Using Time-Resolved Experiments and Coherent Control to

Letter pubs.acs.org/JPCL

Using Time-Resolved Experiments and Coherent Control to Determine the Phase of Transition Dipole Moments between Individual Energy Eigenstates Cian Menzel-Jones*,† and Moshe Shapiro†,‡ †

Departments of Physics and Chemistry, The University of British Columbia, Vancouver, V6T1Z1 Canada Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 7632700, Israel

‡

ABSTRACT: We develop an imaging method which uses bichromatic coherent control (BCC) in conjunction with time-resolved fluorescence to extract the complex amplitudes of individual transition dipole matrix elements (TDMs) as well as the amplitude of time-evolving wave packets. The method relies on determining the phase relation between the BCC fields, which look to deplete the population of different pairs of excited energy eigenstates, through the computation of a Fourier integral of the time-resolved fluorescence at the beat frequencies of these pairs of states. We illustrate our procedure by determining the amplitudes of the TDM’s linking the vibrational states of the A1∑+u and those of the X1∑+u electronic states of Na2. Furthermore, we demonstrate its broad applicability by extracting the expansion coefficients of a wave packet in the basis of vibrational energy eigenstates in the strong spin−orbit coupling potentials in RbCs. The approach, which is found to be quite robust against errors can be readily generalized to the imaging of wave packets of polyatomic molecules. SECTION: Molecular Structure, Quantum Chemistry, and General Theory packet amplitudes and phases by taking a series of finite time Fourier transforms at the various (ωs′,s) beat frequencies16 contained in the time-resolved fluorescence of the entire wave packet. The way BCC helps obviate the need to know the constituent energy eigenstates is by inducing stimulated emission from a chosen pair of states comprising the wave packet to a predetermined ground state. By tuning the relative phase between two external light fields to match the relative phase of two TDM’s linked to a common ground state, one can attain maximum depletion of the population of the pair of states (s,s′) whose Fourier transform at the beat frequency ωs,s′ is being computed. We consider exciting χg(r), a vibrational energy eigenstate belonging to the ground electronic state, with r being a (collective) nuclear coordinate, by a short laser pulse to form a wave packet Ψ(r,t) moving on an unknown excited PES, given as

M

olecules interact with light via their electric (or magnetic) multipole moments; however, largely we may use the dipole approximation, and only consider their transition dipole moments (TDM). Whereas the magnitudes of the TDM matrix elements (TDMs) between energy eigenstates can be deduced from the strength of frequency-resolved spectral lines, yielding the absolute-value-squared of the TDMs, it is much more difficult to use such data to determine the phases of the TDMs. The determination of the TDMs phases is highly desirable for such applications as wave packet dynamics1,2 and quantum computation operations on atomic/ molecular systems.3−8 Although we have shown recently that for an electronic transition between two Born−Oppenheimer potential energy surfaces (PES), knowledge of one PES allows one to derive the other PES, and the TDMs’ phases,9−11 as well as imaging of unknown time-evolving wave packets,12−14 the method requires data of a large number of spectral lines of sufficient quality and completeness, which at this stage are simply not available. In contrast to frequency-resolved experiments, time-resolved experiments1 that might in principle contain information about many spectral transitions, appear to be available. In the present paper we show how to use such data to derive the phase information for individual transitions between energy eigenstates. The method uses an approach developed for the imaging of molecular wave packets whose constituent energy eigenstates are known,14 in conjunction with bichromatic coherent control (BCC),15 thereby obviating the need to know the energy eigenstates. In this approach,14 one extracts the excited wave © 2013 American Chemical Society

ψ (r , t ) =

∑ asψs(r) exp(−iEst /ℏ − γst /2) s

(1)

Our objective is to determine as, the expansion coefficients of Ψ in terms of ψs(r), the excited vibrational eigenstates. Using frequency-resolved spectroscopic data, we may assume that we Received: July 10, 2013 Accepted: August 22, 2013 Published: August 22, 2013 3083

dx.doi.org/10.1021/jz4014272 | J. Phys. Chem. Lett. 2013, 4, 3083−3088

The Journal of Physical Chemistry Letters

Letter

Cs , s asas*′ (exp[i(ωs , s − ω)T ] − 1) ′ ′ − ω ω s , s s,s′ ′ = 6(1/T ) for ω ≠ ωs , s ′ Thus, in order to filter out all the Fourier components save for the ωs,s′ one, we need to choose T ≫ 1/min|ωs,s′ − ωs,s′|, where ωt,t′ ≠ ωs,s′. The relatively long integration times (typically T ∼ 1 ps) tend to average out random measurement errors in the time-resolved fluorescence, rendering the method quite robust. Since only the exponential contributes to the complex nature of R(ω), we can use the outcome of eq 6 to determine δs,s′ for all states s and s′ up to ξs,s′ = 0,π. In order to fix the sign of Cs,s′, (or ξs,s′) thereby resolving the π uncertainty in the phase of as, we use BCC15 to extract the individual TDM’s, from which we can, using eq 5, calculate Cs,s′. With knowledge of both Cs,s′ and δs,s′ we have already shown14 how to determine the magnitudes of as through solving a series of equations for each set of states (s,s′). The present application of BCC18−20 consists of applying shortly after the formation of the excited wave packet of eq 1, a single broadband laser pulse ε(ω), containing the ε(ωs,f) and ε(ωs′,f) Fourier components that couple a pair of excited states ψs and ψs′ to a single ground state |f⟩. By manipulating the magnitudes |ε(ω s,f )| and phases ϕ(ω s,f ) of these two components, we can coherently control the population transferred from the ψs and ψs′ components of Ψ(t) to |f⟩. The maximum effect will occur when the ε(ωs,f)asμs,f product will be identical to the ε(ωs′,f)as′μs′,f product, which will then interfere constructively. Under these conditions, knowledge of μs,f will enable us to extract μs′,f according to

know the energy Es, and the (spontaneous emission) decay rate, γs, for each ψs state, written as2

γs =

R(ω) =

∑ 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)

with |μf,s| ≡ |⟨χf |μ|ψs⟩| being a set of transition dipole matrix elements (TDMs) linking the electronically excited energy eigenstates and the ground energy eigenstates, χf. In the past we have shown9−11,17 that knowledge of |μf,s|2 and the ground PES, enable the extraction of the phases of μf,s as well as the excited PES, knowing which, we can calculate the excited state eigenstates ψs(r). This information, together with knowledge of the excitation laser field, is sufficient for the determination of the expansion coefficients as, from which we can obtain the entire excited state wave packet Ψ(r,t). There are, however, difficulties in implementing this procedure in practice due to the scarcity of frequency-resolved studies that record all the transitions mandated by the completeness requirement of the method. For this reason we now introduce an alternative approach that uses time-resolved data for the determination of the phases of μf,s. The method, which by its very nature requires a less complete set of measurements, harnesses a technique recently developed by us for the imaging of excited wave packets moving on known PES.14 It is based on writing the rate of the time-resolved fluorescence from the excited state wave packet of eq 1 as2 R(t ) = ∑ asas*′ Cs , s exp( −iωs , s t ) ′ ′ s,s′

μs

(4)

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

1 T

∫0

(5)

T

dtR(t ) exp(iωt )

≈ |Cs , s || as || as ′| exp[i(δs , s + ξs , s )] for ω = ωs , s ′ ′ ′ ′

∑

asε(ωs , f ) as ′ε(ωs , f ) ′ asε(ωs , f )

μs , f

exp[i(δs , s + ϕs , s ; f )]μs , f ′ ′ as ′ε(ωs , f ) (7) ′ Due to the reality of μs′,f, ϕs,s′;f = ϕ(ωs,f)−ϕ(ωs′,f) must be equal to −δs,s′ or to π − δs,s′, depending on the sign of μs′,f relative to μs,f. Although at this point we do not know the magnitudes |as| and |as′|, provided that neither |as| ≫ |as′| nor |as| ≪ |as′|, the population transfer resulting from complete constructive interference will be distinguishable from the case of destructive interference occurring when ϕs,s′;f → ϕs,s′;f + π. Noting that we are allowed to choose the phases of all (Nf) | f⟩ states and the phases of all (Ns) |ψs⟩ states at will, then of the Ns × Nf transitions, we are free to choose the phases (or signs) of Ns + Nf − 1 of the TDMs. Therefore, we take all the μs,f1 associated with one, |f1⟩, state and all the μs1,f associated with one, |ψs1⟩, state to be positive. The phases of the remaining TDM’s depend, in addition to the overall phase factor multiplying each individual |ψs⟩ or |f⟩ wave function, on their detailed shape (e.g. nodal structure). Given this choice, it follows from eq 7 that the sign of every μs,f TDMs relative to the sign of μs,f1 is determined by whether ϕs,s1;f1−ϕs,s1;f is 0 or π. Once the TDM signs are obtained in this manner, we use eq 5 to compute the Cs,s′ amplitudes. This allows us to use eq 6 to lift the π ambiguity in δs,s′ and obtain the complex as amplitudes. We have thus chartered a way, using BCC and fFT, of obtaining both the amplitudes of the TDMs and the characterization of the time evolving wave packet Ψ(t), without ever having computed the basis wave functions ψs!

In the above we have neglected the decay due to spontaneous emission because for τ (the time scale of interest) of ∼1 ps, γsτ ≪ 1. In other words, due to the relatively long spontaneous emission lifetimes of 10−8−10−6 seconds, only 10−6 − 10−5 of the excited molecules decay over the 1−10 ps time span of interest. We now select14 a specific (s,s′) component of the R(t) signal by calculating over time T the finite time Fourier transform (fFT) of R(t) at each ωs,s′ beat frequency. In this way we filter out all other ω ≠ ωs,s′ components, which decay as 1/T. We have that, R(ω) ≡

=

=

where ωs,s′ = (Es − Es′)/ℏ are “beat” frequencies,16 and Cs,s′ is a time-independent molecular matrix, given as 2 Cs , s = 3 ′ 3c

′

,f

i T

(6)

where ξs,s′(= 0,π) is the phase of the real matrix Cs,s′, and δs,s′ = δs−δs′ is a result of expressing as as as = |as| exp(iδs). In contrast, for ω ≠ ωs,s′, 3084



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∼60 fs over a period of ∼10 ps, a resolution that is attainable experimentally using up-conversion techniques.25−27 As an example, we choose a particular case of {s = 3, s′ = 4, f = 5}, and μ3,5μ4,5 > 0. In this case, ϕ3,4;5 = −δ3,4 corresponds to constructive interference, and thereby maximizes the |R(ω3,4)| fluorescence depletion rate, whereas ϕ3,4;5 = π − δ3,4 minimizes the fluorescence depletion. In Figure 2a we compare the timeresolved R(t) fluorescence rate of eq 4 with BCC relative phase of ϕ3,4;5 = −δ3,4, to the ϕ3,4;5 = π − δ3,4 case. As shown in the Figure 2a, the result is a small (∼ 5%) reduction in the fluorescence signal which would be difficult to detect experimentally. In contrast, as shown in the Figure 2b, the variation of |R(ω3,4)| of eq 6 with ϕ3,4;5 is very significant. When we ignore the ancillary transitions (black line), the fluorescence minimum, occurring exactly at ϕ3,4;5 = −δ3,4, is 250% smaller than the fluorescence maximum, occurring at ϕ3,4;5 = π − δ3,4. The relative difference remains high even when we do take into account the ancillary transitions (blue dashed line), although the maximum gets shifted slightly to the left. In order to quantify the effect, we plot in Figure 2c the contrast ratio

The maximum in the population depletion resulting from the above application of BCC is detected as a minimum in | R(ωs,s′)|, the fluorescence Fourier component of eq 6 . We choose this method in preference to other experimental techniques for population detection, such as laser-induced fluorescence (LIF),21 resonantly enhanced multiphoton ionization (REMPI),22 or, cross-correlation frequency-resolved optical coherent anti-Stokes Raman scattering (XFROG CARS),23,24 because each of these approaches requires additional laser fields and might be further complicated by other excitation pathways. One possible complication that might result is that due to its large bandwidth, the BCC pulse might affect other states that are close in energy to the targeted ones. In practice however, as we show below, these ancillary stimulated emission processes, which occur at different frequencies of the pulse, do not affect the signs of the μs,f derived from the maximal depletion in | R(ωs,s′)|. We demonstrate our imaging procedure using time-resolved fluorescence for an excited sodium dimer (Na2) wave packet, in a process depicted schematically in Figure 1, according to which

C(0, π ) =

|R(ωs , s )0 − R(ωs , s )π | ′ ′ |R(ωs , s )0 + R(ωs , s )π | ′ ′

(8)

where subscript 0 denotes the ϕ3,4;5 = −δ3,4 phase choice, and subscript π denotes the ϕ3,4;5 = π−δ3,4 phase choice. In Figure 2c, we display the contrast C(0,π) as a function of the |ε(ωs,f)/ ε(ωs′,f)| ratio. Since we know from eq 7 that the constructive interference occurs when ε(ωs,f1)asμs,f1 = ε(ωs′,f1)as′μs′,f1, the contrast maximum as a function of |ε(ωs,f1)/ε(ωs′,f1)| immediately yields the |as/as′| ratio (relative to |μs,f1/μs′,f1|). Once the | as/as′| ratios are known, we can, by varying the state |f⟩, obtain all other |μs,f/μs′,f | TDM ratios. This procedure is illustrated in Figure 2d where we present | R(ω3,4)0 − R(ω3,4)π| for various final states |f⟩, for a fixed |ε(ω3,5)/ε(ω4,5)| ratio. The variation in the contrast with |f⟩ yields the variation in the |μs,f/μs′,f | TDM ratios. A case of great importance and much greater complexity is a wave packet evolving on two coupled electronic states. Here the frequency-resolved methods used by us in the past to extract the TDMs9−11 do not work because they rely on the existence of only one excited PES. In this section we demonstrate the viability of the present time-resolved method for such a case by determining the TDMs for the spin−orbit (SO)-coupled RbCs system. Here the coupling occurs between the two lowest excited electronic states.28−30 We simulate the intersystem dynamics assuming that we know the PES of the excited A1Σ+ singlet state and the b3Π excited triplet state.31 In addition we assume that the electronic transition-dipole32 and the A1Σ+/b3Π SO coupling (SOC)33,34 functions are known. The above information allows us to calculate the frequency-resolved fluorescence line strengths |μs,f |2 of RbCs, where s denotes the vibrational states of the SOcoupled A1Σ+/b3Π excited states, and f denotes the vibrational states of the X1Σ+ ground state35 (or possibly the a3Σ+ lowest triplet state). In order to produce the input time-resolved fluorescence, we envision using an ultrashort broad-band pulse to excite the X1Σ+(|v = 0⟩) state to a wave packet composed of the |s = 24− 33⟩ vibrational eigenstates of the SO-coupled A1Σ+/b3Π electronic states. As shown in Figure 3a, these eigenstates straddle the region of crossing between the A1Σ+ and b3Π PES, and in Figure 3b we see the effect of the SOC on the time-

Figure 1. (Main drawing) The ground, X1∑+g , and first excited, A1∑+u , potential energy surfaces of Na2 and a schematic description of the light pulses. Marked as (1) is the pulse exciting the ground vibrational eigenstates |0⟩ to a set of excited vibrational eigenstates |s⟩; (2) the BCC pulse coupling two excited state |s⟩ and |s′⟩ to a ground state |f⟩; (3) the spontaneous emission from the |s⟩ vibrational states of A1∑+u to the |f⟩ vibrational states of X1∑+g . (Inset) The BCC stimulated emission process which couples states |s⟩ and |s′⟩ to |f⟩. Some ancillary couplings of adjacent |s′ + 1⟩ and |s − 1⟩ states to different final states | f⟩ may also result.

a molecule, initially in the X1∑+g ground rovibrational state (|0⟩ = |v = 0;J = 0⟩), gets excited by a short (δt ∼ 20 fs) laser pulse to a wave packet of vibrational eigenstates belonging to the A1∑+u electronic state. As illustrated in the inset of Figure 1, after the excitation pulse is over (at t ∼ 300 fs), we activate a weaker and longer BCC pulse, dominated by two distinct frequencies that mainly couple two vibrational levels (|s⟩,|s′⟩) in A1∑+u to the vibrational state |f⟩ of X1∑+g . After the BCC pulse, which we take to be a Gaussian of timeaveraged intensity of 3 × 1010W/cm2 and bandwidth of 88 cm−1, is over (at t ∼ 500 fs), we collect the time-resolved fluorescence resulting from the decay of the population still residing in the excited states.The fluorescence is recorded every 3085



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Figure 2. (a) R(t) captured over 10 ps and strobed every 60 fs in the presence and absence of the BCC field, which removes populations from the excited state. (b) |R(ω3,4)| as a function of the relative phase ϕ3,4;5 ranging over −π to π. Solid black line: f including only states |3⟩ and |4⟩; dashed blue line: the result of including in addition to states |3⟩ and |4⟩ all ancillary states. (c) The contrast, C(0,π), of eq 8, as a function of |ε(ω3,5)/ ε(ω4,5)|, the ratio between the two interfering components of the BCC field, for fixed |ε(ω3,5)|2+|ε(ω4,5)|2 sum of intensities. The line codes are as in (b). The maximum contrast marks the point at which |ε(ωs,f)/ε(ωs′,f)| = |as′μs′,f/asμs,f |, allowing us to extract |as′/as| and (by varying |f⟩) |μs′,f/μs,f |. (d) | R(ω3,4)| at different ϕ3,4;0 − ϕ3,4;f phase differences for various final states |f⟩.

Figure 3. (a) Schematic of the RbCs system showing the four PES and the location of some vibrational eigenstates. (b) The resulting temporal fluorescence captures over several picoseconds for when the SOC is excluded (blue-square) and included (black-circle).

s⟩ and |s′⟩ excited pair and the |f = 20−29⟩ final vibrational states for which we extract the TDMs. We demonstrate, in Figure 4b, a very successful extraction of the SO-coupled singlet/triplet components (as) of the excited wave packet for two time points at the excitation pulse center t = 0 (blue points), and after 10 ps (red points). The imaged values (thick black line) appear to faithfully reproduce the “true” values, where we have assumed knowledge of the SOcoupled eigenfunctions for presentational purposes. Figure 4b exhibits a transition, taking place over the 10 ps time span, from

dependent fluorescence (black-circle) in contrast to the outcome when this coupling is omitted (blue-squares). We next simulate applying a BCC pulse whose ϕs,s′;f phase differences between the ωs,f and ωs′,f frequency components affect the fluorescence to one of the |f = 20−29⟩ vibrational states of the X1Σ+ PES. In Figure 4a we display as blue circles the BCC pulse bandwidth (ranging over ∼14−22 cm−1), and as black bars, the intensities (ranging between 106−1012 W/cm2) used for each | 3086



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Figure 4. (a) The BCC pulse intensity (black bars) and bandwidth (blue circles) used to achieve sufficient C(0,π) contrast (of at least 20%). The range of the pulse intensity used was varied as a function of the final X1Σ+ state ( f = 20−29) probed. (b) Extraction of the singlet and the triplet components of an excited wave packet Ψ(r,t) at t = 0, the excitation pulse center (“true” wave packet - blue points) and after 10 ps (“true” wave packet - red points). The imaged wave packet, given at both times as a thick black line, faithfully reproduces the “true” values. Shown also are the A1∑+u (thin black line) and b3Π (dashed line) PES.

a wave packet (blue curve) centered about the singlet potential minimum, to a wave packet centered about the triplet potential minimum (red curve). A behavior of this sort is only possible if the blue curve is mostly singlet and the red curve is mostly triplet. The wave packet, composed of the SO coupled eigenstates, starts out as a singlet because the selection rules of the optical transition from the ground singlet create at t = 0 only the singlet part of the wave packet. The temporal evolution of the wave packet which changes the extracted as expansion coefficients to as exp(iEst) gradually builds up a triplet component which, as shown in Figure 4b, becomes very prominent at t = 10 ps. We have shown how time-resolved fluorescence data in conjunction with BCC can be used to derive the phases as well as the amplitudes of the as expansion coefficients of a wave packet ψ(r,t) = ∑sasψs(r) exp(−iEst/ℏ), where ψs(r) are vibrational eigenfunctions. The method can also be used to extract the amplitudes and phases of μs,f, the individual transition dipole matrix-elements (TDMs) between energy eigenstates, ψs and ψf. The extraction of as and μs,f does not necessitate having prior knowledge of the PES and/or SOC terms, nor do we need to know the ψs(r). Imaging of Ψ(r,t) in coordinate space, would appear to necessitate knowledge the PES because it necessitates knowing ψs(r), but as we showed in the past,9−14 it is possible, using the |μs,f | magnitudes, to extract the excited states PES from which we can calculate ψs(r). We are currently extending this approach to multidimensional problems and applying this procedure to obtain transition dipole moments associated with situation involving “internal conversions” (transitions between different electronic states of the same spin multiplicity).

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ACKNOWLEDGMENTS

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REFERENCES

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, and by an NSERC Discovery Grant.

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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. 3087



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Using Time-Resolved Experiments and Coherent Control to

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