Measurement-assisted quantum dynamics control of 5-level system using intense CW-laser fields

Chemical Physics Letters 428 (2006) 457–460 www.elsevier.com/locate/cplett

Measurement-assisted quantum dynamics control of 5-level system using intense CW-laser fields M. Sugawara

*

Department of Fundamental Science and Technology, Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan Received 16 May 2006; in final form 21 July 2006 Available online 29 July 2006

Abstract A new quantum control scheme using intense CW-laser fields together with quantum measurement is applied to Kobrak–Rice 5-level (KR5) quantum system. Population dynamics of the KR5 system under frequent measurements is clarified by effective Liouville equation. It is found that one can control the stationary population distribution that appears in t ! 1 limit by varying laser intensities.  2006 Elsevier B.V. All rights reserved.

1. Introduction The recent progress of laser technology has stimulated various proposals of quantum control schemes, such as p-pulse, optimal control theory, local control theory, and stimulated Raman adiabatic passage (STIRAP) [1–8]. In those control procedures, quantum dynamics is basically manipulated through coherent interactions between the laser fields and molecules. On the other hand, it is known that frequent quantum measurements can hinder the coherent dynamics driven by the laser fields, which is called quantum Zeno effect (QZE) [9–12]. Since the measurement process plays an important role in the quantum dynamics, the possibility of utilizing it from the viewpoint of quantum control needs full exploration. Gong and Rice have recently proposed the STIRAP-based measurementassisted control scheme on the Kobrak–Rice 5-level (KR5) system [7,8,13]. We have developed the effective Liouville equation (ELE), which determines the population dynamics driven by the intense CW-laser fields under the frequent measurements [14]. The purpose of the present Letter is to propose an alternative scheme of population control for the KR5 system *

Fax: +81 45 566 1697. E-mail address: [email protected].

0009-2614/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.07.069

utilizing the quantum measurements. To do so, we apply the ELE to the KR5 model system and seek the possibility of a measurement-assisted control. This Letter is organized as follows: In Section 2, we briefly review the ELE-based analysis on the quantum dynamics under the frequent measurements. In Section 3, we clarify the laser intensity dependence of the population dynamics of the KR5 system by applying the ELE. Summary is given in Section 4. 2. Theory We first consider the KR5 system shown in Fig. 1 [7,8]. Five CW-lasers are tuned to be resonant to the corresponding transition frequencies. The total Hamiltonian matrix under the dressed state picture is given as [7,8,15] 1 0 0 X12 0 0 0 C B X23 X24 0 C B X12 0 C B H¼B ð1Þ X23 0 0 X35 C C: B0 C B X24 0 0 X45 A @0 0 0 X35 X45 0 Here, the basis set {jiæ} (i = 1, 2, . . ., 5) consists of the direct product of the field number state and the molecular eigenstate satisfying the energy conservation, while Xij denotes

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M. Sugawara / Chemical Physics Letters 428 (2006) 457–460

where t = Ns. In the case that s is very small, the system dynamics obeys the ELE [14]

5 Ω 45

Ω 35

c_ ðtÞ ¼ iLeff ðsÞ  cðtÞ;

4 3 Ω 24

Ω 23

2 Ω12

1 Fig. 1. Schematic diagram of the Kobrak–Rice 5-level model. Each CWlaser is tuned to the corresponding transition frequency.

the system–field interaction with respect to the transition between jiæ and jjæ. In order to treat the system dynamics within the double space (DS) representation [16,17], we define the 25 · 25 Liouvillian matrix L as LIJ ¼ Lij;kl  ðHik djl  Hjl dik Þ= h:

ð2Þ

Here, I corresponds to the combination of subscripts (i, j). Under the DS representation, the system state is given by the vector c(t) = (q11(t), q12(t), . . ., q55(t)), where qij(t) is the density matrix element. Using Eq. (2), time evolution of the system vector c(t) is formally given as cðtÞ ¼ eiLt  cð0Þ:

ð3Þ

b In the Next, we define the measurement super-operator O. case that the states j1æ and j2æ are the current measurement b onto the 5 · 5 density matrix can target, the operation of O be defined as [14] 1 0 q11 q12 q13 q14 q15 C B B q21 q22 q23 q24 q25 C C B b ¼O b B q31 q32 q33 q34 q35 C Oq C B C B @ q41 q42 q43 q44 q45 A q51 q52 q53 q54 q55 1 0 q11 0 0 0 0 C B q22 0 0 0 C B0 C B ¼B ð4Þ 0 q33 q34 q35 C C: B0 C B 0 q43 q44 q45 A @0 0 0 q53 q54 q55 Here, we call the space that j1æ and j2æ span P-space, and its complementary space Q. Note that the off-diagonal elements related to the P-space are eliminated by the operab One should also note that the above operation tion of O. can be described by the matrix-vector operation in the DS as O Æ c. Here, O is the diagonal matrix, whose diagonal elements corresponding to the coherence related to the Pspace are zero while others are unity. Now, we consider the system dynamics under N-times repeated measurements with interval s. Overall time evolution is given by N

cðtÞ ¼ ðO  eiLs  OÞ  cð0Þ;

ð5Þ

ð6Þ

where Leff(s) ” O Æ L Æ O  iG(s). Here, we introduce the matrix G(s) ” O Æ L Æ L Æ Os, which plays an important role on the population kinetics under the frequent measurements. In the limit of s ! 0, incoherent population transfer due to G(s) disappears and the dynamics within the P-space becomes completely frozen [12,14], which corresponds to the QZE. As s increases, the incoherent population dynamics within the P-space and the population transfer between P and Q spaces appear in an early time stage. In many cases, however, the system settles in the stationary distribution in the limit of t ! 1. This stationary feature of the population dynamics can be clarified by the eigenvector analysis as follows. Note that the eigenvector c of Leff(s) with the eigenvalue zero satisfies Leff ðsÞ  c ¼ 0, which implies that c does not evolve as long as Eq. (6) is valid. In the case that other eigenvectors have finite lifetimes, the stationary state with respect to an arbitrary initial condition c(0) is given by X lim cðtÞ ¼ ðcn  cð0ÞÞcn ; ð7Þ t!1

n

where cn denotes the nth eigenvector of Leff(s) with the eigenvalue zero. Note that the stationary distribution given by Eq. (7) depends on the laser field intensities because Leff(s) contains Xij. The result of Eq. (7) also depends on the measurement target, i.e., how the P-space or the O matrix is chosen. We aim to utilize these dependencies for controlling the population dynamics in the following section. 3. Results We focus on the case where the state j2æ is frequently measured, although many other measurement conditions are possible. The initial state is fixed to j1æ throughout the present study. Shown in Fig. 2 is the population dynamics under the frequent measurements on j2æ with

Fig. 2. Population dynamics under frequent measurements on j2æ. The exact population dynamics calculated by Eq. (5) for j1æ, j2æ and j5æ are shown as gray filled squares, gray filled circles and light gray filled circles, whereas those of j3æ and j4æ are shown as light gray triangles. The solid lines shown together denote the population dynamics calculated by ELE.

M. Sugawara / Chemical Physics Letters 428 (2006) 457–460

the measurement interval of s = p/15, whereas the laser intensity parameters are equally taken to be X12 = X23 = X24 = X35 = X45 = 1. Here, units for Xij and s are taken to be  hx and x1, respectively, where x corresponds to the transition frequency between j1æ and j2æ. The solid lines shown in Fig. 2 are the population dynamics calculated by the ELE. This diagram demonstrates that ELE is effective for the present measurement condition, s = p/15, since the calculated lines agree well with the exact values obtained numerically. Note that there is very little oscillatory population dynamics in Fig. 2; this indicates that the coherent time evolution is suppressed by the frequent quantum measurements. As is expected from the formulation in Section 2, the system settles in the stationary population distribution in the limit of t ! 1. In the present case, the stationary population on jiæ {Pi} (i = 1, 2, . . ., 5), are P1 = P2 = P5 = 0.25, P3 = P4 = 0.125. These values can be theoretically predicted by Eq. (7), as given by   ð8Þ P 1 ¼ 4A4 þ B4 X412 =D;  2  2 2 2 2 ð9Þ P 2 ¼ B X12 A þ B X12 =D;  2 2 2  2 2 2 2 P 3 ¼ X12 B X12 X35 þ A ðB þ 4X45 Þ =D; ð10Þ  2 2 2  2 2 2 2 ð11Þ P 4 ¼ X12 B X12 X45 þ A ðB þ 4X35 Þ =D;  2  2 2 2 2 P 5 ¼ B X12 A þ B X12 =D; ð12Þ where A ¼ X24 X35  X23 X45 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ X235 þ X245 ;

ð13Þ ð14Þ

2

D ¼ 4ðA2 þ B2 X212 Þ :

ð15Þ

Here, we consider two cases, B  A and A  B X212 . 2 2 2 In the case of B X12  A , Eqs. (8)–(12) can be simplified as P1 = P2 = P5  1/4, P 3  X235 =ð4B2 Þ, and P 4  X245 = ð4B2 Þ. Note that this is the case of Fig. 2, X12 = X23 = X24 = X35 = X45 = 1, which corresponds to A = 0. Under such condition in which A2 can be neglected compared to B2X12, the branching ratio between j3æ and j4æ in the final distribution R34, defined as R34 ” P3/P4, is simply given as X235 =X245 . Thus, one can control the selective excitation by changing these two parameters. One of the possible settings for achieving such selectivity is X12, X35, X45  X23, X24, which realizes the condition B2X12  A2. In the second case A2  B2 X212 , we obtain the final distribution,  P1  1; P 2 ¼ P5 ¼ B2 X212 =A2 ; P 3 ¼ X212 X235 þ 5X245 ; P 4 ¼  X212 X245 þ 5X235 . Under this condition, substantial yields for excited states in the final distribution cannot be expected since P1  1. However, if one needs selective excitation between j3æ and j4æ while suppressing the excitation onto j2æ and j5æ, this condition becomes preferable since P2 and P5 takes small  values. In this  case, thebranching ratio is given as R34 ¼ X235 þ 5X245 = X245 þ 5X235 . Thus, the laser condition X35 = 0 leads to the maximum value R34 = 5 while we obtain the opposite selectivity R34 = 1/5 with X45 = 0. In order to explore the possibility for controlling the population in t ! 1, we have calculated its laser intensity 2

X212

2

2

2

459

dependence. Here, the laser intensities are parameterized as X12 = 1, X23 = 2c, X24 = 2(1  c), X35 = 2d, X45 = 2(1  d). Shown in Fig. 3 is the c and d-dependence of the stationary distribution. Note that the dependence of j5æ is identical to that of j2æ shown in Fig. 3b, while those of j3æ and j4æ are symmetrical to each other with respect to the point c = d = 0.5 as shown in Fig. 3c,d. The results imply that it is difficult to realize 100% selective population of a single state. As is discussed above, however, one can utilize the dependence shown in Fig. 3 for controlling the branching ratio R34. A possible approach is to use the c = d line, which corresponds to A = 0, i.e. the case of B2X12  A. For maximizing R34, one should choose pffiffiffi the laser condition c = d = 1, which brings P 3 ¼ 1=4 2; P 4 ¼ 0, while c = d = 0 is pfavorable for minimizing R34, i.e., P 3 ¼ 0; ffiffiffi P 4 ¼ 1=4 2. These two cases, shown in Fig. 3 as white circles, are intuitively understandable because they correspond to the conditions that the unfavorable transitions are simply closed. On the contrary, if one aims to selectively populate the state j3æ (or j4æ) whereas the populations on j2æ and j5æ are relatively suppressed, one should choose the condition c = 0.5 and d  0 (or c = 0.5 and d  1), which is indicated as triangle in Fig. 3. Note that those two conditions are hardly obtained intuitively. We have calculated the stationary distributions for more intense laser parameters, X23, X24, X35, X45, with respect to X12, keeping the relative conditions determined by c and d. As for the parameters within Xij/X12  10, ({ij} = 23, 24, 35, 45), the general feature including the maximum yield of each state is unchanged, but the landscape of Fig. 3 looks slightly different. Generally speaking, all the ridges become steeper as the laser intensities increase. Finally, we emphasize the differences between the present approach and that of Ref. [13]. One of the advantages of the use of the ELE scheme is that one can determine the stationary distribution through the eigenvalue analysis of the effective Liouvillian. The obtained stationary distributions is realized by CW-laser irradiation under repeated quantum measurements, whereas s can be taken arbitrarily as long as it is so small that the ELE stands. This feature can facilitate the experimental preparation. In contrast, the control approach of Ref. [13] utilizes non-adiabatic transition, which is controlled sophisticatedly by changing the measurement strength. Thus, accurate pulse shaping and precise control of the measurement frequency are required to experimentally realize such a control condition. Another significant difference is that the control mechanism is clearly stated in analytical expressions in Ref. [13], whereas that of the present approach is implicitly included in the process of the eigenvalue analysis. Therefore, it is difficult to clarify the details of the control mechanism unless the analytical expressions of the eigenvalues and the eigenstates are at hand, as we have shown for the KR5 system. It should be noted, however, that the present approach should be applicable to any arbitrary system, despite that the designing procedure of the control condition has to be a trial and error approach with numerical calculations.

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M. Sugawara / Chemical Physics Letters 428 (2006) 457–460

a

b

c

d

Fig. 3. Laser intensity dependence of the stationary distribution for the measurement on j2æ. See text for the details of parameters c and d. (a), (c) and (d) denote the c and d dependencies of the stationary distribution of j1æ, j3æ and j4æ, respectively, while those of j2æ and j5æ, which are identical, are shown in (b).

4. Summary

References

We have clarified the population dynamics driven by the CW-lasers irradiated to the KR5 system under the frequent measurements condition. It is found that the stationary distribution obtained in the limit of t ! 1 can be controlled by varying the intensities of lasers applied, although the realizable distribution is restricted to the results shown in Fig. 3. We note that the crucial control input to the system is not only the laser intensity but also the measurement target state. The stationary distribution depends on the initial condition as well. A combination of other coherent control schemes and the present method is also possible. Thus, there still remain various cases that one needs to investigate in future studies.

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Acknowledgment This work was supported, in part, by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.