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Surface-Plasmon-Induced Modification on the Spontaneous Letter pubs.acs.org/NanoLett Surface-Plasmon-Induced Modification on the Spontaneous Emission Spectrum via Subwavelength-Confined Anisotropic Purcell Factor † † † ‡ ‡ § Ying Gu,*, Luojia Wang, Pan Ren, Junxiang Zhang, Tiancai Zhang, Olivier J. F. Martin, † and Qihuang Gong*, † State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China ‡ State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China § Nanophotonics and Metrology Laboratory, Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-STI-NAM, ELG Station 11, CH-1015 Lausanne, Switzerland *S Supporting Information ABSTRACT: The mechanism of using the anisotropic Purcell factor to control the spontaneous emission linewidths in a four-level atom is theoretically demonstrated; if the polar- ization angle bisector of the two dipole moments lies along the axis of large/small Purcell factor, destructive/constructive interference narrows/widens the fluorescence center spectral lines. Large anisotropy of the Purcell factor, confined in the subwavelength optical mode volume, leads to rapid spectral line narrowing of atom approaching a metallic nanowire, nanoscale line width pulsing following periodically varying decay rates near a periodic metallic nanostructure, and dramatic modification on the spontaneous emission spectrum near a custom-designed resonant plasmon nanostructure. The combined system opens a good perspective for applications in ultracompact active quantum devices. KEYWORDS: Surface plasmon, spontaneous emission, Purcell factor, quantum emitter, quantum interference ecent developments in nanotechnology and information molecules and semiconductor quantum dots can be controlled − R technologies have made nanoscale light-matter interaction well.27 32 By confining the light into nanoscale volumes, a tremendous research focus.1 Small optical mode area in plasmonic elements allow for a nanoscale realization of Mollow nanofiber-based photonic structures plays a significant role triplet of emission spectra and antibunching of emission allowing low-light level quantum optical phenomena, such as photons of single molecules that traditional technique can not electromagnetically induced transparency in the nanowatt be accessible.1,33,34 Through the nanoscale coupling between 2,3 4 regime, four wave mixing with great gain, and two-photon the surface plasmon modes and single quantum emitter, the 5 − absorption with sharp peaks in the Rubidium vapor. Ultrasmall directional and efficiency generation of single photons17 19 and 6 optical mode volume in plasmon nanostructures leads to entanglement of two qubits35 were proposed. These advan- strong coupling between surface plasmons and quantum tages, thanks to rapidly developed fabrication techniques, make emitters, which enables the vacuum Rabi splitting,7,8 the Fano 9−12 the plasmonic nanostructure a promising candidate for lineshapes in the absorption spectrum, and its obvious applications in ultracompact active quantum devices, for fl 13 − in uence on the two-photon statistics. example, single photons sources17 19 and atomic spectroscopy Superior to many available photonic nanostructures, 36 6 on a chip. associated with ultrasmall optical mode volume, plasmonic Crossing damping between two closely lying upper states of structures present the key advantage of a large subwavelength- atoms, originated from the interaction with the common fi con ned anisotropic vacuum, that is, large anisotropic Purcell vacuum of electromagnetic fields, has led to ultranarrow 14,15 factor, which originates from an anisotropic electric mode spectral line in spontaneous emission,37,38 which is potentially 16−19 density of collective oscillations of free electrons in metals. of interest in spectroscopy. Except for nearly parallel dipoles, Another advantage is strong evanescent field of metallic such interesting interference phenomenon can not occur in a nanostructures, which has promoted many applications, for vacuum due to small or zero value of crossing damping terms. A example, SERS,20 nanometer biosensors and waveguides,21 natural atom generally has arbitrary polarized dipoles, so these − nonlinear optical frequency mixing,22 24 solar cell,25,26 and so forth. Through modifying the population of excited states and Received: February 16, 2012 decay rate of quantum emitters near plasmon structure, the Revised: April 8, 2012 fluorescence enhancement and quenching of fluorescent Published: April 18, 2012 © 2012 American Chemical Society 2488 dx.doi.org/10.1021/nl300655n | Nano Lett. 2012, 12, 2488−2493 Nano Letters Letter ff − A t e ects are scarcely observed experimentally. Anisotropic Purcell and Weisskopf Wigner approximations, the state vectors 1( ), A t B t 45−47 factor can provide nonzero or large crossing damping even with 2( ), and ( ) obey the Schrodinger equations orthogonal dipoles. Hence various photonic structures, for γ κ 39,40 d 1 12 itω12 itΔ 1 example, photonic band gap structures, layered dielectric At1()=− At1() − A21() te +Ω e Bt () waveguides,41 left-handed material,42 and plasmonic struc- dt 2 2 (1a) 43 ff tures, were proposed to enhance this interference e ect via d γ κ 44 2 21 −Δitω12 it 2 the anisotropic Purcell factor. However, the mechanism At2()=− At2() − Ate12() +Ω e Bt () dt 2 2 underlying the anisotropic Purcell factor control of the spectral (1b) lines associated with spontaneous emissions remains unclear. To address this issue, in this Letter we begin by theoretically d −Δit −Δ i t Bt()=−Ω* e12 A () t−Ω* e A () t resolving the underlying mechanism, specifically concentrating dt 11 2 2 (1c) on four-level atoms with closely lying upper levels. As γ γ |a ⟩ |a ⟩ |c⟩ κ where 1 and 2 are decay rates from 1 and 2 to , and 12 illustrated in Figure 1, if the angle bisector of two arbitrarily κ |a ⟩ |a ⟩ and 21 are crossing damping terms between 1 and 2 , |a ⟩ |a ⟩ κ κ κ respectively. For the closing states 1 and 2 , 12 = 21 = .If y-axis is allowed to be the quantum axis and the values of dipole μ μ μ θ moments 1 = 2 = , and 1,2 to be the intersection angles μ⃗ x γ Γ between 1,2 and -axis, in the anisotropic vacuum, 1,2 = xx 2 θ Γ 2 θ κ Γ θ θ Γ θ cos 1,2 + zz sin 1,2 and = xx cos 1 cos 2 + zz sin 1 sin θ Γ γ λ G Γ γ λ ImG γ μ2ω3 2, where zz/ 0 =3 ac Im zz, xx/ 0 =3 ac xx, 0 (= ac/ πε ℏc G β x y z (3 0 3)) is the decay rate in a vacuum, and ββ with = , , are the Green’s tensor coefficients.33,48 In the following, we will see that the x- and z-direction have different Purcell factors14 Γ γ Γ γ xx/ 0 and xx/ 0, which has guaranteed the quantum interference effects of spectral line width narrowing and widening of spontaneous emission. The spontaneous emission S ω π ∫ ∞ τ ⟨E− t E+ t τ ⟩eiωτ t → ∞ spectra ( ) = (1/ )Re 0 d ( ) ( + ) for , where E−(t) is the field operator of emission photons from the upper levels to |c⟩, were investigated in the framework of − quantum regression theorem.45 47 We use the dressed state analysis49 to develop a mechanism that controls the spectral linewidths of spontaneous emission via the anisotropic Purcell factor (Figure 1a). Writing the interaction Hamiltonian of the system as Figure 1. (a) The schematics of a four-level atomic system and its dressed states. (b) Narrowing (red curve) and broadening (green H =ℏΔ|11aa ⟩⟨|+ℏΔ| 1 2 aa 2 ⟩⟨ 2 | curve) of the spectral lines associated with spontaneous emission via +ℏΩ|⟩⟨|+ℏΩ|⟩⟨|+(..)ab ab Hc anisotropic decay rates, compared with lines (black curve) for a 11 2 2 (2) vacuum. Two dipole moments lie at an angle of 0.2π. Rabi frequencies then taking Ω = Ω = Ω and Δ = ω /2, we obtain Ω Ω γ 1 2 1 12 are 1 = 2 = 3.5 0. expressions for the eigenvalues and eigenvectors Ω Ω polarized dipole moments lies along the major/minor axis of R R λ+−= ,0,λλ0 ==− the effective Purcell factor ellipse, destructive/constructive 2 2 (3) interference narrows/widens the center spectral lines of and fluorescence. Using the mechanism, we study the quantum ⎛ 1 +−ε 1 ε ⎞ interference behavior in the spontaneous emission spectrum via η fi ⎛|+⟩⎞ ⎜ 2 ⎟⎛|⟩a ⎞ the subwavelength-con ned anisotropic Purcell factor: atomic ⎜ ⎟ ⎜ 2 2 ⎟⎜ 1 ⎟ spectral line rapid narrowing near a metallic nanowire, ⎜ |⟩0 ⎟ = ⎜ −22ηε η⎟⎜ |⟩b ⎟ nanoscale line width pulsing, that is, periodically from ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝|−⟩⎠ ⎜ 1 − ε 1 + ε ⎟⎝|⟩⎠ narrowing to widening, near a periodic metallic nanostructure, ⎜− 2η − ⎟ a2 and dramatic modification, even subnatural line width for two ⎝ 2 2 ⎠ (4) orthogonal dipoles, of spontaneous emission near a custom- Ω2 ω2 Ω2 ε ω Ω η Ω Ω designed resonant plasmon nanostructure. These results lay the where R = 12 +8 , = 12/ R, and = / R. The foundation of the light-atom interaction within the ultrasmall equations of motion for the density matrices of dressed states are given in the Supporting Information. The modified decay optical mode area. Also, the combined system of atoms and the ρ ρ metallic nanostucture may be integrated into ultracompact rates of their diagonal matrix elements, 00 and ±±, are given active quantum devices. by the expressions Mechanism of the Spontaneous Emission Spectrum 2 Γ=0 (2)4γγ + − κη (5) Control. Consider a closed four-level atomic system with two 12 |a ⟩ |a ⟩ |b⟩ closely lying upper levels 1 and 2 and two ground states 1 + ε2 ε 1 − ε2 and |c⟩ (Figure 1a). The transitions |a ⟩ ↔ |b⟩ and |a ⟩ ↔ |b⟩ Γ=()γγ + ±−()γγ +κ 1 2 ± 12 12 (6) are driven by a pump field with frequency ν and Rabi 4 2 2 Ω Ω Δ ω − ν Δ frequencies 1 and 2, and the detunings 1 = a1 and 2 which determine separately the linewidths of the central peak ω − ν ω − ω ω = a2 where a1 a2 = 12.
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