Chemical Physics Letters 499 (2010) 161–163

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Chemical Physics Letters

journal homepage: www.elsevier .com/ locate /cplet t

Features of quantum control in the sudden regime

Michael Klein, Vincent Beltrani, Herschel Rabitz ⇑Princeton University, Department of Chemistry, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 July 2010In final form 16 August 2010Available online 21 August 2010

0009-2614/$ - see front matter � 2010 Published bydoi:10.1016/j.cplett.2010.08.025

⇑ Corresponding author. Fax: +1 609 258 0967.E-mail address: hrabitz@princeton.edu (H. Rabitz)

Although quantum dynamics in the sudden regime is generally not fully controllable, the simple depen-dence of the observable yield on the applied field enables efficient searches for a control and also admitsgenerally many solutions. This flexibility allows for simultaneous optimization of ancillary objectivesincluding the minimization of competing pathways throughout the control period. These features areillustrated with simple models and simulations.

� 2010 Published by Elsevier B.V.

Controlling quantum phenomena through the application ofshaped laser pulses is an active area of research [1,2]. The dynam-ics are considered sudden or impulsive when the duration, T, of theapplied field satisfies Txjk � 1, where xjk is the largest transitionfrequency amongst the system’s relevant energy levels. Quantumcontrol in the sudden regime has been investigated theoreticallyfor molecular rotation [3], molecular alignment [4], and Rydbergexcitations [5], as well as in many experiments [6,7]. The presentanalysis further explores the characteristics of quantum controlin this regime, where the quantum dynamics depends only onthe transition dipole and the DC component of the applied electricfield. As a consequence of the simplified dynamics, the level ofattainable control will generally be diminished. Balancing this lim-itation is the reduction of search effort for finding a control solu-tion, while still retaining generally many to choose from. Thislatter feature also opens up the flexibility to simultaneously opti-mize multiple control objectives.

Consider an N-level quantum system described byHðtÞ ¼ H0 � l�ðtÞ, where H0 is the internal Hamiltonian, l is the di-pole and �ðtÞ is the electric field. In the sudden regime H0 may beignored and the time evolution operator is simply Uðt;0Þ ¼ eilnðtÞ,where nðtÞ ¼

R t0 �ðt

0Þdt0 and we have chosen units such that �h ¼ 1.As a simple illustration of the principles of controlled suddendynamics, this work will be concerned with the primary goal ofmaximizing the probability of making a transition from state i tostate f at time T

Pi!f ½�ðtÞ� ¼ jhf jUðT;0Þjiij2 ¼ jhf jeilbjiij2; ð1Þ

where b ¼ nðTÞ. The analysis below may be readily extended tomore general control objectives as well. Since Pi!f just dependson the single parameter b, the search for an optimal control solution

Elsevier B.V.

.

(i.e., a field, �ðtÞ, or value of b that gives the best yield) is greatlysimplified. Pi!f ½b� will generally have many suboptimal extrema,dPi!f =db ¼ 0, corresponding to Pi!f values less than 1.0. This canformally be expressed as seeking the local maximal roots over bof the equation

dPi!f

db¼ �2Imfhije�ılbjf ihf jeılbljiig ¼ 0: ð2Þ

At a solution, b, satisfying Eq. (2), the curvature of Pi!f ,

d2Pi!f

db2 ¼ 2fjhf jeılbljiij2 � Refhije�ılbjf ihf jeılbl2Þjiigg; ð3Þ

specifies the robustness to variations, b! bþ db. In the sudden re-gime the structure of l will dictate whether a value of b existswhere Eq. (2) permits Pi!f ¼ 1:0. There is also the possibility of‘‘engineering” the system dipole through special choice of the mol-ecule, and in the case that Pi!f ¼ 1:0, Eq. (3) reduces to

d2Pi!f

db2 ¼ �2fhijl2jii � hijljiiÞ2g: ð4Þ

Thus, at Pi!f ¼ 1, it is not possible to also attain ‘‘full” robustness,with d2Pi!f =db2 ¼ 0, but some systems can approach this limit asshown below.

An extensive set of simulations was run with systems having di-verse dipole structures up to dimension N ¼ 30. These studies em-ployed random complex Hermitian N � N dipole matrices wherethe real and imaginary parts of the elements were drawn from azero mean normal distribution with unit standard deviation. Thetransition probability as a function of b, Pi!f ðbÞ, was generallyfound to be quasi-periodic with many extrema. For systems withN ¼ 3;4 there typically were �40 local maxima over the rangeb 2 ½0;50�. As the dimension increases, the average number of max-

162 M. Klein et al. / Chemical Physics Letters 499 (2010) 161–163

ima over the same b range increases modestly so that for N ¼ 20there were on average �100 maxima.

Although control in the sudden regime operates with just oneparameter b, the significance of even this single control variableis evident from statistical results generated with random choicesof l and b. For example, the distribution of P1!10 values is shownin Figure 1a for N ¼ 10, from ten thousand random dipole matrices,generated as described above and evaluated over the rangeb 2 ½0;1000�. Figure 1b shows the distribution arising from search-ing for the maximum value of P1!10ðbÞ for each dipole matrixacross this range. Optimizing b dramatically alters the distribution,shifting the mean P1!10 value from 0.09 to 0.52. The same optimi-zation was performed for systems of different dimensions, rangingfrom N ¼ 3 to N ¼ 30 (not shown). The distributions of both thenon-optimized and optimized cases across this range of dimen-sions are qualitatively similar to the distributions in Figure 1 forN ¼ 10. In agreement with previous results [8], the mean of thenon-optimized distributions was approximately N�1. The opti-mized distributions appeared to qualitatively approach a normaldistribution. Figure 2 shows that both the mean and the standarddeviation of the optimized distributions decreases with N. Forlow dimensional systems, there is a strong likelihood of attainingvery high yields.

0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

7

8

9

Freq

uenc

y x

106

P1→ 10

a

Figure 1. Distributions of P1!10 generated from ten thousand random N ¼ 10 dimensionwithout optimization. (b) The distribution of the maximum value of P1!10 after optimoptimization over b in (b).

0 5 10 10

0.2

0.4

0.6

0.8

1

Dime

P 1→

N

Figure 2. Mean yield and standard deviation of P1!N optimized with respect to b for syschosen and optimized over b in the same way as those in Figure 1b.

The dependence of Pi!f on a single parameter, b, allows for greatflexibility in choosing a particular field, �ðtÞ, as it is easy to satisfythe constraint

R T0 �ðtÞdt ¼ b. This freedom facilitates seeking multi-

objective quantum control with additional constraints imposed on�ðtÞ. For example, besides maximizing Pi!f at time T, one can max-imize Pi!j, for some other state j, with b0 ¼

R T 0

0 �ðtÞdt at an earliertime T 0 < T. This corresponds to finding an electric field that alsosatisfies b ¼ b0 þ

R TT 0 �ðtÞdt to assure that Pi!f is maximized at time

T. Optimization with respect to the dynamics throughout the inter-val t 2 ½0; T� may also be accomplished. As an illustration, considerminimizing

J1 ¼1T

Z T

0Pi!jðtÞdt ¼ 1

T

Z T

0jhjjeıl�t jiij2 dt; j – f ; ð5Þ

in addition to maximizing Pi!f at final time T. Eq. (5) aims to steerthe dynamics away from state j throughout the control period0 < t 6 T. As an illustration, this optimization was performed onan N ¼ 4 dimensional system, using a stochastic search algorithm[9]. The algorithm minimized J1 for the transition probability P1!3

while maintaining P1!4 ¼ 0:94 at a predetermined value of b atT ¼ 7. The entries of the dipole matrix of the four-dimensionalsystem were chosen as follows: ljj ¼ 0; j ¼ 1; . . . ;4, l12 ¼ 4:0,

0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

250

300

Freq

uenc

y

P1→ 10

b

al Hermitian l matrices. (a) The distribution of P1!10 for all values of b 2 ½0;1000�izing over b 2 ½0;1000�. The mean yield increases from 0.09 in (a) to 0.52 after

5 20 25 30nsion (N)

tems of dimension N ¼ 3—30. For each N value, 5000 random dipole matrices were

M. Klein et al. / Chemical Physics Letters 499 (2010) 161–163 163

l13 ¼ 1:75, l1;4 ¼ 2:0, l23 ¼ 2:5, l24 ¼ 1:5, l34 ¼ 4:0. J1 decreasedfrom 0.243 to 0.017 through optimization of �ðtÞ. Moreover, thepopulation of state 3 with the optimized field remained below0.15 over the entire duration out to time T. This type of competitiveoptimization may be relevant for applications seeking to avoid par-ticular deleterious pathways.

As Figure 1 shows, statistically, it is very unlikely that perfectcontrol can be achieved for a given dipole operator even with opti-mization of b, especially for larger values of N. However, recentwork has explored the possibility of using the time-independentHamiltonian structure as controls, (e.g., by engineering moleculesto have desired properties) [10]. For control in the sudden regime,this would entail manipulation of the dipole, l, while the dynamicsremain sudden. With control over the entries of l and the value ofb, a simple analysis [11] shows that there is sufficient freedom toreach Pi!f ¼ 1. As an illustration of what might be achieved, con-sider maximizing Pi!f while enhancing robustness to control noise,through minimization of jd2Pi!f =db2j. This optimization was per-formed using a stochastic search algorithm [9] to seek b and lfor a system with N ¼ 3 and the goal being P1!3. Many excellentsolutions were found, with the best achieving Pi!f ¼ 1:0 up toten decimal places, along with jd2Pi!f =db2j < 10�6. This result sug-gests the intriguing possibility of designing materials that achievenear-perfect control while being extremely robust to control noise.

Current ultra-fast laser technology routinely operates in thefemtosecond regime with recent experiments pushing into theattosecond domain. These capabilities open up quantum controlin the sudden limit for dynamics involving rotational, vibrationaland even possibly electronic degrees of freedom for many systems.Molecular alignment experiments are already performed in thesudden regime [6,7] and rotational population transfer of heavymolecules using femtosecond laser pulses would satisfy the sud-den criterion. Another candidate is manipulation of high lying elec-tronic Rydberg states [5,12], which also have the advantage of largetransition dipoles allowing for control by low-intensity fields.

Although propogating optimal electric fields do not have a DCcomponent, there are various ways of creating the desired b valuefor sudden control. For example, a carefully timed pump–probetechnique could be utilized to interrogate Pi!f during the timewhen the field is on so as to create the desired b value. Another ap-proach is to use sub-cycle field shaping to divide the AC field intoshaped DC pulse components with relatively long delays betweenthem. Recent studies [4,5] established the possibility of achievingcontrol using sub-cycle pulses of ps duration, and current technol-ogy can generate sub-cycle fs pulses with ps delays between pulses

[13]. These ultra-short DC pulses can be used to achieve rotationaland possibly vibrational control. Some experiments utilize Ramanexcitation to accomplish population transfer via the DC componentof the electric field envelope [14]. There are many technical issuesto consider with these schemes and possibly others for achievingthe desired b value. Although b is just a single variable, lasers withpulse shaping naturally operate with many variables for adjust-ment suggesting that adaptive feedback control of some formmay be beneficial to find viable solutions. The freedom inherentwith pulse shaping could be exploited to search for the best b valuealong with targeting other objectives.

In summary, ultra-fast lasers enable experimental realization ofcontrol in the sudden regime of many physical phenomena. Trade-offs emerge in this regime between control fidelity and experimen-tal flexibility. There is also the prospect of engineering the systemdipole in cases where the goal is to reach a particular state withless concern about the nature of the physical system used thanthe ability to switch from one state to another. Quantum controlin the sudden regime offers an attractive simple domain of opera-tion for a growing range of experimental systems and objectives aslaser technology advances toward ever shorter pulse durations.

Acknowledgement

The authors wish to acknowledge support from ARO and DOE.

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