J. Phys. Chem. A 2010, 114, 817–832
817
Fully and Partially Coherent Pathways in Multiply Enhanced Odd-Order Wave-Mixing Spectroscopy Nathan A. Mathew, Lena A. Yurs, Stephen B. Block, Andrei V. Pakoulev, Kathryn M. Kornau, Edwin L. Sibert III, and John C. Wright* Department of Chemistry, UniVersity of Wisconsin-Madison, Madison, Wisconsin 53706 ReceiVed: September 11, 2009; ReVised Manuscript ReceiVed: October 30, 2009
Nuclear magnetic resonance spectroscopy relies on using multiple excitation pulses to create multiple quantum coherences that provide great specificity for chemical measurements. Coherent multidimensional spectroscopy (CMDS) is the optical analogue of NMR. Current CMDS methods use three excitation pulses and phase matching to create zero, single, and double quantum coherences. In order to create higher order multiple quantum coherences, the number of interactions must be increased by raising the excitation intensities high enough to create Rabi frequencies that are comparable to the dephasing rates of vibrational coherences. The higher Rabi frequencies create multiple, odd-order coherence pathways. The coherence pathways that involve intermediate populations are partially coherent and are sensitive to population relaxation effects. Pathways that are fully coherent involve only coherences and measure the direct coupling between excited quantum states. The fully coherent pathways are related to the multiple quantum coherences created in multiple pulse NMR methods such as heteronuclear multiple quantum coherence (HMQC) spectroscopy with the important difference that HMQC NMR methods have a defined number of interactions and avoid dynamic Stark effects whereas the multiply enhanced odd-order wave-mixing pathways do not. The difference arises because CMDS methods use phase matching to define the interactions and at high intensities, multiple pathways obey the same phase matching conditions. The multiple pathways correspond to the pathways created by dynamic Stark effects. This paper uses rhodium dicarbonyl chelate (RDC) as a model to demonstrate the characteristics of multiply enhanced odd-order wave-mixing (MEOW) methods. Dynamic Stark effects excite vibrational ladders on the symmetric and asymmetric CO stretch modes and create a series of multiple quantum coherences and populations using partially and fully coherent pathways. Vibrational quantum states up to V ) 6 are excited. A series of spectra provides different two-dimensional cross sections through the multidimensional parameter space involving two excitation frequencies, the frequency of the output coherence, and the excitation pulse time delays. The spectra allow the identification of 18 different overtone and combination band states. Comparison with a local mode model with two anharmonic Morse oscillators with interbond coupling shows excellent agreement. Introduction Multiple pulse nuclear magnetic resonance (NMR) methods such as heteronuclear multiple quantum coherence (HMQC) NMR achieve high spectral selectivity by forming higher order coherences with a series of spin transitions on separate coupled nuclei so the coherences are selective enough to resolve individual residues in complex proteins.1-5 Coherent multidimensional spectroscopy (CMDS) methods are the optical analogues of these NMR methods, and it is important to develop multiple pulse CMDS methods that increase the spectral selectivity by forming similar multiple quantum coherences (MQCs).6-9 Currently, frequency domain CMDS methods such as doubly vibrationally enhanced (DOVE)7,10-12 and triply vibrationally enhanced (TRIVE)6,13-16 spectroscopy and time domain 2D-IR17-21 methods use third order nonlinear processes where three optical pulses excite three successive molecular transitions. The molecular quantum states resulting from the interactions depend on the time ordering and frequencies of the pulses. Pairs of the quantum states created by the successive interactions radiate * To whom correspondence should be addressed. E-mail: wright@ chem.wisc.edu.
new fields that form the output signal. The a,b pairs of states are coherences and are defined by Fba density matrix elements. The output signals appear as coherent beams whose direction is defined by momentum conservation as expressed through the phase matching condition for the excitation and output beam wave-vectors: b kout ) ( b k1 ( b k2 ( b k3. Frequency domain multidimensional spectroscopy measures the signal intensity as a function of the excitation frequencies and the output frequency.13 If the frequencies are resonant with several molecular transitions, the output signal is multiplicatively enhanced by each resonance. The output beam frequency depends both on the driven signal created at ωout ) |(ω1 ( ω2 ( ω3| during the excitations and on the subsequent free induction decay of the output coherence at ωbasthe frequency difference between the two states forming the coherence. Multidimensional spectra result from scanning the excitation and output frequencies while measuring the output signal intensity. Scanning the time delays between each excitation pulse measures the dynamics of the coherences and/or populations. These methods use excitation pulse widths that are comparable to the coherences’ dephasing times so that they are long enough to excite single quantum states but short enough to measure intramolecular dynamics.
10.1021/jp9088063 2010 American Chemical Society Published on Web 12/01/2009
818
J. Phys. Chem. A, Vol. 114, No. 2, 2010
We have recently demonstrated the feasibility of achieving multiply enhanced odd-order wave-mixing (MEOW) CMDS using dynamic Stark effects to create a series of ladder-climbing transitions.22 The transitions form a series of intermediate populations in successively higher vibrational states.23-25 Processes involving up to 11 interactions and populations in the V ) 6 state have been observed. We have also demonstrated that mixed frequency/time domain CMDS methods can create high order MQCs using coherence pathways that are fully coherent and avoid any intermediate populations.26 In this paper, we explore the characteristics of these partially and fully coherent pathways for achieving multiply enhanced odd-order wavemixing (MEOW) CMDS in a rhodium dicarbonyl (RDC) chelate. RDC has served as a model system for the development of a number of new nonlinear methods because of its large transition moment and well-defined vibrational states.27-34 The experiments use two independently tunable excitation beams (labeled ω1 and ω2) to excite two different types of modes. Using the dynamic Stark effects that occur at high intensities, we excite a series of ladder-climbing transitions involving the asymmetric and/or symmetric (designated V and V′, respectively) CO stretch modes. The ω2 beam is split to make a third beam labeled ω2′. A monochromator measures the output frequency (ωm). Changing the time delays (τ21 and τ2′1) between the excitation pulses permits choice of different coherence pathways. The paper describes different strategies for obtaining two-dimensional spectra from cross sections in the five dimensional data space (ω1, ω2, ωm, τ21, and τ2′1). The first strategy fixes τ21 and τ2′1 and constrains ωm so ωm ) ω1 while scanning ω1 and ω2. The second strategy fixes τ21, τ2′1, and ω2 while scanning ωm and ω1. These experiments are performed for both partially and fully coherent pathways and for different excitation pulse intensities. The third strategy fixes ω1, ω2, and ωm while scanning either τ21 and/or τ2′1. Each strategy provides complementary information about the states, coherences, populations, coherence pathways, and dynamic Stark effects. Although the fully coherent pathways generate MQCs that are analogous to those in HMQC-NMR, they have important differences. NMR involves only transitions between two spin states whereas CMDS involves multiple transitions between many vibrational and/or electronic states. The number of interactions in HMQC-NMR is defined by the choice of the pulse sequence. The number of interactions in MQC CMDS is defined by the phase matching geometry and the excitation intensity. If the excitation intensity is low, the Rabi frequencies are smaller than the dephasing, and lower order processes always dominate over higher order processes. In this limit, phase matching can uniquely define the number of interactions. However, low excitation intensities make it difficult to observe higher order MQCs. When the excitation intensities become comparable to the dephasing rates, higher order processes become comparable to lower order processes and are easy to detect. A particular phase matching geometry can still define the lowest number of interactions, but higher order processes must still be considered. Since there are many higher order Liouville pathways that obey any particular phase matching condition when the excitation intensities are high, it is not possible to isolate a particular Liouville pathway with a defined number of interactions. A nonperturbative treatment is necessary to accurately describe this approach. However, we will use the lowest order Liouville pathway that still corresponds to individual spectral features as a representative pathway to aid in the understanding of the experimental spectra.
Mathew et al. Changing the time ordering of the excitation pulses changes the allowed coherence pathways between those that are partially coherent and involve intermediate populations and those that are fully coherent and involve only coherences. The cross-peaks in the partially coherent pathways arise from two different mechanisms. Direct coupling creates cross-peaks between quantum states when excitation of one quantum state directly perturbs the other. Alternatively, population relaxation creates additional cross-peaks when an excited population relaxes to lower energy states.33,35-37 This mechanism requires excitation time orderings and frequencies that create excited state populations. When these populations relax to lower energy states, new coherent pathways become possible and create new cross-peaks involving states that are coupled to the newly populated states. The number of new cross-peaks increases as the relaxation randomizes and populates lower energy modes throughout a molecule. Fully coherent multiple quantum coherence pathways involve neither intermediate populations nor population relaxation pathways so the direct coupling between quantum states can be isolated and identified. These pathways result when the time ordering of the excitation pulses create higher order coherences rather than populations. The mixed frequency/time domain approach is well adapted for MEOW spectroscopy because it allows the creation of MQCs involving quantum states with disparate energies.13 Unlike time domain methods where long-term phase coherence is crucial, these methods require only short-term coherence during the MQC dephasing times. This difference allows mixed frequency/ time domain approaches to use independently tunable excitation pulses that do not have any long-term phase relationship between them so the excitation frequency can be matched to excite different states, regardless of the frequencies. Time domain methods rely on the broad bandwidth of the excitation pulses to excite the states of interest. Theory Coherent multidimensional spectroscopy (CMDS) measures the intensity resulting from the induced polarization in the sample by the excitation electric fields. The polarization is a function of the excitation electric fields and is often described phenomenologically by the Taylor series expansion:38
P ) Eχ(1) + E2χ(2) + E3χ(3) + E4χ(4) + ...
(1)
When the excitation field is weak, the linear term dominates. With focused high energy pulsed lasers, the intensities become high enough that the higher order terms and dynamic Stark effects become significant. Since the even-order terms do not create signals in isotropic samples, only the odd-order terms contribute.39 In this work, three beams create the excitation fields E2 + b E2′, where the subscripts indicate the so b E ) b E1 + b frequencies of the three excitation beams, 1, 2, and 2′. Phase matching isolates the particular types of interactions that each k1 - b k2 + excitation field can create. This work uses the b kout ) b 16 b k2′ TRIVE phase-matching condition. Although this phase matching condition is usually used for four wave mixing (FWM) experiments involving three excitation events, it can also allow bi bi - ((N - 1)/2)k any odd number of interactions if ((N + 1)/2)k b ) ki where N is an odd number of interactions of beam i.26 Thus, the Nth order multiply enhanced beam is generated in the same direction as a triply enhanced beam by adding pairs of interactions to the conservation of momentum expression. k2 - b k2 k1 - b k2 + b One such fifth order process could be b k4 ) b b + k2′. Likewise, momentum conservation expressions for even
Multiply Enhanced Odd-Order Wave-Mixing Spectroscopy
J. Phys. Chem. A, Vol. 114, No. 2, 2010 819
Figure 2. Liouville diagrams of example coherence pathways for crosspeaks where excitation 1 pulse arrives first, and excites the b state while excitation pulse 2′ arrives later and excites the a state. Both pulses are strong. Excitation pulse 2 arrives last and excites the a state. It is not shown on the diagram.
Figure 1. Liouville diagrams of example coherence pathways for (a) cross-peaks where excitation pulses 2 and 2′ are strong and excite the a state and excitation 1 pulse is weak and excites the b state; (b) diagonal peaks where all excitation pulses excite the a state. Pulses 2 and 2′ are strong and 1 is weak. The diagrams assume excitation pulses 2 and 2′ occur first and are temporally overlapped while excitation pulse 1 occurs later. The single arrows denote transitions induced by the superscripted excitation pulse, double-body arrows denote population transfer, double-headed arrows denote transitions in each direction, and double-headed arrows with two excitation pulses designated represent two interactions, one with each excitation pulse. The letters designate the density matrix element defining the amplitude for the pairs of states forming each coherence or population. The processes start in the ground state population denoted with a box.
higher order processes can be written for the same phase matched direction. Liouville diagrams provide convenient visualizations of the different coherence pathways that are possible for any particular phase matching condition.40-43 They contain the same information as the popular Feynman diagrams but they are more compact and show the relationships between multiple pathways such as those in Figures 1-3.10 For example, the Liouville diagram -2
2′
1
ggfgafggfbg corresponds to a Feynman diagram with two bra-side interacb2 and then with b k2′. They excite state g to a tions, first with -k and a back to g. The last is a ket-side interaction with b k1 exciting state g to b and creating the bg output coherence. The higher order wave mixing and dynamic Stark effects described in this paper involve many coherence pathways that create new signals, and since all of the pathways are quantum mechanically equivalent, they can interfere with each other. A detailed treatment of dynamic Stark effects using multiple Liouville pathways appears in a number of earlier publications.44-52 Figures 1-3 show example Liouville pathways for the nonlinear processes that are important for this work. Double headed arrows represent the forward and reverse transitions that characterize dynamic Stark effects.49-52 In the limit where the fields become very strong, nonperturbative methods must be employed.53,54 Figure 1 shows an example of partially coherent Liouville pathways for excitation pulse time orderings where pulses 2 and 2′ are intense and temporally overlapped and pulse 1 arrives later and is weak. Pulses 2 and 2′ excite multiple transitions while pulse 1 excites one transition. Pulses 2 and 2′ excite a
Figure 3. Liouville diagrams of example coherence pathways for diagonal peaks where excitation 1 pulse arrives first while excitation pulse 2′ arrives later. Both pulses are strong and both excite the a state. Excitation pulse 2 arrives last. It is not shown on the diagram.
series of populations and pulse 1 creates the output coherence. We use g, a, and b to designate the ground state and any two excited states. For cross-peaks, we use a to designate states resonant with ω2 and b for states resonant with ω1. For diagonal peaks where ω1 ∼ ω2, both frequencies are resonant with state a. We will later use V and V′ to designate the asymmetric and symmetric RDC carbonyl stretch modes, respectively. Singleheaded arrows represent single field-induced transitions while double-headed arrows represent multiple transitions. The arrows are labeled with the pulses responsible for the excitation. Arrows with two labels are a shorthand notation to represent two successive transitions involving either 2 or 2′. These diagrams must be modified for those experiments where all three fields are intense. In the weak field limit and the phase-matched direction of interest, the lowest order nonlinear process is FWM. If the time ordering of the interactions is -2, 2′, 1, the FWM Liouville pathways are -2
2′
1
ggfgafggfbg and -2
2′
1
ggfgafaafa + b,a where g, a, and b represent the ground state and any two excited states, respectively, and the two letters represent the density matrix ket and bra states describing a coherence or population.14 Pathways VR and Vβ result in rephasing and inhomogeneous linenarrowing if the broadening of states a and b is correlated.10,55-58 If the time ordering is 2′, -2, 1, then the pathways are labeled
820
J. Phys. Chem. A, Vol. 114, No. 2, 2010
Mathew et al.
VIR and VIβ and they are nonrephasing pathways. If the excitation pulses are resonant with the same state, the a and b states become the same and additional pathways become possible. For FWM, the additional pathway is -2
2′
1
ggfgafaafag It is a rephasing pathway and it is labeled Vγ. Interchanging the time ordering of 2′ and -2 gives a nonrephasing pathway labeled VIγ. The higher order pathways create successively higher order coherences from ma,ma populations created by 2m interactions with 2 and 2′. Pulse 1 then interacts with the ma,ma population to create b+ma,ma coherences that emit the output beam. The excited state populations in Figure 1 can also relax to lower states and create new coherence pathways. Consider first the pathways in Figure 1a for creating cross-peaks. The doublebody arrows labeled PT designate population transfer. The ω1 pulse interaction now interacts with the [(m - 1)a + b],[(m 1)a + b] population to create [(m - 1)a + 2b],[(m - 1)a + b] and [(m - 1)a + b], (m - 1)a coherences that emit the output beam. Further population transfer can also occur and is not represented in Figure 1. For example, the 4a,4a population can undergo the following relaxation processes PT
PT
PT
4a,4a⇒(3a+b),(3a+b)⇒(2a+2b),(2a+2b)⇒... and new output coherences will be created from each population. Note that the [(m - 1)a + b], (m - 1)a coherences create crosspeaks that are not distinguishable from the cross-peaks formed without population relaxation, but the [(m - 1)a + 2b],[(m 1)a + b] coherences create cross-peaks that can only come from population relaxation. The pathways in Figure 1b create diagonal peaks. Note that the diagonal peak coherences formed from the direct pathways are different from those created by population transfer. In practice, many of these coherences have similar frequencies but can still be distinguished by the differences in their temporal dependences. Figures 2 and 3 show examples of fully coherent Liouville pathways that assume the excitation pulses ω1 and ω2′ are strong and create n and m transitions, respectively. Note that none of the intermediate steps involve a population, so population relaxation is not expected or observed in this approach.22 The ω1 interaction precedes the ω2′ interaction, and the ω2 pulse that arrives last is sufficiently weak that it causes a single transition. Figure 2 assumes that ω1 and ω2 are resonant with the g f b and g f a transitions, respectively, while Figure 3 assumes that both frequencies are resonant with the g f a transition. The diagram does not show the last interaction with the ω2 pulse because it interacts with all the [ma + nb],[(m 1)a + (n - 1)b] coherences to create either [(m - 1)a + nb],[(m - 1)a + (n - 1)b] or [ma + nb],[ma + (n - 1)b] output coherences for the cross-peaks or the ma,(m - 2)a coherences to create either ma,(m - 1)a or (m - 1)a,(m - 2)a output coherences for the diagonal peaks. Experimental Section The experiments are carried out with a mode-locked Ti: sapphire oscillator/regenerative amplifier that excites two independently tunable optical parametric amplifiers (OPAs). Difference frequency generation in each OPA using AgGaS2 creates mid-infrared beams with frequencies ω1 and ω2. A beam splitter creates two beams from the latter beam. They are labeled ω2 and ω2′ and have 65 and 35% of the original intensity, respectively. Typical pulse energies are 0.2-1.5 µJ for ω1 and
2.3 and 1.2 µJ for ω2 and ω2′, respectively. The pulses are ∼900 fs long and have a bandwidth of ∼20 cm-1. The excitation pulses also have two satellite features that are
