Influence of laser fluctuations and spontaneous emission on the ring ...

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PHYSICAL REVIEW A

VOLUME 54, NUMBER 3

SEPTEMBER 1996

Influence of laser fluctuations and spontaneous emission on the ring-shaped atomic distribution in a magneto-optical trap F. Dias Nunes, J. F. Silva, S. C. Zilio, and V. S. Bagnato ˜o Carlos, Universidade de Sa ˜o Paulo, Caixa Postal 369, Departamento de Fı´sica e Cieˆncia dos Materiais, Instituto de Fisica de Sa ˜o Carlos, Sa ˜o Paulo, Brazil 13560-970 Sa ~Received 16 April 1996! We present results of numerical simulations to justify the origin of the width in ring-shaped atomic distributions of cold atoms held in a magneto-optical trap. We have considered the effects of laser fluctuations ~phase and intensity! and the presence of recoil due to spontaneous emission. The results show that the occurrence of spontaneous emission accounts for the existence of a width in the atomic trajectory while laser fluctuations only change the average value of the atomic orbit. @S1050-2947~96!07809-2# PACS number~s!: 32.80.Pj

I. INTRODUCTION

The spatial distribution of atoms in a magneto-optic trap ~MOT! can have different shapes, depending on external parameters such as the laser intensity, detuning, misalignment, and magnetic field gradient. Structures such as ball, ring, and ring with core can be generated when the laser beams are arranged to form a racetrack configuration @1,2#. A theoretical interpretation for such structures, based on the vortex force, which arises due to the misaligned Gaussian laser beams, was recently reported @2# but the model was not complete since it could not explain the existence of a finite width in the ring-shaped distribution. At the time of publication of Refs. @2# and @3# we tried to justify the width of the observed rings using simple arguments, without success. The present work presents an explanation for the existence of the ring width, based on a numerical simulation where we have included the existence of laser fluctuations and the occurrence of spontaneous emission, which could, in principle, explain the observations. Our results show that the existence of phase and amplitude fluctuations just modifies the average radius of the atomic orbit without causing any significant effect on its variance. However, it cannot explain the existence of a finite width. On the other hand, the presence of spontaneous emission has considerable influence on the variance of the radius without change in its mean value. This explains well the experimental observations. This work starts by presenting a brief description of the experimental observations for a ring-shaped trap and how the vortex force model explains it. Next, we obtain the radiation force for the case where fluctuations are included. Such an expression is then used in the numerical integration of the atomic motion equation. Spontaneous emission is introduced through random kicks in the momentum of the atom. The numerical results are presented and interpreted, and the comparison with the experimental observations is made. These results should be interpreted as complementary to our previous works @2,3# where the question about the ring width was left out. II. RING-SHAPED ATOMIC DISTRIBUTION OF TRAPPED ATOMS

For the experimental observations reported in previous works @2,3# we have used a MOT @4,5# that captures sodium 1050-2947/96/54~3!/2271~4!/$10.00

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atoms from a vapor held at about 50 °C. Three orthogonal retroreflected Gaussian laser beams of about 1.2 cm in diameter and carrying opposite circularly polarized light intercept in a region with a magnetic field having a constant gradient of about 10 G/cm in the axial direction and half of that in the other two orthogonal directions. The field is produced by a pair of anti-Helmholtz coils. The vapor is contained in a stainless steel chamber with several entrances used as windows for the trapping beams and for observation, and also as ports for vacuum pumps. The background pressure is kept below 1029 Torr. The laser beam, provided by a ring-dye laser, is tuned to about 210 MHz of the transition 3S 1/2(F51)→3 P 3/2 (F50) ~type II tuning, as described in Ref. @5#! carrying a sideband resonant with the 3S 1/2(F52)→3 P 3/2(F52) transition, which works as a repumper to prevent depletion in the F51 ground-state population. The trap and the repumper beams have almost 100 mW/cm2 each, before splitting to the three directions. The sideband is produced by a homemade electro-optical modulator. The determination of the number of atoms in the trap is done through the measurement of fluorescence of the atoms by imaging the trap onto a calibrated photomultiplier tube. The image of the atomic cloud is made using a homemade imaging system and a CCD ~charge coupled device! camera. The images are recorded on conventional VHS tapes for further analysis. Dimensions of the atomic cloud are determined either by using the CCD camera or through a survey telescope. When all laser beams are well aligned with respect to the retroreflection and magnetic field, as presented in Fig. 1~a!, we observe a bright ball of atoms at the center of the trap as shown in Fig. 1~b!. Under these conditions the spatial atomic distribution is close to spherical. When the laser beams are slightly misaligned, as shown in Fig. 1~c!, different spatial distributions, mainly contained in the x-y plane, are observed @2#. The displacement (s'1 to 2 mm! is normally much smaller than the radius of the Gaussian beams (w'6 mm!. The most common distribution is a ring, as presented in Fig. 1~d!. As we can see, the ring has an average radius and a finite width. In the present case, the ring-shaped atomic distribution is independent of the number of trapped atoms and it has been interpreted by us as the result of a new manifestation of the radiative force, referred to here as a 2271

© 1996 The American Physical Society

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DIAS NUNES, SILVA, ZILIO, AND BAGNATO

scribe the radiation force and the effect of laser field fluctuations was taken into account following a procedure similar to that of Sharma and Prasad @6#, Cook @7#, and already used by Napolitano and Bagnato @8# for analyzing deceleration of atoms with the Zeeman tuning technique. The spontaneous emission effect on the atomic motion was taken into account by adding a variation ~Dv! to the atomic velocity due to the recoil caused by the atomic emission. The recoil was assumed constant in intensity but randomly oriented in space. Next, we will present a resume of the equations that describe the forces for a two-level atom of mass m, resonance frequency v0 and transition dipole moment m in the presence of a radiation field. The laser field with fluctuation in amplitude and phase is given by E~ r,t ! 5 FIG. 1. ~a! Geometry of the wall aligned trapping laser beams in the x-y plane, ~b! almost spherical cloud of cold atoms, ~c! misalignment of the trapping beams in the x-y plane, and ~d! corresponding ring-shaped distribution of cold atoms. ~b! and ~d! are not in the same scale and the structure seen is due to the TV monitor pixels.

coordinate-dependent vortex force @2#. When Gaussian laser beams are used in the configuration of Fig. 1~c!, the total force acting on the atom located at the x-y plane is given by F52Kr2 a r˙1 z ~ x,y !~ yex 2xey ! ,

~1!

where K is the trap spring constant, a is the damping constant, and z (x,y)(yex 2xey ! is the vortex force arising from the misalignment of the Gaussian beams forming the race track around B50. In Eq. ~1!, ex and ey are unit vectors along the directions indicated by the index. A full deduction of Eq. ~1! is provided in Ref. @2#. An atom under the action of the force in Eq. ~1! is accelerated by the vortex force until its velocity becomes large enough that this force is balanced by the damping. This situation may produce a stable circular trajectory for the atoms. The stability of the orbits has already been analyzed in Ref. @2#. Investigations on the dependence of the ring radius with the parameters involved in the experiment ~misalignment of the beams, magnetic-field gradient, laser detuning, and intensity! have been carried out, showing good agreement with the model based on the vortex force @3#. However, we could not explain the width of the ring at the time of that earlier work. The explanation for this is the aim of this paper. Many causes can be suggested as being responsible for the finite width of the ring-shaped atomic distribution. Here we will examine two of them, namely, fluctuation of the laser field ~both amplitude and phase! and the spontaneous emission of the atoms. III. EQUATION OF THE ATOMIC MOTION AND NUMERICAL SIMULATION

In this section we will present the equations used to describe the motion of one atom in a MOT with the influence of laser field fluctuations and spontaneous emission of photons by trapped atoms. No approximation was used to de-

\ @ V ~ r! 1F ~ t !# cos@ u ~ r! 1 v t1 f ~ t !# e, m

~2!

where V~r! is the Rabi frequency, e the polarization state, and F(t) and f (t) are, respectively, the fluctuations in the intensity and phase originated from stochastic processes. If no fluctuation exists we have the condition F(t)5 f (t)50. Using an approach introduced by Cook @6# to determine the radiation force on a two-level atom, we have mr¨5

\ @ U¹V1V ~ V1F ! ¹u # , 2

~3!

where r is the atomic position vector, and U and V are components of the Bloch vector (U,V,W) determined by the set of optical Bloch equations. In the rotating-wave approximation these components must satisfy the following system of differential equations ˙ 5 ~ D1 u˙ 1 ˙f ! V5 1 AU, U 2 ˙V 52 ~ D1 u˙ 1 ˙f ! U1 ~ V1F ! W2 1 AV, 2

~4!

˙ 52 ~ VF ! V2A ~ W11 ! , W where A is the spontaneous emission rate, D5v2v0 the laser detuning from the atomic resonance frequency, and (d u /dt)5(dr/dt)¹ u ~r!. From the theory of multiplicative stochastic processes @9,10# the equation of motion for the ensembled-averaged Bloch vector is m ^ r¨& 5

\ @ ^ U & ¹V1 ^ V & ~ V1F ! ¹ u # , 2

~5!

where the first term inside brackets accounts for the dipole force while the second one gives the spontaneous force. The ensemble-averaged system of equations is then obtained as @8#

S

D

A 1G 1 ^ U & , 2 A ^ V˙ & 52 ~ D1 u˙ ! ^ U & 1V ^ W & 2 1G 1 1G 2 ^ V & , 2 ˙ W 5V V 2 A1G W 2A, ^ & ^ & ~ 2 !^ &

^ U˙ & 5 ^ D1 u˙ &^ V & 2

S

D

~6!

Here the parameters 2G1 and 2G2 are defined as the spectral densities of the stochastic processes associated, respectively, to the phase and amplitude of the electric field in Eq. ~1! and

INFLUENCE OF LASER FLUCTUATIONS AND . . .

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FIG. 2. Simulated trajectories with ~a! amplitude and ~b! phase fluctuations for one atom starting from the origin with v 0 515 m/s. Dependence of the trajectory radius on the time when ~c! amplitude and ~d! phase fluctuations are present.

their definitions are @9# ^ f (t) f (t 8 ) & 52G 1 d (t2t 8 ) and ^ F(t)F(t 8 ) & 52G 2 d (t2t 8 ). In the steady state, the equation of motion ~4! is easily solved. Neglecting the dipole force and taking f (r)52k•r, the solution is given by AV 2 \k mr¨5 , 4 ~ e 1 / e 2 !~ D2k•v! 2 1 e 2 e 3 A 2 12V 2

~7!

where

e 1 511

2G 1 , A

e 2 511

2G 2 , A

e 3 511

2G 1 2G 2 1 . A A

~8!

As can be easily seen, if no fluctuation occurs in the laser field we have e15e25e351 and the conventional radiation force is obtained. In our simulations Eq. ~6! was split in two equations, for the x and y directions, and numerically solved. Here we have considered a two-dimensional motion to simplify the numerical analysis. The addition of the z component does not affect the results qualitatively; it just gives a width to the ring in that direction when spontaneous emission is considered. This effect occurs as a consequence of the recoil in the atomic motion produced by the photon emission. In order to include it in our simulations we adopt the following procedure: we add at equal intervals of time Dt a variation Dv ~with a constant module and a randomized orientation in the x-y plane! to the velocity v of the atom in movement in the MOT. The numerical results obtained with the assumptions presented in this section are discussed below. Our numerical simulations were performed on a Pentium microcomputer using NAG subroutines to solve the equations of atomic motion along the x and y directions as well as to generate random numbers to provide the direction of the recoil. The simulations were carried out for misalignments in the x-y plane only and typical experimental values were assumed for

the parameters used in the calculations. These parameters are s51 mm, V520 MHz, D5210 MHz, and G510 MHz.

IV. THE EFFECT OF THE LASER FIELD FLUCTUATION

Figures 2~a! and 2~b! show the trajectories of one atom in a MOT calculated with fluctuations in the laser intensity and phase, respectively. In both cases it is observed that the radius of the orbit tends to a constant mean value around which occurs a small periodical variation associated to the nonperfectly circular movement of the atom in the trap as a result of the existence of only two pairs of counterpropagating beams along the orthogonal x and y directions. This is shown in Figs. 2~c! and 2~d! where the time dependence of the trajectory radius is presented. It is also observed that the intensity and phase fluctuations cause opposite effects on the atomic trajectory since increasing fluctuations in the laser intensity reduces the mean radius of the trajectory while the same kind of behavior as the phase fluctuations increases the mean radius. One fundamental result is that only one stable orbit is asymptotically achieved by the atomic motion and no width comes from considering the laser field fluctuations. Our calculations have shown that this fact occurs independently of the initial conditions used to start the atomic movement. Experimentally we have G 1 /A'0, due to the characteristics of our laser system. In this situation the trajectory is expected to be stable and cannot account for the observed width. The fluctuations on the phase will introduce instability in the orbit at the level of G 1 A'1. In fact, we have observed that when the laser is not stable, the ring atomic distribution cannot sustain its form. Fluctuations on the phase present orbits with a higher average radius. On the other hand, amplitude fluctuations decrease the averaged radius and if G 2 /A'1 the ring may collapse.

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DIAS NUNES, SILVA, ZILIO, AND BAGNATO

FIG. 3. ~a! Simulated atomic trajectory with the occurrence of spontaneous emission and ~b! time evolution of the trajectory radius.

V. THE EFFECT OF THE SPONTANEOUS EMISSION RECOIL

Figure 3~a! presents the trajectory for one atom in the MOT when spontaneous emission is considered. The average radius as a function of time is shown in Fig. 3~b!. It is well known that each photon spontaneously emitted is able to transfer about 3 cm/s to the sodium atom in an average period of time 32 ns @3#. Here we assume that the atom spends the same amount of time in the excited and ground states. The variation of the atomic velocity, following the transference of momentum from a spontaneous emission, must occur in a spatial random way. Taking this into account, a subroutine was called at periods of time of 32 ns to generate a random number between 0 and 1 in our simulations. This number times 2p gives a random value for the angle between the instantaneous velocity v of the atom and the variation Dv caused by the photon spontaneously emitted. In our simulations only two-dimensional spontaneous emission is considered. With these values of D v x and D v y , the trajectory calculations were restarted from the point where they had been stopped for the addition of the recoil. A small time increment was used in our calculations, namely, 1 ns, for a total period of integration of 0.5 ms, corresponding to the average period of time during which one atom is trapped in a MOT. As can be observed in Fig. 3, the trajectory does not tend to a stable orbit but rather to a random-walk orbit as a consequence of the spread of the atomic momentum caused by the photon spontaneous emission. It is interesting to note that the small value of momentum transferred by the emitted photon is able to affect the atomic motion with tangential velocities of the order of hundreds of centimeters per second. It is also interesting to point out that with spontaneous emission the atomic

@1# I. Guedes, M. T. de Araujo, D. M. B. P. Milori, G. I. Surdutovich, V. S. Bagnato, and S. C. Zilio, J. Opt. Soc. Am. B 11, 1935 ~1994!. @2# V. S. Bagnato, L. G. Marcassa, M. Oria, G. I. Surdutovich, R. Vitlina, and S. C. Zilio, Phys. Rev. A 48, 3771 ~1993!. @3# M. T. Araujo, R. J. Horowicz, D. Milori, A. Tuboy, R. Kaiser, S. C. Zilio, and V. S. Bagnato, Opt. Commun. 15, 85 ~1995!. @4# C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571 ~1990!.

motion in the MOT is also independent of the initial conditions used to start the integration of the motion equation. This is an obvious consequence of the random character of the recoil caused by the spontaneous emission. The results observed in Fig. 3 refer to only one atom. However, if we consider that a large number of atoms are trapped, having started their movements in the MOT at different instants of time, we can easily understand that the cloud of atoms observed in the ring-shaped distribution is the image of the randomly positioned atoms in the MOT and the net result is the finite width of the ring. VI. CONCLUSIONS

In this work we have performed a numerical simulation for the motion of an atom in a ring-shaped trap, including fluctuations in the laser field and spontaneous emission. Although we have found that fluctuations produce considerable variation in the dynamics of atomic motion, modifying the stationary radius, spontaneous emission is the effect responsible for the existence of a finite width in the ring-shaped trap. Fluctuations in the laser intensity and phase just cause a variation in the mean radius of the nearly circular atomic trajectory. ACKNOWLEDGMENTS

The authors acknowledge the financial support from ˜o de Amparo a` Pesquisa do Estado de Sa ˜o FAPESP ~Fundaça Paulo! and CnPq/RHAE ~Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico!. We are grateful to Humberto F. da Silva for helpful suggestions in routine manipulations.

@5# V. Bagnato, G. Lafyatis, W. Phillips, and D. Wineland, Rev. Phys. Appl. Instrum. 8, 24 ~1993!. @6# M. Sharma and S. Prasad, Phys. Rev. A 31, 3988 ~1985!. @7# R. J. Cook, Phys. Rev. A 21, 268 ~1980!. @8# R. J. Napolitano and V. S. Bagnato, J. Mod. Phys. 40, 329 ~1993!. @9# K. Wodkiewicz, Phys. Rev. A 19, 1686 ~1979!. @10# R. F. Fox, J. Math. Phys. 13, 1196 ~1972!.