Dynamical decoupling and isolation in modulated coupled pendulums an b are negligible for the chosen parameters.The shim ch-s og of theu 0w三nt=ne-2i"(2o/m) (13) The shift Eg.(17)can be taken into account in Egs.(15) (16)by replacing o with o in terms of the to zero rder Bessel function.I The mod n+△wo.n this way n exc ulus and the phase of this quantity are plotted as a solid agre ent for the analyti spec ctra (d line in Fig.5. is found in E to extract a timate of the effective coupling from the numerica applied to the amplitude erms of th of the external driving forc in Eq.(11),which gives the ia6 effective averaged driving force =2(@+iy-x)(32/3)” (18) The r in Eq.(18)the B,with the that,for the specific modulation in Eq.(8),has the form: compared with the analytical RW prediction a.13】 f=fes e-olw Jo(lo/w) (14) nitu agree appears to e very for both th and cou the d al de olin =0.The small discrepancies cur when both nu Explicit forms for the steady oscillation gime can be merator and denominator of Eq.(18)go to zero and the derived by setting the time derivatives in Eqs.(11)-(12) s more sens e t sim In this way,one obtains the follov performed in Fig.4.where the numerical results are com- pared to RW prediction of Eq.(14).The agreement is 2 (o-i-we) gain very good,in parti ular for what conc (ex)= (15) hich 4(0o-iy-wex)-of and both fo B2(aex） gets worse when the inequality chain of Eg.(9)is only 4(@o-ir-wexP-l2e下 (16) marginally satisfied. While a qualitative agreement with the numerical predic- o to the oe the d tions shown in Fig.3 is already present at this level,there is still an ove f the This shift of the RW description of the system of coupled pendu ssipative version of the Bose-Hubbard tion to the Rw ar f qua vesmali counter-rotaing part evoing writing a(t)= scribed by the Hamiltonian: s in Fas are coupled by Rw e,+h+∑aa四 i=- co-rotating contributions appear in the equation for a from th ms of a gives, tot where a;and are bosonic on-site operators.The first he bosonic particle the hop into Eg.(10)and isolating the non- term describes hopping e Is phase pij desc bes a no aging over the period of the considered modulation,the ms do not effective frequency shift is pear in the standard condensed matter BH model Hamiltonian in Eq(19)as they would cor △wo--((t)2)r/（2ao）--16/(4oo). (17) spond to processes where the total number part ve This is the principal effect of the counter-rotating terms e and lost fro m the m the the bevond the rw.and it is more and more important for oretical description requires a driving term of the form growing o.All other non-RW contributions involving =()+h.c.as well as damping terms,to be typically included at the level of the Master Equation [33] rep h p-5 Dynamical decoupling and isolation in modulated coupled pendulums For the specific modulation considered here, the effective coupling has the simple expression: Ωeff ≡ Ω eff 12 = Ω e−2iI0/w J0 (2I0/w) (13) in terms of the J0 zero-order Bessel function. 1 The mod￾ulus and the phase of this quantity are plotted as a solid line in Fig. 5. In particular, the modulus |Ωeff| shows a se￾ries of zeros, which are indicative of a complete dynamical decoupling between the two pendulums. The same procedure must be applied to the amplitude of the external driving force in Eq. (11), which gives the effective averaged driving force: f eff ex ≡ fex T Z T 0 dt e i R t 0 vi(t ′ ) dt ′ that, for the specific modulation in Eq. (8), has the form: f eff ex = fex e −iI0/w J0(I0/w). (14) The zeros of f eff ex determine the parameters at which com￾plete dynamical isolation from the external force occurs. Explicit forms for the steady oscillation regime can be derived by setting the time derivatives in Eqs. (11)-(12) to zero. In this way, one obtains the following analytical form of the resonance curves as function of ωex: β1(ωex) = 2 f eff ex (¯ω0 − iγ − ωex) 4(¯ω0 − iγ − ωex) 2 − |Ωeff| 2 (15) β2(ωex) = f eff ex Ω ∗ eff 4(¯ω0 − iγ − ωex) 2 − |Ωeff| 2 . (16) While a qualitative agreement with the numerical predic￾tions shown in Fig. 3 is already present at this level, there is still an overall global shift of the resonances. This shift is easily explained by including the leading order correc￾tion to the RW approximation. We allow for the αi to also have a small counter-rotating part evolving at fre￾quency −ωex by writing αi(t) = α rw i e −iωext +δαnrw i e iωext . Since the α and the α ∗ are coupled by the non-RW terms in Eqs. (10) and their complex conjugate equations, co-rotating contributions appear in the equation for α from the counter-rotating terms of α ∗ . A straightfor￾ward calculation gives, to the leading order: δαnrw i ≃ −(α rw i ) ∗ vi(t)/(2ω0). After substituting this expression into Eq. (10) and isolating the non-rotating terms, by aver￾aging over the period of the considered modulation, the effective frequency shift is: ∆ω0 = −hvi(t) 2 iT /(2ω0) = −I 2 0 /(4ω0). (17) This is the principal effect of the counter-rotating terms beyond the RW, and it is more and more important for growing I0. All other non-RW contributions involving Ω 1A non-zero phase in the driving (8), or equivalently a temporal shift in the stroboscopic sampling, will result in an extra phase factor in Ωeff [17]. and γ are negligible for the chosen parameters. The shift can be interpreted as a classical analog of the quantum Bloch-Siegert shift of nuclear magnetic resonance [31]. The shift Eq. (17) can be taken into account in Eqs. (15)- (16) by replacing ω¯0 with ω˜0 = ¯ω0 + ∆ω0. In this way, an excellent agreement for the analytical spectra (solid lines) with the result of the numerical simulations (dots) is found in Fig. 3. This suggests a way to extract an es￾timate of the effective coupling Ωeff from the numerical results. An explicit expression of it in terms of the oscilla￾tion amplitudes is obtained by taking the ratio of Eq. (15) and Eq. (16): Ωeff = 2 (˜ω0 + i γ − ωex) (β2/β1) ∗ . (18) The result of replacing in Eq. (18) the βi with the numer￾ically calculated Ai is shown by the dots in Fig. 5 and is compared with the analytical RW prediction of Eq. (13). The agreement appears to be very good for both the mag￾nitude and the phase of the coupling, in particular the position of the dynamical decoupling points at which Ωeff = 0. The small discrepancies occur when both nu￾merator and denominator of Eq. (18) go to zero and the procedure is more sensitive to numerical errors. A similar comparison for the effective driving force is performed in Fig.4, where the numerical results are com￾pared to RW prediction of Eq. (14). The agreement is again very good, in particular for what concerns the posi￾tion of the dynamical isolation points for which f eff ex = 0 and both max(|A1,2|) = 0. Of course, the agreement gets worse when the inequality chain of Eq. (9) is only marginally satisfied. Connection to the Bose-Hubbard model. – Before concluding, it is worth to highlight the direct connection of the RW description of the system of coupled pendu￾lums to a driven-dissipative version of the Bose-Hubbard (BH) model of quantum condensed-matter. In the pres￾ence of an external gauge field [32], the BH model is de￾scribed by the Hamiltonian: Hˆ = − X hiji h J e iφij aˆ † i aˆj + h.c.i + X i U 2 aˆ † i aˆ † i aˆiaˆi (19) where aˆi and aˆ † i are bosonic on-site operators. The first term describes hopping of the bosonic particles: the hop￾ping amplitude is J and the phase φij describes a non￾trivial tunneling phase. Of course, non-rotating wave terms do not appear in the standard condensed-matter BH model Hamiltonian in Eq. (19) as they would cor￾respond to processes where the total numbers of parti￾cles is not conserved. If particles are instead injected from a coherent source and lost from the system, the the￾oretical description requires a driving term of the form Hˆ F = P i [fex,i(t) ˆa † i +h.c.] as well as damping terms, to be typically included at the level of the Master Equation [33]. Under the classical approximation where operators are replaced by C numbers, equations of motion analogous to p-5