Nonreciprocal transmission and fast-slow light effects in a cavity optomechanical system
aa r X i v : . [ qu a n t - ph ] J a n Nonreciprocal transmission and fast-slow light effectsin a cavity optomechanical system
Jun-Hao Liu, Ya-Fei Yu, , ∗ and Zhi-Ming Zhang , † Guangdong Provincial Key Laboratory of Nanophotonic Functional Materialsand Devices (School of Information and Optoelectronic Science and Engineering),and Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,South China Normal University, Guangzhou 510006, China compiled: January 7, 2019We study the nonreciprocal transmission and the fast-slow light effects in a cavity optomechanical system, inwhich the cavity supports a clockwise and a counter-clockwise circulating optical modes, both the two modesare driven simultaneously by a strong pump field and a weak signal field. We find that when the intrinsicphoton loss of the cavity is equal to the external coupling loss of the cavity, the system reveals a nonreciprocaltransmission of the signal fields. However, when the intrinsic photon loss is much less than the externalcoupling loss, the nonreciprocity about the transmission properties almost disappears, and the nonreciprocityis shown in the group delay properties of the signal fields, and the system exhibits a nonreciprocal fast-slowlight propagation phenomenon.
OCIS codes: (270.0270) Quantum optics; (120.4880) Optomechanics; (230.0230) Optical devices.http://dx.doi.org/10.1364/XX.99.099999
1. Introduction
In recent years, optical nonreciprocity has got a lot ofattentions for its important applications in photonic net-work, signal processing, and one-way optical communi-cation protocols. In the common nonreciprocal devices,such as, isolator, circulator, nonreciprocal phase shifter,the transmission of the information is not symmetric. Atpresent, the researches about the optical nonreciprocitymainly focused on two aspects: one is the transmissionproperties of the signal fields, another is the photonicstatistical properties of the signal fields.For the first aspect, scientists have demonstrated thatmany physical effects and physical systems, such as Fara-day rotation effect in the magneto-optical crystals [1–3], optical nonlinearity [4], spatial-symmetry-breakingstructures [5, 6], optoacoustic effects [7, 8], the parity-time-symmetric structures [9–12], can be used to real-ize the optical nonreciprocal transmission. Efforts havealso been made to study the nonreciprocal transmis-sion in cavity optomechanical systems [13–17]. Mani-patruni et al. demonstrated that the optical nonrecip-rocal transmission was based on the momentum differ-ence between the forward and backward-moving lightbeams in a Fabry-Perot cavity with one moveable mir-ror [18]. Hafezi et al. proposed a scheme to achieve the ∗ [email protected] † [email protected] nonreciprocal transmission in a microring resonator byusing an unidirectional optical pump [19]. Metelmannand Clerk discussed a general method for nonrecipro-cal photon transmission and amplification via reservoirengineering [20]. Peterson et al. demonstrated an effi-cient frequency-converting microwave isolator based onthe optomechanical interactions [21]. Mirza et al. stud-ied the optical nonreciprocity and slow light propagationin coupled spinning optomechanical resonators [22].For the second aspect, the researches on the photonicstatistic properties of the transmitted fields in nonrecip-rocal devices are fewer. At present, the relevant theo-retical works include the nonreciprocal photon blockade[23], the authors discussed how to create and manipu-late nonclassical light via photon blockade in rotatingnonlinear devices. They found that the light with sub-Poissonian or super-Poissonian photon-number statisticscan emerge when driving the resonator from its left orright side. Subsequently, Xu et al. proposed a scheme tomanipulate the statistic properties of the photons trans-port nonreciprocally via quadratic optomechanical cou-pling [24].In this paper, we study the nonreciprocal transmissionand the fast-slow light effects in a cavity optomechanicalsystem, as shown in Fig. 1. We show that when the in-trinsic photon loss of the cavity equals the external cou-pling loss of the cavity, we can achieve the nonreciprocaltransmission of the signal fields with the red-sidebandpumping or the blue-sideband pumping. We also showthat when the intrinsic photon loss is much less than the ̂ ( ) = 1= 0 ( ) - = 1, < 0= 1, > 0 ( ) Fig. 1. (a) Schematic diagram of our proposed model. An optomechanical microtoroid cavity supports a clockwise circulatingmode (ˆ a ) and a counter-clockwise circulating mode (ˆ c ), both the two cavity modes couple with the mechanical mode (ˆ b ) via theradiation pressure. The pump fields ( ε ap , ε cp ) and signal fields ( ε as , ε cs ) couple with the cavity modes by an optical fiber. (b)The nonreciprocal transmission: the right-moving signal field is completely transmitted ( T a = 1), while the left-moving signalfield is blocking-up ( T c = 0). (c) The nonreciprocal fast-slow light: both the right-moving field and the left-moving signal fieldare transmitted ( T a = T c = 1). However, the group delay of the right-moving signal field is negative ( τ a < τ c > external coupling loss, the nonreciprocity of the systemabout the optical transmission properties almost disap-pears, now the system exhibits a nonreciprocal fast-slowlight propagation phenomenon, i.e., the group velocityof the right-moving signal field will be speed up (fastlight), while the group velocity of the left-moving signalfield will be slowed down (slow light), or vice versa.
2. Model and Hamiltonian
Our system model is shown in Fig. 1(a). We consideran optomechanical microtoroid cavity, which supports aclockwise circulating mode (ˆ a ) and a counter-clockwisecirculating mode (ˆ c ), both the two cavity modes couplewith the mechanical mode (ˆ b ) via the radiation pres-sure. The cavity mode ˆ a (ˆ c ) is driven simultaneouslyby a strong pump field ε ap ( ε cp ) and a weak signal field ε as ( ε cs ). The total Hamiltonian of the system can beexpressed as H total = H om + H aps + H cps + H ac , (1)where H om = ~ ω ˆ a † ˆ a + ~ ω ˆ c † ˆ c + ~ ω m ˆ b † ˆ b + ~ g (ˆ a † ˆ a +ˆ c † ˆ c )(ˆ b † + ˆ b ) is the Hamiltonian of the cavity optome-chanical system. ˆ a (ˆ c ) and ˆ b are the annihilation op-erators of the clockwise (counter clockwise) circulatingcavity mode and the mechanical mode with frequency ω and ω m , respectively. g is the optomechanical couplingstrength between the cavity modes and the mechanicalmode. H aps = i ~ ε ap (ˆ a † e − iω ap t − H.c. )+ i ~ ε as (ˆ a † e − iω as t − H.c. ) describes the interactions of the cavity mode ˆ a with the pump field of amplitude ε ap = p κP ap / ~ ω ap and the signal field of amplitude ε as = p κP as / ~ ω as ,respectively, in which κ is the coupling decay rate of the cavity, and P ap ( P as ) is the laser power. Similarly, H cps = i ~ ε cp (ˆ c † e − iω cp t − H.c. ) + i ~ ε cs (ˆ c † e − iω cs t − H.c. )describes the interaction Hamiltonian of cavity mode ˆ c with the pump field of amplitude ε cp = p κP cp / ~ ω cp and the signal field of amplitude ε cs = p κP cs / ~ ω cs ,respectively. The last term H ac = ~ J (ˆ a † ˆ c + ˆ c † ˆ a ) repre-sents the interaction between the two cavity modes withthe strength J .For simplicity, we assume that the two pump fieldshave the same frequency, i.e., ω ap = ω cp = ω p . Inthe rotation frame with H r = ω p (ˆ a † ˆ a + ˆ c † ˆ c ), the sys-tem Hamiltonian can be written as H = ~ ∆ˆ a † ˆ a + ~ ∆ˆ c † ˆ c + ~ ω m ˆ b † ˆ b + ~ g (ˆ a † ˆ a + ˆ c † ˆ c )(ˆ b † + ˆ b )+ i ~ ε ap (ˆ a † − H.c. ) + i ~ ε as (ˆ a † e − iδ as t − H.c. )+ i ~ ε cp (ˆ c † − H.c. ) + i ~ ε cs (ˆ c † e − iδ cs t − H.c. )+ ~ J (ˆ a † ˆ c + ˆ c † ˆ a ) , (2)where ∆ = ω c − ω p is the frequency detuning betweenthe cavity field (ˆ a, ˆ c ) and the pump field ( ε ap , ε cp ), and δ as = ω as − ω p ( δ cs = ω cs − ω p ) is the frequency detuningbetween the signal field ε as ( ε cs ) and the pump field ε ap ( ε cp ). The system dynamics is fully described by the setof quantum Heisenberg-Langevin equations d ˆ adt = − ( i ∆ + κ t )ˆ a − ig ˆ a (ˆ b † + ˆ b ) − iJ ˆ c + ε ap + ε as e − iδ as t + √ κ ˆ a in ,d ˆ cdt = − ( i ∆ + κ t )ˆ c − ig ˆ c (ˆ b † + ˆ b ) − iJ ˆ a + ε cp + ε cs e − iδ cs t + √ κ ˆ c in ,d ˆ bdt = − ( iω m + γ )ˆ b − ig (ˆ a † ˆ a + ˆ c † ˆ c ) + p γ ˆ b in , (3)where the cavity has the damping rate κ t = κ in + κ ,which are assumed to be due to the intrinsic photon lossand external coupling loss, respectively, and the mechan-ical mode has the damping rate γ . ˆ a in (ˆ c in ), and ˆ b in arethe δ -correlated operators of the input noises for the cav-ity mode ˆ a (ˆ c ) and the mechanical mode ˆ b , respectively.These noise operators satisfy h ˆ a in i = h ˆ c in i = h ˆ b in i = 0.In this model, we are interested in the mean responseof the system. Thus, in the following, we turn to calcu-late the evolutions of the expectation values of ˆ a , ˆ c , ˆ b ,and we denote h ˆ a i ≡ A , h ˆ c i ≡ C , h ˆ b i ≡ B , (cid:10) ˆ a † (cid:11) ≡ A ∗ , (cid:10) ˆ c † (cid:11) ≡ C ∗ , h ˆ b † i ≡ B ∗ . By using the mean-field assump-tion h ˆ a ˆ b ˆ c i = h ˆ a i h ˆ b i h ˆ c i , we can write the equations forthe mean values as dAdt = − ( i ∆ + κ t ) A − igA ( B + B ∗ ) − iJC + ε ap + ε as e − iδ as t ,dCdt = − ( i ∆ + κ t ) C − igC ( B + B ∗ ) − iJA + ε cp + ε cs e − iδ cs t ,dBdt = − ( iω m + γ ) B − ig ( | A | + | C | ) . (4)Equations (4) can be solved by using the perturbationmethod in the limit of the strong pump fields, while tak-ing the signal fields to be weak. Using the linearizationapproximation, we make the following ansatz [25] X = X + X a + e − iδ as t + X a − e iδ as t + X c + e − iδ cs t + X c − e iδ cs t , (5)where X can be any one of the quantities A , B , C , ortheir complex conjugates A ∗ , C ∗ , B ∗ . X representsthe steady-state mean value of the corresponding sys-tem mode, and X a + , X a − , X c + , X c − are the additionalfluctuations. By substituting Eq. (5) into Eqs. (4), andkeeping only the first-order in the small quantities andneglecting the nonlinear terms like A a + C a + , A a + B c − , B c − C a + , · · · , we can obtain the steady-state mean valueequations, and the fluctuation equations for the cavitymode components A a + and C c + (see the appendix). Bysolving these equations, we find that A a + = η ( δ as ) ε as , C c + = ξ ( δ cs ) ε cs , the concrete forms of the coefficients η ( δ as ) and ξ ( δ cs ) are tediously long, and we will notwrite them out here.The relation among the input, internal, and outputfields is given as [26] X out = X in − κX . By us-ing the ansatz again, we write the output field X out as X out + X outa + e − iδ as t + X outa − e iδ as t + X outc + e − iδ cs t + X outc − e iδ cs t . Then we can obtain the output field compo-nents A outa + = ε as − κA a + and C outc + = ε cs − κC c + . Thetransmissivities can be written as t a ( δ as ) = A outa + /ε as , t c ( δ cs ) = C outc + /ε cs . The nonreciprocal transmission isthen described by the normalized transmissivities (trans-mission spectra) T a = | t a ( δ as ) | = | − κη ( δ as ) | ,T c = | t c ( δ cs ) | = | − κξ ( δ cs ) | . (6) What’s more, in the resonant region of the trans-mission spectra, the output signal fields have thephase dispersions φ a ( ω as ) = arg[ T a ( ω as )] and φ c ( ω cs )= arg[ T c ( ω cs )], which can cause the group delay [27] τ a = dφ a ( ω as ) dω as , τ c = dφ c ( ω cs ) dω cs . (7)The group delay τ a ( τ c ) > τ a ( τ c ) < T a = 1, T c = 0 or T a =0, T c = 1) and the nonreciprocal fast-slow light effects( τ a > τ c < τ a < τ c > ω m = 2 π ×
10 MHz and γ = 2 π × Hz (quality factor Q m = 10 ), the equivalentmass of the mechanical resonance m = 5 ng, and theequivalent cavity length l = 1 mm. The damping rateof the optical cavity κ = 2 π × λ = 1064 nm. The other parameters are J = 2 π × Hz, κ in = 2 π ×
3. Nonreciprocal transmission
In this section, we numerically evaluate the transmissionspectra T a and T c to show the possibility of achieving thenonreciprocal transmission of the signal fields.Firstly, we assume that the system works near thered sideband (∆ = ω m ). In Fig. 2 we plot T a and Fig. 2. The transmission spectra T a (red solid lines) and T c (blue dashed lines) as a function of δ as /ω m and δ cs /ω m ,respectively. The system works near the red sideband (∆ = ω m ). The parameters are: (a) P a = P c , (b) P a = 10 P c ,(c) P a = 10 P c , (d) P a = 10 P c . The other parameters arestated in the text. Fig. 3. The transmission spectra T a (red solid lines) and T c (blue dashed lines) as a function of δ as /ω m and δ cs /ω m ,respectively. The system works near the blue sideband (∆ = − ω m ). The parameters are: (a) P a = P c , (b) P a = 6 P c , (c) P a = 8 . P c , (d) P a = 9 . P c , (e) P a = 5 × P c , (f) P a =10 P c . The other parameters are stated in the text. T c as a function of δ as /ω m and δ cs /ω m , respectively.Here we hold the pump power P c constant, P c = 100nW[30]. We can see that the transmission of the left-movingsignal field is simply that of a bare resonator, and T c exhibits a dip near δ cs = ω m ( T c ≈ P a , we find that the transmission ofthe right-moving signal field can be obviously modified.Near δ as = ω m , T a will exhibit a very narrow peak andgradually increase with the increase of P a . When P a =10 P c , we have T a ≈
1. If we continue to increase P a ,the spectrum will exhibit a split, and this is associatedwith the normal mode splitting [31]. The above featurescan result in a nonreciprocal transmission of the signalfields in our system, i.e., the right-moving signal fieldis completely transmitted, while the left-moving signalfield is blocking-up.Then we consider that the system works near the bluesideband (∆ = − ω m ), and we also choose P c = 100nW.In Fig. 3, we can see that T c exhibits a dip near δ cs = − ω m ( T c ≈ P a , T a will exhibit a very narrow peak near δ as = − ω m , withthe increase of P a , the peak value will first increase andthen decrease. When P a = 9 . P c , we have T a > T c <
1, now the right-moving signal field can be amplified,while the left-moving signal field cannot be amplified.When P a = 10 P c , we have T a ≈ T c ≈
0. Nowthe system can also be used to realize the nonreciprocaltransmission of the signal fields.In addition, in our system, the transmissive directionof the signal field can be changed by adjusting the ratioof P a and P c . For example, when ∆ = ω m , P a = 100nWand P c = 10 P a , now the left-moving signal field is com-pletely transmitted while the right-moving signal field isblocking-up.
4. Nonreciprocal fast-slow light effects
In this section, we show how to realize the nonreciprocalfast-slow light propagation of the signal fields, i.e., boththe right-moving and left-moving signal fields can becompletely transmitted, while the group velocity of theright-moving signal field will be speed up and the left-moving signal field will be slowed down, or vice versa.In Fig. 4, we plot T a and T c for different intrinsic pho-ton loss rate κ in under the unbalanced-pumping condi-tion ( P a = 10 P c , P c = 100nW). We find that with thedecrease of κ in the transmission of the left-moving signalfield T c will gradually increase near δ cs = − ω m . How-ever, the transmission of the right-moving signal field Fig. 4. The transmission spectra T a (red solid lines) and T c (blue dashed lines) as a function of δ as /ω m and δ cs /ω m , re-spectively, for different intrinsic photon loss rate κ in . Thesystem works near the blue sideband (∆ = − ω m ). The pa-rameters are: (a) κ in = κ , (b) κ in = 10 − κ ,, (c) κ in = 10 − κ ,(d) κ in = 10 − κ . The other parameters are stated in the text. Fig. 5. The group delay τ a (red solid lines) and τ c (bluedashed lines) as a function of δ as /ω m and δ cs /ω m , respec-tively. The system works near the blue sideband (∆ = − ω m ).The parameters are: (a) P a = 1 × P c , (b) P a = 5 × P c ,(c) P a = 1 × P c , (d) P a = 2 × P c , (e) P a = 5 × P c .The other parameters are stated in the text. T a ≈ δ as = − ω m . When κ in = 10 − κ , we have T a ≈ T c ≈ .
7, now the nonreciprocity of the sys-tem about the optical transmission is weakened. When κ in = 10 − κ , we have T a ≈ . T c ≈ . T a ≈ T c ).However, in this situation ( κ in ≪ κ ), the nonreciproc-ity of system is shown in the group delay properties ofthe signal fields. In Fig. 5, we plot τ a and τ c as a func-tion of δ as /ω m and δ cs /ω m , respectively. We can seethat in the range of the parameters we considered (wehave plotted the transmission spectra T a and T c and wecan guarantee that T a ≈ T c ≈ δ as = − ω m (the group veloc-ity will be speed up), that corresponds to the fast lightpropagation. While the group delay of the left-movingsignal field is positive near δ cs = − ω m (the group veloc-ity will be slowed down), that corresponds to the slowlight propagation. This shows that the system can ex-hibit a nonreciprocal fast-slow light propagation of thesignal fields.Furthermore, we can change the propagation directionof the fast-slow light by adjusting the ratio of of P a and P c . For example, in Fig. 5(c), we have τ a ≈ − . µs and τ c ≈ . µs . However, if we choose P a = 100nW and P c = 10 P a , then we have τ a ≈ . µs and τ c ≈ − . µs ,now the right-moving signal field is slow light and theleft-moving signal field is fast light.
5. Conclusion
In summary, we have studied the nonreciprocal trans-mission and the fast-slow light effects in a cavity optome-chanical system. We have shown that for both the red-sideband pumping or the blue-sideband pumping, thesystem can act as an optical unidirectional isolator. Wehave also shown that if the intrinsic photon loss is muchless than the external coupling loss, the nonreciprocityof the system on the optical transmission almost dis-appears, now the system reveals an interesting nonre-ciprocal fast-slow light propagation phenomenon. Ourproposed model might have applications in the photonicnetwork.
Fundings.
This work was supported by the NationalNatural Science Foundation of China (Nos. 11574092,61775062, 61378012, 91121023); the National Basic Re-search Program of China (No. 2013CB921804).
Appendix
By substituting Eq. (5) into Eqs. (4), we can obtain thethe steady-state mean value equations0 = − ( i ∆ + κ t ) A − igA ( B + B ∗ ) − iJC + ε ap , − ( i ∆ + κ t ) C − igC ( B + B ∗ ) − iJA + ε cp , − ( iω m + γ ) B − ig ( | A | + | C | ) . (A1)In this system, we are interested on the dynamics ofthe cavity mode components A a + e − iδ as t and C c + e − iδ cs t which are resonance with the corresponding signal fields ε as e − iδ as t and ε cs e − iδ cs t , respectively. We can obtainΦ a B a + = − ig ( A ∗ A a + + A ∗ a + A + C ∗ C a + + C ∗ a + C ) , Ω a A a + = − igA ( B a + + B ∗ a + ) − iJC a + + ε as , Ω a C a + = − igC ( B a + + B ∗ a + ) − iJA a + , (A2)Φ c B c + = − ig ( A ∗ A c + + A ∗ c + A + C ∗ C c + + C ∗ c + C ) , Ω c C c + = − igC ( B c + + B ∗ c + ) − iJA c + + ε cs , Ω c A c + = − igA ( B c + + B ∗ c + ) − iJC c + , (A3)where Φ k = i ( ω m − δ ks ) + γ and Ω k = i [∆ + g ( B + B ∗ ) − δ ks ] + κ t , k = a, c . References [1] R. L. Espinola, T. Izuhara, M. C. Tsai, R. 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