Diagrammatic Perturbation Theory

p. 10-15 10.2. DIAGRAMMATIC PERTURBATION THEORY In practice, the nonlinear response functions as written above provide little insight into what the molecular origin of particular nonlinear signals is. These multiply nested terms are difficult to understand when faced the numerous light-matter interactions, which can take on huge range of permutations when performing experiments on a system with multiple quantum states. The different terms in the response function can lead to an array of different nonlinear signals that vary not only microscopically by the time-evolution of the molecular system, but also differ macroscopically in terms of the frequency and wavevector of the emitted radiation. Diagrammatic perturbation theory (DPT) is a simplified way of keeping track of the contributions to a particular nonlinear signal given a particular set of states in H0 that are probed in an experiment. It uses a series of simple diagrams to represent the evolution of the density matrix for H0, showing repeated interaction of ρ with the fields followed by time-propagation under H 0 . From a practical sense, DPT allows us to interpret the microscopic origin of a signal with a particular frequency and wavevector of detection, given the specifics of the quantum system we are studying and the details of the incident radiation. It provides a shorthand form of the correlation functions contributing to a particular nonlinear signal, which can be used to understand the microscopic information content of particular experiments. It is also a bookkeeping method that allows us to keep track of the contributions of the incident fields to the frequency and wavevector of the nonlinear polarization. There are two types of diagrams we will discuss, Feynman and ladder diagrams, each of which has certain advantages and disadvantages. For both types of diagrams, the first step in drawing a diagram is to identify the states of H0 that will be interrogated by the light-fields. The diagrams show an explicit series of absorption or stimulated emission events induced by the incident fields which appear as action of the dipole operator on the bra or ket side of the density matrix. They also symbolize the coherence or population state in which the density matrix evolves during a given time interval. The trace taken at the end following the action of the final dipole operator, i.e. the signal emission, is represented by a final wavy line connecting dipole coupled states. p. 10-16 Feynman Diagrams1 Feynman diagrams are the easiest way of tracking the state of coherences in different time periods, and for noting absorption and emission events. 1. Double line represents ket and bra side of ρ . 2. Time-evolution is upward. 3. Lines intersecting diagram represent field interaction. Absorption is designated through an inward pointing arrow. Emission is an outward pointing arrow. Action on the left line is action on the ket, whereas the right line is bra. 4. System evolves freely under H0 between interactions, and density matrix element for that period is often explicitly written. Ladder Diagrams2 Ladder diagrams are helpful for describing experiments on multistate systems and/or with multiple frequencies; however, it is difficult to immediately see the state of the system during a given time interval. They naturally lend themselves to a description of interactions in terms of the eigenstates of H0. 1. Multiple states arranged vertically by energy. 2. Time propagates to right. 3. Arrows connecting levels indicate resonant interactions. Absorption is an upward arrow and emission is downward. A solid line is used to indicate action on the ket, whereas a dotted line is action on the bra. 4. Free propagation under H0 between interactions, but the state of the density matrix is not always obvious. p. 10-17 For each light-matter interactions represented in a diagram, there is an understanding of how this action contributes to the response function and the final nonlinear polarization state. Each light-matter interaction acts on one side of ρ , either through absorption or stimulated emission. Each interaction adds a dipole matrix element μij that describes the interaction amplitude and any orientational effects.3 Each interaction adds input electric field factors to the polarization, which are used to describe the frequency and wavevector of the radiated signal. The action of the final dipole operator must return you to a diagonal element to contribute to the signal. Remember that action on the bra is the complex conjugate of ket and absorption is complex conjugate of stimulated emission. A table summarizing these interactions contributing to a diagram is below. Diagrammatic Representation Interaction contrib. to ( n) contribution to μba ⋅ ˆε n +k n + ωn μba ⋅ ˆε n −k n − ωn μ*ba ⋅ ˆε n −k n − ωn μ*ba ⋅ ˆε n +k n + ωn R k sig & ωsig KET SIDE Absorption (μ b b En a a En* b a a b ⎡ ⎤ ba ⋅ En ) exp ⎣ik n ⋅ r − iωn t ⎦ Stimulated Emission ( μba ⋅ E ) exp ⎣⎡−ikn ⋅ r + iωnt ⎦⎤ * n BRA SIDE Absorption (μ * ba ⋅ En* ) exp ⎣⎡ −ikn ⋅ r + iωn t ⎦⎤ Stimulated Emission (μ * ba ⋅ En ) exp ⎣⎡ikn ⋅ r − iωnt ⎦⎤ b b a En* a b En a a b SIGNAL EMISSION: b a (Final trace, convention: ket side) a b μba ⋅εˆ an p. 10-18 Once you have written down the relevant diagrams, being careful to identify all permutations of interactions of your system states with the fields relevant to your signal, the correlation functions contributing to the material response and the frequency and wavevector of the signal field can be readily obtained. It is convenient to write the correlation function as a product of several factors for each event during the series of interactions: 1) Start with a factor pn signifying the probability of occupying the initial state, typically a Boltzmann factor. 2) Read off products of transition dipole moments for interactions with the incident fields, and for the final signal emission. 3) Multiply by terms that describe the propagation under H 0 between interactions. As a starting point for understanding an experiment, it is valuable to include the effects of relaxation of the system eigenstates in the time-evolution using a simple phenomenological approach. Coherences and populations are propagated by assigning the damping constant Γ ab to propagation of the ρ ab element: Ĝ (τ ) ρ ab = exp [ −iωabτ − Γ abτ ] ρ ab . (10.1) * Note Γ ab = Γba and Gab = Gba . We can then recognize Γii =1 T1 as the population relaxation rate for state i and Γij = 1 T2 the dephasing rate for the coherence ρij . 4) Multiply by a factor of ( −1) where n is the number of bra side interactions. This factor n accounts for the fact that in evaluating the nested commutator, some correlation functions are subtracted from others. 5) The radiated signal will have frequency ωsig = ∑ ωi and wave vector k sig = ∑ ki i i Example: Linear Response for a Two-Level System Let’s consider the diagrammatic approach to the linear absorption problem, using a two-level system with a lower level a and upper level b. There is only one independent correlation function in the linear response function, p. 10-19 C ( t ) = Tr ⎡⎣ μ ( t ) μ ( 0 ) ρ eq ⎤⎦ (10.2) = Tr ⎡⎣ μ Gˆ ( t ) μ ρeq ⎤⎦ This does not need to be known before starting, but is useful to consider, since it should be recovered in the end. The system will be taken to start in the ground state ρaa. Linear response only allows for one input field interaction, which must be absorption, and which we take to be a ket side interaction. We can now draw two diagrams: (4) Act on ket with μ and take trace. (3) Propagate under H0: −iω τ −Γbaτ Gab (τ ) = e ba . (2) Act on ket with μ(0) to create ρba. (1) Start in ρaa (add factor of pa when reading). With this diagram, we can begin by describing the signal characteristics in terms of the induced polarization. The product of incident fields indicates: E1 e−iω1t +ik1⋅r so that ⇒ P (t ) e −iωsig t +iksig ⋅r ωsig = ω1 ksig = k . (10.3) (10.4) As expected the signal will radiate with the same frequency and in the same direction as the incoming beam. Next we can write down the correlation function for this term. Working from bottom up: (1) (2) (3) (4) C ( t ) = pa [ μba ] ⎡⎣e −iωbat −Γbat ⎤⎦ [ μab ] (10.5) = pa μba e−iωbat−Γbat 2 More sophisticated ways of treating the time-evolution under H0 in step (3) could take the form of some of our earlier treatments of the absorption lineshape: p. 10-20 Gˆ (τ ) ρ ab ~ ρ ab exp [ −iωabτ ] F (τ ) = ρ ab exp ⎡−i ⎣ ωabτ − g ( t ) ⎤⎦ (10.6) Note that one could draw four possible permutations of the linear diagram when considering bra and ket side interactions, and initial population in states a and b: However, there is no new dynamical content in these extra diagrams, and they are generally taken to be understood through one diagram. Diagram ii is just the complex conjugate of eq. (10.5) so adding this signal contribution gives: C ( t ) − C *( t ) = 2i pa μba sin(ωba t) e−Γbat . 2 (10.7) Accounting for the thermally excited population initially in b leads to the expected two-level system response function that depends on the population difference R (t ) = 2 2 ( pa − pb ) μba sin(ωbat) e−Γbat . h (10.8) Example: Second-Order Response for a Three-Level System The second-order response is the simplest nonlinear case, but in molecular spectroscopy is less commonly used than third-order measurements. The signal generation requires a lack of inversion symmetry, which makes it useful for studies of interfaces and chiral systems. However, let’s show how one would diagrammatically evaluate the second order response for a very specific system pictured at right. If we only have population in the ground state at equilibrium and if there are only resonant interactions allowed, the permutations of unique diagrams are as follows: p. 10-21 From the frequency conservation conditions, it should be clear that process i is a sum-frequency signal for the incident fields, whereas diagrams ii-iv refer to difference frequency schemes. To better interpret what these diagrams refer to let’s look at iii. Reading in a time-ordered manner, we can write the correlation function corresponding to this diagram as C2 = Tr ⎡⎣ μ (τ ) ρeq μ ( 0 ) ⎤⎦ ∗ . = ( −1) μbc Ĝcb (τ 2 ) μca Ĝab (τ 1 ) ρ aa μba 1 = − pa μab μbc μca e −iωabτ1 −Γabτ1 e −iωcbτ 2 −Γcbτ 2 Note that a literal interpretation of the final trace in diagram iv would imply an absorption event – an upward transition from b to c. What does this have to do with radiating a signal? On the one hand it is important to remember that a diagram is just mathematical shorthand, and that one can’t distinguish absorption and emission in the final action of the dipole operator prior to taking a trace. The other thing to remember is that such a diagram always has a complex conjugate associated with it in the response function. The complex conjugate of iv, a Q2∗ ket/bra term, shown at right has a downward transition –emission– as the final interaction. The combination Q2 − Q2∗ ultimately describes the observable. (10.9) p. 10-22 Now, consider the wavevector matching conditions for the second order signal iii. Remembering that the magnitude of the wavevector is k = ω c = 2π λ , the length of the vectors will be scaled by the resonance frequencies. When the two incident fields are crossed as a slight angle, the signal would be phase-matched such that the signal is radiated closest to beam 2. Note that the most efficient wavevector matching here would be when fields 1 and 2 are collinear. p. 10-23 Third-Order Nonlinear Spectroscopy Now let’s look at examples of diagrammatic perturbation theory applied to third-order nonlinear spectroscopy. Third-order nonlinearities describe the majority of coherent nonlinear experiments that are used including pump-probe experiments, transient gratings, photon echoes, coherent anti-Stokes Raman spectroscopy (CARS), and degenerate four wave mixing (4WM). These experiments are described by some or all of the eight correlation functions contributing to R (3) : ⎛i⎞ R ( 3) = ⎜ ⎟ ⎝h⎠ 3 4 ⎡⎣ Rα − Rα ⎤⎦ ∑ α * (10.10) =1 The diagrams and corresponding response first requires that we specify the system eigenstates. The simplest case, which allows us discuss a number of examples of third-order spectroscopy is a two-level system. Let’s write out the diagrams and correlation functions for a two-level system starting in ρ aa , where the dipole operator couples b and a . R1 ket/ket/ket E3 R2 bra/ket/bra R3 bra/bra/ket R4 ket/bra/bra b a τ3 b a b a b a a a τ2 b b a a b b b a τ1 a b a b b a a a a a a a −ω1 + ω2 + ω3 − k1 + k2 + k3 +ω1 − ω2 + ω3 + k1 − k2 + k3 E2 a a E1 b a +ω1 − ω2 + ω3 k sig = + k1 − k2 + k3 −ω1 + ω2 + ω3 − k1 + k2 + k3 p. 10-24 As an example, let’s write out the correlation function for R2 obtained from the diagram above. This term is important for understanding photon echo experiments and contributes to pump-probe and degenerate four-wave mixing experiments. ( * R2 = ( −1) pa ( μba ) ⎡⎣e−iωabτ1 −Γabτ1 ⎦⎤ ( μba ) e −iωbbτ 2 −Γbbτ 2 2 )(μ * ab ) ⎡⎣e−iω τ −Γ ba 3 baτ 3 ⎤ ⎦ ( μab ) = pa μab exp ⎡⎣ −iωba (τ 3 − τ 1 ) − Γba (τ 1 + τ 3 ) − Γbb (τ 2 ) ⎤⎦ 4 (10.11) The diagrams show how the input field contributions dictate the signal field frequency and wave( 3) vector. Recognizing the dependence of Esig ~ P ( 3) ~ R2 ( E1 E2 E3 ) , these are obtained from the product of the incident field contributions ( E1 E2 E3 = E1* e + iω1t −ik1 ⋅r ⇒ E1* E2 E3 e )( E e 2 −iω2t +ik2 ⋅r )(E e 3 +iω3t −ik3 ⋅r3 ) −ωsig t + iksig ⋅r ∴ ωsig 2 = − ω1 + ω2 + ω3 ksig 2 = − k1 + k2 + k3 . (10.12) (10.13) Now, let’s compare this to the response obtained from R4 . These we obtain R4 = pa μab exp ⎡− ⎣ iωba (τ 3 + τ 1 ) − Γba (τ 1 + τ 3 ) − Γbb (τ 2 ) ⎤⎦ 4 ωsig 4 = + ω1 − ω2 + ω3 (10.14) (10.15) ksig 4 = + k1 − k2 + k3 Note that R2 and R4 terms are identical, except for the phase acquired during the initial period: exp [iφ ] = exp [ ±iωbaτ 1 ] . The R2 term evolves in conjugate coherences during the τ1 and τ3 periods, whereas the R4 term evolves in the same coherence state during both periods: Coherences in τ 1 and τ 3 Phase acquired in τ 1 and τ 3 R4 b a → b a e R2 a b → b a e − iωba (τ1 +τ 3 ) − iωba (τ1 −τ 3 ) The R2 term has the property of time-reversal: the phase acquired during τ1 is reversed in τ3. For that reason the term is called “rephasing.” Rephasing signals are selected in photon echo experiments and are used to distinguish line broadening mechanisms and study spectral diffusion. For R4 , the phase acquired continuously in τ1 and τ3, and this term is called “non­ p. 10-25 rephasing.” Analysis of R1 and R3 reveals that these terms are non-rephasing and rephasing, phase acquired respectively. φ e 0 e −iωbaτ 3 −iωbaτ1 non-rephasing e +iωbaτ 3 τ1 τ3 rephasing P (t ) t1 t t2 t3 For the present case of a third-order spectroscopy applied to a two-level system, we observe that the two rephasing functions R2 and R3 have the same emission frequency and wavevector, and would therefore both contribute equally to a given detection geometry. The two terms differ in which population state they propagate during the τ2 variable. Similarly, the non­ rephasing functions R1 and R4 each have the same emission frequency and wavevector, but differ by the τ2 population. For transitions between more than two system states, these terms could be separated by frequency or wavevector (see appendix). Since the rephasing pair R2 and R3 both contribute equally to a signal scattered in the −k1 + k2 + k3 direction, they are also referred to as SI. The nonrephasing pair R1 and R4 both scatter in the + k1 − k2 + k3 direction and are labeled as SII. Our findings for the four independent correlation functions are summarized below. SI rephasing SII non-rephasing ωsig k sig τ2 population R2 −ω1 + ω2 + ω3 −k1 + k2 + k3 excited state R3 −ω1 + ω2 + ω3 −k1 + k2 + k3 ground state R1 +ω1 − ω2 + ω3 + k1 − k2 + k3 ground state R4 +ω1 − ω2 + ω3 + k1 − k2 + k3 excited state p. 10-26 Frequency Domain Representation4 A Fourier-Laplace transform of P ( 3) ( t ) with respect to the time intervals allows us to obtain an expression for the third order nonlinear susceptibility, χ (3) (ω1 , ω2 , ω3 ) : P (3) (ωsig ) = χ (3) (ωsig ; ω1 , ω2 , ω3 ) E1 E2 E3 where ∞ ∞ (10.16) χ ( n ) = ∫ dτ n eiΩ τ L ∫ dτ 1 eiΩ τ R ( n ) (τ 1 ,τ 2 ,Kτ n ) . 0 (10.17) 1 1 n n 0 Here the Fourier transform conjugate variables Ωm to the time-interval τ m are the sum over all frequencies for the incident field interactions up to the period for which you are evolving: m Ωm = ∑ ωi (10.18) i=1 For instance, the conjugate variable for the third time-interval of a + k1 − k2 + k3 experiment is the sum over the three preceding incident frequencies Ω3 = ω1 − ω2 + ω3 . In general, χ(3) is a sum over many correlation functions and includes a sum over states: χ ( 3) 1 i (ω1 , ω2 , ω3 ) = ⎛⎜ ⎞⎟ 6⎝h⎠ 3 4 ∑ p α∑ ⎡⎣ χα − χα ⎤⎦ a abcd * (10.19) =1 Here a is the initial state and the sum is over all possible intermediate states. Also, to describe frequency domain experiments, we have to permute over all possible time orderings. Most generally, the eight terms in R (3) lead to 48 terms for χ ( 3) , as a result of the 3!=6 permutations of the time-ordering of the input fields.5 Given a set of diagrams, we can write the nonlinear susceptibility directly as follows: 1) Read off products of light-matter interaction factors. 2) Multiply by resonance denominator terms that describe the propagation under H 0 . In the frequency domain, if we apply eq. (10.17) to response functions that use phenomenological time-propagators of the form eq. (10.1), we obtain Ĝ (τ m ) ρ ab ⇒ 1 ( Ωm − ωba ) − iΓba . (10.20) 3) Ωm is defined in eq. (10.18). n As for the time domain, multiply by a factor of ( −1) for n bra side interactions. 4) The radiated signal will have frequency ωsig = ∑ ωi and wavevector ksig = ∑ ki . i i p. 10-27 As an example, consider the term for R2 applied to a two-level system that we wrote in the time domain in eq. (10.11) χ 2 = μba 4 = μba 4 ( −1) ( −1) 1 ⋅ ⋅ ωab − ( −ω1 ) − iΓ ab ωbb − (ω2 − ω1 ) − iΓbb ωba − (ω3 + ω2 − ω1 ) − iΓba 1 ⋅ 1 ⋅ 1 (10.21) ω1 − ωba − iΓba − (ω2 − ω1 ) − iΓbb − (ω3 + ω2 − ω1 − ωba ) − iΓba The terms are written from a diagram with each interaction and propagation adding a resonant denominator term (here reading left to right). The full frequency domain response is a sum over multiple terms like these. p. 10-28 Appendix: Third-order diagrams for a four-level system The third order response function can describe interaction with up to four eigenstates of the system Hamiltonian. These are examples of correlation functions within R(3) for a four-level system representative of vibronic transitions accompanying an electronic excitation, as relevant to resonance Raman spectroscopy. Note that these diagrams present only one example of multiple permutations that must be considered given a particular time-sequence of incident fields that may have variable frequency. R1 ket / ket / ket E3 R2 bra/ket/bra R3 bra/bra/ket R4 ket/bra/bra d a τ3 d c d c b a c a τ2 d b a c b b b a τ1 a b a b b a a a a a a a E2 a a E1 d d d d b b b b c a c a c a c a μba μcb μdc μad μba* μda μcb* μcd μba* μcb* μda μcd ωsig = +ω1 − ω2 + ω3 −ω1 + ω2 + ω3 −ω1 + ω2 + ω3 = ωba + ωcb + ωdc = ωda −ωba + ωda − ωcb = ωdc −ωba − ωcb + ωda = ωdc * μba μda μcd* μcb +ω1 − ω2 + ω3 ωba − ωda − ωcd = ωbc The signal frequency comes from summing all incident resonance frequencies accounting for the sign of the excitation. The products of transition matrix elements are written in a time-ordered fashion without the projection onto the incident field polarization needed to properly account for orientational effects. The R1 term is more properly written ( μba ⋅ εˆ1 )( μcb ⋅ εˆ2 )( μdc ⋅ εˆ3 )( μad ⋅ εˆan ) . Note that the product of transition dipole matrix elements obtained from the sequence of interactions can always be re-written in the cyclically invariant form μab μbc μcd μda . This is one further manifestation of closed loops formed by the sequence of interactions. p. 10-29 Appendix: Third-order diagrams for a vibration The third-order nonlinear response functions for infrared vibrational spectroscopy are often applied to a weakly anharmonic vibration. For high frequency vibrations in which only the v = 0 state is initially populated, when the incident fields are resonant with the fundamental vibrational transition, we generally consider diagrams involving the system eigenstates v = 0, 1 and 2, and which include v=0-1 and v=1-2 resonances. Then, there are three distinct signal contributions: Signal ksig SI −k1 + k2 + k3 Diagrams and Transition Dipole Scaling rephasing μ10 SII 4 μ10 4 2 μ10 μ21 2 + k1 − k2 + k3 non-rephasing μ10 SIII R/NR 4 μ10 4 2 μ10 μ21 + k1 + k2 − k3 2 non-rephasing 2 μ10 μ21 2 2 μ10 μ21 2 p. 10-30 Note that for the SI and SII signals there are two types of contributions: two diagrams in which all interactions are with the v=0-1 transition (fundamental) and one diagram in which there are two interactions with v=0-1 and two with v=1-2 (the overtone). These two types of contributions have opposite signs, which can be seen by counting the number of bra side interactions, and have emission frequencies of ω10 or ω21. Therefore, for harmonic oscillators, which have ω10 = ω21 and 2 μ10 = μ21 , we can see that the signal contributions should destructively interfere and vanish. This is a manifestation of the finding that harmonic systems display no nonlinear response. Some deviation from harmonic behavior is required to observe a signal, such as vibrational anharmonicity ω10 ≠ ω21, electrical anharmonicity 2 μ10 ≠ μ21 , or level-dependent damping Γ10 ≠ Γ21 or Γ00 ≠ Γ11 . 1 . Yee, T. K. & Gustafson, T. K. Diagrammatic analysis of the density operator for nonlinear optical calculation: Pulsed and cw responses. Phys. Rev. A 18, 1597 (1978) Fujimoto, J.G. and Yee, T.K., Diagrammatic Density Matrix Theory of Transient FourWave Mixing and the Measurement of Transient Phenomena. IEEE J. Quantum Electron. QE-22, 1215 (1986) Druet, S. A. J., Taran, J.-P. E. & Bordé, C. J. Line shape and Doppler broadening in resonant CARS and related nonlinear processes through a diagrammatic approach. J. Phys. 40, 819 (1979). Mukamel, S. Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995). Shen, Y. R. The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984). 2. D. Lee and A. C. Albrecht, “A unified view of Raman, resonance Raman, and fluorescence spectroscopy (and their analogues in two-photon absorption).” Adv. Infrared and Raman Spectr. 12, 179 (1985). 3. To properly account for all orientational factors, the transition dipole moment must be projected onto the incident electric field polarization εˆ leading to the terms in the table. This leads to a nonlinear polarization that can have x, y, and z polarization components in the lab frame. The are obtained by projecting the matrix element prior to the final trace onto the desired analyzer axis εˆan . 4. Prior, Y. A complete expression for the third order susceptibility χ ( 3) -perturbative and diagramatic approaches. IEEE J. Quantum Electron. QE-20, 37 (1984). See also, Dick, B. Response functions and susceptibilities for multiresonant nonlinear optical spectroscopy: Perturbative computer algebra solution including feeding. Chem. Phys. 171, 59 (1993). 5. Bloembergen, N., Lotem, H. & Lynch, R. T. Lineshapes in coherent resonant Raman scattering. Indian J. Pure Appl. Phys. 16, 151 (1978).