Quantum state tomography of dissociating molecules

4 Pages·0.83 MB·English

by  Esben Skovsen

Quantum state tomography of dissociating molecules
Esben Skovsen1, Henrik Stapelfeldt1, Søren Juhl2 and Klaus Mølmer3
1. Department of Chemistry, University of Aarhus, DK 8000 Aarhus C., Denmark
2. Niels Bohr Institute, University of Copenhagen, DK 2100 Copenhagen K, Denmark
3. QUANTOP, Danish National Research Foundation Center for Quantum Optics,
Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark
Using tomographic reconstruction we determinethe complete internuclearquantumstate, repre-
sented by the Wigner function, of a dissociating I2 molecule based on femtosecond time resolved
positionandmomentumdistributionsoftheatomicfragments. Theexperimentaldataarerecorded
by timed ionization of the photofragments with an intense 20 fs laser pulse. Our reconstruction
method,whichreliesonJaynes’maximumentropyprinciple,willalsobeapplicabletotimeresolved
position or momentum dataobtained with other experimental techniques.
3
0
0 Quantum state tomography derives its name from the of the iodine atoms of a dissociating molecule is ionized
2 tomographic technique in medical diagnostics by which the I+ velocity is equal to half of the internuclear veloc-

n three-dimensional images of the inner parts of an object ity, approximately 4.0 ˚A/ps. On the 2-D image, shown
a
orapersoncanbederivedfromtwo-dimensionalNMRor inFig. 1,theseionsconstitute theinnermostpairofhalf
J
X-raypictures obtainedfromdifferentdirections,a tech- rings (I+-I). If, instead, both iodine atoms are ionized
4
nique for which Cormack and Hounsfield were awarded the velocity of the I+ ions is increased due to their in-
2
the Nobel prize in physiology and medicine in 1979. In ternal Coulomb repulsion. On the 2-D image these ions
1 quantumphysics,weoftendealwithaphasespace,char- constitute the outermost pair of half rings (I+-I+). Us-
v acterizing the position and velocity distribution of par- ing Coulomb’s law the I+ velocity distribution from this
5
ticles, and the aim of quantum state tomography is to channel gives directly the internuclear distribution [9].
3
determine this distribution from only position and mo-
1
1 mentum measurements on the particles. As an example, Ppruolsbee(cid:13)
0 the oscillatory motion of a particle in a quadratic po-
3 tential is described as a simple rotation in phase space,
0 Molecular(cid:13)
and hence the mathematical Radon-transformtechnique Pump(cid:13) beam
/ pulse
h [1], used in physiology and medicine can be used to ex-
p tract the phase space distribution of such a particle if I
- 2
only position measurements are made sufficiently often
t
n
overa time correspondingto the oscillationperiod. This
a
technique has been succesfully applied to analyze bound t
u
q molecular states, where the internuclear distance oscil-
: lates harmonically [2], and it has been used for non-
v
i classical states of light [3, 4], which are formally de- I+
X
scribed as harmonically trapped particles. A number of
r extensions have been made to incorporate anharmonici-
a ties in the trapping potential, and to use mathematical ++II- +II- 2-D ion detector
reconstructiontechniqueswhichmatchspecificdetection
schemes[5,6,7,8]. Inthepresentworkwedeterminethe
internuclear quantum state of dissociating I2 molecules.
The atomic fragments are created in a state with posi- FIG. 1: Diagram of the experimental setup showing the
o
tive energy, and they fly apart with no confining force molecular beam crossed at 90 by the pump and the probe
between them. The situation thus differs from the one laser beams. Both laser beams are polarized vertically. The
I+ionsproducedarepushedbyaweakelectrostaticfield(not
of trapped motion, and it calls for a new theoretical ap-
shown) towards a 2-D ion detector, where theirpositions are
proach.
usedtodeterminetheinternuclearvelocitydistribution(I+-I
Experimentally, I2 molecules in a molecular beam are channel) and the internuclear separation distribution (I+-I+
photodissociated by irradiation with a 100-fs-long pump channel). The image shown on the detector is recorded for a
pulse centered at 488 nm (see Fig. 1). To determine delay of 5 psbetween thepump and theprobe pulse.
the internuclear separation (position) and velocitiy (mo-
mentum) of the dissociating molecules at time τ after The atomic density is sampled on a discrete position
excitation the atomic I fragments are ionized by a 20- grid with N grid points, where N = 51 in the example
fs-long probe pulse, and the velocities of the I+ ions are shown in Fig. 2. For the data presented here, the recon-
measuredbya2-dimensionaliondetector[9]. Ifonlyone structionisbasedonpositiondistributionsrecordedatτ
2
=2,3,4,5picoseconds(ps)andthemomentumdistribu- Boththespatialdistributionatvarioustimesandthemo-
tion (which is identical at the four different delays). In mentum distribution, which is independent of time, are
quantum tomography on harmonic oscillator states the measured,and the observablesA in Eq.(1)are precisely
i
fast and slow velocity components are slushing back and the operators |x,tihx,t| and |kihk|, which are now given
forth,andthetimedependentpositiondistributionmaps explicity as N by N matrices in a discrete basis. The
the phase space distribution from all angles. In free ex- exponential in Eq.(1) is understood as a matrix power
pansion, slow particles will not be able to catch up on series,andithastobeevaluatednumericallybecausethe
fast ones, and we have access to only sheared pictures momentumprojectionoperatorsandthepositionprojec-
of the distribution where the high velocity components tion operators at different times do not commute.
areshiftedtowardslargerdisplacementsthanthelowve- In our numerical reconstruction we used four position
locity components. This implies that a range of angular distributions and one momentum distributions, and we
viewinganglesismissing,andwecannotapplytheRadon thus identified 5·51 variables λ so that the expectation
i
transform to reconstruct the phase space distribution. valueshA i=Tr(ρA )areascloseaspossibletothe mea-
i i
Furthermore, the data-sets are noisy due to the finite sureddistributions. We quantify the agreementwith the
counts in different position intervals, and our task is to measured data a by the sum of the squares of all devia-
i
establish the most reliable estimate of ρ in best possi- tions∆= (a −hA i)2. Anumericalroutineidentifies
i i i
ble (rather than perfect) agreement with the observed the globalPminimum of ∆ as a function of the λ vari-
i
data. FollowingDrobnyandBuzek[10],weapplyJaynes’ ables, and we thus obtain the explicit expression for the
maximumentropy principle which atthe same time pro- density matrix in Eq.(1).
vides the smallest deviation from the measured data To illustrate our reconstruction of the internuclear
and the largest possible von Neumann entropy [10, 11]: quantum state of the dissociated molecules we use the
S(ρ) = −Tr(ρlnρ). Drobny and Buzek have used the Wigner function W(x,p) [13] rather than the N by N
maximum entropy principle for quantum state tomogra- complex density matrix, discussed above. The Wigner
phy on harmonically trapped atoms detected at too few function is obtained by a Fourier transformwith respect
instants of time to allow a normal reconstruction [12], to the difference between the two momentum arguments
andwe havereformulatedtheir method to dealwith free in the density matrix. While still fully characterizing
atoms. the quantum state the Wigner function is a real func-
To calculate the von Neumann entropy, one may use tion,andithastheadvantageofpresentingatoneglance
the familiar expression in terms of the eigenvalues pj of the joint position and momentum distribution. Figure 2
the density matrix, S = − jpjlnpj, but as it was shows the Wigner function, W(x,p), at the earliest (2
shown by Jaynes [11], it is Pnot necessary to compute ps) and the latest time (5 ps) for which position distri-
the entropy of ρ in order to use the maximum entropy butionswererecorded. Notethatwepresentthemomen-
principle: Thedensitymatrixwithlargestentropywhich tum dependence in units of the velocity v =p/M of the
conforms best with measured values ai for a set of ob- atomicfragmentstofacilitatecomparisonofthetwopan-
servables Ai, can be written on the form els. The two Wigner functions in Fig. 2 a) and b) have
the samevelocitydistributions,butthey aresheareddif-
1
ρ= exp(− λiAi), (1) ferently in phase space due to the free motion of the two
Z
Xi iodine photofragments. On the side panels the marginal
position and velocity distribution, obtained by integrat-
whereλ arevariablesthathavetobeadjustedtosatisfy
i
ing W(x,p) over velocity and position, respectively, are
theagreementwiththemeasureddata,andZ normalizes
compared with the experimental results (open circles).
the density matrix to unit trace.
Theagreementisexcellent,andsimilargoodagreementis
A position distribution is the set of expectation val-
found between the measured data and the reconstructed
uesofprojectionoperatorsonpositioneigenstates|xihx|.
state at the intermediate times at 3 and 4 ps.
Wedeterminethepositiondistributionatdifferenttimes,
We stress that the Wigner function contains more in-
and we need a formal representation of the projection
formation than what can be obtained directly from a
operator |x,tihx,t| where |x,ti is a position eigenstate
single measured position and velocity distribution. For
at time t. For free particles the momentum p = h¯k is
instance,both the measuredpositionandvelocity distri-
a conserved quantity, and it is convenient to represent
bution at 2 ps consists of two peaks (side panels of Fig.
the projection operators in a momentum representation,
2(A)), butwithoutfurther informationitis notpossible
since momentum eigenstates accumulate only a trivial
phase factor with time, |k,ti=exp(i¯hk2t/2M)|k,t=0i: to determine if the molecules at the larger internuclear
separation (the peak around 13.6 ˚A ) are also the ones
moving with the largest internuclear velocity (the peak
|xihx|=Z dk1dk2|k1ihk1|xihx|k2ihk2| around 4.6 ˚A/ps). The tomographic reconstruction of
1 the Wigner function makes use of our knowledge of the
= 2π Z dk1dk2exp(i(k2−k1)x)|k1ihk2|. (2) position distributions measured at 3, 4 and 5 ps, and,
3
dissociation by the pump pulse [9].
t = 2 ps
It has been emphasized in the literature [13] that the
1.2 Wignerfunctionisnotarealprobabilitydistributionbe-
cause, due to complimentarity and Heisenbergs’s uncer-
1 tainty relation,itis notmeaningful to assignprecise val-
ues for both the position and velocity of a particle. This
0.8
has as its most striking consequence that states exist for
0.6 which W(x,p) attains negative values, but these nega-
tivevaluesalwaysoccurinphasespaceregionswitharea
0.4 less than Planck’s constant h¯ = h/2π. Dissociation of
molecules by two phase-locked femtosecond laser pulses
0.2
will produce photofragments in a coherent superposition
0 of two localized wave packets [9], and we are currently
workingonimprovementoftheexperimentstoenablere-
−0.2
constructionoftheWignerfunctionwithasufficientres-
20 18 2 1 olution to display the negativities expected in this case.
x [Å]16 14 12 4 3 v [Å/ps] Such an experiment constitutes a fs time resolved ana-
10 5 logue to double-slit atom interferometer studies [14].
t = 5 ps Quantum state tomography has been established in
quantumopticsasaremarkablediagnosticstoolthathas
1.2 providedclearillustrationsofthe experimentalabilityto
control and manipulate selected states of fundamental
1
quantum systems such as a single field mode and single
0.8 ions and atoms [4, 6, 7, 8, 12]. Our experimentally re-
constructed density matrix or Wigner function provides
0.6 the complete informationaboutthe non-trivialquantum
0.4 state of the internuclear motion of the dissociating I2
molecules. Although our experimental method is lim-
0.2 ited to small molecules, we believe that quantum state
tomographywill be applicable also to study chemicalre-
0 actionsinlargemoleculesthroughtime resolvedposition
data obtained, e.g., by the emerging femtosecond elec-
−0.2
tron and X-ray diffraction techniques [18, 19, 20, 21].
32 1
30 2 This will open for comparisons with theoretical calcu-
28 3
x [Å] 26 24 4 v [Å/ps] lations at a much more detailed level than probability
22 5
distributions of single observables, which have formed
the basis of comparison so far in femtosecond time re-
FIG. 2: Three dimensional surface plots showing the phase
space distribution function W(x,p) (Wigner function) at solvedchemicalreactiondynamics. Finally,wenotethat
two different times after the pump pulse dissociates the I2 modern femtosecond laser technology enables controlled
molecule (tau=2ps in part (A) and tau = 5 ps in part (B)). shapingofboththeelectronicandtheatomicstructureof
The marginal position and momentum distributions of the molecules through irradiation with sequences of tailored
dissociated molecules are shown on the side panels: The full
laser pulses [15, 16, 17]. To fully exploit the capabilities
curves are the reconstructed distributions (integrals over the
ofsuchquantummanipulationitis crucialtocompletely
Wignerfunctionwithrespecttomomentumandposition, re-
characterize the molecular quantum states formed.
spectively), and the open circles are the measured distribu-
tions. To provide a more intuitive view the momentum vari- WeacknowledgethesupportfromtheCarlsbergFoun-
able is presented in velocity units. dation and The Danish Natural Science Council.
indeed, the two separatedpeaks in W(x,p) at 2 ps show
that the iodine atoms roughly 13.6 ˚A apart have a ve-
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