), 3 P =α 8

1

du[Ps (u) + u2 Pp (u)].

(21)

0

7

nature physics | www.nature.com/naturephysics

7

supplementary information

doi: 10.1038/nphys1961

If the latter equation is compared to Eq. (2), it follows that 3 P0 = α 8  1

3 A = α Abs 4

0

1 0

  du Is + u2 Ip , 

du As + u2 Ap





.

(22)

(23)

Finally, the visibility of the measured interference pattern is given by V = A/P0 .

References [1] Di Stefano, O., Savasta, S. & Girlanda, R. Three-dimensional electromagnetic field quantization in absorbing and dispersive bounded dielectrics. Phys. Rev. A 61, 023803 (2000). [2] Breuer, H.-P. & Petruccione, F. The Theory of Open Quantum Systems (Oxford University Press, 2006).

8

8

nature physics | www.nature.com/naturephysics