Decoupling light and matter: permanent dipole moment induced collapse of Rabi oscillations Denis G. Baranov,1, 2, ∗ Mihail I. Petrov,3 and Alexander E. Krasnok3, 4 1Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden 2Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny 141700, Russia 3ITMO University, St. Petersburg 197101, Russia 4Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, Texas 78712, USA (Dated: June 29, 2018) Rabi oscillations is a key phenomenon among the variety of quantum optical effects that manifests itself in the periodic oscillations of a two-level system between the ground and excited states when interacting with electromagnetic field. Commonly, the rate of these oscillations scales proportionally with the magnitude of the electric field probed by the two-level system. Here, we investigate the interaction of light with a two-level quantum emitter possessing permanent dipole moments. The semi-classical approach to this problem predicts slowing down and even full suppression of Rabi oscillations due to asymmetry in diagonal components of the dipole moment operator of the two- level system. We consider behavior of the system in the fully quantized picture and establish the analytical condition of Rabi oscillations collapse. These results for the first time emphasize the behavior of two-level systems with permanent dipole moments in the few photon regime, and suggest observation of novel quantum optical effects. I. INTRODUCTION ment modifies multi-photon absorption rates. Emission spectrum features of quantum systems possessing perma- Theory of a two-level system (TLS) interacting with nent dipole moments were studied in Refs. [17, 27], where electromagnetic field is of prime importance for a wide it was shown that such a system can radiate at Rabi fre- spectrum of applied problems, including laser science quency and serve as an emitter in the THz range. More [1, 2], fluorescent spectroscopy [3], nano-imaging [4{6], recently, it was demonstrated that a two-level system design of single photon sources [7{10] and efficient light with permanent dipoles can be inversely populated in the emitting devices [8, 11]. It also plays the central role in steady state if acted by two monochromatic fields [28]. the quantum information theory in the context of coher- Here, we investigate in more details the Rabi oscilla- ent qubits control [12{15]. Due to its very general formal- tions of a TLS with PDM. One of the main effects in such ism, it is equally applicable to the wide range of differ- systems is that PDM enables multi-photon Rabi oscilla- ent electronic systems: electric and magnetic atomic and tions and strongly affects the Rabi frequency for single molecular transitions, quantum dots and quantum wells, photon process. Since the pioneering paper of Meth et. superconducting Josephson junctions, defect centers in al. [22], it was shown that the Rabi oscillations in prin- nanocrystals and others. ciple may be suppressed by tuning the magnitude of the In the theoretical description of light-matter interac- emitter PDM. However, the treatment of this problem tion, it is often assumed that dynamics of the system has been mostly limited to the semiclassical approach. is governed by the non-diagonal matrix element of the dipole moment operator. However, certain systems pos- sess non-zero permanent dipole moments (PDM). An ex- ample of such system is a polar molecule, an atom polar- d ized by static electric field [16] or an asymmetric quan- ee tum dot [17]. Magnetic dipole atomic transitions, e.g., TLS deg in rare-earth ions of Eu3+ [18{20] may also have non- dgg zero permanent magnetic dipoles, in contrast to the case arXiv:1611.06897v1 [physics.optics] 21 Nov 2016 of electric dipole transitions, where diagonal elements of the electric dipole moment operator are always zero for dif = <i|qr|f> atomic eigenstates [21], what follows from the parity of EM field the wave function. Quantum systems with electric PDM have been widely investigated in the context of multi-photon processes [22{ FIG. 1. A schematic of the system under study. A two- 26]. It was shown that presence of permanent dipole mo- level system modeling a quantum emitter interacts with elec- tromagnetic field of a cavity resulting in Rabi oscillations of the two-level system population inversion. Inset on the right shows the internal structure of the two-level system with per- ∗ [email protected] manent dipole moments. 2 In this work, we address the fully quantum picture of (b) (a) Ωp = 0.1 ω0 Ωp = ω0 interaction of light with a permanent dipole TLS. By 1.0 1.0 means of numerical simulations and theoretical analysis, 0.5 0.5 we find new eigenstates of the PDM TLS{single mode cavity Hamiltonian and observe the collapse of Rabi os- 0.0 0.0 Inversion cillations in the few photon regime. Our findings demon- Inversion - 0.5 - 0.5 strate previously unexplored regime of light-matter in- - 1.0 - 1.0 teraction and provide novel tools for coherent control of 0 510 15 20 0 510 15 20 quantum emitters possessing PDM. ΩRt ΩRt (c) 0.5 - 0.90 Ωp = 0 - 0.95 II. SEMICLASSICAL DESCRIPTION 0.4 - 1.00 Inversion - 1.05 Ωp = 2ΩR We start from a brief overview of the semiclassical de- 0.3 - 1.10 0 2 4 6 8 10 scription of the dynamics of a TLS with PDM in the pres- Ωp = 4ΩR Ω t ence of light field. The system under study is schemat- 0.2 R Ω = 6Ω ically depicted in Fig. 1. It consists of a generalized Rabi frequency p R TLS interacting with electromagnetic field. The TLS can 0.1 make transition between the ground jgi and excited jei Ωp = 10ΩR collapse states, respectively, separated by the energy ¯h!0. 0.0 To begin with, we consider the simplest scenario of 0.0 0.1 0.2 0.3 0.4 0.5 a TLS driven by a classical monochromatic electromag- μegE0 netic wave E (t) = E0 cos (!0t). In what follows we will use the notation E (t) for description of the field com- FIG. 2. (a, b) Population inversion of a TLS with non-zero ponent interacting with the TLS, implying the electric permanent dipoles (solid) and an equivalent TLS without per- field. However, this formalism is equally applicable to manent dipoles (dashed) in a monochromatic electric field at different values of Ωp. Significant delay becomes visible in any other TLS interacting with either electric or mag- the permanent dipoles TLS dynamics. (c) Modified Rabi fre- netic oscillating field and therefore this does not limit the quency of the TLS with permanent dipole moments for differ- ubiquity of our results. Despite simplicity of this model, ent Ωp=ΩR as a function of applied electric field E0. Dashed it captures important signatures in the dynamics of a line shows unchanged Rabi frequency of a TLS without per- TLS with PDM. The Hamiltonian of the system has the manent dipoles. Inset: temporal dynamics of the population standard form H^ = H^TLS + H^int, where the TLS part is inversion at the collapse point corresponding to the first zero y H^ =h! ¯ 0σ^ σ^ and the interaction part is H^int = −E (t) d^. of the first order Bessel function J1,Ωp ≈ 3:8!0. The dipole moment operator d^ has the form y y y TLS in the interaction representation: d^ = dgeσ^ + degσ^ + dggσ^σ^ + deeσ^ σ;^ (1) Ω ∗ p Hereσ ^ = jgi hej is the lowering operator, d = d is c_g(t) = iΩR cos(!t) exp −i !0t − sin(!t) ce(t); eg ge ! the transition dipole moment, and d and d are the ee gg permanent dipole moments which are equal to zero in Ωp c_e(t) = iΩR cos(!t) exp i !0t − sin(!t) cg(t): the common Jaynes-Cummings model [2]. Without loss ! of generality we will assume the that deg along with dee (3) and dgg are all real-valued. Here we have introduced the quantity ¯hΩp = With the use of expression (1) for the dipole moment E0 (dee − dgg), which will be referred to as the perma- operator we write the Hamiltonian in the symmetric nent coupling constant. It shows the strength of asym- form: metry between the dipole moment elements in ground and excited states. ^ ¯h!0 y ¯h!0 y y Figures 2(a),(b) show the TLS population inversion H = − σ^σ^ + σ^ σ^ − ¯hΩR cos (!0t)σ ^ σ^ 2 2 2 2 (2) W (t) = jce (t)j − jcg (t)j as a function of time obtained y y −dggE0 cos (!0t)σ ^σ^ − deeE0 cos (!0t)σ ^ σ^ from the numerical solution of Eq. 3 for different ratios Ωp=!0. For comparison we also plot the population inver- where ¯hΩR = E0deg is the Rabi frequency. In the ab- sion of a pure non-diagonal system with dee − dgg = 0. sence of permanent dipoles, the population inversion of The Rabi frequency is fixed at value ΩR = 0:01!0. The the TLS oscillates in time with frequency ΩR. TLS is assumed to be initially in the ground state jgi. It Representing the TLS wave function in the general is seen that, in the regime of weak permanent coupling, form jΨ(t)i = ce (t) jei + cg (t) jgi and substituting into Ωp !0, the TLS dynamics is unaffected by permanent the Schroedinger equation, we arrive at the following dipoles [Fig. 2(a)]. However, in the strong permanent equations of motion describing temporal evolution of the coupling regime, Ωp ∼ !0, the effect of permanent dipoles 3 on the dynamics of the TLS becomes clearly visible, Fig.