## Abstract

We demonstrate a novel variable beam splitter using a tripod-linkage of atomic states, the physics of which is based on the laser control of the non-adiabatic coupling between two degenerate dark states. This coupling and the splitting ratio is determined by the time delay of the interaction induced by two of the laser beams.

©1999 Optical Society of America

## 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.

The initially populated quantum state
(2*p*
^{5}3*s*)
^{3}
*P*
_{0} is coupled by
*π*-polarized light to an intermediate state
(2*p*
^{5}3*p*)
3*P*
_{1} (*M* = 0) from where coupling occurs
via *σ*
^{+} and
*σ*
^{-} light to two final magnetic substates
*M* = -1 and *M* = +1, respectively, of
(2*p*
^{5}3*s*)
^{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

$$\mid 2\u3009={\mid}^{3}{P}_{1},M=0,{p}_{x},{p}_{y},{p}_{z}\u3009$$

$$\mid {3}^{-}\u3009={\mid}^{3}{P}_{2},M=1,{p}_{x},{p}_{y},{p}_{z}-\mathit{\u0127}{k}_{\sigma}\u3009$$

$$\mid {3}^{+}\u3009={\mid}^{3}{P}_{2},M=-1,{p}_{x},{p}_{y},{p}_{z}+\mathit{\u0127}{k}_{\sigma}\u3009.$$

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

$$\mid {\Phi}_{2}\u3009=\mathrm{cos}\phi \mid {3}^{-}\u3009-\mathrm{sin}\phi \mid {3}^{+}\u3009$$

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 ∣
Φ_{2}⟩ depends on the Rabi frequencies of the
circularly polarized Stokes lasers only. The composition of state ∣
Φ_{1}⟩ 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 ∣Φ_{1}⟩ coincides initially
with state ∣ 1⟩ while
⟨1∣Φ_{2}⟩ = 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 ∣
Φ_{1}⟩ 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
∣ Φ_{1}⟩ since non-adiabatic coupling to
the dark state ∣ Φ_{1}⟩, degenerate with
∣ Φ_{2}⟩ will not be negligibly small. The
state vector ∣ Ψ⟩ will be of the form [10,11]

In fact, the coupling between the dark states ∣
Φ_{1}⟩ and ∣
Φ_{2}⟩ 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
(2*p*
^{5}3*s*) 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
±04*v*_{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) *μ*m are observed. This separation
corresponds to a difference in transverse momentum in the direction of Stokes
propagation of 2*ħk*_{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.

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 2*ω*
_{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*/(2*ω*
_{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.

For a displacement *D*/(2*ω*
_{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.

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