Complex Wave Function Reconstruction and Direct Electromagnetic

Letter pubs.acs.org/JPCL

Complex Wave Function Reconstruction and Direct Electromagnetic Field Determination from Time-Resolved Intensity Data Cian Menzel-Jones*,† and Moshe Shapiro†,‡,§ †

Department of Physics and Astronomy and ‡Department of Chemistry, The University of British Columbia, Vancouver, V6T1Z1 Canada § Department of Chemical Physics, The Weizmann Institute, Rehovot 76100, Israel ABSTRACT: We present a reference-free robust method for the nondestructive imaging of complex time-evolving molecular wave functions using as input the timeresolved fluorescence signal. The method is based on expanding the evolving wave function in a set of bound stationary states and determining the set of complex expansion coefficients by calculating a series of Fourier integrals of the signal. As illustrated for the A1∑+u electronic state of Na2, the method faithfully reconstructs the time-dependent complex wave function of the nuclear motion. Moreover, using perturbation theory to connect the excitation pulse and the material expansion coefficients, our method is used to determine the electromagnetic field of the excitation pulse, thus providing a simple technique for pulse characterization that obviates the additional measurements and iterative solutions that beset other techniques. The approach, which is found to be quite robust against errors in the experimental data, can be readily generalized to the reconstruction of polyatomic vibrational wave functions. SECTION: Molecular Structure, Quantum Chemistry, and General Theory s the field of quantum information processing progresses into the experimental domain, it becomes increasingly necessary to develop techniques for the nondestructive detection and reconstruction of (complex) wave functions. Such reconstruction is needed for the verification of the initial preparation, the detection of errors due to decoherence and noise, the determination of the fidelity of the logical operations, and the read-out of the final state. In the past few years, molecules have emerged as promising media for large-scale quantum computations, partially due to the existence of strong dipole−dipole interactions that serve as a resource for twoqubit operations.1−6 A number of theoretical and experimental publications demonstrating feasible read/write operations7−12 and logic gates13−19 have been published, with the read-out amounting to the retrieval of both the amplitude and the phase of a molecular wave packet.20−24 There have been several techniques for imaging molecular wave packets, each with their own drawbacks, already suggested. One such technique based on quantum state tomography,7,20,21,25 which has been mainly applied to photonic states, uses a sequence of identical measurements within a series of different bases to reconstruct the quantum wave function completely. Unfortunately, molecular imaging tomography requires a very large number of measurements, such as different phase-space orientations, and is restricted to working well only in the harmonic approximation. Another technique22−24 involves the (algebraic) inversion of the timedependent fluorescence intensity, given in general as R(t) = ∑s′,sa*s′ asCs′,s exp(−iωs′,st), where as are the desired expansion coefficients, ωs′,s are (known) beat frequencies, and Cs′,s are

A

© 2012 American Chemical Society

molecular response matrix elements (to be discussed in detail below), to obtain the bilinear as′*as products. The main limitation of this method is that the conversion of the as′*as products to individual as coefficients is very sensitive to experimental errors. This inordinate sensitivity to experimental errors can be overcome by linearizing the problem, as done in the “quantum state holography” method26,27 and other linear interferometric approaches.11,12,28−30 However, these linear wave-packet interferometry (WPI) techniques encounter limitations related to evolving the target and reference wave packets on similar potential energy surfaces (PESs).31 In addition, they are insensitive to any target probability amplitudes resulting from nonlinear excitation process involving eigenstates with small Franck−Condon factors to the initial states.31 Using a nonlinear WPI approach,32,33 Cina and Humble proposed a remedy to these problems and a resolution to the hindrance of linear-WPI methods associated with the reconstruction of unknown target wave packets on illcharacterized high-lying potentials. Their approach is based on using a pair of phase-related pulse pairs that produce a signal that is quadrilinear in the incident field amplitudes, although the success of their approach relies on the existence, and detailed information, of an auxilary intermediate electronic state used for the propagation of the reference wave packet. Moreover, it is not clear how uncertainties in this reference wave packet affect the fidelity of their reconstruction. Thus, we Received: September 27, 2012 Accepted: November 1, 2012 Published: November 1, 2012 3353

dx.doi.org/10.1021/jz3015327 | J. Phys. Chem. Lett. 2012, 3, 3353−3359

The Journal of Physical Chemistry Letters

Letter

find that hampering all WPI approaches remains the necessity, and accurate characterization, of some reference state. Correlation functions, wave packet overlaps, or both have also been accessed without the need of reference wave packets, instead through the use of coherent anti-Stokes Raman (CARS) scattering.34−39 In particular, Avisar and Tannor37−39 have used CARS to image multidimensional wave packets without knowledge of the excited-state potential. They relate a wave packet’s cross-correlation function with the vibrational eigenfunction of the ground state PES to the three-pulse CARS signal. However, because this relations involves a direct correspondence with elements of the raw CARS signal, significant discrepancies arise once reasonable errors are included in the experimentally data.40 In addition, their method has so far been applied only to the special case of a δ(t − t0) pulse, where the wavepacket Ψ(r, t0) is known at t0 as being a simple replica of the ground energy eigenstate, χg(r). This is only reasonable once a large number of excited states have been included, which is at odds with the solution they provide to the unknown phase problem of the dipole matrix elements that requires exponential time with the number of states. Lastly, kinetic energy distributions of nuclei and photoelectrons or coulomb explosion41−45 techniques provide a good basis for wave function reconstruction.46,47 These methods are, however, destructive, as they involve the ionization and the break-up of molecules. In this letter, we describe a method that extracts the excited wave-packet amplitudes and phases from the time-dependent intensity of fluorescence. The method works by taking a series of finite time Fourier transforms at the (ωs′,s) beat frequencies of the data.48 In this way, one performs a one-by-one extraction of the expansion coefficients of the unknown wave packet. The method requires no reference state yet; because of the relatively long time averaging, it is extremely robust and is much less sensitive to experimental errors compared with the other mentioned schemes. Our approach also avoids the application of the Franck−Condon approximation, which is a necessary assumption in nearly all tomographic, interferometric, and CARS imaging proposals. In contrast with destructive methods, because typically spontaneous emission lifetimes are relatively long (10−8 to 10−6 seconds), only 10−6 to 10−5 of the excited molecules fluoresce over the 1−10 ps time span of interest, thus making fluorescence-based methods essentially nondestructive over such short time spans. In addition to molecular wave-packet imaging, by using the well-known perturbative regime connection between the field and the material expansion coefficients, our method (following a single calibration run) can be used to extract any electromagnetic (e.m.) field whose time-dependent intensity function is known. The problem of extracting the phases of the e.m. fields has received much attention in recent years. There now exists a panoply of methods49−52 that can be roughly categorized as interferometric53−56 and noninterferometric.57−60 These methods, which involve nontrivial nonlinear mixing and iterative solutions of integral equations, work best for ultrashort laser pulses of large bandwidths. They do not work as well for relatively longer (10−11 to 10−10 seconds) pulses of narrower bandwidths, where our method remains successful. We consider the short laser pulse excitation of χg(r), a rovibrational energy eigenstate belonging to the ground electronic state, where r is a (collective) nuclear coordinate, to form a wave packet Ψ(r,t) moving on an excited electronic state, given as

Ψ(r, t ) =

∑ asϕs(r) exp(−iEst /ℏ − γst /2) s

(1)

In the above, ϕs(r) are the excited rovibrational eigenstates of energies Es, as are the expansion coefficients of Ψ in terms ϕs, and γs are the (spontaneous emission) decay rate of each ϕs state, which are expressed as22

γs =

∑ A f ,s (2)

f

where Af,s are Einstein A-coefficients, given, in atomic units, as A f ,s =

4ωf3, s 3c 3

|μf , s |2

(3)

and μf,s ≡ ⟨χf |μ|ϕs⟩ are the dipole matrix elements between the electronically excited energy eigenstates, ϕs, and the ground energy eigenstates, χf. We have shown in the past61−64 that knowledge of |μf,s|2, the magnitude-squared of the dipole matrix elements, which can be determined directly from the frequency-resolved fluorescence (absorption) spectrum, coupled to knowledge of the groundstate potential to (from) which emission (absorption) occurs, enables the extraction of the phases of μf,s. Thus, the only remaining unknowns are the amplitudes |as| and phases δs of the as expansion coefficients. As shown below, these quantities can be extracted from the time-dependent fluorescence of Ψ(t) into various rovibrational ground states. Following ref 22, we write the rate of the time-dependent fluorescence as R (t ) =

∑ asas ′Cs ,s ′ exp(−iωs ,s ′t − (γs + γs ′)t /2) s,s′

(4)

where ωs,s′ are “beat” frequencies, defined as, ωs,s′ = (Es − Es′)/ℏ, and Cs,s′ is a time-independent molecular matrix, given as 48

2 Cs , s = 3 ′ 3c

∑ (ωf3,s + ωf3,s ′)⟨ϕs|μ|χf ⟩⟨χf |μ|ϕs ′⟩ f

(5)

Over typical short time scales of interest (e.g., ∼1 ps) the decline in population due to fluorescence is negligible, allowing us to disregard γs. Writing as explicitly as as = |as| exp(iδs) we obtain that R (t ) =

∑ Cs ,s ′|as||as ′|exp(iωs ,s ′t + iδs ,s ′) s,s′

(6)

where δs,s′ = δs−δs′. We now select a specific (s,s′) component of the R(t) signal by calculating at each beat frequency its “fFT”, the ωs,s′ Fourier integral calculated over long finite time T. The filtering method works because all of the ω ≠ ωs,s′ components decay as 1/T. Explicitly, (1/T)∫ 0T dt R(t) exp(−iωt) = (1/T)∫ T0 dt ∑S,S′CS,S′|as||aS′|(iδS,S′) exp(i(ωS,S′ − ω)t) ⎧Cs , s |as||as ′| exp(iδs , s ) for ωs , s = ω ′ ′ ⎪ ′ ⎪ Cs , s |as||as ′| exp(iδs , s ) =⎨ i ′ ′ for ωs , s ≠ ω ′ ⎪ (ωs , s − ω)T ′ ⎪ ⎩ (exp[i(ωs , s ′ − ω)T ] − 1) (7) Thus, by choosing sufficiently long integration times T, satisfying T ≫ 1/min|ωs,s′ − ω| for ωs,s′ ≠ ω, our method 3354



Letter

Figure 1. (a) Temporal fluorescence captured over 10 ps. (b) Actual and imaged excited-state wave packets on A1∑+u potential of Na2 at t = 0 and 1 ps. (c) Spectrum of the initial excitation pulse, showing absorption lines corresponding to the excited rovibrational eigenstates. (d) Temporal pulse profile of 50 fs width. The displayed real part of the electric field demonstrates good phase extraction.

setup to extract the excitation pulse’s electric field. We again envision a molecule, initially in one of the electronically ground vibrational states χg, with energy Eg, being excited by an electric field

amplifies just one prechosen (s,s′) beat term out of the R(t) signal, while eliminating all others. The absolute value of each (s,s′) term can be taken together with the knowledge of Cs,s′ to generate a set of equations containing products of the unknown amplitudes |as||as′|, which can then be solved, in the leastsquares sense, to obtain the coefficient amplitudes |as|. Using eq 7, we determine δs,s′ for all s and s′, given which, we extract the individual state phases δs (up to an arbitrary overall phase). For diatomic vibrational spacings of ∼1000 cm−1, the integration time must be at least 1 ps to ensure a filtering accuracy of ∼1%. We find that the long integration times tend to wash out measurement errors in the time-dependent fluorescence, rendering the method, as we show in detail below, quite robust against such errors. Because most systems will be overdetermined (having roughly s2/2 equations for s unknowns), it is possible to increase the accuracy and reduce the integration time by using as many redundant equations as possible. In addition, by subtracting out of R(t) the (s,s′) terms whose values have already been determined, we highlight the remaining (possibly weakly contributing) terms, thereby increasing their accuracy. Our ability to determine (the amplitude and phase of) excited-state coefficients makes it possible to use the same

̂ e{,(t ) exp( − iωt )} ε ⃗ (t ) = ε 9

(8)

where ε̂ is the polarization direction. Our aim here is to determine ,(t ), the (complex) pulse envelope, and ω, the carrier frequency of the pulse. Provided that the Rabi angle θ = ∫ dt Ω(t) ≪ π, where the Rabi frequency Ω(t) = μ⃗·ε ,(t )/ℏ, the light-matter interaction can be treated perturbatively. Thus, a time-dependent wave packet will be created in the excited state as in eq 1 with excited-state coefficients given by as = 2πif (ωs , g )μs , g

(9)

where μs,g=⟨ϕs|μ|χg⟩ are the known dipole matrix elements between the ground χg and excited ϕs rovibrational states and f(ωs,g) ≡ ∫ dt ,(t )exp[i(ωs,g − ω)t]. Using the method of the previous section to determine the excited state coefficients as, we obtain the complex values of the frequency spectrum f(ωs,g) from eq 9 as 3355



Letter

Figure 2. (a) Temporal fluorescence captured over 10 ps. (b) “True” (circles) and imaged (full lines) excited-state wave packets moving on the A1∑+u potential of Na2 at t = 0 and 1 ps. (c) Spectrum of the initial excitation pulse, showing absorption lines corresponding to the excited rovibrational eigenstates. (d) Temporal pulse amplitude and the −π and π phase of the field at each instant.

f (ωs , g ) =

as 2πiμs , g

pulses, omitting from the results excited states with populations 0.1%. We have computed the time-resolved fluorescence signal, adding 5% of random Gaussian noise to each time-averaged (∼70 fs) data point, from which, using our method, we have extracted the excited state coefficients. The results of the wave-packet imaging are found to be within 1% rms of the “true” values. Figure 2b compares the “true” (discrete points) wave function at the pulse center (t = 0) and at some later time (t = 1 ps), with the imaged values (solid lines). The f(ωs,g) discrete frequency components of the laser field, extracted from the as coefficients via eq 10, are displayed in Figure 2c. The Fourier transform of these points gives us the temporal shape of the pulse. Figure 2d exhibits an excellent agreement between the “true” and imaged time-dependent pulse amplitude and phase, correctly identifying the linear chirp of the field. We have presented a method that reconstructs excited-state molecular wave packets (and e.m. field amplitudes) by selecting each beat component of the fluorescence signal using finitetime Fourier transforms calculated at each beat frequency. The procedure requires a precalibration involving the frequencyresolved spectrum from which we extract the relevant dipole matrix elements to the final states. The integrating time over the temporal fluorescence is bound between two limits: It must be longer than the inverse of the energy splitting between adjacent vibrational states (∼50 fs), and it must be shorter than the average decay time of the rovibrational levels (∼10 ns). Within this range, our method exhibits impressive robustness to random errors in the experimental signal. Using the acquired knowledge of the excited coefficients to characterize the excitation pulse is a novel approach especially suitable for femtosecond pulses. Contrary to currently available methods, the experimental procedure and mathematical analysis needed to reconstruct the pulse is simple and straightforward. We are currently extending this approach to multidimensional problems and applying this procedure to obtained a detailed image of nuclear motions associated with “internal conversions” (transitions between different electronic states of the same spin multiplicity), and “intersystem crossings” (transitions between electronic states of different spins).

■

■

ACKNOWLEDGMENTS

■

REFERENCES

Letter

This work was supported by a “Major Thematic Grant” from the Peter Wall Institute for Advanced Studies of the University of British Columbia; by the US DoD DTRA program; by NSERC Discovery Grant; by NSERC Alexander Graham Bell CGS award; and by the Walter C. Sumner Memorial Fellowship.

(1) Computational Molecular Spectroscopy; Jensen, P., Bunker, P. R., Eds.; Wiley: Chichester, U.K., 2000. (2) DeMille, D. Quantum Computation with Trapped Polar Molecules. Phys. Rev. Lett. 2002, 88, 067901. (3) de Araujo, L. E. E.; Walmsley, I. A. Analytic Solution for Quantum Control of Atomic and Molecular Wavepackets. J. Opt. B: Quantum Semiclassical Opt. 2003, 5, R27. (4) Micheli, A.; Brennen, G.; Zoller, P. A Toolbox for Lattice-Spin Models with Polar Molecules. Nat. Phys. 2006, 2, 341−347. (5) André, A.; DeMille, D.; Doyle, J. M.; Lukin, M. D.; Maxwell, S. E.; Rabl, P.; Schoelkopf, R. J.; Zoller, P. A Coherent All-Electrical Interface between Polar Molecules and Mesoscopic Superconducting Resonators. Nat. Phys. 2006, 2, 637−642. (6) Hosaka, K.; Shimada, H.; Chiba, H.; Katsuki, H.; Teranishi, Y.; Ohtsuki, Y.; Ohmori, K. Ultrafast Fourier Transform with a Femtosecond-Laser-Driven Molecule. Phys. Rev. Lett. 2010, 104, 180501. (7) Dunn, T. J.; Sweetser, J. N.; Walmsley, I. A.; Radzewicz, C. Experimental Determination of the Dynamics of a Molecular Nuclear Wave Packet via the Spectra of Spontaneous Emission. Phys. Rev. Lett. 1993, 70, 3388−3391. (8) Weinacht, T. C.; Ahn, J.; Bucksbaum, P. H. Measurement of the Amplitude and Phase of a Sculpted Rydberg Wave Packet. Phys. Rev. Lett. 1998, 80, 5508−5511. (9) Katsuki, H.; Chiba, H.; Girard, B.; Meier, C.; Ohmori, K. Visualizing Picometric Quantum Ripples of Ultrafast Wave-Packet Interference. Science 2006, 311, 1589−1592. (10) Monmayrant, A.; Chatel, B.; Girard, B. Quantum State Measurement Using Coherent Transients. Phys. Rev. Lett. 2006, 96, 103002. (11) Ohmori, K.; Katsuki, H.; Chiba, H.; Honda, M.; Hagihara, Y.; Fujiwara, K.; Sato, Y.; Ueda, K. Real-Time Observation of PhaseControlled Molecular Wave-Packet Interference. Phys. Rev. Lett. 2006, 96, 093002. (12) Ohmori, K. Wave-Packet and Coherent Control Dynamics. Annu. Rev. Phys. Chem. 2009, 60, 487−511. (13) Palao, J. P.; Kosloff, R. Quantum Computing by an Optimal Control Algorithm for Unitary Transformations. Phys. Rev. Lett. 2002, 89, 188301. (14) Tesch, C. M.; de Vivie-Riedle, R. Quantum Computation with Vibrationally Excited Molecules. Phys. Rev. Lett. 2002, 89, 157901. (15) de Vivie-Riedle, R.; Troppmann, U. Femtosecond Lasers for Quantum Information Technology. Chem. Rev. 2007, 107, 5082− 5100. (16) Menzel-Jones, C.; Shapiro, M. Robust Operation of a Universal Set of Logic Gates for Quantum Computation using Adiabatic Population Transfer between Molecular Levels. Phys. Rev. A 2007, 75, 052308. (17) Gollub, C.; Kowalewski, M.; de Vivie-Riedle, R. Monotonic Convergent Optimal Control Theory with Strict Limitations on the Spectrum of Optimized Laser Fields. Phys. Rev. Lett. 2008, 101, 073002. (18) Bomble, L.; Lauvergnat, D.; Remacle, F.; Desouter-Lecomte, M. Vibrational computing: Simulation of a Full Adder by Optimal Control. J. Chem. Phys. 2008, 128, 064110. (19) Zaari, R. R.; Brown, A. Effect of Laser Pulse Shaping Parameters on the Fidelity of Quantum Logic Gates. J. Chem. Phys. 2012, 137, 104306.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1 604 822 8375. Fax: +1 604 822 5324. Notes

The authors declare no competing financial interest. 3357



Letter

from Arrays of Subwavelength Apertures. Phys. Rev. B 2004, 70, 033404. (43) Ergler, T.; Rudenko, A.; Feuerstein, B.; Zrost, K.; Schröter, C. D.; Moshammer, R.; Ullrich, J. Spatiotemporal Imaging of Ultrafast Molecular Motion: Collapse and Revival of the D2+ Nuclear Wave Packet. Phys. Rev. Lett. 2006, 97, 193001. (44) Bocharova, I. A.; Alnaser, A. S.; Thumm, U.; Niederhausen, T.; Ray, D.; Cocke, C. L.; Litvinyuk, I. V. Time-resolved Coulombexplosion Imaging of Nuclear Wave-packet Dynamics Induced in Diatomic Molecules by Intense Few-cycle Laser Pulses. Phys. Rev. A 2011, 83, 013417. (45) Schmidt, L. P. H.; Jahnke, T.; Czasch, A.; Schöffler, M.; Schmidt-Bö cking, H.; Dö r ner, R. Spatial Imaging of the H 2+ Vibrational Wave Function at the Quantum Limit. Phys. Rev. Lett. 2012, 108, 073202. (46) Fischer, I.; Villeneuve, D. M.; Vrakking, M. J. J.; Stolow, A. Femtosecond Wave-packet Dynamics Studied by Time-resolved Zerokinetic Energy Photoelectron Spectroscopy. J. Chem. Phys. 1995, 102, 5566−5569. (47) González-Castrillo, A.; Palacios, A.; Bachau, H.; Martín, F. Clocking Ultrafast Wave Packet Dynamics in Molecules through UVInduced Symmetry Breaking. Phys. Rev. Lett. 2012, 108, 063009. (48) Lefebvre-Brion, H.; Field, R. W. The Spectra and Dynamics of Diatomic Molecules, 1st ed.; Elsevier/Academic: Amsterdam, 2004. (49) Gallmann, L.; Sutter, D. H.; Matuschek, N.; Steinmeyer, G.; Keller, U. Techniques for the Characterization of Sub-10-fs Optical Pulses: A Comparison. Appl. Phys. B: Laser Opt. 2000, 70, S67−S75. (50) Stibenz, G.; Ropers, C.; Lienau, C.; Warmuth, C.; Wyatt, A.; Walmsley, I.; Steinmeyer, G. Advanced Methods for the Characterization of Few-cycle Light Pulses: A Comparison. Appl. Phys. B: Laser Opt. 2006, 83, 511−519. (51) Birger, S.; Heinrich, S. A Method for Unique Phase Retrieval of Ultrafast Optical Fields. Meas. Sci. Technol. 2009, 20, 015303. (52) Ratner, J.; Steinmeyer, G.; Wong, T. C.; Bartels, R.; Trebino, R. Coherent Artifact in Modern Pulse Measurements. Opt. Lett. 2012, 37, 2874−2876. (53) Wong, V.; Walmsley, I. A. Ultrashort-pulse Characterization from Dynamic Spectrograms by Iterative Phase Retrieval. J. Opt. Soc. Am. B 1997, 14, 944−949. (54) Trebino, R. Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses; Kluwer Academic Publishers: Boston, 2002. (55) Wu, T.-W.; Tang, J.; Hajj, B.; Cui, M. Phase Resolved Interferometric Spectral Modulation (PRISM) for Ultrafast Pulse Measurement and Compression. Opt. Express 2011, 19, 12961−12968. (56) Wong, T. C.; Ratner, J.; Chauhan, V.; Cohen, J.; Vaughan, P. M.; Xu, L.; Consoli, A.; Trebino, R. Simultaneously Measuring Two Ultrashort Laser Pulses on a Single-shot using Double-blind Frequency-resolved Optical Gating. J. Opt. Soc. Am. B 2012, 29, 1237−1244. (57) Iaconis, C.; Walmsley, I. Spectral Phase Interferometry for Direct Electric-field Reconstruction of Ultrashort Optical Pulses. Opt. Lett. 1998, 23, 792−794. (58) Amat-Roldán, I.; Cormack, I. G.; Loza-Alvarez, P.; Artigas, D. Measurement of Electric Field by Interferometric Spectral Trace Observation. Opt. Lett. 2005, 30, 1063−1065. (59) Witting, T.; Frank, F.; Arrell, C. A.; Okell, W. A.; Marangos, J. P.; Tisch, J. W. G. Characterization of High-intensity Sub-4-fs Laser Pulses using Spatially Encoded Spectral Shearing Interferometry. Opt. Lett. 2011, 36, 1680−1682. (60) Pasquazi, A.; Peccianti, M.; Park, Y.; Little, B. E.; Chu, S. T.; Morandotti, R.; Azaña, J.; Moss, D. J. Sub-picosecond Phase-sensitive Optical Pulse Characterization on a Chip. Nat. Photonics 2011, 5, 618−623. (61) Li, X.; Menzel-Jones, C.; Shapiro, M. Spectroscopic Phase and the Extraction of Excited-State Potentials from Fluorescence Data. J. Phys. Chem. Lett. 2010, 1, 3172−3176. (62) Li, X.; Menzel-Jones, C.; Avisar, D.; Shapiro, M. Solving the Spectroscopic Phase: Imaging Excited Wave Packets and Extracting

(20) Dunn, T. J.; Walmsley, I. A.; Mukamel, S. Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography. Phys. Rev. Lett. 1995, 74, 884−887. (21) Skovsen, E.; Stapelfeldt, H.; Juhl, S.; Mølmer, K. Quantum State Tomography of Dissociating Molecules. Phys. Rev. Lett. 2003, 91, 090406. (22) Shapiro, M. Imaging of Wave Functions and Potentials from Time-resolved and Frequency-resolved Fluorescence Data. J. Chem. Phys. 1995, 103, 1748−1754. (23) Shapiro, M. Spectroscopic Wave Function Imaging and Potential Inversion. J. Phys. Chem. 1996, 100, 7859−7866. (24) Shapiro, M. Spectroscopic Density-matrix Imaging and Potential Inversion: The Rotational Effect. Chem. Phys. 1996, 207, 317−329. (25) Coffey, T. M.; Wyatt, R. E.; Schieve, W. C. Reconstruction of the Time-Dependent Wave Function Exclusively from Position Data. Phys. Rev. Lett. 2011, 107, 230403. (26) Leichtle, C.; Schleich, W. P.; Averbukh, I. S.; Shapiro, M. Quantum State Holography. Phys. Rev. Lett. 1998, 80, 1418−1421. (27) Leichtle, C.; Schleich, W. P.; Averbukh, I. S.; Shapiro, M. Wave Packet Interferometry without Phase-locking. J. Chem. Phys. 1998, 108, 6057−6067. (28) Bienert, M.; Haug, F.; Schleich, P.; W. Raizen, G. M. State Reconstruction of the Kicked Rotor. Phys. Rev. Lett. 2002, 89, 050403. (29) Tekavec, P. F.; Dyke, T. R.; Marcus, A. H. Wave Packet Interferometry and Quantum State Reconstruction by Acousto-optic Phase Modulation. J. Chem. Phys. 2006, 125, 194303. (30) Hasegawa, H.; Ohshima, Y. Quantum State Reconstruction of a Rotational Wave Packet Created by a Nonresonant Intense Femtosecond Laser Field. Phys. Rev. Lett. 2008, 101, 053002. (31) Cina, J. A. Wave-Packet Interferometry and Molecular State Reconstruction: Spectroscopic Adventures on the Left-Hand Side of the Schrödinger Equation. Annu. Rev. Phys. Chem. 2008, 59, 319−342. (32) Humble, T. S.; Cina, J. A. Molecular State Reconstruction by Nonlinear Wave Packet Interferometry. Phys. Rev. Lett. 2004, 93, 060402. (33) Humble, T. S.; Cina, J. A. Nonlinear Wave-Packet Interferometry and Molecular State Reconstruction in a Vibrating and Rotating Diatomic Molecule. J. Phys. Chem. B 2006, 110, 18879− 18892. (34) Kiviniemi, T.; Aumanen, J.; Myllyperkio, P.; Apkarian, V. A.; Pettersson, M. Time-resolved Coherent Anti-stokes Raman-scattering Measurements of I2 in Solid Kr: Vibrational Dephasing on the Ground Electronic State at 2.6−32 K. J. Chem. Phys. 2005, 123, 064509. (35) Akopyan, M. E.; Novikova, I. Y.; Poretsky, S. A.; Pravilov, A. M.; Smolin, A. G.; Tscherbul, T. V.; Buchachenko, A. A. Collision-induced Nonadiabatic Transitions in the Second-tier Ion-pair States of Iodine Molecule: Experimental and Theoretical Study of the I2(f0g+) Collisions with Rare Gas Atoms. J. Chem. Phys. 2005, 122, 204318. (36) Karavitis, M.; Kumada, T.; Goldschleger, I. U.; Apkarian, V. A. Vibrational Dissipation and Dephasing of I2(v = 1−19) in Solid Kr. Phys. Chem. Chem. Phys. 2005, 7, 791−796. (37) Avisar, D.; Tannor, D. J. Complete Reconstruction of the Wave Function of a Reacting Molecule by Four-Wave Mixing Spectroscopy. Phys. Rev. Lett. 2011, 106, 170405. (38) Avisar, D.; Tannor, D. J. Wavepacket and Potential Reconstruction by Four-wave Mixing Spectroscopy: Preliminary Application to Polyatomic Molecules. Faraday Discuss. 2011, 153, 131−148. (39) Avisar, D.; Tannor, D. J. Multi-dimensional Wavepacket and Potential Reconstruction by Resonant Coherent Anti-Stokes Raman Scattering: Application to H2O and HOD. J. Chem. Phys. 2012, 136, 214107. (40) Wilkinson, J. Modern Error Analysis. SIAM Rev. 1971, 13, 548− 568. (41) Weber, T.; et al. Complete Photo-fragmentation of the Deuterium Molecule. Nature 2004, 431, 437−440. (42) Egorov, D.; Dennis, B. S.; Blumberg, G.; Haftel, M. I. Twodimensional Control of Surface Plasmons and Directional Beaming 3358



Letter

Excited State Potentials from Fluorescence Data. Phys. Chem. Chem. Phys. 2010, 12, 15760−15765. (63) Menzel-Jones, C.; Li, X.; Shapiro, M. Extracting Double Minima Excited State Potentials from Bound-continuum Spectroscopic Data. J. Mol. Spectrosc. 2011, 268, 221−225. (64) Li, X.; Shapiro, M. Inversion of Two-dimensional Potentials from Frequency-resolved Spectroscopic Data. J. Chem. Phys. 2011, 134, 094113. (65) Koyama, T.; Nakajima, M.; Suemoto, T. Nuclear Wave-packet Oscillations at the F Center in KCl and RbCl. Phys. Rev. B 2008, 78, 155126. (66) Kim, C. H.; Joo, T. Ultrafast Time-resolved Fluorescence by Two Photon Absorption Excitation. Opt. Express 2008, 16, 20742− 20747.

3359


Complex Wave Function Reconstruction and Direct Electromagnetic

Recommend Documents