General theory of photoexcitation induced photoelectron circular dichroism
aa r X i v : . [ phy s i c s . a t m - c l u s ] F e b General theory of photoexcitation inducedphotoelectron circular dichroism
Alex G Harvey , Zdeněk Mašín and Olga Smirnova , Max-Born-Institut, Max-Born-Str. 2A, 12489 Berlin Technische Universität Berlin, Ernst-Ruska-Gebäude, Hardenbergstr. 36A,10623, Berlin,Germany.
Abstract.
The photoionization of chiral molecules prepared in a coherent superposition ofexcited states can give access to the underlying chiral coherent dynamics in a procedure knownas photoexcitation induced photoelectron circular dichroism (PXECD) [1, 2, 3]. This exclusivedependence on coherence can also be seen in a different part of the angular spectrum, whereit is not contingent on the chirality of the molecule, thus allowing extension of PXECD’ssensitivity to tracking coherence to non-chiral molecules. Here we present a general theory ofPXECD based on angular momentum algebra and derive explicit expressions for all pertinentasymmetry parameters which arise for arbitrary polarisation of pump and probe pulses.The theory is developed in a way that clearly and simply separates chiral and non-chiralcontributions to the photoelectron angular distribution, and also demonstrates how PXECDand PECD-type contributions, which may be distinguished by whether pump or ionizing probeenables chiral response, are mixed when arbitrary polarization is used.
1. Introduction
Chirality is associated with mirror-symmetry breaking. It is ubiquitous in natureand fundamental to the understanding of natural processes. For chiral molecules,this mirror-symmetry breaking leads to two versions of a molecule, the left andright enantiomers.Today, characterising molecular chirality is a dynamic and multidisciplinaryresearch field with an expanding arsenal of techniques. In the gas phase, theseinclude techniques such as Coulomb explosion imaging [4], microwave detection [5],the combination of mass spectrometry with multiphoton and vibrational excitationtechniques [6, 7], high harmonic spectroscopy [8, 9, 10], and photoelectron circulardichroism (PECD) [11, 12, 12, 13]. The growing interest in the response of chiral eneral theory of PXECD % of the total signal [19], thantechniques reliant on magnetic dipole effects. In both single and multi-photonPECD, the highest PECD signal is seen in the low-energy region of the spectrum.PECD shows a strong dependence on both the initial, intermediate and final statesand is a structurally sensitive probe, as seen in the striking difference observedbetween camphor and fenchone due to the methyl group substitution, althoughthe involved bound states and photoelectron spectra hardly change [20], and inthe pronounced dependence of PECD signal on molecular geometry and sensitivityto non-Frank-Condon effects [13].In contrast to conventional PECD, PXECD requires the coherent populationof multiple states, and hence the dichroic signal displays quantum beating withrespect to the delay between excitation and ionization pulses [2, 3]. PXECD isthus a form of time-resolved photoelectron spectroscopy (TRPES) and can beused to investigate the time evolution of various intramolecular processes (for areview see [21]). TRPES has its origins in studies in which atomic hyperfinelevels were coherently excited and probed by ionization at nanosecond delay usinglinear pulses [22, 23, 24]. This lead to the observation of quantum beats inthe photoelectron angular distributions and allowed information on the ionizationcontinuum and the hyperfine interaction to be extracted. Later work extended thisconcept to the hyperfine levels of the NO molecule [25]. As shorter pulses becameavailable, experimental and numerical studies involving the coherent excitationof rotational states in the first step examined the influence of rotation-vibrationcoupling [26, 27], and non-adiabatic dynamics [28] in small molecules at pico tofemtosecond time resolution. Recent TRPES studies include joint experimentaland theoretical work to time-resolve valence electron dynamics during a chemicalreaction [29], and a theoretical study of non-adiabatic dynamics in the vicinity of aconical intersection [30]. We anticipate PXECD to be a similarly useful tool withthe added bonus of sensitivity to the chirality of the studied system.In this paper we extend and generalise our previous theoretical descriptions ofPXECD, combining the best aspects of our initial angular algebra based approach eneral theory of PXECD
2. Theory
As in our previous works [1, 2, 3], we model the interaction between the electricfield and the molecule using first order perturbation theory and the dipoleapproximation.We define the pump field in the laboratory reference frame as: E L ( t ) = √ F ( t ) ˆ ε L e − i ( ωt + δ ) + c . c . (1)where ω is the carrier frequency, F ( t ) includes the field amplitude and the envelope,and the carrier-envelope phase δ determines the orientation of the electric fieldvector at the moment t = . Finally, the helicity σ = ± determines whether thefield is left or right polarized and the polarization of the field is expressed in thespherical basis ˆ ε L − = √ ( ˆ x L − iˆ y L ) , ˆ ε L0 = ˆ z L ˆ ε L + = − √ ( ˆ x L + iˆ y L ) (2)The superscripts L and M indicate vectors are in the laboratory and molecularframe respectively. The transformation of vectors from the lab frame to themolecular frame is performed according to v M = D † ( ̺ ) v L where ̺ ≡ ( α, β, γ ) theEuler angles in the active z - y - z convention. In the angular momentum basis, thisrotation operator corresponds to the Wigner rotation matrix, we use its complextranspose here to account for the usual convention that the Wigner rotation matrixtransforms basis vectors covariantly. Using perturbation theory, after the end ofthe pump pulse of duration T , we find the wave function at a time τ : ψ ̺ ( τ ) = c ψ e − i ω τ + ∑ i = c i ( ̺ ) ψ e − i ω i τ , (3)where c ≈ and the expressions for the excitation amplitudes are standard: c i ( ̺ ) = i [ d M i ⋅ D † ( ̺ ) ˆ ε L ] E ( ω i ) (4) i labels the intermediate excited states, d M i are the transition dipole matrixelements to these states from the ground state, in the molecular frame sphericalbasis. The excitation amplitude is proportional to the spectral component of thepump at the corresponding transition frequency ω i , E ( ω i ) . eneral theory of PXECD ˆ ξ L . The population amplitude of a continuumstate k M after the end of the probe pulse, assuming that the pump and the probedo not overlap, is c ( k M ; ̺ ) = i ∑ i c i ( ̺ ) e − i ω i τ [ d M i ( k M ) ⋅ D † ( ̺ ) ˆ ξ L ] E ′ ( ω ′ k i ) (5)where E ′ ( ω k ,i ) is the spectral amplitude of the probe at the required transitionfrequency and d M i ( k ) are bound-free transition dipoles in the molecular frame. Inthis work we will consider electronic states only. The molecular frame PAD isproportional to dσd k M ( ̺, τ ) ∝ ∣ ∑ i e − i ω i τ [ d M i ( k M ) ⋅ D † ( ̺ ) ˆ ξ L ] [ d M i ⋅ D † ( ̺ ) ˆ ε L ]∣ (6)Performing a partial wave expansion for the photoelectron and writing component-wise dσd ˆk M ( E, τ ; ̺ ) ∝ RRRRRRRRRRR ∑ i e − i ω i τ ∑ lmp q D ∗ p q ˆ ξ L p d M i,q ,lm ( E ) Y lm ( ˆk M ) ∑ p q D ∗ p q ˆ ε L ∗ p d Mi ,q RRRRRRRRRRR , (7)We note that we have absorbed a factor of i − l e iσ l , where σ l is the Coulombphase, into the dipole matrix elements in contrast to how they are usually written.Expanding the modulus square we get, dσd ˆk M ( E, τ ; ̺ ) ∝ ∑ ii ′ e − i ω ii ′ τ ∑ K e M e ∑ lmp q D ∗ p q d M i,q ,lm ( E ) d M ∗ i ′ ,q ′ ,l ′ m ′ ( E )D p ′ q ′ ρ ξ L p p ′ Y K e M e ( ˆk M )× (− ) m ′ + M e [ ˜ l ˜ l ′ ˜ K e π ] / ( l l ′ K e − m m ′ M e ) ( l l ′ K e )∑ p q D ∗ p q d Mi ,q d M ∗ i ′ ,q ′ D p ′ q ′ ρ ε L p p ′ , (8)where the product of polarization vectors ˆ ε L p ˆ ε L ∗ p ′ = ρ ε L p p ′ and ˆ ξ L p ˆ ξ L ∗ p ′ = ρ ξ L p p ′ giveelements of the polarization density matrix for the first and second photon, andthe product of spherical harmonics has been contracted using the identity Y lm ( ˆk ) Y ∗ l ′ m ′ ( ˆk ) = ∑ K e M e ( − ) m [ ˜ l ˜ l ′ ˜ K e π ] / ( l l ′ K e − m m ′ M e ) ( l l ′ K e ) Y K e M e ( ˆk ) . (9)involving the − j symbols and where ˜ l = l + . At this point it is useful tointroduce some of the properties of the − j symbol, as they will be crucial eneral theory of PXECD ∑ abc ∣ Aa ⟩ ∣ Bb ⟩ ∣ Cc ⟩ ( A B Ca b c ) = ∣ ⟩ . An important symmetry property is thata − j symbol is unchanged after even permutation of its column, and acquiresa phase ( − ) ( A + B + C ) under odd permutations, the same phase is acquired if thebottom row is multiplied by − (equivalent to inversion in 3D). From this it canbe seen that if the sum of the top row is odd (and the three vectors are polar) thenthe scalar invariant is a pseudo-scalar. This is the hall mark of a chiral quantityand we will now proceed to transform the equation for the PAD into a form inwhich this can be seen explicitly.With this in mind we observe that the product of dipoles and the product ofthe polarization density matrix and spherical harmonic are themselves elementsof tensors that can be put in spherical tensor form. The general form ofthis transformation is d Cc = ∑ ab ( − ) a ˜ C ( A B C − a b c ) d Aa,Bb
We also rotate theoutgoing electron direction into the lab frame where it is detected. dσd ˆk L ( E, τ ; ̺ ) ∝ ∑ ii ′ e − i ω ii ′ τ ∑ K K e K J M J N J D M ii ′ , ( K K e ) K J M J ( E ) D K J ∗ N J M J Z ( K K e ) K J N J ( ˆk L )∑ p q p ′ q ′ D ∗ p q d Mi ,q d M ∗ i ′ ,q ′ D p ′ q ′ ρ ε L p p ′ , (10)where, D M ii ′ , ( K K e ) K J M J ( E ) = ∑ M e M ( − ) M ˜ K J ( K K e K J − M M e M J ) ∑ q q ′ ( − ) q ˜ K ( K − q q ′ M )∑ ll ′ mm ′ ( − ) m ˜ K e ( l l ′ K e − m m ′ M e ) d M i,q ,lm ( E ) d M ∗ i ′ ,q ′ ,l ′ m ′ ( E ) ( l l ′ K e ) ( ˜ l ˜ l ′ π ) (11)and Z ( K K e ) K J N J ( ˆk L ) = ∑ N e N p p ′ ( − ) N + p ( ˜ K ˜ K J ) ( K K e K J − N N e N J ) ( K − p p ′ N ) ρ ξ L p p ′ Y K e N e ( ˆk L ) (12) eneral theory of PXECD dσd ˆk L ( E, τ ) = π ˆ dσd ˆk L ( E, τ ; ̺ ) d̺ (13)giving dσd ˆk L ( E, τ ) ∝ ∑ ii ′ e − i ω ii ′ τ ∑ K K e K J N J p p ′ ⎧⎪⎪⎨⎪⎪⎩ ∑ M J q q ′ D M ii ′ , ( K K e ) K J M J ( E ) d Mi ,q d Mi ′ , − q ′ ( K J M J q − q ′ )⎫⎪⎪⎬⎪⎪⎭( − ) p ′ ( K J N J p − p ′ ) ρ ε L p p ′ Z ( K K e ) K J N J ( ˆk L ) , (14)The PAD has been separated into two parts: outside the braces are the lab framequantities (the photon polarizations and the outgoing electron direction), insidethe braces we get a scalar invariant involving the molecular frame quantities only,namely the transition and ionization dipoles. We denote this invariant scalar α ( K K e ) K J where K J can take the values { , , } , and also transform the photondensity matrices into their irreducible spherical tensor form. The non-vanishingcomponents of the first photon density matrix are: ρ ε L00 = − √ / , ρ ε L10 = − √ / C , ρ ε L20 = − √ / and ρ ε L22 = ρ ε L ∗ − = ( / ) L . Here − ≤ C ≤ defines the the amount ofcircular polarization and ≤ L ≤ the amount of linear polarization. L + C isunity for pure polarization, less than 1 for partial polarization and 0 for unpolarized(in the x − y plane) light. The major axis of polarization defines the x -directionand the propagation direction is z . dσd ˆk L ( E, τ ) ∝ ∑ ii ′ e − i ω ii ′ τ ∑ K K e K J N J α ( K K e ) K J ρ ε L K J N J Z ( K K e ) K J N J ( ˆk L ) , (15)We can write this in vector form as α ( K K e ) = D M † ii ′ , ( K K e ) ( E ) ⋅ A ii ′ , = − √ D M ii ′ , ( K K e ) ( E ) d Mi ⋅ d Mi ′ α ( K K e ) = √ D M † ii ′ , ( K K e ) ( E ) ⋅ A ii ′ , = √ D M † ii ′ , ( K K e ) ( E ) ⋅ ( d Mi × d Mi ′ ) α ( K K e ) = √ D M † ii ′ , ( K K e ) ( E ) ⋅ A ii ′ , , (16)where A ii ′ ,K J M J = ∑ q q ′ ( − ) q ′ ˜ K J d Mi ,q d M ∗ i ′ ,q ′ ( K J M J q − q ′ ) . (17) eneral theory of PXECD A ii ′ ,K J describes isotropy/orientation/alignment of the systeminduced by the first pulse, while D M † ii ′ , ( K K e ) K J ( E ) describes the furtherorientation/alignment induced by the second pulse and detection of thephotoelectron. PXECD arises from the α ( K K e ) coefficient, therefore we seethat orientation of the ensemble by the pump pulse is integral to PXECD. Thisorientation creates an induced net dipole in the ensemble that oscillates withangular frequency ω ii ′ as discussed in [2, 3].It is easy to see that α ( K K e ) exist only when the bound transition dipoles arenon-parallel and hence only exists for the interference terms (involving differentexcited states) not the direct terms. This implies it requires coherent populationof multiple states to be observed. One might be tempted to say that α ( K K e ) K J isscalar for even values of K J and pseudoscalar for odd values, but some care mustbe taken here, the d Mi are polar vectors, however D M ii ′ , ( K K e ) K J ( E ) can be either apolar or pseudovector, examination of the j symbols in eqn. 11 shows that underinversion iD M ii ′ , ( K K e ) K J ( E ) = ( − ) K e + K J D M ii ′ , ( K K e ) K J ( E ) i.e. it is a pseudovectorwhen K e + K J is odd. Hence α ( K K e ) K J is a pseudoscalar only when K e is odd(note: the dot product of a vector and a pseudovector is a pseudoscalar while thedot product of a pseudovector and a pseudovector is a scalar).It is straightforward to demonstrate that pseudoscalar α ( K K e ) K J exist only inchiral molecules. Consider first the reflection of a randomly oriented ensemble ofchiral molecules. This operation changes the sign of pseudoscalar α ( K K e ) K J butalso changes the enantiomer in the sample. Pseudoscalar α ( K K e ) K J is therefore thesource of asymmetry in photoelectron emission that changes sign with enantiomer.For a non-chiral ensemble, reflection does not change the ensemble, therefore α ( K K e ) K J = . Interestingly, it can be seen that α ( K K e ) can exist in non-chiralmolecules for even values of K e .We now expand the summation over K J , N J and insert the explicit density eneral theory of PXECD dσd ˆk L ( E, τ ) ∝ ∑ ii ′ e − i ω ii ′ τ ∑ K K e − [ √ α ( K K e ) Z ( K K e ) ( ˆk L ) + √ α ( K K e ) Z ( K K e ) ( ˆk L )] − C √ α ( K K e ) Z ( K K e ) ( ˆk L ) + L α ( K K e ) [ Z ( K K e ) − ( ˆk L ) + Z ( K K e ) ( ˆk L )] , (18)We remind the reader that this is still the general form for any polarization andpropagation direction of the two photons. In the above equation the terms in thefirst square bracket do not depend on polarization, they exists for an unpolarizedpump pulse (note: for a completely unpolarized pump pulse where there is alsono preferred propagation direction e.g. produced by three orthogonal beams, onlythe Z ( K K e ) term survives). The term mulitplied by C depends on the sign anddegree of circular polarization, while the last term depends on the degree of linearpolarization.We now extend the idea of PXECD as described [2, 3]. We associate C α ( K K e ) as the fundamental quantity describing PXECD, all terms involvingit change sign with a change in helicity of the pump pulse, and it only existswhen multiple states are coherently populated. It does not necessarily change signwith enantiomer, and therefore can also exist in non-chiral molecules. We will seelater, when we consider the polarization of the second photon, that the PECDterms arise from C α ( K e ) K J and do not require coherently populated states withnon-co-linear dipoles.To determine the PAD we need to examine Z ( K K e ) K J N J ( ˆk L ) = ∑ N e N N ′ ( − ) N ( ˜ K J ) ( K K e K J − N N e N J ) ¯ ρ ξ L K N ′ D K N ′ N ( µ, ν, η ) Y K e N e ( ˆk L ) , (19)where we have written ρ ξ L K N = ∑ N ′ ¯ ρ ξ L K N ′ D K N ′ N ( µ, ν, η ) and the Euler angles ( µ, ν, η ) define the rotation between the coordinate frame of the first and secondphoton.We now look at the specific case of co-propagating pump and probe pulseswhere the coordinate frame of the two photons coincide, i.e. µ = ν = η = . eneral theory of PXECD We obtain the following, dσd ˆk L ( E, τ ) ∝ ∑ ii ′ e − i ω ii ′ τ [ α ( ) ii ′ + α ( ) ii ′ + C C α ( ) ii ′ + L L α ( ) ii ′ , S ] S ( ˆk L ) − ⎡⎢⎢⎢⎢⎣ C ⎛⎝ α ( ) ii ′ √ − α ( ) ii ′ √ ⎞⎠ − C ⎛⎝ α ( ) ii ′ √ − α ( ) ii ′ √ ⎞⎠⎤⎥⎥⎥⎥⎦ S ( ˆk L ) + ⎡⎢⎢⎢⎢⎣ α ( ) ii ′ √ + α ( ) ii ′ √ − α ( ) ii ′ √ − C C α ( ) ii ′ √ + L L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) − ⎡⎢⎢⎢⎢⎣ C α ( ) ii ′ √ − C α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) + ⎡⎢⎢⎢⎢⎣ α ( ) ii ′ √ + L L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) − i √ ⎡⎢⎢⎢⎢⎣ C L α ( ) ii ′ √ + C L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S − ( ˆk L ) − √ ⎡⎢⎢⎢⎢⎣ L ⎛⎝ α ( ) ii ′ √ + α ( ) ii ′ √ ⎞⎠ + L ⎛⎝ α ( ) ii ′ √ + α ( ) ii ′ √ ⎞⎠⎤⎥⎥⎥⎥⎦ S ( ˆk L ) − i √ ⎡⎢⎢⎢⎢⎣ L α ( ) ii ′ √ − L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S − ( ˆk L ) − √ ⎡⎢⎢⎢⎢⎣ C L α ( ) ii ′ √ − C L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) − √ ⎡⎢⎢⎢⎢⎣ L α ( ) ii ′ √ + L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) (20)We can group the terms into 5 classes: Not dependent on light polarizationor chirality of the molecule, dependent on circular polarization and chirality,dependent on linear polarization and not chirality, dependent on circularpolarization but not chirality, and dependent on linear polarization and chirality.The last two categories are particularly interesting, examples are found,respectively, in the coefficients of S − ( ˆk L ) which require orientation from thepump(/probe) and alignment from the probe(/pump) and quadrupole emission, eneral theory of PXECD S − ( ˆk L ) which require alignment of both pump and probe, and octupoleemission. We also see that there are two terms that require both pulses to havecircular components involving α ( ) ii ′ in the isotropic part of the emission (henceseen in the total cross section) and α ( ) ii ′ seen in S ( ˆk L ) , these change sign withchange of relative sign between the circularly polarized components of pump andprobe pulses, exist for non-chiral molecules, and correspond to both pulses inducingnet dipoles in the system, which then couple to give either isotropic or quadrupoleemission.It is also interesting to examine the various coefficients to see their dependenceon the orientation/alignment state of the component spherical vectors. We noticethat terms that change sign due to the circular polarisation of the pump(/probe)pulse always correspond to the orientation vector component induced by thepump(/probe) pulse i.e K J (/ K ) = . All terms not dependent on circularpolarization have the spherical vectors related to pump and probe pulses as eitherisotropic or aligned.We see that asymmetry in the photoemission (corresponding to odd order realspherical harmonics) comes from orientation by the first pulse for PXECD andfrom the second ionizing pulse for PECD. We observe that α ( ) ii , correspondingto the isotropic part of the first pulse, corresponds to standard one photon PECDfrom the excited state i up to a constant given by bound transition strength, andso we see that ionization where the second pulse is also circular is not exclusivelycontingent on coherent population of multiple states.We can also observe that two-photon PECD (i.e. two circular pulses) mixesPXECD terms with PECD terms in the coefficient of S ( ˆk L ) .We now look at the PXECD experimental setup as described in [2]. Setting L = and C = gives the full angular distribution for PXECD asdescribed in [2]. The following result is obtained. eneral theory of PXECD dσd ˆk L ( E, τ ) ∝ ∑ ii ′ e − i ω ii ′ τ [ α ( ) ii ′ + α ( ) ii ′ ] S ( ˆk L ) − ⎡⎢⎢⎢⎢⎣ C ⎛⎝ α ( ) ii ′ √ − α ( ) ii ′ √ ⎞⎠⎤⎥⎥⎥⎥⎦ S ( ˆk L ) + ⎡⎢⎢⎢⎢⎣ α ( ) ii ′ √ + α ( ) ii ′ √ − α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) − ⎡⎢⎢⎢⎢⎣ C α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) + ⎡⎢⎢⎢⎢⎣ α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) − i √ ⎡⎢⎢⎢⎢⎣ C L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S − ( ˆk L ) − √ ⎡⎢⎢⎢⎢⎣ L ⎛⎝ α ( ) ii ′ √ + α ( ) ii ′ √ ⎞⎠⎤⎥⎥⎥⎥⎦ S ( ˆk L ) + i √ ⎡⎢⎢⎢⎢⎣ √ L α ( ) ii ′ ⎤⎥⎥⎥⎥⎦ S − ( ˆk L ) + √ ⎡⎢⎢⎢⎢⎣ C L α ( ) ii ′ √ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) − √ ⎡⎢⎢⎢⎢⎣ √ L α ( ) ii ′ ⎤⎥⎥⎥⎥⎦ S ( ˆk L ) (21) eneral theory of PXECD dσd ˆk L ( E, τ ) ∝ ∑ ii ′ e − i ω ii ′ τ [ − √ D M ii ′ , ( ) ( E ) d Mi ⋅ d Mi ′ + √ D M † ii ′ , ( ) ( E ) ⋅ A ii ′ , ] S ( ˆk L ) − C [ ( D M † ii ′ , ( ) ( E ) − √ D M † ii ′ , ( ) ( E )) ⋅ ( d Mi × d Mi ′ )] S ( ˆk L ) + [( √ D M † ii ′ , ( ) ( E ) − √ D M † ii ′ , ( ) ( E )) ⋅ A ii ′ , − √ D M ii ′ , ( ) ( E ) d Mi ⋅ d Mi ′ ] S ( ˆk L ) − C [ √ D M † ii ′ , ( ) ( E ) ⋅ ( d Mi × d Mi ′ )] S ( ˆk L ) + [ √ D M † ii ′ , ( ) ( E ) ⋅ A ii ′ , ] S ( ˆk L ) − iC L [ √ D M † ii ′ , ( ) ( E ) ⋅ ( d Mi × d Mi ′ )] S − ( ˆk L ) + L [( √ √ D M ii ′ , ( ) ( E ) d Mi ⋅ d Mi ′ − √ √ D M † ii ′ , ( ) ( E ) ⋅ A ii ′ , )] S ( ˆk L ) + iL [ √ D M † ii ′ , ( ) ( E ) ⋅ A ii ′ , ] S − ( ˆk L ) + C L [ √ D M † ii ′ , ( ) ( E ) ⋅ ( d Mi × d Mi ′ )] S ( ˆk L ) − L [ √ √ D M † ii ′ , ( ) ( E ) ⋅ A ii ′ , ] S ( ˆk L ) (22)Here we can connect back to the results of [2] where the photoelectron currentin the z -direction was shown to be a triple product in the Cartesian basis byrecognising that S ( ˆk L ) ∝ k z is responsible for the chiral current in the z -direction. ( D M † ii ′ , ( ) ( E ) − √ D M † ii ′ , ( ) ( E )) is equivalent, up to a constant, to the Raman typephotoionization vector defined in [2]. The triple product can be transformed fromthe spherical basis to the Cartesian basis by using the usual unitary transformationbetween the two, this preserves the triple product up to a phase e − i π comingfrom the determinant of the transformation matrix. The same transformationcan, of course, be applied to all other terms, remembering to multiply by theappropriate phase for transformation of pseudovectors. D M † ii ′ , ( ) ( E ) depends onlyon the isotropic part of the probe pulse and hence survives even for a completelyunpolarized probe pulse, while D M † ii ′ , ( ) ( E ) does not.
3. Conclusions
We presented a general theory of PXECD for arbitrary polarization of both thepulse that prepares the molecule in a superposition of excited states, and theionizing pulse. A conventional way of analysing angular and energy resolved eneral theory of PXECD ω ii ′ as discussed in [2, 3]. The induced chiral dipole underlies the PXCD(photoexcitation circular dichroism) phenomenon introduced in [2, 3]. This isin contrast to one-photon PECD where chiral asymmetric emission emerges as aresult of the orientation imposed by the ionizing pulse. In PXECD all asymmetryin the forwards/backwards direction (coefficients of and S ( ˆk L ) and S ( ˆk L ) )is contingent on both chirality and coherent population of multiple states. Incontrast, in two-photon PECD there is a mixing of PXCD terms, that requirecoherent population of excited states, with PECD-like terms that do not rely onsuch coherencies.We have identified the terms uniquely related to PXECD in chiral molecules.We have also shown that PXECD recorded in polarization frame of the pumppulse contains asymmetry parameters, which are exclusively sensitive to coherence,but not associated with chiral response. These terms always arise for fieldconfigurations leading to cylindrical symmetry breaking and inducing extrinsicchirality in the polarization plane.Thus, PXECD is a background-free probe of coherent bound dynamicsproviding individual access to its chiral and non-chiral contributions. Acknowledgements
We would like to thank useful communications with Andres Ordonez. AHacknowledges support from DFG project number HA 8552/2-1
References [1] A. G. Harvey, Z. Masin, and O. Smirnova (In preparation) .[2] S. Beaulieu, A. Comby, D. Descamps, B. Fabre, G. A. Garcia, R. Geneaux, A. G.Harvey, F. Legare, Z. Masin, L. Nahon, A. F. Ordonez, S. Petit, B. Pons, Y. Mairesse,O. Smirnova, and V. Blanchet, “Photoexcitation Circular Dichroism in Chiral Molecules,” arXiv:1612.08764 [physics] , 2016.[3] S. Beaulieu, A. Comby, D. Descamps, B. Fabre, G. A. Garcia, R. Geneaux, A. G. Harvey, eneral theory of PXECD F. Legare, Z. Masin, L. Nahon, A. F. Ordonez, S. Petit, B. Pons, Y. Mairesse, O. Smirnova,and V. Blanchet, “Photoexcitation Circular Dichroism in Chiral Molecules,”
NaturePhysics , Dec. 2018, in press. DOI: 10.1038/s41567-017-0038-z.[4] M. Pitzer, M. Kunitski, A. S. Johnson, T. Jahnke, H. Sann, F. Sturm, L. P. H. Schmidt,H. Schmidt-Böcking, R. Dörner, J. Stohner, J. Kiedrowski, M. Reggelin, S. Marquardt,A. Schießer, R. Berger, and M. S. Schöffler, “Direct Determination of Absolute MolecularStereochemistry in Gas Phase by Coulomb Explosion Imaging,”
Science , vol. 341, no. 6150,pp. 1096–1100, 2013.[5] D. Patterson, M. Schnell, and J. M. Doyle, “Enantiomer-specific detection of chiral moleculesvia microwave spectroscopy,”
Nature , vol. 497, no. 7450, pp. 475–477, 2013.[6] C. Lux, M. Wollenhaupt, T. Bolze, Q. Liang, J. Köhler, C. Sarpe, and T. Baumert, “CircularDichroism in the Photoelectron Angular Distributions of Camphor and Fenchone fromMultiphoton Ionization with Femtosecond Laser Pulses,”
Angew. Chem. Int. Ed. , vol. 51,no. 20, pp. 5001–5005, 2012.[7] H. Rhee, Y.-G. June, J.-S. Lee, K.-K. Lee, J.-H. Ha, Z. H. Kim, S.-J. Jeon, and M. Cho,“Femtosecond characterization of vibrational optical activity of chiral molecules,”
Nature ,vol. 458, no. 7236, pp. 310–313, 2009.[8] R. Cireasa, A. E. Boguslavskiy, B. Pons, M. C. H. Wong, D. Descamps, S. Petit, H. Ruf,N. Thiré, A. Ferré, J. Suarez, J. Higuet, B. E. Schmidt, A. F. Alharbi, F. Légaré,V. Blanchet, B. Fabre, S. Patchkovskii, O. Smirnova, Y. Mairesse, and V. R. Bhardwaj,“Probing molecular chirality on a sub-femtosecond timescale,”
Nat. Phys. , vol. 11, no. 8,pp. 654–658, 2015.[9] O. Smirnova, Y. Mairesse, and S. Patchkovskii, “Opportunities for chiral discriminationusing high harmonic generation in tailored laser fields,”
Journal of Physics B: Atomic,Molecular and Optical Physics , vol. 48, no. 23, p. 234005, 2015.[10] D. Ayuso, P. Decleva, S. Patchkowskii, and O. Smirnova, “Chiral dichroism in bi-ellipticalhigh-order harmonic generation,”
Journal of Physics B: Atomic, Molecular and OpticalPhysics , 2018.[11] B. Ritchie, “Theory of the angular distribution of photoelectrons ejected from optically activemolecules and molecular negative ions,”
Phys. Rev. A , vol. 13, no. 4, pp. 1411–1415, 1976.[12] I. Powis, “Photoelectron circular dichroism of the randomly oriented chiral moleculesglyceraldehyde and lactic acid,”
The Journal of Chemical Physics , vol. 112, no. 1, pp. 301–310, 2000.[13] G. A. Garcia, L. Nahon, S. Daly, and I. Powis, “Vibrationally induced inversionof photoelectron forward-backward asymmetry in chiral molecule photoionization bycircularly polarized light,”
Nat. Commun. , vol. 4, p. 2132, 2013.[14] P. Fischer and F. Hache, “Nonlinear optical spectroscopy of chiral molecules,”
Chirality ,vol. 17, no. 8, pp. 421–437, 2005.[15] D. Abramavicius, W. Zhuang, and S. Mukamel, “Probing molecular chirality via excitonicnonlinear response,”
J. Phys. B: At. Mol. Opt. Phys. , vol. 39, no. 24, p. 5051, 2006.[16] J.-H. Choi, S. Cheon, H. Lee, and M. Cho, “Two-dimensional nonlinear optical activityspectroscopy of coupled multi-chromophore system,”
Phys. Chem. Chem. Phys. , vol. 10,no. 26, pp. 3839–3856, 2008.[17] A. F. Fidler, V. P. Singh, P. D. Long, P. D. Dahlberg, and G. S. Engel, “Dynamic localizationof electronic excitation in photosynthetic complexes revealed with chiral two-dimensional eneral theory of PXECD spectroscopy,” Nat. Commun. , vol. 5, p. 3286, 2014.[18] J. R. Rouxel, M. Kowalewski, and S. Mukamel, “Photoinduced molecular chirality probedby ultrafast resonant x-ray spectroscopy,”
Structural Dynamics , vol. 4, no. 4, p. 044006,2017.[19] I. Powis, “Photoelectron Spectroscopy and Circular Dichroism in Chiral Biomolecules: l-Alanine,”
J. Phys. Chem. A , vol. 104, no. 5, pp. 878–882, 2000.[20] I. Powis, C. J. Harding, G. A. Garcia, and L. Nahon, “A Valence Photoelectron ImagingInvestigation of Chiral Asymmetry in the Photoionization of Fenchone and Camphor,”
ChemPhysChem , vol. 9, no. 3, pp. 475–483, 2008.[21] A. Stolow and J. G. Underwood, “Time-Resolved Photoelectron Spectroscopy ofNonadiabatic Dynamics in Polyatomic Molecules,” in
Advances in Chemical Physics (S. A.Rice, ed.), pp. 497–584, John Wiley & Sons, Inc., 2008.[22] M. P. Strand, J. Hansen, R.-L. Chien, and R. S. Berry, “Influence of nuclear spin on angulardistribution and polarization of photoelectrons: Resonant two-photon ionization of Na,”
Chemical Physics Letters , vol. 59, no. 2, pp. 205–209, 1978.[23] G. Leuchs, S. J. Smith, E. Khawaja, and H. Walther, “Quantum beats observed inphotoionization,”
Optics Communications , vol. 31, no. 3, pp. 313–316, 1979.[24] R.-l. Chien, O. C. Mullins, and R. S. Berry, “Angular distributions and quantum beats ofphotoelectrons from resonant two-photon ionization of lithium,”
Phys. Rev. A , vol. 28,no. 4, pp. 2078–2084, 1983.[25] K. L. Reid, S. P. Duxon, and M. Towrie, “Observation of time- and angle-resolvedphotoelectron flux from an optically prepared state of a molecule. Hyperfine depolarizationin NO (A 2 Σ +),” Chemical Physics Letters , vol. 228, no. 4, pp. 351–356, 1994.[26] K. L. Reid, T. A. Field, M. Towrie, and P. Matousek, “Photoelectron angular distributionsas a probe of alignment evolution in a polyatomic molecule: Picosecond time- and angle-resolved photoelectron spectroscopy of S1 para-difluorobenzene,”
The Journal of ChemicalPhysics , vol. 111, no. 4, pp. 1438–1445, 1999.[27] S. C. Althorpe and T. Seideman, “Predictions of rotation–vibration effects in time-resolvedphotoelectron angular distributions,”
The Journal of Chemical Physics , vol. 113, no. 18,pp. 7901–7910, 2000.[28] J. G. Underwood and K. L. Reid, “Time-resolved photoelectron angular distributions as aprobe of intramolecular dynamics: Connecting the molecular frame and the laboratoryframe,”
The Journal of Chemical Physics , vol. 113, no. 3, pp. 1067–1074, 2000.[29] P. Hockett, C. Z. Bisgaard, O. J. Clarkin, and A. Stolow, “Time-resolved imaging of purelyvalence-electron dynamics during a chemical reaction,”
Nat Phys , vol. 7, no. 8, pp. 612–615, 2011.[30] K. Bennett, M. Kowalewski, and S. Mukamel, “Nonadiabatic Dynamics May Be Probedthrough Electronic Coherence in Time-Resolved Photoelectron Spectroscopy,”
J. Chem.Theory Comput. , vol. 12, no. 2, pp. 740–752, 2016.[31] D. M. Brink and G. R. Satchler,
Angular Momentum . Oxford University Press., 1994.[32] U. Fano, “Geometrical characterization of nuclear states and the theory of angularcorrelations,”
Phys. Rev. , vol. 90, pp. 577–579, May 1953.[33] K. Blum,