Momentum Transfer Using Chirped Standing Wave Fields: Bragg Scattering

Home , Atomic mirror

arXiv:quant-ph/0209108v1 19 Sep 2002 xie tts tm eani hi rudsaedur- state ground their in populating remain avoid atoms to states; one excited to allows interaction and off-resonant Both atom-field The scattering with fields. Bragg atoms standing-wave two-level frequency-chirped order of passage. interaction frequency-chirped high off-resonant rapid an on using utilize based adiabatic mirror are and schemes a waves and standing splitter beam result a pop- as never are degraded states be excited ulated. cannot the tech- as 20] decay, STIRAP spontaneous 19, Further-of 18, the 17, on [14, parameters. based nique is worth- pulse transfer method is momentum in passage an It more, changes adiabatic create the against [14]. to that in robust method mention proposed this to was while of and splitter use Bergmann beam by The atom observed de- Stim- first [16]. using was by co-workers pulses (STIRAP) transfer laser Passage Population layed Adiabatic Raman 15]. have ulated [14, methods Passage proposed Adiabatic been Rapid areas, specified transitions. power with the the decreases of but resolution order splitting, increasing the angle and large increase produce requirements to also used Bragg can be [13] [12]. transitions fields multiphoton wave involving standing scattering strong Al- and bichro- forces, potentials, split- magneto-optical matic precision. beam of use high angle the large involve fairly ters producing a for control to methods to duration ternative one or require power experiments pulse These 11]. sub- [10, state levels ground different involving states superposition π ileeet fa tmitreoee.Optical essen- are interferometer. atom that an splitters such of beam elements devices, atom tial practical ex- and of Momentum mirrors basis atom the fields. as as used the momen- be and can of atoms change exchange the an is between motion tum atomic of control 9]. tical 8, 7, [6, problem way. this stud- to controllable experimental and devoted a theoretical ies of in number a motion abil- are the atomic There from manipulate substantially to benefit quantum ity can for schemes 5] certain [4, [1], and computing sensing 3] inertial [2, to holography optics atom atom of Applications years. nti ae epooeshmst raea50/50 a create to schemes propose we paper this In having pulses with involved difficulties the avoid To h neligpyia ehns epnil o op- for responsible mechanism physical underlying The recent in advances rapid experienced has optics Atom usshv enue ocet n eetcoherent deflect and create to used been have pulses oetmtase sn hre tnigwv ed:Bragg fields: wave standing chirped using transfer Momentum ASnmes 37.g 28.t 42.50.Vk atom 32.80.-t, the 03.75.Dg, to numbers: transferred PACS pr be are can scattering momenta Bragg photon on of number based sta mined schemes frequency-chirped mirror using and splitter transfer momentum consider We nvriyo ihgn n ro,M 48109-1120 MI Arbor, Ann Michigan, of University ldmrS aiosyadPu .Berman R. Paul and Malinovsky S. Vladimir & ihgnCne o hoeia Physics Theoretical for Center Michigan OU etr eateto Physics, of Department Center, FOCUS π/ and 2 pnaeu msinpasangiil oe euse momentum transfer We having momentum states efficient to role. produce sequentially to negligible pulses a and chirped plays interaction atom-field emission the of spontaneous evolution entire the ing adro rg tts h rgno hssetu is spectrum this of degenerate origin doubly The a have below. we states. discussed that Bragg (see in of somewhat [13]), differs ladder ref. scheme of level 3 the pas- Fig. but adiabatic in 23], chirped transfer. transfer [22, Raman sage population momentum using to systems related of multilevel closely dynamics is problem the The understand to optical in trapped are rapid that adiabatic [21]. com- atoms of potentials is accelerate use work to the This involving passage state. that are target to duration final plementary pulse the and control rate to chirp used The integer. positive a ω k frequency. nrytr iie by divided term energy mltd.Teeuto fmto o h rudstate ground the for motion of equation The state excited amplitude. the eliminate adiabatically and proximation ag oprdwith compared large aoaoyfae n iedpnetphases time-dependent and frame, velocity with laboratory moving frame rest atomic cies aevectors wave ie.Tefilscul h tmcgon state ground atomic the state couple excited an fields quan- The is transverse vectors) propagation the tized. field but the to classically, (parallel treated motion is motion the tudinal in counterpropagate fields tmi the in atom lmn and element where eciigteao editrcinis interaction field atom the describing ˆ x 0 stepoaainvco foeo h ed and fields the of one of vector propagation the is naoi emhvn ogtdnlvelocity longitudinal having beam atomic An us allows which mirror, atom an first consider We suigta h euig = ∆ detuning, the that Assuming t ieto rse edrgo nwihtooptical two which in region field a crosses direction + H + b | Ω δ µ = 1 p ± b 12 ( k stecne-fms oetmoeao fthe of operator momentum center-of-mass the is t 2 E p ( b dn aefils oe atom-beam Novel fields. wave nding ), sne.I ssonta predeter- a that shown is It esented. m t 2 − ) ω | + k ˆ z 2 m = ( ± nasnl neato zone. interaction single a in s ( t E t direction, cos( ) k | = ) steaoi as h ae ed have fields laser The mass. atomic the is 21 µ | ˆ z 2 12 us envelopes pulse , | i 2 E ω aigenergy having ih ω | 0 ± 2 d 2 t k − | ( + [ ~ t ( ω ) t emk h oaigwv ap- wave rotating the make we , δ µ 2 ) − 2 h / ( 12 ( t ~ ) t kz µ | )+ n h rnvrekinetic transverse the and , − sadpl oetmatrix moment dipole a is 12 ) ω E ˆ z δ | 0 1 1 k t ( E scattering ieto.Telongi- The direction. ih ( where , t E t )] 21 2 cos( ) ± | /dt h Hamiltonian The . k + ( u t E | h.c. sse nthe in seen as ) aifrequen- Rabi , ω ± eaiet the to relative 21 ω 1 n 0 ( / i t 2 ~ sacentral a is + ) , ~ k − u ω where , kz 1 ω nthe in ( | 1 t 0 ) = ) i n is , (1) to is 2 wave function in the momentum representation takes the 5 4 −2 10 3 (a) following form 2 ) k

ω −1 2 0 p ~ 0 ia˙(p,t)= 2m~ a(p,t) − Ωe(t) [exp[iφ(t)]a(p − 2 k,t) (0,1) 1 energy ( (1,2) + exp[−iφ(t)]a(p +2~k,t)] , −10 (2,3) (2) where Ωe(t)=Ωk(t)Ω−k(t)/(4∆) is an eﬀective Rabi fre- τΒ (b) )

R 3 quency, φ(t) = δ1(t) − δ2(t) − δ0t. We have omitted a 2 factor depending on the light shift Ω∆(t) = (Ωk(t) + 2 2 Ω−k(t))/(4∆) in the Eq. (2) since this corresponds sim- ply to a redeﬁnition of the ground state energy. 1 From Eq. (2) it is clear that states with momentum p mean velocity (v 0 ~ 0 1 2 3 4 couple only to the neighboring states p ± 2 k. Using the time (τ ) initial condition a(p,t =0)= δ(p) one can set B

∞ FIG. 1: (a) Dressed states as a function of time for the case of an atomic mirror. Dark solid lines show diabatic states of a(p,t)= an(t)δ(p − 2n~k) exp[inφ(t)] X the positive branch, dashed lines show diabatic states of the n=−∞ negative branch. The dotted line shows the lowest adiabatic and write Eq. (2) in matrix form for the amplitudes a (t) state. Several of the closest adiabatic states (dot-dashed lines) n and a few avoided crossings are shown. (b) Mean velocity with the Hamiltonian 2 vs time. Dotted line shows Bloch oscillations, α = 0.01ωk , 2 2 Ω0 = 0.15ωk. Dashed (α = 0.01ωk) and solid (α = 0.1ωk) .. ..  . . 0  lines show two different regimes at Ω0 = 0.7ωk.    .. ~   . E−1(t)/ −Ω−n(t) 0   ∗ ~  H(t)=  0 −Ω−n(t) E0(t)/ −Ω+n(t) 0  ,   Another way to consider momentum transfer in an op- ∗ .. 0 −Ω (t) E+1(t)/~ .  +n  tical lattice is to describe motion of an atom in the peri-  . .  odic potential under the influence a constant force using  0 .. ..    the Bloch formalism [24]. In this picture one can observe  (3) Bloch oscillations of the mean atomic velocity as shown where Ω− (t)= Ω+ (t)= Ω (t). The quasi-energies n n e in Fig. 1(b) by dotted line. The adiabatic sequential mo- ~ 2 ˙ in Eq. (3) are given by En(t)= nn ωk + nφ(t)o, where mentum transfer mentioned above corresponds to Bloch 2 ωk =2~k /m is a (two-photon) recoil frequency, and n = oscillations in the lowest band. The amplitude of these 0, ±1, ±2, ··· . This expression for the quasienergies is oscillations is suppressed with increasing Rabi frequency easy to understand. The Bragg levels have energy En = (dashed line). To satisfy adiabatic conditions one must 2 ~n ωk and the ground state (n = 0) is coupled to state n use a very small chirp rate, α, resulting in a long period by an n-(two)-photon process having effective frequency for Bloch oscillations, τB = 2ωk/α, and, consequently, φ˙(t). Thus the field energy is lowered by nφ˙(t) when the in a long time for momentum transfer. We show below atom is excited to state n and this loss is reflected in the a possibility to reduce considerably the total absolute quasienergies of the atom+field. The Hamiltonian (3) time of momentum transfer by increasing the chirp rate. 2 describe dynamics in momentum space and |an(t)| are An increasing chirp rate breaks adiabaticity at the time the probability for an atom to have momentum 2n~k. of the first avoided crossing, (0, 1) (see Fig. 1(a)) and In the case of an linear-chirped standing wave field nonadiabatic couplings became important in this regime. However, one can still have efficient transfer to the target φ˙(t) = α(t − tc) − δ0 , where α is the chirp rate and tc is a constant. It is clear that in a diabatic represen- state provided α is not too large. The oscillations in the tation there are many sequential crossings between the solid curve of Fig. 1 (b) are evidence for nonadiabatic En(t). They become avoided crossings owing to the in- effects; nevertheless, the transfer to the target state in teraction with the laser fields (see Fig. 1). The resonances nearly 100 % for these parameters, as is shown below. can be viewed as sequential two-photon Bragg resonances According to the structure of Eq. (3) the initial state, equally spaced in time with period ∆t =2ωk/α. The dot- a0(t), is connected to the ±1 states. From the ex- ted line in Fig. 1(a) represents the lowest adiabatic state. pression for the quasi-energies, En(t), it follows that a This state correlates with the zero momentum state as positive chirp will sequentially bring the effective fre- the pulse arrives and with the target state following the quency into resonance with transition frequencies be- pulse. When adiabatic conditions are satisfied for all se- tween states of the positive branch while states of the quential crossings, atoms remain in this instantaneous negative branch will be shifted more and more from res- eigenstate which evolves into the state having momen- onance (see Fig. 1(a)). Consequently for successful mo- tum 2n~k. mentum transfer we have to make sure that transient 3 population of the −2~k state due to the off-resonant in- In our example of positive chirp, population is transferred teraction with the laser field is minimal. As we show in to the positive n-mode state (the +25th state, Fig. 2). By the example below this can be accomplished by adjusting changing the chirp sign we are able to switch the direction a switching-on stage of the pulse and the chirp rate. of the population transfer and populate the −25th state at a later time. 1 n=0 (a) n=25 It is also possible to coherently split an atomic beam 0.8 using a frequency chirped standing wave. One way to 0.6 accomplish this task is to use an additional laser pulse of opposite chirp. The idea is to create simultane- 0.4 ously two frequency-chirped standing waves. A positively population 0.2 chirped standing wave provides a momentum transfer in ~ 0 the +n k branch of momentum states while negatively 0.03 chirped wave works in parallel on the −n~k branch. n=26 n=−1 (b) In this case the modified equation for the ground state 0.02 wave function in the momentum representation takes the form 0.01 population 2 p ~ ia˙ (p,t)= 2m~ a(p,t) − Ωe(t)a(p − 2 k,t) 0 (4) 0.8 ∗ ~ −Ωe(t)a(p +2 k,t) , 0.6 (c) + −

(t) where Ωe(t) = Ωe (t) exp[iφ1(t)] + Ωe (t) exp[iφ2(t)], e 0.4 ′ Ω φ1(t) = δ1(t) − δ2(t) − δ0t, φ2(t) = δ1(t)+ δ3(t)+ δ0t, 0.2 ˙ ± δ3(t) represents chirp of an additional pulse, Ωe (t) = ± ~ ± 0 Ωk(t)Ω−k(t)/(4∆), Ωk(t) = µ12Ek(t)/ , Ω−k(t) = 0 100 200 300 400 500 ± ~ time (1/ω ) µ12E−k(t)/ . In Eq. (4) we have omitted the common k light shift. Equation (4) can be rewritten as a linear system FIG. 2: Population dynamics for an atomic mirror. We choose 2 with the Hamiltonian as in Eq. (3). However in this tc = 10/ωk, α = 0.1ωk. (a) Solid lines show the population case the coupling between momentum state wave func- flow to the target state n = 25. (b) Population of the n = −1 tions is different for the negative and positive branches: and of the n = 26 states. (c) Shape of the laser pulse. + − ′ Ω−n(t) = Ωe (t)+Ωe (t)exp {−iδ23(t)}, Ω+n(t) = + ′ − ′ Ωe (t)exp {iδ23(t)}+Ωe (t), where δ23(t)= δ2(t)+δ3(t)+ Figure 2 shows the momentum transfer to the 50~k ′ [δ0 + δ0] t. More important is that the quasi-energies state for the initial condition a0(t = 0) = 1. This E−n(t) and E+n(t) are controlled by the different chirps, figure demonstrates almost 100% efficiency of adiabatic ˙ ˙ ~ 2 ˙ momentum transfer and corresponds to an atom mirror. φ1(t) and φ2(t): E−n,+n(t)= nωkn + nφ1,2(t)o. Asa As a target we chose the 50~k state; however, in prin- result one can coherently control wave function dynamics ciple, there is no limit to the number of the momentum in both branches of momentum states. quanta 2~k which can be transferred to the atoms. As Figure 3 illustrates the dynamics for the case of an long as we approximately satisfy an adiabatic condition atom beam splitter. We choose φ˙1(t) = −φ˙2(t) = + − ′ at the beginning of the pulse, when the field is not so α(t − tc), Ek(t)= E−k(t)= E−k(t), δ0 = δ0 = 0, and the 2 ~ strong, momentum transfer is nearly 100% efficient for chirp rate α = 0.1ωk. Our target states are +50 k and the sequential Bragg resonances. The pulse duration is −50~k. At later time we have almost perfect splitting the parameter that determines how many transitions take of the population between our target states. By adjust- place. After a target state is chosen the turn-off stage of ing the efficiency of the first avoided crossing taking place the pulse can be adjusted as at the beginning of the pulse between the 0 and ±1 states in such a way that the proba- to avoid population of higher states. bility of a population transfer in both directions is equal There is enough freedom to change the time interval we achieve symmetrical beam splitting. If the driving between sequential momentum transfers by adjusting the fields derived from the same laser, a coherent superposi- chirp rate. In the case of the atom mirror (see Fig. 2) tion of the ±50~k states can be created. Note that the the time interval between crossings, ∆t = 2ωk/α = 20, non-adiabatic couplings at the time of the first avoided −1 in units of ωk . At the same time the Landau-Zener crossings are responsible for small amount of population transition time for one of the crossings tLZ ≈ Ωeff (ti)/α (≈ 0.5%) remaining in low lying levels. We note that is of order 7, where ti is the crossing time. All that is this scheme is very robust as well as selective and con- required is that the transition time, tLZ , be less than the trollable. By increasing or decreasing the pulse duration time between crossings, ∆t. It is clear that the direction by N∆t = N2ωk/α (N is an integer), one can create a of population flow is controlled by the sign of the chirp. beam splitter of larger or smaller angle. 4

1 (a) we ﬁnd that to make a beam splitter (n = ±25) one 0.8 n=0 n=25 has to tune the frequency diﬀerence in the range of 0.6 50ωk/2π = 2.5 MHz. That means a transform limited pulse of ≈ 60 nsec duration should be used to produce the 0.4 2

population chirp α/2π =0.1 ωk ≈ 1.6 kHz/µsec (see Fig. 2 and 3). 0.2 One method of producing appropriate chirp rates is 0 1 to use acousto-optical modulators (AOM) as in studies n=0 (b) of Landau-Zener tunneling [21]. In this case one must 0.8 n=−25 devise a method to transfer the temporal frequency chirp 0.6 to a spatial one as an atomic beam passes through the 0.4 interaction zone. One possibility for producing a spatial population 0.2 chirp directly is to use the Doppler shift associated with curved wave fronts [25, 26]. 0 If a Bragg beam splitter of this type is to be used 0.3 (c) as an element of an atom interferometer, the transverse

(t) 0.2 momentum spread of the atomic beam must be less than +,− e some critical ∆pc of order ~k/ (2nωkT ) where T is the Ω 0.1 total pulse duration. Preselection of this narrow velocity 0 range by a preparation pulse may be needed. This se- 0 100 200 300 400 500 vere restriction on the momentum spread is based on the time (1/ω ) k assumption that the different momentum components in the beam are uncorrelated. However, if a coherent source FIG. 3: Population dynamics for an atomic beam-splitter. such as a BEC is used, our wave packet simulations show 2 We choose tc = 20/ωk, α = 0.1ωk. Solid lines show the that our schemes are valid even for a transverse momen- population flow from initial state n = 0 to the target state tum spread as large as 0.5~k. n = 25 - (a) and to the target state n = −25 - (b). (c) Shape of the laser pulse. We express our appreciation for many useful discus- sions with P. Bucksbaum, G. Raithel, B. Dubetsky. This work is supported by the U. S. Office of Army Research Let us give preliminary values of the pulse parame- under Grant No. DAAD19-00-1-0412 the Michigan Cen- ters which can be used in experiments. All frequen- ter for Theoretical Physics, and the National Science cies in our simulations have been normalized to the re- Foundation under Grant No. PHY-9800981, Grant No. coil frequency. Assuming that ωk/2π is about 50 kHz PHY-0098016, and the FOCUS Center Grant.

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