Perfect Optical Nonreciprocity in a Double-Cavity Optomechanical System
aa r X i v : . [ qu a n t - ph ] S e p Perfect optical nonreciprocity in a double-cavity optomechanical system
Xiao-Bo Yan, ∗ He-Lin Lu, Feng Gao, and Liu Yang † College of Electronic Science, Northeast Petroleum University, Daqing 163318, China Department of Physics, Yunnan Minzu University, Kunming 650500, China College of Science, Shenyang Aerospace University, Shenyang 110136, China College of Automation, Harbin Engineering University, Harbin 150001, China (Dated: September 12, 2019)Nonreciprocal devices are indispensable for building quantum networks and ubiquitous in moderncommunication technology. Here, we propose to take advantage of the interference between optome-chanical interaction and linearly-coupled interaction to realize optical nonreciprocal transmission ina double-cavity optomechanical system. Particularly, we have derived essential conditions for per-fect optical nonreciprocity and analysed properties of the optical nonreciprocal transmission. Theseresults can be used to control optical transmission in quantum information processing.
PACS numbers: 42.65.Yj, 03.65.Ta, 42.50.Wk
I. INTRODUCTION
Nonreciprocal optical devices, such as isolators and cir-culators, allow transmission of signals to exhibit differentcharacteristics if source and observer are interchanged.They are essential to several applications in quantum sig-nal processing and communication, as they can suppressspurious modes and unwanted signals [1]. For example,they can protect devices from noise emanating from read-out electronics in quantum superconducting circuits. Toviolate reciprocity and obtain asymmetric transmission,breaking time-reversal symmetry is required in any suchdevice. Traditionally, nonreciprocal transmission has re-lied on applied magnetic bias fields to break time-reversalsymmetry and Lorentz reciprocity [2, 3]. These conven-tional devices are typically bulky and incompatible withultra-low loss superconducting circuits. Many alternativemethods recently have been proposed to replace conven-tional nonreciprocal schemes, such as usage of coupled-mode systems [4, 5], Brillouin scattering [6], and spa-tiotemporal modulation of the refractive index [7]. Theseschemes are particularly promising because they can beintegrated on-chip with existing superconducting tech-nology.In recent years, the rapidly growing field of cavityoptomechanics [8–10], where optical fields and mechan-ical resonators are coupled through radiation pressure,has shown promising potential for applications in quan-tum information processing and communication. So far,many interesting quantum phenomena have been stud-ied in this field, such as mechanical ground-state cooling[11–13], optomechanically induced transparency [14–20],entanglement [21–28], nonlinear effects [29, 30], and co-herent perfect absorption [31–33]. Very recently, it hasbeen realized that optomechanical coupling can lead tononreciprocal transmission and optical isolation. In Refs. ∗ Electronic address: [email protected] † Electronic address: [email protected] [34–38], nonreciprocal optical responses are theoreticallypredicted through optomechanical interactions, and non-reciprocal transmission spectra were recently observedin Refs. [39–45]. In Refs. [46–49], it was recognizedthat the mechanically-mediated quantum-state transferbetween two cavity modes can be made nonreciprocalwith suitable optical driving. In Ref. [50], nonreciprocalquantum-limited amplification of microwave signals hasbeen proposed in an optomechanical system. Besides, inRefs. [51, 52], phonon circulators and thermal diodes aretheoretically predicted through optomechanical coupling.In most of these references, perfect optical nonreciprocitycan be achieved under the conditions of equal dampingrate (mechanical damping rate γ is equal to cavity damp-ing rate κ ) or nonreciprocal phase difference θ = ± π .Here, we show that perfect optical nonreciprocity canbe achieved under more general conditions, using the ex-ample of a double-cavity system in Fig. 1. This setup hasbeen realized in several recent experiments [53–55], andquantum nonlinearity [56] has been theoretically stud-ied in this setup. With this simple model, we can eas-ily capture the essential mechanisms about nonreciproc-ity, i.e., quantum interference of signal transmission be-tween two possible paths corresponding to two interac-tions (optomechanical interaction and linearly-coupledinteraction). From the expressions of output fields, wederive essential conditions to achieve perfect optical non- , L p e w , c c e w , d c e w , c w , c w , R p e w † †1 2 2 1 ( )
J c c c c + FIG. 1: A double-cavity optomechanical system with a me-chanical resonator interacted with two cavities. Two strongcoupling fields (probe fields) with amplitudes ε c and ε d ( ε L and ε R ) are used to drive the system from the left and rightfixed mirror respectively. Meanwhile, the two cavities are lin-early coupled to each other with coupling strength J . reciprocity, and find some interesting results. One ofthem is that mechanical decay rate does not influencethe appearance of perfect optical nonreciprocity, whichmeans perfect optical nonreciprocity can still occur inthe realistic parameter regime ( γ ≪ κ ) in cavity optome-chanics. Another interesting result is that perfect opticalnonreciprocity can be achieved with any phase difference θ ( θ = 0 , π ) as long as rotating wave approximation isvalid. We believe the results of this paper can be usedto control optical transmission in modern communicationtechnology. II. SYSTEM MODEL AND EQUATIONS
The system we considered here is depicted in Fig. 1,where a mechanical membrane is placed in the middleof an optical cavity. The operators c and c denotegeometrically distinct optical modes with same frequency ω and decay rates κ and κ respectively. The coupling J describes photon tunneling through the membrane, andthe interaction between the two cavities is described by ~ J ( c † c + c † c ). The mechanical resonator with frequency ω m and decay rate γ is described by operator b . Twostrong coupling fields (probe fields) with same frequency ω c ( ω p ) and amplitudes ε c and ε d ( ε L and ε R ) are usedto drive the double-cavity system from the left and rightfixed mirror respectively. Then the total Hamiltonian inthe rotating-wave frame of coupling frequency ω c can bewritten as ( ~ = 1) H = ∆ c ( c † c + c † c ) + ω m b † b + g ( c † c − c † c )( b † + b )+ J ( c † c + c † c ) + i ( ε c c † − ε ∗ c c ) + i ( ε d c † − ε ∗ d c )+ iε L ( c † e − iδt − c e iδt ) + iε R ( c † e − iδt − c e iδt ) . (1)Here, ∆ c = ω − ω c ( δ = ω p − ω c ) is the detuning betweencavity modes (probe fields) and coupling fields, and g isthe single photon coupling constant between mechanicalresonator and optical modes.The dynamics of the system is described by the quan-tum Langevin equations for the relevant operators of themechanical and optical modes˙ c = − [ i ∆ c + κ − ig ( b † + b )] c + ε c + ε L e − iδt − iJc , ˙ c = − [ i ∆ c + κ ig ( b † + b )] c + ε d + ε R e − iδt − iJc , ˙ b = − iω m b − γ b − ig ( c † c − c † c ) . (2)In the absence of probe fields ε L , ε R and with the fac-torization assumption h bc i i = h b ih c i i , we can obtain the steady-state mean values h b i = b s = − ig ( | c s | − | c s | ) γ + iω m , h c i = c s = ( κ + i ∆ ) ε c − iJε d J + ( κ + i ∆ )( κ + i ∆ ) , h c i = c s = ( κ + i ∆ ) ε d − iJε c J + ( κ + i ∆ )( κ + i ∆ ) (3)with ∆ , = ∆ c ∓ g ( b s + b ∗ s ) denoting the effective de-tunings between cavity modes and coupling fields. In thepresence of both probe fields, however, we can write eachoperator as the sum of its mean value and its small fluctu-ation, i.e., b = b s + δb , c = c s + δc , c = c s + δc to solveEq. (2) when both coupling fields are sufficiently strong.Then keeping only the linear terms of fluctuation opera-tors and moving into an interaction picture by introduc-ing δb → δbe − iω m t , δc → δc e − i ∆ t , δc → δc e − i ∆ t ,we obtain the linearized quantum Langevin equations δ ˙ c = − κ δc + iG ( δb † e i ( ω m +∆ ) t + δbe − i ( ω m − ∆ ) t )+ ε L e − i ( δ − ∆ ) t − iJδc e i (∆ − ∆ ) t ,δ ˙ c = − κ δc − iG e iθ ( δb † e i ( ω m +∆ ) t + δbe − i ( ω m − ∆ ) t )+ ε R e − i ( δ − ∆ ) t − iJδc e i (∆ − ∆ ) t ,δ ˙ b = − γ δb + iG ( δc e i ( ω m − ∆ ) t + δc † e i ( ω m +∆ ) t ) − iG ( e − iθ δc e i ( ω m − ∆ ) t + e iθ δc † e i (∆ + ω m ) t ) (4)with G = g c s and G = g c s e − iθ . The phase differ-ence θ between effective optomechanical coupling g c s and g c s can be controlled by adjusting the couplingfields amplitudes ε c and ε d according to Eq. (3). It willbe seen that the phase difference θ is a critical factor toattain optical nonreciprocity. Without loss of generality,we take G i and J as positive number (not negative toavoid introducing unimportant phase difference π ).If each coupling field drives one cavity mode at themechanical red sideband (∆ ≈ ∆ ≈ ω m ), and the me-chanical frequency ω m is much larger than g | c s | and g | c s | , then Eq. (4) will be simplified to δ ˙ c = − κ δc + iG δb − iJδc + ε L e − ixt ,δ ˙ c = − κ δc − iG e iθ δb − iJδc + ε R e − ixt ,δ ˙ b = − γ δb + iG δc − iG e − iθ δc (5)with x = δ − ω m . For simplicity, we set equal dampingrate κ = κ = κ and equal coupling G = G = G in thefollowing (actually, it can be proven that G must equal G if κ = κ when the system exhibits perfect opticalnonreciprocity).By assuming δs = δs + e − ixt + δs − e ixt ( s = b, c , c ),we can solve Eq. (5) as follows δb + = 4 G [( iκ x − Je − iθ ) ε L + (2 J − iκ x e − iθ ) ε R ]8 G κ x + (4 J + κ x ) γ x + 16 iG J cos θ ,δc = 2(4 G + γ x κ x ) ε L + (8 G e − iθ − iJγ x ) ε R G κ x + (4 J + κ x ) γ x + 16 iG J cos θ ,δc = 2(4 G + γ x κ x ) ε R + (8 G e iθ − iJγ x ) ε L G κ x + (4 J + κ x ) γ x + 16 iG J cos θ (6)where γ x = γ − ix , κ x = κ − ix , and δs − = 0.To study optical nonreciprocity, we must study theoutput optical fields ε outL and ε outR which can be obtainedaccording to the input-output relation [31, 32, 57] ε outL + ε inL e − ixt = √ κδc ε outR + ε inR e − ixt = √ κδc , (7)here, ε inL,R = ε L,R / √ κ . Still following the assumption δs = δs + e − ixt + δs − e ixt , the output fields can be obtainedas ε outL + = √ κδc − ε L / √ κε outR + = √ κδc − ε R / √ κ (8)and ε outL − = ε outR − = 0. III. PERFECT OPTICAL NONRECIPROCITY
Perfect optical nonreciprocity can be achieved if trans-mission amplitudes T i → j ( i, j = L, R ) satisfy T L → R = (cid:12)(cid:12)(cid:12)(cid:12) ε outR ε inL (cid:12)(cid:12)(cid:12)(cid:12) ε inR =0 = 1 , T R → L = (cid:12)(cid:12)(cid:12)(cid:12) ε outL ε inR (cid:12)(cid:12)(cid:12)(cid:12) ε inL =0 = 0 , (9a)or T L → R = (cid:12)(cid:12)(cid:12)(cid:12) ε outR ε inL (cid:12)(cid:12)(cid:12)(cid:12) ε inR =0 = 0 , T R → L = (cid:12)(cid:12)(cid:12)(cid:12) ε outL ε inR (cid:12)(cid:12)(cid:12)(cid:12) ε inL =0 = 1 . (9b)It means that the input signal from one side can be com-pletely transmitted to the other side, but not vice versa.What the Eq. (9a) and (9b) represent is the two differentdirections of isolation. Here, we just discuss the case ofEq. (9a), as the case of Eq. (9b) is similar. The sub-script ε inR/L = 0 indicates there is not signal injected intothe system from right/left side. We omit the subscriptsbecause, in general, nonreciprocity is only related to one-way input, and write transmission amplitudes T i → j as T ij for simplicity in the following.According to Eq. (6) and Eq. (8), the two opticaloutput fields can be obtained as ε outR + ε inL = 4 κ (2 G e iθ − iJγ x )8 G κ x + (4 J + κ x ) γ x + 16 iG J cos θ ,ε outL + ε inR = 4 κ (2 G e − iθ − iJγ x )8 G κ x + (4 J + κ x ) γ x + 16 iG J cos θ . (10) When θ = nπ ( n is an integer), the two output fields areequal, which indicates the photon transmission is recipro-cal. But in the other cases, where θ = nπ , the system willexhibit a nonreciprocal response. It can be clearly seenfrom the numerator of Eq. (10) that the optical nonre-ciprocity comes from quantum interference between theoptomechanical interaction G and the linearly-coupledinteraction J .With Eq. (10), we find perfect optical nonreciprocityEq. (9a) can be achieved only when J = − e ∓ iθ ( γ cot θ ± iκ )2 (11)which can take positive real number only if θ = − π , (12a)or κ = γ. (12b)In the following, we will discuss perfect optical nonre-ciprocity in two cases, Eq. (12a) and (12b), respectively. A. Phase difference θ = − π With nonreciprocal phase difference θ = − π , the twooptical output fields in Eq. (10) now become ε outR + ε inL = − iκ (2 G + Jγ x )8 G κ x + (4 J + κ x ) γ x ,ε outL + ε inR = 4 iκ (2 G − Jγ x )8 G κ x + (4 J + κ x ) γ x . (13) FIG. 2: Transmission amplitudes T LR (red line) and T RL (black line) are plotted vs normalized detuning x/κ for dif-ferent mechanical decay rate: (a) γ/κ =2, (b) γ/κ =1, (c) γ/κ =1/5, and (d) γ/κ =1/100. The coupling strengths J = κ and G = √ κγ according to Eq. (14). According to Eq. (13), the perfect optical nonreciprocityEq. (9a) can be achieved only when x = 0 ,J = κ ,G = √ κγ . (14)It is surprising that there is not any restriction on me-chanical decay rate γ in Eq. (14). In other words, me-chanical decay rate γ does not influence perfect opticalnonreciprocity, which means that perfect optical nonre-ciprocity can still occur even in the case of γ/κ → γ is muchless than cavity decay rate κ in cavity optomechanics. Inaddition, even with very weak optomechanical coupling( G ≪ κ ), perfect optical nonreciprocity can still occur as γ ≪ κ according to Eq. (14).In Fig. 2(a)–2(d), we plot transmission amplitudes T LR (red line) and T RL (black line) vs normalized detun-ing x/κ with J = κ , G = √ κγ for γ/κ = 2, 1, 1 /
5, 1 / γ indeed does not affect the appear-ance of perfect optical nonreciprocity, but can stronglyaffect the width of transmission spectrum, especially forthe case of γ ≪ κ . The two curves of transmission ampli-tudes T LR and T RL tend to coincide outside the vicinityof resonance frequency ( x = 0) when γ/κ →
0. But thesystem always exhibits optical nonreciprocity near res-onance frequency in the case, such as γ/κ = 1 /
100 [seeFig. 2(d)]. By the way, the perfect optical nonreciprocityEq. (9b) will occur if θ = π . FIG. 3: Normalized coupling strengths
G/γ ( J = G ) (blueline) and detuning x/γ (yellow line) are plotted vs phase dif-ference θ according to Eq. (16). B. Equal damping rate κ = γ With equal damping rate κ = γ , the two optical outputfields Eq. (10) now become ε outR + ε inL = 4 γ (2 e iθ G − iJγ x )(8 G + 4 J + γ x ) γ x + 16 iG J cos θ ,ε outL + ε inR = 4 γ (2 e − iθ G − iJγ x )(8 G + 4 J + γ x ) γ x + 16 iG J cos θ . (15)From Eq. (15), we can obtain the conditions for perfectoptical nonreciprocity as follows x = ± γ cot θ ,J = ± γ csc θ ,G = ± γ csc θ θ ∈ ( π, π ) meet Eq. (9a),and positive sign and θ ∈ (0 , π ) meet Eq. (9b). It meansthat we can change the direction of isolation by adjustingthe nonreciprocal phase difference θ ∈ (0 , π ) or ( π, π ).In Fig. 3, we plot the normalized coupling strengths G/γ , J/γ ( J = G ) (blue line) and detuning x/γ (yellowline) vs phase difference θ according to Eq. (16). Forthe special case of θ = ± π , the coupling strength G ( J )takes the minimum value γ and detuning x = 0 (see Fig.3), and the transmission spectrums T LR and T RL takea symmetric form with respect to detuning x [see Fig.2(b)].From Eq. (16), we can see that perfect optical nonre-ciprocity can occur with any phase θ ( θ = nπ ) as longas G ≪ ω m ( | sin θ | ≫ γ ω m ) where rotating wave ap-proximation is valid. It means the strongest quantum FIG. 4: Transmission amplitudes T LR (red line) and T RL (black line) are plotted vs normalized detuning x/γ for dif-ferent phase difference: (a) θ = − π , (b) θ = − π , (c) θ = π ,and (d) θ = π . The coupling strengths G = ± γ csc θ ( J = G )according to Eq. (16). interference takes place at detuning x = ± γ cot θ in thecase of κ = γ . In Fig. 4(a)–4(d), we plot the transmis-sion amplitudes T LR (red line) and T RL (black line) vsnormalized detuning x/γ with G = ± γ csc θ ( J = G ) for θ = − π , − π , π , π , respectively. It can be seen fromFig. 4, the transmission spectrums T LR and T RL willnot take the symmetric form anymore as θ = ± π , and T LR > T RL for θ ∈ ( − π, T LR < T RL for θ ∈ (0 , π ). IV. CONCLUSION
In summary, we have theoretically studied how toachieve perfect optical nonreciprocity in a double-cavityoptomechanical system. In this paper, we focus on theconditions where the system can exhibit perfect opticalnonreciprocity. From the expressions of condition, wecan draw three important conclusions: (1) when nonre-ciprocal phase difference θ = ± π , the mechanical damp-ing rate has no effect on the appearance of perfect op- tical nonreciprocity as long as Eq. (14) is satisfied; (2)even with very weak optomechanical coupling ( G ≪ κ ),perfect optical nonreciprocity can still occur accordingto Eq. (14); (3) the system can exhibit perfect opticalnonreciprocity with any nonreciprocal phase difference θ ( θ = 0 , π ) if κ = γ and Eq. (16) is satisfied. Our re-sults can also be applied to other parametrically coupledthree-mode bosonic systems, besides optomechanical sys-tems. Acknowledgments