Research Article Molecular Tomography of the Quantum State...

Hindawi Publishing CorporationPhysics Research InternationalVolume 2013 Article ID 236743 8 pageshttpdxdoiorg1011552013236743

Research ArticleMolecular Tomography of the Quantum State byTime-Resolved Electron Diffraction

A A Ischenko

Moscow Lomonosov State University of Fine Chemical Technologies Vernadskogo prosp 86 Moscow 119589 Russia

Correspondence should be addressed to A A Ischenko aischenkoyasenevoru

Received 18 March 2013 Revised 11 July 2013 Accepted 14 October 2013

Academic Editor Ashok Chatterjee

Copyright copy 2013 A A IschenkoThis is an open access article distributed under theCreativeCommonsAttributionLicense whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A procedure is described that can be used to reconstruct the quantum state of a molecular ensemble from time-dependentinternuclear probability density functions determined by time-resolved electron diffractionTheproceduremakes use of establishedtechniques for evaluating the density matrix and the phase-space joint probability density that is the Wigner function A novelexpression for describing electron diffraction intensities in terms of the Wigner function is presented An approximate variant ofthe method neglecting the off-diagonal elements of the density matrix was tested by analyzing gas electron diffraction data forN2in a Boltzmann distribution and TRED data obtained from the 193 nm photodissociation of CS

2to carbon monosulfide CS at

20 40 and 120 ns after irradiation The coherent changes in the nuclear subsystem by time-resolved electron diffraction methoddetermine the fundamental transition from the standard kinetics to the dynamics of the phase trajectory of the molecule and thetomography of molecular quantum state

1 Introduction

In accordance with basic quantum principles the state ofan individual molecule cannot be determined experimentally[1] However for an ensemble of similarly prepared systemsit is possible to determine their state operator the so-called density matrix Knowing the state of a system meanshaving the maximum possible information about all physicalquantities of interest available [2] The density matrix andthe joint phase-space probability density orWigner function[3 4] have a one-to-one correspondence [5] that describesthe maximal statistical information available Thus in thefollowing text when the term molecular quantum state isused we mean the quantum state of an ensemble of similarlyprepared molecular species

In 1933 it was demonstrated by Freenberg [6] (see also[1 page 71]) that in principle a pure quantum state |Ψ⟩

can be reconstructed from the time-dependent coordinateprobability density 119875(119903 119905) = |Ψ(119903 119905)|

2 and its derivative120597119875(119903 119905)120597119905 It was shown by Weigert [7] that a pure quantumstatemay also be reconstructed bymeasuring the distribution119875(119903 119905) at time 119905 andmonitoring its evolution after a short timeinterval Δ119905 that is 119875(119903 119905 + Δ119905) = |Ψ(119903 119905 + Δ119905)|

2

For mixed quantum states the method of optical homo-dyne tomography to measure the Wigner function (anddensity matrix) was first demonstrated by Smithey et al [8 9]for both vacuum and squeezed vacuum states of a singlespatial-temporalmode in an applied electromagnetic field (Anumber of investigations into the preparation and measure-ment of the quantum state of light may be found in [10])

In experiments of Kurtsiefer et al [11 12] it is demon-strated that Wignerrsquos function can be reconstructed fromdouble-slit experiments involving a coherent beam of heliumatoms Their result [11 12] shows that the joint phase-spaceprobability density 119882(119903 119901) where 119901 is the momentumexhibits negative areas indicating purely nonclassical behav-ior such as what might be expected of a particle in an atomicinterferometer when it is in a linear superposition of states attwo separate locations (see also comments in [13])

In studies of molecular states measurements of time-dependent fluorescence spectra make it possible to recon-struct Wignerrsquos function for single molecular vibrationalmodes excited by short optical pulses This technique isreferred to as Molecular Emission Tomography (MET) [14]andwas used byDunn et al in studies of electronically excitedNa2wave packets moving in a suggested harmonic potential

2 Physics Research International

In the current paper it is described how time-resolvedelectron diffraction (TRED) may be used in tomographicstudies of the molecular state TRED [15ndash17] provides exper-imental observations having a direct correspondence tothe molecular time-dependent probability density function119875(119903 119905) In TREDexperiments a pulsed electron source is usedto probe an ensemble of similarly prepared species through-out a sequence in timeThe diagnostic electron pulses and theconcomitant exciting laser pump pulses are synchronized insuch way that stroboscopic diffraction pictures are obtainedIn this way time is introduced as a new variable into theotherwise conventional diffraction technique [15ndash17] Thusreacting laser-excited species transient states of chemicalreactions or molecular wave packets can be investigated[15ndash20] The dynamics of wave packets as the coherentsuperposition of quantum states created by short phase-controlled optical pulses [21ndash24] can also be probed Coher-ent nuclear dynamics can be followed in this way [15 25ndash27]because the elastic scattering of fast electrons proceeds on anultrashort (ultimately attosecond) time scale and represents anondestructive probing technique [15 25 26]

2 Theory

21 General Aspects TRED experiments yield time-depend-ent molecular scattering intensity functions 119872(119904 119905) [15 2526] where |s| = 2|k| sin(1205792) k is the momentum vector ofthe scattered electrons and 120579 is the scattering angle

119872(119904 119905) = (1198680

1198772)119892 (119904) int119875 (119903 119905) exp (119894sr) 119889119903 (1)

In (1) 1198680is the incident electron beam profile 119877 is the scatter-

ing distance 119892(119904) are the scattering functions [28] and sr isthe dot product of the scattering vector s and internucleardistance vector r Integration in (1) and in the followingequations is over the infinite space if not indicated otherwise

In classical mechanics there are no equations thatdescribe the evolution of the probability densities 119875(119903 119905)

(or 119875(119901 119905) where 119901 is momentum) only joint probabilitydensities 119882cl(119901 119903 119905) can be expressed by the Liouvilleequation Therefore corresponding quantum equations for119875(119903 119905) as used in (1) and 119875(119901 119905) cannot exist However thewell known Wigner-Liouville equation [4] can be used todescribe the evolution of the Wigner function 119882(119901 119903 119905)In order to derive an equation that expresses the time-dependent molecular intensities 119872(119904 119905) in terms of theWigner function the marginal properties of 119882(119901 119903 119905) [3 4]must be considered That is

int119882(119901 119903 119905) 119889119901 = 119875 (119903 119905) (2)

int119882(119901 119903 119905) 119889119903 = 119875 (119901 119905) (3)

Hence (1) can be expressed as follows

119872(119904 119905) = (1198680

1198772)119892 (119904) int 119889119901int119882(119901 119903 119905) exp (119894sr) 119889119903 (4)

Equation (4) is the most general representation of TREDmolecular intensities in terms of the Wigner function Inthis representation 119872(119904 119905) can be interpreted as the fil-tered projection of the Wigner function with the scatteringoperator exp(119894sr) as the filter modified by the scatteringfunctions 119892(119904) For a number of problems 119882(119901 119903 119905) canbe evaluated analytically (see eg the review in [4]) orby solving the Wigner-Liouville equation numerically withappropriate molecular potential functions

In (1) and (2) 119875(119903 119905) = |Ψ(119903 119905)|2 The wave function

Ψ(119903 119905) can be expanded in terms of the orthonormal basisfunctions 120595

119899(119903) in the following way [1 2]

Ψ (119903 119905) = sum119899

119862119899120595119899(119903) exp (minus119894120596

119899119905) (5)

where 119899 is the quantum number of the state energy 119864119899 120596119899is

the oscillator frequency and 119862119899is the amplitude Then (1)

can be written as

119872(119904 119905) = (1198680

1198772)119892 (119904)

times sum119898119899

120588119898119899

exp (119894 (120596119898minus 120596119899) 119905)

times ⟨120595119898(119903)

1003816100381610038161003816exp (119894sr)1003816100381610038161003816 120595119899 (119903)⟩

(6)

where the 120588119898119899

are the density matrix elementsFrom (6) it is seen that UEDmolecular intensities depend

explicitly on the quantum state of a molecular systemLikewise the 119875(119903 119905) obtained from TRED data that is theFourier transform of (1)

119875 (119903 119905) = (2

120587)12

(1198772

1198680

)int119872(119904 119905) [119892 (119904)]minus1 exp (minus119894sr) 119889119904

(7)

depends on both the position 119903 and the time 119905 and con-tains implicitly all information on the quantum state ofthe system being a projection or ldquoshadowrdquo [29] of theWigner function A simple sequence of TREDmeasurementsof the position-dependent probability density provides thenecessary information for tomographic reconstruction ofthe Wigner function For this purpose the inverse Radontransformation can be applied [29 30]

Another possibility for molecular quantum state recon-struction is the evaluation of the density matrix elements120588119898119899 which have a one-to-one correspondence [5] toWignerrsquos

function

119882(119903 119901) = (1

120587ℎ)int exp(

2119894119901119909

ℎ) ⟨119903 minus 119909

10038161003816100381610038161205881003816100381610038161003816 119903 + 119909⟩ 119889119909 (8)

where 120588 is the density matrix [2]

120588 = sum119899119898

120588119898119899

1003816100381610038161003816120595119899 (119903)⟩ ⟨120595119898 (119903)1003816100381610038161003816 (9)

22 Approximate Procedures When the TRED measure-ments do not span a complete cycle and the data are incom-plete only the diagonal density matrix elements 120588

119899= 120588119899119899

Physics Research International 3

can be evaluated from the probability density ⟨119875(119903 119905)⟩120591

averaged over the time interval 120591 ≫ sup(|120596119898minus 120596119899|minus1) with

119898 = 119899 as shown by Richter andWunsche [31ndash34] Leonhardt[29 35] Leonhardt andRaymer [36] and others [37ndash39]Thismethod uses the superposition principle combined with thestructure of Schrodingerrsquos equation to reveal the quantumstate of a molecule from the time-averaged coordinate-dependent probability density function In general and fora complete data set

120588119898119899

= int119875 (119903 120596119898minus 120596119899)

120597 [120595119898(119903) 120593119899(119903)]

120597119903 119889119903 (10)

where 119875(119903 120596119898

minus 120596119899) is the Fourier transform of the time-

dependent probability density 119875(119903 119905) [29 equation 559]the 120595119899(119903) are the normalized regular solutions of the time-

independent Schrodinger equation and the 120593119899(119903) are the

linearly independent nonnormalizable irregular second solu-tions [33 36 40]

In the particular case of incomplete data one can expressthe diagonal density matrix elements as

120588119899119899

= int ⟨119875(119903 119905)⟩120591 120597 [120595119899(119903) 120593119899(119903)]

120597119903 119889119903 (11)

Time-dependent probability density function 119875(119903 119905) aver-aged over electron pulse profile of duration 120591 provides evolu-tion in time of the diagonal density matrix elements 120588

119899119899in

(11)The terms state density matrix or Wigner function are

for most practical purposes synonymous Thus when theavailable data are incomplete and only the diagonal densitymatrix elements are used one has to keep in mind that itis not possible to determine the complete Wigner functionHowever it is possible to use the diagonal density matrixelements to calculate an approximate or truncated Wignerfunction which contains useful information when the off-diagonal elements are smallThe truncatedWigner functionscan be used to calculate approximate vonNeumann entropiesand purity parameters which nonetheless may containmeaningful information on the quantum state of a system

3 Experiment

Instrumentation for TRED developed at the University ofArkansas spanning the time domain from microseconds topicoseconds has been described before [41] A UV laser-driven tantalum photocathode in a Pierce-type [42] electrongun assembly was used for generation of the probe electronpulses The mechanics of the photoemission process suggestthat the electron pulse should follow the temporal profile ofthe incident optical pulse [43 44] In the experiments usedfor this study the output from an excimer laser (QuestekModel 2260 193 nm (ArF) pulse duration sim15 ns) was usedto generate electron pulses (duration sim15 ns FWHM sim1010electrons per pulse [41]) which were accelerated through apotential of sim40 keV For the current study carbon disulfideCS2 target molecules were irradiated with similar 193 nm

laser pulses from a second excimer at 50ndash130mJ per pulse

120 140 160 180r (pm)

Time-dependent P(r td)

P(rtd)

for C equiv S

td

20ns40ns120ns

(a)

Sampling functions (n = 3 4 5)2015

00

0 1 2 3

3

4

4

5

5

0510

minus05

minus10

minus15

minus20

minus5 minus4 minus3 minus2 minus1

X

(b)

Figure 1 Experimental probability densities for CS at 20 40 and120 ns following irradiation of CS

2at 193 nm as calculated from the

stochastic approach [45] and selected sampling functions (lowercurves) The sampling procedure was applied for 119899 lt 10 and thedimensionless variable 119909 = (2120587120583120596

119890ℎ)12

(119903 minus 119903119890) where 120583 is the

reduced mass 120596119890the harmonic frequency and 119903

119890the equilibrium

internuclear distance Sampling functions for CS (bottom) aredefined in (14)

Delay times between excitation and electron scattering variedfrom 20 to 120 ns and the diffraction intensities were recordedonline using a phosphor screen-photodiode array detector[41]The target gaseswere admitted into the vacuumchamberthrough a nozzle orifice (ID 200 120583m) at a pushing pressureof 30ndash50 Torr The electron scattering occurred in a volumeimmediately adjacent to the nozzle exit in which the gaspressure is estimated to besim10 Torr Such conditions precludethe formation of clusters and the number 119885 of collisions perparticle in the target gas is estimated to be in the range of100lt 119885 lt 10

2 in 15 ns Also time averaging is determined bythe electron pulse profile and averaging time 120591 sim 15 ns Theexperimental TREDdata sets [45] and sampling functions arepresented in Figure 1


Table 1 Diagonal density matrix elements for N2 and CS

119899120588119899119899

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14N2(exp)a 09938 00011 00102 minus0000 0001 0005 minus0009 0012 minus0011 0009 minus0005 mdash mdash mdash mdash

N2(theor)b 09935 00064 00000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

CS (nasc)c 0000 0049 0098 0156 0101 0077 0061 0070 0073 0077 0077 0073 0067 0021 0000CS (nasc)d 0000 0074 0148 0236 0153 0116 0083 0042 0042 0032 0023 0023 0028 mdash mdashCS (20 ns)e 0349 0260 0172 0099 0053 0025 0013 0000 minus0001 minus0003 0001 mdash mdash mdash mdashCS (40 ns)e 0204 0190 0160 0129 0097 0075 0058 0033 0005 minus0015 minus0015 mdash mdash mdash mdashCS (120 ns)e 0173 0165 0147 0124 0100 0080 0058 0040 0029 0021 0013 mdash mdash mdash mdashaValues obtained from experimental electron diffraction intensities of N2 in an equilibrium Boltzmann distribution recorded at 673K Estimated uncertaintiesare 001 bValues calculated for a Boltzmann distribution at119879 = 673K cValues derived from the nascent ensemble of CS reported in [46] dValues derived fromthe nascent ensemble of CS reported in [47] eDerived from TRED data of photogenerated CS recorded at 20 40 and 120 ns after the 193 nm photodissociationof CS2

To further illustrate the described procedures N2in an

equilibrium Boltzmann distribution was chosen as a test casebecause its potential energy function is well known fromhigh-resolution spectroscopic measurements Data wererecorded throughout the temperature range from 298K to773Kwith a continuous electron beam [15 41] Experimentaldata and sampling functions are presented in Figure 2

4 Results and Discussion

The photochemistry of CS2has been thoroughly studied in

the absorption region from 180 to 210 nm [45ndash47] At 193 nmand in single-photon processes the photodissociation pro-ceeds according to the following two-channel scheme

CS2(1198831Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS(119883 1Σ+

119892)+119878 (

3119875119869)

(12)

CS2(1198831

Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS (119883 1Σ+

119892)+119878 (

1119863119869)

(13)

The structure of excited state of CS2is believed to be quasi-

linear but with different geometrical parameters comparedto the ground state further predissociation occurs on a timescale of sim1 ps

In supersonic molecular beams of 193 nm irradiated CS2

it was found [46 47] that the nascent vibrational distribu-tion of CS is strongly inverted for both the 1119863

2and 3119875

2

dissociation channels exhibiting a bimodal [46] or broad [47]distribution with low occupancy at 119899 = 0 and 119899 ge 10 assummarized in Table 1

To apply the quantum sampling procedure to N2as

a test equilibrium case position-probability densities werefirst derived from the diffraction intensities by using thestochastic approach to data analysis [45] This approach[45] allows one to refine TRED data of laser-excited specieswithout imposing any model restrictions on intramolecularmotion It is an advantage of the procedure that the requisitemolecular parameters may be refined from the experimentaldata without any information on the potential energy surface

90 95 100 105 110 115 120 125 130r (pm)

N2 P(r) versus r

(a)


00

0 1 2 3

0

4

1

5

2

0510

minus05

minus10

minus15

minus20


X

(b)

Figure 2 Experimental probability density for N2(upper curve) in

an equilibrium distribution at 673K note the slight asymmetry ofthe 119875(119903) due to the anharm1onicity of the molecular vibration Typ-ical sampling functions (lower curves) selected for illustration Thesampling procedure was applied for 119899 lt 10 and the dimensionlessvariable 119909 = (2120587120583120596

119890ℎ)12

(119903 minus 119903119890) where 120583 is the reduced mass 120596

119890

is the harmonic frequency and 119903119890is the equilibrium internuclear

distance Sampling functions for N2(bottom) are defined in (14)


Momentum

Distance (pm)

60

50 4030 20

10 080 100 120 140

160

minus10minus20

minus30minus40

minus50

N2 Boltzmann (673K)

Momentum

Di

0

4030 20

10 080 100 1200 140

minus10minus20

minus30minus40

minus

Figure 3 Wigner function for N2in a Boltzmann distribution

reconstructed from the experimental electron diffraction results of[41 45] The marginal 119875(119903) and 119875(119901) at the left and right edges ofthe graph respectively are plotted on an arbitrary vertical scalemomentum scale is in arbitrary units For this equilibrium case thediagonal density matrix elements and Wigner function adequatelydescribe the full quantum state of the system

The probability density functions obtained in this way werethen sampled with the functions 119865

119898119899(119903) [34]

119865119898119899

(119903) =120597 [120595119898(119903) 120593119899(119903)]

120597119903 (14)

where 120595119898(119903) and 120593

119899(119903) are respectively the normalized reg-

ular and nonnormalizable irregular wave functions referredto above The sampling functions for the first 3 levels areshown in Figure 2 Calculated densitymatrix elements for theBoltzmanndistribution ofN

2at119879 = 673Kare comparedwith

the experimentally determined values in Table 1The truncated Wigner function together with the

associated marginal position- and momentum-dependentprobability density functions ((2) and (3)) is shown inFigure 3 The results represent for the nearly harmonic one-dimensional oscillator at equilibrium a Gaussian-like func-tion and Gaussian-like marginal position- and momentum-probability densities

For CS we derived time-dependent probability densityfunctions ⟨119875(119903 119905)⟩

120591= 119875(119903 119905

119889) at several delay times from the

experimental time-dependent molecular intensity functionsusing the stochastic approach [15 45] To calculate thesampling functions of (14) the Dunham vibrational potentialof CS as determined spectroscopically up to sixth order [48]was used The regular wave functions 120595

119899(119903) were evaluated

by solving Schrodingerrsquos equation for the same Dunhampotential [48] using the variational technique of [49]

The values of the diagonal density matrix elements 120588119899119899

obtained by applying (11) to the experimental 119875(119903 119905119889) at

different delay times are represented in Table 1 TruncatedWigner functions for 119905

119889= 20 ns 40 ns and 120 ns are shown

in Figure 4 and compared with those for the nascent statedistribution of [47] (see also Table 1) For 119905

119889= 40 and

120 ns the truncatedWigner functions and the corresponding

marginal density functions 119875(119903 119905) and 119875(119901 119905) ((2) and (3))are similar In contrast these functions exhibit significantchanges between 119905

119889= 0 20 and 40 ns We interpret these

changes in the following wayChanges in the first 20 ns of the evolution of the sys-

tem are indicative of rapid equilibration predominantlyby noncollisional intermolecular vibrational energy transferinvolving for example dipole-dipole interactions Duringthis time span bimodal distributions both in position andmomentum space evolve to monomodal and narrower dis-tributions Between 20 and 40 ns collision-induced energytransfer becomes more important than collisionless energytransfer processesThus we believe that the changes between20 and 40 ns indicate electronic to vibrational energy transferinduced by collisions between 119878(

1119863119869) atoms and CS(119883 1Σ+

119892)

molecules Between 40 and 120 ns this process shows featuresof saturation

It can be inferred from Table 1 that keeping the foregoingapproximations in mind there is significant depopulationof the first three vibrational levels of CS between 20 and120 ns very likely indicating the transfer of vibrational energyto rotational and translational degrees of freedom Thisinterpretation is supported by the fact that the rotationaltemperature of CS in this process increases from119879rot = 4200plusmn

300K at 20 ns to 5000 plusmn 600K at 40 ns and to 5200 plusmn 800Kat 120 ns [45] The truncated Wigner functions at 20 40and 120 ns are positive throughout the entire phase space inagreement with the hypothesis that the molecular quantumstate of CS after 20 ns can be characterized as a classical-like statistical mixture rather than a coherently preparedsuperposition of the lower vibrational states This confirmsthe increasing importance of collisions after 20 ns and is incontrast to the nascent molecular distribution for whichthe complete Wigner function exhibits significant negativeareas (not visible in Figure 4) Furthermore in this case thebimodality of the marginal distributions 119875(119903 119905) and 119875(119901 119905)reflects the fact that the 193 nm photodissociation of CS

2

proceeds via two different channels producing 119878(3119875119869) and

119878(1119863119869) The symmetric nature of the bimodality shown in

Figure 4 is an artifact due to the truncation of the Wignerfunction

We have also investigated the processes described abovein terms of the von Neumann entropy 119878 [29]

119878 = minus tr 120588 ln120588 (15)

where tr119876 is the trace of the operator119876 and 120588 is the densitymatrix The von Neumann entropy vanishes for pure statesexceeds zero for mixed states and is an extensive quantity fornonentangled subsystems thus it is a fundamental measureof the preparation impurity of a quantum state [29] It isalso informative to consider the so-called purity parameter119878pur which represents a lower bound for the von Neumannentropy that is 119878pur le 119878

119878pur

= 1 minus tr 1205882 = 1 minus 2120587ℎ∬[119882(119901 119903 119905)]2

119889119901119889119903 (16)

In the current investigation the diagonal density matrixelements were used to derive approximate estimates for 119878


MomentumDistance (pm)

104

5030

10124144 164 184

204

minus10

minus30

minus50

CS nascent

MomentumD

430

10124144 164 184

minus10

minus30

minus

(a)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30minus50

CS 20ns

Momentum

D

0430

10124 144 164 184

minus10

minus30minus5

(b)

CS 40ns

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

Momentum430

10124 144 164

minus10

minus30

minus

(c)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

CS 120ns

Momentum0430

10124 144 164

minus10

minus30

minus

(d)

Figure 4Wigner functions for CS in a nascent distribution (upper left) and at 20 40 and 120 ns following photodissociation of CS2by 193 nm

laser pulses For the nascent distribution the diagonal density matrix was obtained from the time-of-flight measurements of [47] negativeareas are present which are not visible from the chosen perspective The experimental time-dependent Wigner functions were calculatedusing the diagonal density matrix elements of Table 1 The marginal probability density functions are shown at the left and right sides of eachgraph scales are arbitrary

and 119878pur In spite of their approximate nature it is seen from

Table 2 that the estimates obtained are in agreement withprevious descriptions [45] of the 193 nm photodissociation ofCS2 That is at 20 ns both the approximated von Neumann

entropy and purity parameters are smaller than at 119905 = 0

(nascent CS) indicating the transition from bimodal tomonomodal distributions At times later than 20 ns both ofthese parameters increase due to increasing populations ofvibrational states above 119899 = 3

5 Conclusion

The material presented above shows that time-resolved elec-tron diffraction measurements offer a new avenue for quan-tum state reconstruction yielding the Wigner function froma tomographically complete set of time-dependent position-probability densities When the TRED data are incompletea truncated Wigner function can be determined from thediagonal density matrix elements derived from the time-averaged experimental probability density ⟨119875(119903 119905)⟩ Even

though the diagonal terms do not describe the full quantumstate of a system they still contain useful information

Deriving the truncated Wigner function for N2in an

equilibrium Boltzmann distribution the expected Gaussian-like features were found In the case of the 193 nm photodis-sociation of CS

2 the analysis indicates collisionless vibra-

tional energy transfer mechanisms in the photogeneratedensemble of nascent CS and a collision-induced electronic tovibrational energy transfer process from 119878(

1119863) to CS(119883 1Σ+

119892)

at later times Apart from quantum state reconstruction(4) represents a novel expression for analyzing diffractionintensities directly in terms of the Wigner function

Molecular quantum state measurements by TRED arecomplementary to quantum state preparation and controlof matter by laser fields Coherent nuclear dynamics can befollowed in this way [15 25ndash27] because the elastic scatteringof fast electrons proceeds on an ultrashort (ultimately attosec-ond) time scale and represents a nondestructive probingtechnique Adequately new avenues in physical chemistry


Table 2 Sum of the diagonal density matrix elements (sum119899120588119899)

approximate von Neuman entropies (119878) and approximate purityparameters (119878pur) for N2 and CS

sum119899120588119899

119878 119878pur

N2(exp)a 103 019 0012

N2(theor)b 100 0039 0013

CS (nasc)c 100 249 091CS (nasc)d 100 221 087CS (20 ns)e 097 156 077CS (40 ns)e 095 192 086CS (120 ns)e 090 211 088aValues obtained from experimental electron diffraction intensities of N2in an equilibrium Boltzmann distribution recorded at 673K Estimateduncertainties are 001 bValues calculated for a Boltzmann distribution at119879 = 673K cValues derived from the nascent ensemble of CS reported in[46] dValues derived from the nascent ensemble of CS reported in [47]eValues derived from TRED data of photogenerated CS recorded at 20 40and 120 ns after the 193 nm photodissociation of CS2

are in development based on studies of coherent nucleardynamics [15 16 50]

Since the 1980 the scientific world made intensive effortsin order to register a movie about the coherent nucleidynamics in the molecules the fast dynamic processes inbiological tissues and cells and the structure dynamics ofthe solid in nanovolumes in time The observed coherentchanges in the nuclear subsystem by time-resolved electrondiffraction method determine the fundamental transitionfrom the standard kinetics to the dynamics of the phasetrajectory of the molecule and the tomography of molecularquantum state

Acknowledgments

Academician of the RAS A L Buchachenko Professor GV Fetisov (Chemistry Department of Moscow LomonosovStateUniversity) Professor L Schafer (University ofArkansasat Fayetteville USA) Professor E A Ryabov (Institute ofSpectroscopy RAS) and Professor V N Bagratashvili (IPLITRAS) are greatly thanked for their helpful discussions Thiswork was supported by RFBR Grants 12-02-00840-a and 12-02-12048-ofi m

References

[1] E C KembleThe Fundamental Principles of QuantumMechan-ics McGraw-Hill London UK 1937

[2] L E BallentineQuantumMechanics Prentice Hall EnglewoodCliffs NJ USA 1990

[3] E P Wigner Perspectives in Quantum Theory edited by WYorgrau andA van derMerveDoverNewYorkNYUSA 1979

[4] M Hillery R F OrsquoConnell M O Scully and E PWigner ldquoDis-tribution functions in physics fundamentalsrdquo Physics Reportsvol 106 no 3 pp 121ndash167 1984

[5] K E Cahill and R J Glauber ldquoDensity operators andquasiprobability distributionsrdquo Physical Review vol 177 no 5pp 1882ndash1902 1969

[6] E Freenberg The scattering of slow electrons in neutral atoms[PhD thesis] Harvard University 1933

[7] S Weigert ldquoHow to determine a quantum state by measure-ments the Pauli problem for a particle with arbitrary potentialrdquoPhysical Review A vol 53 pp 2078ndash2083 1996

[8] D T Smithey M Beck M G Raymer and A FaridanildquoMeasurement of the Wigner distribution and the densitymatrix of a light mode using optical homodyne tomographyapplication to squeezed states and the vacuumrdquo Physical ReviewLetters vol 70 no 9 pp 1244ndash1247 1993

[9] D T SmitheyM Beck J Cooper andMG Raymer ldquoCompleteexperimental characterization of the quantum state of a lightmode via the Wigner function and the density matrix applica-tion to quantum phase distributions of vacuum and squeezed-vacuum statesrdquo Physica Scripta vol 48 p 35 1993

[10] W P Schleich and M G Raymer ldquoQuantum state preparationand measurementrdquo Journal of Modern Optics vol 44 no 11-121997

[11] C Kurtsiefer T Pfau and J Mlynek ldquoMeasurement of thewigner function of an ensemble of helium atomsrdquo Nature vol386 no 6621 pp 150ndash153 1997

[12] T Pfau and C Kurtsiefer ldquoPartial reconstruction of themotional Wigner function of an ensemble of helium atomsrdquoJournal of Modern Optics vol 44 no 11-12 pp 2551ndash2564 1997

[13] M Freyberger and W P Schleich ldquoTrue vision of a quantumstaterdquo Nature vol 386 no 6621 pp 121ndash122 1997

[14] T J Dunn I A Walmsley and S Mukamel ldquoExperimentaldetermination of the quantum-mechanical state of a molecularvibrational mode using fluorescence tomographyrdquo PhysicalReview Letters vol 74 no 6 pp 884ndash887 1995

[15] A A Ischenko G V Girichev and I Y Tarasov ldquoElectronDiffraction structure and dynamics of free molecules andcondensed matterrdquo in Proceedings of the Femtosecond ElectronImaging and Spectroscopy Fizmatlit 2013

[16] R Srinivasan V A Lobastov C-Y Ruan and A H ZewailldquoUltrafast electron diffraction (UED) a new developmentfor the 4D determination of transient molecular structuresrdquoHelvetica Chimica Acta vol 86 no 6 pp 1763ndash1838 2003

[17] W E King G H Campbell A Frank et al ldquoUltrafast electronmicroscopy in materials science biology and chemistryrdquo Jour-nal of Applied Physics vol 97 no 11 pp 1ndash27 2005

[18] J R Dwyer C T Hebeisen R Ernstorfer et al ldquoFemtosecondelectron diffraction lsquomaking the molecular moviersquordquo Philosoph-ical Transactions of the Royal Society A vol 364 no 1840 pp741ndash778 2006

[19] G Sciaini andR J DMiller ldquoFemtosecond electron diffractionheralding the era of atomically resolved dynamicsrdquo Reports onProgress in Physics vol 74 Article ID 09610 2011

[20] C-Y Ruan Y Murooka R K Raman R A Murdick R JWorhatch and A Pell ldquoThe development and applications ofultrafast electron nanocrystallographyrdquo Microscopy and Micro-analysis vol 15 no 4 pp 323ndash337 2009

[21] D J Tannor and S A Rice ldquoControl of selectivity of chemicalreaction via control of wave packet evolutionrdquo The Journal ofChemical Physics vol 83 no 10 pp 5013ndash5018 1985

[22] J K Krause RMWhitnell K RWilson and Y J Yan Femtos-econd Chemistry edited by J Manz and L Woste SpringerWeinheim Germany 1995

[23] Y J Yan and K R Wilson ldquoOptimal control of moleculardynamics via two-photon processesrdquo Journal of Chemical Phys-ics vol 100 p 1094 1994


[24] J L Krause K J Schafer M Ben-Nun and K R WilsonldquoCreating and detecting shaped rydberg wave packetsrdquo PhysicalReview Letters vol 79 pp 4978ndash4981 1997

[25] H Ihee V A Lobastov U M Gomez et al ldquoDirect imagingof transient molecular structures with ultrafast diffractionrdquoScience vol 291 no 5503 pp 458ndash462 2001

[26] AH Zewail and JMThomas 4DElectronMicroscopy Imagingin Space and Time Imperial College Press London UK 2010

[27] J C Williamson and A H Zewail ldquoUltrafast electron diffrac-tion 4Molecular structures and coherent dynamicsrdquo Journal ofPhysical Chemistry vol 98 no 11 pp 2766ndash2781 1994

[28] RA BonhamandM FinkHighEnergy Electron Scattering VanNostrand Reinhold New York NY USA 1974

[29] U Leonhardt Measuring the Quantum State of Light Cam-bridge University Press 1997

[30] F NattererTheMathematics of Computerized Tomography JohnWiley amp Sons Chichester UK 1986

[31] T Richter and A Wunsche ldquoDetermination of occupationprobabilities from time-averaged position distributionsrdquo Physi-cal Review A vol 53 pp R1974ndashR1977 1996

[32] T Richter ldquoPattern functions used in tomographic reconstruc-tion of photon statistics revisitedrdquo Physics Letters A vol 211 pp327ndash330 1996

[33] T Richter and A Wunsche ldquoDetermination of quantum statefrom time-dependent position distributionsrdquo Acta Physica Slo-vaca vol 46 pp 487ndash492 1996

[34] AWunsche ldquoRadon-transform and pattern functions in quan-tum tomographyrdquo Journal of Modern Optics vol 44 pp 2293ndash2331 1997

[35] U Leonhardt ldquoState reconstruction in quantum mechanicsrdquoActa Physica Slovaca vol 46 no 3 pp 309ndash316 1996

[36] U Leonhardt and M G Raymer ldquoObservation of moving wavepackets reveals their quantum staterdquo Physical Review Lettersvol 76 no 12 pp 1985ndash1989 1996

[37] U Leonhardt and S Schneider ldquoState reconstruction in one-dimensional quantum mechanics the continuous spectrumrdquoPhysical Review A vol 56 no 4 pp 2549ndash2556 1997

[38] U Leonhardt M Munroe T Kiss T Richter and M GRaymer ldquoSampling of photon statistics and densitymatrix usinghomodyne detectionrdquo Optics Communications vol 127 no 1ndash3pp 144ndash160 1996

[39] U Leonhardt ldquoQuantum and classical tomography with equi-distant reference anglesrdquo Journal of Modern Optics vol 44 pp2271ndash2280 1997

[40] A Messiah Quantum Mechanics vol 1 North-Hollandl Ams-terdam The Netherlads 1965

[41] V A Lobastov J D Ewbank L Schafer and A A IschenkoldquoInstrumentation for time-resolved electron diffraction span-ning the time domain from microseconds to picosecondsrdquoReview of Scientific Instruments vol 69 no 7 pp 2633ndash26431998

[42] J R PierceTheory andDesign of ElectronBeams Van-NostrandPrinceton NJ USA 1954

[43] J R Helliwell and P M Rentzepis Eds Time-Resolved Diffrac-tion Oxford University Press 1997

[44] ldquoTime-resolved electron and x-ray diffractionrdquo P M RentzepisEd vol 2521 of Proceedings of SPIE SPIE Bellingham WaUSA 1995

[45] A A Ischenko J D Ewbank and L Schafer ldquoStructural andvibrational kinetics by time-resolved gas electron diffraction

stochastic approach to data analysisrdquo Journal of Physical Chem-istry vol 99 pp 15790ndash15797 1995

[46] V R McCrary R Lu D Zakheim J A Russell J B Halpernand W M Jackson ldquoCoaxial measurement of the translationalenergy distribution of CS produced in the laser photolysis ofCS2 at 193 nmrdquo The Journal of Chemical Physics vol 83 no 7pp 3481ndash3490 1985

[47] W-B Tzeng H-M Yin W-Y Leung et al ldquoA 193 nm laserphotofragmentation time-of-flight mass spectrometric study ofCS2 and CS2 clustersrdquoThe Journal of Chemical Physics vol 88no 3 pp 1658ndash1669 1988

[48] T R Todd and W B Olson ldquoThe infrared spectra of 12C32S12C34S 13C32S and 12C33Srdquo Journal of Molecular Spectroscopyvol 74 pp 190ndash202 1979

[49] I Suzuki ldquoGeneral anharmonic force constants of carbondisulfiderdquo Bulletin of the Chemical Society of Japan vol 48 pp1685ndash1690 1975

[50] A L Buchachenko ldquoChemistry on the border of two centuriesachievements and prospectsrdquo Russian Chemical Reviews vol68 no 2 p 85 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


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AstrophysicsJournal of


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Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


In the current paper it is described how time-resolvedelectron diffraction (TRED) may be used in tomographicstudies of the molecular state TRED [15ndash17] provides exper-imental observations having a direct correspondence tothe molecular time-dependent probability density function119875(119903 119905) In TREDexperiments a pulsed electron source is usedto probe an ensemble of similarly prepared species through-out a sequence in timeThe diagnostic electron pulses and theconcomitant exciting laser pump pulses are synchronized insuch way that stroboscopic diffraction pictures are obtainedIn this way time is introduced as a new variable into theotherwise conventional diffraction technique [15ndash17] Thusreacting laser-excited species transient states of chemicalreactions or molecular wave packets can be investigated[15ndash20] The dynamics of wave packets as the coherentsuperposition of quantum states created by short phase-controlled optical pulses [21ndash24] can also be probed Coher-ent nuclear dynamics can be followed in this way [15 25ndash27]because the elastic scattering of fast electrons proceeds on anultrashort (ultimately attosecond) time scale and represents anondestructive probing technique [15 25 26]

2 Theory

21 General Aspects TRED experiments yield time-depend-ent molecular scattering intensity functions 119872(119904 119905) [15 2526] where |s| = 2|k| sin(1205792) k is the momentum vector ofthe scattered electrons and 120579 is the scattering angle

119872(119904 119905) = (1198680

1198772)119892 (119904) int119875 (119903 119905) exp (119894sr) 119889119903 (1)

In (1) 1198680is the incident electron beam profile 119877 is the scatter-

ing distance 119892(119904) are the scattering functions [28] and sr isthe dot product of the scattering vector s and internucleardistance vector r Integration in (1) and in the followingequations is over the infinite space if not indicated otherwise

In classical mechanics there are no equations thatdescribe the evolution of the probability densities 119875(119903 119905)

(or 119875(119901 119905) where 119901 is momentum) only joint probabilitydensities 119882cl(119901 119903 119905) can be expressed by the Liouvilleequation Therefore corresponding quantum equations for119875(119903 119905) as used in (1) and 119875(119901 119905) cannot exist However thewell known Wigner-Liouville equation [4] can be used todescribe the evolution of the Wigner function 119882(119901 119903 119905)In order to derive an equation that expresses the time-dependent molecular intensities 119872(119904 119905) in terms of theWigner function the marginal properties of 119882(119901 119903 119905) [3 4]must be considered That is

int119882(119901 119903 119905) 119889119901 = 119875 (119903 119905) (2)

int119882(119901 119903 119905) 119889119903 = 119875 (119901 119905) (3)

Hence (1) can be expressed as follows

119872(119904 119905) = (1198680

1198772)119892 (119904) int 119889119901int119882(119901 119903 119905) exp (119894sr) 119889119903 (4)

Equation (4) is the most general representation of TREDmolecular intensities in terms of the Wigner function Inthis representation 119872(119904 119905) can be interpreted as the fil-tered projection of the Wigner function with the scatteringoperator exp(119894sr) as the filter modified by the scatteringfunctions 119892(119904) For a number of problems 119882(119901 119903 119905) canbe evaluated analytically (see eg the review in [4]) orby solving the Wigner-Liouville equation numerically withappropriate molecular potential functions

In (1) and (2) 119875(119903 119905) = |Ψ(119903 119905)|2 The wave function

Ψ(119903 119905) can be expanded in terms of the orthonormal basisfunctions 120595

119899(119903) in the following way [1 2]

Ψ (119903 119905) = sum119899

119862119899120595119899(119903) exp (minus119894120596

119899119905) (5)

where 119899 is the quantum number of the state energy 119864119899 120596119899is

the oscillator frequency and 119862119899is the amplitude Then (1)

can be written as

119872(119904 119905) = (1198680

1198772)119892 (119904)

times sum119898119899

120588119898119899

exp (119894 (120596119898minus 120596119899) 119905)

times ⟨120595119898(119903)

1003816100381610038161003816exp (119894sr)1003816100381610038161003816 120595119899 (119903)⟩

(6)

where the 120588119898119899

are the density matrix elementsFrom (6) it is seen that UEDmolecular intensities depend

explicitly on the quantum state of a molecular systemLikewise the 119875(119903 119905) obtained from TRED data that is theFourier transform of (1)

119875 (119903 119905) = (2

120587)12

(1198772

1198680

)int119872(119904 119905) [119892 (119904)]minus1 exp (minus119894sr) 119889119904

(7)

depends on both the position 119903 and the time 119905 and con-tains implicitly all information on the quantum state ofthe system being a projection or ldquoshadowrdquo [29] of theWigner function A simple sequence of TREDmeasurementsof the position-dependent probability density provides thenecessary information for tomographic reconstruction ofthe Wigner function For this purpose the inverse Radontransformation can be applied [29 30]

Another possibility for molecular quantum state recon-struction is the evaluation of the density matrix elements120588119898119899 which have a one-to-one correspondence [5] toWignerrsquos

function

119882(119903 119901) = (1

120587ℎ)int exp(

2119894119901119909

ℎ) ⟨119903 minus 119909

10038161003816100381610038161205881003816100381610038161003816 119903 + 119909⟩ 119889119909 (8)

where 120588 is the density matrix [2]

120588 = sum119899119898

120588119898119899

1003816100381610038161003816120595119899 (119903)⟩ ⟨120595119898 (119903)1003816100381610038161003816 (9)

22 Approximate Procedures When the TRED measure-ments do not span a complete cycle and the data are incom-plete only the diagonal density matrix elements 120588

119899= 120588119899119899





120588119898119899

= int119875 (119903 120596119898minus 120596119899)

120597 [120595119898(119903) 120593119899(119903)]

120597119903 119889119903 (10)

where 119875(119903 120596119898






120588119899119899

= int ⟨119875(119903 119905)⟩120591 120597 [120595119899(119903) 120593119899(119903)]

120597119903 119889119903 (11)


119899119899in



3 Experiment



120 140 160 180r (pm)


P(rtd)

for C equiv S

td

20ns40ns120ns

(a)


00

0 1 2 3

3

4

4

5

5

0510

minus05

minus10

minus15

minus20


X

(b)




119890ℎ)12









119899120588119899119899


N2(theor)b 09935 00064 00000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000







CS2(1198831Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS(119883 1Σ+

119892)+119878 (

3119875119869)

(12)

CS2(1198831

Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS (119883 1Σ+

119892)+119878 (

1119863119869)

(13)





2and 3119875

2




90 95 100 105 110 115 120 125 130r (pm)

N2 P(r) versus r

(a)


00

0 1 2 3

0

4

1

5

2

0510

minus05

minus10

minus15

minus20


X

(b)



119890ℎ)12


119890




Momentum

Distance (pm)

60

50 4030 20

10 080 100 120 140

160

minus10minus20

minus30minus40

minus50

N2 Boltzmann (673K)

Momentum

Di

0

4030 20

10 080 100 1200 140

minus10minus20

minus30minus40

minus




119898119899(119903) [34]

119865119898119899

(119903) =120597 [120595119898(119903) 120593119899(119903)]

120597119903 (14)

where 120595119898(119903) and 120593







120591= 119875(119903 119905










119889= 40 and






1119863119869) atoms and CS(119883 1Σ+

119892)




2





119878 = minus tr 120588 ln120588 (15)


119878pur

= 1 minus tr 1205882 = 1 minus 2120587ℎ∬[119882(119901 119903 119905)]2

119889119901119889119903 (16)




104

5030

10124144 164 184

204

minus10

minus30

minus50

CS nascent

MomentumD

430

10124144 164 184

minus10

minus30

minus

(a)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30minus50

CS 20ns

Momentum

D

0430

10124 144 164 184

minus10

minus30minus5

(b)

CS 40ns

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

Momentum430

10124 144 164

minus10

minus30

minus

(c)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

CS 120ns

Momentum0430

10124 144 164

minus10

minus30

minus

(d)







5 Conclusion







1119863) to CS(119883 1Σ+

119892)






sum119899120588119899

119878 119878pur

N2(exp)a 103 019 0012

N2(theor)b 100 0039 0013




Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







120588119898119899

= int119875 (119903 120596119898minus 120596119899)

120597 [120595119898(119903) 120593119899(119903)]

120597119903 119889119903 (10)

where 119875(119903 120596119898






120588119899119899

= int ⟨119875(119903 119905)⟩120591 120597 [120595119899(119903) 120593119899(119903)]

120597119903 119889119903 (11)


119899119899in



3 Experiment



120 140 160 180r (pm)


P(rtd)

for C equiv S

td

20ns40ns120ns

(a)


00

0 1 2 3

3

4

4

5

5

0510

minus05

minus10

minus15

minus20


X

(b)




119890ℎ)12









119899120588119899119899


N2(theor)b 09935 00064 00000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000







CS2(1198831Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS(119883 1Σ+

119892)+119878 (

3119875119869)

(12)

CS2(1198831

Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS (119883 1Σ+

119892)+119878 (

1119863119869)

(13)





2and 3119875

2




90 95 100 105 110 115 120 125 130r (pm)

N2 P(r) versus r

(a)


00

0 1 2 3

0

4

1

5

2

0510

minus05

minus10

minus15

minus20


X

(b)



119890ℎ)12


119890




Momentum

Distance (pm)

60

50 4030 20

10 080 100 120 140

160

minus10minus20

minus30minus40

minus50

N2 Boltzmann (673K)

Momentum

Di

0

4030 20

10 080 100 1200 140

minus10minus20

minus30minus40

minus




119898119899(119903) [34]

119865119898119899

(119903) =120597 [120595119898(119903) 120593119899(119903)]

120597119903 (14)

where 120595119898(119903) and 120593







120591= 119875(119903 119905










119889= 40 and






1119863119869) atoms and CS(119883 1Σ+

119892)




2





119878 = minus tr 120588 ln120588 (15)


119878pur

= 1 minus tr 1205882 = 1 minus 2120587ℎ∬[119882(119901 119903 119905)]2

119889119901119889119903 (16)




104

5030

10124144 164 184

204

minus10

minus30

minus50

CS nascent

MomentumD

430

10124144 164 184

minus10

minus30

minus

(a)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30minus50

CS 20ns

Momentum

D

0430

10124 144 164 184

minus10

minus30minus5

(b)

CS 40ns

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

Momentum430

10124 144 164

minus10

minus30

minus

(c)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

CS 120ns

Momentum0430

10124 144 164

minus10

minus30

minus

(d)







5 Conclusion







1119863) to CS(119883 1Σ+

119892)






sum119899120588119899

119878 119878pur

N2(exp)a 103 019 0012

N2(theor)b 100 0039 0013




Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119899120588119899119899


N2(theor)b 09935 00064 00000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000







CS2(1198831Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS(119883 1Σ+

119892)+119878 (

3119875119869)

(12)

CS2(1198831

Σ+

119892)997888rarrCS

2(11198612

1

Σminus

119906)997888rarrCS (119883 1Σ+

119892)+119878 (

1119863119869)

(13)





2and 3119875

2




90 95 100 105 110 115 120 125 130r (pm)

N2 P(r) versus r

(a)


00

0 1 2 3

0

4

1

5

2

0510

minus05

minus10

minus15

minus20


X

(b)



119890ℎ)12


119890




Momentum

Distance (pm)

60

50 4030 20

10 080 100 120 140

160

minus10minus20

minus30minus40

minus50

N2 Boltzmann (673K)

Momentum

Di

0

4030 20

10 080 100 1200 140

minus10minus20

minus30minus40

minus




119898119899(119903) [34]

119865119898119899

(119903) =120597 [120595119898(119903) 120593119899(119903)]

120597119903 (14)

where 120595119898(119903) and 120593







120591= 119875(119903 119905










119889= 40 and






1119863119869) atoms and CS(119883 1Σ+

119892)




2





119878 = minus tr 120588 ln120588 (15)


119878pur

= 1 minus tr 1205882 = 1 minus 2120587ℎ∬[119882(119901 119903 119905)]2

119889119901119889119903 (16)




104

5030

10124144 164 184

204

minus10

minus30

minus50

CS nascent

MomentumD

430

10124144 164 184

minus10

minus30

minus

(a)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30minus50

CS 20ns

Momentum

D

0430

10124 144 164 184

minus10

minus30minus5

(b)

CS 40ns

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

Momentum430

10124 144 164

minus10

minus30

minus

(c)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

CS 120ns

Momentum0430

10124 144 164

minus10

minus30

minus

(d)







5 Conclusion







1119863) to CS(119883 1Σ+

119892)






sum119899120588119899

119878 119878pur

N2(exp)a 103 019 0012

N2(theor)b 100 0039 0013




Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Momentum

Distance (pm)

60

50 4030 20

10 080 100 120 140

160

minus10minus20

minus30minus40

minus50

N2 Boltzmann (673K)

Momentum

Di

0

4030 20

10 080 100 1200 140

minus10minus20

minus30minus40

minus




119898119899(119903) [34]

119865119898119899

(119903) =120597 [120595119898(119903) 120593119899(119903)]

120597119903 (14)

where 120595119898(119903) and 120593







120591= 119875(119903 119905










119889= 40 and






1119863119869) atoms and CS(119883 1Σ+

119892)




2





119878 = minus tr 120588 ln120588 (15)


119878pur

= 1 minus tr 1205882 = 1 minus 2120587ℎ∬[119882(119901 119903 119905)]2

119889119901119889119903 (16)




104

5030

10124144 164 184

204

minus10

minus30

minus50

CS nascent

MomentumD

430

10124144 164 184

minus10

minus30

minus

(a)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30minus50

CS 20ns

Momentum

D

0430

10124 144 164 184

minus10

minus30minus5

(b)

CS 40ns

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

Momentum430

10124 144 164

minus10

minus30

minus

(c)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

CS 120ns

Momentum0430

10124 144 164

minus10

minus30

minus

(d)







5 Conclusion







1119863) to CS(119883 1Σ+

119892)






sum119899120588119899

119878 119878pur

N2(exp)a 103 019 0012

N2(theor)b 100 0039 0013




Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





104

5030

10124144 164 184

204

minus10

minus30

minus50

CS nascent

MomentumD

430

10124144 164 184

minus10

minus30

minus

(a)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30minus50

CS 20ns

Momentum

D

0430

10124 144 164 184

minus10

minus30minus5

(b)

CS 40ns

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

Momentum430

10124 144 164

minus10

minus30

minus

(c)

Momentum

Distance (pm)

104

5030

10124 144 164 184 204

minus10

minus30

minus50

CS 120ns

Momentum0430

10124 144 164

minus10

minus30

minus

(d)







5 Conclusion







1119863) to CS(119883 1Σ+

119892)






sum119899120588119899

119878 119878pur

N2(exp)a 103 019 0012

N2(theor)b 100 0039 0013




Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






sum119899120588119899

119878 119878pur

N2(exp)a 103 019 0012

N2(theor)b 100 0039 0013




Acknowledgments


References


























































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Research Article Molecular Tomography of the Quantum State...

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