Nonadiabatic transitions in finite-time adiabatic rapid passage

T. Lu,1 X. Miao,2 and H. Metcalf2

1Computational Science Center, Brookhaven National Laboratory, Upton, New York 11973, USA2Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA

Received 20 April 2007; published 26 June 2007

To apply the adiabatic rapid passage process repetitively T. Lu, X. Miao, and H. Metcalf, Phys. Rev. A 71,061405R 2005, the nonadiabatic transition probability of a two-level atom subject to chirped light pulsesover a finite period of time needs to be calculated. Using a unitary first-order perturbation method in therotating adiabatic frame, an approximate formula has been derived for such transition probabilities in the entireparameter space of the pulses.

DOI: 10.1103/PhysRevA.75.063422 PACS numbers: 32.80.Pj, 42.50.Vk

I. BACKGROUND

Adiabatic rapid passage ARP is a long-studied methodof inverting the population of a two-level system that hasbeen well known since the early days of magnetic resonance.The population inversion occurs as the result of a very largefrequency sweep through the resonance condition in a verylong time, so the sweeps are typically approximated as infi-nite. By contrast, exploitation of ARP for producing largeoptical forces requires repeated sweeps of both the opticalfrequency and the amplitude of the light field so they arenecessarily finite in both frequency range and duration. Ineither case the population inversion may be incomplete, andwe denote the probability for not making the desired adia-batic transition as Pnad.

When the light beams are oppositely directed, a pair ofappropriately timed nonoverlapping sweeps coherently ex-changes momentum between them, imparting the difference2k to the atoms. When these cycles are repeated at ratem, the force is FARP=km /k /2, the usual radia-tive force on the atoms 1,2. Here =2 /k is the wave-length of the transition to an excited state of lifetime 1/. Inthis paper we show that the details of both frequency andamplitude sweep wave forms can have a dramatic impact onthe parameter dependence of Pnad. If the counterpropagatinglight pulses overlap in time in the interaction region, themomentum exchange between each pair can exceed 2k dueto multiphoton transitions 3. However, this force is not di-rectly related to Pnad, and it is beyond the scope of this paper.

Numerical calculations for repetitive sweeps 1 haveshown that sinusoidal wave forms result in Pnad distributionsthat have desirable consequences with readily accessible pa-rameters, and experiments confirm these results 2. How-ever, there are modest differences between the measurementsand the calculations, and these may be partially attributed tosweep wave forms that are not precisely sinusoidal. To ex-plore this, we have recalculated the ARP map of Ref. 1with different wave forms, and have found very significantqualitative differences among the results. In addition, wehave developed an analytic approximation for Pnad that isappropriate for a large class of finite wave forms.

We consider the reference frame that rotates at the fre-quency of the light field so that the Hamiltonian HR can bewritten as 4

HRt =

2 t t

t − t , 1

where −a is the detuning of the light at frequency

from the atomic resonance at a, and egE ·re / is theRabi frequency that characterizes the on-resonance, electricdipole interaction between the light and atoms. The ARPprocess can be envisioned in a dressed-atom view of theenergies of the two-level system see Refs. 1,5. An alter-

native view is the rotation of the Bloch vector R t= t t, where the is an artificial vector whosecomponents are the Pauli matrices, on the Bloch sphere un-der the influence of the modulated light field. The verticalaxis z of the sphere is the population difference term andthe horizontal axes are related to the relative phase of theatomic superposition see Ref. 4. The equation of motion,

dR /dt= tR , can be derived from Eq. 1. Here tis an artificial “torque” vector having componentst ,0 ,t.

The process of ARP in this view involves a synchronizedsweep of both the amplitude and frequency of the light sothat the torque vector sweeps along a meridian from one poleto the other. As illustrated in Fig. 1a, if the initial state lies

0.5

0

0.5 0.50

0.5

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

yx

z

0.04 0.02 0 0.02 0.04

0.1

0.08

0.06

0.04

0.02

0

x

y

O

(a) (b)

FIG. 1. Adiabatic following of the Bloch vector along the trackof the torque vector of a modulated light pulse. a The wavy curveconnecting two poles is a typical trace of the Bloch vector on theBloch sphere. The meridian close to it is the trace of the torquevector. b The same pair of traces in the adiabatic frame. Themeridian reduces to a fixed point at origin in the adiabatic frame.

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in an eigenstate, R will start from a pole and precess aroundthe torque vector to the other pole. As long as the sweep isslow enough, the population will be completely inverted atthe end. Figure 1b shows the trace of the Bloch vector inthe adiabatic frame. The adiabatic frame rotates togetherwith the torque vector along the meridian but without pre-cession, so that the torque vector is fixed at the origin,which is point O in Fig. 1b, while the Bloch vector startsfrom and ends at the same point O over the duration of apulse.

If the sweep is not slow enough, the Bloch vector does notfollow the torque vector adiabatically, and there is a smallfraction of the population not inverted at the end of eachsweep. The amount of the small fraction is called the nona-diabatic transition probability, denoted by Pnad. In order todesign sweep schemes with vanishingly small Pnad, we needto calculate Pnad for various pulse schemes and parameters.The ARP process is usually used for one-time populationinversions, where the sweep time can be regarded as infinite.Extensive research has been done in this area 6–10, whilethe focus of this paper is on the class of finite-time sweeps.

II. CHARACTERISTICS OF THE Pnad DISTRIBUTION

The value of Pnad resulting from a chirped light pulsedepends on the maximum pulse intensity, denoted by 0, themaximum detuning, denoted by 0, and the pulse profile. Westart from a sinusoidally varying light pulse, whose time de-pendence is described by

t = 0 cos mt, t = 0 sin mt , 2

where m= /T and −T /2 tT /2. The pulse profile isplotted in Fig. 2a. The special case of 0=0 has beenstudied analytically in some detail 11. Assuming that theatom starts in the ground state, we integrate the Schrödingerequation numerically from −T /2 to T /2 and calculate theremaining ground state population at the end, which givesPnad for the process. From dimensional analysis, Pnad is afunction of two dimensionless parameters 0 /m and 0 /m.

For the case of 0=0, we reproduce the results of Ref. 11.The two-dimensional map of Pnad is plotted in Fig. 2b. Thelower-left corner of Fig. 2b has been corroborated experi-mentally 2.

The Pnad map is divided into two regions—oscillatory andnonoscillatory. Just as pointed out in Ref. 10 for infinite-time transitions, each oscillation is connected to an integernumber of precessions of the Bloch vector during the timeevolution. For example, Fig. 1 shows the trace of the Blochvector under the influence of a sinusoidal light pulse withparameters at point A in Fig. 2b. Point A in Fig. 2b lies onthe 11th curve counted from the origin along the diagonal.Correspondingly, Fig. 1b shows that in the adiabatic frame,the Bloch vector precesses exactly 11 cycles while the torquevector is fixed at the origin. Along each “loop” of zeros ofPnad the darkest regions in Fig. 2b, there is a special pointconnecting the upper arc and lower arc, whose precessionnumber differs by one.

Roughly speaking, in the nonoscillatory region 00,the nonadiabatic transition is dominated by the barelyavoided crossing at resonance, and the Landau-Zener for-mula applies,

Pnad exp−

2

02

0 = exp−

2

02

0m . 3

In the oscillatory region, the oscillation amplitude of Pnaddecreases from 1 to 0 as 0 /0 increases. For an unchirpedpulse, i.e., 0=0,

Pnad = cos20

T/2

tdt = cos20

m . 4

It agrees with the area theorem 4. To attain complete popu-lation inversion, i.e., Pnad=0, with an unchirped pulse, theintegral of the eigenfrequency t over the entire pulsemust be an odd multiple of . In the adiabatic limit, i.e.,0→ and 0→ simultaneously with their ratio fixed, orequivalently, →0, we will show later that

- δ0

Ω0

δ0

T/2 0 T/2

5 10 15 20 25

5

10

15

20

25

δ0/ω

m

Ω0/ω

m

A

(a) (b) (c)

5 10 15 20 25

5

10

15

20

25

δ0/ω

m

Ω0/ω

m

FIG. 2. Contour map of Pnad for sinusoidal pulses. a The profiles of the pulse detuning and Rabi frequency. b The contour curves ofPnad=0.1,0.01, . . .. The darkest regions represent zeros of Pnad. The point A with coordinates 25, 18.724 in part b corresponds to the traceof Bloch vector shown in Fig. 1. c The contour plot of the approximate Pnad.

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Pnad 02

T

2

−2

sin20

T/2

tdt= 0

2

0m−2

sin20

T/2

tdt , 5

where t=2t+2t is the generalized Rabi fre-quency. To attain complete population inversion in the adia-batic limit, the integral of the eigenfrequency t over theentire pulse must be an integer multiple of 2. All charac-teristics of the Pnad map described above are true for generalpulse profiles. Precise asymptotic forms of Pnad will be pre-sented with detailed derivation in a forthcoming paper.

It is instructive to compare the Pnad map for finite-timesinusoidal pulses, which is not exactly solvable, to that forinfinite-time pulses in the Demkov-Kunike DK model7,10, which has a similar pulse profile but is exactly solv-able. The time dependence of the field in the DK model is

t = 0 secht

2, t = 0 tanht

2 . 6

The pulse profile of the DK model, which is plotted in Fig.3a, is similar to that of sinusoidal pulses except for the tailsat large t. The special case of 0

2−02= /22 was studied

in Ref. 4. The nonadiabatic transition probability of the DKmodel was derived in Ref. 10,

PnadDK =

cosh2 02 − 0

2

cosh2 0. 7

The contour map of Pnad for the DK model is plotted in Fig.3b. As in Fig. 2b, it has a nonoscillatory region where theLandau-Zener formula applies; for unchirped pulses the areatheorem also applies. However, the figures differ signifi-cantly in the oscillatory region because Eq. 5 does notagree with Eq. 7 in the adiabatic limit. As a conclusion, theseemingly negligible tails in the pulse profile makes a bigdifference in the map of Pnad. Equation 7 is a poor approxi-mation of Pnad for sinusoidal pulses.

In the following section we will derive a simple formulathat serves as a good approximation for Pnad of finite-timepulses in the entire map.

III. APPROXIMATE FORMULA FOR Pnad

We derive an approximate formula for Pnad for finite-timepulses with large intensity and detuning. It is obtained by afirst order perturbation calculation in the appropriate refer-ence frame. The rotating frame is not appropriate because theHamiltonian in it see Eq. 1 is large and so the atomic statechanges quickly. Since the atomic state follows the energyeigenstate adiabatically under such a pulse, the adiabaticframe is more appropriate. The transformation matrix fromthe rotating frame to the adiabatic frame is formed by instan-taneous energy eigenstates, which are called the dressedstates of the atom in the field 12,

UAt = cos/2 sin/2− sin/2 cos/2

, 8

where 0 with tan t=t /t. The Hamiltonianin the adiabatic frame is

HAt = UAHRUA† − iUA

d

dtUA

† =

2 t it

− it − t .

9

The large diagonal elements of HA indicate that the atomicstate has a rapidly changing phase as it follows the energyeigenstate. In the Bloch sphere view, it means that the Blochvector precesses rapidly around the torque vector. Only in areference frame that rotates at the same frequency about theeigenstate the torque vector does the atomic state theBloch vector change slowly enough so that a perturbationmethod can be applied. Such a frame is called the rotatingprecessing adiabatic frame 8,10. The transformation ma-trix associated with the rotation is

(a) (b) (c)

- δ0

Ω0

δ0

0- τ τ

5 10 15 20 25

5

10

15

20

25

δ0τ

Ω0τ

0 5 10 15 20 250

5

10

15

20

25

δ0τ

Ω0τ

FIG. 3. Contour map of Pnad for the Demkov-Kunike model. a The profiles of the detuning and Rabi frequency. b The contour curvesof Pnad=0.1,0.01, . . .. The darkest regions represent zeros of Pnad. c The contour plot of the approximate Pnad.

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URt = eist/2 0

0 e−ist/2 , 10

where st=0t d. The Hamiltonian in the rotating adia-

batic frame is

Ht = URHAUR† − iUR

d

dtUR

† =

2 0 iteist

− ite−ist 0 .

11

The absolute value of Ht is not small for highly detunedintense pulses, but due to the rapidly changing phase st, theevolution of the atomic state in this frame is indeed slow. Ingeneral the Schrödinger equation in the rotating adiabaticframe is not exactly solvable. A formal solution can be writ-ten as the Dyson expansion that involves nested time inte-grations and time-ordered products 13,14.

Here we apply the perturbation method in the rotatingadiabatic frame. Denote the propagation matrix for time from0 to t by Ot. The first order perturbation gives

Ot I −i

0

t

Hd . 12

However, this approximation is not unitary, which can leadto unphysical results if used repeatedly. A better option is touse

Ot = exp−i

0

t

Hd , 13

which is called the unitary first order approximation. Equa-tion 13 ignores the noncommutativity between the Hamil-tonians at different times, therefore it is exact only for com-mutative Hamiltonians. Nevertheless, as demonstratedbelow, Eq. 13 serves as a good approximation for the map-ping of Pnad resulting from a general chirped light pulse.Substituting Eq. 11 into Eq. 13, we can obtain the explicit

form of Ot,

Ot = exp− i 0 izt− iz*t 0

= coszt

ztzt

sinzt

−z*tzt

sinzt coszt , 14

where zt= 120

t deis and z*t is the complex conju-gate of zt.

In spite of the unitarity, Eq. 13 is still only of first orderaccuracy. Higher order unitary approximations for Ot canalso be derived from the Dyson expansion. However, theirexpressions and evaluations are more complicated and willnot be discussed in this paper.

The nonadiabatic transition amplitude is the off-diagonalelement of the propagation matrix in the rotating adiabaticframe for the entire pulse −T /2 ,T /2. Successive applica-tion of the unitary approximation to the propagation matrix

on each half pulse −T /2 ,0 and 0,T /2 enhances the ac-

curacy of O for the entire pulse, from which Pnad is derived.For simplicity, we only consider symmetric pulses in thispaper, i.e., −t=−t, −t=t. For such pulses thepropagation matrices for the two half pulses are related in asimple way. Denote the propagation matrix Ot by

Ot = *t − t*t t

. 15

For a symmetric pulse we have

O−T/2,T/2 = O0,T/2O−T/2,0

= *T/2 − T/2*T/2 T/2

*T/2 − *T/2T/2 T/2

.

16

So

Pnad = T/2T/2 + *T/2*T/22. 17

Comparison between Eqs. 14 and 15 gives the approxi-mation for t and t. Substituting the approximations

t and t into Eq. 17, we have the approximate formulafor Pnad,

Pnad = sin20

T/2

dtteistcos2arg 0

T/2

dtteist .

18

The contour map of Pnad for sinusoidal pulses is plotted inFig. 2c. It is in good quantitative agreement with the con-tour map of the exact Pnad in Fig. 2b. The maximum erroris 0.176 at a point near origin. To demonstrate the universal-ity of Eq. 18, we did the same comparison for triangularpulses. The time dependence of the intensity and the fre-quency sweep for a triangular pulse is

t = 01 − 2t/T, t = 02t/T . 19

The pulse profile is plotted in Fig. 4a. In the Bloch sphereview of the problem, the torque vector moves along astraight line with uniform speed for each half pulse. Theproblem reduces to the Landau-Zener model in a rotatedframe, and thus is solvable in terms of confluent geometricfunctions. The map of the exact Pnad is plotted in Fig. 4b.The corresponding Pnad is plotted in Fig. 4c. Again thedifferences are small, and there is good quantitative agree-ment. The maximum error is 0.172.

Equation 18 can also be applied to infinite-time pulses.The approximate Pnad obtained from Eq. 18 for theDemkov-Kunike model is plotted in Fig. 3c. Once againthey agree quantitatively with the exact form, Eq. 7, and itscontour plot in Fig. 3b. The maximum error is 0.168.

From Eq. 18 we can derive the asymptotic form of Pnadin the adiabatic limit,

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Pnad = sin20

T/2

dtteistcos2arg 0

T/2

dtteist

0

T/2

dtteist2

cos2arg 0

T/2

dtteist= Re

0

T/2

dtteist2

Reteist

it

0

T/22

= 02

T

2

−2

sin20

T/2

tdt . 20

Since Pnad is obtained by perturbation calculation in the ro-tating adiabatic frame, its asymptotic form in the adiabaticlimit is the same as that of the exact Pnad, which is given inEq. 5.

IV. CONCLUSION

By applying unitary first order perturbation in the rotatingadiabatic frame, we have derived an approximate formula forthe nonadiabatic transition probability Pnad for the finite-time

ARP process in the entire parameter space of pulse intensityand detuning. We showed that Pnad for finite-time pulses andinfinite-time pulses of similar profile has characteristicallydifferent distribution in the parameter space. For finite-timepulses, the map of Pnad in the pulse parameter space is di-vided into oscillatory and nonoscillatory regions. TheLandau-Zener formula only applies in the nonoscillatory re-gion. In the oscillatory region, the oscillation amplitude andthe phase of Pnad have different asymptotic forms dependingon the region in the parameter space.

The map of Pnad in the pulse parameter space provides thecritical guidance in the selection of optimal parameters toachieve large optical force using the ARP scheme 2. Thefailure of the Landau-Zener formula in the oscillatory regionnecessitates an approach that is appropriate for finite-timepulses. The agreement between the exact Pnad and the ap-

proximate formula Pnad obtained here has proven that theunitary first order perturbation in the rotating adiabatic frameis a successful approach.

ACKNOWLEDGMENT

This work was supported by ONR, BNL, and ARO.

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New York, 1994, Sec. 5.6.

(a) (b) (c)

- δ0

Ω0

δ0

T/2 0 T/2

0 5 10 15 20 250

5

10

15

20

25

δ0T/4

Ω0T

/4

0 5 10 15 20 250

5

10

15

20

25

δ0T/4

Ω0T

/4

FIG. 4. Contour map of Pnad for triangular pulses. a The profiles of the pulse detuning and Rabi frequency. b The contour curves ofPnad=0.1,0.01, . . .. The darkest regions represent zeros of Pnad. c The contour plot of the approximate Pnad.

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