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Current vs charge density contributions to nonlinear X-ray spectroscopy Jeremy R. Rouxel, Markus Kowalewski, and Shaul Mukamel J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00279 • Publication Date (Web): 27 Jun 2016 Downloaded from http://pubs.acs.org on June 28, 2016

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Journal of Chemical Theory and Computation

Current vs charge density contributions to nonlinear X-ray spectroscopy Jérémy R. Rouxel,∗ Markus Kowalewski,∗ and Shaul Mukamel∗ Department of Chemistry, University of California, Irvine, California 92697-2025, USA E-mail: [email protected]; [email protected]; [email protected] Abstract Stimulated (coherent) and spontaneous (incoherent) nonlinear X-ray signals are expressed using a spatially non-local response tensor which directly connects them to the time evolving current j and charge σ densities rather than to electric and magnetic multipoles. The relative contributions of the σA2 and j · A minimal coupling terms, where A is the vector potential, are demonstrated. The two dominate off-resonant and resonant scattering respectively and make comparable contributions at near resonant detunings.

1 Introduction Recent development of bright ultrafast coherent X-ray sources, such as free electron lasers (XFEL) 1 or high harmonic generation (HHG), 2 has triggered an intense research activity aimed at the experimental realization of all-X-ray nonlinear spectroscopy. 3–5 Combined with other advances in single molecule spectroscopy 6 and 1

ACS Paragon Plus Environment


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ultrafocusing of beams, 7,8 this opens up new opportunities to create real time movies of elementary chemical events. This article applies a non-local response formalism based on the minimal coupling Hamiltonian of light-matter interaction to the X-ray regime. Extending the standard response function theories 9–11 using the minimal coupling Hamiltonian permits one to include the spatial variation of the exciting electric and magnetic fields. The non-local response approach has several merits. First, in the hard Xray regime, the dipole (long wavelength) approximation may fail for certain material transitions. Deviation from the dipole approximation may also be observed at longer wavelengths when the current densities are highly delocalized. A non-local description that implicitly accounts for all higher multipoles is then called for. Expansion of the current into electric and magnetic multipole moments is possible but becomes tedious when higher multipoles are needed, while the nonlocal description leads to expressions that are only marginally more complicated than the dipole case. Second, the non-local response tensors are expressed in terms of physically intuitive elementary quantities: the time dependent current and charge densities. Third, non-local response tensors encompass both the contribution of current j and charge σ densities while the σ term is neglected is the dipole approximation. This allows to describe pure σ interaction pathways, usually used in off-resonance diffraction applications, or pure resonant j pathways that, when integrated over space, correspond to the standard resonant nonlinear optical pathways in the dipole approximation. Pathways that depend on both j and σ, and are relevant at intermediate detunings, can be accounted for. In near resonant diffraction, these mixed charge/current contributions are known as anomalous diffraction. 12 The outline of this paper is as follow. First, we review the non-local response 2


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description of nonlinear spectroscopy. Formal expressions are derived for heterodyne detected (coherent, stimulated) signals as well as for spontaneous emission (incoherent) signals. In section 3, we use this formalism to simulate various heterodyne (stimulated) signals for thiophene at the sulfur K-edge. These include X-ray absorption (XAS) and stimulated X-ray Raman spectroscopy (SXRS). Spontaneous scattering signals are then discussed and formally linked to two-photon absorption in section 4.

2 Nonlocal description of X-ray spectroscopy The nonlocal response formalism 13,14 starts with the minimal coupling field-matter interaction Hamiltonian 15 :

Hint (t) = −

Z

dr j(r) · A(r, t) −

e σ(r)A2 (r, t) 2mc

(1)

e and m are the electron charge and mass, A(r) is the vector potential. e |r ihr | p + p† |r ihr | 2m e¯h = ψ† (r)∇ψ(r) − (∇ψ(r)† )ψ(r) 2mi

j(r) =

(2)

is the current density operator and σ (r) = eψ† (r)ψ(r)

(3)

is the charge density operator. ψ† (r) and ψ(r) are the electron field fermion creation and annihilation operators at position r. p is the momentum operator. Note that 3



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the required molecular poperties, namely the charge and the current densities are independent of the choice of origin. We shall consider signals given by the rate of change of the photon occupation number Ns in the detected field mode s : S(ks , {αi }) =

Z

dthh N˙ s |ρ T (t)ii

(4)

Here, Ns is the number operator for photon in the mode s. |Oii denotes Liouville

space vectors and hh A| Bii= ˆ Tr( A† B) is the scalar product in Liouville space. 9 ρ T (t)

denotes the total (field and matter) density matrix and ρ(t) is the matter density matrix. The set αi represents the parameters used in a specific signal (time delays, incoming wave vectors...). These will be specified in the following applications. In appendix A, we employ a quantum description of the detected field modes to obtain : i S(ks , {αi }) = h¯

Z

drdthhJ(r) · A(r)|ρ T (t)ii

(5)

The physical matter quantity that determines this signal is the gauge invariant electron current density 13

J(r) = j(r) −

e A(r) σ (r) mc

(6)

Assuming a classical optical field, we can recast eq 5 as : i S(ks , {αi }) = h¯

Z

dr hhJ(r)|ρ(t)ii · A(r, t)

4


(7)

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h

hhJ(r)|ρ(t)ii = Tr J(r) exp+ − i

Z

Lint (t) ρ−eq

i

(8)

Here, ρ−eq is the stationary density matrix and the interaction Liouvillian Lint (r, t) = eint (r, t) is : H

Lint (t) = −

Z

e e σ(r)A2 (r, t) drej(r) · A(r, t) − 2mc

(9)

The tilde denote the commutation superoperator associated with a Hilbert space e ≡ [O, X ]. Eq. 7 is known as the heterodyne detected signal, which operator O, OX

represents the change of transmission of an incoming probe field.

Specific signals can be obtained by expanding the exponent in equation 8 per(n)

tubatively in Lint (t). We will refer to the n’th order term as Lint . We should then collect terms in powers of A. Since Lint (t) (eq. 9) is both linear and quadratic in A, (n)

Lint will contain contributions to all orders between An to A2n . The common prac-

tice is to use the e σA2 coupling for off-resonant processes and the ej · A coupling for

resonant processes. 16 Hereafter, we examine the relative contributions of both terms, discuss the validity of the above approximations and present some four-wave mixing signals that involve both resonant and off resonant contributions. To first order in A, the gauge invariant current density (eq. 8) is :

J

(1)

i (r, ω ) = − h¯

Z

e dr1 dt1 hhj(r )|G(t1 )ej(r1 ) · A(r1 , ω )eiωt1 |ρ−∞ ii − hhσ(r )|ρ−eq iiA(r, ω ) mc (10)

5



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where

G(t) = θ (t) ∑ |νν′ iie−iωνν′ t−Γνν′ t hhνν′ |

(11)

νν′

A sum-over-states expression for J(1) (r, ω ) for an incoming Gaussian pulse, is given in appendix B. By displaying at the total current density J(r, t) that generates the detected field, we can visualize the contribution of different regions in the molecule to the response. In fig. 1, the first order current density J(1) (r, ω ) given in eq. 10 is displayed for various excitation wavelengths. The total current density is delocalized for optical transitions and becomes more localized for deeper core transitions. This is connected to the localization of the molecular orbitals of the states involved in the transition: the j term in eq. 10 has a spatial extent comparable to the underlying core orbital while the σ term, negligible at resonance, has a contribution that extends over the molecule. Higher order nonlinear current densities J(2) , J(3) , etc, which depend on various interaction delays between pulses, can provide a real space visualization of multidimensional signals.

2.1 Stimulated (heterodyne-detected) third-order signals Below, we give the expressions for time and wavevector resolved signals for impulsive pulses based on the results of references. 13,14 Indices i, ii, iii will be used to denote the first, second and third order contribution in Lint respectively. The third order non-local response function is given by : S(3) (ks , k1 k2 k3 , t1 t2 t3 ) = Si + Sii + Siii 6


(12)

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For classical optical fields, the heterodyne signal is given by : i Shet (ks , {αi }) = h¯

Z

dks dk3 dk2 dk1 dtdt3 dt2 dt1 (3) S (ks , k1 k2 k3 , t1 t2 t3 ) · As (ks , t) ⊗ A3 (k3 , t − t3 ) ⊗ A2 (k2 , t − t3 − t2 ) ⊗ A1 (k1 , t − t3 − t2 − t1 )

(16)

where {αi } stands for all pulse parameters. Assuming that the incoming pulses are temporally well separated, the interactions with the various pulses have a controlled time ordering. In this case, there are 22 possible interaction pathways (8 from Siii , 12 from Sii and 2 from Si ) that can be represented by ladder diagrams. Adding the possibility to interact with either the positive or the negative frequency component of the vector potential for each interaction yields 22×16 ladder diagrams. The rules

for calculating these diagrams in the dipole approximation are given in Ref. 9 Some additional rules are needed to extend them to the minimal coupling Hamiltonian and allow the possibility to interact either with σ or j (eq. 1).

1. Each interaction with the current density brings a factor of −i/¯h, each interaction with the charge density brings a factor of −e/2mc (−e/mc if this interaction is the last). 2. Each interaction with the charge density σ (ri ) is accompanied by δ(ri+1 − r i ) δ ( t1 ). Depending on the level scheme and field frequencies, some diagrams may be neglected by invoking the rotating wave approximation (RWA), which simplifies the calculation. Moreover, the position of the detector will select only a subset of diagrams for a given phase matching direction. In the dipole approximation, the signal 8


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is generated along the directions ±k1 ± k2 ± k3 . In the non-local formalism, the pulses can come with a finite k space envelope (ultra focused fields) and each diagram represents an emission in a cone. It is then necessary to sum over all the modes that the detector receives taking into account its position and spatial extent. For plane wave fields, one can eliminate the k space integrals in eq. 16 but the signal will still depend on the incoming wave vector. This applies for example to diffraction.

2.2 Incoherent scattering signals Fluorescence and diffraction are spontaneously emitted (scattering) signals. We assume that the detected field mode is initially in the vacuum state and the field+matter density matrix prior to detection is |ρ T (t)ii = |ρ M (t)ii ⊗ |0s ii. If the matter is initially in its ground state, only the J(r) as contribution in eq. 31 (appendix A) survives the RWA. A zero order expansion of the density matrix in eq. 5 gives a vanishing signal. The density matrix must be expanded at least to first order in the interaction with the spontaneous mode s. The second interaction with the detected mode yields a non-vanishing field density matrix element and the trace over the field degrees of freedom no longer vanishes.

Ssle (ks ) =

Z i i e es (r ) · A(r, t) G(t1 ) hh Ns | ejs (r ) − σ − h¯ h¯ 2mc ejs (r1 ) − e σ es (r1 ) · A(r1 , t − t1 ) |ρ T (t − t1 ) ⊗ 0s , 0s ii (17) 2mc

where js (r ) = j(r ) · As (r ) and σs (r ) = σ (r ) · As (r ). A with no subscript stands for externally applied field modes.

We have calculated all elements of this expansion using the matrix elements 9



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given in appendix A. By tracing out the detected mode, the signal is finally recast in terms of purely matter correlation functions :

Ssle (ks ) =

+ + +

Z

h¯ 2 drs dr′s dtdts hhj(rs ) · As |G(ts )ej(r′s ) · As |ρ(t − ts )ii 2ǫ0 L3 ωs h¯ 2 e hhj(rs ) · As |G(ts )e σ(r′s )|ρ(t − ts )iiAs · A(r′s , t − ts ) − mc e hhσ(rs )|G(ts )ej(r′s ) · As |ρ(t − ts )iiAs · A(rs , t) − 2mc −e −e hhσ(rs )|G(ts )e σ(r′s )|ρ(t − ts )iiAs · A(rs , t)As · A(r′s , t −(18) ts ) 2mc mc

The density matrix may be expanded further in the incoming classical fields as it was done in the previous section. The signal is finally obtained by replacing the R discrete sum over field modes by its mode density ∑i −→ L3 dωs (ωs2 /π 2 c3 ) : ωs Ssle (ks ) = 2ǫ0 π 2 c3 h¯

Z

drdr′s dtdts

1 Ssle

2 + Ssle

3 + Ssle

(19)

1 Ssle = hhj(rs ) · ǫs eiks ·rs |G(ts )ej(r′s ) · ǫs |ρ(t − ts )ii (20) e 2 hhj(rs ) · ǫs |G(ts )e σ(r′s )|ρ(t − ts )iiǫs · A(r′s , t − ts ) Ssle = − 2mc e hhσ(rs )|G(ts )ej(r′s ) · ǫs |ρ(t − ts )iiǫs · A(rs , t) (21) + − 2mc e 2 3 Ssle = − hhσ(rs )|G(ts )e σ (r′s )|ρ(t − ts )iiǫs · A(rs , t)ǫs · A(r′s , t − ts ) (22) 2mc

ǫs is the polarization vector of the detected photon. In the absence of a polarized detection, one has to sum over polarizations.

Eqs. 19-22 may be used as a starting point for a pertubative expansion of the matter density matrix |ρ(t − t1 )ii in the incoming fields thus yielding various types of multidimensional signals. The system may be excited by an arbitrary sequence of 10


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pulses and then evolves freely until the time t − t1 where the measurement process starts as given by eq. 22.

3 Nonlinear spectroscopy at the sulfur K-edge of thiophene Core hole states in molecules can be calculated with e.g. the ∆SCF method, time dependent density functional theory (TDDFT) and Complete Active Space Self Consistent Field (CASSCF) based methods (for a broader overview see 5 references therein). While TDDFT is very efficient for larger molecules, CASSCF based methods can be readily applied to smaller molecules, providing a good balance between accuracy, flexibility and calculation time. As a multi-configuration method, it allows to capture the involved valence states properly. Moreover a CASSCF calculation can be augmented with a multi reference pertubative expansion to obtain highly accurate x-ray spectra. 17 The low lying core states of the sulfur atom of thiophene make it a good marker for hard X-ray spectroscopy. The valence states are calculated with the package MOLPRO 18 at the CASSCF(6/5)/cc-pVTZ level of theory. To take into account the necessary relativistic corrections for the sulfur 1s electrons a second order DouglasKroll-Hess Hamiltonian 19,20 is employed. The sulfur 1s core excited states are then obtained in a subsequent restricted active space calculation (RASSCF): the S(1s) orbital is rotated into the active space resulting in a RASSCF(8/6)/cc-pVTZ calculation, where the S(1s) orbital is restricted to single occupancy. The S(1s) molecular orbital coefficients are frozen in the wave function optimization to guarantee convergence to a core-hole state rather than a valence state. The resulting lowest lying 11



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core-hole transition is 2485 eV (compared to experiment, 21 2471 eV ), which corresponds to an error < 1%. All relevant quantities necessary to compute the nonlocal response are directly obtained from the MCSCF wave functions as described in appendix C.

3.1 X-ray absorption of oriented thiophene We have simulated the transmission from an oriented thiophene at the sulfur K-edge (2.48 keV) : i SXAS (ks , ω ) = h¯

Z

dkdk1 hhj(k) · A1 (−k)|G(ω )ej(k1 ) · A1 (−k1 )|ρ−eq ii e − mc

Z

dkdk1 hhσ (k + k1 )|ρ−eq iiAs (−k) · A1 (−k1 ) (23)

The ks dependence comes from the fact that the vector potential of a plane wave with wavevector ks : A1 (k) = A1 δ(k − ks ). From this expression, one can readily see that σ only contributes a frequency-independent background. However, its variation with ks can still provide information about the molecular structure in the off-resonance regime where the current densities can be neglected. Figure 2 depicts the signal (eq. 23) versus the frequency and the direction of an incoming Gaussian beam. The beam central frequency is tuned at the sulfur Kedge, 2.48 keV, and is assumed to be ultrabroadband (tens of eV are achievable 22 ). In the impulsive limit, the signal is given in eq. 23. We used the sum-over-state expression given in appendix D, eq. 45. The transition at 2488 eV possesses a strong maximum when the incoming x polarized beam is incident with an angle of π/4. The symmetry of the angular dependence is due to the σν symmetry plane. The signal depends on the molecular orientation. 12


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STPA jjσ (ks , τ )

1 = 2¯hǫ0 c|ks |

Z

drs ...dr1 dt...dt1

′ hh js (rs )eiks ·rs |G(ts )ejs∗ (r′s )eiks ·rs G(τs )ej(r2 )G(t2 )ej(r2′ )G(τ )e σ(r1 )|ρ−eq ii

A(r2 , t − ts − τs ) · A(r2′ , t − ts − τs )A(r1 , t − ts − τs − τ ) · A(r1 , t − ts − τs − τ ) (28) In this example the molecule was excited by two-photon absorption prior to the fluorescence. However, the general spontaneous signals presented in Eq. 22 can be used to calculate other spontaneous signals such as homodyne detected for samples much larger than the wavelength of the incoming beams or diffraction type when one focus on the k space distribution of the emission.

4 Discussion We have implemented a nonlocal response formulation of nonlinear X-ray signals based on the minimal coupling Hamiltonian for the light-matter interaction. The matter enters through the charge and current densities thus offering a real-space picture of the electron dynamics in the molecule. The focus has been put on Xray spectroscopies because the dipole approximation is more likely to fail when the wavelength is comparable to the molecular size. In this article, we have demonstrated that the extension of the transition current densities scales with the incoming wavelength. However, the dipole approximation fails to consider non-trivial beam spatial variation and the charge densities contributions. The non-local formalism, that avoids the multipolar expansion, may be used to describe both stimulated heterodyne-detected or spontaneous incoherent signals. In this paper, we have calculated X-ray absorption, stimulated X-ray Raman spec19



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troscopy and two-photon induced fluorescence signals. The standard resonant and off-resonant treatments that only retain the current densities or the charge density respectively are recovered. In anomalous X-ray scattering, 12 both quantities must be retained and are naturally included in the nonlocal formalism. Both current and charge densities then contribute substantially to the signals.

The nonlocal description of light-matter interaction offers many future possibilities. An obvious application would be the design of interactions of matter with a strongly spatially varying field. Such fields could be achieved experimentally at short wavelength, using the progresses of beam ultra-focusing or of non-trivial polarization state (such as radial or azimuthal polarization). The important progress in near-field optics can also lead to new exciting signals, using the near-field at the vicinity of a nanoparticle for example. We note that nonlocal effects already occur in spectroscopies that depends on the magnetic dipole, e.g. circular dichroism. Additionally, joint current and charge density interactions that can occur in nonlinear optical experiment can be accounted for in the nonlocal description.

Appendix A

Derivation of equation 5

Using the Liouville-Von Neumann equation, the rate of change of the occupation number is readily expressed as N˙ s = i/¯hLint (r ) a†s as . Lint (r ) was defined in eq. 9 with classical vector potential that are now replaced by their quantum operators:

A(r) =

∑ Ai ai + Ai∗ ai†

(29)

i

20


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Ai ( r ) =

s

h¯ εi eiki ·r 2ǫ0 ωi L3

(30)

We detect the field s mode, and εi is the polarization of this mode. We then define j+ = ja†s , j− = jas , σ2+ = σa†s a†s , σ0 = σ, σ2− = σas as and the corresponding Liouville space operators. Using these definitions, the photon rate of change becomes i N˙ s = − h¯

Z

drej+ (r ) · A∗s (r) a†s as + ej− (r ) · As (r) a†s as

−

e e σ0 (r)As · A∗s + e σ 2− ( r ) As · As + e σ2+ (r)A∗s · A∗s a†s as (31) 2mc

These operator products can be simplified using :

ej+ (r ) · A∗s (r) a†s as = −j+ · A∗s (r) e σ2+ (r) a†s as = −2σ2+

(32) (33)

and the following matrix elements :

hh as |e as |0s ii = −1

hh a†s |e a†s |0s ii = 1

hh as as |e σ2− |0s ii = −2σ

hh a†s a†s |e σ2+ |0s ii = 2σ

Inserting these expressions in eq. 4 leads to the signal given in eq. 5.

21


(34) (35) (36) (37)


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Appendix B Expressions for J (r, t) and J (r, ω ) In this appendix, we provide a sum-over-state expression for J (r, ω ) defined in eq. 10. At the linear order, the time-dependent current density given in eq. 8 is:

J(1) (r, t) = −

i h¯

Z

dr1 dt1 hhj(r )|G(t1 )ej(r1 ) · A(r1 , t − t1 )|ρ−eq ii −

e hhσ(r )|ρ−eq iiA(r, t) mc (38)

If we assume an ultrashort pulse (Dirac δ function in time), the integral over t1 is simplified and the current density becomes : 1 J(1) (r, t) = ∑ jgm (r ) h¯ m

R

dr1 jmg (r1 ) · A(r1 , ω ) + ω − ωmg + iΓmg

R

dr1 j∗mg (r1 ) · A(r1 , ω ) ω + ωmg + iΓmg

!

−

e σgg (r)A(r, ω ) mc (39)

This is the formula used to compute the current densities displayed in fig. 1. Alternatively, we can consider a temporal Gaussian profile for the incoming pulse A(r1 , t − t1 ) = A(r1 )e

−

( t − t1 )2 2σ2

e−iω L t1 + c.c. The current density is given by :

i √ J(1) (r, t) = − σ 2π ∑ jgm (r) h¯ m Re ei(ω L −ωmg )t−Γmg t e

Z

dr1 jmg (r1 ) · A(r1 )×

1 2 2 2 σ ( Γmg −i ( ω L − ωmg ))

t σ 1 + erf( √ − (Γmg − i (ω L − ωm g)) 2σ 2

! (40)

22


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Appendix C

Electron currents from MCSCF wave functions

In this section we briefly review how the electron currents from eq. 2 necessary for the spectra calculation can be obtained from a configuration interaction (CI) type wave function. The electron current j(r) can be expressed in terms of the full real valued electronic eigenstates Ψi ≡ Ψi (r, r2 , . . . , r N ): 1 M 1 j(r) = ∑ ρij jij (r) = ∑ ρij 2 i,j6=i 2 i,j6=i

Z

dr 2 . . . dr N Ψi ∇Ψ j − Ψ j ∇Ψi

(41)

where the sum runs over all electronic states involved. Note that the diagonal elements (i = j) do not contribute as the current for any real valued, non-degenerate eigenstate vanishes. Here ρij is the density of matrix element of the electronic states. The wave function is then further expanded in a multi-configuration self consistent field (MCSCF) wave function

Ψi =

∑ Ami Φm

(42)

m

such that the transition current reads:

jij (r) =

Z

dr 2 . . . dr N

∑ m,n6=m

Ami Anj (Φm ∇Φn − Φn ∇Φm )

where Ami is the CI coefficients of state i, configuration m, and Φm its corresponding Slater determinant. The transition currents are one-electron properties, meaning Brillouins theorem 26 applies and only combinations of determinants Φm and Φn

23



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differing in exactly one orbital need to be considered. The contributing determinants can be defined through:

|Φn i = a†µ aν |Φm i

(43)

where a†µ , aν create and annihilate an electron in spin orbital µ and ν respectively. The current for an MCSCF wave functions is then determined by the molecular or(m)

(n)

bital pairs ψµ (r) and ψν (r) generated from configurations m and n: j(r) =

1 (m) (n) (n) (m) ρ A A ψ ∇ ψ − ψ ∇ ψ µ ν ν µ ij ∑ mi nj 2 i,j∑ 6 =i m,n6=m

(44)

The time dependence of j enters through the coherences ρij , whereas the time independent quantities (A and ψ) have been obtained from the MCSCF calculation.

Appendix D

Sum-over-states expressions for X-ray signals

In this section, we provide explicit expression for the different signals considered in this paper. The linear X-ray absorption signal is defined in eq. 23. We assume an x polarized plane wave excitation A1 (k) = ex A1 δ(k − ks ). We vary the incident angle on the incoming pulse according to fig. 2. In the RWA approximation, the signal is given by : jgm ( R x (θ )|ks |ey ) · A1 j∗mg ( R x (θ )|ks |ey ) · A1 i e SXAS (θ, ω ) = − ∑ − σgg ( R x (θ )|ks |ey )A1 · A1 h¯ m ω − ωmg + iΓmg mc (45)

24


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Where R x (θ ) is the rotation matrix around the x axis by an angle θ. We give sum-over-state expression for the complete third order response function. ii ii ii i S(3) (r, r1 r2 r3 , t1 , t2 , t3 ) = Siii jjjj + S jjσ + S jσj + Sσjj + Sσσ

i Sσσ =

Siijjσ

(46)

−e −e ∗ + σgm (r)Gmg,mg (t2 )σmg (r2 ) − σmg (r)G gm,gm (t2 )σmg (r2 ) 2mc mc (47)

−i 2 −e = h¯ 2mc

+ jgm (r)Gmg,mg (t3 ) jma (r3 )G ag,ag (t2 )σag (r1 ) ∗ − jma (r)G am,am (t3 ) jmg (r3 )G ag,ag (t2 )σag (r1 ) ∗ − jam (r)Gma,ma (t3 ) jmg (r3 )G ga,ga (t2 )σag (r1 ) ∗ ∗ − jmg (r)G gm,gm (t3 ) jma (r3 )G ga,ga (t2 )σag ( r1 )

Siijσj =

−i 2 −e h¯ 2mc

(48)

(49)

+ jgm (r)Gmg,mg (t3 )σma (r3 )G ag,ag (t1 ) jag (r1 ) ∗ − jma (r)G am,am (t3 )σmg (r3 )G ag,ag (t1 ) jag (r1 ) ∗ − jam (r)Gma,ma (t3 )σmg (r3 )G ga,ga (t1 ) jag ( r1 ) ∗ ∗ − jmg (r)G gm,gm (t3 )σma (r3 )G ga,ga (t1 ) jag (r1 )

25



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ii Sσjj

=

−i 2 −e h¯

mc

Page 26 of 34

+σgm (r)Gmg,mg (t2 ) jma (r2 )G ag,ag (t1 ) jag (r1 ) ∗ −σma (r)G am,am (t2 ) jmg (r2 )G ag,ag (t1 ) jag (r1 ) ∗ −σam (r)Gma,ma (t2 ) jmg (r2 )G ga,ga (t1 ) jag ( r1 ) ∗ ∗ −σmg (r)G gm,gm (t2 ) jma (r2 )G ga,ga (t1 ) jag (r1 )

Siii jjjj =

− i 3 h¯

(50)

+ jgm (r)G gm,gm (t3 ) jma (r3 )G ag,ag (t2 ) jac (r2 )Gcg,cg (t1 ) jcg (r1 )

∗ − jcm (r)Gcm,cm (t3 ) jma (r3 )G ac,ac (t2 ) jag (r2 )G gc,gc (t1 ) jcg (r1 ) ∗ − jam (r)G am,am (t3 ) jmc (r3 )Gca,ca (t2 ) jag (r2 )Gcg,cg (t1 ) jcg (r1 ) ∗ − jma (r)Gma,ma (t3 ) jmg (r3 )G ag,ag (t2 ) jac (r2 )Gcg,cg (t1 ) jcg (r1 ) ∗ ∗ + jam (r)G am,am (t3 ) jmg (r3 )G ga,ga (t2 ) jac (r2 )G gc,gc (t1 ) jcg (r1 ) ∗ ∗ ( r1 ) + jma (r)Gma,ma (t3 ) jmc (r3 )G ac,ac (t2 ) jag (r2 )G gc,gc (t1 ) jcg ∗ ∗ + jmc (r)Gmc,mc (t3 ) jma (r3 )Gca,ca (t2 ) jag (r2 )Gcg,cg (t1 ) jcg (r1 ) ∗ ∗ ∗ − jmg (r)Gmg,mg (t3 ) jmc (r3 )G ga,ga (t2 ) jac (r2 )G gc,gc (t1 ) jcg (r1 )

(51)

In section 3.2, the SXRS signal is calculated using the following expressions, derived from the diagrams in fig. 4. jjjj

SSXRS =

∑′ jgc′ (r) · A2∗ (r, t − τ )jc′ v (r3 ) · A2 (r3, t3 − τ )e−iωL (t3 −t)

cc v

jvc (r2 ) · A1∗ (r2 , t2 )jcg (r1 ) · A2 (r1 , t1 )e

−iωc′ g t+iωc′ v t3 −iωcv t2 +iωcg t1 −iω L (t1 −t2 )

jvc (r2 ) · A1 (r2 , t2 )jcg (r1 ) · A1∗ (r1 , t1 )e

+

−iωcv t+iωcg t3 +iωc′ v t2 −iωc′ g t1 +iω L (t1 −t2 )

26


(52)

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jjσ

SSXRS =

∑ cv

−

e σgv (r )δ(r − r3 )δ(t − t3 ) A2∗ (r, t − τ ) · A2 (r3 , t3 − τ mc

jvc (r2 ) · A1∗ (r2 , t2 )jcg (r1 ) · A2 (r1 , t1 )e

jvc (r2 ) · A1 (r2 , t2 )jcg (r1 ) · A1∗ (r1 , t1 )e

σjj

SSXRS =

−iωc′ g t+iωc′ v t3 −iωcv t2 +iωcg t1 −iω L (t1 −t2 )

+

−iωcv t+iωcg t3 +iωc′ v t2 −iωc′ g t1 +iω L (t1 −t2 )

(53)

∑′ jgc′ (r) · A2∗ (r, t − τ )jc′ v (r3 ) · A2 (r3, t3 − τ )e−iωL (t3 −t) cv

e −iω t+iωc′ v t3 −iωcv t2 +iωcg t1 σvg (r )δ(r2 − r1 )δ(t2 − t1 ) A1∗ (r2 , t2 ) · A2 (r1 , t1 )e c′ g + − 2mce −iω t+iωcg t3 +iωc′ v t2 −iωc′ g t1 − σvg (r )δ(r2 − r1 )δ(t2 − t1 ) A1 (r2 , t2 ) · A1∗ (r1 , t1 )e cv (54) 2mc

e ∗ σ ( r ) δ ( r − r ) δ ( t − t ) A ( r, t − τ ) · A ( r , t − τ gv 3 3 2 3 3 ∑ 2 mc v e −iω t+iωc′ v t3 −iωcv t2 +iωcg t1 − σvg (r )δ(r2 − r1 )δ(t2 − t1 ) A1∗ (r2 , t2 ) · A2 (r1 , t1 )e c′ g + 2mce −iωcv t+iωcg t3 +iωc′ v t2 −iωc′ g t1 ∗ σvg (r )δ(r2 − r1 )δ(t2 − t1 ) A1 (r2 , t2 ) · A1 (r1 , t1 )e (55) − 2mc σσ SSXRS =

−

Finally, we give the sum-over-state expression of the two-photon absorption fluorescence defined in equation 26.

ωs STPA jjjj (ks , τ ) = 2¯hǫ0 π 2 c3

∑ d f ikn

Z

′

drs ...dr1′

s (r ) eiks ·rs js∗ (r′ ) eiks ·rs jnk s nf s

(ωkn + iΓkn )

θ (τs )e−iωk f τs −Γk f τs

jki (r2 ) · A(r2 , τ )jig (r2′ ) · A(r2′ , τ ) θ (τ )e−iωg f τ −Γg f τ j∗f d (r1 ) · A(r1 , 0)j∗dg (r1′ ) · A(r1′ , 0) (56)

27



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Page 28 of 34

Appendix E Impulsive Gaussian beams The monochromatic linearly polarized Gaussian beam propagating along the z axis in the paraxial approximation is given by : 2

w0 − w2r (z) −i(kz−ωt)−ik 2Rr2(z) +iζ (z) A(r, z, ω ) = A0 e e + c.c. w(z)

(57)

where w(z) is the beam waist, R(z) is the radius of curvature, ζ (z) is the Gouy phase. 27 We further assumed an infinite Rayleigh length. Thus, the beam divergence along the z axis is neglected. This is justified by the small extension of a molecule compared to a typical Rayleigh length (few microns). The Gaussian beam has a much simpler expression : 2

A(r, z, ω ) = A0 e

− r2 w0

e−i(kz−ωt)) + c.c.

(58)

An Gaussian time envelope for the Gaussian beam is obtained by choosing a Gaussian frequency distribution and carrying out the Fourier transform of the previous equation : 2

A0 − r 2 − 4 ln 2 (t−τ − zc )2 −i(kz−ωt)) e + c.c. A(r, z, t) = √ e w0 e ∆t2 2π

(59)

Where ∆t is the FWHM and τ is the retardation of the pulse. In the impulsive limit, the beam takes the form used throughout this article to calculate integrated components of the current and charge densities :

A(r, z, t) =

√

2

2π A0 e

− r2 w0

z δ(t − τ − ) c 28


(60)

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Appendix F Final expressions for SXRS In this appendix, we give the SXRS signal (eq. 24) using eq. 52-55 in appendix D and the definition of the vector potential given in appendix E, eq. 59 for a pulse with a finite duration. SSXRS (τ ) = −πσ12 σ22 ℜ ∑

cvc′ µ2gc′ · A2∗ µ2c′ v · A2

+

µ1vc · A1∗ µ1cg · A1 e−iωvg τ −Γvg τ (ωvg +i (Γ ′ +Γ ′ ))2 +(2ω2 −ω ′ −ω ′ +i (Γ ′ +Γ ′ ))2 2ω2 − ωc′ v − ωc′ g + i (Γc′ v + Γc′ g ) c v c g c v c g c v c g −σ22 4 e erfc − iσ2 2 (ω gv +i (Γcg +Γcv ))2 +(2ω1 −ωcg −ωcv +i (Γcg +Γcv ))2 2ω − ω − ω + i (Γcg + Γcv ) 2 cg cv 1 4 e−σ1 erfc − iσ1 2

µ2vc′ · A2∗ µ2c′ g · A2 µ1vc · A1 µ1cg · A1∗ e−iωgv τ −Γgv τ (ω gv +i (Γ ′ +Γ ′ ))2 +(2ω2 −ω ′ −ω ′ +i (Γ ′ +Γ ′ ))2 2ω2 − ωc′ g − ωc′ v + i (Γc′ g + Γc′ v ) c g c v c g c v c g c v −σ22 4 e erfc − iσ2 2 (ω gv +i (Γcg +Γcv ))2 +(2ω1 −ωcg −ωcv +i (Γcg +Γcv ))2 2ω − ω − ω + i (Γcg + Γcv ) 2 cg cv 1 4 erfc + iσ1 (61) e−σ1 2

√ Where σi is related to the FWHM of the ith pulse by ∆ti = 2 ln 2σi . The spatially inR tegrated current densities µimn are given by µimn = drAi (r)jmn (r) where Ai (r) is the

spatial profile of the incoming beam. For example, for a Gaussian beam incoming 2 − r2 √ w from the z axis, discussed in appendix E, Ai (r) = 1/ 2π e i e−iki z .

Acknowledgement The support of the Chemical Sciences, Geosciences, and Biosciences division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy through Award No. DE-FG02-04ER15571 and of the National Science Foundation (Grant No 29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

CHE-1361516) is gratefully acknowledged. J.R was supported by the DOE grant. M.K gratefully acknowledges support from the Alexander von Humboldt foundation through the Feodor Lynen program.

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Current vs Charge Density Contributions to ... - Shaul Mukamel - UCI

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