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Momentum Transfer Using Chirped Standing Wave Fields: Bragg Scattering

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Momentum Transfer Using Chirped Standing Wave Fields: Bragg Scattering Momentum transfer using chirped standing wave fields: Bragg scattering Vladimir S. Malinovsky and Paul R. Berman Michigan Center for Theoretical Physics & FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120 We consider momentum transfer using frequency-chirped standing wave fields. Novel atom-beam splitter and mirror schemes based on Bragg scattering are presented. It is shown that a predeter- mined number of photon momenta can be transferred to the atoms in a single interaction zone. PACS numbers: 03.75.Dg, 32.80.-t, 42.50.Vk Atom optics has experienced rapid advances in recent ing the entire evolution of the atom-field interaction and years. Applications of atom optics to inertial sensing [1], spontaneous emission plays a negligible role. We use atom holography [2, 3] and certain schemes for quantum chirped pulses to produce efficient momentum transfer computing [4, 5] can benefit substantially from the abil- sequentially to states having momentum ±n2~k, where ity to manipulate atomic motion in a controllable way. k is the propagation vector of one of the fields and n is There are a number of theoretical and experimental stud- a positive integer. The chirp rate and pulse duration are ies devoted to this problem [6, 7, 8, 9]. used to control the final target state. This work is com- The underlying physical mechanism responsible for op- plementary to that involving the use of adiabatic rapid tical control of atomic motion is an exchange of momen- passage to accelerate atoms that are trapped in optical tum between the atoms and the fields. Momentum ex- potentials [21]. change can be used as the basis of practical devices, such We consider first an atom mirror, which allows us as atom mirrors and atom beam splitters that are essen- to understand the dynamics of momentum transfer. tial elements of an atom interferometer. Optical π/2 and The problem is closely related to population transfer in π pulses have been used to create and deflect coherent multilevel systems using Raman chirped adiabatic pas- superposition states involving different ground state sub- sage [22, 23], but the level scheme differs somewhat (see levels [10, 11]. These experiments require one to control Fig. 3 of ref. [13]), in that we have a doubly degenerate pulse power or duration to a fairly high precision. Al- ladder of Bragg states. The origin of this spectrum is ternative methods for producing large angle beam split- discussed below. u ters involve the use of magneto-optical potentials, bichro- An atomic beam having longitudinal velocity in the ˆx matic forces, and strong standing wave fields [12]. Bragg direction crosses a field region in which two optical ˆz scattering involving multiphoton transitions [13] can also fields counterpropagate in the direction. The longi- be used to produce large angle splitting, but the power tudinal motion is treated classically, but the transverse requirements increase and the resolution decreases with motion (parallel to the field propagation vectors) is quan- increasing order of the transitions. tized. The fields couple the atomic ground state |1i to an excited state |2i having energy E21. The Hamiltonian To avoid the difficulties involved with pulses having describing the atom field interaction is specified areas, Rapid Adiabatic Passage methods have been proposed [14, 15]. Population transfer by Stim- p2 H = 2 + E21|2ih2|− µ12Ek(t)cos(ω1(t)+ kz) bm h arXiv:quant-ph/0209108v1 19 Sep 2002 ulated Raman Adiabatic Passage (STIRAP) using de- b layed laser pulses was first observed by Bergmann and co-workers [16]. The use of this method to create an + µ12E−k(t)cos(ω2(t) − kz)|1ih2| + h.c.i , atom beam splitter was proposed in [14]. It is worth- (1) while to mention that the adiabatic passage method is where p is the center-of-mass momentum operator of the robust against changes in pulse parameters. Further- ˆz atom inb the direction, µ12 is a dipole moment matrix more, momentum transfer based on the STIRAP tech- element and m is the atomic mass. The laser fields have nique [14, 17, 18, 19, 20] cannot be degraded as a result wave vectors ±kˆz, pulse envelopes E±k(t) as seen in the of spontaneous decay, as the excited states are never pop- atomic rest frame moving with velocity u relative to the ulated. laboratory frame, and time-dependent phases ω1(t) = In this paper we propose schemes to create a 50/50 ω0t+δ1(t), ω2(t)= ω0t+δ2(t)+δ0t, where ω0 is a central beam splitter and a mirror using frequency-chirped frequency. standing waves and adiabatic rapid passage. Both Assuming that the detuning, ∆ = E21/~ − ω0, is schemes are based on high order Bragg scattering and large compared with |d [ω2(t) − ω1(t)] /dt|, Rabi frequen- utilize an off-resonant interaction of two-level atoms with cies |Ω±k(t)| = |µ12E±k(t)/~|, and the transverse kinetic frequency-chirped standing-wave fields. The off-resonant energy term divided by ~, we make the rotating wave ap- atom-field interaction allows to one to avoid populating proximation and adiabatically eliminate the excited state excited states; atoms remain in their ground state dur- amplitude. The equation of motion for the ground state 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.
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