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J Chem Phys. Author manuscript; available in PMC 2010 July 9.

Published in final edited form as:

PMCID: PMC2901102

NIHMSID: NIHMS212348

Shaul Mukamel, Department of Chemistry, University of California, Irvine, California 92697, USA;

The publisher's final edited version of this article is available at J Chem Phys

See other articles in PMC that cite the published article.

The correlated behavior of electrons determines the structure and optical properties of molecules, semiconductors, and other systems. Valuable information on these correlations is provided by measuring the response to femtosecond laser pulses, which probe the very short time period during which 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 by many 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-electron systems. The proposed signal pulse is generated only when the electrons are correlated, which gives rise to an extraordinary sensitivity. The peak pattern in two-dimensional plots, obtained by displaying the signal versus two frequencies conjugated to two pulse delays, provides a direct visualization and specific signatures of the many-electron wave functions.

Predicting the energies and wave functions of interacting electrons lies at the heart of our understanding of all structural, optical, and transport properties of molecules and materials.^{1}^{–}^{10} The Hartree–Fock (HF) approximation provides the simplest description of interacting fermions.^{1}^{,}^{5} At this level of theory each electron moves in the average field created by the others. This provides a numerically tractable, uncorrelated-particle picture for the electrons, which approximates many systems well and provides a convenient basis for higher-level descriptions. Electronic dynamics is described in terms of orbitals, one electron at a time. Correlated *n*-electron wave functions, in contrast, live in a high (3*n*) dimensional space and may not be readily visualized. Deviations from the uncorrelated picture (correlations) are responsible for many important effects. Correlation energies are comparable in magnitude to chemical bonding energies and are thus crucial for predicting molecular geometries and reaction barriers and rates with chemical accuracy. These energies can be computed for molecules by employing a broad arsenal of computational techniques such as perturbative corrections,^{10} configuration interaction,^{9} multideterminant techniques,^{3} coupled cluster theory,^{2} and time dependent density functional theory (TDDFT).^{1}^{,}^{4}^{,}^{11} Correlation effects are essential in superconductors^{7}^{,}^{8}^{,}^{12} and can be manipulated in artificial semiconductor nanostructures.^{13}^{–}^{15} The fields of quantum computing and information are based on manipulating correlations between spatially separated systems, this is known as entanglement.^{16}

In this article we propose a nonlinear optical signal that provides a unique probe for electron correlations. The technique uses a sequence of three optical pulses with wave vectors **k**_{1}, **k**_{2}, and **k**_{3}, and detects the four wave mixing signal generated in the direction **k*** _{S}*=

Starting with the HF ground-state (*g*) of the system, each interaction with the laser fields can only move a single electron from an occupied to an unoccupied orbital. The first interaction generates a manifold (*e*) of single electron-hole (e-h) pair states. A second interaction 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 generate manifolds 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 the Feynman diagrams^{17} shown in Fig. 1. Each diagram shows the sequence of interactions of the system with the various fields and the state of the electron density matrix during each delay period. We shall display the signal as *S*_{CI}(Ω_{3},Ω_{2},*t*_{1}) where Ω_{3} and Ω_{2} are frequency variables conjugate to the delays *t*_{3} and *t*_{2} (Fig. 1) by a Fourier transform

$${S}_{\mathrm{CI}}({\Omega}_{3},{\Omega}_{2},{t}_{1})={\int}_{0}^{\infty}{\int}_{0}^{\infty}d{t}_{2}d{t}_{3}{S}_{\mathrm{CI}}({t}_{3},{t}_{2},{t}_{1})\times \mathrm{exp}(i{\Omega}_{2}{t}_{2}+i{\Omega}_{3}{t}_{3}),$$

with *t*_{1} fixed. This yields an expression for the exact response function

$${S}_{\mathrm{CI}}({\Omega}_{3},{\Omega}_{2},{t}_{1}=0)=\sum _{e,{e}^{\prime},f}\frac{1}{{\Omega}_{2}-{\omega}_{fg}}\left[\frac{{\mu}_{ge}{\mu}_{ef}{\mu}_{f{e}^{\prime}}{\mu}_{{e}^{\prime}g}}{{\Omega}_{3}-{\omega}_{{e}^{\prime}g}}-\frac{{\mu}_{g{e}^{\prime}}{\mu}_{{e}^{\prime}f}{\mu}_{fe}{\mu}_{eg}}{{\Omega}_{3}-{\omega}_{fe}}\right],$$

(1)

where for simplicity we set *t*_{1}=0. Two-dimensional correlation plots of Ω_{2} versus Ω_{3} then reveal a characteristic peak pattern, which spans the spectral region permitted by the pulse bandwidths. 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 energies between electronic states, shifted by the pulse carrier frequency ω

In both diagrams, during *t*_{2} the system is in a coherent superposition (coherence) between the doubly excited state *f* and the ground state *g*. This gives the common prefactor (Ω_{2}–ω* _{fg}*)

The remarkable point that makes this technique so powerful is that the two terms in Eq. (1) interfere in a very special way. For independent electrons, where correlations are totally absent, the two e-h pair state *f* is simply given by a direct product of the single pair states *e* and *e*’, and the double-excitation energy is the sum of the single-excitation energies * _{f}* =

The following simulations carried out for simple model systems, which contain a few orbitals and electrons, illustrate the power of the proposed technique. Doubly excited states can be expressed as superpositions of products of two e-h pair states. Along Ω_{2} we should see the various doubly excited states at ω* _{fg}*, whereas along Ω

$${H}_{0}=\sum _{{m}_{1},{n}_{1}}{t}_{{m}_{1},{n}_{1}}{c}_{{m}_{1}}^{\u2020}{c}_{{n}_{1}}+\sum _{{m}_{2},{n}_{2}}{t}_{{m}_{2},{n}_{2}}{d}_{{m}_{2}}^{\u2020}{d}_{{n}_{2}}.$$

where *c _{n1}* and

$${H}_{C}=\frac{1}{2}\sum _{{m}_{1},{n}_{1}}{V}_{{m}_{1}{n}_{1}}^{\mathrm{ee}}{c}_{{m}_{1}}^{\u2020}{c}_{{n}_{1}}^{\u2020}{c}_{{n}_{1}}{c}_{{m}_{1}}+\frac{1}{2}\sum _{{m}_{2},{n}_{2}}{V}_{{m}_{2}{n}_{2}}^{\mathrm{hh}}{d}_{{m}_{2}}^{\u2020}{d}_{{n}_{2}}^{\u2020}{d}_{{n}_{2}}{d}_{{m}_{2}}-\sum _{{m}_{1},{m}_{2}}{V}_{{m}_{1}{m}_{2}}^{\mathrm{eh}}{c}_{{m}_{1}}^{\u2020}{d}_{{m}_{2}}^{\u2020}{d}_{{m}_{2}}{c}_{{m}_{1}}$$

contains only direct Coulomb couplings. The electron-electron, hole-hole, and electron-hole interactions are denoted *V*^{ee}, *V*^{hh}, and *V*^{eh}, respectively. Values of *t* and the Coulomb integrals *V*^{eh}_{00}, *V*^{eh}_{01} (subscripts 0 and 1 denote the sites) were derived by fitting emission spectra of coupled quantum dots.^{18} Owing to the nature of quantum dots states, we can assume that *V*^{eh}*V*^{ee}*V*^{hh}.^{19} We choose *V*^{ee}=*V*^{hh} = 1.2*V*^{eh} and use these values for all orbitals. *H _{L}* describes the dipole interaction with the laser pulses,

Even though our parameters are fitted to quantum dots, the overall picture emerging from the calculations can be applied to a wider class of systems, whose optical response is determined by correlated e-h pairs. We have employed an equation of motion approach for computing the signal. Many-body states are never calculated explicitly in this algorithm. Instead, we obtain the signal directly by solving the nonlinear exciton equations (NEE).^{20}^{,}^{21} These equations describe the coupled dynamics of two types of variables representing single e-h pairs: *B _{m}*=

In general the TDHF signal contains a different number of resonances along Ω_{2} than the exact one. Their positions, ω* _{fg}*=

We first consider a simple model, consisting of a single site with one valence orbital and one conduction orbital [Fig.2(A)]. The energy of the (spin-degenerate) single-pair state is * _{e}*=–

(Color) 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 transitions. Each system has *N*(*N*+1)/2 doubly excited levels, where *N* is the number **...**

This high sensitivity to correlation effects is general and is maintained in more complex systems. In Fig. 2(B) we consider a system with two valence orbitals with a splitting Δ and one conduction orbital. It has two single e-h pair transitions *e*_{1}, *e*_{2} with energies _{1}=–*V*^{eh} and _{2}=–*V*^{eh} +Δ. The exact spectrum contains eight peaks, with * _{f}* energies being sums of all quasiparticle interactions and hole level energies:

The simple energy-level structure of the two systems [Figs. 2(A) and 2(B)], whereby ${\stackrel{\u2012}{\omega}}_{fe}={\omega}_{eg}$, allows an insight into the differences in predictions of the two response functions. We recast Eq. (1) for the exact signal in a slightly different form: (Ω_{2}–ω* _{fg}*)

Figure 2(C) shows the signal from two coupled quantum dots, each hosting one valence and one conduction orbital. The *S*_{CI} signal contains a rich peak structure, reflecting the four (ten) many body levels in the single- (double-) excited manifold. Again, the TDHF method misses many peaks. In this case, unlike the two previous systems, TDHF does not show all possible resonances along the Ω_{2} axis. This is because one of the single-excited states (*e*_{1}) is not optically allowed. In TDHF any *f* state, constructed as a direct product of *e*_{1} with another state *e _{i}* (

Computing electron correlation effects, which are neglected by HF theory, constitutes a formidable challenge of many-body theory. Each higher-level theory for electron correlations^{1}^{,}^{4} is expected to predict a distinct two-dimensional signal, which will reflect the accuracy of its energies and many-body wave functions. The proposed technique thus offers a direct experimental test for the accuracy of the energies as well as the many-body wave functions calculated by different approaches. TDDFT within the adiabatic approximation extends TDHF to better include exchange and correlation effects.^{4}^{,}^{11} However, the two are formally equivalent and yield a similar excited-state structure.^{22} The two-dimensional peak pattern of TDDFT will suffer from the same limitations of TDHF.

We can summarize our findings as follows: At the HF level which assumes independent electrons, the *S*_{CI} signal vanishes due to interference. TDHF (or TDDFT) goes one step further and provides a picture of independent transitions (quasiparticles). Here the signal no longer vanishes but shows a limited number of peaks. When correlation effects are fully incorporated, the many-electron wave functions become superpositions of states with different numbers and types of e-h pairs. The Ω_{2} and Ω_{3} axes will then contain many more peaks corresponding to all many body states (in the frequency range spanned by the pulse bandwidths), which project into the doubly excited states. Thus, along Ω_{2} the peaks will be shifted, reflecting the level of theory used to describe electron correlations. Along Ω_{3}, the effect is even more dramatic and new peaks will show up corresponding to splittings between various levels. This highly resolved two-dimensional spectrum provides an invaluable direct dynamical probe of electron correlations (both energies and wave functions).

Signals obtained from a similar pulse sequence, calculated for electronic transitions in molecular aggregates^{23} and molecular vibrations,^{24} show the role of coupling between Frenkel excitons. A conceptually related nuclear magnetic resonance technique known as double quantum coherence reveals correlation effects among spins. The technique showed unusual sensitivity for weak couplings between spatially remote spins and has been used to develop new magnetic resonance imaging techniques.^{25}^{,}^{26} Here we have extended this idea to all many-electron systems. The proposed technique should apply to molecules, atoms, quantum dots, and highly correlated systems such as superconductors. It has been recently demonstrated that two-exciton couplings can be controlled in onionlike semiconductor nanoparticles with a core and an outer shell made of different materials.^{13} Nonlinear spectroscopy of the kind proposed here could provide invaluable insights into the nature of such two-exciton states.

This research was supported by the National Science Foundation Grant No. CHE-0446555 and the National Institute of Health Grant No. GM59230.

Shaul Mukamel, Department of Chemistry, University of California, Irvine, California 92697, USA.

Rafał Oszwałdowski, Instytut Fizyki, Uniwersytet Mikołaja Kopernika, Grudziadzka 5/7, 87-100, Toruń, Poland and Department of Chemistry, University of California, Irvine, California 92697, USA.

Lijun Yang, Department of Chemistry, University of California, Irvine, California 92697, USA.

1. Giuliani GF, Vignale G. Quantum Theory of the Electron Liquid. Cambridge University Press; Cambridge: 2005.

2. Bartlett RJ, Musial M. Rev. Mod. Phys. 2007;79:291.

3. Roos B. Acc. Chem. Res. 1999;32:137.

4. Marques MAL, Ullrich CA, Nogueira F, Rubio A, Burke K, Gross EKU, editors. Time-Dependent Density Functional Theory. Springer; Berlin: 2006.

5. Fulde P. Electron Correlations in Molecules and Solids. 3rd ed Springer; Berlin: 1984.

6. Wilson AK, Peterson KA, editors. Recent Advances in Electron Correlation Methodology. American Chemical Society; Washington, DC: 2007. (ACS Symposium Series).

7. Lee PA, Nagaosa N, Wen X-G. Rev. Mod. Phys. 2006;78:17.

8. Kotliar G, Savrasov SY, Haule K, Oudovenko VS, Parcollet O, Marianetti CA. Rev. Mod. Phys. 2006;78:865.

9. Sherrill CD, Schaefer HF. Advances in Quantum Chemistry. Vol. 34. Academic; New York: 1999.

10. Curtiss LA, Raghavachari K, Redfern PC, Rassolov V, Pople JA. J. Chem. Phys. 1998;109:7764.

11. Onida G, Reining L, Rubio A. Rev. Mod. Phys. 2002;74:601.

12. Kaindl RA, Woerner M, Elsaesser T, Smith DC, Ryan JF, Farnan GA, McCurry MP, Walmsley DG. Science. 2000;287:470. [PubMed]

13. Klimov VI, Ivanov SA, Nanda J, Achermann M, Bezel I, McGuire JA, Piryatinski A. Nature. Vol. 447. London: 2007. p. 441. [PubMed]

14. Rossi F, Kuhn T. Rev. Mod. Phys. 2002;74:895.

15. Chemla DS, Shah J. Nature. Vol. 411. London: 2001. p. 549. [PubMed]

16. Nielsen MA, Chuang LI. Quantum Computation and Quantum Information. Cambridge University Press; Cambridge: 2000.

17. Mukamel S. Principles of Nonlinear Optical Spectroscopy. Oxford University Press; New York: 1995.

18. Bayer M, Hawrylak P, Hinzer K, Fafard S, Korkusinski M, Wasilewski ZR, Stern O, Forchel A. Science. 2001;291:451. [PubMed]

19. Bayer M, Stern O, Hawrylak P, Fafard S, Forchel A. Nature. Vol. 405. London: 2000. p. 923. [PubMed]

20. Chernyak V, Zhang WM, Mukamel S. J. Chem. Phys. 1998;109:9587.

21. Axt VM, Mukamel S. Rev. Mod. Phys. 1998;70:145.

22. Berman O, Mukamel S. Phys. Rev. A. 2003;67:042503.

23. Mukamel S. Annu. Rev. Phys. Chem. 2000;51:691. [PubMed]

24. Zhuang W, Abramavicius D, Mukamel S. Proc. Natl. Acad. Sci. U.S.A. 2005;102:7443. [PubMed]

25. Richter W, Warren WS. Concepts Magn. Reson. 2000;12:396.

26. Mukamel S, Tortschanof A. Chem. Phys. Lett. 2002;357:327.

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