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1 May 1997

OPTICS COMMUNICATIONS

ELSEVIER Optics Communications 137 (1997) 269-275

Atomic juggling using feedback

K.S. Wong ‘, M.J. Collett ‘, D.F. Walls 3

Department of Physics, Unit,ersiq of Auckland, Priratr Bag 92019. Auckland. News Zealand

Received 6 August 1996; revised 4 November 1996: accepted 6 November 1996

Abstract

The position of an atom inside a cavity can be estimated by measuring the phase shift of the homodyne current leaking

out from the cavity. By continuously monitoring the position of the atom inside the cavity, the phase of the standing wave may be adjusted so that the dipole force always keeps the atom from falling out of the cavity due to the gravitational force.

We have called this technique atomic juggling.

1. Introduction

Advances in the laser cooling of atoms and in the construction of microcavities have lead to significant de- velopments in the field of cavity quantum electrodynamics [l-6]. One of the goals of cavity quantum electrodynamics is to study the interaction of a single atom with the cavity field. Current experiments which pass a weak beam of

atoms through the cavity suffer from the problem of atomic number fluctuations [5]. An ideal situation would

be one where one atom was trapped in the cavity. This may be possible with cold atoms in a standing wave field which provides an upward dipole force which opposes the gravitational force. This scheme requires that the atom to be kept away from a node of the standing wave where the intensity of the field is zero. This may be accomplished if the position of the atom is monitored. This information may then be used to adjust the phase of the standing wave so that the atom is always kept away from the nodes of the standing wave field.

The position of an atom in a standing wave field may be determined by measuring the position dependent phase shift imparted to the field [7,8]. Continuous measurements of the quadrature phase of the output light from the cavity

will monitor the transverse motion of the atom [9]. We propose to use this measurement scheme to estimate the position of the atom then feed back this information to change the phase of the cavity field so that the atom is always kept away from a node (Fig. 1). This manipulation of the atom is in a sense similar to juggling and hence we have called it utomic juggling.

In this paper, we consider three possible schemes to feed the atomic position information back into the system.

These are (i) the modulating scheme, which gives a contin- uous estimate of the atomic position, (ii) the binary scheme, which only estimates if the atom is near or far from the

node. and (iii) a combined scheme.

2. Localization scheme

We consider a two-level atom inside a one-dimensional cavity as indicated in Fig. 1. The Hamiltonian for the combined atom-field system is

H = Ku”, + Hneid + HI,, 3 (1)

where

2

H Clf”Ill = 2 + h wouz, Hfie,‘, = h o,a+a,

’ E-mail: ksw@phy.auckland.ac.na.

’ E-mail: mjc@phy.auckland.ac.nz.

A E-mail: dfw@phy.auckland.ac.nz.

Hi,,= fhO(a~~~+acr+)cosk(x-x,).

Here n and ut are the annihilation and creation opera- tors for the cavity field, and u:, u, and at are the

0030.4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved.

P/l SOO30-4018(96)007 18-3

270 K.S. Wong et al. /Optics Communications 137 (19971 269-275

Homodyne

Detector

Fig. I, Schematic diagram of the feedback scheme.

internal atomic operators. o, and k are the frequency and wave number of the cavity mode. oa is the atomic transi- tion frequency, which is detuned from the cavity frequency

by an amount A = q, - coo. f2 is the single-photon Rabi frequency. x0 is the position of the antinode of the stand- ing cavity field. We consider here only the motion along the x-direction (cavity axis); in the vertical (z) direction, it is assumed that the gravitational force is balanced by the dipole force, provided that the atom is not too close to a node 4 (where the dipole force is zero).

For the case in which the atomic transition frequency is

highly detuned from the cavity frequency, the probability that the field will induce an atomic transition between the ground and the excited states is small. Changing to the interaction picture, the effective Hamiltonian in the regime

of large detuning, obtained by adiabatically eliminating the excited state is

2

P.Z z-

2m - ;fi A - $ Ka+u cos*k( x - x0).

Here K = 0*/A and we assume that K( ata))/A +c 1. In order to monitor the phase of the field, we have to

allow it to leak out of the cavity at a rate y,,,. Hence it is necessary to have a driving input field in order to maintain

4 It can be shown that if we generalize our potential to include

the :-dependence, i.e. H,,, = fRfi(a’g + acr+)cosk(x -

x,)exp(- ;‘/~a). with the waist ~1~ = 50 km, and use the

parameters that we are going to use in our simulations, then the upward dipole force ( = 2 X lo-” N) is about 100 times greater

than the gravitational force (= 2X lo-l4 N) at the position

(Z=M’“,X=X”).

a constant intensity of light inside the cavity. The driving

field can be modelled by [ 101

t&(t) = 96 + u(t).

Here (Y is a real coherent amplitude, and v(r) is a white

noise operator of zero mean obeying

[ u(t),u~(t~)] = S(t- t') = (u(t)u’(t’)>.

The white noise operator v can be interpreted as vacuum

fluctuations, and can be decomposed into two independent but non-commuting quadratures

5(r) = u(t) + u+(t), l(t) = -i[ u(t) - v+(d)

where t(t) and l(t) satisfy

(5(1)5(t’)) = <L(r)s(r’)> = S(t-r’). (5)

The Hamiltonian including the coupling of the cavity

mode to the external field is

7

H=~-fhd-th~n’n~~~‘k(x-x,)

- in&r_ [ bi”( t)a’ + dq”( t)] . (6)

The Heisenberg equation of motion for the field operator a is

b = ;KlZCOS’k(X-X,) - &b,,(f) - +a. (7)

Assuming y,,, to be large compared to the time scale of the motion of the atom so that we can set the time derivative to zero, the stationary solution for a is

a= 1 - iKCOS’h_( X -X,)/2&

(8)

K.S. Wong et al. /Optics Communications 137 f IY97J 269-275 271

Hence ut, can be approximated by

a+a z ff2 - 2+)&z

1 + [ KCOS’k( X - X0)/2Y,,,] *

if K +=sc 2y,.,. Substituting this into the effective Hamilto- man equation (Eq. (2)). we have

H P.Z

eff = G -;hA-pi K( a2-F)

Xco?k(x-x0). (10) The output field of the damped cavity is given by

bO~i(t) = hin(t) + aa

-cos’k(x -X0)

(11) where Eqs. (3) and (8) have been used.

The position of the atom inside the cavity can be estimated by measuring the quadrature phase of the output field, which is proportional to the scaled homodyne pho- tocurrent

L,,(r) = \/r,, [bout(t) + b;,,(r)]

= -KNOS’k(X-X,) - &t(f). (12)

Consider the case in which we measure the homodyne photocurrent continuously from t = 0 to t = T. The aver-

age photocurrent in the time interval is

(13)

If we assume that in the time interval 7, the displacement

of the atom is small compared to the wavelength of the standing wave, and r is long enough so that the average of l(t) is negligible. then

+,,,,(t)dt = - KWOS’k( X -X,,)_ (‘4) 7 0

Now because cos’k(.u - x0) is an even function, it only gives us information about the absolute displacement of the atom away from the antinode. It does not however tell us which side of the antinode the atom is on. Therefore if we try to shift the phase of the standing wave, there is a 50% chance that the atom would be closer to (rather than further away from) the node. Hence our aim of keeping the atom away from the node cannot be achieved in this fashion.

One solution to this problem is to introduce a modula- tion on the signal. By demodulating the output signal, we

can not only deduce the absolute displacement of the atom

from the antinode, but also deduce which side of the antinode the atom is on. The other possible scheme uses

the fact that the dipole force of the standing light field on the atom is typically much greater that the gravitational

force. It turns out that the absolute displacement of the atom from the antinode is all we need to know in order to keep the atom away from the node. A discussion on a

scheme which combines these two schemes will also be included for completeness.

3. Juggling schemes

3. I. Modulating scheme

Consider the position of the antinode of the standing light field to be modulated by an amount ~coswt. Physi- cally we can think of the mirrors of the cavity being

moved backwards and forwards with amplitude E and angular frequency o. The position of the atom can be estimated by the following iterative process.

First of all, we start with a guess of the atomic position,

R, and shift the antinode of the standing wave to R. The demodulated homodyne photocurrent is given by

I”,,,(t) = ;),&)coswtdt

1 T =-

/[ 7 0

- K(YCOS’k( X - 2 - ECOSOt)

-&W,] cosotdt (15)

= -$Kaeksin2k(s-P), (16)

where we assume that kecoswt -=z 1 and r is long enough so that the average of t(t) is negligible.

Hence the estimate of the atomic position is given by

I x = -sin-’

2k (17)

We can use this estimate as our next guess of the atomic position, 2, and shift the antinode to P. It should be noted that, unlike Eq. (14). the sine term in Eq. (16) is an odd function. This implies that we are not only able to estimate

the absolute displacement of the atom from the antinode, but also know which side of the antinode the atom is on.

From Eq. (I 5), we can see that the noise term t(t) limits our ability to estimate the atomic position. The squared signal-to-noise ratio (SNR) in this case is propor- tional to the coupling constant K’, and therefore higher SNR can be obtained in principle by increasing the single- photon Rabi frequency 0. However J2 cannot be too high for two reasons. Firstly, we do not want R to be much greater than an experimentally feasible value. Secondly, if R is too high, then the atomic excitation cannot be

272 K.S. Wang rr rd. / 0ptic.s Commcnicntinns 17 I I997) 269-275

neglected even for large detuning. The atomic excitation not only gives random momentum kicks to the atom due to spontaneous emissions, but also raises another problem by

introducing two potentials to the system: the ground state atom sees one standing wave potential while the excited state atom sees the opposite one. Therefore we would like

to avoid atomic excitation, but at the same time have a reasonable SNR in the atomic position estimation.

To demonstrate how the modulating scheme works, we simulate the time evolution of an atom inside a cavity. The time evolution is determined using the split operator method. We choose the following parameters in our simu-

lations: 0 = 700 MHz, A = 7000 MHz, (Y = 3.5, A =

2a/k = 8.52 nm, yCdy = 90 MHz, IPI = 2.2 X IO-” kg and

the feedback time scale r = 45 ns. Fig. 2 shows a simula-

I I I I I 1 1 -20000 -1 0 1

0

-5000

-10000

-15000

I I I I 1 ' 1 -20000

-1 0 1

Fig. 2. A simulation of the time evolution of an atom inside a

cavity without feedback. The horizontal axis is the x-axis (in

microns). The left-hand vertical axis is the square of the probabil-

ity amplitude. and the right-hand vertical axis is the standing wave potential (in the unit of the recoil energy). The atom (solid line)

continuously moves to the right of the potential (dotted line) until

it reached the node (hence would fall out of the cavity).

‘li-::i:I:: -1 0 1

0.04, I

0

-5000

10000

-15000

-20000

0

-5000

-10000

-15000

-20000

0

-5000

-i 0000

-15000

-20000

I I I I I 1 1 -20000 -1 0 1

Fig. 3. A simulation of the time evolution of an atom inside a

cavity with feedback using the modulating scheme. The position

of the atom is continuously monitored and the phase of the

standing wave is shifted such that the atom is alway\ near the

antinode. i.e., the minimum of the potential for a red detuned light

field. Therefore the atom will remain in the cavity for a long time.

tion of the time evolution of an atom in the case of no feedback. The scalar atom (solid line) in the simulation is represented by a Gaussian wave packet and started initially at the bottom of the standing wave potential (dotted line). The atom has a non-zero initial velocity and travels to the right of the potential. The total time of the simulation is 0.4 ps. We can see that the atom will eventually reach the node and fall out of the cavity in about 0.4 ~_LLS in the case of no feedback. The dispersion of the wave packet shown in the figure is expected but is enhanced by the nonlinear- ity of the standing wave potential, Eq. (IO). This disper- sion effect is unwanted because it reduces the efficiency of the juggling process and we will see below how the juggling scheme itself can minimize it.

What happens if we introduce feedback to the system’? Fig. 3 shows a simulation of the time evolution of an atom

K.S. Wang et al./ Optics Conununicntions 137 (1997) 269-275 273

in the case of feedback. The total time of the simulation is 3.6 us. The atomic position is estimated every 4.5 ns in the

simulation and fed back to the system. We can see that

although the atom is not always at the antinode, because the estimations are not perfect due to the existence of noise, the feedback does ensure that the atom is always kept away from the node. Fig. 3 also shows another interesting effect of this juggling scheme - the cooling

effect. We can see in the figure that the atom slows down continuously. The origin of this cooling effect is due to the

fact that the atom spends most of its time climbing up the potential (between the shifts of the phase of the potential),

and hence it loses energy and decelerates continuously. This cooling effect is an advantage of the juggling scheme because the slower the atom moves, the easier the atomic

position estimation process. We also notice. from Fig. 3, that the dispersion of the

wave packet is less severe than the previous simulation (despite the fact that the simulation time is 9 times longer than the previous simulation), and the wave packet stops spreading later in the simulation. There are three effects contributing to this observation. First of all, as we men- tioned before, the dispersion is enhanced by the nonlinear- ity of the standing wave potential. In this juggling scheme,

the atom is always kept near the antinode of standing wave which implies that the effective potential seen by the atom is always approximately harmonic. As a result the spread- ing due to the nonlinearity is minimized. Secondly, the homodyne measurement itself provides us the information

about the atomic position and hence helps keeping the wave packet from continuous spreading. The third explana- tion is the cooling effect that we have just discussed. The atom loses its energy by this cooling effect and hence eventually ends up in the ground state of the potential well with certain width.

Another possible way of juggling the atom is the binary scheme. This scheme is simpler than the modulating

;w; 1 ‘..,’ \_

Fig. 4. Schematic diagram of the binary scheme.

0

-5coo

-10000

-15000

-20000

Fig. 5. A simulation of the time evolution of an atom inside a

cavity with feedback using the binary scheme.

scheme in the sense that we do not need the modulation for our localization scheme. As mentioned before, the

dipole force on the atom due to the standing wave is typically much greater than the gravitational force. In another words, the atom will not fall out of the cavity unless it is very close to the node at which the intensity of the light field is zero. Hence it is only necessary to know the absolute displacement of the atom from the antinode. If the atom is estimated to be close to the antinode, then the phase of the potential does not need to be adjusted. On the other hand, if the atom is estimated to be far away from the antinode, then what we need to do is to flip the standing wave potential (by shifting the phase of the standing wave by r/2). So this is essentially a binary system. In practice, one can take the mid-point between the maximum and the minimum of the potential as the reference point, ~,,r (Fig. 4). If the atom is determined to be in the interval [ xref, -.q,,], this means the atom is

reasonably close to the antinode, then the phase of the potential does not need to be adjusted. On the other hand,

274 KS. Wang et al./Optics Communications 137 (IYY7I 269-275

if the atom is determined to be outside the interval [ x,,~, -_+I, then we can flip the potential so that the atom is reasonably close to the antinode again.

The binary scheme is demonstrated in Fig. 5. The total time of the simulation is 3.6 ps, and the same parameters as for the modulating scheme are used in the simulation.

The situation is similar to the modulating scheme except we now only have two choices: either flip or do not flip the potential. (It should be noticed that there are some flips

of the potential between figures that are not shown.) Again we can see that the dispersion is less severe compared to

the case of no feedback due to the fact that the cooling process also happens in the binary scheme. However the dispersion is a little bit faster than the modulating scheme because the atom spends some of its time in the nonlinear region of the potential. The simulation shows that the atom is kept safely away from the nodes of the standing wave

field.

3.3. Combined scheme

The third possible scheme combines the modulating

and binary schemes. As mentioned before, the SNR (and hence 0) in the modulating scheme must be reasonably

high in order to have accurate estimation of the atomic position. However, if we are only interested in the siglt of the estimate, then the SNR can be reduced. In another words, we want to extract the information about which side of the antinode the atom is on using the modulating scheme. The absolute magnitude of the displacement from the antinode can then be estimated using the binary scheme. Thus we can use lower 0 for this combined scheme

compared to the previous two. The basic principle of the combined scheme is outlined as follows.

From Eq. (16), if rho,,,(t) > 0, then sin2k(x - 2) < 0 and hence the atom is on the left-hand side of the antinode. Similarly &,,(r) < 0 implies the atom is on the right-hand

side of the antinode. For the estimation of the absolute displacement, from

Eq. (lb),

i,,,(t) = T/:;[l +cos2x-(X-x,- ECOS or)] dr

1 + c0s2k(x-x0)c0s(2k~c0sot)

+sin2k( x-x,)sin(2lrecoswt)]df. (18)

In the case of integrating over one cycle of modulation,

Ln<~> = +E[ 1 + cos2k( X - xo)Jo(2ke)], (19)

where the integral representation of the Bessel function,

7~.f& z> = /lcos( xos 0 ) d 0, has been used. Hence the ab- solute displacement is given by

1 1 ~x--‘iol= zkcos-’ ~

[ ( 2&0,(t)

J,,W~)

-------1 KCY II

We can then feed the information back to the system

and adjust the phase of the cavity field as before.

4. Conclusion

In this paper we have demonstrated the possibility of trapping a cold atom inside an optical cavity using the dipole force to combat gravity. The position of the atom is determined by continuously measuring the phase of the output cavity field. This information is fed back to adjust

the phase of the cavity field so that the atom is kept away from a node, so called atomic juggling. We have discussed three possible juggling schemes: (i) a modulating scheme, (ii) a binary scheme and (iii) a combined scheme. The binary scheme is simpler than the modulating scheme

since no modulation of the photocurrent is required. The combined scheme has the advantage of requiring lower

single-photon Rabi frequencies.

Acknowledgements

K.S. Wong would like to thank Scott Parkins for valuable discussions. This research has been supported by the Marsden Fund of the Royal Society of New Zealand, the University of Auckland Research Committee and the

New Zealand Lotteries’ Grants Board.

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