doi:10.1016/j.cplett.2007.04.049 
Copyright © 2007 Elsevier B.V. All rights reserved.
Quantum state reconstruction for rigid rotors
Sarin A. Deshpandea and Gregory S. Ezra
, a, 
aDepartment of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, NY 14853, United States
Received 26 December 2006;
revised 23 March 2007.
Available online 19 April 2007.
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Abstract
We describe a quantum state reconstruction scheme for dipolar rigid rotors based on determination of the expectation value of the molecular orientation. A key feature is the use of half-cycle pulses to excite the rotor prior to the orientation measurement. The set of expectation values obtained by varying the intensity and polarization of the laser and the time interval between excitation and measurement can be inverted directly to yield the rotor density operator. When the density operator corresponds to the admixture of relatively few rotor states, our procedure successfully reconstructs both pure and mixed states.
Graphical abstract
Convergence of the trace of the reconstructed rigid rotor density operator as a function of assumed size.
Fig. 1. Trace 
of the reconstructed density operator 
as a function of the assumed density operator rank N′, for ν = 0–4. The maximum intensity parameter βmax = 1.5. This plot indicates that the reconstruction is essentially converged at jmax = N′ − 1 = 4.
Fig. 3. Norm squared 
(cf.Eq. (13)) of the difference between the density operator ρ(ν) and the reconstructed operator 
as a function of the assumed density operator rank N′, for ν = 0–4. Note the log scale. The maximum intensity parameter βmax = 1.5.
Fig. 4. (a) Log of the condition number of the measurement matrix 
as a function of the log of the intensity parameter βmax. (b) log (
ρ − ρrec2) versus log βmax obtained using either straightforward matrix inversion (filled circles) or the pseudoinverse (triangles).
Table 1.
Values of time tk, direction 
of the polarization laser, and laser intensity parameter βk used in the inversion calculation for the case j = 2 discussed in Section 3.2
The angle 
is fixed at value 
.
Table 2.
Properties of the reconstructed density operator 
calculated for density operators based on the rotor coherent state Eq. (17) with j = 2, ν = 0–5
Table 3.
Reconstruction in the presence of random observational errors for the two-pulse excitation scheme discussed in Section 4
A random Gaussian distributed error on the rotor orientation of magnitude A is added to the orientation expectation values.
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