Analysis of Kapitza-Dirac diffraction patterns beyond the Raman-Nath regime
Bryce Gadway, Daniel Pertot, Rene Reimann, Martin G. Cohen, Dominik Schneble
AAnalysis of Kapitza-Dirac diffractionpatterns beyond the Raman-Nath regime
Bryce Gadway, Daniel Pertot, Ren´e Reimann, Martin G. Cohen, andDominik Schneble
Department of Physics and Astronomy, Stony Brook University,Stony Brook, NY [email protected]
Abstract:
We study Kapitza-Dirac diffraction of a Bose-Einstein con-densate from a standing light wave for a square pulse with variable pulselength but constant pulse area. We find that for sufficiently weak pulses, theusual analytical short-pulse prediction for the Raman-Nath regime contin-ues to hold for longer times, albeit with a reduction of the apparent modu-lation depth of the standing wave. We quantitatively relate this effect to theFourier width of the pulse, and draw analogies to the Rabi dynamics of acoupled two-state system. Our findings, combined with numerical model-ing for stronger pulses, are of practical interest for the calibration of opticallattices in ultracold atomic systems.
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References and links
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1. Introduction
The diffraction of matter-waves from a standing light wave is a fundamental concept in atomoptics [1, 2]. Originally predicted by Kapitza and Dirac [3] for electrons more than 75 years ago(and recently also observed for these [4]), it was first demonstrated in the 1980s with an atomicbeam [5], and has since become a standard tool in atom interferometry for the coherent mixingof momentum modes [6, 2]. The advent of Bose-Einstein condensates in the 1990s [7] hasmade it possible to directly observe the dynamics of matter-wave diffraction in time-of-flightimages [8, 9, 10]. Moreover, the diffraction of condensate atoms from standing light wavesfinds applications in high-resolution spectroscopy [11] and metrology [12, 13], and plays afundamental role in superradiance [14, 15].Two diffraction scenarios can generally be distinguished. In the resonant Bragg case, theatoms oscillate between two resonantly coupled plane-wave momentum states depending onthe strength and duration of the interaction with the light field [16]. If the Bragg condition isnot met, atoms can nevertheless be diffracted into a number of off-resonant momentum states,provided that the interaction is sufficiently short and strong [5]. In accordance with a commonconvention in atom optics [17] we refer to this case as Kapitza-Dirac diffraction, but we extendit to include times beyond the Raman-Nath regime. In this context, it is interesting to note asimilar discussion for the diffraction of light from sound waves in acousto-optic devices [18].In the Raman-Nath regime, i.e. when atomic motion during the interaction with the light fieldcan be neglected, the populations of the diffracted states exclusively depend on the product ofthe strength V and the duration τ of the interaction, i.e. the area of the applied pulse [5].Outside the Raman-Nath regime, the diffraction dynamics exhibits collapses and revivals forconstant interaction strength [8, 9, 10]. In this paper, we now specifically analyze the case ofa pulsed interaction of variable duration but constant pulse area. This allows for a study of thebreakdown of the Raman-Nath prediction and, in particular, for a quantification of deviationswhen the system is close to, but not deep into, the Raman-Nath regime.The findings of our study are of direct interest for the elimination of systematic errors incalibration measurements for experiments with ultracold atoms in optical lattices [19, 20] whenthe lattice depth is determined via Kapitza-Dirac diffraction.This paper is organized as follows: Section 2 reviews general aspects of the matter-wavediffraction from a one-dimensional optical lattice, while Section 3 briefly describes our experi-mental system. In Section 4 we analyze diffraction patterns for weak pulses, and give an analyt-cal modification to the Raman-Nath diffraction formula, which is motivated by a comparisonof the diffraction dynamics with that of a coupled Rabi system, as well as by considering thespectral properties of the pulse. Section 5 discusses the more complicated dynamics of strongpulses, and Section 6 deals with calibration methods for the depth of optical lattices, comparingnumerical simulations with single-shot diffraction patterns not restricted to the Raman-Nathregime.
2. Raman-Nath regime
We first briefly review general aspects of the diffraction of a condensate from a standing lightwave with wavenumber k = π / λ that is switched on for a duration τ . The standing wavegives rise to a sinusoidal optical potential V cos kz [19]. The evolution of the condensate inthe standing wave can then be modeled [21, 22] (neglecting mean-field interactions) as that ofa matter wave ψ subject to the Hamiltonianˆ H = − ( ¯ h / m ) ∂ z + V cos kz (1)where m is the atomic mass. Expanding the condensate wave function in the basis of planewaves populated by diffraction from a standing wave as ψ ( t ) = ∑ n c n ( t ) e i nkz (where n = , ± , ± , . . . and c n ( t = ) = δ n , ) and introducing the dimensionless parameters α = ( E ( ) r / ¯ h ) τ (2) β = ( V / ¯ h ) τ , (3)(where E ( n ) r = ( n ¯ hk ) / m denotes the n -photon recoil energy, with E ( ) r ≡ E r ), transforms thetime-dependent Schr¨odinger equation into a set of coupled differential equations i d c n d t = α n τ c n + β τ ( c n − + c n + c n + ) (4)for the amplitudes c n ( t ) of the diffracted orders n . For a given lattice depth, the highest momen-tum order ( ± n ¯ hk ) capable of being populated is given by the cutoff¯ n = (cid:112) β / α (5)for which the potential energy is fully converted into kinetic energy (cf. [10]). Note that in theseequations, α is the pulse duration τ in units of the 2-photon recoil time τ ( ) r = ¯ h / E ( ) r , and β measures the area of the pulse.The dynamics of the condensate in the standing wave depends on the ratio between α n and β , i.e. between the kinetic energies acquired during diffraction and the depth of the potential.The Raman-Nath approximation consists of neglecting the α n terms in Eqs. (4), which isjustified if τ is much shorter than the harmonic oscillation period in a potential well [8], suchthat τω ho (cid:28)
1, with ω ho = [ V E ( ) r ] / / ¯ h , or equivalently β α (cid:28)
1. In this case, the solutionof Eqs. (4) is c n ( t ) = ( − i ) n e − i β t / τ J n ( β t / τ ) , such that the population P n = | c n | of the n thdiffracted order after application of the pulse is given by P n = J n (cid:18) β (cid:19) , (6)where the J n are Bessel functions of the first kind. . Experimental procedure In the experiments described in this paper, we subject an optically trapped Rb Bose-Einsteincondensate to a vertically-oriented, pulsed standing light wave at 1064 nm, for which the 2-photon recoil time τ ( ) r ≈ µ s. A description of our apparatus for the production of conden-sates is given in ref. [23]. In brief, in a nearly isotropic crossed-beam optical dipole trap weproduce condensates typically containing 5 × atoms in the | F = , m F = − (cid:105) hyperfineground state without a discernible thermal fraction. The standing light wave, with a Gaussian1 / e radius of 130 µ m at the position of the condensate, is derived from a single-frequencyytterbium fiber laser with a linewidth of 70 kHz and can be switched off within 1 . µ s usingan acousto-optic modulator. Immediately after application of the pulse, the atoms are releasedfrom the optical trap and are imaged after 15 ms time-of-flight using near-resonant absorptionimaging on the cycling transition.A series of images taken for pulses of variable duration τ but with constant pulse area β = . ħ k+4 ħ k+2 ħ k0 ħ k-2 ħ k-4 ħ k-6 ħ k µs Fig. 1. Time-of-flight (TOF) absorption images of a condensate diffracted from a 1064nm standing-wave optical pulses of constant area β = V τ / ¯ h = .
5, and durations rangingfrom τ = µ s ( α = τ / τ ( ) r ≈ .
2) to 360 µ s ( α ≈ τ (cid:38) µ s (forwhich [ β / α ] / (cid:46) . ( β / ) sinc ( α / ) (see text). The populations of the 0¯ hk and ± hk momentum orders in the series of Fig. 1 are shownin Figs. 2 (A) and (B), together with corresponding data for a weaker standing wave with β = .
5, for which diffracted orders 2 n ¯ hk with | n | > β = .
5, this is only the case for pulse durationsexceeding 75 µ s due to contributions of the ± hk orders at shorter durations, as shown ininset (C).
4. Weak-pulse dynamics
For sufficiently weak pulses with a cutoff ¯ n ≈ ± hk arepopulated, Eqs. (4) can be reduced to three coupled equations; this approximation is valid forshallow potentials V (cid:46) E r . The solutions c and c ± to Eqs. (4), which can be obtained in atraightforward way, then lead to P ± = β β + α sin (cid:32) (cid:112) β / + α (cid:33) (7)and P = − P ± . This can be cast into the standard form of a Rabi oscillation P ± = (cid:16) χ Ω (cid:17) sin (cid:18) Ω τ (cid:19) (8)with generalized Rabi frequency Ω = (cid:2) χ + ∆ (cid:3) / , resonant coupling χ = V / √ h and detun-ing ∆ = E ( ) r / ¯ h = ω ( ) r between the atomic momentum states 0¯ hk and ± hk . Another way tointerpret this result is the following: each scattering event that changes the atomic momentumfrom 0 to ± hk by energy conservation leads to a frequency mismatch of the scattered photonby ∆ with respect to the standing wave, requiring a corresponding Fourier width of the pulse forstimulated scattering to be able to occur. Consequently, the drop-off of populations in higherorders with increasing α can be seen as resulting from the decrease in the Fourier width of thepulse, which motivates a more general modification of Eq. (6) to P n = J n (cid:18) β α (cid:19) , (9)where the sinc function arises from the Fourier transform of the square pulse. Indeed, the pop-ulations P , ± predicted by Eq. (9) agree with those of Eq. (7) up to O ( α β ) , independentof the pulse duration α or pulse area β . For [ β / α ] / ≤
1, the agreement of Eq. (9) with theexperimentally observed patterns is excellent, as can be seen in Fig. 2 (A) and (B).It is interesting to note the connection of Eq. (9) to the resonant case of n th order Braggdiffraction from a moving standing wave (with a relative detuning δ ω = n ∆ between the twobeams). In this case, the Fourier transform of the applied potential V cos ( kz + δ ω t / ) withduration τ would lead to a frequency distribution ∝ sinc [( ω − δ ω ) τ / ] (i.e. identical to that ofthe standing wave case, but shifted by δ ω ) making available the resonant frequency component,independent of the pulse duration.The form of Eq. (9) shows that although the populations P n in a spectrum may follow aBessel distribution as predicted by the Raman-Nath Eq. (6), a fit will not necessarily returnthe correct modulation depth. The apparent pulse area in Eq. (9) is given by ( β / ) sinc ( α / ) ,which means that for a pulse of duration τ , the apparent modulation depth of the standing waveextracted from a diffraction spectrum is reduced from V to V , eff = V sinc (cid:16) τ / τ ( ) r (cid:17) . (10)The apparent pulse area returned by a fit of Eq. (6) to the diffraction patterns is shown in Fig.2 (D) for both the β = . β = . | n | > β = .
5, the observed behavior still shows qualitative agreement with revivals fromsampling of the sinc-shaped frequency distribution with increasing pulse duration τ , but clearlyexhibits deviations when higher-order momentum states are present.
5. Strong-pulse dynamics
For more intense pulses with ¯ n >
1, the simple generalization of Eq. (9) no longer holds, as isevident by the large discrepancies in Fig. 2 for τ < µ s in the β = . P k P k β eff β ABD P k C Fig. 2. Suppression and revival of atomic diffraction from a constant-area standing wavelight pulse with increasing pulse length. (A and B) Relative populations of the central con-densate 0¯ hk and orders ± hk vs. normalized pulse duration α = τ / τ ( ) r ∼ τ / µ s. Filledblue dots and open red squares refer to pulses of area β = . J n [( β / ) sinc ( α / )] . The dashed black lines cor-respond to numerical fits for β = .
5. (C) Decay of the ± hk orders in the β = . J [( β / ) sinc ( α / )] .(D) Behavior of β eff / β , where the atomic diffraction patterns are fit with the distribution J n ( β eff / ) . The black solid line is sinc ( α / ) , and the dashed black line corresponds to anumerical fit using β = . The presence of higher diffraction orders leads to population decays that are faster than thosedetermined by the two-photon timescale in Eqs. (9) and (10). This is shown in inset (C) of Fig.2 for the ± hk ( n =
2) orders, which require the exchange of four photons in the transitionfrom 0¯ hk . In general, the presence of many higher momentum orders will complicate analyticdescriptions of the diffraction dynamics.Nevertheless, it is possible to accurately describe the observed dynamics by numerical inte-gration of the coupled differential Eqs. (4), truncated for unpopulated higher orders beyond ¯ n .This is shown in Figs. 2 and 3 (A).
6. Depth calibration of optical lattices
In experiments with ultracold atoms in optical lattices, the tunneling rate depends exponentiallyon the lattice depth V [19], for which an accurate knowledge thus is essential. Unlike mostother methods (cf. [19]), applying a “short” Kapitza-Dirac diffraction pulse can convenientlyreveal the lattice depth in a single-shot measurement. However, the application of well-definedpulses that are short enough to be deeply in the Raman-Nath regime, yet strong enough toyield significant diffraction, can be technically challenging. The present study quantifies howthe results of such a calibration measurement need to be corrected if pulses of finite length areused. • • •• • • • • • • • • • • • • • •• • • • • • • • • • • • • •• • • •• • • • • • • • • • • • • • τ = 25 µsτ = 50 µsτ = 8 µs Lattice Depth [E rec ] M o m e n t u m S t a t e P o p u l a t i o n s
20 30 40 50 600102030405060 F i t D e p t h [ E r e c ] f r o m µ s d a t a Expected Lattice Depth [E rec ]from 8µs data BA Fig. 3. (A) Breakdown of the analytical description. For each data set, numerical simula-tions (using Eqs. (4)) of the momentum state distribution are shown as solid curves. Thered crosses are data for the 0¯ hk order, and the open squares (filled dots) are data averagedfor the ± hk ( ± hk ) orders. (B) Comparison of lattice depth calibrations from relativelyshort (8 µ s) and long (50 µ s) optical lattice pulses for various intensities of the standinglight wave. The results for numerically-fit lattice depths agree to within 4% (the solid linehas a slope of 1). In Fig. 3 we present the comparison of experimentally observed momentum state distribu-tions to numerical simulations, for three different pulse lengths (8, 25, and 50 µ s), as the latticedepth is varied from 0 to 65 E r . While the observed momentum state distributions are close tothe typical Raman-Nath form Eq. (6) for the 8 µ s pulses, this is no longer the case for longerpulses. For all pulse lengths, numerical simulations agree well, as can be seen in Fig. 3 (A).The lattice depth calibrations for the three pulse durations, determined by comparison of exper-imental data to numerically simulated patterns, agree to within 4%, as can be seen in Fig. 3 (B)for the extreme cases of 8 and 50 µ s. Despite the complicated form the dynamics take, onecan thus obtain a reliable lattice depth calibration even with longer pulse durations. We haveindependently verified the method described here with another single-shot calibration methodfor deep lattices (adiabatic lattice rampup, followed by a sudden projection of the ground bandpopulation onto the ± hk plane-wave states [24]), for which we found comparable agreementto within 10%.
7. Conclusion