Collective spontaneous emission and quantum - optical nanoantennas

Transcription

1 Collective spontaneous emission and quantum - optical nanoantennas Gregory Slepyan, School of Electrical Engineering, Tel-Aviv University, Tel-Aviv, Israel. Sergey Maksimenko, Research Institute for Nuclear Problems of Belarussian State University, Minsk, Belarus Salman Mokhlespour, Jos E. M. Haverkort, Eindhoven University of Technology, Physics Department, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Axel Hoffmann Institut für Festkörperphysik, Technische Universität Berlin Hardenbergstr. 36, D Berlin, Germany

2 Quantum Theory is a complex thing: it has both real and imaginary parts - folklore

3 1. Epoch-making discoveries of quantum optics: Parcell effect and super-radiance

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6 2. Optical nano-antennas

7 3. Collective spontaneous emission in the dense atomic cloud: effect of timing Timing

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9 4. Collective spontaneous emission in the periodic array of quantum dots: statement of the problem Fig. 1. Schematic illustration of a 2D rectangular lattice of QDs

10 5. Theory H ˆ = H ˆ + H ˆ I 0 H S aa N ˆ ˆ z ˆ ˆ 0 = ω0 i + ωk k k +, i= 1 k 2 i= N ˆ ω ( )( ˆ ˆ ) k + H ˆ Ï = μu i Si + Si a + H. c. k r k, 2ε k

11 ˆ ρ t = ˆ ρ t ˆ ρ 0 ( ) ( ) ( ) QF Q F Master equation ˆ ρ ( t) = Tr ˆ ρ ( t) Q F QF ˆ ρ 1 = i S, i Ω S S, Γ ( S S + S S 2 S S ), t N N N ˆz ˆ+ ˆ ˆ+ ˆ ˆ ˆ ˆ ˆ+ ˆ ˆ ˆ ˆ ˆ+ ω 0 i ρ ij i j ρ ij ρ i j i jρ jρ i i i, j, i j 2 i, j Timed Dicke basis + ˆb g = a ˆb a = g ˆ b g bˆ+ = a = 0 Sˆ ± 1 i i ˆ ± i = e r b. 2π N 1 ˆ ± Se i 2π N i= 1 ˆ ± i b = ± r, i Timing ˆ ρ ˆ ˆ ˆ Γ = iω b, ˆ ρ i Ω b b, ˆ ρ ˆ ρbˆ bˆ + bˆ bˆ ˆ ρ 2 bˆ ˆ ρbˆ, ( ) z t 2

12 D bˆ t z ˆ z Γ = Γ b. 2 6ω 1 ( ) lim 1 Im { G( r,0; ω )}, 2 eff 0 ω0 = πc r k0 2 x 0 G 2 πω 0µ Γ = 3 ε 0 D eff ( ω ), exp 1 ik ( x pa) + ( y qa) + z,0;. 0 iqak0 cosθinc ( r ω ) = e 0 Spontaneous emission decay 4 π pq, = ( x pa) + ( y qa) + z Photonic density of states 0 ( δ ) 2 µ 2 m0 2 2πm Γ = k 2 0 ε 0a mn, { } kmn a πm 2πn 0 0 cosθinc kmn = k + k a a

13 6. Numerical results and discussion Fig. 1 The spontaneous emission rate of a rectangular lattice of QDs normalized with the spontaneous emission rate of a single QD for normal incidence of pump excitation pulse.

14 Fig. 2 The spontaneous emission rate of a rectangular lattice of QDs normalized with the spontaneous emission rate of a single QD as a function of the lattice period for different angles of pump pulse.

15 Fig. 3 The spontaneous emission rate of a rectangular lattice of QDs normalized to the spontaneous emission rate of a single QD as a function of the angle of the pump excitation pulse for lattice periodicities much smaller than the emission wavelength.

16 Fig. 5 The spontaneous emission rate of a rectangular lattice of QDs normalized with the spontaneous emission rate of a single QD as a function of the incident angle of the pump excitation pulse, for lattice periodicities larger than the emission wavelength.

17 7. Quantum antennas: general concept 2 ], the spontaneous emission probability is proportional to ξ k ξ k i( ) = θ e k r sin( ) i, j ij,

18 ξ ( ) 3 sin( θ N i N ) k r e d = sin( ) ( 2 ) 3 ( ) V θ V π δ k r k V Radiation pattern ( AF) N N e i( N 1)( k + k k cos θ ) a/2 x y 0 inc x y 0 inc = sin( Nk a / 2) sin[ Nk ( kcos θ ) a/ 2] sin( ka/ 2) sin[( k k cos θ ) a/ 2] x y 0 inc 2 2π ξ sin( θ) δ ( pq ) k = + a k k pq, k = + k pq

19 8. Conclusion 1). The influence of the dipole-dipole interaction in the array of emitters is equal to the action of an effective structured photonic reservoir on a single QD emitter within a pure state. As a result, the spontaneous emission decay rate for a QD-array is presented in a form similar to the theory of the Purcell effect in terms of a photonic density of states. In contrast to the Purcell effect, the decay rate in the lattice of QDs is related to a collective entangled state, and not to an excited state of a single particle. 2). The decay rate of the timed Dicke state strongly depends on the frequency and timing (via the pump excitation wave direction). Both superradiant and subradiant regimes are possible. The strong narrow peaks of the decay rate along the frequency axis represent the Van-Hove singularities of the photonic density of states. 3). The collective spontaneous emission strongly manifests itself in the directional properties of the emission due to diffraction processes in the array. The radiation pattern consists of a set of strong radiation peaks, implying a finite probability for photon emission either along the pump excitation wave direction, or in any other direction (spatial harmonics) which satisfies the diffraction condition of the periodic QD-array.. 4). The direction of the diffractive ray s are dictated by timing. Therefore, digital directional tuning of the radiation pattern become accessible via optical means (by variation of the pump wave direction). The collective spontaneous emission in arrays looks promising for fabrication of optically tuned phased-array nanoantennas intended for the directive transmission of digital optical signals in quantum informatics.

20 Quantum Theory is a complex thing: it has both real and imaginary parts -folklore Thank you very mach for your attention!