- Email: [email protected]
Diffraction of atoms from a standing light wave
Physica B 151 (1988) 255-261 North-Holland, Amsterdam
DIFFRACTION OF ATOMS FROM A STANDING LIGHT WAVE Peter J. MARTIN*, Phillip L. GOULD**, Bruce G. O L D A K E R , Andrew H. M I K L I C H and David E. P R I T C H A R D Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA This paper presents experiments concerning momentum transfer to atoms by a standing light wave in the absence of spontaneous decay. This momentum transfer, which occurs in discrete units of 2 hk along the k-vector of the standing light wave, can be viewed as absorption/stimulated emission of photon pairs from the counterpropagating traveling waves which make up the standing light wave. In a dual sense, this phenomenon can also be viewed as diffraction of an atomic de Broglie wave from the periodic intensity grating of the standing light wave. In addition, we address how the inherent Heisenberg uncertainty between the focussed waist of the standing light wave and the angular spread of the k-vectors of photons traveling through this waist affects momentum transfer to the atoms by the light. For large widths of the standing light wave, the reduction of the uncertainty in the direction of the photons results in resonances for the momentum transfer only for discrete values of atomic momentum along the k-vector of the standing light wave which satisfy the Bragg condition. We present experimental data which display the Pendellrsung effect for Bragg scattering of atomic de Broglie waves from a standing light wave. Finally, we discuss the possibility of exploiting these phenomena to build an atomic interferometer, one that interferes atomic de Broglie waves.
I. Introduction
Experiments on wave-like diffraction of particles such as electrons and neutrons from crystals were among the initial confirmation of the concept of quantum mechanical wave-particle dualism. This paper presents experimental data of diffraction of atoms from a standing light wave. These experiments constitute a full reversal of the ordinary roles of waves and particles in that we are diffracting matter waves from the periodic structure of a standing light wave. These experiments also represent a major breakthrough in the coherent manipulation of atoms, completing the technology necessary to construct an atomic interferometer; a discussion of this possibility will be given at the end of this paper. Kapitza and Dirac [1] predicted in 1933 that an electron beam would diffract from a standing *Current address: Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, CO 803090440, USA. ** Current address: Electricity Division, Center for Basic Standards, National Bureau of Standards, Gaithersburg, MD 20899, USA.
light wave as a result of stimulated Compton scattering from a periodic grating. This idea attracted much interest because it illuminated the dual nature of both matter and light (i.e. it is diffraction of matter waves from a light grating). It was conceded by Kapitza and Dirac, however, that this phenomenon would be very difficult, if not impossible, to observe because of the extremely weak interaction between the electrons and the light. The Kapitza-Dirac effect is much easier to observe with neutral atoms than with electrons [2]. Because atoms have internal structure, there is an enormous enhancement of the interaction strength when the frequency of the standing light wave is tuned close to an atomic transition frequency (i.e. a transition between different electronic orbital states). Kapitza-Dirac scattering of neutral atoms can be viewed classically as arising from an interaction of the atom's induced dipole moment with the intensity gradient of the standing light wave. The major complexity of using atoms instead of electrons, however, is the possibility of spontaneous decay in atoms. Spontaneous decay is a stochastic process and tends to
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P.J. Martin et al. / Diffraction o f atoms from a standing light wave
destroy the coherence of the light-atom interaction necessary for diffraction of atoms by a standing light wave. The way to circumvent the problems associated with spontaneous decay is to avoid tuning the light too close to the atomic transition frequency. In this case, the probability that the atom is in the excited state, hence the probability of spontaneous decay, remains small. Thus, the coherence of the light-atom interaction is preserved even when the light-atom interaction time is many spontaneous lifetimes [3]. Quantum mechanically, momentum transfer to atoms by a standing light wave (i.e. KapitzaDirac scattering) in the absence of spontaneous emission can be viewed in two ways. If one views the standing light wave as being composed of two counterpropagating traveling waves, then the momentum transfer process can be viewed as absorption of a photon and its momentum, I hk, from one traveling wave followed by stimulated emission of this photon into the counterpropagating traveling wave. This process results in momentum transfer to the atom in discrete units of 2 hk along the k-vector of the standing light wave. In this view, both the atom and the light are considered to have purely corpuscular properties. In a dual sense, one can view this phenomenon as diffraction of an atomic de Broglie wave, with wavelength AD.B. = h/p, from the intensity grating of the standing light wave which has spatial periodicity dlight = Alight/2. Thus, constructive interference occurs at discrete angles given by ~b = AD.B./dlight which again results in momentum transfer to the atom in discrete units of 2 hk along the k-vector of the standing light wave. The experimental apparatus, described in refs. [3-6], was designed and built to make measurements of momentum transfer to atoms by a standing light wave under controlled conditions to facilitate quantitative comparison between experiment and theory. A monoenergetic sodium beam (Av/v = 11% F W H M ) is highly collimated with two 10 Ixm slits spaced 0.9 m apart. In addition, each sodium atom is prepared in such a way that it can be thought of as a two-state internal system (i.e. the ground and the excited state). The light-atom interaction region, displayed in
fig. 1, is constructed by reflecting a laser beam from a mirror such that this standing light wave intersects the highly collimated atomic beam. The angle 0 between the atomic beam and the standing wave nodes, which needs to be smaller than 2 mrad to even observe diffraction, can be systematically varied with a high degree of accuracy by tilting the mirror. The final momentum distributions of the atomic beam are detected 1.2 m downstream from the interaction region by a 25 p~m scanning hot-wire detector (this detector counts atoms hitting a thin wire which can be precisely translated). The overall momentum resolution of this apparatus is 0.8 hk (FWHM), less than the momentum of a single photon of light. Experimental data [4] of Kapitza-Dirac diffraction of atoms from a standing light wave are shown in fig. 2. If the light wave is blocked, then all of the atoms are in the undiffracted state at p =Ohk. When the atoms interact with the standing light wave, momentum transfer occurs in discrete units of 2 hk, as predicted. Increased light power manifests itself as larger intensity gradients between the nodes and antinodes of the standing light wave. These larger intensity gradients result in a larger dipole force, hence the rms width of the diffraction patterns increases as the light power is increased.
/
X
otomic Na beam
Fig. 1. Diagram of interaction region. Standing light wave has a Gaussian intensity profile (a e x p [ - ½(y/Am)2]) in direction perpendicular to the k-vector. Aw is the rms width of the intensity profile of the standing light wave.
P.J. Martin et al. / Diffraction o f atoms from a standing light wave
o)
.I
b)
.18
c)
• 25
0 -12
6
-6
12
p/hk Fig. 2. Experimental data of K a p i t z a - D i r a c scattering for different angles 0 between the atomic b e a m and the standing wavefronts. (a) 0 = 0; (b) O = 1.2 mrad; (c) 0 = 1.7 mrad. For all scans, the light power is P = 2 . 2 m W , the detuning is A = 280 MHz, and the waist size of the standing light wave is Aw = 80 ~m.
The greatest amount of momentum transfer, for a given light power and detuning, occurs when the standing wavefronts are exactly parallel to the initial atomic beam. The rms width of these diffraction patterns falls off monotonically as the angle 0 between the standing-wave nodes and the initial atomic beam is increased. This monotonic decrease can be viewed classically as due to an averaging of the dipole force as the atom travels over "hill and dale" of the nodal structure of the standing light wave. In addition, the Kapitza-Dirac diffraction patterns remain symmetric with respect to the undiffracted momentum state even when the initial atomic beam is not parallel to the nodes of the standing light wave. We have developed a theory for momentum transfer to atoms by a standing light wave in the absence of spontaneous decay which is based on
257
a one-dimensional, time-dependent Schr6dinger equation [3-5]. This theory, which contains no adjustable parameters, predicts that the probability amplitudes of the various momentum states are given by a Bessel function distribution of a single argument that depends on the radiative power of the light and also the detuning of the light frequency from the atomic transition frequency. The argument of the Bessel function also depends on the initial velocity component, vx = v sin 0, of the atom along the k-vector of the standing light wave. The theoretical prediction convolved with the experimental resolution is also displayed in fig. 2 (dashed lines). We find no disagreement between experiment and theory [4]. A particularly interesting phenomenon concerns how the atomic motion along the k-vector of the standing light wave affects momentum transfer to the atom by the light. A simple physical picture of this phenomenon is presented here. The focussed waist of a Gaussian light beam, such as we use, has a minimum Heisenberg uncertainty between the rms waist size Aw and the rms angular spread A~b of the k-vectors of photons traveling through this waist, a w h p = A w . h k . Aq~ = h / 2 . For tightly focussed waists (Aw--50/zm) used for Kapitza-Dirac diffraction, the uncertainty in the directions of the k-vectors of photons traveling through this waist is much larger than the diffraction angle between different orders /~D.B./dlight= 60 Izrad. The momentum transfer via absorption/stimulated emission of photon pairs changes the momentum but not the laboratory kinetic energy of the atom, thus constraining the various final momentum vectors to a circle in momentum space. In the case of Kapitza-Dirac scattering, the uncertainty in the direction of the k-vectors is so large that the atom can scatter into many different orders (i.e. final momentum states) and still conserve energy and momentum, as shown in fig. 3. In addition, because of this uncertainty, the diffraction patterns remain symmetric even when the standing light wave is tilted with respect to the atomic beam. If the waist size of the Gaussian light beam is
258
P.J. Martin et al. / Diffraction o f atoms from a standing light wave
\
AW~ 50/~ /-
where the incident angle 0 satisfies the Bragg condition
. . . .
mAD.B. ----2dlight sin 0 .
.. l-~_:-:TZZZ~ ....
///,~/"
/'x,~..
~
~W~
5mm .
.... i ///
Pf~P~
.
.
.
.
.
.
L( \x
Fig. 3. Comparison of Kapitza-Dirac scattering and Bragg scattering. A tightly focussed waist (top) has a large angular uncertainty in the direction of its photons, thus allowing energy conservation for many final momentum states pf - this is the Kapitza-Dirac regime. The observation of Bragg scattering requires a much larger waist for the light where the photons are highly collimated. In this case, the only process which can conserve energy and momentum is Bragg scattering.
large enough ( A w - 3 mm) then the angular uncertainty of the k-vectors of the photons is less than the diffraction angle q5 =AD.B./dlight b e t w e e n orders. In this case, the only momentum transfer process which can conserve both momentum and energy is Bragg scattering,
In this expression, AD.B. is the de Broglie wavelength of the atom, dlight = )tlight/2 is the periodic intensity spacing of the standing light wave and m = 1, 2, 3 . . . . is called the order. The Bragg condition dictates that momentum transfer can only occur for discrete initial values Px = rn h k of atomic momentum along the k-vector of the standing light wave. Energy and momentum conservation then allows momentum transfer only to the state Px = - m h k as shown in fig. 3. For the Bragg scattering experiment, the rms waist size of the standing light wave was 3 mm, approximately two orders of magnitude larger than the waist size used for Kapitza-Dirac scattering. Although the transit time of the atom though this waist was approximately a thousand spontaneous lifetimes, the effects of spontaneous decay were again greatly suppressed by detuning the frequency of the light far from the resonance frequency of the internal atomic transition. Typically, the detunings used to observe Bragg scattering are two orders of magnitude larger than the spontaneous linewidth, F. Experimental data [5] of first-order Bragg scattering (m = 1) are shown in figs. 4a and 4b. When the light wave is blocked, all atoms are in the undiffracted state p = 0 h k . When the atoms interact with the standing light wave and the angle of the mirror, hence the standing wavefronts, is positioned at 0 = 30 txrad with respect to the atomic beam, then population is seen in the diffracted momentum state p = - 2 h k . As the laser power is increased, the population in the diffracted momentum state first increases and then decreases, with the majority of the population returning to the undiffracted momentum state p = 0 h k . In addition, higher-order Bragg scattering has been observed for orders m = 2 ( 0 = 6 0 p . r a d ) , m = 3 (0=901xrad) and m = 4 (0 = 120 i~rad). Fig. 4c displays data of secondorder Bragg scattering. We have developed a theory [5] which yields an analytic expression for the probability that the
P.J. Martin et al. / Diffraction o f atoms from a standing light wave
259
1.0
• 54
•27 ~
a)
.8
.4 b)
•2
~
.32
k
0
l
i
Power [mW] c)
•
25
-B
-4
0
4
pl~k Fig. 4. Experimental data of Bragg scattering. (a) First-order Bragg scattering, P = 6 mW, zl = 800 MHz, Aw = 3.2 ram; (b) first-order Bragg scattering, P = 10 roW, zl = 800 MHz, Aw = 3.2 m m ; (c) second-order Bragg scattering, P = 4 mW, zl = 5 0 0 M H z , Aw = 1 . 6 m m . T h e angle between the standing wavefronts and the atomic b e a m was 30 i~rad times the order, m.
atom exits the standing light wave in the diffracted momentum state (p = - 2 hk) in the case of first-order Bragg scattering. This probability is given by
P(t = oo) = sin •
[-~--
In this expression, g20 is called the Rabi frequency, ~-/2 is the transit time for the rms width Aw of the standing light wave, and zl is the detuning. The quantity/202~ - is proportional to the radiative power of the standing light wave. This expression for the probability of the atom to be in the diffracted momentum state convolved with the experimental resolution is plotted in fig. 5 as a function of radiative power of the light• As shown in fig. 5, the results of the convolution is a reduction in the total amount of population
Fig. 5. Plot of probability to be in the diffracted m o m e n t u m state ( p = - 2 hk) as a function of light power for first-order Bragg scattering (m = 1). zi = 800 M H z and Aw = 2.5 m m for these plots. Also shown is experimental data of the population in the diffracted state for various light powers.
transfer to the diffracted momentum state, although the probability still varies sinusoidally with light power. Experimental data of the population in the diffracted momentum state for various light powers is also displayed in fig. 5. The experiment agrees qualitatively with theory in that the functional form of momentum transfer as a function of laser power and detuning seems to be correct. However, imperfections in the experimental apparatus (i.e. polarization imperfections, light beam aberrations and finite velocity resolution of the atomic beam) are assumed to lead to a reduction in the amount of momentum transfer to the diffracted peak and also from the diffracted peak back to the undiffracted peak. In addition, although spontaneous decays were suppressed by detuning far from resonance [3] (the average number of spontaneous decays in the standing light wave, N < 0.1 for these experiments) the residual effect of spontaneous decay could account for some of the discrepancy between theory and experiment. The measurements shown in fig. 5 display the analog of the Pendellrsung effect, previously observed in Bragg scattering of neutrons from crystals [7]. In the case of Bragg scattering of X-rays, neutrons or atoms from a periodic potential, the dynamical theory of diffraction [8] pre-
260
P.J. Martin et al. / Diffraction o f atoms f r o m a standing light wave
dicts a coherent splitting of the incident wave into four components, with two traveling wave components passing within the crystal in the Bragg (diffracted) direction and two components in the incident direction. These components are, in general, described by different wavevectors, differing both in magnitude and direction. For Bragg scattering of neutrons from crystals, the energy splitting between the two components of either the diffracted or undiffracted results in a beating of these two components at different depths of the crystal. This beating results in a sinusoidally varying probability for the final diffracted wave as the length of the crystal is increased. The crystal length at which this probability amplitude reaches its first maximum is called half the Pendell6sung length. For first-order Bragg scattering of atoms from a standing light wave, the Pendell6sung effect results in a sinusoidally varying probability amplitude of the diffracted atomic wave as the width • of the standing light wave is increased 2 (for fixed light intensity, g20~-). For our experiment, it was easier to keep the length ~- constant and vary the light power (i.e. the interaction potential) as shown in fig. 4. We define a half Pendell6sung interaction strength g2~-= C-8--~za as the total interaction stength needed to achieve maximum momentum transfer to the diffracted state for first-order Bragg scattering of atoms from a standing light wave. The observation and control of Bragg scattering of atomic de Broglie waves provides a major breakthrough in the development of an atomic
interferometer [9]. Scan (a) of fig. 4 basically displays an atomic "beam s p l i t t e r " - a device which coherently splits an atomic de Broglie wave into two spatially distinct waves. One can reflect these beams and then recombine them with another Bragg scattering interaction region, thus constructing a M a c h - Z e n d e r atomic interferometer. This device could be used to measure ground state phase shifts of an atomic beam. Possible sources of this phase shift are electric or magnetic fields, black-body radiation, the Casimir shift [10] or scattering from atoms (where the real part of the forward scattering cross section could be measured).
References [1] P.L. Kapitza and P.A.M. Dirac, Proc. Cambridge Philos. Soc. 29 (1933) 297. [2] S. Altshuler, L.M. Frantz and R. Braunstein, Phys. Rev. Lett. 17 (1966) 231. [3] P.L. Gould, G.A. Ruff and D.E. Pritchard, Phys. Rev. Lett. 56 (1986) 327. [4] P.J. Martin, P.L. Gould, B.G. Oldaker, A.H. Miklich and D.E. Pritchard, Phys. Rev. A 36 (1987) 2495. [5] P.J. Martin, B.G. Oldaker, A.H. Miklich and D.E. Pritchard, accepted by Phys. Rev. Lett. [6] D.E. Pritchard and P.L. Gould, J. Opt. Soc. Am. B 2 (1985) 1799. [7] C.G. Shull, Phys. Rev. Lett. 21 (1968) 1585. [8] R.W. James, Solid State Physics. (Academic Press, New York, 1963), vol. 15, pp. 53-220. [9] V.P. Chebotayev, B.Y. Dubetsky, A.P. Kazantsev and V.P. Yakovlev, J. Opt. Soc. Am. B 2 (1985) 1791. [10] H.B.G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360.
DISCUSSION (Q) H. Rauch: Can you give numbers for the electric (or magnetic) field strength at the nodes of the standing light wave? (A) P.J. Martin: T h e standing light wave has a Gaussian intensity profile in the y-direction and a sinusoidally varying intensity profile in the x-direction (see fig. 1 of paper). The "potential" seen by the atom is (in atomic physics terminology the ac Stark shift of the internal ground state of the atom)
V(x, y) = +-h{A 2 + 122oexp[- ~(y/aw)=l
cos=kx) 'j= .
A: detuning of the standing light wave from the two-state resonance frequency, 12o =- E d / h Rabi frequency, E: peak electric field strength of one of the two counterpropagating travelling waves which make up the standing light wave, d=- ( e l d [ g ) : matrix element of the atomic transition,
261
P.J. Martin et al. / Diffraction o f atoms f r o m a standing light wave
l l
excited s t a t e (e) ....
energy
t
hC°lase r
incredible advantage of changing the potential merely by turning up the laser power. In the case of neutron scattering in a crystal this cannot be easily done. From solving Schr6dinger's equation I arrived at a solution for the probability of being in the diffracted peak: p = sin2[ Laser P o w e r . N u m e r i c a l Factors] Detuning For constant laser intensity, laser power scales linearly with "thickness" of the beam.
g r o u n d s t a t e (g) Fig. D. 1.
AW: rms width of the intensity profile of the standing wave. Because the atom enters the field adiabatically from the ground state and is assumed not to experience spontaneous emission while in the field, the atom stays in the potential with the minus sign. Therefore,
(C) J. Clauser: Actually the stability problems for atoms are much easier than those of neutrons since the diffraction angle is so small. As a result the transverse standing matter wave has just the wavelength of the l a s e r - t h o u s a n d s of Angs t r o m s - not a few Angstroms as one gets with wide angles and neutrons. Hence the transverse bending problems are a thousand times easier.
V(x, y) = - h { A 2 +/22 e x p [ - ½(y/Aw) 2] cos 2 kx) 1/2
(A) P.J. Martin: I am not at all familar with transverse bending problems. However, your comment about the transverse de Broglie (standing) matter wave is definitely true. A~B = )tO.B./0 (transverse de Broglie wavelength),
We are in the regime where A >> 120 (A//20 ~ 100) and therefore can expand the square root:
"~DB 0 -= 0Bragg 2(Alight/2 ) '
V(x, y) = - h A -
(h/2~/2A) e x p [ - ½( y / A w ) 2] cos 2 k x .
T AD.B = Alight= 0.589 × 10 -6 m for our case.
This is the effective potential seen by the atom.
Numbers f o r our experiment (sodium-atom): For convenience, all energies (frequencies) are in units of the spontaneous frequency linewidth of the two level atomic transition frequency (F~-~ "-~, F : FWHM, ~': natural spontaneous lifetime of the excited state). Typical values: F / 2 ~ = 10 MHz for this transition in sodium. A --~ 1 G H z . 2~r = 2~- 100F. Aw = 3 mm, k = 2"if/Alight ()[light = 0.589 x 10 -6 m). The detuning is very small compared to the frequency of the light: A/to~.... = 10 6. Atomic sodium beam (de Broglie wave): v = 1 × 105 cm/s, A v / v = 11% (FWHM). M = 22 amu. ~ AD.B. = 0.2/~, E = 8 X 10-22 J. For O 0 = F and A = 1 0 0 F ~ V = h / 2 ~ / 2 A = 3 x 10 -29 J. (Q) G. Greene: Is the variation in intensity between the forward and once diffracted beam as a function of power analogous to thickness Pendell6sung in crystal diffraction? (A) P.J. Martin: After talking with Horne and Greenberger the answer to your question is yes. In hindsight, we have an
~li~ht
:'DB Fig. D.2. Scattering of atoms from a standing light wave.
( 0 ) S. lida: In solid state physics, we usually understand the atom by the electron cloud or the bound state electron field and the point-like nucleus. May I understand that your experiment shows clearly that there is another state function of the whole atom having the de Broglie wavelength, which can interfer with the uncertainty relation for the number of the atoms and the phase, so that the world has a complicated multi-state function structure? (A) P.J. Martin: No answer to S. Iida's question.
DIFFRACTION OF ATOMS FROM A STANDING LIGHT WAVE Peter J. MARTIN*, Phillip L. GOULD**, Bruce G. O L D A K E R , Andrew H. M I K L I C H and David E. P R I T C H A R D Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA This paper presents experiments concerning momentum transfer to atoms by a standing light wave in the absence of spontaneous decay. This momentum transfer, which occurs in discrete units of 2 hk along the k-vector of the standing light wave, can be viewed as absorption/stimulated emission of photon pairs from the counterpropagating traveling waves which make up the standing light wave. In a dual sense, this phenomenon can also be viewed as diffraction of an atomic de Broglie wave from the periodic intensity grating of the standing light wave. In addition, we address how the inherent Heisenberg uncertainty between the focussed waist of the standing light wave and the angular spread of the k-vectors of photons traveling through this waist affects momentum transfer to the atoms by the light. For large widths of the standing light wave, the reduction of the uncertainty in the direction of the photons results in resonances for the momentum transfer only for discrete values of atomic momentum along the k-vector of the standing light wave which satisfy the Bragg condition. We present experimental data which display the Pendellrsung effect for Bragg scattering of atomic de Broglie waves from a standing light wave. Finally, we discuss the possibility of exploiting these phenomena to build an atomic interferometer, one that interferes atomic de Broglie waves.
I. Introduction
Experiments on wave-like diffraction of particles such as electrons and neutrons from crystals were among the initial confirmation of the concept of quantum mechanical wave-particle dualism. This paper presents experimental data of diffraction of atoms from a standing light wave. These experiments constitute a full reversal of the ordinary roles of waves and particles in that we are diffracting matter waves from the periodic structure of a standing light wave. These experiments also represent a major breakthrough in the coherent manipulation of atoms, completing the technology necessary to construct an atomic interferometer; a discussion of this possibility will be given at the end of this paper. Kapitza and Dirac [1] predicted in 1933 that an electron beam would diffract from a standing *Current address: Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, CO 803090440, USA. ** Current address: Electricity Division, Center for Basic Standards, National Bureau of Standards, Gaithersburg, MD 20899, USA.
light wave as a result of stimulated Compton scattering from a periodic grating. This idea attracted much interest because it illuminated the dual nature of both matter and light (i.e. it is diffraction of matter waves from a light grating). It was conceded by Kapitza and Dirac, however, that this phenomenon would be very difficult, if not impossible, to observe because of the extremely weak interaction between the electrons and the light. The Kapitza-Dirac effect is much easier to observe with neutral atoms than with electrons [2]. Because atoms have internal structure, there is an enormous enhancement of the interaction strength when the frequency of the standing light wave is tuned close to an atomic transition frequency (i.e. a transition between different electronic orbital states). Kapitza-Dirac scattering of neutral atoms can be viewed classically as arising from an interaction of the atom's induced dipole moment with the intensity gradient of the standing light wave. The major complexity of using atoms instead of electrons, however, is the possibility of spontaneous decay in atoms. Spontaneous decay is a stochastic process and tends to
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P.J. Martin et al. / Diffraction o f atoms from a standing light wave
destroy the coherence of the light-atom interaction necessary for diffraction of atoms by a standing light wave. The way to circumvent the problems associated with spontaneous decay is to avoid tuning the light too close to the atomic transition frequency. In this case, the probability that the atom is in the excited state, hence the probability of spontaneous decay, remains small. Thus, the coherence of the light-atom interaction is preserved even when the light-atom interaction time is many spontaneous lifetimes [3]. Quantum mechanically, momentum transfer to atoms by a standing light wave (i.e. KapitzaDirac scattering) in the absence of spontaneous emission can be viewed in two ways. If one views the standing light wave as being composed of two counterpropagating traveling waves, then the momentum transfer process can be viewed as absorption of a photon and its momentum, I hk, from one traveling wave followed by stimulated emission of this photon into the counterpropagating traveling wave. This process results in momentum transfer to the atom in discrete units of 2 hk along the k-vector of the standing light wave. In this view, both the atom and the light are considered to have purely corpuscular properties. In a dual sense, one can view this phenomenon as diffraction of an atomic de Broglie wave, with wavelength AD.B. = h/p, from the intensity grating of the standing light wave which has spatial periodicity dlight = Alight/2. Thus, constructive interference occurs at discrete angles given by ~b = AD.B./dlight which again results in momentum transfer to the atom in discrete units of 2 hk along the k-vector of the standing light wave. The experimental apparatus, described in refs. [3-6], was designed and built to make measurements of momentum transfer to atoms by a standing light wave under controlled conditions to facilitate quantitative comparison between experiment and theory. A monoenergetic sodium beam (Av/v = 11% F W H M ) is highly collimated with two 10 Ixm slits spaced 0.9 m apart. In addition, each sodium atom is prepared in such a way that it can be thought of as a two-state internal system (i.e. the ground and the excited state). The light-atom interaction region, displayed in
fig. 1, is constructed by reflecting a laser beam from a mirror such that this standing light wave intersects the highly collimated atomic beam. The angle 0 between the atomic beam and the standing wave nodes, which needs to be smaller than 2 mrad to even observe diffraction, can be systematically varied with a high degree of accuracy by tilting the mirror. The final momentum distributions of the atomic beam are detected 1.2 m downstream from the interaction region by a 25 p~m scanning hot-wire detector (this detector counts atoms hitting a thin wire which can be precisely translated). The overall momentum resolution of this apparatus is 0.8 hk (FWHM), less than the momentum of a single photon of light. Experimental data [4] of Kapitza-Dirac diffraction of atoms from a standing light wave are shown in fig. 2. If the light wave is blocked, then all of the atoms are in the undiffracted state at p =Ohk. When the atoms interact with the standing light wave, momentum transfer occurs in discrete units of 2 hk, as predicted. Increased light power manifests itself as larger intensity gradients between the nodes and antinodes of the standing light wave. These larger intensity gradients result in a larger dipole force, hence the rms width of the diffraction patterns increases as the light power is increased.
/
X
otomic Na beam
Fig. 1. Diagram of interaction region. Standing light wave has a Gaussian intensity profile (a e x p [ - ½(y/Am)2]) in direction perpendicular to the k-vector. Aw is the rms width of the intensity profile of the standing light wave.
P.J. Martin et al. / Diffraction o f atoms from a standing light wave
o)
.I
b)
.18
c)
• 25
0 -12
6
-6
12
p/hk Fig. 2. Experimental data of K a p i t z a - D i r a c scattering for different angles 0 between the atomic b e a m and the standing wavefronts. (a) 0 = 0; (b) O = 1.2 mrad; (c) 0 = 1.7 mrad. For all scans, the light power is P = 2 . 2 m W , the detuning is A = 280 MHz, and the waist size of the standing light wave is Aw = 80 ~m.
The greatest amount of momentum transfer, for a given light power and detuning, occurs when the standing wavefronts are exactly parallel to the initial atomic beam. The rms width of these diffraction patterns falls off monotonically as the angle 0 between the standing-wave nodes and the initial atomic beam is increased. This monotonic decrease can be viewed classically as due to an averaging of the dipole force as the atom travels over "hill and dale" of the nodal structure of the standing light wave. In addition, the Kapitza-Dirac diffraction patterns remain symmetric with respect to the undiffracted momentum state even when the initial atomic beam is not parallel to the nodes of the standing light wave. We have developed a theory for momentum transfer to atoms by a standing light wave in the absence of spontaneous decay which is based on
257
a one-dimensional, time-dependent Schr6dinger equation [3-5]. This theory, which contains no adjustable parameters, predicts that the probability amplitudes of the various momentum states are given by a Bessel function distribution of a single argument that depends on the radiative power of the light and also the detuning of the light frequency from the atomic transition frequency. The argument of the Bessel function also depends on the initial velocity component, vx = v sin 0, of the atom along the k-vector of the standing light wave. The theoretical prediction convolved with the experimental resolution is also displayed in fig. 2 (dashed lines). We find no disagreement between experiment and theory [4]. A particularly interesting phenomenon concerns how the atomic motion along the k-vector of the standing light wave affects momentum transfer to the atom by the light. A simple physical picture of this phenomenon is presented here. The focussed waist of a Gaussian light beam, such as we use, has a minimum Heisenberg uncertainty between the rms waist size Aw and the rms angular spread A~b of the k-vectors of photons traveling through this waist, a w h p = A w . h k . Aq~ = h / 2 . For tightly focussed waists (Aw--50/zm) used for Kapitza-Dirac diffraction, the uncertainty in the directions of the k-vectors of photons traveling through this waist is much larger than the diffraction angle between different orders /~D.B./dlight= 60 Izrad. The momentum transfer via absorption/stimulated emission of photon pairs changes the momentum but not the laboratory kinetic energy of the atom, thus constraining the various final momentum vectors to a circle in momentum space. In the case of Kapitza-Dirac scattering, the uncertainty in the direction of the k-vectors is so large that the atom can scatter into many different orders (i.e. final momentum states) and still conserve energy and momentum, as shown in fig. 3. In addition, because of this uncertainty, the diffraction patterns remain symmetric even when the standing light wave is tilted with respect to the atomic beam. If the waist size of the Gaussian light beam is
258
P.J. Martin et al. / Diffraction o f atoms from a standing light wave
\
AW~ 50/~ /-
where the incident angle 0 satisfies the Bragg condition
. . . .
mAD.B. ----2dlight sin 0 .
.. l-~_:-:TZZZ~ ....
///,~/"
/'x,~..
~
~W~
5mm .
.... i ///
Pf~P~
.
.
.
.
.
.
L( \x
Fig. 3. Comparison of Kapitza-Dirac scattering and Bragg scattering. A tightly focussed waist (top) has a large angular uncertainty in the direction of its photons, thus allowing energy conservation for many final momentum states pf - this is the Kapitza-Dirac regime. The observation of Bragg scattering requires a much larger waist for the light where the photons are highly collimated. In this case, the only process which can conserve energy and momentum is Bragg scattering.
large enough ( A w - 3 mm) then the angular uncertainty of the k-vectors of the photons is less than the diffraction angle q5 =AD.B./dlight b e t w e e n orders. In this case, the only momentum transfer process which can conserve both momentum and energy is Bragg scattering,
In this expression, AD.B. is the de Broglie wavelength of the atom, dlight = )tlight/2 is the periodic intensity spacing of the standing light wave and m = 1, 2, 3 . . . . is called the order. The Bragg condition dictates that momentum transfer can only occur for discrete initial values Px = rn h k of atomic momentum along the k-vector of the standing light wave. Energy and momentum conservation then allows momentum transfer only to the state Px = - m h k as shown in fig. 3. For the Bragg scattering experiment, the rms waist size of the standing light wave was 3 mm, approximately two orders of magnitude larger than the waist size used for Kapitza-Dirac scattering. Although the transit time of the atom though this waist was approximately a thousand spontaneous lifetimes, the effects of spontaneous decay were again greatly suppressed by detuning the frequency of the light far from the resonance frequency of the internal atomic transition. Typically, the detunings used to observe Bragg scattering are two orders of magnitude larger than the spontaneous linewidth, F. Experimental data [5] of first-order Bragg scattering (m = 1) are shown in figs. 4a and 4b. When the light wave is blocked, all atoms are in the undiffracted state p = 0 h k . When the atoms interact with the standing light wave and the angle of the mirror, hence the standing wavefronts, is positioned at 0 = 30 txrad with respect to the atomic beam, then population is seen in the diffracted momentum state p = - 2 h k . As the laser power is increased, the population in the diffracted momentum state first increases and then decreases, with the majority of the population returning to the undiffracted momentum state p = 0 h k . In addition, higher-order Bragg scattering has been observed for orders m = 2 ( 0 = 6 0 p . r a d ) , m = 3 (0=901xrad) and m = 4 (0 = 120 i~rad). Fig. 4c displays data of secondorder Bragg scattering. We have developed a theory [5] which yields an analytic expression for the probability that the
P.J. Martin et al. / Diffraction o f atoms from a standing light wave
259
1.0
• 54
•27 ~
a)
.8
.4 b)
•2
~
.32
k
0
l
i
Power [mW] c)
•
25
-B
-4
0
4
pl~k Fig. 4. Experimental data of Bragg scattering. (a) First-order Bragg scattering, P = 6 mW, zl = 800 MHz, Aw = 3.2 ram; (b) first-order Bragg scattering, P = 10 roW, zl = 800 MHz, Aw = 3.2 m m ; (c) second-order Bragg scattering, P = 4 mW, zl = 5 0 0 M H z , Aw = 1 . 6 m m . T h e angle between the standing wavefronts and the atomic b e a m was 30 i~rad times the order, m.
atom exits the standing light wave in the diffracted momentum state (p = - 2 hk) in the case of first-order Bragg scattering. This probability is given by
P(t = oo) = sin •
[-~--
In this expression, g20 is called the Rabi frequency, ~-/2 is the transit time for the rms width Aw of the standing light wave, and zl is the detuning. The quantity/202~ - is proportional to the radiative power of the standing light wave. This expression for the probability of the atom to be in the diffracted momentum state convolved with the experimental resolution is plotted in fig. 5 as a function of radiative power of the light• As shown in fig. 5, the results of the convolution is a reduction in the total amount of population
Fig. 5. Plot of probability to be in the diffracted m o m e n t u m state ( p = - 2 hk) as a function of light power for first-order Bragg scattering (m = 1). zi = 800 M H z and Aw = 2.5 m m for these plots. Also shown is experimental data of the population in the diffracted state for various light powers.
transfer to the diffracted momentum state, although the probability still varies sinusoidally with light power. Experimental data of the population in the diffracted momentum state for various light powers is also displayed in fig. 5. The experiment agrees qualitatively with theory in that the functional form of momentum transfer as a function of laser power and detuning seems to be correct. However, imperfections in the experimental apparatus (i.e. polarization imperfections, light beam aberrations and finite velocity resolution of the atomic beam) are assumed to lead to a reduction in the amount of momentum transfer to the diffracted peak and also from the diffracted peak back to the undiffracted peak. In addition, although spontaneous decays were suppressed by detuning far from resonance [3] (the average number of spontaneous decays in the standing light wave, N < 0.1 for these experiments) the residual effect of spontaneous decay could account for some of the discrepancy between theory and experiment. The measurements shown in fig. 5 display the analog of the Pendellrsung effect, previously observed in Bragg scattering of neutrons from crystals [7]. In the case of Bragg scattering of X-rays, neutrons or atoms from a periodic potential, the dynamical theory of diffraction [8] pre-
260
P.J. Martin et al. / Diffraction o f atoms f r o m a standing light wave
dicts a coherent splitting of the incident wave into four components, with two traveling wave components passing within the crystal in the Bragg (diffracted) direction and two components in the incident direction. These components are, in general, described by different wavevectors, differing both in magnitude and direction. For Bragg scattering of neutrons from crystals, the energy splitting between the two components of either the diffracted or undiffracted results in a beating of these two components at different depths of the crystal. This beating results in a sinusoidally varying probability for the final diffracted wave as the length of the crystal is increased. The crystal length at which this probability amplitude reaches its first maximum is called half the Pendell6sung length. For first-order Bragg scattering of atoms from a standing light wave, the Pendell6sung effect results in a sinusoidally varying probability amplitude of the diffracted atomic wave as the width • of the standing light wave is increased 2 (for fixed light intensity, g20~-). For our experiment, it was easier to keep the length ~- constant and vary the light power (i.e. the interaction potential) as shown in fig. 4. We define a half Pendell6sung interaction strength g2~-= C-8--~za as the total interaction stength needed to achieve maximum momentum transfer to the diffracted state for first-order Bragg scattering of atoms from a standing light wave. The observation and control of Bragg scattering of atomic de Broglie waves provides a major breakthrough in the development of an atomic
interferometer [9]. Scan (a) of fig. 4 basically displays an atomic "beam s p l i t t e r " - a device which coherently splits an atomic de Broglie wave into two spatially distinct waves. One can reflect these beams and then recombine them with another Bragg scattering interaction region, thus constructing a M a c h - Z e n d e r atomic interferometer. This device could be used to measure ground state phase shifts of an atomic beam. Possible sources of this phase shift are electric or magnetic fields, black-body radiation, the Casimir shift [10] or scattering from atoms (where the real part of the forward scattering cross section could be measured).
References [1] P.L. Kapitza and P.A.M. Dirac, Proc. Cambridge Philos. Soc. 29 (1933) 297. [2] S. Altshuler, L.M. Frantz and R. Braunstein, Phys. Rev. Lett. 17 (1966) 231. [3] P.L. Gould, G.A. Ruff and D.E. Pritchard, Phys. Rev. Lett. 56 (1986) 327. [4] P.J. Martin, P.L. Gould, B.G. Oldaker, A.H. Miklich and D.E. Pritchard, Phys. Rev. A 36 (1987) 2495. [5] P.J. Martin, B.G. Oldaker, A.H. Miklich and D.E. Pritchard, accepted by Phys. Rev. Lett. [6] D.E. Pritchard and P.L. Gould, J. Opt. Soc. Am. B 2 (1985) 1799. [7] C.G. Shull, Phys. Rev. Lett. 21 (1968) 1585. [8] R.W. James, Solid State Physics. (Academic Press, New York, 1963), vol. 15, pp. 53-220. [9] V.P. Chebotayev, B.Y. Dubetsky, A.P. Kazantsev and V.P. Yakovlev, J. Opt. Soc. Am. B 2 (1985) 1791. [10] H.B.G. Casimir and D. Polder, Phys. Rev. 73 (1948) 360.
DISCUSSION (Q) H. Rauch: Can you give numbers for the electric (or magnetic) field strength at the nodes of the standing light wave? (A) P.J. Martin: T h e standing light wave has a Gaussian intensity profile in the y-direction and a sinusoidally varying intensity profile in the x-direction (see fig. 1 of paper). The "potential" seen by the atom is (in atomic physics terminology the ac Stark shift of the internal ground state of the atom)
V(x, y) = +-h{A 2 + 122oexp[- ~(y/aw)=l
cos=kx) 'j= .
A: detuning of the standing light wave from the two-state resonance frequency, 12o =- E d / h Rabi frequency, E: peak electric field strength of one of the two counterpropagating travelling waves which make up the standing light wave, d=- ( e l d [ g ) : matrix element of the atomic transition,
261
P.J. Martin et al. / Diffraction o f atoms f r o m a standing light wave
l l
excited s t a t e (e) ....
energy
t
hC°lase r
incredible advantage of changing the potential merely by turning up the laser power. In the case of neutron scattering in a crystal this cannot be easily done. From solving Schr6dinger's equation I arrived at a solution for the probability of being in the diffracted peak: p = sin2[ Laser P o w e r . N u m e r i c a l Factors] Detuning For constant laser intensity, laser power scales linearly with "thickness" of the beam.
g r o u n d s t a t e (g) Fig. D. 1.
AW: rms width of the intensity profile of the standing wave. Because the atom enters the field adiabatically from the ground state and is assumed not to experience spontaneous emission while in the field, the atom stays in the potential with the minus sign. Therefore,
(C) J. Clauser: Actually the stability problems for atoms are much easier than those of neutrons since the diffraction angle is so small. As a result the transverse standing matter wave has just the wavelength of the l a s e r - t h o u s a n d s of Angs t r o m s - not a few Angstroms as one gets with wide angles and neutrons. Hence the transverse bending problems are a thousand times easier.
V(x, y) = - h { A 2 +/22 e x p [ - ½(y/Aw) 2] cos 2 kx) 1/2
(A) P.J. Martin: I am not at all familar with transverse bending problems. However, your comment about the transverse de Broglie (standing) matter wave is definitely true. A~B = )tO.B./0 (transverse de Broglie wavelength),
We are in the regime where A >> 120 (A//20 ~ 100) and therefore can expand the square root:
"~DB 0 -= 0Bragg 2(Alight/2 ) '
V(x, y) = - h A -
(h/2~/2A) e x p [ - ½( y / A w ) 2] cos 2 k x .
T AD.B = Alight= 0.589 × 10 -6 m for our case.
This is the effective potential seen by the atom.
Numbers f o r our experiment (sodium-atom): For convenience, all energies (frequencies) are in units of the spontaneous frequency linewidth of the two level atomic transition frequency (F~-~ "-~, F : FWHM, ~': natural spontaneous lifetime of the excited state). Typical values: F / 2 ~ = 10 MHz for this transition in sodium. A --~ 1 G H z . 2~r = 2~- 100F. Aw = 3 mm, k = 2"if/Alight ()[light = 0.589 x 10 -6 m). The detuning is very small compared to the frequency of the light: A/to~.... = 10 6. Atomic sodium beam (de Broglie wave): v = 1 × 105 cm/s, A v / v = 11% (FWHM). M = 22 amu. ~ AD.B. = 0.2/~, E = 8 X 10-22 J. For O 0 = F and A = 1 0 0 F ~ V = h / 2 ~ / 2 A = 3 x 10 -29 J. (Q) G. Greene: Is the variation in intensity between the forward and once diffracted beam as a function of power analogous to thickness Pendell6sung in crystal diffraction? (A) P.J. Martin: After talking with Horne and Greenberger the answer to your question is yes. In hindsight, we have an
~li~ht
:'DB Fig. D.2. Scattering of atoms from a standing light wave.
( 0 ) S. lida: In solid state physics, we usually understand the atom by the electron cloud or the bound state electron field and the point-like nucleus. May I understand that your experiment shows clearly that there is another state function of the whole atom having the de Broglie wavelength, which can interfer with the uncertainty relation for the number of the atoms and the phase, so that the world has a complicated multi-state function structure? (A) P.J. Martin: No answer to S. Iida's question.