Spontaneous emission of a two-level system near the interface of topological insulators

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Spontaneous emission of a two-level system near the interface of topological insulators View the table of contents for this issue, or go to the journal homepage for more 2014 EPL 105 64001 (http://iopscience.iop.org/0295-5075/105/6/64001) Home Search Collections Journals About Contact us My IOPscience iopscience.iop.org/page/terms http://iopscience.iop.org/0295-5075/105/6 http://iopscience.iop.org/0295-5075 http://iopscience.iop.org/ http://iopscience.iop.org/search http://iopscience.iop.org/collections http://iopscience.iop.org/journals http://iopscience.iop.org/page/aboutioppublishing http://iopscience.iop.org/contact http://iopscience.iop.org/myiopscience March 2014 EPL, 105 (2014) 64001 www.epljournal.org doi: 10.1209/0295-5075/105/64001 Spontaneous emission of a two-level system near the interface of topological insulators Ge Song, Jing-ping Xu(a) and Ya-ping Yang Key Laboratory of Advanced Micro-structure Materials, Ministry of Education, School of Physics Science and Engineering, Tongji University - Shanghai 200092, PRC received 18 October 2013; accepted in final form 19 February 2014 published online 26 March 2014 PACS 42.50.Ct – Quantum description of interaction of light and matter; related experiments PACS 78.20.-e – Optical properties of bulk materials and thin films PACS 03.65.Vf – Phases: geometric; dynamic or topological Abstract – Spontaneous emission of a two-level system near the interface of topological insulators (TIs) is investigated. Because of the topological magnetoelectric (TME) effect of the TI, the decay rates near a TI interface are substantially inhibited in comparison with that near a dielectric interface, especially for parallel dipole. The influences of topological magnetoelectric polarizability on decay rates through the radiative mode and decay rates through the evanescent mode have been analyzed in detail. Decay through dissipation has also been considered. We provide a new way to control the spontaneous emission by using TIs and this could be used to confirm the TME effect. Copyright c© EPLA, 2014 Introduction. – Significant efforts have been devoted to a new type of material, namely the topological insula- tors (TIs), which have bulk insulating energy gaps with gapless surface states on the boundary protected by time- reversal symmetry [1–5]. Experimentally, the topological state has been observed in the HgTe quantum well [6], the BixSb1−x alloy [7], and the stoichiometric crystals Bi2Se3, Bi2Te3, TlBiSe2 and Sb2Te3 [8–10]. Theoretically, three- dimensional TIs exhibit the topological magnetoelectric (TME) effect [2], which is characterized by the topologi- cal magnetoelectric polarizability (TMEP) Θ. A magne- toelectric coupling can be induced using a time-reversal breaking perturbation [2,11] (e.g., a thin magnetic coat- ing on the surface of the TI or an applied perpendicular magnetic field). Specifically, the TME effect means an applied electric field generates a magnetization and vice versa [11]. Therefore, the electromagnetic response of the TI shows significant differences from that of a dielectric. As a result, the Fresnel formula, Brewster angle and the Goos-Hänchen effect [12], as well as the Imbert-Fedorov shift [13] have been investigated. At the same time, the magneto-optical Faraday effect and Kerr effect in the TI system have been studied theoretically [14–16] and exper- imentally [17], and the quantum effects in the TI system, e.g. the Casimir effect, has also attracted considerable attention [18–22]. (a)E-mail: xx jj [email protected] (corresponding author) It is well known that the spontaneous emission is dependent not only on the property of the emitter it- self, but also on the environment surrounding it [23]. Therefore, in recent decades, materials or structures with unusual electromagnetic properties have been widely used to control the spontaneous emission [24–34], and the dy- namics of spontaneous emission in disordered systems is also extensively studied experimentally [35–37]. Because of the special electromagnetic response of the TI, the spon- taneous emission near it must be significantly different from that near a dielectric, and is still not well studied. Thus, this study considers the spontaneous emission of a two-level system near the TI interface and the results are analyzed in detail. Model and decay rates. – Before the calculation of decay rate, the influence of TI on the electromag- netic model should be made clear. Based on the topo- logical field theory [2], the electromagnetic response of a three-dimensional TI is well described by appending the topologically quantized magnetoelectric term SΘ =( αΘ/4π2 ) ∫ dx3dtE ·B to the usual electromagnetic ac- tion S0 = ∫ dx3dt [ εE2 − (1/μ)B2 ] , where dx3dt is the volume element of space and time. E and B are the elec- tric and magnetic fields, respectively. Here α = e2/�c (where c is the speed of light) is the fine-structure constant and Θ, which is known as the topological magnetoelec- tric polarizability (axion field), describes the topological 64001-p1 Ge Song et al. Fig. 1: (Colour on-line) The configuration of the TI with a mag- netic coating of thickness δ � λ0 � d. Bold arrows indicate a two-level system, which is placed in vacuum. The origin of the z coordinate is taken at the interface which is in the x-y plane. magnetoelectric (TME) effect, which can induce mixed polarizations of the electric field and magnetic field at the interface [2,18]. It is known that all time-reversal invariant insulators fall into two general classes, described by Θ = 0 (trivial vacuum) or Θ = π (nontrivial TI). In this case, all physically measurable quantities are invariant when Θ is under the 2π shift [11]. To achieve the controllability of spontaneous emission, a time-reversal symmetry breaking perturbation including a thin magnetic coating (thickness δ) on the surface of the TI is applied to remove the de- generacy of the surface states, which is shown in fig. 1. In this case, physically measurable quantities are no longer periodic in Θ [11]. The magnetic coating leads to a quan- tized Θ with odd integer values of π as Θ = (2n+ 1)π, where n ∈ Z. Here positive or negative values are related to different signs of the magnetization on the surface [11] and n is determined by the nature of the coating but in- dependently of the absolute value of the magnetization of the coating [18,19]. It is stressed that the thickness of magnetic coating is much less than the wavelength λ0 (corresponding to the transition frequency), which is much less than the thickness of TI d, that is δ � λ0 � d. There- fore, an effective action S0 + SΘ including the boundary surface is derived. Furthermore, the axion coupling term SΘ does not modify the conventional Maxwell’s equations with modified constituent relations D = εE + α (Θ/π)B and H = (B/μ)−α (Θ/π)E [2,17,18]. Therefore, from the two constituent relations and usual Maxwell’s equations, the reflection matrix describing the relation between the incident and reflective field from semi-infinite dielectric into semi-infinite TI can be written as [38] R = ( rTETE r TM TE rTETM r TM TM ) = 1 Δ ( nL 2−nT 2−ᾱ2+nLnT ξ− 2ᾱnL 2ᾱnL nT 2−nL2+ᾱ2+nLnT ξ− ) , (1) where nL (nT ) is the refractive index of vacuum (TI), ᾱ = αΘ/(πε0c), Δ = nL2 + nT 2 + (ᾱ/ε0c)2 + nLnT ξ+, and ξ± = (KLzKT ) / (KTzKL) ± (KTzKL)/ (KLzKT ). Here KL (KT ) is the wave number in vacuum (TI), KL = nLω/c, KT = nTω/c and KLz (KTz) is its z com- ponent. The corresponding matrix element (the reflection coefficient) rQq (q,Q = TE, TM) represents the ratio of the reflected part of polarization Q to the incident field of polarization q at the surface. Here we stress that, for a dielectric the off-diagonal elements rTMTE and r TE TM are zero, which means the polarization of the reflective field is the same as the incident field and the diagonal elements re- duce to the usual Fresnel coefficients with Θ = 0. Because of the TME effect, those terms are modulated so that TI can change the electromagnetic environment around it es- sentially different in comparison with a dielectric. We restrict our discussion to the case in which a two- level system with transition frequency ω0 (in the low- frequency limit ω0 < Es, Eg, where Es is the gap of the surface states while Eg is the TI’s bulk gap) and the dipole moment Pa is placed at an arbitrary position on the nega- tive z-axis ra = (0, 0, za). Assuming the system is initially in the upper level, following the previous works steps un- der the rotation wave approximation and Markovian ap- proximation [29,30], we can derive the expression of the spontaneous decay rate at the transition frequency [39] Γ = 2 �ε0 ω0 2 c2 Pa · Im ↔ G (ra, ra, ω0) ·Pa, (2) where Im ↔ G(ra, ra, ω0) is the imaginary part of the classical Green’s tensor at transition frequency ω0 and ra is the position of the emitter. Following the method in ref. [40], we deduce the Green’s tensor concerning our model in fig. 1, in which the unit source dipole (position ra) and the detection point (posi- tion r) are both in vacuum, as ↔ G (r, ra, ω0) = i 8π2 ∫ d2K‖ KLz eiK‖·(ρ−ρa) × ∑ q=TE,TM [ eiKLz(z−za)ê+qLê + qLsgn (z−za) +e−iKLz(z−za)ê−qLê − qLsgn (za − z) + ∑ Q=TE,TM rQq e −iKLz(z+za)ê−QLê + qL ] , (3) where sgn(z) is the sign function, K‖ is the component of the wave vector parallel to the interface (x-y plane), and the equation K‖ 2 + KLz2 = KL2 = ω02/c2 is satisfied. Here ê+qL (ê − qL) denotes the unit vector of the q-polarized electric field incident along the positive (negative) z-axis in vacuum. Because of the TME effect of TI, the reflective field will contain both the TE and TM polarization field no matter the incident field is TE or TM polarized, which is implicit in the Green’s tensor. Different from the Green’s tensor of the usual dielectric case [31,32], eq. (3) con- tains the contribution of topological magnetoelectric effect which is embedded by the reflection matrix element rQq . 64001-p2 Spontaneous emission of a two-level system near the interface of topological insulators -0.2 -0.1 0.0 0 2 4 6 8 10 Θ 0 Θ π Θ 5π Θ 9π Θ 29π Γ ⊥ /Γ 0 z a /λ(a) -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0 2 4 6 8 10 -0.2 -0.1 0.0 0 1 2 3 4 5 6 7 Γ | |/Γ 0 z a /λ Θ 0 Θ π Θ 5π Θ 9π Θ 29π -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0 1 2 3 4 5 6 7 (b) Fig. 2: (Colour on-line) The decay rates of a two-level system near different samples interfaces with εT = 41.5 as functions of the position. The dipole is perpendicular (a) or parallel (b) to the interface. Insets: enlarged views focusing on positions within one wavelength. With the Green’s tensor concerning TI eq. (3) in hand, it is straightforward to calculate the decay rate of a two- level system near the TI interface with the dipole momen- tum parallel or perpendicular to the interface. After a rigourous derivation, the decay rates are expressed as Γ‖ = Γ0 3 4 Re ∫ ∞ 0 dK‖ KL K‖ KLz {[ 1 + rTETEe −2iKLzza] + ( KLz KL )2 [ 1− rTMTM e−2iKLzza ]} , (4) Γ⊥ = Γ0 3 2 Re ∫ ∞ 0 dK‖ 1 KLz K‖ 3 KL 3 [ 1 + rTMTM e −2iKLzza] , (5) where Γ0 = pa2ω03/ ( 3πε0�c3 ) is the decay rate in free space and the reflection coefficients can be got from eq. (1). Although the TME effect greatly affects the reflective be- haviours, specific to the decay rates, they are only related to the diagonal elements of the reflection matrix. Since po- larization conversion has no effect on the decay rates, the influence on decay is just embodied in TMEP Θ, which is precisely the unusual electromagnetic responses of three- dimensional TIs. Therefore, we will study the influence of TMEP Θ on the decay rates. Calculation and analysis. – In this section, the nu- merical calculations and analysis are given. At first, we compare the difference of decay rates between dielectric (Θ = 0) and TI (Θ = (2n+ 1)π, n ∈ Z). Since Θ is determined by the nature of TI, in order to emphasize the influence of Θ we need to get several TI samples with dif- ferent TMEPs. Most candidates of TI are alloys or crystals so the permittivity is always high, for Bi1−xSbx it is εT ∼= 100 [11,12] and for the Bi2X3 (X = Te, Se) class of mate- rials it is εT ∼ 30–80 [14] in the infrared frequency area. Therefore, for simplicity we suppose there are several TI samples with the same permittivity εT = 41.5 (dielectric LiTaO3 at infrared frequency [41]) and μT = 1 with differ- ent TMEPs. In experiment it is convenient to use Rydberg atom or quantum dot because their transition frequencies can be prepared in the infrared frequency area. By mea- suring the emission spectrum, one can get the decay rates. Now we show the results of the calculations for the decay rates. Decay rates as a function of position in the vicinity of TI samples surfaces with different TMEPs have been plotted in fig. 2. Insets are enlarged views focusing on po- sitions within one wavelength. Solid black lines indicate the decay rates near a dielectric (Θ = 0) while other colour lines mean the decay rates near TIs (Θ �= 0). Obviously, all rates oscillate and tend to Γ0 far away from the interface as shown in the insets. Besides, there exist some different behaviors between different samples. When Θ = π, the discrepancy for the decay rates near a dielectric and a TI interface is very tiny, and magnitudes are nearly equiva- lent. However, with increasing Θ, the decay rates near the surface are substantially inhibited. Especially for parallel dipole, the inhibiting effect is significant. Behaviors of the decay rates with larger Θ seem that the two-level system is placed near a perfect mirror. Based on these reasons, we choose an extremely large value of TMEP, i.e., Θ = 29π to calculate the decay rates, which are shown as the violet dotted lines in fig. 2. It is easy to find that, Γ⊥ ≈ 2Γ0 and Γ‖ ≈ 0 at the interface. This is the signature of a two-level system near a perfect mirror, which is shown in fig. 4(a) of ref. [26]. Different actions of a two-level system near the TI interface have been attributed to the TME effect. In order to clarify the reason for the decay behaviours, it is nessessary to analyze the reflection matrix. After some deduction, the diagonal elements of the reflection matrix can be rewritten as rTETE = − εT (1− KLzKTz ) + ( KTz KLz − 1) + ᾱ2 εT (1 + KLzKTz ) + ( KTz KLz + 1) + ᾱ2 , (6) rTMTM = εT (1 + KLzKTz )− ( KTz KLz + 1) + ᾱ2 εT (1 + KLzKTz ) + ( KTz KLz + 1) + ᾱ2 . (7) Since ᾱ = αΘ/(πε0c), it is obvious that lim Θ→∞ rTETE → −1 and lim Θ→∞ rTMTM → 1. It is easy to get Γ‖|za=0 = 0 and 64001-p3 Ge Song et al. -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.8 1.2 1.6 2.0 (a) Θ 0 Θ π Θ 5π Θ 9π Θ 29π Γ ⊥ ra d/Γ 0 z a /λ -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.0 0.5 1.0 1.5 (b) Θ 0 Θ π Θ 5π Θ 9π Θ 29π Γ | |ra d/Γ 0 z a /λ -0.20 -0.15 -0.10 -0.05 0.00 0.0 0.5 1.0 1.5 2.0 Θ 0 Θ π Θ 5π Θ 9π Θ 29π Γ ⊥ ev a/Γ 0 z a /λ(c) -0.20 -0.15 -0.10 -0.05 0.00 0.0 0.5 1.0 1.5 2.0 Θ 0 Θ π Θ 5π Θ 9π Θ 29πΓ | |ev a/Γ 0 z a /λ(d) Fig. 3: (Colour on-line) Radiative decay rates ((a), (b)) and evanescent decay rates ((c), (d)) as functions of the position with different TMEPs. Γ⊥|za=0 = 2Γ0 by substituting rTETE = −1, rTMTM = 1 and za = 0 into eqs. (4) and (5). This is the reason for radiative decay rates at a TI interface like that at a perfect mirror with increasing TMEP, which is shown in fig. 2 (violet dotted lines). On the contrary, when Θ → 0, we can get rTETE → (KLz −KTz) / (KLz +KTz) and rTMTM → (εTKLz −KTz) / (εTKLz +KTz), which are the ordinary Fresnel coefficients. This is the reason for the tiny discrepancy between the black solid lines (dielec- tric) and the red dashed lines (TI) in fig. 2. Therefore, TI could be used to control the decay rates. By means of controlling TMEP, decay behaviors can be moved from the dielectric interface case to the perfect mirror case. In addition, this could be an evidence to verify the TME ef- fect. Moreover, as increasing TMEP, the angle satisfying rTMTM = 0 is increasing, which means the Brewster angle tending to π/2. It is well known that the decay contains radiative de- cay and evanescent decay [42]. Specifically, when KL < K‖ < KT , KL always has imaginary components KLz and these modes have evanescent forms in vacuum, whilst KT always has real components and these modes are propa- gating wavelike forms in the medium. The contribution of other modes (K‖ > KT ) to decay rates is zero. Therefore, in eqs. (4) and (5) the decay rates can be divided into two parts: the radiative decay rate Γrad and the evanes- cent decay rate Γeva, and the relation Γ = Γrad + Γeva is satisfied. Radiative decay refers to decay through propa- gating wavelike forms in both media, which is represented by integration over K‖ from 0 to KL. Evanescent decay indicates decay through modes evanescent in vacuum but propagating in dielectric, which is represented by integra- tion over K‖ from KL to KT . These decay rates have been distinguished in fig. 3. For a dielectric interface (solid black lines), when the two-level system is extremely near the interface (|za| < 0.2λ0), the decay rates mainly originate from evanescent modes. Propagating modes will play an important role to decay rates with increasing dis- tance. Specifically, when za = 0, Γ⊥/‖ ≈ nTΓ0. Equiv- alent results have been given previously by Khosravi and Loudon [42]. Moreover, the functions of TMEP on dif- ferent modes are also shown in fig. 3. With increasing TMEPs, Γ⊥eva is inhibited but Γ⊥rad is enhanced. In con- trast, both Γ‖eva and Γ‖rad are substantially inhibited. As a result, Γ‖ is more sensitive to TMEP, which is shown in fig. 2. Next we will analyze those phenonena in detail. From the physical point of view spontaneous emission is modified by the surrounding environment which is influ- enced by reflected electromagnetic waves. When z0 = 0, eqs. (4) and (5) are reduced to Γ‖ = Γ0 3 4 Re ∫ ∞ 0 dK‖ KL K‖ KLz × [( 1 + rTETE ) + K2Lz K2L ( 1− rTMTM )] , (8) Γ⊥ = Γ0 3 2 Re ∫ ∞ 0 dK‖ KLz K‖ 3 KL 3 [1 + r TM TM ]. (9) Decay rates are related to the real parts of the integrands. The interval has been divided into three parts, i.e., 0 → KL, KL → KT = KL √ εT and KT → ∞. 64001-p4 Spontaneous emission of a two-level system near the interface of topological insulators -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0 2 4 6 8 10 -0.10 -0.05 0.00 0 2 4 6 8 10 Γ ⊥eva Γ 0 δ 0.5 0.05 0.01 Γ ⊥ di ss i Γ 0 z a λ 0 (a) -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0 2 4 6 8 10 Θ 0 Θ π Θ 5π -0.0 01 .00 0 2 4 6 8 10 Γ ⊥ di ss i Γ 0 z a λ 0 (b) Fig. 4: (Colour on-line) The decay rates through dissipation Γ⊥dissi as functions of position. (a) For losses TIs: εT = 41.5 + iδ and Θ = π with different imaginary δ. (b) For different TMEPs with εT = 41.5 + i0.5. Firstly, when 0 < K‖ < KL (radiative decay), every terms of the integrands in eqs. (8) and (9) are real and they all affect the decay rates. From eqs. (6) and (7), it is obvious to get lim Θ→∞ rTETE → −1 and lim Θ→∞ rTMTM → 1. Therefore, with increasing TMEPs, Γ⊥rad is enhanced but Γ‖rad is inhibited, as is shown in figs. 3(a) and (b). Secondly, when KL < K‖ < KL √ εT (evanescent de- cay), KLz becomes a pure imaginary number. In order to get decay rates, the imaginary parts of rTETE and r TM TM are necessary. After simple deduction, these imaginary parts can be expressed as ImrTETE = 2εTa3ᾱ2+2εT 2a3+2a a2ᾱ4+2εT a2ᾱ2+2a2ᾱ2+εT 2a4+εT 2a2+a2+1 , (10) ImrTMTM = 2aᾱ2+2εTa3+2εTa a2ᾱ4+2εTa2ᾱ2+2a2ᾱ2+εT 2a4+εT 2a2+a2+1 , (11) where a is the imaginary part of KLz/KTz. Straightfor- wardly, we can get lim Θ→∞ ImrTETE = lim Θ→∞ ImrTMTM = 0. As a result, both integrands in eqs. (8) and (9) tend to 0, so do Γ⊥eva and Γ‖eva, which are shown in figs. 3(c) and (d). Thirdly, when K‖ > KL √ εT , rTETE and r TM TM become real but KLz is still a pure imaginary number. Therefore, the real parts of the integrands in eqs. (8) and (9) are 0. This is the reason why the contribution of other modes is 0, which is mentioned above. Based on the above analysis, it is easy to explain why the parallel decay rate is sensitive to the value of TMEP. Besides, it is obvious that decay rates do not vary with the sign of TMEP. It is easy to find that the expressions of rTETE and r TM TM in eqs. (8) and (9) contain the square of TMEP so that these reflection coefficients will not vary when Θ changes sign. The sign of TMEP only changes the direction of the axion field but there is no effect on spontaneous emission rates. Finally, we consider the loss of materials. The indexes of samples (Θ = 0, π, 5π) are set as εT = 41.5+iδ. The dipole is perpendicular to the interface and its position changes from −0.05λ0 to 0. The decay rate through dissipation can be defined as Γ⊥dissi = Γ⊥Total−Γ⊥rad−Γ⊥eva, where Γ⊥rad and Γ⊥eva take the values in fig. 3 approximately. The dependence of Γ⊥dissi on the position with Θ = π is plotted in fig. 4(a). It is clear that Γ⊥dissi decreases with decreasing δ and becomes negligible with increasing za. The inset of fig. 4(a) presents the comparison between the decay through dissipation and the decay through evanes- cent modes. Though Γ⊥eva is larger than Γ⊥dissi when za < −0.005λ0, both of them are inhibited with lager za, and Γ⊥rad becomes the largest among all channels. In fig. 4(b), we take εT = 41.5 + i0.5 and compare Γ⊥dissi for different values of Θ and the inset is the magnifica- tion of the region with za from −0.01λ0 to 0. It is clear that TMEP can significantly inhibit the decay through dissipation. Conclusion. – In this paper, we analyze the sponta- neous emission of a two-level system near the surface of topological insulators in detail. Decays both through ra- diative mode and through evanescent mode are consid- ered. Due to the topological magnetoelectric effect which is characterized by TMEP (Θ), the reflection properties of the TI are unusual and the decay near the topological insu- lator has different properties in comparison with that near the dielectric. The decay is substantially inhibited and the parallel dipole is more sensitive to the value of TMEP. When TMEP tends to zero, the TI’s influence on decay rates is just like that of a dielectric. Instead, when TMEP approaches infinity, the decay behavior can be modulated just like that near a perfect mirror. Moreover, the sign of TMEP changes the direction of the axion field and the absolute value determines the spontaneous emission rates. Dissipation has also been considered for perpen- dicular dipole. The decay rate through dissipation can be extracted out and is inhibited significantly with increasing value of Θ. Our results show a new way to control sponta- neous emission by using TIs and provide a new application of TI in the quantum optics field. Furthermore, our method could be used to confirm the TME effect. 64001-p5 Ge Song et al. ∗ ∗ ∗ The authors are grateful to Dr Zeng Ran for useful discussions. 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