A Coherent Nonlinear Optical Signal Induced by Electron Correlations
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov A Coherent Nonlinear Optical Signal Induced by ElectronCorrelations
Shaul Mukamel , Rafa l Oszwa ldowski , , and Lijun Yang Department of Chemistry, University of California, Irvine, CA 92697, and Instytut Fizyki, Uniwersytet Miko laja Kopernika,Grudzi¸adzka 5/7, 87-100, Toru´n, Poland
Abstract
The correlated behavior of electrons determines the structure and optical properties of molecu-les, semiconductor and other systems. Valuable information on these correlations is provided bymeasuring the response to femtosecond laser pulses, which probe the very short time period duringwhich the excited particles remain correlated. The interpretation of four-wave-mixing techniques,commonly used to study the energy levels and dynamics of many-electron systems, is complicated bymany competing effects and overlapping resonances. Here we propose a coherent optical technique,specifically designed to provide a background-free probe for electronic correlations in many-electronsystems. The proposed signal pulse is generated only when the electrons are correlated, whichgives rise to an extraordinary sensitivity. The peak pattern in two-dimensional plots, obtainedby displaying the signal vs. two frequencies conjugated to two pulse delays, provides a directvisualization and specific signatures of the many-electron wavefunctions.
PACS numbers: 71.35.Cc, 73.21.La, 78.47.+p . The Hartree-Fock (HF) approximation provides the simplest de-scription of interacting fermions . At this level of theory each electron moves in the aver-age field created by the others. This provides a numerically tractable, uncorrelated-particlepicture for the electrons, that approximates many systems well and provides a convenientbasis for higher- level descriptions. Electronic dynamics is described in terms of orbitals,one electron at a time. Correlated n-electron wavefunctions, in contrast, live in a high (3n)dimensional space and may not be readily visualized. Deviations from the uncorrelatedpicture (correlations), are responsible for many important effects. Correlation energies arecomparable in magnitude to chemical bonding energies and are thus crucial for predictingmolecular geometries and reaction barriers and rates with chemical accuracy. These energiescan be computed for molecules by employing a broad arsenal of computational techniquessuch as perturbative corrections , configuration interaction , multideterminant techniques ,coupled cluster theory and time dependent density functional theory TDDFT . Cor-relation effects are essential in superconductors and can be manipulated in artificialsemiconductor nanostructures . The fields of quantum computing and information arebased on manipulating correlations between spatially separated systems, this is known asentanglement .In this article we propose a nonlinear optical signal that provides a unique probe forelectron correlations. The technique uses a sequence of three optical pulses with wavevec-tors k , k and k , and detects the four wave mixing signal generated in the direction k S = k + k − k by mixing it with a fourth pulse (heterodyne detection). We show that thiscorrelation-induced signal S CI ( t , t , t ), which depends parametrically on the consequentivedelays t , t , t between pulses, vanishes for uncorrelated systems, providing a unique indi-cator of electron correlations. This technique opens up new avenues for probing correlationeffects by coherent ultrafast spectroscopy.Starting with the HF ground-state ( g ) of the system, each interaction with the laserfields can only move a single electron from an occupied to an unoccupied orbital. Thefirst interaction generates a manifold ( e ) of single electron-hole (e-h) pair states. A secondinteraction can either bring the system back to the ground state or create a second e-h pair.We shall denote the manifold of doubly excited states as f (Fig. 1). We can go on to generate2anifolds of higher levels. However, this will not be necessary for the present technique.The quantum pathways (i) and (ii) contributing to this signal can be represented by theFeynman diagrams shown in Fig. 1. Each diagram shows the sequence of interactions ofthe system with the various fields and the state of the electron density matrix during eachdelay period. We shall display the signal as S CI (Ω , Ω , t ), where Ω , Ω are frequencyvariables conjugate to the delays t and t (Fig. 1) by a Fourier transform S CI (Ω , Ω , t ) = Z ∞ Z ∞ dt dt S CI ( t , t , t ) exp( i Ω t + i Ω t ) , with t fixed. This yields an expression for the exact response function S CI (Ω , Ω , t = 0) (1)= X e,e ′ ,f − ω fg (cid:20) µ ge µ ef µ fe ′ µ e ′ g Ω − ω e ′ g − µ ge ′ µ e ′ f µ fe µ eg Ω − ω fe (cid:21) , where for simplicity we set t = 0. Two-dimensional correlation plots of Ω vs. Ω thenreveal a characteristic peak pattern, which spans the spectral region permitted by the pulsebandwidths. The two terms in the brackets correspond respectively to diagrams (i) and(ii) of Fig. 1. Here µ νν ′ are the transition dipoles and ω νν ′ are the transition energiesbetween electronic states, shifted by the pulse carrier frequency ω , i.e., ω e ′ g = ǫ e ′ − ǫ g − ω ,ω fe = ǫ f − ǫ e − ω and ω fg = ǫ f − ǫ g − ω . This shift eliminates the high optical frequencies.The carrier frequency of the three beams, ω , is held fixed and used to select the desiredspectral region. In Eq. (1) we have invoked the rotating wave approximation (RWA), andonly retained the dominant terms where all fields are resonant with an electronic transition.In both diagrams, during t the system is in a coherent superposition (coherence) betweenthe doubly excited state f and the ground state g . This gives the common prefactor (Ω − ω fg ) − . As Ω is scanned, the signal will thus show resonances corresponding to the differentdoubly-excited states f . However, the projection along the other axis (Ω ) is different forthe two diagrams. In diagram (i) the system is in a coherence between e ′ and g during t . As Ω is scanned, the first term in brackets reveals single excitation resonances whenΩ = ω e ′ g . For the second diagram (ii) the system is in a coherence between f and e ′ during t . This gives resonances at Ω = ω fe in the second term in the brackets. Many new peakscorresponding to all possible transitions between doubly and singly excited states ω fe shouldthen show up. 3he remarkable point that makes this technique so powerful is that the two terms inEq. (1) interfere in a very special way. For independent electrons, where correlations aretotally absent, the two e-h pair state f is simply given by a direct product of the singlepair states e and e ′ , and the double-excitation energy is the sum of the single-excitationenergies ǫ f = ǫ e + ǫ e ′ , so that ω e ′ g = ω fe and the two terms in the brackets exactly cancel.Density functional theory , when implemented using the Kohn-Sham approach, gives aset of orbitals that carry some information about correlations in the ground state. Thesignal calculated using transitions between Kohn-Sham orbitals will vanish as well. Thiswill be the case for any uncorrelated-particle calculation that uses transitions between fixedorbitals, no matter how sophisticated was the procedure used to compute these orbitals. Weexpect the resonance pattern of the two-dimensional S CI signal to provide a characteristicfingerprint for electron correlations.The following simulations carried out for simple model systems which contain a feworbitals and electrons, illustrate the power of the proposed technique. Doubly excitedstates can be expressed as superpositions of products of two e-h pair states. Along Ω weshould see the various doubly excited states at ω fg , whereas along Ω we observe the variousprojections of the f state onto single pair states ω e ′ g and the differences ω fe = ω fg − ω eg . The2D spectra thus provide direct information about the nature of the many body wavefunctionsthat is very difficult to measure by any other means. The patterns predicted by differentlevels of electronic structure simulations provide a direct means for comparing their accuracy.We used a tight-binding Hamiltonian H = H + H C + H L . The single-particle contribution H contains orbital energies and hoppings H = X m ,n t m ,n c † m c n + X m ,n t m ,n d † m d n , where c n and d n are electron and hole annihilation operators respectively, and the sum-mations run over spin-orbitals. We assume equal hopping t for electrons and holes. Themany-body term responsible for correlations H C = 12 X m ,n V eem n c † m c † n c n c m + 12 X m ,n V hhm n d † m d † n d n d m − X m ,m V ehm m c † m d † m d m c m , contains only direct Coulomb couplings. The electron-electron, hole-hole and electron-holeinteraction are denoted V ee , V hh , and V eh , respectively. Values of t and Coulomb integrals V eh , V eh (subscripts 0 and 1 indicate the sites) were derived by fitting emission spectra of4oupled quantum dots . Owing to the nature of quantum dots states, we can assume that V eh ≃ V ee ≃ V hh . We choose V ee = V hh = 1 . V eh and use these values for all orbitals. H L describes the dipole interaction with the laser pulses, H L = − E ( t ) µ m m d m c m + h.c ,where E ( t ) is the light field and µ m m are local dipole moments of various orbitals m , m . ω was tuned to the single-particle optical gap energy (1 3 eV).Even though our parameters are fitted to QDs, the overall picture emerging from thecalculations can be applied to a wider class of systems, whose optical response is determinedby correlated e-h pairs. We have employed an equation of motion approach for computingthe signal. Many-body states are never calculated explicitly in this algorithm. Instead, weobtain the signal directly by solving the Nonlinear Exciton Equations (NEE) . Theseequations describe the coupled dynamics of two types of variables representing single e-hpairs: B m = h d m c m i (here m = ( m , m ) stands for both the electron index m and thehole index m ) and two pairs Y mn = h d m c m d n c n i . For our model the NEE is equivalentto full CI, and yield the exact signal with all correlation effects fully included. This signalprovides a direct experimental test for many-body theories, which use various degrees ofapproximations to treat electron correlations. We compare the exact calculation (NEE)with the time dependent Hartree Fock (TDHF) theory, which is an approximate, widely usedtechnique for treating correlations by factorizing the Y variables into h d m c m i h d n c n i −h d n c m i h d m c n i . This assumes that two e-h pairs are independent and we only need tosolve the equations for h d n c n i . Correlation within e-h pairs is nevertheless retained by thislevel of theory, as evidenced by the finite S CI signal. The equations of motion derived usingboth levels of factorization are solved analytically, yielding the exact and the TDHF S CI signals. The TDHF solutions have the following structure: a set of single-particle excitationswith energies ǫ α and the corresponding transition dipole moments are obtained by solvingthe linearized TDHF equations. Many-particle state energies are given by sums of theseelementary energies. Two-particle energies are of the form ǫ α + ǫ β . This approximation isthe price we pay for the enormous simplicity and convenience of TDHF. Correlated many-electron energies computed by higher level techniques do not possess this additivity property.In general the TDHF signal contains a different number of resonances along Ω than theexact one. Their positions, ω fg = ǫ f − ǫ g , also differ since the former uses the additiveapproximation for the energies ǫ f . The Ω value of each resonance in the exact simulationis given by either ω e ′ g (first term in brackets in Eq. 1) or ω fe (second term in the brackets).5he simulations of the TDHF response function presented below show fewer peaks than inthe exact calculation. This dramatic effect reflects direct signatures of the correlated twoe-h pair wavefunction, which are only revealed by the S CI technique.We first consider a simple model, consisting of a single site with one valence orbital andone conduction orbital (Fig. 2A). The energy of the (spin-degenerate) single-pair state is ǫ e = − V eh . The only two-pair state has energy ǫ f = − V eh + V ee + V hh , compared with¯ ǫ f = 2 ǫ e in the TDHF approximation (the TDHF double-excited energies and frequencies willbe marked with a bar: ¯ ǫ f , ¯ ω fe ). Thus, the exact signal has two peaks at (Ω , Ω ) = ( ω eg , ω fg )and ( ω fe , ω fg ), while TDHF predicts only one peak at (Ω , Ω ) = ( ω eg , ¯ ω fg ) = (¯ ω fe , ¯ ω fg ).This high sensitivity to correlation effects is general and is maintained in more complexsystems. In Fig. 2B we consider a system with two valence orbitals with a splitting ∆ andone conduction orbital. It has 2 single e-h pair transitions e , e with energies ǫ = − V eh and ǫ = − V eh + ∆. The exact spectrum contains 8 peaks, with ǫ f energies being sums ofall quasiparticle interactions and hole level energies: ǫ f = − V eh + V ee + V hh , ǫ f = ǫ f + ∆, ǫ f = ǫ f + 2∆. Within TDHF we find ¯ ǫ f = 2 ǫ , ¯ ǫ f = ǫ + ǫ , ¯ ǫ f = 2 ǫ and the 2D spectrumonly shows 4 peaks.The simple energy-level structure of the two systems (Fig. 2A,B), whereby ¯ ω fe = ω eg , allows an insight into the differences in predictions of the two response func-tions. We recast Eq. (1) for the exact signal in a slightly different form:(Ω − ω fg ) − (Ω − ω e ′ g ) − (Ω − ω fe ) − (all dipoles for this system are equal µ ge ′ = µ ge = µ e ′ f etc.). The corresponding TDHF expression is: (Ω − ¯ ω fg ) − (Ω − ω e ′ g ) − . The differentΩ dependencies reflect different numbers of resonances with different FWHM along the Ω axis. The double resonance in TDHF is split into two resonances in the exact expression.Fig. 2C shows the signal from two coupled quantum dots, each hosting one valence andone conduction orbital. The S CI signal contains a rich peak structure, reflecting the 4(10) many body levels in the single- (double-) excited manifold. Again, the TDHF methodmisses many peaks. In this case, unlike the two previous systems, TDHF does not show allpossible resonances along the Ω axis. This is because one of the single-excited states ( e )is not optically allowed. In TDHF any f state, constructed as a direct product of e withanother state e i ( i = 1 , . . . ,
4) is forbidden, thus we have only 6 resonances. In the exactcalculation, the f states are not direct products, so we see all possible resonances along Ω .The differences between the TDHF and exact spectra in all these examples, illustrate the6ensitivity of the proposed signal to the correlated wavefunction.Computing electron correlation effects, which are neglected by HF theory, consti-tutes a formidable challenge of many-body theory. Each higher-level theory for electroncorrelations is expected to predict a distinct two-dimensional signal, which will reflectthe accuracy of its energies and many-body wavefunctions. The proposed technique thusoffers a direct experimental test for the accuracy of the energies as well as the many-bodywavefunctions calculated by different approaches. Time dependent density functional theoryTDDFT within the adiabatic approximation extends TDHF to better include exchange andcorrelation effects . However, the two are formally equivalent and yield a similar excited-state structure . The two-dimensional peak pattern of TDDFT will suffer from the samelimitations of TDHF.We can summarize our findings as follows: at the HF level which assumes independentelectrons , the S CI signal vanishes due to interference. TDHF (or TDDFT) goes one stepfurther and provides a picture of independent transitions (quasiparticles). Here the signalno longer vanishes, but shows a limited number of peaks. When correlation effects arefully incorporated, the many-electron wavefunctions become superpositions of states withdifferent numbers and types of e-h pairs. The Ω and Ω axes will then contain many morepeaks corresponding to all many body states (in the frequency range spanned by the pulsebandwidths), which project into the doubly-excited states. Thus, along Ω the peaks will beshifted, reflecting the level of theory used to describe electron correlations. Along Ω , theeffect is even more dramatic and new peaks will show up corresponding to splittings betweenvarious levels. This highly-resolved two dimensional spectrum provides a invaluable directdynamical probe of electron correlations (both energies and wavefunctions).Signals obtained from a similar pulse sequence, calculated for electronic transitions inmolecular aggregates and molecular vibrations show the role of coupling between Frenkelexcitons. A conceptually related NMR technique known as double quantum coherence re-veals correlation effects among spins. The technique showed unusual sensitivity for weakcouplings between spatially remote spins and has been used to develop new MRI imagingtechniques . Here we have extended this idea to all many-electron systems. The proposedtechnique should apply to molecules, atoms, quantum dots and highly correlated systemssuch as superconductors. It has been recently demonstrated that two-exciton couplings canbe controlled in onion-like semiconductor nanoparticles with a core and an outer shell made7f different materials . Nonlinear spectroscopy of the kind proposed here could provideinvaluable insights into the nature of such two-exciton states. Acknowledgments
This research was supported by the National Science Foundation Grant No. CHE-0446555and the National Institutes of Health Grant No. GM59230. G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge UniversityPress, 2005). R. J. Bartlett and M. Musial, Rev. Mod. Phys. 79, 291 (pages 62) (2007). B. Roos, Accounts of Chemical Research 32, 137 (1999), ISSN 0001-4842. M. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and et al., eds., Time-Dependent Density Functional Theory (Springer-Verlag, Berlin, 2006). P. Fulde, Electron Correlations in Molecules and Solids (Springer-Verlag, Berlin, 1984), 3rd ed. A. K. Wilson and K. A. Peterson, eds., Recent Advances in Electron Correlation Methodology(Acs Symposium Series, An American Chemical Society Publication, 2007). P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (pages 69) (2006). G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti,Rev. Mod. Phys. 78, 865 (pages 87) (2006). C. D. Sherrill and H. F. Schaefer, Advances in Quantum Chemistry, 34 (Academic Press, 1999). L. A. Curtiss, K. Raghavachari, P. C. Redfern, V. Rassolov, and J. A. Pople, J. Chem. Phys.109,7764 (1998). G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002). R. A. Kaindl, M. Woerner, T. Elsaesser, D. C. Smith, J. F. Ryan, G. A. Farnan, M. P. McCurry,and D. G. Walmsley, Science 287, 470 (2000). V. I. Klimov, S. A. Ivanov, J. Nanda, M. Achermann, I. Bezel, J. A. McGuire, and A. Piryatinski,Nature 447, 441 (2007), ISSN 0028-0836. F. Rossi and T. Kuhn, Rev. Mod. Phys. 74, 895 (2002). D. S. Chemla and J. Shah, Nature 411, 549 (2001). M. A. Nielsen and L. I. Chuang, Quantum Computation and Quantum Information (CambridgeUniversity Press, Cambridge, 2000). S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, NewYork,1995). M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern, andA. Forchel, Science 291, 451 (2001). M. Bayer, O. Stern, P. Hawrylak, S. Fafard, and A. Forchel, Nature 405, 923 (2000), ISSN0028-0836. V. Chernyak, W. M. Zhang, and S. Mukamel, J. Chem. Phys. 109, 9587 (1998). V. M. Axt and S. Mukamel, Rev. Mod. Phys. 70 (1), 145 (1998). O. Berman and S. Mukamel, Phys. Rev. A 67, 042503 (2003). S. Mukamel, Annu. Rev. Phys. Chem. 51, 691 (2000). W. Zhuang, D. Abramavicius, and S. Mukamel, Proc. Nat. Acad. Sci. USA 102, 7443 (2005). W. Richter and W. S. Warren, Concepts in Magnetic Resonance 12, 396 (2000). S. Mukamel and A. Tortschanof, Chem. Phys. Lett. 357, 327 (2002). igure Captions Fig. 1 Left: many-body states connected by transitions dipoles, including the groundstate g , the manifold of single e-h pairs e and the manifold of two-pair states f . Right: thetwo Feynman diagrams contributing to the correlation-induced signal S CI (Eq 1). t i are thetime delays between laser pulses. For independent electrons ω fe = ω e ′ g and diagrams of type(i) and (ii) cancel in pairs.Fig. 2 Absolute value of the exact and the TDHF S CI signals for three model systems.Energies on the axes are referenced to the carrier frequency, which excites interband tran-sitions. Each system has N ( N + 1) / N is the number ofsingle-excited levels. (A) 2 orbital system with N = 1 ( V eh = 194 . − ), (B) 3 orbitalsystem with N = 2 ( V eh = 194 . − ), (C) 2 coupled systems of type (A), but with dif-ferent gaps, V eh = 193 . − for one dot and 78 . − for the other and t = 59 . − , N = 4. The corresponding orbitals and splittings are given schematically below each panel.For systems (A) and (B), the TDHF misses half of the resonances along Ω , while for (C) itmisses 4 out of 10 resonances along Ω . 10. 10