1. Introduction

One of the important elements in atom optics is a beam splitter [1] which separates the atomic wave function into a superposition state of two components with different linear momentum [2–4]. A common approach is to apply π/2 pulses [5]. However, this technique is not robust because it requires a carefully controlled duration and power of the pulse in order to assure a pulse area of π/2. Alternatively, as suggested by [6] and verified by [7], one can modify the ordinary 3-state stimulated Raman adiabatic passage (STIRAP) technique [8,9] to implement an atomic beam splitter. Rather than having the ratio of the relevant Rabi frequencies increased from zero to infinity one assures that this ratio approaches a constant value at late times.

Here we discuss a novel concept of a robust and variable beam splitter, which is based on the laser control of two degenerate dark states, and verify it experimentally for metastable neon (Ne ^*) atoms in a beam. The relevant atomic levels and transitions are shown in Fig. 1a.

 

Figure 1. (a) The relevant level scheme for the realization of the beam splitter using the tripod-linkage system. Initially only the state ³ P ₀ is populated. (b) The geometry of the setup.

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The initially populated quantum state (2p ⁵3s) ³ P ₀ is coupled by π-polarized light to an intermediate state (2p ⁵3p) 3P ₁ (M = 0) from where coupling occurs via σ ⁺ and σ ^- light to two final magnetic substates M = -1 and M = +1, respectively, of (2p ⁵3s) ³ P ₂. The atoms cross the laser beams at right angle. The sequence of interaction with the three laser beams is controlled by the spatial displacement of their axes. The σ ⁺- and σ ^--beams propagate in opposite direction, while the axis of the π-polarized beam is at right angle to the others (see Fig. 1b).

2. Theory

The fields associated with the three laser beams (see Fig. 1b) are

the π- and σ ^±-laser beam axes cross the atomic beam axis at x ₀, ±x ₁ with width ω _0,1. The atoms have initially a momentum P = {p_x ,p_y ,p_z }. After the interaction with the laser beams, the atoms have acquired additional momentum depending on their quantum state according to

The Hamiltonian has a set of four adiabatic eigenstates. Two of these, displayed in eq. (5), are orthogonal degenerate dark states, i.e. states which have no components of state ∣ 2⟩, which are therefore immune to loss of coherence and population due to spontaneous emission. For on resonance excitation, the two dark states are

where the two mixing angles are defined as

and Ω_±π are the Rabi frequencies related to various lasers. The mixing angle φ which determines the evolution of ∣ Φ₂⟩ depends on the Rabi frequencies of the circularly polarized Stokes lasers only. The composition of state ∣ Φ₁⟩ depends on all three Rabi frequencies, including the linearly polarized pump pulse. When the Stokes pulse precedes the pump pulse, as required for coherent population transfer [8,9] from state ∣ 1⟩ to state ∣ 3^-⟩ and ∣3⁺⟩, the mixing angle θ is initially zero. Therefore the trapped state ∣Φ₁⟩ coincides initially with state ∣ 1⟩ while ⟨1∣Φ₂⟩ = 0. When the pulse areas are sufficiently large, Ω_i T ≫ 1, where T is the interaction time, non-adiabatic coupling to state ∣ 2⟩ is small. According to eq. (5) the dark state ∣ Φ₁⟩ evolves into ∣ 3^-⟩ or ∣ 3⁺⟩, depending on the evolution of the mixing angle φ. Toward the end of the interaction, the latter will be π/2 when the interaction with the σ ^- beam starts and ends earlier than that with the σ ⁺ beam, and ∣ 3^-⟩ will be populated. It will approach zero, resulting in a population of ∣ 3+⟩, if the sequence is the other way around.

However, the population will not necessarily follow the evolution of state ∣ Φ₁⟩ since non-adiabatic coupling to the dark state ∣ Φ₁⟩, degenerate with ∣ Φ₂⟩ will not be negligibly small. The state vector ∣ Ψ⟩ will be of the form [10,11]

In fact, the coupling between the dark states ∣ Φ₁⟩ and ∣ Φ₂⟩ is controlled by the displacement of the laser beams and determines the evolution of the system from the initial state ∣1⟩ into a superposition state of ∣ 3^-⟩ and ∣ 3⁺⟩. Substituting this expansion into the Schrödinger equation, taking the scalar product with the adiabatic states, and using the fact that the ∣ Φ_i⟩ are orthonormal, we find with the initial condition ∣B ₁ (-∞)∣ = 1 and B ₂ (-∞) = 0 the solution for the amplitudes to be

where the angle γ is given by

According to eq. (9) and eq. (6), the angle γ depends on the shape of the Stokes pulses, the delay between them and the delay of the pump pulse. When the Stokes pulses are identical and overlap fully we have γ = 0 since φ ^· = 0. The value of γ increases monotonically with the delay between the two Stokes pulses. When the Stokes pulses precede the pump laser we have θ = π/2 towards the end of the interaction while φ = 0 or φ = π/2 depending on which of the Stokes pulses is the leading one. Therefore the state vector will emerge into one of the superposition states

It has been shown in Ref. [10,11] that the angle γ is independent of the pulse areas Ω_i T. Thus the angle γ is insensitive to the longitudinal velocity distribution of the atoms in the beam.

3. Experimental

A beam of metastable Neon atoms emerges from a liquid nitrogen cooled cold cathode discharge. The metastable states ³ P ₀ and ³ P ₂ with the electronic configuration (2p ⁵3s) are populated with an efficiency of the order of 10^-4. The mean longitudinal velocity is 600 ms^-1 with a full width of half maximum of 200 ms^-1. The on-axis beam intensity is increased by two dimensional polarization gradient cooling of the transversal velocity components in a zone a few cm downstream of the nozzle. The cooling enhances the on-axis intensity of the ³ P ₂ metastable atoms by a factor of 27.

Next the atoms in state ³ P ₂ are transferred to the ³ P ₀ state by optical pumping. An excitation laser (λ = 588 nm) drives the transition to the 3Pi level which has a lifetime of 18 ns. A fraction of 27.6 % of the atoms in the ³ P ₁ state reaches the ³ P ₀ metastable state by spontaneous decay.

The atoms pass two collimation slits, 141 cm apart, with a width of 50 μm and 10 μm, respectively. The resulting beam is collimated to 1 : 47000, which is equivalent to a transverse velocity component of ±1.3 cms^-1 or ±04v_recoil where v_recoil is the recoil velocity related to the transfer of one photon momentum ħk. This highly collimated atomic beam is manipulated by three laser fields as shown in Fig. 1b. The magnetic field is reduced to less than 1 μT in the relevant region using the Larmor velocity filter setup [12]. The transverse atomic beam profile is monitored further downstream with a channeltron behind a 25 μm slit driven perpendicularly to the atomic beam axis by a stepper motor.

Three independent continuous single mode dye lasers (Coherent 699) are used in this experiment. The cooling laser operates at 640.402 nm. The laser which increases the population of the ³ P ₀ state by optical pumping and both Stokes beams are provided by the same dye laser (λ = 588.350 nm). The third laser generates the 616.530 nm radiation needed for the ³ P ₀ ↔ ³ P ₁ transition. This is the pump laser in the STIRAP process. All laser beams are delivered to the apparatus by single mode fibers. The state of polarization is controlled by fiber polarizers at the fiber exits followed by Glan-Thompson prisms. The Stokes laser passes through a λ/4 waveplate, interacts with the atomic beam and is back reflected by a cats eye retro reflector with an integrated λ/4 retarder plate [12]. The translation of the cats eye parallel to the atomic beam axis allows precise adjustment of the spatial displacement of the two Stokes lasers.

4. Results

The application of two Stokes and one pump laser in STIRAP configuration (Stokes precedes pump) leads to population transfer from ³ P ₀ to ³ P ₂ (M = ±1). Since the Stokes beams (with different circular polarization) propagate in opposite direction, the momentum transfer to the M = +1 and M = - 1 states have opposite signs, resulting in coherent beam splitting.

Fig. 2 shows examples of atomic beam profiles recorded for different displacements of the Stokes lasers. Two maxima separated by (122 ± 2) μm are observed. This separation corresponds to a difference in transverse momentum in the direction of Stokes propagation of 2ħk_Stokes . The momentum which is accumulate by an atom during the transfer process is ħ(${\overrightarrow{k}}_{\mathit{\text{Pump}}}$ ± ${\overrightarrow{k}}_{\mathit{\text{Stokes}}}$ ). Since the beam is collimated by slits and is detected behind a narrow slit, which is parallel to the π-polarized beam, only the component of the momentum parallel to the Stokes laser beam axis is observed.

The data shown in Fig. 2 demonstrate, that the splitting ratio can be smoothly controlled by the delay of the Stokes laser interactions. When the axes of the Stokes beams coincide, we observe a 50 : 50 beam splitting, as expected.

The experimental setup assures that the variation of the relative phase of the σ ⁺- and σ ^--Stokes beams is negligibly small during the interaction time with these lasers which is of the order of 1 μs. Although there is little, if any, doubt, that the beam splitting is indeed coherent, experiments to prove the coherence directly are under way.

 

Figure 2. Variation of the profiles of the metastable neon atomic beam with the relative displacement of the σ-polarized Stokes-laser beams. From top to bottom the displacement is +250 μm, +50 μm, 0 μm, -150 μm and -250 μm. The diameter of the laser beams is 0.6 mm. The solid lines result from a fit to the data.

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In Fig. 3 we show the variation of the maximum of the peaks, related to M = +1 and M = - 1, with the displacement D of the Stokes lasers which is measured in units of the laser beam width 2ω ₀. It reveals again the essential properties of the beam splitter. In the absence of losses, the sum of the population in the level ³ P ² should be constant.

For an displacement D/(2ω ₀) ≈ 0, the population in the, J = 2 level is indeed nearly constant (see Fig. 3a) while the relevant height of the maximum for M = +1 and M = - 1 depends on the relative ordering of the circularly polarized Stokes beams. Here, the axis of the σ ⁺-beam remains unchanged while the axis of the σ ^--beam is displaced.

The experimental data are nicely reproduced by the results of numerical density matrix calculations shown in Fig. 3b. For very large negative displacement, the atoms interact firstly with the σ ⁺-light, next with the π-light and finally with the σ ^--light. At some time (early on) the σ ⁺ and π-light act simultaneously on the atoms, leading to population transfer from ∣1⟩ to state ∣ 3⁺⟩ (M = -1) by a conventional STIRAP process. The π- and σ ^--beams also overlap partially but there is no overlap between the σ ⁺- and σ ^-- beams. Optical selection rules prevent the σ ^--light from interaction with the population of the ³ P ₂(M = -1) level. Therefore, all the population of state ∣ 1⟩ reaches the level M = - 1 and remains there.

 

Figure 3. (a) Intensity of the two components of the coherently split atomic beam as a function of the displacement D of the σ ^--Stokes laser measured in units of the laser beam width 2ω ₀ ≈ 0.6mm. The open circles give the flux of atoms in state ∣3^-⟩ (M = +1) while the dots refer to state ∣3⁺⟩ (M = -1). The sum of both count rates is also shown (triangles). The relevant laser powers are P_P = 46 mW, P_S = 34 mW corresponding to Ω_P ≈ 100 MHz and Ω_± ≈ 30 MHz. (b) Numerical simulation for the intensities of both beam splitter channels as a function of D. The gray line gives the total population in the ³ P ₂ state.

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For a displacement D/(2ω ₀) ≈ - 1 the σ ⁺-beam profile still extend beyond that of the π-beam. There is, however, some overlap with the σ ⁺-beam. Therefore, all three laser beams act simultaneously on the atom for a certain period of time and the tripod-mechanism begins to work. A fraction of the population is transferred to state ∣ 3^-⟩ (M = +1). Since the σ ^--light is the last one to interact with the atoms, the population which has reached M = +1 is depleted by optical pumping.

The measured and calculated transfer efficiency decreases slightly as the displacement D increases from -0.5 to +0.5. This is related to small losses by non-adiabatic coupling to state ∣ 2⟩. Since the overlap of the π-polarized beam with the σ-polarized one increases as D is varied from D = +0.5 to D = -0.5, the lossrate does not vary symmetrically with D in the range near D = 0.

Closer inspection of the data shown in Fig. 3a reveals a small difference of the maximum population in the states M = +1 and M = -1. We attribute this to a small deviation from perfect alignment. Although the axes of the Stokes laser beams where parallel to within 2 mrad or better, the small two-photon linewidth of the order of a few MHz may result in a deviation from perfect two-photon resonance for one of the beams.

5. Conclusion

We have demonstrated a laser controlled variable atomic beam splitter using an extension of 3-level STIRAP to a tripod-linkage system. The splitting ratio is controlled by the overlap of the three lasers. Good agreement of the experimental and numerical results is found. This beam splitter promises to be a versatile tool for atom optics experiments.

We thank B.W. Shore for enlightening discussions. RU thanks the Alexander von Humbold Foundation. This work was supported by the “Deutsche Forschungsge-meinschaft” and by the EU network “Laser Controlled Dynamics of Molecular Processes and Applications”, ERB-CH3-XCT-94-0603.