Atomic Raman scattering: Third-order diffraction in a double geometry
Sabrina Hartmann, Jens Jenewein, Sven Abend, Albert Roura, Enno Giese
AAtomic Raman scattering: Third-order diffraction in a double geometry
Sabrina Hartmann, Jens Jenewein, Sven Abend, Albert Roura, and Enno Giese Institut f¨ur Quantenphysik and Center for Integrated Quantum Science andTechnology (IQ ST ), Universit¨at Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, Welfengarten 1, D-30167 Hannover, Germany Institute of Quantum Technologies, German Aerospace Center (DLR), S¨oflinger Straße 100, D-89077 Ulm, Germany
In a retroreflective scheme atomic Raman diffraction adopts some of the properties of Braggdiffraction due to additional couplings to off-resonant momenta. As a consequence, double Ramandiffraction has to be performed in a Bragg-type regime. Taking advantage of this regime, doubleRaman allows for resonant higher-order diffraction. We study theoretically the case of third-orderdiffraction and compare it to first order as well as a sequence of first-order pulses giving rise tothe same momentum transfer as the third-order pulse. In fact, third-order diffraction constitutes acompetitive tool for the diffraction of ultracold atoms and interferometry based on large momentumtransfer since it allows to reduce the complexity of the experiment as well as the total duration ofthe diffraction process compared to a sequence.
I. INTRODUCTION
Higher-order Bragg diffraction [1–5] in combinationwith sequential pulses [6, 7] has become a standard toolfor large-momentum-transfer (LMT) techniques to en-hance the sensitivity of light-pulse atom interferometers[8, 9]. However, with Raman diffraction [8, 10, 11], thethe other main mechanism, only sequential pulses [12–14]have routinely been employed so far. In this article, weextend Raman in a double-diffraction geometry [15–17]to also allow for higher-order diffraction and study theefficiency compared to a standard first-order sequence.Such a setup retains the possibility of state-selective de-tection, while being more efficient and less complex thana sequence of first-order pulses for narrow momentumdistributions.Bloch oscillations [18–22], higher-order diffraction [1–5], and sequential pulses [6, 7, 12–14, 23] are some ofthe most commonly used techniques used for LMT ap-plications based on Bragg diffraction. They are com-plemented by double diffraction [2, 6, 15–17], where anatom at rest diffracts from two counterpropagating lightgratings in two opposite directions. The latter is par-ticularly well suited for experiments under microgravityconditions [24–30] or for horizontal geometries [31, 32].Due to its symmetry, laser phases are not imprintedon the two branches of the interferometer, and similarnoise sources are intrinsically suppressed [2, 6, 12]. Eventhough many applications of double diffraction focus onBragg, the geometry was first pioneered for Raman andis still used to date as one of the few LMT techniquesfor Raman diffraction, together with sequential pulses.However, one of the benefits of double Raman diffractionhas not been explored so far, namely the possibility toscatter into higher diffraction orders.In contrast to single Raman, which can be described asa closed two-level system, off-resonant couplings appearin single Bragg diffraction [33, 34], limiting the opera-tion to the Bragg regime but at the same time allowingfor higher-order diffraction [35]. The additional grating in double diffraction induces further off-resonant transi-tions for both Raman and Bragg diffraction. As a conse-quence, the application of Raman diffraction is restrictedto a Bragg-type regime as well. In double Bragg diffrac-tion resonant and off-resonant couplings at the same mo-mentum state appear causing a more complex diffractionbehavior [35, 36]. However, these features do not appearin double Raman diffraction, which therefore constitutesa simpler diffraction mechanism.In this article we demonstrate that third-order dou-ble Raman diffraction with high efficiency is possible, al-though it is more velocity selective than its first-ordercounterpart. However, for narrow momentum distribu-tions like the ones associated with Bose-Einstein conden-sates (BECs) it can be a competitive alternative to apulse sequence when the duration of the beam splittingprocess is limited.In Sec. II we recall first-order double Raman diffrac-tion with a Gaussian pulse shape as well as sequen-tial Doppler-detuned single-diffraction with typical box-shaped pulses to calculate the efficiency of an LMT beamsplitter. Such a combination of Gaussian and box-shapedpulses constitutes a good compromise between diffrac-tion efficiency and overall duration of the sequence. Wethen perform in Sec. III an analysis of third-order Ramandiffraction and show that even though its efficiency is in-herently worse than a comparable first-order pulse, it canbe better than that of the sequence. We conclude with abrief discussion in Sec. IV. For completeness, the generalset of differential equations for double Raman diffractionis given in the Appendix.
II. FIRST-ORDER DIFFRACTIONA. Double Raman diffraction
An atom at rest interacts with two strongly detunedoptical gratings (with a detuning much larger than thelinewidth) that move in opposite directions, each one a r X i v : . [ qu a n t - ph ] J u l FIG. 1. Schematic setup with an atom at rest ( p = 0) ina retroreflective geometry built by a λ/ ω b and emits a photon ω r in opposite direction from each grating. This process causesa total recoil of ± (cid:126) K with K = ( ω b + ω r ) /c and by that leadsto a gain of kinetic energy (cid:126) ω K . The energy-momentum dia-gram in (b) shows that such a process is resonant if the energydifference between the light fields (cid:126) ∆ ω ≡ (cid:126) ( ω b − ω r ) equalsthe energy difference (cid:126) ω eg between the atomic ground | g (cid:105) andexcited state | e (cid:105) in addition to the recoil energy (cid:126) ω K . Reso-nant processes start and end on the parabola (solid arrows),off-resonant processes are denoted by dashed arrows. generated by counterpropagating light fields of frequen-cies ω b and ω r , see Fig. 1(a). The gratings can be dis-tinguished by their polarization [12], so that within aretroreflective setup, where both light fields are guidedfrom one side to the atom and retroreflected at theother side, the polarizations have to be rotated by a λ/ ω b and subsequentlyemitting a photon with frequency ω r in the oppositedirection from each grating. This process causes a to-tal momentum recoil of ± (cid:126) K for the two gratings, with K ≡ ( ω b + ω r ) /c , and the atom gains a kinetic energy (cid:126) ω K , in terms of the recoil frequency ω K = (cid:126) K M , (1)where M is the atomic mass.The diffraction process is determined by the trans-ferred energy, i. e. by the difference of the laser fre-quencies ∆ ω ≡ ω b − ω r . A transition that is resonantfor first-order diffraction corresponds in Fig. 1(b) to the case where the solid arrows start and end on a parabola,the kinetic energy of an internal state. This is possible if (cid:126) ∆ ω equals the kinetic energy (cid:126) ω K gained through recoilplus the energy difference (cid:126) ω eg between internal ground | g (cid:105) and excited state | e (cid:105) , i.e.∆ ω = ω eg + ω K . (2)Since the AC Stark shift can in principle be compensated,we refrain from including it in the subsequent discussionor the resonance condition.The two gratings allow simultaneous diffraction in op-posite directions but also enable spurious off-resonanttransitions to higher diffraction orders denoted by dashedarrows. Additional couplings through polarization im-perfections are neglected throughout this article. More-over, we assume plane waves and neglect wave front dis-tortions.The diffraction process depicted in Fig. 1(b) is de-scribed by the truncated system of differential equations˙ g = iΩ e − i ω D t e + iΩ e i ω D t e − (3a)˙ e ± = iΩ e ∓ i ω D t e − i4 ω K t g ± + iΩ e ± i ω D t g , (3b)coupling the ground state probability amplitudes g n ≡ g ( p + n (cid:126) K ) for the momentum eigenstate | p + n (cid:126) K (cid:105) tothe excited state amplitudes e n ≡ e ( p + n (cid:126) K ). The sys-tem of equations is derived from the generalized versionof the differential equations describing double Ramandiffraction presented in the Appendix. Rabi oscillationstake place between the probability amplitude g of theground state and those of the excited state with two dif-ferent momenta, e and e − . At the same time, the prob-ability amplitudes of the excited states e ± couple off-resonantly to g ± indicated with a detuning 4 ω K . Thesekind of transitions are prominent in the Raman-Nath ( Kapitza-Dirac ) regime [35, 37] where Ω /ω K (cid:38)
1, butare suppressed in the
Bragg -type regime with Ω /ω K (cid:28) e ± couples further to higher diffraction orders, but thesetransitions are even more off-resonant and therefore sup-pressed in the Bragg-type regime. The Doppler frequency ω D = pK/M corresponds to the deviation from the res-onant momentum p = 0 within a wave packet and actsas a detuning to the resonant transition, leading to theeffect of velocity selectivity [10, 11, 38–40]. As couplingstrength Ω( t ) ∝ Ω exp[ − t / (2∆ τ )] we consider a Gaus-sian function of width ∆ τ .The coupling strength is connected to the pulse area A via A = (cid:90) d t √ t ) . (4)An area of A = π/ | g, (cid:105) → ( | e, (cid:126) K (cid:105) + | e, − (cid:126) K (cid:105) ) / √
2, creating a superposition of left-and right-moving wave-packet components and thereforecorresponds to a double-Raman beam splitter. . . . . .
20 ∆ τ (µs) ∆ ℘ / ( (cid:126) K ) . . . . . . τ opt FIG. 2. Efficiency E ± for a first-order double-diffractionbeam splitter as a function of the width of the initial wavefunction ∆ ℘ and the pulse duration ∆ τ for a Gaussian pulseshape. Higher order diffraction appears in the Raman-Nathregime (drop in efficiency for small ∆ τ ), for higher ∆ τ lossesare determined by velocity selectivity.
1. Numerical treatment
We numerically solve the system of differential equa-tions Eq. (18) using the corresponding resonance condi-tion Eq. (2) for Rb with Matlab ’s ODE45, a Runge-Kutta algorithm, with relative accuracy 10 − and ab-solute accuracy 10 − . We calculate a transition func-tion G ∆ τ ( p f , p i ) which connects the initial and final mo-mentum eigenstates [36]. The transition function canbe applied to the initial Gaussian wave packet ψ i ( p i ) ∝ exp[ − ( p i − p ) / (4∆ ℘ )] with p = 0 to obtain the finalwave function ψ f ( p f ) = (cid:90) d p i G ∆ τ ( p f , p i ) ψ i ( p i ) . (5)We truncate the range of momenta so that the solutionfor the diffraction efficiency (discussed in the followingparagraph) obtained with n max and n max+1 is at most ofthe same magnitude as the solver accuracy.
2. Diffraction Efficiency
We define the efficiency of an n th-order symmetricdiffraction process between the momenta |± n (cid:126) K (cid:105) and |± ( n + n ) (cid:126) K (cid:105) as E n ± n = (cid:90) p + p − d p f | ψ f ( p f ) | + (cid:90) − p − − p + d p f | ψ f ( p f ) | (6)with the integration range p ± = ( n + n ± / (cid:126) K and n, n ∈ N . Even though the expression works for ar-bitrary initial momenta, we have restricted ourselves tointeger momenta p = n (cid:126) K that are relevant for se-quences of pulses. FIG. 3. Energy-momentum diagram and resonant transitionsfor an atom with initial momentum p = (cid:126) K (solid arrows)and initially in the excited state. The Doppler detuning of thespurious grating suppresses off-resonant transitions (dashedarrows), turning double into single diffraction. The initialconditions are chosen so that they correspond to a resonantsequential pulse following a double-diffraction beam splitter. The efficiency E ± of the first-order double Ramanbeam-splitter process sketched in Fig. 1(b) is shown inFig. 2 as a function of the pulse duration ∆ τ and thewidth of the initial wave function ∆ ℘ . For short pulse du-rations (i. e. in the Raman-Nath regime) diffraction intohigher off-resonant orders becomes important and the ef-ficiency of the beam-splitting process drops. For longerpulses, an efficiency close to unity demonstrates thatdiffraction in the Bragg-type regime leads to the targetedbeam splitter. However, the longer the pulse, the moredominant the Doppler detuning becomes, which leads tovelocity selectivity and the diffraction efficiency drops forbroad momentum distributions. The red dashed line de-notes the optimal pulse duration ∆ τ opt at intermediatetimes in the quasi-Bragg regime [35] and with highest ef-ficiency for a broad range of different momentum widths∆ ℘ . For that, we determine for each value ∆ ℘ the pulseduration at which the maximal efficiency occurs and cal-culate the median over all ∆ ℘ . It will later be used for acomparison between diffraction schemes. B. Doppler-detuned Raman diffraction
Atoms in a retroreflective setup with initial momen-tum p interact predominantly with only one of the twolaser pairs because the other pair is Doppler-detunedby p K/M . Consequently, the double-diffraction pro-cess turns into a single-diffraction process, shown bysolid arrows in Fig. 3. Note that if the atom is in asuperposition of momenta ± p , two opposite but inde-pendent single-diffraction processes occur. However, theoff-resonant Doppler-detuned transitions (dashed arrowsfrom | e, ± (cid:126) K (cid:105) to | g, (cid:105) in Fig. 3) are still present andcause a shift of the addressed atomic energy levels and bythat detuned Rabi oscillations. A small detuning leads tothe two-photon light shift [41, 42], while a large detuningreduces the diffraction efficiency. In contrast to Braggdiffraction, for Raman diffraction adiabatic eliminationallows to identify the differential energy shift ∆ E/ (cid:126) fortime-independent pulse shapes [43]. For the momenta p = n (cid:126) K with n (cid:15) N that are of interest to our studyof sequential pulses, one obtains for the widely used box-shaped pulses with Rabi frequency Ω the following dif-ferential energy shift∆ E/ (cid:126) ≡ ω K δ = ± Ω ω K n + 14 n ( n + 1) . (7)The negative sign corresponds to transitions from | g, n (cid:126) K (cid:105) to | e, ( n + 1) (cid:126) K (cid:105) while the positive sign corre-sponds to transitions from | e, n (cid:126) K (cid:105) to | g, ( n + 1) (cid:126) K (cid:105) .The detuning caused by this shift can be compensatedby modifying accordingly the resonance condition fromwhich ∆ ω is obtained. Box-shaped pulses are commonlyemployed for sequential pulses, as they are easy to im-plement experimentally and have shorter durations com-pared to Gaussian pulses, while they can maintain a highdiffraction efficiency.To demonstrate this effect, we consider in the following p = ± (cid:126) K , depicted in Fig. 3, and p = ± (cid:126) K for boxshaped pulses i.e. Ω( t ) = Ω . The pulse duration τ corresponds to the temporal length of the box and differssignificantly from the width ∆ τ of a Gaussian pulse.The resonance condition for transition | e, ± (cid:126) K (cid:105) →| g, ± (cid:126) K (cid:105) , i.e. p = ± (cid:126) K as depicted in Fig. 3 is givenby ∆ ω = ω eg − ω K + ∆ E/ (cid:126) = ω eg − (3 − δ ) ω K (8)with δ = 3Ω / (8 ω K ). The system of differential equa-tions in an appropriate rotating frame reduces then toan effective two-level system without light shifts: (cid:18) ˙ e ± ˙ g ± (cid:19) = iΩ (cid:18) ∓ i ω D t e ± i ω D t (cid:19) (cid:18) e ± g ± (cid:19) . (9)Keeping in mind the differences between single and dou-ble diffraction, we now investigate π pulses by choosing A = π = 2 Ω τ. (10)Equation (9) is analytically solvable, but to also calcu-late loss to off-resonant states that inevitably appearsbeyond the Bragg-type regime, we resort to a numericaltreatment.Similarly, the resonance condition for the transition | g, ± (cid:126) K (cid:105) → | e, ± (cid:126) K (cid:105) with p = ± (cid:126) K takes the from∆ ω = ω eg + (5 + δ ) ω K (11)with δ = − / (24 ω K ). It can be reduced to a two-level-system between | g, ± (cid:126) K (cid:105) and | e, ± (cid:126) K (cid:105) similar toEq. (9). . . . . . ∆ ℘ / ( (cid:126) K ) . . . . τ (µs) ∆ ℘ / ( (cid:126) K ) . . . . . . (a)(b) τ opt FIG. 4. Efficiency for Doppler-detuned box shaped pulses forvarying width of the initial wave function ∆ ℘ and pulse du-ration τ . In panel (a) the process | e, ± (cid:126) K (cid:105) → | g, ± (cid:126) K (cid:105) andin panel (b) the process | g, ± (cid:126) K (cid:105) → | e, ± (cid:126) K (cid:105) is shown. Inthe Raman-Nath regime both processes show transitions intoother diffraction orders that reduce for increasing p , whichmakes the pulse in panel (b) more efficient. Using the resonance condition Eq. (8), we calculate thediffraction efficiency E ± for the Doppler-detuned transi-tion | e, ± (cid:126) K (cid:105) to | g, ± (cid:126) K (cid:105) and using the resonance con-dition Eq. (11) to calculate the efficiency E ± for thetransition | g, ± (cid:126) K (cid:105) and | e, ± (cid:126) K (cid:105) with an analogousnumerical treatment as discussed in Sec. II A 1. The onlydifferences are the modified resonance conditions and boxshaped pulses i.e. Ω( t ) = Ω . Moreover, the initial wavepacket is a superposition of two Gaussians centered at ± p described by ψ i ( p i ) ∝ exp (cid:20) − ( p i − p ) (4∆ ℘ ) (cid:21) + exp (cid:20) − ( p i + p ) (4∆ ℘ ) (cid:21) . (12)Figure 4(a) shows the efficiency for p = ± (cid:126) K andFig. 4(b) the efficiency for p = ± (cid:126) K defined throughEq. (6) as a function of the width of the initial wave func-tion ∆ ℘ and the pulse duration τ . Although using differ-ent pulse shapes, we observe similar to Fig. 2 diffractionto spurious orders in the Raman-Nath regime and there-fore a significant loss of efficiency for short pulses. Sincethe spurious grating is increasingly off-resonant the largerthe initial momentum [36], the Raman-Nath regime isless important for the transition | g, ± (cid:126) K (cid:105) → | e, ± (cid:126) K (cid:105) compared to the transition | e, ± (cid:126) K (cid:105) → | g, ± (cid:126) K (cid:105) .We compare in Fig. 5 the efficiency obtained withthe optimal pulse duration τ opt ∼ = 30 . µ s for the two .
05 0 .
10 0 .
15 0 . . . . . ℘ / ( (cid:126) K ) e ffi c i e n c y E ± E ± E ± E seq FIG. 5. Diffraction efficiency for the three individual first-order pulses and their sequential application, obtained withthe optimal pulse durations. The efficiency E ± for thedouble-Raman beam splitter with Gaussian pulse shape cor-responds to the cut along the red dashed line in Fig. 2, theefficiencies of the Doppler-detuned and box-shaped single-diffraction mirror pulses E ± and E ± to the cuts along the reddashed lines in the two panels of Fig. 4. They differ becauseof the different diffraction geometries (Doppler-detuned sin-gle or double diffraction) as well as the pulse shape employed(Gaussian or box). The efficiency E seq of the sequential ap-plication of the three pulses is lower than the efficiency of theindividual processes. effective single-diffraction pulses to that of the double-diffraction beam splitter (i.e., the cuts along the reddashed lines in Figs. 2 and 4). Since the Raman-Nathregime is suppressed for Gaussian pulses, we observethat the double-diffraction beam splitter has the best effi-ciency for all momentum widths. Off-resonant couplingsare suppressed by a Doppler detuning that scales withthe initial momentum [36] and therefore affect the tran-sition | e, ± (cid:126) K (cid:105) → | g, ± (cid:126) K (cid:105) more than the subsequentprocess with higher initial momentum. Hence, the firstsequential pulse has the lowest efficiency of the individualpulses. However, these two diffraction types differ signifi-cantly in their geometry (single versus double diffraction)as well as in the applied pulse shape, which makes a di-rect comparison difficult. C. Three sequential pulses
In this section we use the diffraction processes dis-cussed in Sections II A and II B to perform a pulse se-quence transferring population from the state | g, (cid:105) toan equal-amplitude superposition of | e, ± (cid:126) K (cid:105) . Ramanpulses for a double geometry have already been experi-mentally realized, however only for the transition from | (cid:105) to | (cid:126) K (cid:105) [12]. A double-diffraction beam splitterwith a Gaussian pulse shape transfers the initial wavefunction from | g, (cid:105) to | e, ± (cid:126) K (cid:105) . Two subsequent box- FIG. 6. Energy-momentum diagram that shows the resonantprocesses of a sequence consisting of a double-diffraction beamsplitter (green arrows) and two subsequent Doppler-detunedsingle-diffraction pulses (blue and red arrows). The initialwave packet is transferred from | g, (cid:105) to | e, ± (cid:126) K (cid:105) via thestates | e, ± (cid:126) K (cid:105) and | g, ± (cid:126) K (cid:105) . shaped and Doppler-detuned effective single-diffractionpulses transfer the population further to | g, ± (cid:126) K (cid:105) and | e, ± (cid:126) K (cid:105) see Fig. 6. The combination of Gaussian andbox-shaped pulses in the sequence allows to benefit fromtheir particular advantages regarding experimental du-ration and transfer efficiency. Each pulse induces first-order diffraction and requires an adjustment of the laserfrequencies to fulfill the corresponding resonance condi-tions from Eqs. (2), (8) and (11). We use the optimalpulse durations ∆ τ opt ∼ = 8 . µ s and τ opt ∼ = 30 . µ s ob-tained in Section II A and Section II B for the individualpulses.This sequence of optimal pulses leads to a momentumtransfer of 3 (cid:126) K and its efficiency E seq is shown in Fig. 5.It is obtained from Eq. (6) by integrating over the pop-ulation in the states | e, ± (cid:126) K (cid:105) after the sequence andit is slightly larger than the product of the individualefficiencies E ± E ± E ± .Compared to the three individual pulses shown in thefigure, the efficiency of the sequence E seq is lower. Infact, it is mainly limited by the lowest efficiency E ± ofthe first sequential pulse. Moreover, the width of thediffracted wave function after each step of the sequenceis iteratively reduced due to velocity selectivity. III. THIRD-ORDER DIFFRACTION
Instead of three sequential pulses we focus in this sec-tion on only one pulse that relies on third-order diffrac-tion to achieve the same momentum transfer of ± (cid:126) K .As Fig. 7 shows, the two laser pairs with frequencies ω b and ω r induce a six-photon diffraction process and trans-fer the population from | g, (cid:105) to | e, ± (cid:126) K (cid:105) . The interme- FIG. 7. Energy-momentum diagram for a double Raman six-photon diffraction process. Two laser pairs (red an blue ar-rows) induce the transition. The first and second scatteringprocess, each a two-photon process, are off-resonant. Thus,the transition from | g, (cid:105) to | e, ± (cid:126) K (cid:105) occurs by populatingthe states | e, ± (cid:126) K (cid:105) and | g, ± (cid:126) K (cid:105) only virtually. diate two-photon processes are off-resonant and thus, thestates | e, ± (cid:126) K (cid:105) and | g, ± (cid:126) K (cid:105) are only virtually popu-lated.
1. Resonance condition and pulse area
In such a third-order process, the atom gains due to itsquadratic dispersion relation a kinetic energy of 9 (cid:126) ω K ,which leads for p = 0 to the following modified reso-nance condition:∆ ω = ω eg + (9 + δ ) ω K . (13)Here, we included the factor ω K δ to compensate for pos-sible energy shifts similar to the Doppler-detuned diffrac-tion processes in Section II B. While these shifts can bederived for higher-order single-Bragg diffraction with box-shaped pulses through conventional adiabatic eliminationof the intermediate states, the two counterpropagatingoptical lattices in double Raman diffraction prevent astraightforward application of the procedure [44], eventhough the technique can be generalized [45] to our caseusing Floquet theory. Similarly, we apply the methodof averaging [46, 47] that has already proven useful fordouble Bragg diffraction [2] to arrive at ω K δ = − ω K . (14)Through this procedure, we also find an effective Rabifrequency for the third-order process that scales withΩ /ω K and connect it to the effective pulse area AA = π √ ω K τ. (15) . . . . . ∆ ℘ / ( (cid:126) K ) . . . .
15 ∆ τ (µs) ∆ ℘ / ( (cid:126) K ) . . . . . . (a)(b) ∆ τ opt FIG. 8. Comparison of the efficiency for a first-order beamsplitter E ± and a third-order beam splitter E ± for differentwidths of the initial wave function ∆ ℘ and pulse durations∆ τ . Panel (a) recalls the results for the first-order efficiencyfor times up to 60 µ s from Fig. 2 and panel (b) shows thesimulated efficiency for third-order diffraction on the sametime scale. While first-order diffraction is more efficient for abroad range of pulse durations in a Bragg-type regime, third-order diffraction is limited by two main effects: In the Bragg-type regime, higher-order diffraction is intrinsically limited byvelocity selectivity, while for small pulse durations losses intointermediate state appear. However, adiabtic-elimination like the method of aver-aging cannot be trivially extended to other pulse shapes.Since we focus in this article on
Gaussian pulse shapesfor double diffraction, we determine the energy shifts ω K δ = β Ω ω K (16)as well as the connection to the modified Rabi frequencyand pulse area A = (cid:90) d t α Ω ( t ) ω K . (17)through a numerical optimization of the diffraction effi-ciency with the Matlab function fminsearch by deter-mining the optimization parameters β and α . For ourrange of initial momentum widths and pulse durations,we find that β ∈ [ − . , − .
42] and α ∈ [0 . , . FIG. 9. Efficiency at optimal pulse duration for differentwidths of the initial wave function ∆ ℘ for the first- and third-order beam splitter as well as the sequence. Due to differentregimes of pulse durations the first-order beam splitter is lessvelocity selective than its third-order counterpart. For small∆ ℘ the third-order pulse is more efficient than the sequence,which makes it an interesting alternative for the diffraction ofnarrow wave packets like BECs.
2. Comparison to first-order and sequential pulses
We recall in Fig. 8(a) the efficiency of the first-orderbeam-splitter pulse E ± from Fig. 2 and compare it to thecorresponding third-order beam splitter efficiency E ± inFig. 8(b) for different widths of the initial wave function∆ ℘ and pulse durations ∆ τ . As expected, third-orderdiffraction requires longer pulse durations, or, higher in-tensities for an efficient transfer since the population hasto overcome two intermediate and off-resonant states.Moreover, velocity selectivity increases with the order ofthe diffraction process. Indeed, for n th-order diffractionthe velocity spread associated with velocity-selectivity ef-fects is proportional to 1 /n because the effective Dopplerdetuning is given in that case by nω D = npK/m . Forsmall pulse durations losses into the intermediate statesappear, especially into | g, ± (cid:126) K (cid:105) since it is the least off-resonant intermediate state as shown by Fig. 7, whilefor larger pulse durations the loss of the efficiency ofthe diffracted population is mainly caused by velocityselectivity. Again, there exists a pulse duration ∆ τ opt at which the atoms are diffracted most efficiently (reddashed line). When comparing these graphs, it seemsthat third-order diffraction seems to be less efficient thanthe first-order pulse.In Fig. 9 we compare the efficiencies with optimal pulseduration for the first-order (∆ τ opt ∼ = 8 . µ s) and third-order beam splitter (∆ τ opt ∼ = 13 . µ s), as well as the se-quence introduced in Section II C as a function of thewidths of the initial wave function ∆ ℘ . As already ob-served above, the first-order beam splitter has a muchhigher efficiency than its third-order counterpart, which can be understood in terms of velocity selectivity and lossto intermediate states. However, if the targeted states are | e, ± (cid:126) K (cid:105) , the third-order pulse has to be compared tothe sequence of three first-order pulses rather than justthe initial beam splitter. Indeed, the third-order pulseshows high efficiency for small momentum distributions,that exceeds the efficiency of the sequential applicationof three individual pulses. Even though the efficiency ofthe sequence could be improved by using Gaussian pulsesthroughout the sequence instead of only for the initialbeam splitter, this would come at the cost of an evenlonger duration of the whole sequence. Consequently,third-order diffraction might be an interesting tool forthe diffraction of wave packets with a narrow momentumdistribution like BECs, since it allows to reduce the com-plexity of the experiment. In general, each transition ofa sequence might introduce spurious phase contributions[48] and using less pulses may facilitate the suppressionof some uncertainties connected to frequency chirps [49–51]. Furthermore, the overall duration of a single pulsecan become shorter than than that of a correspondingsequence of pulses, which might be particularly appeal-ing for very compact set-ups [52] intended for real-worldapplications [53]. IV. CONCLUSIONS
Double Raman diffraction allows in principle for reso-nant diffraction of odd orders, i.e. of order 2 n + 1 with n = 0 , , , . . . . Higher diffraction orders come along witha higher velocity selectivity than first-order pulses. More-over, higher intensities are necessary to overcome the in-termediate states to achieve the optimal diffraction effi-ciency. However, when comparing third-order diffractionwith a sequence consisting of three first-order pulses wefind that third-order pulses diffract narrow momentumdistributions like the ones associated with BECs moreefficiently. The efficiency of our sequence, consisting ofone Gaussian and two box-shaped pulses, could be im-proved by using Gaussian pulses only but at the cost ofa significantly higher duration of the sequence.Third-order Raman mirrors can also be realized butsuffer further limitations like losses into the central state | g, (cid:105) , a feature intrinsic to double-diffraction mirrors[36]. However, the difficulties can be overcome by replac-ing the mirror pulse through Bragg diffraction of sixthorder from a standing wave, a scheme not investigated inthis study.In addition to the possibility of higher-order diffrac-tion, the symmetry of double-Raman pulses suppresseslaser phase noise. Thus, it can be applied within LMTsequences together with Bragg diffraction or combinedwith Bloch oscillations.Hence, double Raman diffraction is a versatile tool forLMT techniques with the same flexibility and limitationsas double Bragg diffraction. Not only does it occur nat-urally in microgravity or horizontal setups, it can also becombined perfectly with other LMT applications [22] toenhance the sensitivity of atom interferometers. ACKNOWLEDGMENTS
We thank M. Gebbe, M. Gersemann, C. M. Carmesin,A. Friedrich and the whole QUANTUS group in Ulm aswell as our partners of the QUANTUS collaboration forhelpful discussions. This work is supported by the Ger-man Aerospace Center (Deutsches Zentrum fr Luft- undRaumfahrt, DLR) with funds provided by the FederalMinistry for Economic Affairs and Energy (Bundesmin-isterium f¨ur Wirtschaft und Energie, BMWi) due to anenactment of the German Bundestag under Grant Nos.DLR 50WM1556 (QUANTUS IV), DLR 50WM1956(QUANTUS V), DLR 50WP1700 and 50WP1705 (BEC-CAL), 50RK1957 (QGYRO) as well as the Associationof German Engineers (Verein Deutscher Ingenieure, VDI)with funds provided by the Federal Ministry of Educa-tion and Research (Bundesministerium fr Bildung undForschung, BMBF) under Grant No. VDI 13N14838(TAIOL). E. G. thanks the German Research Foun-dation (Deutsche Forschungsgemeinschaft, DFG) for aMercator Fellowship within CRC 1227 (DQ-mat). Wethank the Ministry of Science, Research and Art Baden-W¨urttemberg (Ministerium f¨ur Wissenschaft, Forschungund Kunst Baden-W¨urttemberg) for financially support-ing the work of IQ ST . APPENDIX: GENERAL EQUATIONS
In the following we discuss the differential equationsfor double Raman diffraction in their most general form, i.e. we do not focus on a specific resonance condition. Atruncated version can also be found in Ref [54].The differential equations are derived within a rotatingwave approximation [55] and the optically excited stateis eliminated by adiabatic elimination [56–58]. Moreover,the equations are in an interaction picture with respectto the free evolution of the atoms and we assume thatthe laser phases vanish. They read˙ g n = iΩ e − i[ ω D + ω eg − ∆ ω + ω AC +(1+2 n ) ω K ] t e n +1 + iΩ e − i[ − ω D + ω eg − ∆ ω + ω AC +(1 − n ) ω K ] t e n − (18a)˙ e n +1 = iΩ e − i[ ω D − ω eg +∆ ω − ω AC +(3+2 n ) ω K ] t g n +2 + iΩ e − i[ − ω D − ω eg +∆ ω − ω AC − (1+2 n ) ω K ] t g n (18b)Hence, the probability amplitudes of the ground g n ≡ g ( p + n (cid:126) K ) and excited state e n ≡ e ( p + n (cid:126) K ) forma system of coupled differential equations. 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