Sarin A. Deshpande and Gregory S. Ezra- Quantum state reconstruction for rigid rotors

8/3/2019 Sarin A. Deshpande and Gregory S. Ezra- Quantum state reconstruction for rigid rotors

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Quantum state reconstruction for rigid rotors

Sarin A. Deshpande, Gregory S. Ezra *

Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, NY 14853, United States

Received 26 December 2006; in final form 23 March 2007Available online 19 April 2007

Abstract

We describe a quantum state reconstruction scheme for dipolar rigid rotors based on determination of the expectation value of themolecular orientation. A key feature is the use of half-cycle pulses to excite the rotor prior to the orientation measurement. The set ofexpectation values obtained by varying the intensity and polarization of the laser and the time interval between excitation and measure-ment can be inverted directly to yield the rotor density operator. When the density operator corresponds to the admixture of relativelyfew rotor states, our procedure successfully reconstructs both pure and mixed states. 2007 Elsevier B.V. All rights reserved.

1. Introduction

The general question of inferring or reconstructing the

quantum state of a system from measurement of expecta-tion values of observables was raised early on by Pauli[14]. There has been renewed recent interest in the prob-lem, particularly in the development of quantum tomogra-phy (see, for example, Refs. [57]).

In this Letter we consider the problem of reconstructionof the quantum state (density operator) of a dipolar rigidrotor molecule using as input the expectation values of rel-evant observables. Any density operator of finite rank Ncan be written as a linear combination of N2 Hermitianoperators that form a basis in an operator inner-productspace [8]. The expectation values of the N2 basis operators

then determine the density operator uniquely. From thepractical point of view, the problem is that the relevantbasis operators will in general correspond to complicatedobservables whose physical measurement may be difficultto perform. In the case of a rotor, for example, the opera-tor basis will consist of irreducible spherical tensor opera-

tors of high rank [9,10]. A previously suggested approachfor reconstruction of rotor states implicitly requires themeasurement of tensor operators of high rank [11] (see also

Ref. [12]).Here we describe a reconstruction scheme for the rigid

rotor that relies solely on determination of the expectationvalue of a low rank tensor, in this case the molecular orien-tation. Previous work on the determination of spin quan-tum states has shown that a set of 4F+ 1 SternGerlachmeasurements with respect to suitably chosen quantizationaxes serve to define the quantum state of a spin F [13,2,14](see also [15]). A key feature of our approach is the use ofone or more ultrashort half-cycle pulses (HCPs) [1620] toexcite the rotor prior to the time at which the measurementis made. The action of the pulse on the rotor is assumed

known; we shall use the sudden approximation for simplic-ity [21,22]. By varying the intensity and polarization of thelaser, and the time interval between excitation and mea-surement, we obtain a set of expectation values that canbe inverted directly to yield the rotor density operator. Inthe case that the density operator corresponds to theadmixture of relatively few rotor states, our procedure isshown to successfully reconstruct both pure and mixeddensity operators. Possible extensions of our approach tomore complicated cases are discussed in the Summarysection.

0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2007.04.049

* Corresponding author. Fax: +1 607 255 4137.E-mail address: [email protected] (G.S. Ezra).

www.elsevier.com/locate/cplett

Chemical Physics Letters 440 (2007) 341347
mailto:[email protected]:[email protected]


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2. Quantum state reconstruction via matrix inversion

2.1. General principles

A general state of an N-state quantum system isdescribed by the N N density matrix qij, i,j= 1, . . . , N,

where, if q is normalized (tr[q] = 1), a set of (N2

1) realparameters characterize the matrix q [8]. The density oper-ator q can in general be expressed as a linear combinationof (N2 1) operators that form a basis in a space of oper-ators [8], and the coefficients in the expansion can then bedetermined by determining expectation values (traces) ofthe N2 1 operators in the state q [8].

For a problem involving rotor states a natural operatorbasis consists of irreducible spherical tensor operators con-structed by coupling the ket and bra sides of the densityoperator [9,10]. Complete determination of the densityoperator then in general requires the evaluation of expecta-tion values of tensor operators of large rank. Essentially

such an approach has been proposed by Mouritzen andMolmer, who have shown that the density operator of arigid rotor can in principle be determined by measurementof the orientational distribution function of the rotor axis[11]. Resolution of fine details of the angular distributionthen corresponds to measurement of the expectation valuesof high-order tensor operators.

It is natural to ask whether it is possible to determine thedensity operator for a rigid rotor using a set of measure-ments of simple observables (i.e., tensor operators of lowrank), such as orientation (expectation value of cos h,where h is the angle between the rotor dipole axis and the

lab-fixed z-direction). The measurements are to be madeon an ensemble of identically prepared systems describedby the density operator q.

Suppose the system, initially in state q, is subjected to anultrashort pulse at t = 0, which transforms the densitymatrix essentially instantaneously from q ! q. For animpulsive perturbation of intensity Ip, the transformed den-sity operator is

qp bVpqbVpy; 1aor, in some orthonormal basis

q

p

ij

Xi0j0 Vpii0 qi0j0 Vpy

j0j;

1b

where the transformation matrix V(p)ij is assumed known(see below). The state at time t = tn after the pulse is then

qp; n U0tnbVpqbVpyU0tny 2where bU0tn is the operator representing free propagationfor time tn. If observable X is measured at time tn, theexpectation value X(k) X(p, n) is

hXik hXip; n trbXbU0tnbVpqbVpy bU0tny 3aX

ij

bVpy

bU0tny

bX

bU0tn

bVpijqji

3b

X

ij

Mkij~qij 3c

Mk ~q 3dwhere in the last line we have written the expectation value X(k) as the inner-product of the N2-dimensional vectors

M(k) and ~q. If we assume that the density operator is effec-tively a finite-dimensional N N matrix, then by takingvarious combinations of intensity Ik and post-pulse propa-gation times tn, we can in principle obtain a set of linearequations for the initial state density matrix elements qij,which can be solved to determine q. If q is effectively anN N matrix, then taking N2 combinations (rather thanN2 1, as we shall not explicitly impose a normalizationcondition) of pulse intensity Ip and observation time tnwe obtain an N2 N2 measurement supermatrix

Mk; Mkij; 4

where k (p, n) = 1, . . . , N2

, (ij) = 1, . . . , N2

. The matrixM is independent of the initial state q; ifM is invertible(full rank), then we can solve the set of N2 linear equations

hXik XN21Mk;~q k 1; . . . ;N2 5

directly to determine the matrix elements qij. Note that inthe practical implementation of the theory it is not neces-sary to impose any constraints on the matrix elements qij;both normalization and pure/mixed character of the den-sity operator are completely determined by the algorithm.

We demonstrate below that it is possible to obtaininvertible measurement supermatrices for dipolar rigidrotors subject to HCP followed by measurement of orien-tation. In that case, we can invert a set of Eq. 5 to recon-struct the quantum state q, provided the effective size ofq is not too large.

2.2. Quantum state reconstruction for dipolar rigid rotors

In our approach we imagine the rigid rotor to be excitedusing an ultrashort HCP at t = 0 [1720], and expectationvalues of the orientation X of the molecule evaluated atvarious times. Recent experiments have shown the feasibil-ity of measuring the orientation cosh of dipolar moleculeslike OCS on femtosecond timescales [23,24]. In general weneed to consider a set of expectation values obtained usingdifferent combinations of laser polarization vector n, laserintensity Ip, and measurement time tn.

We consider excitation pulses for which the pulse dura-tion Tp is much shorter than the rotational period Trot [1720], so that the interaction can be treated using the suddenapproximation. This approximation is not essential, butprovides an analytical model for the rotor excitationdynamics [21]. If the sudden approximation is not applica-ble, it is necessary to integrate the quantum mechanical

equations of motion for the time-dependent Hamiltonian

342 S.A. Deshpande, G.S. Ezra / Chemical Physics Letters 440 (2007) 341347


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Eq. (6) over the duration of the pulse. The molecule-laserHamiltonian isbHt bH0 bH0t 6with free rotor Hamiltonian

bH0 B^J2; 7a(rotational constant B) and molecule-laser interactionbH0t E0d ngt 7bwhere n is the laser polarization vector, dis the rotor dipoleoperator, and the magnitude of the laser electric field isE(t) = E0g(t). We take the pulse profile g(t) to be a cosinelobe of width Tp/2, where Tp . 1 ps is the period of the fullpulse cycle, corresponding to a frequency mp = 1 THz [16].In the sudden approximation, matrix elements betweenfree-rotor states jjm and jj0m 0 of the complete propagatorover the duration of the pulse are approximated as follows[21]:

j0m0j exp ih

ZTp=4Tp=4

dtbHt" #jjm* +

j0m0j exp ih

ZTp=4Tp=4

dtbH0t" #jjm* + 8a j0m0j exp ibd n

h ijjm

D E8b

where the parameter b is

b dE0h Z

Tp=4

Tp=4dt gt dE0

h ZTp=4

Tp=4dtcos2pmpt 2dE0

hmp

9and d is a unit vector along the molecular dipole axis.

The matrix V that transforms q to q in the free-rotorbasis is then

hj0m0jVjjmi hj0m0j expibd njjmi 10a hj0m0jbR/n; hn; 0 expibd zbRy/n; hn; 0jjmi 10b hj0m0jbR/n; hn; 0 expib cos hbRy/n; hn; 0jjmi 10c

where bR is the unitary rotation operator that rotates thelab-fixed z-axis into the laser polarization axis n [10]. Therotation operator depends on only two Euler angles(h

n, /

n), the spherical polar angles defining the direction

of the polarization axis n.If the rotor density operator is effectively an N N

matrix q, N2 measurements must be made to define theN2 N2 measurement supermatrix M. For a tractablereconstruction problem, the effective size of the state spaceN should not be too large. Moreover, for theoretical anal-ysis the pulse at t = 0 should not excite too large a numberof excited rotor states, as we wish to ensure that the quan-tum mechanical calculation of the excitation process andsubsequent time evolution is converged. The pulse shouldnot be too weak, however, as this can lead to numerical dif-

ficulties inverting M to obtain q (see below).

For experimentally available HCP, Tp/2 . 500 fs [16],so that for most molecules (except hydrides such as HFand HCl) the condition Tp/2 ( Trot is satisfied, and thesudden approximation provides a reasonably accuratedescription of the excitation process. Attainable maximumfield strengths are of order E0 $ 1.5 107 V/m [16], so thatfor a molecule with a large dipole moment, such as LiCl(d= 7.13 D [25]), the corresponding value of b is b . 1.5.In the calculations presented below we implement ourscheme for laser intensities corresponding to b values inthe range 0 < b < bmax = 1.5. Reducing the value of bmaxleads to a larger condition number forM, and possibly lessaccurate reconstruction of the density matrix (see below).For molecules having appreciably smaller dipole momentsthan LiCl, either larger HCP peak field strengths E0 mustbe used to obtain the same value of bmax, or the schemecan be extended to allow for a sequence of HCPs actingon the molecule before measurement of orientation (seeSection 4).

3. Theoretical implementation of reconstruction scheme

3.1. Mixture of j states at constant m = 0

Consider the case of a mixture of rotor states with fixedm, m = 0 say. For a z-polarized laser field, we have the rel-atively simple expression for the matrix elements (10) [21]

Vjj0 hj00jVjj0i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij 1j0 1

p Xliljlbhj0j00jl0i2; 11

where j1, m1,j2, m2jj3, m3 are ClebschGordan (vector cou-pling) coefficients [10] and jl(b) are spherical Bessel func-tions, with intensity parameter b being related to thelaser field strength by Eq. (9).

To illustrate our reconstruction method, an initial den-sity operator q is obtained as follows. A rotor in theground state q0 = j0, 00, 0j is subject to a z-polarizedHCP with b = 1.0. This excitation process defines a pureinitial state q bVq0bVy with m = 0 for subsequent analysis.To obtain a set of mixed density operators for analysis, theoff-diagonal elements ofq in the free-rotor basis are multi-plied by an attenuation factor cm, where c

1ffiffi2p and m =1, . . . ,4. The set of five density operators q(m), m =

0,1, . . . , 4 is then used to compute experimental cos hvalues for a combination of different times (tn) and intensi-ties Ip(b), (0 6 b 6 bmax = 1.5), and the quantum statedetermination scheme applied to this numerically gener-ated data for each of the initial states q(m).

The size of the measurement supermatrix M is deter-mined by the maximum assumed value of the rotor quan-tum number, jmax. For each value of jmax N

0 1 weobtain an N

02 N02 matrix MN0 by taking suitable com-

binations of intensity and measurement times; if the matrixM

N0

can be inverted, we obtain the (transpose of) the

reconstructed density matrix

S.A. Deshpande, G.S. Ezra / Chemical Physics Letters 440 (2007) 341347 343


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~qrecN0 MN01 XN0: 12where X(N

0

) represents the N02-dimensional vector of mea-

sured orientations, cos h.We have explored numerically several excitation/mea-

surement schemes. For example, matricesM were obtainedusing N

02 values of tn for fixed intensity parameter b. Also,

calculations have been carried out with the observationtime tn fixed and N

02 different b values. In both cases it isfound that accurate inversion of the measurement matrixis only possible using unphysically large values of bmax.These excitation/measurement schemes are however foundto work for physically more reasonable values of b whenmore than one excitation pulse is used (cf. Section 4).

For the calculations reported here, we used a set of N02

randomly chosen pairs of values {tk, bk} to obtain theobservation matrix M, taking observation times in therange 0 6 tn 6 tmax = Trot, and laser intensities such that0 6 b 6 bmax = 1.5.

The convergence of the reconstructed density operatorqrec(N

0

) to the original q can be studied in several ways.In Fig. 1 we plot the trace tr[qrec(N

0

)] as a function of N0

;this plot indicates that the reconstruction is essentially con-verged at jmax = 4. (It should be noted that convergence ofall calculations is also checked with respect to the total

rotor basis size Ntot; for the present example, use of a basisof rotor states with Ntot = 13 ensures that all expectationvalues are converged with respect to basis size.) Fig. 2shows tr[qrec(N

0

)2] as a function of N0

. For the mixed states,tr[qrec

2] < 1. Again we see convergence by jmax = 4. Finally,Fig. 3 shows convergence of the norm of the difference

kq qrecN0k2 XN0i;j1

jq qrecN0ijj2 13

with increasing N0.We now consider in more detail the dependence of the

accuracy of the reconstruction procedure on the maximumintensity parameter bmax for jmax = 6. In Fig. 4a we showthe condition number of the measurement matrix M(defined as the square root of the absolute magnitude ofthe ratio of the largest to the smallest eigenvalue ofM

yM

fM

M) as a function ofbmax. Although the matrix

M is quite poorly conditioned even for bmax = 1.5,

straightforward numerical inversion succeeds in reproduc-ing the density operator to acceptable accuracy for bmaxvalues down to .0.05. This is seen in Fig. 4b, which plotslog(iq qreci2) versus log bmax. For smaller values of bmax,use of the pseudoinverse [26] yields a reasonably accuratereconstructed density matrix even for the poorly condi-

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

2 3 4 5 6 7

tr[rec

() (N)]

N

0

1

2

3

4

Fig. 1. Trace trqmrecN0 of the reconstructed density operator qmrecN0 as afunction of the assumed density operator rank N

0

, for m = 04. Themaximum intensity parameter bmax = 1.5. This plot indicates that the

reconstruction is essentially converged at jmax = N0

1 = 4.

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

2 3 4 5 6 7

tr[rec

() (N)2]

N

0

1

2

3

4

Fig. 2. Trace trqmrecN02 of the square of the reconstructed density matrixqmrecN0 as a function of the assumed density operator rank N

0

, for m = 04. The maximum intensity parameter bmax = 1.5. For mixed states, mP 1,

trqm2rec < 1. Again we see convergence by jmax = 4.



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tioned measurement matrices obtained at bmax values aslow as 0.03. (For use of the pseudoinverse in density matrixreconstruction, see also Ref. [14].)

It is shown in Section 4 that use of a multi-pulse excita-tion scheme serves to stabilize the inversion procedure.

3.2. Mixture of m states at constant j

For states that are mixtures of rotor states with fixed j,

possible m values are m = +j,j 1, . . . ,j, so that thedimension of the state space is fixed in this case,N= 2j+ 1. The measurement supermatrix M is then a(2j+ 1)2 (2j+ 1)2 matrix.

Expressions for the matrix elements V (10a) are morecomplicated than in the previous case with m = 0. We have

Vj1m1 ;jm hj1m1jVjjmi 14a

X

m

1mei/nmm1dj1m1 mhn

2

djmm

hn

2

X

l

iljlbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

j1 1j 1p

hj1 mjmjl0ihj10j0jl0i 14b

where we have used the fact that hj1 m1jmjl0i dm1 m.The symmetry property of the rotation matrices [10]

djm1mh

2

1m1mdjm1m

h

2

15

implies certain relations between elements of theMmatrix,

which lead to difficulties in the implementation of thereconstruction method. For example, for any laser polari-zation vector n and for bX cos h it can be shown thatMjm;jm Mjm;jm: 16

The measurement matrix M therefore does not have fullrank (it has two identical columns) and cannot be invertedto obtain q. In order to get around this problem we con-sider expectation values of two observables, bX1 zcos h and bX2 x sin h cos /, corresponding to measure-ment of the rotor orientation along the z- or the x-axis,respectively. If the observation set includes measurements

of orientation along both the x- and z-axes, then an invert-ible matrix M can be obtained.

As an example, consider the case j= 2. As before weconstruct a pure state density operator and obtain a setof mixed states by multiplying off-diagonal density matrixelements by an attenuation factor. The initial pure stateis taken to be the rotor coherent state [27]

jj; abi Xj

mjjj; micm

Xjmj

jj; miDjmja; b; 0 17

specified by the Euler angles a; b; 0. Matrix elements ofthe corresponding (2j+ 1) (2j+ 1) density matrix areqmm0 cmcm0 . As previously, the pure state density matrixq was used to generate a set of mixed state density opera-tors q(m) by multiplying off-diagonal matrix elements bythe factor 1=

ffiffiffi2

p m, m = 1, . . . , 5.

Again, each of the matrices q(m) is used to obtain exper-imental data for a combination of (2j+ 1)2 = 25 differentlaser polarization directions, intensities, observation times,and orientation axes. The calculations were performed onthe density matrix derived from the coherent state witha p=ffiffiffi2p and b p=ffiffiffi5p . We chose to vary the time ofobservation tn, the laser polarization angle hn (the angle/n

is held fixed, /n

p=ffiffiffi3p

), and the intensity. The recon-struction calculation was carried out in two different ways:first, 25 random values for time and polarization directionwere chosen while keeping the intensity parameter b fixed,b = 1.41. Second, five observation times and five pairs ofpolarization angle and laser intensity were chosen, and 25expectation values calculated for the direct product set ofparameter values. Several such parameter sets were investi-gated, and a representative set of values is shown in Table1. Using either approach the reconstructed density opera-tor matches the original exactly. Results for the firstmethod described above are given in Table 2. It can be seenthat all the density matrices q(m) can be reconstructed essen-

tially exactly by inverting Eq. 12 with N0

= 25.

1e-009

1e-008

1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

1

2 3 4 5 6 7

||

() -rec

() (N)||2

N

0

1

2

3

4

Fig. 3. Norm squared kqm qmrecN0k2 (cf.Eq. (13)) of the difference

between the density operator q

(m)

and the reconstructed operator qm

recN0as a function of the assumed density operator rank N0, for m = 04. Notethe log scale. The maximum intensity parameter bmax = 1.5.



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4. Multi-pulse excitation

As shown in the preceding section, although reconstruc-tion of the density operator for rigid rotors is numericallyfeasible, the measurement supermatrixM can be ill-condi-tioned, especially for small values of bmax. This results inextreme sensitivity of the inversion procedure to randomerrors in the measurement of rotor orientation.

The first and simplest way to stabilize the inversion pro-cedure is to increase the value of bmax, corresponding phys-ically to increasing the intensity of the exciting HCP.Another method is to excite the rotor with more thanone pulse before making measurements of molecular orien-tation. In this case it is not necessary to increase the valueof bmax, so that b can be restricted to the range0 6 b 6 bmax = 1.5. Multi-pulse excitation corresponds toeffectively larger values ofbmax, and leads to smaller condi-tion numbers for the measurement matrix M.

To illustrate, consider two-pulse excitation with N0

= 7and Ntot = 13. Calculations were performed by choosingN

02 = 49 triples fbk; t1k ; t2k g, where 0 6 tmk 6 Trot is thetime at which pulse m interacts with the system, m = 1,2,and 0 6 bk6 bmax = 1.5. The Log10[condition number]

for M is reduced to .4. With multi-pulse excitation, we

6

7

8

9

10

11

12

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Log10

[conditionnumber]

Log10

[max]

-9

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Log10

[||-

rec

||2]

Log10

[max]

Inverse

PseudoInverse

a b

Fig. 4. (a) Log of the condition number of the measurement matrixMbmax as a function of the log of the intensity parameter bmax. (b) log(iq qreci2)versus log bmax obtained using either straightforward matrix inversion (filled circles) or the pseudoinverse (triangles).

Table 1Values of time tk, direction nhk; / of the polarization laser, and laserintensity parameter bk used in the inversion calculation for the case j= 2discussed in Section 3.2

tk hk/p bk

0.5272 0.2276 0.24220.9133 0.7904 0.63710.6276 0.8441 1.33280.5752 0.1739 1.24560.0948 0.9745 1.2625

The angle / is fixed at value /

p=ffiffiffi3p .

Table 2Properties of the reconstructed density operator qmrec calculated for densityoperators based on the rotor coherent state Eq. (17) with j= 2, m = 05

mtr q

mrec

h iq

mrec qm

2 tr qm2rech i0 1.00 0.00 1.0001 1.00 0.00 0.6392 1.00 0.00 0.4583 1.00 0.00 0.3684 1.00 0.00 0.3235 1.00 0.00 0.300



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are able to add random Gaussian distributed errors dX tothe computed orientation values and still reconstruct thedensity matrix to a reasonable degree of accuracy (seeTable 3).

5. Summary and conclusion

We have described and implemented a simple quantumstate reconstruction scheme for rigid dipolar rotors. In con-trast to other proposals for rotor state reconstruction [11],our method involves measurement of the expectation valueof a low-rank tensor operator, the molecular orientation.Our approach requires a set of experiments to be per-formed in which the initial rotor state is excited by oneor more ultrashort half-cycle laser pulses, followed by mea-surement of molecular orientation after a known timedelay. Variation of the pulse intensities and polarization,the time delay and the axis along which the orientation ismeasured provides a set of measured values that are linearcombinations of matrix elements of the rotor density

matrix [8]. We have shown that it is possible to define asuitable set of measurements for which this relation canbe inverted to yield the density matrix directly, at least inthe case where q corresponds to the admixture of relativelyfew states. Further study of the invertibility of the measure-ment supermatrixM for different excitation schemes wouldbe desirable.

Our method is not necessarily restricted to rigid rotors,as rotor nonrigidity can be accounted for by includinghigher order centrifugal terms in the effective rotorHamiltonian.

We now mention some limitations of our approach andpossible avenues for future investigation. First, the possibleextension of our scheme to nonlinear rotors (symmetricand asymmetric tops) and to nonpolar linear moleculesand would be of interest. In present form our method isnot practicable if a large number of levels are initially pop-ulated. (The number of density matrix elements to be deter-

mined scales as the square of the number of levelspopulated.) If many levels are populated, then it will benecessary to parametrize the density operator in some fash-ion. A maximum entropy approach [8] to determination ofthe density operator subject to constraints provided byobservables of the kind discussed here is one possibility.

The effects of decoherence and/or dissipation on the pres-ent scheme are also of great interest. Some explicit model-ling of the effects of dephasing/dissipation will presumablybe necessary, but this lies outside the scope of the presentpaper.

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Table 3Reconstruction in the presence of random observational errors for thetwo-pulse excitation scheme discussed in Section 4

A tr[qrec] iqrec qi2 tr q2rec trq2

106 0.9999 2.62 107 3.61 106

105 0.9999 2.59 105 1.25 105

104 0.9991 2.59 103 2.2 103

103 0.9913 0.2591 0.2553

A random Gaussian distributed error on the rotor orientation of magni-tude A is added to the orientation expectation values.


Sarin A. Deshpande and Gregory S. Ezra- Quantum state reconstruction for rigid rotors

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