VOLUME 80, NUMBER 17 PHYSICAL REVIEW LETTERS 27 APRIL 1998 Selection Rules for the High Harmonic Generation Spectra Ofir E. Alon, 1 Vitali Averbukh, 1 and Nimrod Moiseyev 1,2 1 Department of Chemistry, Technion-Israel Institute of Technology, Haifa 32000, Israel 2 Minerva Center for Non-Linear Physics of Complex Systems, Technion-Israel Institute of Technology, Haifa 32000, Israel (Received 25 August 1997; revised manuscript received 3 November 1997) For three-dimensional many-electron time periodic Hamiltonians which are invariant under dynamical symmetry of order N we prove that only the nN 6 1th, n  1, 2, . . . , harmonics are generated. We discuss the application of the dynamical symmetry based selection rules to the generation of high harmonics by thin crystals. The derived selection rules are demonstrated numerically for a one- dimensional model, showing that the dynamically symmetric systems can be used not only as “filters” of the very high harmonics but also as their “amplifiers.” [S0031-9007(98)05899-2] PACS numbers: 42.65.Ky, 33.80.Wz, 42.25.Ja, 78.90. + t Numerous experimental and theoretical investigations of harmonic generation spectra (HGS) of noble gases in intense linearly polarized laser fields were stimulated by the interest in short-wavelength sources [1]. For example, by using ultrahigh powerful lasers Sarukura et al. [2] found the relative intensities of 9th to 23rd harmonics of He, and more recently Preston et al. [3] published new experimental data with HGS extending up to the 35th harmonic. Most recently Moiseyev and Weinhold have shown that the HGS in He can be treated as a single Floquet state phenomenon [4]. Calculations show that even for nonperiodic Hamiltonians the HGS can be obtained from the Fourier analysis of the time dependent dipole moment for a single Floquet state, provided the duration of the pulse is sufficiently long (see, for example, Fig. 7 of Ref. [5]). We, therefore, study the HGS employing Floquet formalism. Using the extended Hilbert space formalism of Sambe and Howland [6] the probability to get the nth harmonic from a system found in a Floquet state, C ´  exp2i ´t  ¯ hF ´ , is given by s n ´ ~ n 4 j†F ´ j ˆ me 2invt jF ´ ‡j 2 , (1) where the double bra-ket notation, † ··· ‡, stands for the integration over spatial variables and over time, ˆ m for the dipole moment operator, and v for the laser frequency. It is well known that the HGS of atoms in linearly polarized fields are composed only of odd harmonics [7]. A nonperturbative proof valid for quasienergy eigenstates was given by Ben-Tal, Beswick, and Moiseyev [8]. The proof can be reformulated in the following way: Suppose that no Floquet states are degenerate (accidental degeneracies of these states may occur at specific values of the field parameters [9], but generically the Floquet states of the systems of interest in this work are nondegenerate). Then jF ´ ‡ are simultaneous eigenfunctions of the second order dynamical sym- metry (DS) operator, ˆ P 2  x ! 2x, t ! t 1p v, with eigenvalues 61 and of the Floquet Hamilton- ian (here we assume that the field is polarized in ˆ x direction). The nth harmonic is emitted if and only if †F ´ j ˆ f n jF ´ ‡  † ˆ P 2 F ´ j ˆ P 2 ˆ f n ˆ P 21 2 j ˆ P 2 F ´ ‡ fi 0, where ˆ f n  ˆ mxe 2invt . Since in our case ˆ P 2 F ´  6F ´ , it implies that ˆ P 2 ˆ f n ˆ P 21 2  ˆ f n , i.e., ˆ f n belongs to the trivial representation of the DS group generated by ˆ P 2 . Consequently, the nonzero values of s n ´ are ob- tained if and only if ˆ mxe 2invt  ˆ m2xe 2invt 1pv , that is, for odd n’s. This result holds for HGS of any many-electron three-dimensional (3D) system of the second order DS. It will be extended below to the case of a DS of an arbitrary order N . The question we address here is how one can use such selection rules for the HGS to choose systems and the proper field polarization in order to filter out all high harmonics up to the nth one. The next question we shall answer is how stable are the results to perturbations which break the DS. Of course, it is interesting to know whether the selected system acts not only as a “filter” but also as an “amplifier” of the high harmonics. For the sake of clarity and without loss of generality (with regard to many dimensions and many-electron sys- tems), let us consider first the following effectively one- dimensional Hamiltonian which describes an electron’s motion in a circle under the influence of a time indepen- dent potential, V w, and the circularly polarized time de- pendent electric field: ˆ Hw, t   ˆ P 2 w 2mr 2 0 1 V w 1 eE 0 r 0 cosw2vt  . (2) The circle plane is assumed to be perpendicular to the field propagation direction. Suppose that V w possesses an N -fold symmetry axis. In such a case, the Hamiltonian of Eq. (2) is invariant under the following DS operator (written symbolically and not explicitly), ˆ P N  μ w ! w1 2p N , t ! t 1 2p N v ∂ . (3) Thus, the eigenfunctions of the Floquet Hamiltonian, ˆ H f w, t   2i ¯ h ≠ ≠t 1 ˆ Hw, t , are eigenfunctions of ˆ P N as well, ˆ H f w, t F ´ w, t   ´F ´ w, t , ˆ P N F ´ w, t   N p 1 F ´ w, t  , 0031-9007 98 80(17) 3743(4)$15.00 © 1998 The American Physical Society 3743