Attosecond Control of Restoration of Electronic Structure Symmetry

PHYSICAL REVIEW LETTERS 121, 173201 (2018) Attosecond Control of Restoration of Electronic Structure Symmetry † ChunMei Liu,1 Jörn Manz,1,2,3,* Kenji Ohmori,4,5, Christian Sommer,4,5,6 Nobuyuki Takei,4,5 Jean Christophe Tremblay,1 and Yichi Zhang2,3,4 1Freie Universität Berlin, Institut für Chemie und Biochemie, 14195 Berlin, Germany 2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China 3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China 4Institute for Molecular Science, National Institutes of Natural Sciences, Myodaiji, Okazaki 444-8585, Japan 5SOKENDAI (The Graduate University of Advanced Studies), Myodaiji, Okazaki 444-8585, Japan 6Max-Planck-Institut für die Physik des Lichts, 91058 Erlangen, Germany (Received 23 March 2018; published 25 October 2018) Laser pulses can break the electronic structure symmetry of atoms and molecules by preparing a superposition of states with different irreducible representations. Here, we discover the reverse process, symmetry restoration, by means of two circularly polarized laser pulses. The laser pulse for symmetry restoration is designed as a copy of the pulse for symmetry breaking. Symmetry restoration is achieved if the time delay is chosen such that the superposed states have the same phases at the temporal center. This condition must be satisfied with a precision of a few attoseconds. Numerical simulations are presented for 87 the C6H6 molecule and Rb atom. The experimental feasibility of symmetry restoration is demonstrated by means of high-contrast time-dependent Ramsey interferometry of the 87Rb atom. DOI: 10.1103/PhysRevLett.121.173201 The purpose of this Letter is to show that well-designed system is in a quasi-field-free environment—at initial laser pulses can not only break (see, e.g., Refs. [1–7])but (ti <tb), “central” [tc ¼ 0.5 ð tb þ tr Þ¼0], and final also restore the electronic symmetry of atoms and mole- times (tf ¼ −ti). For convenience, all pulses have cules. This presents a new challenge in coherent control Gaussian envelopes, or they are constructed as super- [8–10]. Controlling the electronic symmetry may be used in position of two Gaussian subpulses labeled j ¼ 1, 2, with chemical reaction [11–14] and charge migration [15,16].In ðjÞ ðjÞ ðjÞ the same amplitudes ϵ ¼ ϵ ¼ ϵr , carrier frequencies retrospect, there are already implicit solutions, e.g., in early b ωðjÞ ωðjÞ ωðjÞ τðjÞ τðjÞ τðjÞ ≪ work on Ramsey fringes [17,18], or high harmonic gen- c ¼ cb ¼ cr , and durations ¼ b ¼ r td. The eration [19–23], but the authors did not note the phenome- carrier envelope phases are set equal to zero, ηðjÞ ηðjÞ ηðjÞ ηðjÞ 0 non. We shall first derive the theory for two laser pulses that bx ¼ by ¼ rx ¼ ry ¼ . For example, break and restore symmetry, with applications to two model 2 2 −ðt−tbÞ =2τ systems that have highly symmetric ground states: the ϵ⃗bðtÞ¼ϵe fcos½ωcðt−tbÞe⃗x þsin½ωcðt−tbÞe⃗yg: benzene molecule and the 87Rb atom. Subsequently, we shall demonstrate its feasibility by a high-contrast Ramsey ð2Þ interferometry experiment applied to 87Rb. As by-product, As a consequence, the matrix representation of the the results will add a fascinating effect to attosecond (as) Hamiltonian of the system (s) with semiclassical dipole science [6,7,21–23]. − ⃗ ϵ⃗ As a proof of principle, we consider the scenario coupling to the laser field, HðtÞ¼Hs d · ðtÞ is adjunct where symmetry is broken (b) and restored (r)bytwo upon time reversal (see the Supplemental Material [24]) right circularly polarized laser pulses that are well separated HðtÞ¼HÃð−tÞ: ð3Þ from each other, with maximum intensities at tb < 0 and at tr ¼ −tb. The time delay is td ¼ tr − tb. The electric field Our scenario focuses on the laser driven electron for symmetry restoration ϵr is designed as a copy of the dynamics for fixed nuclei. It starts with the system in its field ϵb for symmetry breaking, ground (g) state, with wave function ΨðtiÞ¼Ψg, energy 0 ϵrðt þ trÞ¼ϵbðt þ tbÞ: ð1Þ Egð¼ Þ, and irreducible representation IRREPg. The first laser pulse breaks symmetry by exciting the system to a Ψ Ψ Ψ Ψ The total electric field is ϵðtÞ¼ϵbðtÞþϵrðtÞ, with intensity superposition ðtÞ¼cgðtÞ g þ ceðtÞ e of g and an 2 IðtÞ¼ð1=2Þcε0jϵðtÞj . It is negligible—that means the excited state Ψe with energy Ee, and with populations 0031-9007=18=121(17)=173201(6) 173201-1 © 2018 American Physical Society PHYSICAL REVIEW LETTERS 121, 173201 (2018) FIG. 1. Evolution of the electronic density upon laser-induced symmetry breaking, charge migration, and symmetry restoration. The central panels show snapshots of the time-dependent part of the electronic density in the superposition state iηg −iEgt=ℏ iηe −iEet=ℏ 2 Cge e ψ g þ Cee e ψ e, jCej ¼ 0.0037.Timetc ¼ 0 is chosen to satisfy condition (5). (a) Application to 87 C6H6∶ D6h → Cs → D6h. (b) Application to Rb: isotropic → anisotropic → isotropic. The laser parameters are in Figs. 2 and 3,legends. 2 2 1 1 PgðtÞ¼jcgðtÞj , PeðtÞ¼jceðtÞj and with different 0 4 − 2π 0 PeðtfÞ¼ PgðtcÞPeðtcÞ cosð t =TÞ : ð6Þ IRREPg ≠ IRREPe—this means symmetry breaking. 2 2 This can be visualizedR by snapshots of the one-electron ρ r Ψ rN sN 2 sN rN−1 rN density ð ;tÞ¼ j ð ; ;tÞj d d (where and The first application is to the benzene molecule. sN denote the positions and spins of the electrons). In the Symmetry breaking in benzene has already been docu- quasi-field-free environment near tc, the superposition mented by Ulusoy and Nest [32] in their fundamental iηg−iEgt=ℏ state evolves with coefficients cgðtÞ¼Cge and approach to laser control of aromaticity—this motivates our iηe−iEet=ℏ ceðtÞ¼Cee , and ρðr;tÞ describes charge migra- choice and adaptation of their molecular model system. tion with period T ¼ h=ΔE where ΔE ¼ Ee − Eg Initially (t ¼ ti), the molecule is in its electronic ground [15,16,25–31]. The related probabilities are nearly con- state S0, with D6h symmetry, see Fig. 1(a). The electric ϵ⃗ ϵ⃗ stant, and the phase difference decreases linearly with time, fields bðtÞ and rðtÞ of the laser pulses that break and restore this molecular symmetry are illustrated in Fig. 2(a). The laser parameters with resonant frequency defined by ΔηðtÞ¼ηe − ηg − ΔE · t=ℏ: ð4Þ ℏωc ¼ Ee − Eg ¼ hc=λ are listed in the Fig. 2 legend. The pulse defined in Eq. (2) excites the benzene This allows us to choose the time delay td such that molecule from its initial (t ¼ ti) electronic ground state Ψg ¼ S0 (IRREP A1g, D6h symmetry) to the excited Δη 0 2π 0 π ðtc ¼ Þ mod ¼f ; Æ g: ð5Þ superposition state Ψðtc ¼ 0Þ¼cgð0ÞΨg þ ceð0ÞΨe (Cs symmetry). Relatively weak pulses with peak intensity I ¼ 2 6 2 As a consequence, the coefficients cgðtcÞ and ceðtcÞ . GW=cm are applied, on a timescale where nuclear − 10 are real valued, except for an irrelevant global phase motion can safely be neglected [33,34], tf ti < fs. 0 ηð¼ ηg ¼ ηe mod πÞ. This yields the maximal excitation probability Peðtc ¼ Þ¼ 0 0037 Ψ The second laser pulse ϵrðtÞ, Eq. (1), deexcites the . Here, the excited state e is a complex-valued superposition state back to the ground state if the two superpositionpffiffiffiffiffiffiffiffi of two real-valued degenerate excited states, — conditions, Eqs. (3) and (5) are satisfied thus it restores Ψe ¼ 1=2ðSx þ iSyÞ (IRREP E1u) with energy Ee ¼ symmetry. In practice, condition (5) must be satisfied with Ex ¼ Ey [16,32]. The corresponding energy gap is precision of a few attoseconds, otherwise symmetry cannot ΔE ¼ 8.21 eV, and the period of charge migration is be restored. For example, if the second pulse is fired with T ¼ 504 as. Subsequently, the pulse with electric field additional time t0 at t0 t t0, that means with increasing r ¼ r þ ϵ⃗rðtÞ restores symmetry (D6h). The population dynamics time delay t0 t t0 that violates condition (5), then the d ¼ d þ PeðtÞ is illustrated in Fig. 2(b). 0 0 final (tf ¼ tf þ t ) population of the excited state varies Figure 1(a) shows snapshots of the initial and periodically, with amplitude 2PgðtcÞPeðtcÞ and period T, final electronic densities of benzene, together with the 173201-2 PHYSICAL REVIEW LETTERS 121, 173201 (2018) (a) (c) (b) (d) FIG. 2. Laser-driven symmetry breaking and restoration in benzene upon application of right circularly polarized pulses with amplitude ϵ ¼ 4.42 × 106 V=cm, wavelength λ ¼ 151.016 nm, and with pulse duration τ ¼ 0.47 fs. (a) Envelope and x component of the laser pulses used to break and restore symmetry at time delay tr − tb. (b) Evolution of the excited state population for various 0 time delays, tr − tb ¼ tr − tb þ kT=16, with tr − tb ¼ 4.282 fs and period T ¼ 504 as and k ∈ f0; …; 16g. (c) Final populations [compare panel (b)] versus time delay between the pulses for symmetry breaking and restoration. (d) Phase difference (modulo 2π) 0 at tc,seeEq.(4). time-dependent part of the density between the two laser ΔE ¼ 4.1688969 eV [35], the period is T ¼ 0.9920292 fs. pulses, which documents circular charge migration. The Excitations of Rydberg states of 87Rb have already circulating electronic structure of the superposition state discovered various fundamental effects [36–40]—this has broken symmetry (Cs) compared to the initial sym- motivates our choice of this atomic model system. metry (D6h) of benzene in its ground state S0.

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