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Reflection of atoms from a dielectric wave guide
15 October 1994 OPTICS COMMUNICATIONS EISEVIER
Optics Communications 111 (1994) 566-576
Full length article
Reflection of atoms from a dielectric wave guide W. Seifert a, R. Kaiser b, A. Aspect b, J. Mlynek a b Institut d’optique,
aFakultbt ftir Physik, Universitiit Konstanz, D-78434 Konstanz, Germany Unit6 Associee au CNRS n”14, CentreScientijqued’Orsay, Bat 503, BP 147, F-91403 Orsay, France Received 6 April 1994; revised manuscript received 16 June 1994
Abstract We report on the reflection of metastable argon atoms from an evanescent wave enhanced by means of a dielectric wave guide by two orders of magnitude in intensity. This allows to reflect atoms with an incidence velocity of up to 3 m/s. We compare the experimental results to a theoretical model by means of a Monte-Carlo simulation.
1. Introduction In the last few years there has been significant progress in atom optics [ 11. Due to breakthroughs in laser cooling and atom interferometry new fields of interesting research have been opened. One useful element for atom optics is an atom mirror. It can be used to bend the atomic paths in an interferometer, to focus an atomic beam onto a surface e.g. in atom lithography or in a scanning atom microscope or to diffract atoms if the reflectivity is spatially modulated [ 2,3 1. A mirror can also be used in an atom resonator [ 4-7 ] where atoms are bottled without permanent interaction with the confining potential or the multiple interference between the de Broglie waves would allow to study quantum statistical effects. For the application in interferometric experiments or as a reflection grating the coherence of the de Broglie wave has to be preserved during the reflection. The most promising physical realisation of an atom mirror uses the dipole force of a blue detuned (light frequency above atomic transition frequency) inhomogeneous light field to reflect the atoms. An evanescent wave obtained by total internal reflection of a
laser beam in a prism gives the required field gradient. This scheme was first proposed by Cook and Hill [ 8 ] and demonstrated by Balykin et al. [ 9 1. In many applications of this type of mirror it is crucial, that the atoms do not spontaneously emit photons. Spontaneous emissions cause momentum diffusion leading to a broadening of the reflected atomic beam and also destroy the mutual coherence between the incident and the reflected de Broglie waves. The probability for spontaneous emissions during the reflection can be reduced by increasing the resonance detuning of the light field, but increasing the detuning reduces the repelling optical potential. Consequently it is desirable to enhance the light intensity to allow for the detuning to be increased. Recently reflection of atoms using a plasmon enhanced evanescent wave has been reported [ lo- 12 1. Additionally the probability for spontaneous emissions is reduced when the time of interaction with the evanescent wave is reduced by decreasing the decay length of the evanescent wave. A promising approach to enhance the evanescent light field and to provide simultaneously a short decay length is a system of dielectric layers. Here a light wave is coupled into a dielectric wave guide via optical tunnelling through a
0030-4018/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved .SsDIOO30-4018(94)00369-6
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W. Seifet et al. /Optics Communications 11I (1994) 566-576
solid gap. This scheme allows to enhance the light intensity by 2-3 orders of magnitude and it is possible to adapt the dielectric layers for different resonance angles of the incident light beam yielding thus different decay lengths of the evanescent wave. In this communication we report on the first reflection of atoms from such a multilayer system, analyse its influence on the reflection of atoms and compare the results with the reflection from a plasmon enhanced evanescent wave. This paper is organised as follows: in Sect. 2 we recall the principles of an atom mirror based on an evanescent wave. In Sect. 3 we briefly describe the dielectric wave guide used for the enhancement of the evanescent wave. The fabrication and the optical properties of our dielectric multilayer system has been reported elsewhere [ 13 1. In Sect. 4 the experimental set-up is described. In Sect. 5 the experimental results are reported and in particular the reflection of atoms bounced off an evanescent wave on a bare prism is compared to the case of an enhanced evanescent wave using our dielectric coating. In Sect. 6 we present Monte-Carlo calculations that simulate our experiment and reproduce the experimental results. These simulations are based on the dressed state picture presented in Sect. 2. They allow us to study separately the different processes appearing during the reflection of the atoms.
2. Principle of the atom mirror The reflection of atoms from an evanescent wave is outlined in Fig. 1. An atomic beam is incident with angle Oi relative to the prism surface and reflected from the evanescent wave. As the spread of the atomic wave packet is small compared to the decay length of the evanescent wave, the atom is treated as a classical particle with a well defined position r. A quantum analysis of the phase of the reflected de Broglie wave will be given elsewhere [ 141. The main features of the atom mirror can be understood by considering a two-level atom (atomic transition frequency mat, atomic dipole moment d, velocity v) interacting with a strong light wave (laser frequency oL). As the maximum atom-laser coupling frequency and the detuning are large compared to the decay rate r of the excited state in our experiment, a convenient approach
\ s’
I/’
Fig. 1. Schematic representation of reflection of an atomic beam from an evanescent wave. The atomic beam is incident at an angle or relative to the surface plane. The incidence angle (Yof the laser beam is measured against the surface normal. The prism (n,= 1.9) is coated with a layer of TiOz (~I,=86 nm, n,=2.37) on spacer layer of SiOl (d,=350 nm, n2= 1.46).
for the theoretical description of the reflection of atoms from an evanescent wave is the dressed atom model [ 151. Although the laser field in the evanescent wave does not need to be quantized for a complete description of our experimental results, it provides a straightforward insight into the time evolution of the atomic degrees of freedom and the trajectories. Fig. 2 shows the energy levels of the dressed states I 1, rt; r) and 12, n; r) . I 1, n; r) means that the atom has the position r and the atom-light system is in the dressed state 1 with n photons in the light mode. Following the notations of Ref. [ 15 ] we write the energies of these states: E In(r)=(n+l)Aw
L
E,,(r)=(n+l)fiw,-
-E
2
+ *n(r) 2’
AA K?(r) 7 - 2,
(la) (lb)
where A is the effective detuning (A=o,-co,,-k-u), k the real part of the wave vector of the evanescent wave, n the number of photons in the laser mode, on the on-resonance Rabi frequency and Q(r) = r + A t h e generalised Rabi frequency. The eix/v genstates of the dressed atom can be written: ll,n;r)=exp(ik-r/2)sinO(r)lg,n+l) +exp( -ik.r/2) (2, n; r) =exp(ik*r/2) -exp(
-ik.r/2)
cos O(r) le, n) cosO(r) Ig, n + 1) sin e(r) le, n) ,
with the angle O(r) defined by
(2)
568
W. Seifert et al. /Optics Communications 11 I (1994) 566-5 76 A
the prism-vacuum interface. This theoretical treatment holds for the coated and for the uncoated prism. For large z and positive detuning, the state 11, n; r) corresponds to the state 1g, n + 1) . An atom arriving in the ground state and following its energy level E,,(r) will consequently see a potential barrier. If its incident velocity v, normal to the plane of the prism is sufficiently small so that the kinetic energy is smaller than the repelling light potential
R
1:’ L Il,n;r)
12,n,r>
2
\I/
Y
Fig. 2. Spatial dependence of dressed state energy levels involved in the reflection process. The sizes of the different frequencies involved are not scaled. A possible sequence of transitions is shown as an example. An atom incoming in state 11,n; r) makes a transition to the attractive state 12, n - 1; r) by spontaneous emission of a photon. A short time later it falls back to a repulsive state, stops and is accelerated away from the surface.
cos 28(r) = - -
A
Q(r)
sin 28(r) = Q(r) wR(r)
’
’
(3)
1g, n) and 1e, n) mean that the atom is in the ground state or excited state respectively and there are n photons in the light mode. The on-resonance Rabi frequency WR(r) = - d.E( r) /#i depends on the electric field amplitude E(r) of the light field. Using an evanescent wave, as shown in Fig. 1, the electric field interacting with the atom is E(r)=E,,(r)
exp( -KZ) cos(k,x-m,t)
,
with the inverse decay length of the evanescent
(4) wave
the refraction index of the prism n and k2- Ic2= tot/c’. E. is the electric field amplitude on
the atom runs up this potential barrier in the evanescent wave, is slowed down by transforming kinetic energy into potential energy, stops and turns around before touching the surface of the prism. The reflection from an evanescent wave realises in this case an ideal mirror for atoms: the mutual coherence between incident and reflected de Broglie wave is preserved and no additional divergence of the atomic beam is introduced. If the initial kinetic energy of the atom is higher than the potential barrier, the atom falls onto the prism surface. As the laser field has a gaussian profile, only the atoms hitting the evanescent wave close enough to the centre of the laser beam will see a light shift large enough to reflect them. From Eq. (6) we obtain that the effective section of the atom mirror is an ellipse on the surface of the prism with the area A depending on the incidence velocity of the atom A= ;wXwYln
C&(x=y=z=O) (m~f/h)~+2Amv3i
>’
Here w, and wY mean the waists of the evanescent wave in x- and y-direction. If projected onto the cross section of the incoming atomic beam the value for the area A has to be multiplied by the sine of the incidence angle @,. When the incidence angle is increased from zero, the number of reflected atoms at first increases as fewer atoms pass by the mirror and then decreases as for more and more atoms the potential is not high enough. Another consequence of the non uniform transverse profile of the laser beam is that it makes the surfaces of constant potential convex (Fig. 1). The force experienced by the atoms during the reflection is consequently not directed exactly along the z-axis due to the x- and y- component of the electric field
W. Seifert et al. /Optics Communications I I1 (I 994) 566-5 76
gradient. This leads to a broadening of the reflected atomic beam. This broadening is however too small (typically 4/~w,=O.l mrad) to be observed in our experimental configuration. The atom can, however, make transitions to other states with different energy levels. These transitions can be caused by nonadiabatic coupling of different states. The probability PNAfor the nonadiabatic transition from state 11, n; r) to state 12, n; r) is according to Ref. [ 161 overestimated by
569
states and is consequently well suited for the MonteCarlo calculations described in Sect. 6. As shown in Fig. 2, spontaneous emission can lead to various transitions. Starting from the energy level 11, n; r) the transition rate PI 1 to the repulsive state I 1, n - 1; r) is PI, =rl
(g, nil,
n-1,
whereas the transition ]2,n-1;r) is
r> (e, 1111,n; r> I*
rate Pzl to the attractive
state
(8) and is negligible in our experiments. Even with worst case assumptions rc-l= 100 nm, v,=3 m/s and A= 0.7 GHz this probability is very small, PNAI 10p5. Losses due to tunnelling through the potential barrier onto the surface are also negligible (lo-“) as the typical approach of the atoms to the prism is about 10 to 50 nm. A more important issue are transitions by spontaneous emission of photons. Eqs. (2) and (3) show, that the atomic state 11, n; r) becomes more and more mixed up with the excited state as the atom approaches the surface. This may lead to the spontaneous emission of a photon. Spontaneous emission is in our experimental configuration the main cause to the non-ideal behaviour of the atom mirror. The spontaneous emissions degrade the coherence of the atomic wave function, because the atom acquires a random phase in the emission process. Consequently the spontaneous emissions decrease the visibility of the interference fringes in all mirror applications such as a Lloyds mirror or an atom cavity. We have checked with the Monte-Carlo simulations that the number of spontaneous emission processes for our experimental parameters varies from 0 to 10. This situation was chosen to see distinctly the influence of the experimental parameters on the number of spontaneous emissions. Consequently we have to follow the time evolution of the internal degrees of freedom of the atoms. As Q(r) >, r, we can use the limit of well resolved lines (secular approximation) which allows to regard the dressed atom to be in one of the eigenstates described in Eq. (2) and not in a superposition of these
r A-Q(r) =-4 ( Q(r)
*
>
d3PWR(r)r
e
z s(r)2 ,
(10)
where s(r) = oi( r) /2A2 is the saturation parameter at position r. Eqs. (9) and ( 10) show that in the large detuning limit transitions from state I 1, n; r) dominantly occur to the repulsive level I 1, n - 1; r) , whereas transitions changing the dressed level type are proportional to s2 and thus less likely if S-=K1. Once a transition to the dressed level 12, n- 1; r) occurs, the atom quickly returns back to a repulsive level 11, n - 2; r) with the rate PL2:
=-
r 4
(11)
The transitions described by Eqs. (9)-( 11) have very different contributions to the broadening of the reflected atomic beam. An atom falling from the level I 1, n; r) to the level I 1, n- 1; r) will get a random recoil of v,,,= hk/m = 1.2 cm/s which will contribute up to vrec/v= 0.02 mrad to the angular broadening of the reflected atomic beam. Transitions from I 1, n; r) to 12, IZ- 1; r) with a change of the sign of the potential, however, will lead to an additional broadening effect. The gradient force is attractive while the atom stays on the level 12, II - 1; r). As the rate to return back to the state I 1, n - 2; r) is very large, the atom light system will fall back quickly to the repelling level. But during the time the atom follows the attractive potential it is accelerated when approaching the surface or decelerated when going away from the sur-
w Seifert et al. /Optics Communications 1 I I (1994) 566-5 76
” 8 vts momentun, c ir r::wxtls a photl3i-i 1 I m Iluctuations, d I ‘cICC’.lead to a brow 11 ‘?om small deflect ,(I: ~-alIue. above whit ,#Iich have not un,del 1I :oergy levels. I:,1:tic sublevels are ; I _III: fluctuations, whl : r 3 involved in a tra ! +J’== 3 transition w r light. Due to the :: I coefficients for tl Lt Zeeman sublevel: ;‘I ~1for the ground si 1I ~~1have different I L:I light shifts fo’r t I t:l i : quantum numbe I : r ergy levels is diffe L s 1#3levels, the atom ( ?i -1:e fluctuations du /’ t I .different magnetic : I I; rlepulsive levels wl: t- t i the transitions I I c i;1 to fluctuations c ,I ti als of the different I I it I:ompared to the ( t /(, :c n repulsive and al I ‘1 I;]t ons will only slig t 1’. tI II tier to avoid spont I I ~I of the atom can be f ‘81i\c detuning A. Fro] ( :,I cl’sn that for large d I ?#;,m .I: I ed state is inverse1 1 I : t.f fective detuning L A2WR ' y' ',
cc -
> n;r)12----'
mges by an amount that :coil by a factor 50. These e to the fluctuations of the background of atoms exin angles to its maximum one can see a peak of atone any transitions to atadditional source of moI different magnetic subsition. In our experiment s excited with linearly poifference in the Clebschtransitions starting from there is not only one ente, but the levels with difght shifts. There are three e different values of the I m,l = 1,2, 3. As the slope :nt for these various magwill experience extra dito transitions among lev;ubstate. Even transitions :h are more likely to hap) attractive energy levels . the dipole force. As the pulsive sublevels are quite fference in the potentials *active levels these force tly broaden the reflected neous emissions the excieduced by increasing the Eqs. (2) and (3) it can unings the population of proportional to the square
:(y) A2
’
(12)
lsing the detuning also rential, that is proportional & ) and inversely proporiing
id;
1
ll,l)
&I ___-A2
dBwn
hI&(
44
1’~r~~~;c:quentlythe inten
Tj(e,nll,w;r),~d~=T~r,,,.
ty of the evanescent
(13) wave
(14)
Here WR,maxis the Rabi frequency at the turning point, where the light shift (Eq. ( 13 ) ) compensates for the incident kinetic energy
(15) and rrefl= ~/KU, is the characteristic time of the atomlaser interaction during the reflection. Note that the right hand side of Eq. ( 14) can be interpreted as the product of the decay constant, the maximum population of the excited state and the characteristic interaction time. Using the parameters of the incident atom the probability of spontaneous emission can be written as p ”
I II t 11 nfortunately, incrl ; 1.1:I,,’; t t e height of the po I 11111 : ight intensity (c I I .: I; 1I o the effective dett
1
has to be increased to allow for a large detuning to reduce the excitation and probability of spontaneous emissions while keeping the same potential barrier for the atoms. The probability Psp for a spontaneous emission event during the reflection of the atom is the time integral of the decay constant rtimes the population of the excited state I (e, n 11, n; r) I ’ during the reflection. If the effective detuning is large compared to the Rabi frequency (A P+ wR (r) ) , this integral can be solved analytically:
=P?!!!c A ?iK ’
Note that the probability for spontaneous emission does not depend on the intensity of the evanescent wave because the repulsive potential depends exponentially on the atom surface separation and, provided the potential is sufficiently high to reflect, an increase in the intensity of the evanescent wave just shifts the turning point of the trajectory away from the surface but has no influence on the field strength seen by the atom. Nevertheless, a higher intensity allows for a higher detuning, decreasing the probability of spontaneous emission. The detuning can be increased until the turning point of the trajectory is just above the prism surface. This maximum value follows from Eq. ( 15 ) with the Rabi frequency just above the prism surface w~,~ as the Rabi frequency at the turning point WR,max.
W.Seifert et al. / OpticsCommunications1: 1(1994) 566-5 76 With this value for the detuning written as p
= 2rm2v: spZG&a
Eq. ( 16) can be
1 Kc&
.
(17)
Increasing the intensity of the evanescent wave ( a e& ) allows for larger detuning. Decreasing the decay length l/~ decreases the characteristic interaction time z,,~. Both reduces the probability for spontaneous emission, reducing the number of transitions with a change of the magnetic substates or the sign of the potential. Increasing the intensity and decreasing the decay length give an appealing future for atom mirrors reflecting de Broglie waves without degrading the coherence. A promising approach to enhance the evanescent light field and to provide simultaneously a short decay length is a system of dielectric layers. According to Eq. (5 ) a large resonance angle (Yprovides the desired short decay length of the evanescent wave. Whereas the enhancement by surface plasmons requires a certain incidence angle a’, dependent on the dielectric constant of the metal used, the dielectric layer enhancement technique allows to design the layer system for large resonance angles cx and thus combines the advantages of enhanced intensity of the evanescent wave with a steep potential barrier. It is also possible to design the dielectric layers for desired polarisations.
3. Dielectric wave guide The prism with the dielectric layer system is sketched in Fig. 1. The prism consists of LaSFN 18 glass with an index of refraction of n, = 1.9. It is coated with a 350 nm thick layer of Si02 with an index of refraction of nz = 1.46 and covered by the wave guide layer of TiO,( n3 = 2.37) which is 86 nm thick. An onresonance enhancement of the evanescent wave intensity of 100 is expected at an incidence angle of 6 1.6 degrees with a full width at half maximum of 0.4 degrees, when light with a wavelength of 8 12 nm and a transverse electric polarisation (the electric wave vector is perpendicular to the plane of incidence; known as TE- or s-polarisation) is used. This leads to a decay length of the evanescent wave of 97 nm. In the simple “zig-zag” model of a wave guide the
571
light is treated as a plane wave being successively reflected between the upper and lower interface of the wave guide. This plane wave interferes constructively with itself if it accumulates between two subsequent reflections on the same boundary a phase of mx2q where m is an integer number. This condition determines the modes of the wave guide. Consequently the thickness of the wave guide layer d3 controls the incidence angle a under which a wave guide mode can be excited and also the decay length of the evanescent wave in the vacuum above the wave guide layer. The coupling between the substrate and the wave guide layer can be adjusted by the thickness d2 of the Si02 layer. An incident light beam decays exponentially away off the prism substrate as the incidence angle of the light beam (Y is higher than the critical angle CX,(sin cry,= n , /n2 ) . When the coupling matches the losses in the wave guide layer, the highest intensity of the light in the wave guide and consequently of the evanescent wave is achieved. The enhancement of evanescent waves by this dielectric wave guide structure has been described in more detail in Ref. [ 13 1.
4. Experimental
set-up
The set up of the experiment is shown in Fig. 3. A supersonic beam of metastable argon atoms in the 1s5 state is reflected. The mean velocity of the atom is 560 m/s and the monochromaticity (fwhm) v/Au is 10. The beam is collimated by two slits of 100 urn (S, ) and 10 urn (S,) in a separation 1, of 0.95 m to a divergence of 110 mad. The prism with the dielectric layers on is placed 7 cm downstream from the 10 urn slit. It can be rotated with a precision of 0.15 mrad around an axis in the middle of the atomic beam and parallel to the slits. The reflecting surface touches this axis, as shown in Fig. 3. Consequently the prism cuts half of the atomic beam when adjusted in parallel position. When it is rotated off the parallel position the number of atoms hitting the reflecting prism surface increases with increasing incidence angle until all atoms passing the 10 urn slit fall on the prism surface. The channeltron detector (ch) placed 0.58 m from the prism and covered by a 50 urn wide entrance slit (S,) can be moved in transverse direction by means
W Seifert et al. / Optics Communications 1I1 (1994) 566-5 76
Ti : sapphire ring laser (Coherent 899-2 1). To determine the detuning we used Doppler free spectroscopy in an argon discharge to find the resonance frequency and detuned the laser from this reference line. The laser beam is transverse electric (TE ) polarised with 1.1 W power and waists of 1.89 mm and 2.6 mm in the plane of incidence and transverse to it. The intensity in the centre of the laser beam is 1.4x 1O5W/ m2.
5. Experimental results Fig. 3. Sketch of the experimenta set up. A beam of metastable argon atoms in the Is, state is co1 imated by two slits, S1 and Sa, with a width of 100 urn and 10 un I respectively and separated by I, =0.95 m. A short distance (lz= 0.07 m) behind the second slit it is reflected from the evanescen wave tuned slightly above the lss-2p, resonance and enhanced by the dielectric layers evaporated on a glass prism. The prism can be rotated around the axis shown. The atomic distribution was detected further downstream (&=O.SS m) with a chanreltron (ch) behind adetection slit S, of 50 urn width, that was scanned across the atomic beam.
The profile of the reflected beam for different angles of incidence is shown in Fig. 4. The incidence angle is determined by the tilt angle @r of the prism relative to the parallel position (see Fig. 1). The zero position of the detection angle scale is given by the
10
0 of a stepper motor. The he ght of the slits is 20 mm except the second one that is 3 mm high. The slits were adjusted parallel to each other with a precision of 1.5 mrad by illuminating them with a HeNe laser and aligning the diffraction patterns. The reflecting prism surface was set parallel to the slits by using the 10 urn slit and the prism st.rface as a Lloyd’s mirror for the light of the HeNe laser. This procedure led to a precision of the alignmen of the prism surface parallel to the slits of around 5 mrad. The tolerance in the angular alignment of the slits relative to each other and relative to the prism a Id their finite width provide a resolution of 0.6 mr; d for the deflection angle @d. We use the 1s5 (J=2)-2p, (J=3) transition of metastable argon with a wavelength ;i of 8 12 nm and a lifetime T of 27 ns. The atomic beam touches only the upper half of the reflel:ting prism surface. This part of the prism is coated with a system of dielectric layers designed to enhance the evanescent field. The excitation of a wave guide mode in the prism was optimised by detecting the stray light from the dielectric layers that is maximal for optimal enhancement and by a m-line type experiment with light initially polarised at 45” from the incidence plane as described in Ref. [ 13 1. The light was produced by a
:
lo t 0
T
10
-
2.
5 =
.u 5 ~, x5.0 k
;
:
:
:
;
:
:
:
,’
:
:
1.0 mrad
1
I\
r_“:\.
;
; : : : : : : 2.0 mrad
A :.:::.:::::::
2.9 mrad
/,
, 3.9 mrad
:; I\
‘4.6 mrad
4
I\
2t
5.6 mrad A -A-
^A
6
deflection
I
\#
6
10
12
angle @)d (mrad)
Fig. 4. Distribution of the reflected atoms for different angles of incidence. The zero position of the detection angle scale is given by the position of the atoms passing by the prism in parallel position when the laser is switched off. The atomic velocity parallel to the prism is v,= 560 m/s and the detuning is A= 2nx 70 MHz. Note the different scales for the vertical axes. The flux of reflected atoms decreases with increasing incident angle, because the area with a potential high enough to reflect the atoms decreases. Note that even for large incidence angles the beam profile is still narrow.
573
W. Seifert et al. /Optics Communications 1 I1 (1994) 566-576
position of the atoms flying past the prism in the parallel position when the laser is switched off. For all incidence angles the profile of the reflected beam is within our experimental accuracy centred around a deflection angle Qd corresponding to twice the angle of incidence as expected for specular reflection. The number of reflected atoms decreases as the incidence angle @i increases, because the effective area where the intensity of the evanescent wave is high enough to reflect the atoms is getting smaller as described in Eq. (7). At a deflection angle of 2.0 mrad 35% of the incident atoms is detected in the reflected beam. The missing atoms hit the prism at the periphery of the evanescent wave where the optical potential is too weak to reflect or are attracted by the dipole force as explained in Sect. 2. Reflected atoms are observed up to an incidence angle of 6 mrad corresponding to a normal velocity v, of 3.3 m/s. There are some atoms found at a deflection angle smaller than twice the angle of incidence but the count rate for much larger deflection angles is just the dark count rate. The reflected beam is well separated from the incident one. Even at a relatively large angle of incidence of 4.8 mrad it has a width of 0.3 mm corresponding to a divergence angle spread of 0.6 mrad. This is just the angular resolution. To obtain a very high repelling potential the detuning was only 27rx 70 MHz in this experiment. As the evanescent wave is running in the same direction as the atoms, the detuning relative to an atom at rest is the detuning d plus the Doppler shift k,v,= 2n x 1.2 GHz. This Doppler shift is larger than for the interaction with a running wave, because the real component of the wave vector of an evanescent wave is larger than the wave vector of a plane wave. Due to this relatively large detuning we could not see any influence of stray light on the atoms. We looked for a possible effect by comparing the position of the atoms that passed by the prism at small incidence angles when the reflection laser was blocked to their position when the reflection laser was on. Fig. 5 displays the dependence of the intensity of the reflected beam on the detuning for a fixed deflection angle Qd of 2.3 mrad. When the effective detuning d is negative the atoms are attracted towards the prism surface. The small peak at - 0.7 GHz is due to a deflection of the atoms by part of the laser beam which is used as an additional frequency marker. When the effective detuning is positive, the atoms are
.o
5m
‘0 0
p I
0
I
I
2 effective
I 4
detuning
...._._ I
I 6
I
8
A/2n (GHz)
Fig. 5. Number of reflected atoms for different effective detunings at a deflection angle of 2.3 mrad. Solid line: intensity of the reflected atomic beam for varying effective detuning of the evanescent wave. With growing detuning the optical potential becomes weaker and only atoms that hit the evanescent wave close enough to its centre see a potential that is high enough for reflection. The dashed line shows the result of the Monte-Carlo simulation.
reflected. With growing effective detuning the intensity of the reflected beam is decreased according to Eq. (7). Reflection of atoms was obtained up to a detuning of 6.2 GHz. To evaluate the field enhancement of the dielectric layers two tests were performed. First we checked the dependence of the enhancement on the polarisation of the incident laser beam by switching the polarisation from TE to TM. Even at a much smaller incidence angle of 0.32 mrad the detuning could be increased only up to 600 MHz before the reflection disappeared. Indeed the evanescent field is for TM polarised light by far weaker, because our dielectric layers have no resonance mode for this polarisation and no enhancement is possible. Second a more quantitative test was performed. An evanescent wave was created by total internal reflection of a TE polarised light beam on a bare glass surface to investigate the enhancement factor due to the dielectric wave guide. The prism with half of the surface coated was turned upside down. The atoms were now reflected from the uncoated half of the reflecting surface. For a fixed deflection angle of 0.9 mrad atoms could be reflected up to a maximum detuning of 2 GHz. And for small detunings reflection was observed up to an incidence angle of 1.7 mrad. From Eq. (6) follows for small detuning (A-K oR), that the maximum incident velocity, that can be reflected, is
574
W. Seifert et al. /Optics Communications 11 I (1994) 566-5 76
proportional to the square r3ot of the Rabi frequency and to the fourth root of the intensity. Consequently one estimates from these data an intensity enhancement of (6 mrad/ 1.7 mrad)4= 150, defined as the ratio between the intensity of the evanescent wave with dielectric layers and without at the same polarisation and incidence angle cr of the light. This value depends strongly on the maximum angles that cannot be determined very sharply and is therefore a rough estimation; it is of the same order of magnitude as the values obtained by the optical measurements in Ref. [ 131 and by the Monte-Ca.rlo simulation described in the following section.
6. Monte-Carlo simulation of the reflection process We simulated the experiment by a Monte-Carlo calculation, because under our experimental conditions neither the steady state approximation nor the approximation that the dressed atom always stays on the initial dressed level can be applied. This simulation helps us to understand the contributions of the various broadening mechanisms on the width of the reflected atomic beam, the role of spontaneously emitted photons, and confirms the factor for the field enhancement by the dielectric layers. The numerical simulation follows the theory and assumptions described in Sect. 2. The trajectory of the atom is calculated by solving the two dimensional equation of motion in the x- and z-direction by a fourth order Runge-Kutta algorithm with variable size of the time step dt. Due to the secular approximation that is made the dressed atom is described as cascading along its energy levels without any coherence between different dressed states. Consequently the potential is the light shift (Eq. ( 1) ) that depends on the position of the atom, on the type of the dressed state (Eqs. (2), (3) ) and on the magnetic sublevel. The atomic transition is a J= 2+J’= 3 transition driven with linearly polarised light. This causes each pair of dressed levels 11, n; r) and 12, n; r) to split into three pairs 11, m, n; r) and 12, m, n; r) with different energy levels for the magnetic sublevels Im I= 0,1, 2, because of the different Clebsch-Gordon coefficients. By comparing the emission probability to a random number it is decided after each time step dt of the Runge-Kutta algorithm, whether the atom-light
system spontaneously emits a photon and decays to a different state. If the atom emits a photon, the new state is determined randomly according to the corresponding transition probabilities. The atom suffers a recoil with random direction due to the emission of the photon and the potential is changed to that of the new dressed state. Thus the fluctuations of the dipole force as described in Sect. 2 are included. To consider properly the influence of the gaussian shape of the evanescent wave, the equation of motion is solved in the z- and x-direction. The motion in ydirection (parallel to the slits) is neglected, because the detector slit is larger than the atomic beam. The whole geometry of the experiment including the sizes and positions of the slits and the shape of the evanescent wave with the waists in x- and y-direction was taken into account. The initial position of an atom in the beam in the direction parallel (y-axis) and vertical (z-axis) to the prism surface and its exact velocity and incidence angle within the narrow distributions are randomly chosen. First the dependence of the number of reflected atoms on the detuning was simulated and compared to the experimental curve in Fig. 5. The calculated curve is shown as a dashed line. Our simulations reveal that the curve depends very critically on the alignment accuracy of the experiment. If the atomic beam does not hit the evanescent wave exactly in its centre the Rabi frequency at the prism surface aa,0 is reduced and the count rate begins to drop off at a smaller detuning. In this simulation only the exact impact position of the incident atomic beam on the evanescent wave has been taken as an adjustable parameter. The best fit is obtained when the atomic beam hits the evanescent wave close to the position where the electric field is l/e times its maximum value E(x=y=z=O).From Eq. (16) follows that the average number of spontaneous emissions was 0.06 for a detuning of 4 GHz. Second the atomic distribution has been simulated for different angles of incidence. The detuning was less than the Rabi frequency, so Eq. (16) cannot be used here to calculate the number of spontaneous emissions and the Monte-Carlo simulation is needed for a quantitative analysis. Fig. 6 shows a MonteCarlo simulation of a reflection experiment at an incidence angle of 4.8 mrad. The detuning was 70 MHz which is small compared to the Rabi frequency at the
W. Seifet et al. /Optics Communications lli(l994)
J
_lLl_L-/ 2 0
4 deflection
a
6
10
angle ad (mrad)
Fig. 6. Spatial distribution of the atoms for an incidence 4.8 mrad (solid line) and the result of the Monte-Carlo tion (dashed line).
4r
1
I
deflection
I
angle of simula-
1 1
angle Q,, (mrad)
Fig. 7. Result of the Monte-Carlo simulation. The dashed line shows the distribution of atoms that have undergone transitions to attractive levels during the reflection. The solid line is the distribution of atoms that have never been on an attractive level. The sum of the two lines is the dashed line in Fig. 6.
turning point. In the simulations the atoms underwent an average of five spontaneous emissions. To match the width of the reflected beam, a small misalignment in the parallelism of the reflecting surface relative to the slits of 6 mrad has been assumed. In this simulation the overlap of the atomic beam with the evanescent wave was supposed to be perfect because in the corresponding experiment it could be controlled much more precisely than in the experiment shown in Fig. 5. To investigate the influence of the geometry and the spontaneous emissions on the width of the reflected beam, the atoms that have never been in an attractive level were counted separately. The resulting calculated distributions are drawn in Fig. 7. The
566-576
575
solid line shows the distribution of those atoms that have never been in an attractive state. The dashed line represents the distribution of atoms that have made at least one transition to an attractive state. The sum of the two lines is the dashed line in Fig. 6. Comparing the two curves we see that transitions to attractive levels lead to momentum diffusion, that is very strong in the steep field gradient, producing a broad background and not stretching the profile of the reflected beam. In addition the simulation was repeated suppressing spontaneous emissions. The distribution of atoms that did not emit any photons is nearly the same as the distributions of those that have emitted on an average three photons spontaneously but have always ended on repulsive states like I 1, n; r) as plotted by the solid line in Fig. 7. This comes because the potential does not change its sign and the atoms are mainly in magnetic substates with very similar potentials as expected from the ratio of the Clebsch-Gordon coefficients. The width of the reflected beam is consequently dominated by geometrical effects as the widths of the slits, their parallelism with the reflecting prism and each other and the convex shape of the evanescent wave. In summary, transitions among repulsive levels have little influence on the width of the reflected beam, whereas transitions to attractive levels lead to a broad background. To reproduce the results obtained by reflection from evanescent waves on the bare glass and on the dielectric layers the intensity of the evanescent wave has to be enhanced by a factor 130 for the dielectric layer experiment. This enhancement factor is close to the value of 100 calculated for the wavelength 8 12 nm from the parameters obtained by optical measurements [ 13 1. The intensity used for the simulation has always been smaller than calculated from the laser beam parameters. A similar discrepancy has been observed in various other experiments [ 6,7,9,10,12 1. This deviation may be explained by atom-surface interactions as the Van der Waals and Casimir Polder force leading to an additional attractive force on the atom and to a shift of the electronic levels when the atom comes close to the surface. These forces are relevant for an atom surface separation below 50 nm, which is half the decay length of the evanescent wave.
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W. Seifert et al. /Optics Communications 1I1 (1994) 566-5 76
7. Conclusion
We have presented in this paper the reflection of metastable argon atoms off an evanescent wave enhanced by a dielectric wave guide coating. In comparison to the enhancement of the evanescent wave by surface plasmons [ IO- 12 ] we see three advantages of the dielectric wave guides. First the decay length of the evanescent wave is a design parameter and can be chosen small to keep the probability for spontaneous emissions low whereas the decay length connected to surface plasmons is determined by material constants. Second, the enhancement is limited by losses in the layers, that can be small and an enhancement of several thousands seems to be realistic. Third the dielectric layers are easy to handle, chemically stable and withstand high laser intensities. We did not see any degradation of the films during various different experiments over several months. We have observed reflection of atoms with an incidence velocity of 3.3 m/s normal to the surface of the prism, corresponding to a deflection angle Dd of 11 mrad. We simulated the experiment in detail by Monte-Carlo techniques to improve the understanding of the experimental results. The dielectric layer system enhanced the evanescent wave by two orders of magnitude compared to the evanescent wave on the bare glass surface. This is in agreement with optical measurements of the enhancement reported in Ref. [ 13 1. The short decay length of the evanescent wave provided a narrow profile of the reflected beam indicating that the number of spontaneous emissions was, despite the small detuning, relatively low in this experiment due to the short interaction time. If the evanescent wave realised here was used to reflect atoms falling from a height of 2 cm in a gravitational cavity, the mean number of spontaneous emissions during the reflection according to Eq. ( 16 ) would be expected to be 2.5 x 1OP4 at the maximum possible detuning. In future experiments we aim at enhancement factors of several thousands and plan to examine the properties of dielectric layers on curved surfaces that are of interest for atom cavities. The dielectric layers make an atom mirror based on an evanescent wave a
more useful optical element, because they allow the suppression of spontaneous emissions and the reflection of atoms incident with higher velocities.
Acknowledgements This work was supported by the European Community’s Science programme (SC 1-CT92-0778)) DRET (grant number 9 1055 ) and the Deutsche Forschungsgemeinschaft. We gratefully acknowledge fruitful discussions with J. Dalibard, C. Westbrook, C.S. Adams, C. Ekstrom, M. Wilkens and S. Herminghaus.
References [ 1] See e.g. Special Issue “Optics
and Interferometry with Atoms”, J. Mlynek, V.I. Balykin and P. Meystre, eds., Appl. Phys. B 54 (1992); C. Adams, M. Sigel and J. Mlynek, Phys. Rep. 240 ( 1994) 143. [2 I J.V. Hajnal, K.G.H. Baldwin, P.T.H. Fisk, H.-A. Bachor and G.I. Opat, Optics Comm. 73 (1989) 331. and W. 13 M. Christ, A. Scholz, M. Schiffer, R. Deutschmann Ertmer, Optics Comm. 107 ( 1994) 2 11. 14 V.I. Balykin and V.S. Letokhov, Appl. Phys. B 48 (1989) 517. [ 51 H. Wallis, J. Dalibard and C. Cohen-Tannoudji, Appl. Phys. B 54 (1992) 407. [6] M.A. Kasevich, D.S. Weiss and S. Chu, Optics Lett. 15 ( 1990) 607. [ 71 C.G. Aminoff, A.M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard and C. Cohen-Tannoudji, Phys. Rev. Lett. 71 (1993) 3083. [ 81 R.J. Cook and R.K. Hill, Optics Comm. 43 ( 1982) 258. [9] V.I. Balykin, V.S. Lethokov, Yu.B. Ovchinnikov and AI. Sidorov, Sov. Phys. JETP Lett. 45 (1987) 353. [lo] T. Esslinger, M. Weidemtiller, A. Hemmerich and T.W. Hansch, Optics Lett. 18 ( 1993) 450. [ 111 S. Feron, J. Reinhardt, S. Le Boiteux, 0. Gorceix, J. Baudon, M. Ducloy, J. Robert, Ch. Miniatura, S. Nit Chormaic, H. Haberland and V. Lorent, Optics Comm. 102 (1993) 83. [ 121 W. Seifert, C.S. Adams, V.I. Balykin, C. Heine, Yu. Ovchinnikov and J. Mlynek, Phys. Rev. A. 49 ( 1994) 38 14. [ 131 R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leioold and J. Mlvnek. Ootics Comm. 104 (1994) 234. 114 C. Henkel, A. Asped, R.‘Kaiser, C. Westbrook and J.-Y. Courtois, Phase Shifts of Atomic de Broglie Waves at an Evanescent Wave Mirror, Laser Physics, accepted for publication. J. Opt. Sot. Am. B 2 I15 J. Dalibard and C. Cohen-Tannoudji, (1985) 1707. Il6 A. Messiah, Mecanique Quantique II (Dunod, Paris, 1964).
Optics Communications 111 (1994) 566-576
Full length article
Reflection of atoms from a dielectric wave guide W. Seifert a, R. Kaiser b, A. Aspect b, J. Mlynek a b Institut d’optique,
aFakultbt ftir Physik, Universitiit Konstanz, D-78434 Konstanz, Germany Unit6 Associee au CNRS n”14, CentreScientijqued’Orsay, Bat 503, BP 147, F-91403 Orsay, France Received 6 April 1994; revised manuscript received 16 June 1994
Abstract We report on the reflection of metastable argon atoms from an evanescent wave enhanced by means of a dielectric wave guide by two orders of magnitude in intensity. This allows to reflect atoms with an incidence velocity of up to 3 m/s. We compare the experimental results to a theoretical model by means of a Monte-Carlo simulation.
1. Introduction In the last few years there has been significant progress in atom optics [ 11. Due to breakthroughs in laser cooling and atom interferometry new fields of interesting research have been opened. One useful element for atom optics is an atom mirror. It can be used to bend the atomic paths in an interferometer, to focus an atomic beam onto a surface e.g. in atom lithography or in a scanning atom microscope or to diffract atoms if the reflectivity is spatially modulated [ 2,3 1. A mirror can also be used in an atom resonator [ 4-7 ] where atoms are bottled without permanent interaction with the confining potential or the multiple interference between the de Broglie waves would allow to study quantum statistical effects. For the application in interferometric experiments or as a reflection grating the coherence of the de Broglie wave has to be preserved during the reflection. The most promising physical realisation of an atom mirror uses the dipole force of a blue detuned (light frequency above atomic transition frequency) inhomogeneous light field to reflect the atoms. An evanescent wave obtained by total internal reflection of a
laser beam in a prism gives the required field gradient. This scheme was first proposed by Cook and Hill [ 8 ] and demonstrated by Balykin et al. [ 9 1. In many applications of this type of mirror it is crucial, that the atoms do not spontaneously emit photons. Spontaneous emissions cause momentum diffusion leading to a broadening of the reflected atomic beam and also destroy the mutual coherence between the incident and the reflected de Broglie waves. The probability for spontaneous emissions during the reflection can be reduced by increasing the resonance detuning of the light field, but increasing the detuning reduces the repelling optical potential. Consequently it is desirable to enhance the light intensity to allow for the detuning to be increased. Recently reflection of atoms using a plasmon enhanced evanescent wave has been reported [ lo- 12 1. Additionally the probability for spontaneous emissions is reduced when the time of interaction with the evanescent wave is reduced by decreasing the decay length of the evanescent wave. A promising approach to enhance the evanescent light field and to provide simultaneously a short decay length is a system of dielectric layers. Here a light wave is coupled into a dielectric wave guide via optical tunnelling through a
0030-4018/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved .SsDIOO30-4018(94)00369-6
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W. Seifet et al. /Optics Communications 11I (1994) 566-576
solid gap. This scheme allows to enhance the light intensity by 2-3 orders of magnitude and it is possible to adapt the dielectric layers for different resonance angles of the incident light beam yielding thus different decay lengths of the evanescent wave. In this communication we report on the first reflection of atoms from such a multilayer system, analyse its influence on the reflection of atoms and compare the results with the reflection from a plasmon enhanced evanescent wave. This paper is organised as follows: in Sect. 2 we recall the principles of an atom mirror based on an evanescent wave. In Sect. 3 we briefly describe the dielectric wave guide used for the enhancement of the evanescent wave. The fabrication and the optical properties of our dielectric multilayer system has been reported elsewhere [ 13 1. In Sect. 4 the experimental set-up is described. In Sect. 5 the experimental results are reported and in particular the reflection of atoms bounced off an evanescent wave on a bare prism is compared to the case of an enhanced evanescent wave using our dielectric coating. In Sect. 6 we present Monte-Carlo calculations that simulate our experiment and reproduce the experimental results. These simulations are based on the dressed state picture presented in Sect. 2. They allow us to study separately the different processes appearing during the reflection of the atoms.
2. Principle of the atom mirror The reflection of atoms from an evanescent wave is outlined in Fig. 1. An atomic beam is incident with angle Oi relative to the prism surface and reflected from the evanescent wave. As the spread of the atomic wave packet is small compared to the decay length of the evanescent wave, the atom is treated as a classical particle with a well defined position r. A quantum analysis of the phase of the reflected de Broglie wave will be given elsewhere [ 141. The main features of the atom mirror can be understood by considering a two-level atom (atomic transition frequency mat, atomic dipole moment d, velocity v) interacting with a strong light wave (laser frequency oL). As the maximum atom-laser coupling frequency and the detuning are large compared to the decay rate r of the excited state in our experiment, a convenient approach
\ s’
I/’
Fig. 1. Schematic representation of reflection of an atomic beam from an evanescent wave. The atomic beam is incident at an angle or relative to the surface plane. The incidence angle (Yof the laser beam is measured against the surface normal. The prism (n,= 1.9) is coated with a layer of TiOz (~I,=86 nm, n,=2.37) on spacer layer of SiOl (d,=350 nm, n2= 1.46).
for the theoretical description of the reflection of atoms from an evanescent wave is the dressed atom model [ 151. Although the laser field in the evanescent wave does not need to be quantized for a complete description of our experimental results, it provides a straightforward insight into the time evolution of the atomic degrees of freedom and the trajectories. Fig. 2 shows the energy levels of the dressed states I 1, rt; r) and 12, n; r) . I 1, n; r) means that the atom has the position r and the atom-light system is in the dressed state 1 with n photons in the light mode. Following the notations of Ref. [ 15 ] we write the energies of these states: E In(r)=(n+l)Aw
L
E,,(r)=(n+l)fiw,-
-E
2
+ *n(r) 2’
AA K?(r) 7 - 2,
(la) (lb)
where A is the effective detuning (A=o,-co,,-k-u), k the real part of the wave vector of the evanescent wave, n the number of photons in the laser mode, on the on-resonance Rabi frequency and Q(r) = r + A t h e generalised Rabi frequency. The eix/v genstates of the dressed atom can be written: ll,n;r)=exp(ik-r/2)sinO(r)lg,n+l) +exp( -ik.r/2) (2, n; r) =exp(ik*r/2) -exp(
-ik.r/2)
cos O(r) le, n) cosO(r) Ig, n + 1) sin e(r) le, n) ,
with the angle O(r) defined by
(2)
568
W. Seifert et al. /Optics Communications 11 I (1994) 566-5 76 A
the prism-vacuum interface. This theoretical treatment holds for the coated and for the uncoated prism. For large z and positive detuning, the state 11, n; r) corresponds to the state 1g, n + 1) . An atom arriving in the ground state and following its energy level E,,(r) will consequently see a potential barrier. If its incident velocity v, normal to the plane of the prism is sufficiently small so that the kinetic energy is smaller than the repelling light potential
R
1:’ L Il,n;r)
12,n,r>
2
\I/
Y
Fig. 2. Spatial dependence of dressed state energy levels involved in the reflection process. The sizes of the different frequencies involved are not scaled. A possible sequence of transitions is shown as an example. An atom incoming in state 11,n; r) makes a transition to the attractive state 12, n - 1; r) by spontaneous emission of a photon. A short time later it falls back to a repulsive state, stops and is accelerated away from the surface.
cos 28(r) = - -
A
Q(r)
sin 28(r) = Q(r) wR(r)
’
’
(3)
1g, n) and 1e, n) mean that the atom is in the ground state or excited state respectively and there are n photons in the light mode. The on-resonance Rabi frequency WR(r) = - d.E( r) /#i depends on the electric field amplitude E(r) of the light field. Using an evanescent wave, as shown in Fig. 1, the electric field interacting with the atom is E(r)=E,,(r)
exp( -KZ) cos(k,x-m,t)
,
with the inverse decay length of the evanescent
(4) wave
the refraction index of the prism n and k2- Ic2= tot/c’. E. is the electric field amplitude on
the atom runs up this potential barrier in the evanescent wave, is slowed down by transforming kinetic energy into potential energy, stops and turns around before touching the surface of the prism. The reflection from an evanescent wave realises in this case an ideal mirror for atoms: the mutual coherence between incident and reflected de Broglie wave is preserved and no additional divergence of the atomic beam is introduced. If the initial kinetic energy of the atom is higher than the potential barrier, the atom falls onto the prism surface. As the laser field has a gaussian profile, only the atoms hitting the evanescent wave close enough to the centre of the laser beam will see a light shift large enough to reflect them. From Eq. (6) we obtain that the effective section of the atom mirror is an ellipse on the surface of the prism with the area A depending on the incidence velocity of the atom A= ;wXwYln
C&(x=y=z=O) (m~f/h)~+2Amv3i
>’
Here w, and wY mean the waists of the evanescent wave in x- and y-direction. If projected onto the cross section of the incoming atomic beam the value for the area A has to be multiplied by the sine of the incidence angle @,. When the incidence angle is increased from zero, the number of reflected atoms at first increases as fewer atoms pass by the mirror and then decreases as for more and more atoms the potential is not high enough. Another consequence of the non uniform transverse profile of the laser beam is that it makes the surfaces of constant potential convex (Fig. 1). The force experienced by the atoms during the reflection is consequently not directed exactly along the z-axis due to the x- and y- component of the electric field
W. Seifert et al. /Optics Communications I I1 (I 994) 566-5 76
gradient. This leads to a broadening of the reflected atomic beam. This broadening is however too small (typically 4/~w,=O.l mrad) to be observed in our experimental configuration. The atom can, however, make transitions to other states with different energy levels. These transitions can be caused by nonadiabatic coupling of different states. The probability PNAfor the nonadiabatic transition from state 11, n; r) to state 12, n; r) is according to Ref. [ 161 overestimated by
569
states and is consequently well suited for the MonteCarlo calculations described in Sect. 6. As shown in Fig. 2, spontaneous emission can lead to various transitions. Starting from the energy level 11, n; r) the transition rate PI 1 to the repulsive state I 1, n - 1; r) is PI, =rl
(g, nil,
n-1,
whereas the transition ]2,n-1;r) is
r> (e, 1111,n; r> I*
rate Pzl to the attractive
state
(8) and is negligible in our experiments. Even with worst case assumptions rc-l= 100 nm, v,=3 m/s and A= 0.7 GHz this probability is very small, PNAI 10p5. Losses due to tunnelling through the potential barrier onto the surface are also negligible (lo-“) as the typical approach of the atoms to the prism is about 10 to 50 nm. A more important issue are transitions by spontaneous emission of photons. Eqs. (2) and (3) show, that the atomic state 11, n; r) becomes more and more mixed up with the excited state as the atom approaches the surface. This may lead to the spontaneous emission of a photon. Spontaneous emission is in our experimental configuration the main cause to the non-ideal behaviour of the atom mirror. The spontaneous emissions degrade the coherence of the atomic wave function, because the atom acquires a random phase in the emission process. Consequently the spontaneous emissions decrease the visibility of the interference fringes in all mirror applications such as a Lloyds mirror or an atom cavity. We have checked with the Monte-Carlo simulations that the number of spontaneous emission processes for our experimental parameters varies from 0 to 10. This situation was chosen to see distinctly the influence of the experimental parameters on the number of spontaneous emissions. Consequently we have to follow the time evolution of the internal degrees of freedom of the atoms. As Q(r) >, r, we can use the limit of well resolved lines (secular approximation) which allows to regard the dressed atom to be in one of the eigenstates described in Eq. (2) and not in a superposition of these
r A-Q(r) =-4 ( Q(r)
*
>
d3PWR(r)r
e
z s(r)2 ,
(10)
where s(r) = oi( r) /2A2 is the saturation parameter at position r. Eqs. (9) and ( 10) show that in the large detuning limit transitions from state I 1, n; r) dominantly occur to the repulsive level I 1, n - 1; r) , whereas transitions changing the dressed level type are proportional to s2 and thus less likely if S-=K1. Once a transition to the dressed level 12, n- 1; r) occurs, the atom quickly returns back to a repulsive level 11, n - 2; r) with the rate PL2:
=-
r 4
(11)
The transitions described by Eqs. (9)-( 11) have very different contributions to the broadening of the reflected atomic beam. An atom falling from the level I 1, n; r) to the level I 1, n- 1; r) will get a random recoil of v,,,= hk/m = 1.2 cm/s which will contribute up to vrec/v= 0.02 mrad to the angular broadening of the reflected atomic beam. Transitions from I 1, n; r) to 12, IZ- 1; r) with a change of the sign of the potential, however, will lead to an additional broadening effect. The gradient force is attractive while the atom stays on the level 12, II - 1; r). As the rate to return back to the state I 1, n - 2; r) is very large, the atom light system will fall back quickly to the repelling level. But during the time the atom follows the attractive potential it is accelerated when approaching the surface or decelerated when going away from the sur-
w Seifert et al. /Optics Communications 1 I I (1994) 566-5 76
” 8 vts momentun, c ir r::wxtls a photl3i-i 1 I m Iluctuations, d I ‘cICC’.lead to a brow 11 ‘?om small deflect ,(I: ~-alIue. above whit ,#Iich have not un,del 1I :oergy levels. I:,1:tic sublevels are ; I _III: fluctuations, whl : r 3 involved in a tra ! +J’== 3 transition w r light. Due to the :: I coefficients for tl Lt Zeeman sublevel: ;‘I ~1for the ground si 1I ~~1have different I L:I light shifts fo’r t I t:l i : quantum numbe I : r ergy levels is diffe L s 1#3levels, the atom ( ?i -1:e fluctuations du /’ t I .different magnetic : I I; rlepulsive levels wl: t- t i the transitions I I c i;1 to fluctuations c ,I ti als of the different I I it I:ompared to the ( t /(, :c n repulsive and al I ‘1 I;]t ons will only slig t 1’. tI II tier to avoid spont I I ~I of the atom can be f ‘81i\c detuning A. Fro] ( :,I cl’sn that for large d I ?#;,m .I: I ed state is inverse1 1 I : t.f fective detuning L A2WR ' y' ',
cc -
> n;r)12----'
mges by an amount that :coil by a factor 50. These e to the fluctuations of the background of atoms exin angles to its maximum one can see a peak of atone any transitions to atadditional source of moI different magnetic subsition. In our experiment s excited with linearly poifference in the Clebschtransitions starting from there is not only one ente, but the levels with difght shifts. There are three e different values of the I m,l = 1,2, 3. As the slope :nt for these various magwill experience extra dito transitions among lev;ubstate. Even transitions :h are more likely to hap) attractive energy levels . the dipole force. As the pulsive sublevels are quite fference in the potentials *active levels these force tly broaden the reflected neous emissions the excieduced by increasing the Eqs. (2) and (3) it can unings the population of proportional to the square
:(y) A2
’
(12)
lsing the detuning also rential, that is proportional & ) and inversely proporiing
id;
1
ll,l)
&I ___-A2
dBwn
hI&(
44
1’~r~~~;c:quentlythe inten
Tj(e,nll,w;r),~d~=T~r,,,.
ty of the evanescent
(13) wave
(14)
Here WR,maxis the Rabi frequency at the turning point, where the light shift (Eq. ( 13 ) ) compensates for the incident kinetic energy
(15) and rrefl= ~/KU, is the characteristic time of the atomlaser interaction during the reflection. Note that the right hand side of Eq. ( 14) can be interpreted as the product of the decay constant, the maximum population of the excited state and the characteristic interaction time. Using the parameters of the incident atom the probability of spontaneous emission can be written as p ”
I II t 11 nfortunately, incrl ; 1.1:I,,’; t t e height of the po I 11111 : ight intensity (c I I .: I; 1I o the effective dett
1
has to be increased to allow for a large detuning to reduce the excitation and probability of spontaneous emissions while keeping the same potential barrier for the atoms. The probability Psp for a spontaneous emission event during the reflection of the atom is the time integral of the decay constant rtimes the population of the excited state I (e, n 11, n; r) I ’ during the reflection. If the effective detuning is large compared to the Rabi frequency (A P+ wR (r) ) , this integral can be solved analytically:
=P?!!!c A ?iK ’
Note that the probability for spontaneous emission does not depend on the intensity of the evanescent wave because the repulsive potential depends exponentially on the atom surface separation and, provided the potential is sufficiently high to reflect, an increase in the intensity of the evanescent wave just shifts the turning point of the trajectory away from the surface but has no influence on the field strength seen by the atom. Nevertheless, a higher intensity allows for a higher detuning, decreasing the probability of spontaneous emission. The detuning can be increased until the turning point of the trajectory is just above the prism surface. This maximum value follows from Eq. ( 15 ) with the Rabi frequency just above the prism surface w~,~ as the Rabi frequency at the turning point WR,max.
W.Seifert et al. / OpticsCommunications1: 1(1994) 566-5 76 With this value for the detuning written as p
= 2rm2v: spZG&a
Eq. ( 16) can be
1 Kc&
.
(17)
Increasing the intensity of the evanescent wave ( a e& ) allows for larger detuning. Decreasing the decay length l/~ decreases the characteristic interaction time z,,~. Both reduces the probability for spontaneous emission, reducing the number of transitions with a change of the magnetic substates or the sign of the potential. Increasing the intensity and decreasing the decay length give an appealing future for atom mirrors reflecting de Broglie waves without degrading the coherence. A promising approach to enhance the evanescent light field and to provide simultaneously a short decay length is a system of dielectric layers. According to Eq. (5 ) a large resonance angle (Yprovides the desired short decay length of the evanescent wave. Whereas the enhancement by surface plasmons requires a certain incidence angle a’, dependent on the dielectric constant of the metal used, the dielectric layer enhancement technique allows to design the layer system for large resonance angles cx and thus combines the advantages of enhanced intensity of the evanescent wave with a steep potential barrier. It is also possible to design the dielectric layers for desired polarisations.
3. Dielectric wave guide The prism with the dielectric layer system is sketched in Fig. 1. The prism consists of LaSFN 18 glass with an index of refraction of n, = 1.9. It is coated with a 350 nm thick layer of Si02 with an index of refraction of nz = 1.46 and covered by the wave guide layer of TiO,( n3 = 2.37) which is 86 nm thick. An onresonance enhancement of the evanescent wave intensity of 100 is expected at an incidence angle of 6 1.6 degrees with a full width at half maximum of 0.4 degrees, when light with a wavelength of 8 12 nm and a transverse electric polarisation (the electric wave vector is perpendicular to the plane of incidence; known as TE- or s-polarisation) is used. This leads to a decay length of the evanescent wave of 97 nm. In the simple “zig-zag” model of a wave guide the
571
light is treated as a plane wave being successively reflected between the upper and lower interface of the wave guide. This plane wave interferes constructively with itself if it accumulates between two subsequent reflections on the same boundary a phase of mx2q where m is an integer number. This condition determines the modes of the wave guide. Consequently the thickness of the wave guide layer d3 controls the incidence angle a under which a wave guide mode can be excited and also the decay length of the evanescent wave in the vacuum above the wave guide layer. The coupling between the substrate and the wave guide layer can be adjusted by the thickness d2 of the Si02 layer. An incident light beam decays exponentially away off the prism substrate as the incidence angle of the light beam (Y is higher than the critical angle CX,(sin cry,= n , /n2 ) . When the coupling matches the losses in the wave guide layer, the highest intensity of the light in the wave guide and consequently of the evanescent wave is achieved. The enhancement of evanescent waves by this dielectric wave guide structure has been described in more detail in Ref. [ 13 1.
4. Experimental
set-up
The set up of the experiment is shown in Fig. 3. A supersonic beam of metastable argon atoms in the 1s5 state is reflected. The mean velocity of the atom is 560 m/s and the monochromaticity (fwhm) v/Au is 10. The beam is collimated by two slits of 100 urn (S, ) and 10 urn (S,) in a separation 1, of 0.95 m to a divergence of 110 mad. The prism with the dielectric layers on is placed 7 cm downstream from the 10 urn slit. It can be rotated with a precision of 0.15 mrad around an axis in the middle of the atomic beam and parallel to the slits. The reflecting surface touches this axis, as shown in Fig. 3. Consequently the prism cuts half of the atomic beam when adjusted in parallel position. When it is rotated off the parallel position the number of atoms hitting the reflecting prism surface increases with increasing incidence angle until all atoms passing the 10 urn slit fall on the prism surface. The channeltron detector (ch) placed 0.58 m from the prism and covered by a 50 urn wide entrance slit (S,) can be moved in transverse direction by means
W Seifert et al. / Optics Communications 1I1 (1994) 566-5 76
Ti : sapphire ring laser (Coherent 899-2 1). To determine the detuning we used Doppler free spectroscopy in an argon discharge to find the resonance frequency and detuned the laser from this reference line. The laser beam is transverse electric (TE ) polarised with 1.1 W power and waists of 1.89 mm and 2.6 mm in the plane of incidence and transverse to it. The intensity in the centre of the laser beam is 1.4x 1O5W/ m2.
5. Experimental results Fig. 3. Sketch of the experimenta set up. A beam of metastable argon atoms in the Is, state is co1 imated by two slits, S1 and Sa, with a width of 100 urn and 10 un I respectively and separated by I, =0.95 m. A short distance (lz= 0.07 m) behind the second slit it is reflected from the evanescen wave tuned slightly above the lss-2p, resonance and enhanced by the dielectric layers evaporated on a glass prism. The prism can be rotated around the axis shown. The atomic distribution was detected further downstream (&=O.SS m) with a chanreltron (ch) behind adetection slit S, of 50 urn width, that was scanned across the atomic beam.
The profile of the reflected beam for different angles of incidence is shown in Fig. 4. The incidence angle is determined by the tilt angle @r of the prism relative to the parallel position (see Fig. 1). The zero position of the detection angle scale is given by the
10
0 of a stepper motor. The he ght of the slits is 20 mm except the second one that is 3 mm high. The slits were adjusted parallel to each other with a precision of 1.5 mrad by illuminating them with a HeNe laser and aligning the diffraction patterns. The reflecting prism surface was set parallel to the slits by using the 10 urn slit and the prism st.rface as a Lloyd’s mirror for the light of the HeNe laser. This procedure led to a precision of the alignmen of the prism surface parallel to the slits of around 5 mrad. The tolerance in the angular alignment of the slits relative to each other and relative to the prism a Id their finite width provide a resolution of 0.6 mr; d for the deflection angle @d. We use the 1s5 (J=2)-2p, (J=3) transition of metastable argon with a wavelength ;i of 8 12 nm and a lifetime T of 27 ns. The atomic beam touches only the upper half of the reflel:ting prism surface. This part of the prism is coated with a system of dielectric layers designed to enhance the evanescent field. The excitation of a wave guide mode in the prism was optimised by detecting the stray light from the dielectric layers that is maximal for optimal enhancement and by a m-line type experiment with light initially polarised at 45” from the incidence plane as described in Ref. [ 13 1. The light was produced by a
:
lo t 0
T
10
-
2.
5 =
.u 5 ~, x5.0 k
;
:
:
:
;
:
:
:
,’
:
:
1.0 mrad
1
I\
r_“:\.
;
; : : : : : : 2.0 mrad
A :.:::.:::::::
2.9 mrad
/,
, 3.9 mrad
:; I\
‘4.6 mrad
4
I\
2t
5.6 mrad A -A-
^A
6
deflection
I
\#
6
10
12
angle @)d (mrad)
Fig. 4. Distribution of the reflected atoms for different angles of incidence. The zero position of the detection angle scale is given by the position of the atoms passing by the prism in parallel position when the laser is switched off. The atomic velocity parallel to the prism is v,= 560 m/s and the detuning is A= 2nx 70 MHz. Note the different scales for the vertical axes. The flux of reflected atoms decreases with increasing incident angle, because the area with a potential high enough to reflect the atoms decreases. Note that even for large incidence angles the beam profile is still narrow.
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W. Seifert et al. /Optics Communications 1 I1 (1994) 566-576
position of the atoms flying past the prism in the parallel position when the laser is switched off. For all incidence angles the profile of the reflected beam is within our experimental accuracy centred around a deflection angle Qd corresponding to twice the angle of incidence as expected for specular reflection. The number of reflected atoms decreases as the incidence angle @i increases, because the effective area where the intensity of the evanescent wave is high enough to reflect the atoms is getting smaller as described in Eq. (7). At a deflection angle of 2.0 mrad 35% of the incident atoms is detected in the reflected beam. The missing atoms hit the prism at the periphery of the evanescent wave where the optical potential is too weak to reflect or are attracted by the dipole force as explained in Sect. 2. Reflected atoms are observed up to an incidence angle of 6 mrad corresponding to a normal velocity v, of 3.3 m/s. There are some atoms found at a deflection angle smaller than twice the angle of incidence but the count rate for much larger deflection angles is just the dark count rate. The reflected beam is well separated from the incident one. Even at a relatively large angle of incidence of 4.8 mrad it has a width of 0.3 mm corresponding to a divergence angle spread of 0.6 mrad. This is just the angular resolution. To obtain a very high repelling potential the detuning was only 27rx 70 MHz in this experiment. As the evanescent wave is running in the same direction as the atoms, the detuning relative to an atom at rest is the detuning d plus the Doppler shift k,v,= 2n x 1.2 GHz. This Doppler shift is larger than for the interaction with a running wave, because the real component of the wave vector of an evanescent wave is larger than the wave vector of a plane wave. Due to this relatively large detuning we could not see any influence of stray light on the atoms. We looked for a possible effect by comparing the position of the atoms that passed by the prism at small incidence angles when the reflection laser was blocked to their position when the reflection laser was on. Fig. 5 displays the dependence of the intensity of the reflected beam on the detuning for a fixed deflection angle Qd of 2.3 mrad. When the effective detuning d is negative the atoms are attracted towards the prism surface. The small peak at - 0.7 GHz is due to a deflection of the atoms by part of the laser beam which is used as an additional frequency marker. When the effective detuning is positive, the atoms are
.o
5m
‘0 0
p I
0
I
I
2 effective
I 4
detuning
...._._ I
I 6
I
8
A/2n (GHz)
Fig. 5. Number of reflected atoms for different effective detunings at a deflection angle of 2.3 mrad. Solid line: intensity of the reflected atomic beam for varying effective detuning of the evanescent wave. With growing detuning the optical potential becomes weaker and only atoms that hit the evanescent wave close enough to its centre see a potential that is high enough for reflection. The dashed line shows the result of the Monte-Carlo simulation.
reflected. With growing effective detuning the intensity of the reflected beam is decreased according to Eq. (7). Reflection of atoms was obtained up to a detuning of 6.2 GHz. To evaluate the field enhancement of the dielectric layers two tests were performed. First we checked the dependence of the enhancement on the polarisation of the incident laser beam by switching the polarisation from TE to TM. Even at a much smaller incidence angle of 0.32 mrad the detuning could be increased only up to 600 MHz before the reflection disappeared. Indeed the evanescent field is for TM polarised light by far weaker, because our dielectric layers have no resonance mode for this polarisation and no enhancement is possible. Second a more quantitative test was performed. An evanescent wave was created by total internal reflection of a TE polarised light beam on a bare glass surface to investigate the enhancement factor due to the dielectric wave guide. The prism with half of the surface coated was turned upside down. The atoms were now reflected from the uncoated half of the reflecting surface. For a fixed deflection angle of 0.9 mrad atoms could be reflected up to a maximum detuning of 2 GHz. And for small detunings reflection was observed up to an incidence angle of 1.7 mrad. From Eq. (6) follows for small detuning (A-K oR), that the maximum incident velocity, that can be reflected, is
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W. Seifert et al. /Optics Communications 11 I (1994) 566-5 76
proportional to the square r3ot of the Rabi frequency and to the fourth root of the intensity. Consequently one estimates from these data an intensity enhancement of (6 mrad/ 1.7 mrad)4= 150, defined as the ratio between the intensity of the evanescent wave with dielectric layers and without at the same polarisation and incidence angle cr of the light. This value depends strongly on the maximum angles that cannot be determined very sharply and is therefore a rough estimation; it is of the same order of magnitude as the values obtained by the optical measurements in Ref. [ 131 and by the Monte-Ca.rlo simulation described in the following section.
6. Monte-Carlo simulation of the reflection process We simulated the experiment by a Monte-Carlo calculation, because under our experimental conditions neither the steady state approximation nor the approximation that the dressed atom always stays on the initial dressed level can be applied. This simulation helps us to understand the contributions of the various broadening mechanisms on the width of the reflected atomic beam, the role of spontaneously emitted photons, and confirms the factor for the field enhancement by the dielectric layers. The numerical simulation follows the theory and assumptions described in Sect. 2. The trajectory of the atom is calculated by solving the two dimensional equation of motion in the x- and z-direction by a fourth order Runge-Kutta algorithm with variable size of the time step dt. Due to the secular approximation that is made the dressed atom is described as cascading along its energy levels without any coherence between different dressed states. Consequently the potential is the light shift (Eq. ( 1) ) that depends on the position of the atom, on the type of the dressed state (Eqs. (2), (3) ) and on the magnetic sublevel. The atomic transition is a J= 2+J’= 3 transition driven with linearly polarised light. This causes each pair of dressed levels 11, n; r) and 12, n; r) to split into three pairs 11, m, n; r) and 12, m, n; r) with different energy levels for the magnetic sublevels Im I= 0,1, 2, because of the different Clebsch-Gordon coefficients. By comparing the emission probability to a random number it is decided after each time step dt of the Runge-Kutta algorithm, whether the atom-light
system spontaneously emits a photon and decays to a different state. If the atom emits a photon, the new state is determined randomly according to the corresponding transition probabilities. The atom suffers a recoil with random direction due to the emission of the photon and the potential is changed to that of the new dressed state. Thus the fluctuations of the dipole force as described in Sect. 2 are included. To consider properly the influence of the gaussian shape of the evanescent wave, the equation of motion is solved in the z- and x-direction. The motion in ydirection (parallel to the slits) is neglected, because the detector slit is larger than the atomic beam. The whole geometry of the experiment including the sizes and positions of the slits and the shape of the evanescent wave with the waists in x- and y-direction was taken into account. The initial position of an atom in the beam in the direction parallel (y-axis) and vertical (z-axis) to the prism surface and its exact velocity and incidence angle within the narrow distributions are randomly chosen. First the dependence of the number of reflected atoms on the detuning was simulated and compared to the experimental curve in Fig. 5. The calculated curve is shown as a dashed line. Our simulations reveal that the curve depends very critically on the alignment accuracy of the experiment. If the atomic beam does not hit the evanescent wave exactly in its centre the Rabi frequency at the prism surface aa,0 is reduced and the count rate begins to drop off at a smaller detuning. In this simulation only the exact impact position of the incident atomic beam on the evanescent wave has been taken as an adjustable parameter. The best fit is obtained when the atomic beam hits the evanescent wave close to the position where the electric field is l/e times its maximum value E(x=y=z=O).From Eq. (16) follows that the average number of spontaneous emissions was 0.06 for a detuning of 4 GHz. Second the atomic distribution has been simulated for different angles of incidence. The detuning was less than the Rabi frequency, so Eq. (16) cannot be used here to calculate the number of spontaneous emissions and the Monte-Carlo simulation is needed for a quantitative analysis. Fig. 6 shows a MonteCarlo simulation of a reflection experiment at an incidence angle of 4.8 mrad. The detuning was 70 MHz which is small compared to the Rabi frequency at the
W. Seifet et al. /Optics Communications lli(l994)
J
_lLl_L-/ 2 0
4 deflection
a
6
10
angle ad (mrad)
Fig. 6. Spatial distribution of the atoms for an incidence 4.8 mrad (solid line) and the result of the Monte-Carlo tion (dashed line).
4r
1
I
deflection
I
angle of simula-
1 1
angle Q,, (mrad)
Fig. 7. Result of the Monte-Carlo simulation. The dashed line shows the distribution of atoms that have undergone transitions to attractive levels during the reflection. The solid line is the distribution of atoms that have never been on an attractive level. The sum of the two lines is the dashed line in Fig. 6.
turning point. In the simulations the atoms underwent an average of five spontaneous emissions. To match the width of the reflected beam, a small misalignment in the parallelism of the reflecting surface relative to the slits of 6 mrad has been assumed. In this simulation the overlap of the atomic beam with the evanescent wave was supposed to be perfect because in the corresponding experiment it could be controlled much more precisely than in the experiment shown in Fig. 5. To investigate the influence of the geometry and the spontaneous emissions on the width of the reflected beam, the atoms that have never been in an attractive level were counted separately. The resulting calculated distributions are drawn in Fig. 7. The
566-576
575
solid line shows the distribution of those atoms that have never been in an attractive state. The dashed line represents the distribution of atoms that have made at least one transition to an attractive state. The sum of the two lines is the dashed line in Fig. 6. Comparing the two curves we see that transitions to attractive levels lead to momentum diffusion, that is very strong in the steep field gradient, producing a broad background and not stretching the profile of the reflected beam. In addition the simulation was repeated suppressing spontaneous emissions. The distribution of atoms that did not emit any photons is nearly the same as the distributions of those that have emitted on an average three photons spontaneously but have always ended on repulsive states like I 1, n; r) as plotted by the solid line in Fig. 7. This comes because the potential does not change its sign and the atoms are mainly in magnetic substates with very similar potentials as expected from the ratio of the Clebsch-Gordon coefficients. The width of the reflected beam is consequently dominated by geometrical effects as the widths of the slits, their parallelism with the reflecting prism and each other and the convex shape of the evanescent wave. In summary, transitions among repulsive levels have little influence on the width of the reflected beam, whereas transitions to attractive levels lead to a broad background. To reproduce the results obtained by reflection from evanescent waves on the bare glass and on the dielectric layers the intensity of the evanescent wave has to be enhanced by a factor 130 for the dielectric layer experiment. This enhancement factor is close to the value of 100 calculated for the wavelength 8 12 nm from the parameters obtained by optical measurements [ 13 1. The intensity used for the simulation has always been smaller than calculated from the laser beam parameters. A similar discrepancy has been observed in various other experiments [ 6,7,9,10,12 1. This deviation may be explained by atom-surface interactions as the Van der Waals and Casimir Polder force leading to an additional attractive force on the atom and to a shift of the electronic levels when the atom comes close to the surface. These forces are relevant for an atom surface separation below 50 nm, which is half the decay length of the evanescent wave.
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W. Seifert et al. /Optics Communications 1I1 (1994) 566-5 76
7. Conclusion
We have presented in this paper the reflection of metastable argon atoms off an evanescent wave enhanced by a dielectric wave guide coating. In comparison to the enhancement of the evanescent wave by surface plasmons [ IO- 12 ] we see three advantages of the dielectric wave guides. First the decay length of the evanescent wave is a design parameter and can be chosen small to keep the probability for spontaneous emissions low whereas the decay length connected to surface plasmons is determined by material constants. Second, the enhancement is limited by losses in the layers, that can be small and an enhancement of several thousands seems to be realistic. Third the dielectric layers are easy to handle, chemically stable and withstand high laser intensities. We did not see any degradation of the films during various different experiments over several months. We have observed reflection of atoms with an incidence velocity of 3.3 m/s normal to the surface of the prism, corresponding to a deflection angle Dd of 11 mrad. We simulated the experiment in detail by Monte-Carlo techniques to improve the understanding of the experimental results. The dielectric layer system enhanced the evanescent wave by two orders of magnitude compared to the evanescent wave on the bare glass surface. This is in agreement with optical measurements of the enhancement reported in Ref. [ 13 1. The short decay length of the evanescent wave provided a narrow profile of the reflected beam indicating that the number of spontaneous emissions was, despite the small detuning, relatively low in this experiment due to the short interaction time. If the evanescent wave realised here was used to reflect atoms falling from a height of 2 cm in a gravitational cavity, the mean number of spontaneous emissions during the reflection according to Eq. ( 16 ) would be expected to be 2.5 x 1OP4 at the maximum possible detuning. In future experiments we aim at enhancement factors of several thousands and plan to examine the properties of dielectric layers on curved surfaces that are of interest for atom cavities. The dielectric layers make an atom mirror based on an evanescent wave a
more useful optical element, because they allow the suppression of spontaneous emissions and the reflection of atoms incident with higher velocities.
Acknowledgements This work was supported by the European Community’s Science programme (SC 1-CT92-0778)) DRET (grant number 9 1055 ) and the Deutsche Forschungsgemeinschaft. We gratefully acknowledge fruitful discussions with J. Dalibard, C. Westbrook, C.S. Adams, C. Ekstrom, M. Wilkens and S. Herminghaus.
References [ 1] See e.g. Special Issue “Optics
and Interferometry with Atoms”, J. Mlynek, V.I. Balykin and P. Meystre, eds., Appl. Phys. B 54 (1992); C. Adams, M. Sigel and J. Mlynek, Phys. Rep. 240 ( 1994) 143. [2 I J.V. Hajnal, K.G.H. Baldwin, P.T.H. Fisk, H.-A. Bachor and G.I. Opat, Optics Comm. 73 (1989) 331. and W. 13 M. Christ, A. Scholz, M. Schiffer, R. Deutschmann Ertmer, Optics Comm. 107 ( 1994) 2 11. 14 V.I. Balykin and V.S. Letokhov, Appl. Phys. B 48 (1989) 517. [ 51 H. Wallis, J. Dalibard and C. Cohen-Tannoudji, Appl. Phys. B 54 (1992) 407. [6] M.A. Kasevich, D.S. Weiss and S. Chu, Optics Lett. 15 ( 1990) 607. [ 71 C.G. Aminoff, A.M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard and C. Cohen-Tannoudji, Phys. Rev. Lett. 71 (1993) 3083. [ 81 R.J. Cook and R.K. Hill, Optics Comm. 43 ( 1982) 258. [9] V.I. Balykin, V.S. Lethokov, Yu.B. Ovchinnikov and AI. Sidorov, Sov. Phys. JETP Lett. 45 (1987) 353. [lo] T. Esslinger, M. Weidemtiller, A. Hemmerich and T.W. Hansch, Optics Lett. 18 ( 1993) 450. [ 111 S. Feron, J. Reinhardt, S. Le Boiteux, 0. Gorceix, J. Baudon, M. Ducloy, J. Robert, Ch. Miniatura, S. Nit Chormaic, H. Haberland and V. Lorent, Optics Comm. 102 (1993) 83. [ 121 W. Seifert, C.S. Adams, V.I. Balykin, C. Heine, Yu. Ovchinnikov and J. Mlynek, Phys. Rev. A. 49 ( 1994) 38 14. [ 131 R. Kaiser, Y. Levy, N. Vansteenkiste, A. Aspect, W. Seifert, D. Leioold and J. Mlvnek. Ootics Comm. 104 (1994) 234. 114 C. Henkel, A. Asped, R.‘Kaiser, C. Westbrook and J.-Y. Courtois, Phase Shifts of Atomic de Broglie Waves at an Evanescent Wave Mirror, Laser Physics, accepted for publication. J. Opt. Sot. Am. B 2 I15 J. Dalibard and C. Cohen-Tannoudji, (1985) 1707. Il6 A. Messiah, Mecanique Quantique II (Dunod, Paris, 1964).