Controlling single-photon transport in waveguides with finite cross-section
aa r X i v : . [ qu a n t - ph ] A ug Controlling single-photon transport in waveguides with finite cross-section
Jin-Feng Huang,
1, 2
Tao Shi, C. P. Sun,
1, 4 and Franco Nori
1, 5 Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198 Japan State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, and University of the Chinese Academy of Sciences, Beijing 100190, China Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Strasse 1, Garching, Germany Beijing Computational Science Research Center, Beijing 100084, China Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA (Dated: February 26, 2018)We study the transverse-size e ff ect of a quasi-one-dimensional rectangular waveguide on the single-photonscattering on a two-level system. We calculate the transmission and reflection coe ffi cients for single incidentphotons using the scattering formalism based on the Lippmann-Schwinger equation. When the transverse size ofthe waveguide is larger than a critical size, we find that the transverse mode will be involved in the single-photonscattering. Including the coupling to a higher traverse mode, we find that the photon in the lowest channel willbe lost into the other channel, corresponding to the other transverse modes, when the input energy is largerthan the maximum bound-state energy. Three kinds of resonance phenomena are predicted: single-photonresonance, photonic Feshbach resonance, and cuto ff (minimum) frequency resonance. At these resonances, theinput photon is completely reflected. PACS numbers: 42.50.Ct, 42.50.Gy, 03.65.Nk
I. INTRODUCTION
Current optical communications use electronic switchingand thus are limited to electronic speeds of a few gigahertz.To reach much higher speeds, various proposals have beenmade including optical networks [1], as well as using all-optical routers [2] and switches [3–8]. Also, quantum opticalnetworks were motivated by quantum information (communi-cation), using elements with quantum coherence (such as su-perposition and entanglement) of photons. Thus the elementaldevice can be implemented as a generalized cavity QED sys-tem: a photon confined to a one-dimensional (1D) waveguide,and controlled by a quantum switch, made of a two (or more)energy-level systems [3–11, 11–23].There have been numerous theoretical [3–8, 24] and exper-imental [25, 26] studies for such a quantum switch, whichcould be realized in various physical systems, e.g., a trans-mission line [6, 27–30] coupled to a charge qubit [31–35] anda defect cavity waveguide coupled to a quantum dot [36–38].Most theoretical studies on these systems are excessively ide-alized, because the experimental system is never one dimen-sional.In order to consider more realistic systems, here we studythe finite cross-section e ff ect of the waveguide on the single-photon transport controlled by a two-level system (TLS). Weconsider the waveguide as a quasi-1D system with a rectan-gular cross-section. It is well known that if a photon could beperfectly transported in a quasi-1D waveguide, its frequencymust be larger than the cuto ff frequency of a certain trans-verse mode. Moreover, to avoid the loss of the photon inci-dent in the lowest transverse mode due to scattering into othermodes, people need to make the cross section of the waveg-uide as small as possible. However, the cross section of real-istic waveguides cannot be infinitely small, and a waveguidewith a finite cross section would allow the photon transit fromone transverse mode to another. Furthermore, if the incident photon frequency is far from the cuto ff frequency, such as xray [39, 40], then the di ff erent transverse modes would be soclose that the incident photon would be inevitably coupled tohigher transverse modes. This consideration motivates us tostudy the incident photon transport in one mode while coupledto another (higher) mode.We solve the Lippmann-Schwinger equation for calculatingthe reflection and transmission coe ffi cients of a single photonscattered by a TLS. Since the exact dispersion relation of aphoton in a waveguide with finite cross section is more like aquadratic one near the cuto ff frequency, quite di ff erent fromthe linear regime, we approximate the exact dispersion rela-tion by a quadratic function of the wave vector of the photonby expanding it to second order in the wave vector. In such aquadratic waveguide, we find that there is a bound state andtwo quasibound states for each scattering channel defined bya certain transverse mode. We note that this bound state doesnot exist in the usual linear waveguides.There are three kinds of resonance phenomena, which cor-respond to the complete reflection of the photon incident ina given channel. One occurs at the single-photon resonance,namely the incident photon energy is resonant with the TLSwithout coupling to the higher transverse mode. Once theincident photon couples to the higher transverse mode, thisresonance phenomenon is replaced by a photonic Feshbachresonance, namely a complete reflection occurs when the inci-dent energy of the photon equals the bound-state energy of thehigher transverse mode. The third type of resonance alwaysoccurs at the minimum frequency of the quadratic waveguide,whether or not the singe photon is coupled to a higher trans-verse mode. This resonance phenomenon is called cuto ff -frequency resonance. We also notice that the transverse modewill lead to an incident photon loss as a result of scatteringinto other higher channels. We also compare in detail the re-sults obtained by the linear and quadratic dispersion relations,respectively.This paper is organized as follows. In Sec. II, we describethe system and the e ff ective Hamiltonian, including two trans-verse modes. We also derive the second-order dispersionrelation. Then, we calculate the single-photon transport inthe higher transverse mode without coupling to the incidentmode in Sec. III. We find a bound state and two quasiboundstates [5, 41, 42] by utilizing the quadratic dispersion rela-tion. In Sec. IV, we obtain the single-photon reflection andtransmission coe ffi cients with coupling to the higher trans-verse mode through the Lippmann-Schwinger equation. Thetransverse e ff ect in both linear and quadratic waveguides arediscussed in Sec. V. Finally, we present our conclusions inSec. VI. II. MODEL
The setup under consideration is a waveguide-QED sys-tem (see Fig. 1) consisting of a quasi-1D rectangular waveg-uide with inner dimensions L x and L y and a two-level atom.The waveguide supports quantum fields of transverse electricwaves TE mn , which are described by the annihilation (cre-ation) operators a ( † ) m , n , k . Here the natural numbers m and n are,respectively, the transverse quantum numbers in the x and y di-rections, while the continuous variable k denotes the wavevec-tor along the z axis. The eigenmode function of the electricfields in the waveguide can be expressed as [43] e u ( x ) m , n , k ( r ) = − i ε k n π k cut L y cos m π L x x ! sin n π L y y ! e ikz , e u ( y ) m , n , k ( r ) = i ε k m π k cut L x sin m π L x x ! cos n π L y y ! e ikz , (1)where we introduce the cuto ff wavenumber k cut = r ( m π/ L x ) + (cid:16) n π/ L y (cid:17) , (2)and the electric field per photon ε k = p ~ ω m , n , k / (2 ǫ V k ), withfrequency ω m , n , k = c r ( m π/ L x ) + (cid:16) n π/ L y (cid:17) + k , (3)and the e ff ective volume V k = L x L y π/ | k | of a segment (withlength 2 π/ | k | ) of the waveguide. The parameter ǫ is the vac-uum permittivity and c is the speed of light in vacuum.When a two-level atom is placed in the waveguide, it willcouple to these quantum fields via the dipole interaction. De-noting the ground and excited states of the atom as | g i (withenergy 0) and | e i (with energy ω ), we can define the atomictransition operators as σ + = | e i h g | and σ − = | g i h e | , and thenthe Hamiltonian (with ~ =
1) of the waveguide-QED systemreads H = ω | e i h e | + Z + ∞−∞ dk X m , n ω m , n , k a † m , n , k a m , n , k + Z + ∞−∞ dk X m , n ( g m , n , k σ + a m , n , k + H.c.) . (4) e g ω , , m n k g FIG. 1: (Color online) Schematic diagram for single-photon trans-port in a quasi-one-dimensional waveguide coupled to a TLS withtransition frequency ω . The cross-section size of the waveguide is L x ( L y ) along the x ( y ) direction. Here, the coupling strength is g m , n , k = − d ( x ) e , g e u ( x ) m , n , k ( r ) − d ( y ) e , g e u ( y ) m , n , k ( r ).Keeping the coupling between photons and atoms g m , n , k nonzero requires mn ,
0. If m =
0, namely the coupling along y direction is zero, then the transverse mode quantum num-ber n should be nonzero, namely n = , , , . . . ; otherwise,if n =
0, namely the coupling along x direction is zero, thenthe transverse mode quantum number m should be nonzero,namely m = , , , . . . . When the transverse sizes satisfy L x = L y , then the modes TE and TE , bearing the samecuto ff frequencies, are degenerate. To mainly show our idea,namely the e ff ect induced by transverse size of the waveguide,we will choose two modes with di ff erent cuto ff frequencies.The relation ω cut m , n ≡ ck cut gives the exact cuto ff frequency ω cut m , n for the transverse mode ( m , n ).As a result of g , , k =
0, we do not consider the TE modewith ω , , k = c | k | . To reduce the energy distribution in thetransverse mode of the transport photon, we assume that thephotons are in the lowest transverse mode TE , which is themain transport channel we will consider here. However, thetransverse mode TE with a little higher energy is very closeto the lowest transverse mode TE for a finite cross sectionof the waveguide, while other transverse modes are far awayfrom TE . Therefore, the finite cross-section e ff ect of thequasi-1D waveguide on photon transport can be mainly char-acterized by the two transverse modes TE and TE . Thenthe Hamiltonian (4) reduces to H = H + V (5)with the free Hamiltonian H of the photon and the two-levelatom H = H w + ω | e i h e | , (6)where H w = Z + ∞−∞ dk ( ω a , k a † k a k + ω b , k b † k b k ) , (7)and the interaction Hamiltonian V between the photon and theatom V = Z + ∞−∞ dk σ + ( g k a k + g k b k ) + H.c. (8)by defining TE as the a mode, and TE as the b mode, thatis a k ≡ a , , k , b k ≡ a , , k , (9) g , , k = id ( x ) e , g s ~ ω ak ǫ V k sin π L x y ! ≡ g k , (10) g , , k = id ( x ) e , g s ~ ω bk ǫ V k cos π L x x ! sin π L x y ! ≡ g k , (11)and ω a , k ≡ ω , , k , ω b , k ≡ ω , , k , (12) ω cut a ≡ ω cut , , ω cut b ≡ ω cut , . (13)In many works related to 1D waveguides, the dispersion re-lation of the photon is approximated up to the first order ofthe photon wave vector [11, 13–15, 18–23]. However, theexact dispersion relation (3) near the cuto ff frequency is morelike a quadratic one, so we expand the frequency ω a , k around( k , ω ) with ω = ω a , k = c q π L − y + k , and ω b , k around( k ′ , ω ) with ω = ω b , k ′ = c q π L − x + π L − y + k ′ , up to sec-ond order in k . After introducing p = k − k (for ω a , k ), and p = k − k ′ (for ω b , k ), the two dispersion relations can be rewrit-ten as ω s , p ≃ ω + v s p + v s p ( s = a , b ) , (14)with the first- and the second-order coe ffi cients given by v a = c δ/ω , v a = ω v a δ − v a ω , (15) v b = | v a | q δ − ω /δ, v b = v a . (16)Here, we have introduced ω ≡ c π/ L x and δ = ± q ω − ω ,which is proportional to the size L x of the cross section. The ± sign represents the sign of k ( k ′ ). The approximatedquadratic dispersion relation (14) shifts the cuto ff frequencyfrom ω cut s (exact) to ω min s = (cid:16) v s ω − v s (cid:17) / (4 v s ) (approxi-mated). Here s = a , b .We assume the photons are entering from the left end of thewaveguide in the a mode; thus for the right-moving photons, k , k ′ >
0, and δ takes the ” + ” sign, while for the left-moving photons, k , k ′ <
0, and δ takes the ” − ” sign. Therefore, thedispersion relations (14) can be rewritten as ω s , k ≃ ω + | v s | k + v s k , k , k ′ > ω − | v s | k + v s k , k , k ′ < s = a , b . We note that the terms in the dispersion rela-tion (14) that depend on the photon wave vector p describethe frequency detuning of the photon from the atom. Later on,we will use the dispersion relations (14) in our derivations. III. SCATTERING AND BOUND STATES IN THE SINGLE b MODE
Since the photon scattering process in the so-called b modemay contribute to the photon transport in the a mode, we firstconsider the photon scattering in a single b mode. We in-ject the photon in the b mode with the atom only coupledto the transverse-mode b mode ( g = b mode. The bound state is alsoobtained by the poles of the T matrix [44].Under the above consideration, the Hamiltonian is directlyobtained by setting g = ω a , k = H b = H b + V b , which includes the free Hamiltonian H b = H bw + ω | e i h e | (18)with H bw = R + ∞−∞ dk ω b , k b † k b k , and the interaction part V b = Z + ∞−∞ dk ( g k σ + b k + H.c.) . (19)We assume the single photon is initially input from the leftend of the waveguide in the b mode b † k |∅i with energy ω b , k ,while the atom is in the ground state | g i , then the scatteringstate is given by the Lippmann-Schwinger equation [44, 45] (cid:12)(cid:12)(cid:12) ϕ ( + ) bk E = b † k |∅i | g i + ω b , k + i + − H b V b (cid:12)(cid:12)(cid:12) ϕ ( + ) bk E . (20)Here, the input state b † k |∅i | g i is the eigenstate of the freeHamiltonian H b with eigenenergy ω b , k , H b b † k |∅i | g i = ω b , k b † k |∅i | g i , (21)and (cid:12)(cid:12)(cid:12) ϕ ( + ) bk E is the eigenstate of the total Hamiltonian H b withthe same eigenenergy ω b , k .We assume that the solution of the scattering state (cid:12)(cid:12)(cid:12) ϕ ( + ) bk E isin the form (cid:12)(cid:12)(cid:12) ϕ ( + ) bk E = (cid:12)(cid:12)(cid:12) φ b , k (cid:11) | g i + β b , k |∅i | e i . (22)Here, (cid:12)(cid:12)(cid:12) φ b , k (cid:11) is the single-photon state after being scattered, and β b , k is the probability amplitude for the atom to be in its ex-cited state. Substituting this solution into Eq. (20), the scatter-ing state is obtained, (cid:12)(cid:12)(cid:12) ϕ ( + ) bk E = b † k |∅i | g i + β b , k |∅i | e i + G bw (cid:0) ω b , k + i + (cid:1) β bk Z + ∞−∞ dk ′ g ∗ k ′ b † k ′ |∅i | g i , (23)where G bw ( z ) = (cid:16) z − H bw (cid:17) − is the free Green operator for the b -mode photon, and β bk = g k ω b , k + i + − ω − Σ b (cid:0) ω b , k (cid:1) , (24)with the self-energy defined by Σ b ( E ) ≡ Z + ∞−∞ dk | g k | E + i + − ω b , k (25) ≈ − i γ b v b q v b + v b ( E − ω ) . (26)Here, if we directly substitute the exact coupling expres-sion (11) into Eq. (25), the divergence of self-energy Σ b oc-cures. In obtaining the result (26), we have assumed g k tobe independent of k , namely g k = g . This assumption isequivalent to the Markov approximation [46].It follows from Eq. (26) that, when E ≥ (4 v b ) − (cid:16) v b ω − v b (cid:17) = ω min bk , Σ b ( E ) is purely imagi-nary, while E < ω min bk , Σ b ( E ) is real. Using the scatteringstate, the T -matrix elements are given by t k ′ k (cid:0) ω b , k (cid:1) = h g | h∅| b k ′ V b | ϕ ( + ) bk i = β bk g ∗ k ′ . (27)The bound state can be obtained by solving the transcen-dental equation [ t k ′ k ( E bs )] − =
0. We directly obtain thebound-state-energy transcendental equation E bs = ω − i γ b v b q v b + v b ( E bs − ω ) (28)by using the result (26). Here we have defined the decay ratefor the atom induced by the b -mode γ b = π | g | / v b . Lateron, we use γ b to denote the coupling strength g .It follows from this result (28) that if v b →
0, which corre-sponds to a linear waveguide, the bound-state-energy solutionis E bs = ω − i | γ b | . (29)The fact that there is no real solution means that there isno bound state in the linear waveguide. However, for thequadratic waveguide, the transcendental equation (28) givesone real solution E bs = ∆ F a + ω , (30)with ∆ F a ≡ u b − u b v b + v b u b v b . (31) This real solution denotes a bound state. Two complex solu-tions E bs = (cid:16) − + i √ (cid:17) u b − v b + (cid:16) − − i √ (cid:17) v b u b + v b ω × (24 v b ) − (32)and E bs = (cid:16) − − i √ (cid:17) u b − v b + (cid:16) − + i √ (cid:17) v b u b + v b ω × (24 v b ) − , (33)correspond to two quasibound states. Each of these is ametastable state that decays on a very long time scale and ap-pears to be a localized bound state in real space [41, 42]. Theparameters therein are defined by u b = l b , l b = − v b − v b v b γ b + √ q v b v b γ b (cid:16) v b + v b γ b (cid:17) . (34)We note that l b is always real, thus there are three values for u b = l / b , l / b e i π , and l / b e i π . However, we can choose u b tobe real. Then there is always a real solution (30) for Eq. (28),which describes a bound state with energy (30). In addition,when the detuning ∆ ak ≡ ω ak − ω satisfies ∆ ak = ∆ F a (35)or equivalently, k = − v a ± q v a + v a ∆ F a v a ≡ k F , (36)the input photon energy ω ak is resonant with the bound statein the b mode. This is the Feshbach resonance. Moreover,lim γ b → ∆ F a = − v b v b = ∆ Fmax . (37)This maximum value ∆ Fmax of ∆ F a versus the coupling strength γ b between the transverse mode and the TLS denotes the max-imum value of the bound-state energy in this transverse mode E max bs = ∆ Fmax + ω . IV. PHOTON TRANSMISSION AND REFLECTION INTHE a MODE WHILE THE ATOM IS COUPLED TO THE b MODE
Now we consider the photon injected in the a mode withthe atom coupled to the a and b modes at the same time. Wecalculate the scattering state of the a -mode photon. Using theLippmann-Schwinger equation, the scattering state is (cid:12)(cid:12)(cid:12) ψ ( + ) k E = a † k |∅i | g i + ω a , k + i + − H V (cid:12)(cid:12)(cid:12) ψ ( + ) k E . (38)By a similar procedure to the last section, the scatteringstate is obtained as (cid:12)(cid:12)(cid:12) ψ ( + ) k E = a † k |∅i | g i + β k |∅i | e i + G w (cid:0) ω a , k + i + (cid:1) β k Z + ∞−∞ dk ′ (cid:16) g ∗ k ′ a † k ′ + g ∗ k ′ b † k ′ (cid:17) |∅i | g i , (39)where the similar free Green operator is G w ( z ) = ( z − H w ) − ,and the excited probability amplitude of the atom is β k = g k ω a , k + i + − ω − Σ a (cid:0) ω a , k (cid:1) − Σ b (cid:0) ω a , k (cid:1) (40)with the self-energy for the a mode defined by Σ a ( E ) ≡ Z + ∞−∞ dk | g k | E + i + − ω a , k (41) ≃ − i γ a v a q v a + v a ( E − ω ) , (42)and Σ b ( E ) defined by Eq. (25). Similarly, if we directly sub-stitute the exact coupling expression (10) into Eq. (41), thedivergence of self-energy Σ a also occurs. In obtaining the re-sult (42), we have also assumed g k to be independent of k ,namely g k = g . Here, the decay rate induced by the a mode γ a = π | g | / v a is introduced. Later on, we also use γ a todenote the coupling strength g .By using the scattering state (39), we obtain the matrix ele-ments of the scattering operator S in k space, S k ′ , k = δ (cid:0) k ′ − k (cid:1) − π i δ ( ω ak ′ − ω ak ) t k ′ k (cid:0) ω a , k + i + (cid:1) , (43)where the T -matrix elements are directly obtained as t k ′ k (cid:0) ω a , k + i + (cid:1) = h g | h∅| a k ′ V (cid:12)(cid:12)(cid:12) ψ ( + ) k E = β k g ∗ k ′ , (44)and the δ function here is defined as δ ( x ) = x =
0; other-wise, δ ( x ) = S k ′ , k = r δ (cid:0) k + k ′ (cid:1) + t δ (cid:0) k − k ′ (cid:1) , (45)we obtain the reflection amplitude r ( k ) = − i | v a k + | v a || γ a v a ∆ ak − Σ a (cid:0) ω a , k (cid:1) − Σ b (cid:0) ω a , k (cid:1) (46)for the input single photon in the a mode. Note that, in ob-taining the result (46), we have discarded the term propor-tional to ¯ δ (cid:2) ∆ ak − Σ a (cid:0) ω a , k (cid:1) − Σ b (cid:0) ω a , k (cid:1)(cid:3) and the principle valuelabel P when using the formula 1 / ( x + i + ) = P / x − i π ¯ δ ( x ). T r a n s m i ss i on Photon-TLS detuning ak b b b (a) Coupling strength b T ( R e s on a n ce ) (b) b b b P r ob a b ilit y l o ss P L Photon-TLS detuning ak (c) (d) P L ( R e s on a n ce ) Coupling strength b FIG. 2: (Color online) Results for linear waveguides. (a) Transmis-sion coe ffi cient T versus detuning ∆ ak and (b) versus the couplingstrength γ b between the transverse mode and the TLS at the single-photon resonance ∆ ak =
0. (c) The single-photon loss probability P L versus detuning ∆ ak and (d) versus the coupling strength γ b be-tween the transverse mode and the TLS at the single-photon reso-nance ∆ ak =
0. Other parameters are γ a /ω = . δ/ω = .
8, and v a =
1. All the parameters are in units of ω . Here the Dirac δ function is defined as ¯ δ ( x ) = ∞ if x = δ ( x ) =
0. This procedure is reasonable becausethe definition of detuning ∆ ak already restricts its regimeto ∆ ak ≥ − v a / (4 v a ) = ∆ min ak , which contradicts the ba-sic condition ∆ ak < − v a / (4 v a ), under which the δ term¯ δ (cid:2) ∆ ak − Σ a (cid:0) ω a , k (cid:1) − Σ b (cid:0) ω a , k (cid:1)(cid:3) may contribute.In terms of the detuning ∆ ak , the reflection amplitude is r ( ∆ ak ) = − i q v a + v a ∆ ak × γ a v a ∆ ak − Σ a ( ∆ ak + ω ) − Σ b ( ∆ ak + ω ) . (47)Then the reflection coe ffi cient R = | r | can be directly ob-tained.The transmission amplitude t is directly obtained through t = + r and the transmitted coe ffi cient is straightforwardlyobtained as T = | t | . Interestingly, we find three reso-nance points where the single-photon transmission amplitudeis zero: (1) t (cid:16) ∆ ak = ∆ F a (cid:17) =
0, where ∆ ak = ∆ F a means thatthe input single-photon energy is resonant with the bound-state energy in the transverse mode ω ak = E bs , namely thephotonic Feshbach resonance; (2) t ( ∆ ak = , γ b = = t ( k = k res , γ b = =
0, with k res = , − v a / v a for v a >
0. This resonance is denoted assingle-photon resonance. (3) lim ∆ ak → ∆ min ak t ( ∆ ak ) =
0. We call thisresonance the cuto ff (minimum) frequency resonance. Underthese three resonances, the transmission T = r = − i γ a ∆ ak + i ( γ a + γ b ) , (48)and the transmission amplitude t = + r = ∆ ak + i γ b ∆ ak + i ( γ a + γ b ) (49)for linear waveguides and add a subscript ”1” to denote thatthis result only applies to linear waveguides. This result is inagreement with Refs. [13, 23] when γ b =
0. Correspondingly,the reflection and transmission coe ffi cients are R = | r | = γ a ∆ ak + ( γ a + γ b ) (50)and T = | t | = ∆ ak + γ b ∆ ak + ( γ a + γ b ) . (51)Here, ∆ ak = v a k is the detuning of the single photon for the a mode in the linear waveguide from the two-level atom.For linear waveguides, it follows from Eq. (50) that thetransverse mode will reduce the reflection of the single pho-ton. For the transmission of the photon, the transverse modewill increase the transmission of the single photon when γ a γ b > ∆ ak ; otherwise, it will decrease its transmission. Inaddition, as a result of the transverse mode, i.e., γ b ,
0, thesingle-photon probability is not conserved in its input mode,that is T + R <
1. The photon is scattered into the transversemode with probability P L ≡ − R − T = γ a γ b ∆ ak + ( γ a + γ b ) . (52)This probability loss has a Lorentz shape centered at thesingle-photon resonance ∆ ak = γ a + γ b . Un-der the resonance condition ∆ ak = a mode) and transverse mode ( b mode),namely, γ a = γ b , the loss probability reaches P L = . V. TRANSVERSE EFFECT IN LINEAR AND QUADRATICWAVEGUIDESA. Transverse e ff ect in linear waveguides To show the transverse e ff ect on single-photon transport,we first consider its e ff ect in linear waveguides. We plot thetransmission coe ffi cient T and P L versus detuning ∆ ak ( = k with v a =
1) under di ff erent transverse coupling strengths γ b /ω = , .
01, and 0 . γ b under the single-photonresonance condition ∆ ak = T =
0) of the sin-gle photon is damaged by the transverse mode, and the widthof the transmission energy band increases as the transverse-mode coupling strength increases. When the transverse-modecoupling strength is strong enough, the perfect reflection be-comes perfect transmission [Fig. 2(b)]. Furthermore, the -5 -4 -3 -2 -1 0 1 -3.37 -3.310.00.51.0 -0.26 -0.200.00.51.0-3.50 -3.35 -3.200.00.51.0 -0.4 -0.2 0.00.00.51.0 k res k C k res T P (a) b F k C k F (b) b F k F k C (c) b Wave-vector k k FIG. 3: (Color online) Results for quadratic waveguides: transmis-sion coe ffi cient T and probability loss P L versus wave vector k when(a) γ b /ω =
0, (b) γ b /ω = .
05, and (c) γ b /ω = .
15. Other pa-rameters are the same as in Fig. 2. transverse mode forces the single photon to leave the inputmode if the input photon is near resonance with the atom.Especially, exactly at the single-photon resonance, the lossprobability reaches its largest value. However, this largestvalue at the single-photon resonance not always increases asthe transverse-mode coupling strength increases, as shown inFig. 2(d). It first increases rapidly to 0 . γ b = γ a , thendecreases gradually as the transverse-mode coupling strengthincreases and finally reaches zero when γ b is strong enough. B. Transverse e ff ect in quadratic waveguides Now we illustrate the transverse e ff ect on the single-photontransport properties in a quadratic waveguide. We plot thetransmission coe ffi cient T versus wave vector k in Fig. 3 andversus the detuning ∆ ak in Fig. 4(a), and the loss probability P L = − T − R versus the detuning ∆ ak in Fig. 4(b).Figure 3(a) shows that the single photon is perfectly re-flected at k = k res and k = k C ; otherwise, it is completelytransmitted without coupling to the transverse mode. How- -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 b b b Fa (a) T r a n s m i ss i on T minak Fmax
Photon-TLS detuning ak b b b b b (b) P r ob a b ilit y l o ss P L FIG. 4: (Color online) Results for quadratic waveguides. (a) Trans-mission coe ffi cient T versus detuning ∆ ak in quadratic waveguide.(b)The single-photon loss probability P L versus detuning ∆ ak . Otherparameters are the same as in Fig. 2. ever, once the TLS is coupled to the transverse mode, theoriginal perfect-reflection points k = k res have been shiftedto k = k F with some probability loss at k = k res . When in-creasing the coupling strength of the transverse mode, the twosides of the perfect reflection peaks at k = k F move toward thecenter peak at k = k C , while the probability loss at k = k F isreduced. We also note that the center perfect-reflection peakat k = k C is not dependent on the transverse-mode coupling;It is decoupled from the transverse mode. This is because itis only determined by the minimum detuning ∆ min ak betweenthe photon and the TLS. Compared with the linear waveg-uide, this phenomenon is more robust against the finite cross-section e ff ect of the waveguide. Also, there are two additionalperfect-reflection peaks. Between the peaks k F < k < k C (or k C < k < k F ), there is a perfect transmission band.Figure 4 shows the single-photon transport properties interms of the input energy. Without coupling to the transversemode, the single photon is perfectly reflected at ∆ ak = ∆ min ak , ∆ ak = ∆ ak = ∆ F a , which denotes that theinput single-photon energy is resonant with the bound-stateenergy in the transverse mode. This is the photonic Fesh-bach resonance [17, 47, 48]. Moreover, the position of theperfect reflection as a result of photonic Feshbach resonance -0.7-0.6-0.5-0.4-0.3-0.2 10 0.5 F e s c hb ac h p ea k F a Coupling strength b FIG. 5: Feschbach peak ∆ F a versus the transverse-mode couplingstrength γ b for waveguides in the quadratic regime. Other param-eters are the same as in Fig. 2. moves away from the single-photon resonance position. Fig-ure 4(b) shows that the photon loss probability only occurs inthe regime ∆ ak > ∆ Fmax . This is becauselim ∆ ak → ∆ Fmax P L = . (53)When ∆ ak ≤ ∆ Fmax , the loss probability becomes zero P L =
0. Therefore, when the single-photon input energy satisfies ∆ min ak ≤ ∆ ak ≤ ∆ Fmax , the transverse mode cannot exert a nega-tive e ff ect on the single-photon transport. We point out that thefeatures for T and P L versus the b mode photon-atom cou-pling γ b at the single-photon resonance remain similar withthat in the linear waveguide [Figs. 2(b) and 2(d)].To show this more explicitly, how the transverse mode playsa role in the photonic Feshbach resonance, we plot the pho-tonic Feshbach resonance peak position ∆ F a in Fig. 4 versus thetransverse mode coupling strength γ b in Fig. 5. As the curveshows, ∆ F a is nearly a linear curve and decreases when increas-ing the transverse-mode coupling strength. This phenomenonagrees with the properties of T shown in Fig. 4. Since ∆ F a isalso a component of the bound-state energy (30) in the trans-verse mode, except for a constant ω , this curve also shows thebound-state-energy dependence on the transverse-mode cou-pling strength.We also find the line shape for the photonic Feshbach reso-nance is very close to Fano line shape [49], which is comparedwith the Fano line in Fig. 6 by defining the Fano function f = (cid:16) ∆ ak − ∆ F a + q (cid:17) (cid:0) ∆ ak − ∆ F a (cid:1) + d . (54)We call our line shape quasi-Fano line. Here, we would like topoint out that the similar resonance originated from a boundstate in higher transverse modes has also been discovered in anelectronic quasi-1D waveguide [50, 51]. The resonance line -0.30 -0.25 -0.20 -0.15 -0.100.00.51.00.00.51.0 Fa (b) ak bb Fa Fano T (a)
FIG. 6: (Color online) Comparison between the quasi-Fano lineshape (red solid line) for transmission coe ffi cient T around Feshbachresonance and the Fano line shape (black dashed line) given by Eq.(54). (a) q = − , d = − ; (b) q = − , d = − / . Otherparameters are the same as in Fig. 2. shape is Fano type for electrons in their results [51]. How-ever, compared with the resonance found in Refs. [50, 51] forelectrons, the similar resonance induced by the bound statefor photons we find occurs exactly at the bound-state energy,while the resonance position for electrons acquires a shift inelectronic waveguides [50, 51].Finally, we would like to estimate some parameters for theconditions when the additional transverse mode is involvedfor the study of single-photon transport. Usually, we can ig-nore the influence of the b mode, when the a mode is closeto resonance of an atomic transition while the b mode is o ff -resonance. We now estimate the quantitative condition by as-suming that the e ff ect of mode a is 100 times that of mode b .Namely, when 100 × g | ω bk − ω | ≤ g | ω ak − ω | (55)or L x ≤ c (cid:16) √ − (cid:17) πω ≡ L c , (56)the transverse mode b cannot a ff ect the single-photon trans-port. To obtain Eq. (56), we have used (cid:12)(cid:12)(cid:12) ω cut b − ω cut a (cid:12)(cid:12)(cid:12) ≥| ω bk − ω ak | and g = g . However, when the transverse size L x of the waveguide is larger than the critical size L c , L x > L c ,the transverse mode should be taken into account. For a 1Dcircuit system with ω ≃
10 GHz [26], then L c ≃ . ω ≃ . × GHz [52],then L c ≃ . ∼ GHz, suchas x rays [39, 40], the TLS with transition energy 14 . F e ), corresponding to ω ≃ . × GHz, then the critical size becomes L c ≃ .
12 Å. Ex-perimentally, this 1 .
12 Å looks too di ffi cult. Therefore, it isvery necessary to consider the transverse-mode e ff ect in thesingle-photon transport in a waveguide with finite cross sec-tion. A finite-cross-section waveguide is closer to our experi-mental quantum coherent device design and fabrication. Tak-ing advantage of the finite-cross-section waveguide will easethe stringent requirements on realizing quantum-coherent de-vices. VI. CONCLUSIONS AND DISCUSSIONS
We studied the finite cross-sectional e ff ect of the waveg-uide on single-photon transport. To mainly characterize thefinite cross-section e ff ect of the waveguide, we pick out oneof the numerous transverse modes, whose eigenfrequency isclosest to that of the transport mode. We consider the trans-port properties of a single photon in such a finite cross-sectionwaveguide by calculating the transmission, reflection coef-ficients and the single-photon loss probability. By using aquadratic dispersion relation, we find a bound state and twoquasibound states [5, 41, 42] emerging in such a waveguidewith a finite cross section, which will not occur in the usuallinear waveguide. Moreover, when the input photon energy isresonant with the bound-state energy in the transverse mode,the photon will be completely reflected. This is the photonicFeshbach resonance. In addition, the input photon is also com-pletely reflected when the input energy of it is at a single-photon resonance with the TLS or at the cuto ff frequency al-lowed by the approximated quadratic waveguide. The pho-tonic Feshbach resonance and the cuto ff frequency resonancephenomena do not occur in a linear waveguide even in an in-finitely idealized 1D waveguide.Furthermore, as a result of transverse-mode coupling, thephoton will be lost when the input energy is above the max-imum bound-state energy regulated by the coupling strengthbetween the transverse mode and the TLS. Therefore, onlywhen the input energy is below this maximum bound-stateenergy, the single photon can safely pass through or be com-pletely reflected by the TLS instead of lost in some other trans-verse mode even though in a finite cross-section waveguide. Acknowledgments
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