Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. B 1743

Dynamics of trapped particle cooling in the Lamb-Dickelimit

Stig Stenholm

Research Institute for Theoretical Physics, University of Helsinki, Helsinki, Finland

Received April 8, 1985; accepted July 2, 1985

This paper reviews our current understanding of the dynamics of a laser-cooled trapped particle in the Lamb-Dickeregime, where the quantum structure of the energy levels cannot be ignored. The derivation and validity of amaster equation are surveyed, and its physical interpretation is discussed in some detail. The structure andphysical nature of the ultimate steady state are discussed. Using a generating function method, we can solve forboth the complete eigenvalue spectrum and the general time-dependent solution of the master equation. Theseresults are derived and interpreted physically. They have earlier been scattered in our various publications and arepresented here in a coherent way for the first time. Also included are some new results and a physical discussion ofthe situation. The paper concludes with a discussion of the validity and limitations of the model as treated so far.

1. INTRODUCTION

With the appearances2 of the experimental reports on lasercooling of trapped ions the importance of the new fieldimmediately became obvious. Further work (see, e.g., Refs.3-5) has proved that the situation contains a lot of interest-ing physics.

A simple harmonic oscillator model for trapped particleswas described in Ref. 6, where its limit for slow motion wastreated. Then the behavior is essentially like that of a freeparticle with the instantaneous motion given by the oscilla-tions in the trap. The same limit was treated by Gordon andAshkin.7 The harmonic trapping potential is assumed to fitthe radio-frequency trap but not the Penning trap.4 In theformer case the particle wave function becomes confined towithin less than a wavelength of the optical cooling radia-tion.2 In this Lamb-Dicke limit the quantum effects of themotion can no longer be neglected, and the velocity cannotbe entered as a mere parameter. The standard methods (seeRefs. 8 and 9) cannot be used in this case. In Refs. 10 and 11we developed the theory of cooling in this limit using theoccupation number state representation of the harmonicmotion. Our approach was, however, an incoherent one,and the rates obtained did not include saturation correctly.Javanainen reformulated the problem using the Wignerfunction,12 "13 but only recently has this formulation'4 lead tothe correct answers. The correct expression for the finalenergy was first derived by Lindberg,1 5 and the correct dy-namic time evolution was derived in Ref. 16. A comparisonamong the various results is given in Ref. 17. From differentpoints of view the unsaturated perturbation results havebeen derived by Neuhauser et al.2 and Winelend andItano.' 8 "19 As far as I know there exist no other detailedcalculations of the trapped-ion cooling situation.

In this paper I give a unified treatment of the whole dy-namic situation in the Lamb-Dicke limit. For this purposeI collect and review results scattered in our earlier publica-tions, extend and unify the treatment by new material, andelucidate the situation by adding physical remarks and com-ments. In this way I hope to be able to show how completely

the dynamic situation is under control in the Lamb-Dickelimit. At this time the experiments do not verify the detailsof the theory, but it is of interest to see how, at least in asimple model, the various theoretical aspects manifestthemselves in physical processes.

In Section 2 I introduce the master equation without anydetailed derivation. This is found in Ref. 16. The equationgoverns the time evolution after an initial transient time ofthe order of the lifetime of the upper level of the cooledsystem. It describes both the dynamics and the finallyemerging steady state. This final state is discussed in Sec-tion 3. Section 4 introduces a useful technical tool, thegenerating function, which is used to obtain both the eigen-value spectrum (Section 5) and the exact time-dependentsolution of the problem (Section 6). This is evaluated ex-plicitly for some particular initial distributions in Section 7.Finally in Section 8 I comment on the model and discuss itslimitations and unsolved problems.

2. THE MASTER EQUATION

The recoil frequency in the spontaneous-emission process ishq2/2M, when hq is the photon momentum, and our smallexpansion parameter is the ratio of this to the oscillationalfrequency v in the trap

2 = hq 2Mv = 2 (1)

where ao is the size of the wave functions for the harmonicmotion. When -q is small the particle becomes confinedwithin an optical wavelength toward the end of the coolingprocess, and the Lamb-Dicke limit prevails.

The upper level decays spontaneously to the lower one atthe rate r, and the induced coherences decay at the rate

1 > Ir.2

(2)

The equality sign is valid if no dephasing processes inter-vene. The two states are coupled by the dipole matrixelement Au, and this causes a Rabi frequency of magnitude

0740-3224/85/111743-08$02.00 © 1985 Optical Society of America

I.a

tM?n

7q

e

Stig Stenholm

1744 J. Opt. Soc. Am. B/Vol. 2, No. 11/November 1985

(,AE/h) in the laser field E. The golden rule of perturbationtheory gives the transition rate at detuning A as

P(A) 27rG(E/2h)26(A)

',2A r A

2 + (3)

where we have introduced the dimensionless power parame-ter

I = ___

ar

n-i n

(a)

(4)

The internal variables of the atomic system rearrangethemselves with the rates F and P(A), which for optimalconditions are of the same order of magnitude. Smallervalues of P lead to slower cooling, whereas values of P >> F donot considerably speed up the process because the spontane-ous emission is the bottleneck. 20 The increased number ofinduced processes, however, shows up as an increase in theattainable final temperature. Under these conditions it ispossible to eliminate adiabatically the internal variables,e.g., the Bloch vector components, and one obtains an equa-

tion for the slow time evolution induced by photon recoil.'6

Its rate is of order q2 F, and the evolution is governed by amaster equation. This describes the behavior of the re-duced density matrix (nlpln'), where n is the label for theharmonic oscillator states in the particle trap.

During time intervals of the order of I-' all correlationsdisappear between states of the type (ni pin + k) with differ-ent k's. The slow time evolution carries out a reorganizationof the different values of n to provide the eigenfunctions ofthe master equation.6 "l7 The general master equation isfound to be

d (nlpln + k) = ikv (nlpln + k)

+ 772 U[(n + 1)(n + 1 + k)]12 A-(n + 11pmn + 1 + k)-(n + 1 + ±k)A+(nlpln + k)

-(n + Ik)A (nlpIn + k)+ [n(n + k)]" 2A+(n- 11pn - 1 + k)j. (5)

The rate coefficients A± are independent of n and k and aregiven by' 6

A_(v) = A+(-v) = I + 2P(A) (a - 1)

ImI r + 2P(A) Im{ 1 + 2yrl

+A2 - (V + i-Y)2

x Frf 1 1 I 2P(A) ( + ir) (V + i-Y)lA [ i Y A -V-iD V A2-(V+i)2

(6)

The quantity a describes the geometry of the spontaneous-emission events.2 1-23 For dipole transitions a = 2/5, butassuming a hypothetical spherically symmetric emission wehave a = 1/3.

To interpret the coefficients in Eq. (5) we specify to the

case of k = 0, which describes the rate of transfer between

r P(A-v) P(Ar

n -1 n n+1

(b)

Fig. 1. To lowest order in the laser intensity we have only two typesof processes transferring population from the level n to levels n + 1.(a) This is a direct induced process followed by spontaneous decay-ing. The process acts as a diffusive spreading. (b) This processcontains an induced process utilizing the vibrational quanta fromthe trap motion followed by spontaneous emission. Depending onthe detuning A this is either a heating (n - n + 1) or a cooling (n - n-1) process.

the oscillator levels n - 1, n, and n - 1. Then the coefficientA+ describes the rate upward and A_ the rate downward,and, evaluating Eq. (6) to lowest order in the laser intensityI, we obtain

A,(v) = aP(A) + P(A F v). (7)

This is easily interpreted in terms of the two elementaryprocesses shown in Fig. 1. In Fig. 1(a) the system is excitedfrom oscillator level n to the upper level at the rate P(A) anddecays spontaneously with the fractional rates a F to levels nd 1. This contributes both to up and down transitionsequally; it acts like a diffusive heating process. In the pro-cesses in Fig. 1(b) the induced excitations go to n + 1 or n - 1at rates P(A + v) given by the resonance condition. This isfollowed by a spontaneous decay at rate F without a changeof n. The processes neglected are either saturation correc-tion to this [see Eq. (6)] or of higher order in the parameter n(see Ref. 17). The rates (7) agree with those derived byNeuhauser et al.2

If we tune to A = v we favor the downward processes and

A_ > A, (8)

which implies cooling. For simplicity we assume the case ofwell-resolved resonances

Y << V = Al

and introduce the detuned intensity I through

17 = P(v) _ (y/i,) 2 1Ir 4rP(A + v).

From Eq. (7) we then obtain

A+ = 1IF + alP,4

- 1=7P ± a7P.

(9)

(10)

(11)

The cooling rate becomes

n+l

Stig Stenholm

Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. B 1745

W = q2 (A- - A+)

= 72r + o(I),and the ratio of the rates is given by

s-A+ = kvIjA_ v (I+a)+...<<1.

3. THE STEADY-STATE DISTRIBUTION

For the diagonal elements

P(n) = (nJpln)

(12)

Using the perturbation theory results, Eq. (7), we obtainthe expression

Efin = hp~i _ v)-P + v)+a() + 'II ~(A p)-P(A ) 2

(13) which was first given by Neuhauser et al.2

In the limit of broadly overlapping resonances

'y >> v,

we can approximate

(14)

the master equation (5) reduces to a rate equation for thepopulation probabilities. It is of the form

P(n) = n2m(n + 1)A-P(n + 1)

- [(n + )A + nA-]P(n) + nA+P(n - 1)1. (15)This is similar to the Scully-Lamb equation2 4 for a laserwithout saturation. Because we have cooling, [see expres-sion (13)], our case corresponds to the laser below threshold.Equation (15) was derived for the Lamb-Dicke limit first inRef. 11 but not with the complete coefficients A+ as given byEq. (6).

The various terms in Eq. (15) can be interpreted as therates indicated in Fig. 2. In steady state there can be no flowof probability between the various oscillator levels. Hencethe rates up must equal the rates down, which gives thedetailed balance condition

AP(n + 1) = A+P(n).

This has the normalized solution

P,(n) = (1 -)Sn

(16)

(17)

where s is defined as in expression (13). This is an accept-able solution of the Planck blackbody type.

The steady-state distribution Eq. (17) gives the final ener-gy

Efi = hv(n + ))= hv P(n)(n + )

P(A cP(A + ) F rI 22 Y

P(A -0 -P(A + ) = 4rI 72A '(12 + A2)2

which, when introduced into Eq. (20), gives the energy

Efi = h(l + a)(+ );

here the zero-point energy can be neglected. This resultagrees with the one for a free particle. 79 2 2 For a trappedparticle it was derived in Ref. 6 for the case a = 1/3. Theminimum is found for the detuning A = y, which optimizesthe overlap with the cooling part of the oscillator spectrumwithout detuning too much. We then obtain the energy

This result is easy to interpret physically. The cooling pro-cess is due to the vacuum fluctuations caused by spontane-ous emission. The decay rate is a measure of the energyper unit frequency range in these fluctuations, and the re-sult, Eq. (25), tells that in equilibrium the oscillator energy isequal to this fluctuation energy density, as we expect from asystem in equilibrium with a heat bath.

In the limit of well-resolved resonances, expression (9), wefind from Eqs. (10), (11), and (20)

Ef hv (s + )

= hv(s + 1) (18)

If we introduce the effective final temperature Tf by setting

s = exp(-hv/kBTf), (19)

we fi:nd the correct equilibrium energy for a harmonic oscil-lator.

n+l

n

n-l

If we introduce the coefficient a = 1/3 we obtain the earlierobtainedll factor 7/12, whereas the more realistic value a =2/5 gives the factor 13/20. A result of the general form (26)was derived also by Neuhauser et al.2 and Itano and Wine-land,'9 and from a slightly different point of view theseauthors obtained an equivalent result earlier in Ref. 18.The numerical factor is of no major consequence for theresult, but it may serve as a check on results derived bydifferent authors with different methods.

If we omit the zero-point energy we obtain the excitationenergy

Eex=( 4) ()A

( n 0l)A (n+l)A-

n A+ nA_

Fig. 2. The rates A± are the quantum rates for harmonic oscillatortransitions up or down the energy-level ladder.

(27)

which seems to contradict what we expect from the argu-ment presented after Eq. (25). The imbalance is, however,compensated by the zero-point energy

E0 = hv >> h-y2

(28)

in this limit.

(20)

(21)

(22)

(23)

(24)

Efn = 2Lh-y(1 + a). (25)

= hv[(y) (a + ) + (26)

Stig Stenholm

1746 J. Opt. Soc. Am. B/Vol. 2, No. 11/November 1985

The anomalously low level for the excitation energy is afeature characteristic of the extreme quantum regime. Instatistical mechanics it is known as the "freezing-out ofdegrees of freedom," and it manifests itself in the third lawof thermodynamics. If we use the relationship (19) to lookat the final temperature

where as before

S= A,<1-A_

(36)

The case k = 0 is of special interest because then Gngenerates the moments

kBTf = hp 1log(s')

we find in the limit (13)

kBT~ 2 log(v/Y)> 2

(29)

Hence, for the temperature, the emission rate cooling limithF still applies. As is to be expected the thermal equilibri-um argument is to be applied to temperatures, not to ener-gies. In the quantum limit the noise energy hy is insuffi-cient to populate even the lowest oscillator level at energy hi.

With the exact expressions (6) for the rates A;, the finalenergy of the cooling can be written in the form' 4"15

,'7

Ef = hp - o(p,Y,AI), (31)A

where (p is a function regular at A = 0. Its limiting form forlow intensity is given by Eq. (20). The singularity at A = 0 isphysical, and it is a precursor of the heating instabilityemerging when the sign of detuning is changed. One couldsay that the dynamic evolution experiences a phase transi-tion from cooling to heating at this point. The function(-yep/A) has been plotted for some approximate treatments inRef. 17.

n= = [(Z z ) Go ' )]z =1n=0

(37)

For p = 1 we obtain the average excitation quantum numbern(t), which determines the instantaneous value of the energyof the harmonic motion. For p = 0 we obtain the normaliza-tion condition for the Drobabilities

G0(1, t) = 1. (38)

5. THE EIGENVALUE SPECTRUM

For no recoil, 1 = 0, the master equation (5) shows that for allvalues of n the reduced density matrix oscillates at the freeoscillator frequency v according to

(nlpln + k) c eikvt, (39)

which shows that the eigenvalue spectrum is denumerablyinfinitely degenerate at the frequencies ikv. This is used inRef. 16 as the basis for the perturbation calculation givingEq. (5). Following our work' 7 we look for eigenvalues of thegenerating function by introducing the ansatz

(40)Gk(z, t) = exp(-n 2 Aet)gk(z)

into Eq. (35). The resulting equation

4. THE USE OF A GENERATING FUNCTION

The master equation (5) looks unwieldy, but it can bebrought into a far simpler form by the transformation

O(nk) = e-ivkt [(n + n)!]'/2 (nlpin + k) (32)

(I -sz)(1g- F)a~ k 1 (1-sz)(1-Z)-- -(1-s)+s(1+k)(1-z)-egk=Oaz L2

(41)

has the solution

gA(Z) = C(l-SZ)-(k + 1 + )(l-ZY,

which applies for k > 0; a similar calculation is possible for k< 0. Inserted into Eq. (5) this gives

dtdd On, k) = r/2$(n + l)AO(n + 1, k)

- (n + 1 + .k)A+O(n, k)

- (n + .k)A_0(n, k)

+ (n + k)A+O(n - 1, k)}, (33)

which has only linear variables in its coefficients. For thiscase the method of a generating function is valuable, and wedefinell" 7

Gk(z, t) = zn0(n, k), (34)n=O

which is found to obey the partial differential equation

where

(43)2 -lk+ e2 1- s

Because the function Gk generates the moments of the dis-tribution for z = 1, we must require that all its derivativesexist at this point. This is possible only if a in Eq. (42) is anonnegative integer N. Because s < 1 the other factor in Eq.(42) always behaves properly. Setting a = N, we obtainfrom Eq. (43) the eigenvalue

(44)

Thus we see that for all k and N the time evolution isdetermined by the rate

72A_(I - s) = W. (45)

(42)

2 f s ) 1 z G k k1A (1- sz) (1-z) d -[2(1 -s) + s(l +k)(1 -z)] Gk

aGh-- , (35

at

as seen already in Eq. (12). Thus the degenerate spectrum(39) is split so that each eigenvalue is replaced by a seriesextended to larger and larger negative real parts with in-creasing N. Thus we find for long time the dominatingbehaviors

Stig Stenholm

f=(1_S)(-!k+N) -2

Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. B 1747

(nl pl n + k) exp(ikvt - Wkt) + . . .

and

P(n) P(n) + O(e-Wt),

46)(X) = CNXN

N=O

(46b)

(53)

where the coefficients CN derive from the initial conditions.When inserted into Eq. (52) the expansion gives

where Po(n) is given in Eq. (17).The diagonal elements thus reorganize themselves toward

the ultimate equilibrium in a time of the order W-1, as statedin Section 2. The off-diagonal elements k F 0 decay to zerobut over a time scale (2/W), which does not differ from thereorganization time scale. Consequently there is sufficienttime for the off-diagonal elements to reorganize within themanifold of given k before they decay. These processescould be of interest in the interpretation of dynamic scatter-ing data, but the theoretical understanding of this field isstill insufficient.

For = 0 the steady-state solution with N = 0 becomesunique. Its generating function (41) becomes

gt(z) = 1 -

which, with the normalization (38) and

C= (1-s),

(47)

(48)

agrees with the result (17). From Eq. (42) we can obtain theother eigenfunctions just as easily, and an arbitrary initialstate can be decomposed, its components decaying at therates given by Eq. (44). This completes the analysis of thespectrum of the master equation.

6. THE DYNAMIC TIME EVOLUTION

It is also possible to solve the time-dependent equation (35)directly. It is a first-order partial differential equation withthe characteristics given by

t1A~dt =(1 - z)(sz - 1)

dGk

[Rk(s -1) +s(1 + k)(z - 1)]Gk(49)

The two ensuing differential equations are easily integrated,and their constants of motion can be written as

1 Z eWt

sz - 1

C2 = Gk(1-Z) (SZ - 1) 2

The most general solution of Eq. (35) is consequently

Gk 'z (lt)= k 1) + I (1-Z ZWt1

(51)

(52)

This solution has been written assuming that k > 0. Thefunction 4' is arbitrary and must be determined by the initialdistribution. For the special case k = 0, this solution wasgiven in Ref. 11.

We can use our knowledge about the spectrum in Eq. (44)to learn about the general form of the function 4'. All timedependence in Gk must be of the form exp[-Wt('/ 2 k + N)],which implies that the function 4' is always of the form

Gk(z, t) = , E CN( 1 -)eNWt(SZ -1)1+ N=O (Z-1

(54)

When only one CN differs from zero we directly obtain theeigenoscillations as given by Eq. (42).

The time evolution of the energy can be obtained exactly"from the expression (52). Setting k = 0 and taking the firstderivative, we obtain from Eq. (37)

n(t) = - 440] - - 1 4,'[]eWt(s'- 1) 2 (s 12

The steady-state value at t -- gives

__ __ Sn(-) = -bp(0) (1 -s)(s -1)which agrees with Eq. (18) when the value

b(0) = (s - 1)

(55)

(56)

(57)

is used. This also ensures the correct normalization Eq. (38)in Eq. (52).

Setting t = 0, we obtain from Eq. (55) the result

n(o) - n(0) = 1 &P (0) (S -1)2

which allows us to write Eq. (55) in the form

n(t) = n(O)e-wt + n(-)(1 -e-wt).

(58)

(59)

Thus the average occupation number interpolates smoothlybetween its initial and final values within a time of orderW-; the same consequently holds for the oscillator energy.This result is independent of the initial distribution.

7. DYNAMICAL EVOLUTION OF SPECIALINITIAL DISTRIBUTIONS

It is possible to evaluate the result Eq. (52) explicitly forseveral initial distributions of physical interest." We con-sider the following situations.

A. A Thermal Distribution(50) We assume that at t = 0 we have an initial thermal distribu-

tion (17) with

s(0) 5d (A+/A_) = s.

The initial generating function is then

(60)

GO(z,0) = 1 - s(0) . (61)1 -zs(0)

From Eq. (52) we obtain the equation

I-zs(0) sz- (s1 - Z) (62)- I1- (0) SZ 1 SZ___I

and setting y = (1 - z)/(sz - 1) and solving for 4' we obtain

-[s - s(0)]y + [1 - s(0)] (63

Stig Stenholm

(46a)

I

I

1748 J. Opt. Soc. Am. B/Vol. 2, No. 11/November 1985

P(nt)

5 10

Fig. 3. The time evolution of an initial Planck-type distributionrelaxing toward a final one without changing its form. The initialaverage excitation energy <n> = 10, and the final one is 1.5.

When this expression is used in Eq. (52) we obtain

G0(z, t) - 1 - Wt1 - s (t)Z

where we have introduced the function

s(t) = [s - s(0)]e-Wt + [s(O) - ]s[s - s(O)]e-Wt + [s(O) - 1]

This is an evolution from the initial s(0) to the final s. Theform of the generating function (64) is preserved and corre-sponds to a time-dependent Planck distribution with thetime-dependent parameter s(t). The shape of the Planckdistribution is hence seen to remain invariant under the timeevolution. It is easy to check that the result (59) followsdirectly from the form (64). The time evolution given by thedistribution function

p = [1 - (t)][s(t)]N (66)

is shown in Fig. 3.

B. An Initial Poisson DistributionWhen the trapped particle is performing a nearly classicalpendulum motion in its trap, we can assume its state to beclose to a coherent one. Then the initial probabilities aregiven by the Poisson distribution

(64)

(65)

where

(71)s(1 -et)1 - seWt

and

N(1 -s) 2ewtP s(se-wt -1)(1 -eWt)

(72)

Recognizing the generating function for the Laguerre poly-nomials in Eq. (70), we can obtain for the probabilities

1 5 (1 set) exp 1 - se-wt ]

I [ N(1 -)eWt s(1 _e-wt ien!fs(se-wt - 1)(1 -et) (1 - se"'-)

(73)

The argument of Ln is singular at t = 0, but because for largearguments Ln(z) = (z)n we find that

limLn[ N(1 - s) 2eWt ] 1seCWtJ = N (74)

and Eq. (73) gives the result (67). For large times, t , weuse Ln(O) = n!, and we find from Eq. (73) the equilibriumresult (17).

The time evolution described by Eq. (73) is shown in Fig.4. The peak of the distribution is shifted to lower values,broadened, and finally flattened into the Planck-type equi-librium distribution.

C. The Initially Pure StateWe finally calculate the time evolution of a single pure stateIN> of the oscillator. This state may be difficult to preparefor a trapped particle, but it is physically interesting andallows us to calculate the time evolution of an arbitraryinitial state by summing the contributions from its pure-state components.

The generating function now becomes simply

GO(z, 0) = zN, (75)

eNPn(0) = N- X

and the generating function is given by

Go(z, 0) = e' n!

n=O

-N(z-1)

(67)

(68)

Inserting this into Eq. (53) and proceeding as above, we find

(P1 2(y) = s- exp M -s) y]Sy + 1[1 +Sy) IJ

(69)

This gives the solution

GO(z, t) = (1 - s) exp (s 1)e-wt exp( x(I- se-Wt) exp -se-Wt j 1 - X

M 7

0U 10 20

n

Fig. 4. The time evolution of an initial Poisson distribution withthe initial excitation level <n> = 10 toward the final value 1.5 as in

0) Fig. 3.

Stig Stenholm

Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. B 1749

0 10 20n

Fig. 5. Relaxing from the initial pure oscillator state with N = 10,the oscillator cools through stages such as the Poissonian case of Fig.4 toward the same final state with <n> = 1.5 as in Figs. 3 and 4.

and, solving for the function 4' as before, we now obtain

=S (-i) (1 + y)N (6

(sy + 1 ) (76)which gives the expression

Gz,0 (s - 1) [(e-Wt - 1) + (s - eWt)ZIN

[(seWt - 1) + s( -e-wt)z]N+1

N~~~NW=S (-1 i ( I)( k 1)z'(e- Wt 1)N-I

1=0 k=0

X (s - eWt(e-wt - l)N--k( - se-Wtk. (77)Collecting terms with k + = n and reordering, the sums, weobtain the probability

PnN(t) = (1 -s) i ( (nk)( k )

X s k( - e Wt)N-n+2k (e-Wt - S)nk(1 - se-Wt)N-k-l.

(78)

This complicated expression is easily evaluated numerically.For N = 10 its evolution is shown in Fig. 5. We see that theinitially sharp delta distribution is broadened to a Poisson-like one in a fraction of the time W', and after that theevolution follows much the same pattern as in Fig. 4. Thefinal distribution at t = is, of course, the same as before.To obtain this from Eq. (78) is awkward, but setting t = inEq. (77) gives directly the result Eqs. (47) and (48).

Knowing PnN from Eq. (78), we can obtain the distributionstarting from any P(N) by the use of it as a Green functionfor the problem. We have

pN

Pn(t) = E Pn (t)PO(N) (79)N=O

This solves the general problem of the distribution as well asthe general solution Eq. (52).

8. SUMMARY AND CONCLUSIONS

In this paper I have reviewed the dynamic time evolution ofa harmonically trapped particle subjected to laser cooling.

In the Lamb-Dicke limit the quantum structure of the trap-ping well cannot be overlooked, but on the other hand thesmall expansion parameter Eq. (1) is proportional toPlanck's constant h, and hence the expansion is of a semi-classical type. The ordinary Fokker-Planck equation isfound to be derived from a similar expansion.25 The masterequation obtained does, however, introduce only one newtime scale instead of the two normally found: one for thelight pressure force and one for the diffusion. The implica-tion of this is that even if the off-diagonal elements decay tozero they reorganize on a time scale similar to that of theapproach to equilibrium for the diagonal occupation proba-bilities. Hence the internal evolution of the coherences maybe an observable effect here. These features seem to begeneral for the type of master equation that we get for a fullyharmonic motion of a quantum-mechanical system.

In the trapped-ion case the potential is not a static oscilla-tor one but a dynamically trapped motion under the influ-ence of a radio frequency. The ensuing micromotion is, ofcourse, neglected in the present treatment. It would be ofinterest to know how this omission affects the results ob-tained here. Because the final state of the ion in the rf traptends toward localization, there is some hope that the pre-sent discussion retains qualitative validity. It appears10that if many quantum states are coupled the cooling processloses its efficiency and many particles are lost from the trapthrough diffusive heating.

Another simplifying feature here is the restriction to onedimension. A real trap must be harmonic in all three direc-tions of space. If the corresponding oscillations are degen-erate, instabilities appear, as a perturbation calculation26

has shown. This is a real phenomenon and indicates theheating of the eigenvibration orthogonal to the laser beam.Cooling in this case may require the use of more complexlaser geometries. When the oscillational frequencies, on theother hand, are greatly different, we expect their coupling tobe minute, and the present treatment can be applied sepa-rately to each mode when the proper geometric factors areintroduced.

The spectrum of the scattered light carries important in-formation about the vibrational state of the trapped parti-cle. Except for a perturbation calculation2 7 there has beenno progress on this problem. We only need to remind our-selves of the intricacies of the strongly driven resonancefluorescence for free atoms to realize the complexity of theproblem for a trapped and cooling particle. We have evenless knowledge about the information possibly carried by thephoton statistics in the scattered radiation. For the freeparticle this proved to be an interesting question on itsown. 23 ,2 5 ,28

Finally I want to mention the fact that our master equa-tion obviously describes the evolution of an ensemble oftrapped particles, whereas the measurements are carried outon a single one. As long as the observations last for macro-scopically long times (minutes to hours) we can invoke anergodic argument and claim to expect the results observed torepresent the ensemble averages. On the other hand, thereis no doubt that we have here an instance of a single isolatedquantum-mechanical system that we can investigate at willfor long time periods. When the detailed dynamics of theion becomes experimentally manageable, the system offers aunique possibility to test fundamental questions of quantummechanics, because we can choose to make either repeated

MnW0a

Otl

P

Stig Stenholm

1750 J. Opt. Soc. Am. B/Vol. 2, No. 11/November 1985

measurements (ensemble averages) or time averages. Herethe use of laser techniques such as saturation spectroscopyor coherent transients may find new areas of application.

In conclusion, I find the physics of the single trapped ionmost interesting both from the point of view of applicationsand in its own right as a tool to apply our understanding ofquantum theory. We also foresee the trapping of singleneutrals in the near future, and many of the present resultsmay be applicable to this situation too.

REFERENCES

1. D. J. Wineland, R. E. Drullinger, and F. L. Walls, Phys. Rev.Lett. 40, 1639 (1978).

2. W. Neuhauser, M. Hohenstatt, P. Toschek, and H. G. Dehmelt,Phys. Rev. Lett. 41, 233 (1978).

3. P. E. Toschek and W. Neuhauser, in Atomic Physics, D.Kleppner and F. M. Pipkin, eds. (Plenum, New York, 1981),Vol. 7, p. 529.

4. H. G. Dehmelt, in Laser Spectroscopy V, A. R. W. McKellar, T.Oka, and B. P. Stoicheff, eds. (Springer-Verlag, Berlin, 1981), p.353.

5. W. M. Itano and D. J. Wineland, in Laser Spectroscopy V, A. R.

W. McKellar, T. Oka, and B. P. Stoicheff, eds. (Springer-Verlag,Berlin, 1981), p. 360.

6. J. Javanainen and S. Stenholm, Appl. Phys. 21, 283 (1980).7. J. P. Gordon and A. Ashkin, Phys. Rev. A 21, 1606 (1980).8. S. Stenholm, Phys. Rep. 43, 152 (1978).9. V. S. Letokhov and V. G. Minogin, Phys. Rep. 73, 2 (1981).

10. J. Javanainen and S. Stenholm, Appl. Phys. 24, 71 (1981).11. J. Javanainen and S. Stenholm, Appl. Phys. 24, 151 (1981).12. J. Javanainen, J. Phys. B 14, 2519 (1981).13. J. Javanainen, J. Phys. B 14, 4191 (1981).14. J. Javanainen, J. Phys. B 18, 1549 (1985).15. M. Lindberg, J. Phys. B 17, 2129 (1984).16. M. Lindberg and S. Stenholm, J. Phys. B 17, 3375 (1984).17. J. Javanainen, M. Lindberg, and S. Stenholm, J. Opt. Soc. Am.

B 1, 111 (1984).18. D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979).19. W. M. Itano and D. J. Wineland, Phys. Rev. A 25, 35 (1982).20. J. Javanainen and S. Stenholm, Appl. Phys. 21, 163 (1980).21. A. Yu. Pusep, Sov. Phys. JETP 43, 441 (1976).22. J. Javanainen and S. Stenholm, Appl. Phys. 21, 35 (1980).23. L. Mandel, J. Opt. (Paris) 10, 51 (1979).24. M. 0. Scully and W. E. Lamb, Jr., Phys. Rev. 159, 208 (1967).25. S. Stenholm, Phys. Rev. A 27, 2513 (1983).26. J. Javanainen, Appl. Phys. 23, 175 (1980).27. J. Javanainen, Opt. Commun. 34, 375 (1980).28. R. J. Cook, Opt. Commun. 35, 437 (1980).

Stig StenholmStig Stenholm was born on February 26,1939. He graduated from the HelsinkiUniversity of Technology and HelsinkiUniversity in 1964. He studied for theDr. Phil. degree at Oxford Universityfrom 1964 to 1967. He worked at Yale

>\0000S :Universit from 1967 to 1968. He hasalso worked at the Helsinki University ofTechnology, the University of Helsinki,and the Academy of Finland. In 1974 heworked at IBM, San Jose, California.From 1974 to 1980 he served as an asso-ciate professor of physics at the Univer-

sity of Helsinki. He worked at the Optical Sciences Center, Tucson,Arizona, from 1982 to 1983. At present he is scientific director atthe Research Institute for Theoretical Physics, University of Helsinki.

Cf)

ca

Stig Stenholin