We theoretically study the dynamics of a pair of coupled pendulums subject to a periodic temporal modulation of their oscillation frequency. Inspired from analogous developments in quantum mechanics, we anticipate dynamical localization and dynamical isolation effects, as well as the occurrence of non-trivial coupling phases. Perspectives in the direction of studying synthetic gauge fields in a classical mechanics context are outlined.

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Dynamical decoupling and dynamical isolation in temporally modulated coupled pendulums 寸 GRAZIA SALERNO and IACOPO CARUSOTTO INO-CNR BEC Center and Dipartimento di Fisica,Universita di Trento,via Sommarive 14,38123 Povo,Italy 5.v quantum mechanics.we antic ate dynamical localization and dyn Introduction.- Dynamical localization is a surpris ate synhe tic gauge field for photons [25,26] ing combi quence quantum mec s applied o par- dy of dynam This effect was first observed as renormalization of the cal this text.The dynamic stabilization of the inverted pendulum when its pivot point is made to oscillate in space is a well elebrated example of nor anical effect stem [2 fal in a stro nside system closely related effect is the coherent destruction of tunnel- and periodic ally varied in time In analogy to 导 made c dict quant dable material y between the pendulums tions a and electronic decoherence.the robust coherence effect,where exchange of ene is suppressed.When the pendulums are driven by an ex- ernal force,we an a nove Further studies of dynamical The system and the theoretical model.- The system were 113,14h of a d induced superfluid to Mott-insulator transition [15]. mi=pI (1) scheme 1--m[1+h(t)x1+k(x2-x1)-11(2) he to gen m2=2 (3) sites 16,17]and then fields for atoms 118-231 Corres ndingly to these advances in 2=-mu1+(t切2+k(1-x2)-2. omicphyic erve dyes light in t从ex,。aiables indicate theo he penc thom the um pos avegu

arXiv:1401.3978v2 [physics.class-ph] 16 Apr 2014 epl draft Dynamical decoupling and dynamical isolation in temporally modulated coupled pendulums GRAZIA SALERNO and IACOPO CARUSOTTO INO-CNR BEC Center and Dipartimento di Fisica, Universit`a di Trento, via Sommarive 14, 38123 Povo, Italy PACS 45.20.D- – Newtonian mechanics PACS 46.40.Ff – Resonance, damping, and dynamic stability PACS 03.65.Vf – Phases: geometric; dynamic or topological Abstract –We theoretically study the dynamics of a pair of coupled pendulums subject to a periodic temporal modulation of their oscillation frequency. Inspired from analogous developments in quantum mechanics, we anticipate dynamical localization and dynamical isolation effects, as well as the occurrence of non-trivial coupling phases. Perspectives in the direction of studying synthetic gauge fields in a classical mechanics context are outlined. Introduction. – Dynamical localization is a surprising consequence of quantum mechanics applied to particles subject to a strong time-dependent external force. This effect was first observed as renormalization of the magnetic response of an atom illuminated by a strong rf field [1]. In a solid state context, dynamical localization was proposed in [2–6] as a dramatic suppression of the d.c. conductivity of a metal in a strong a.c. field. Another closely related effect is the coherent destruction of tunneling in a double-well geometry, first predicted in [7,8] and extensively compared to dynamical localization in [10]. While the experimental study of these effects in solids is made difficult by the unavoidable material imperfections and electronic decoherence, the robust coherence and the clean periodic potential experienced by atomic matter waves in temporally modulated optical lattices has allowed for a clear observation of Bloch band suppression in a new atomic physics context [9]. Further studies of dynamical matter wave localization effects were reported in [11, 12] using Bose-condensed atomic samples. Following the proposal [13, 14], this research line culminated in the observation of a dynamicallyinduced superfluid to Mott-insulator transition [15]. Very exciting further developments of these ideas aim at using more complex modulation schemes to generate non-trivial hopping phases between the lattice sites [16, 17] and then synthetic gauge fields for neutral atoms [18–23]. Correspondingly to these advances in atomic physics, the same ideas are being explored in photonics to observe dynamical localization of light in coupled optical waveguides [24] and, very recently, to generate synthetic gauge field for photons [25, 26]. In this Letter, we report a theoretical study of dynamical localization phenomena in a classical mechanics context. The dynamic stabilization of the inverted pendulum when its pivot point is made to oscillate in space is a well celebrated example of non-trivial mechanical effect stemming from a temporal modulation of the system parameters [27]. Here we consider a system of two coupled pendulums, whose oscillation frequencies are independently and periodically varied in time. In analogy to the coherent destruction of tunneling of a quantum particle in a double-well potential, we predict a dynamical decoupling effect, where exchange of energy between the pendulums is suppressed. When the pendulums are driven by an external force, we anticipate a novel dynamic isolation effect, where the temporal modulation effectively decouples the system from the external force. The system and the theoretical model. – The system of two identical coupled pendulums is modelled as a pair of coupled harmonic oscillators of equal masses m following the motion equations: m x˙ 1 = p1 (1) p˙1 = −mω2 0 [1 + ν1(t)] x1 + k(x2 − x1) − ξ1x˙ 1 (2) m x˙ 2 = p2 (3) p˙2 = −mω2 0 [1 + ν2(t)] x2 + k(x1 − x2) − ξ2x˙ 2 . (4) The x1,2 variables indicate the spatial displacement of the pendulums from the equilibrium position. The linearised form of the motion equations is legitimate in p-1

G.Salerno et al. for the a=12 complex variables defined as: =受+ (7) Cor Physi ex(t) by measuring the instantaneous position and velocity of In order for the coupling between the oscillators to be 以0 effective,the strong coupling condition,,will be assumed. eing the man orms the idal ones Sketch of the physical system ins a magnet and ntatammta2a my the v;(t)=(-1)'Iosin(wt) 8 As we shall see better in the following of this Letter,the rve the on pnys erized by the in 0多多且. (9) the small oscillation regime where the displacements are The latter inequa is es sential to describe e the system ndulms have th 9 might othe vise occur for large values of the modulation The coupling between the pendulums occurs via a spring amplitude,29]. nt k.The key eler ent to achieve the dynamica A ng and isc ated Eq as a temporal modulation of the restorine force stren gths in the vanishing friction case.The integration Kutta of relative amplitude().One of the possible concrete realizations of this model is Fig.1:each pen p ard four Kunge- In Fig. A生 led at th ated be is.In this the 2/w,the system is described by a time gravitational restoring force felt by each pendulum is sup independent discrete evolution law.The different panels plemented by a contribution of magneti c origin,which g.2 correspond the sam ation frequency u an be contro via the (tme-de ent)current 11.2 anel (a)there is no modulation .o =0:the am lation of the natural oscillator frequencies then has the plitudes of the two oscillators exhibit the usual beating form n the o w号2()=话1+h.2(t, tors at uping frequency the frequency of the beat is modified:in particular,for the parameters of panels(b)and(c),the beat frequency is tities nore and more r nl=9,=点 k is observed (not shown). 2 =2m Experimental proposal and dynamical isolation the four equations of motion Egs.(1)-(4)can be summa- In orde to perform quar study of rized in a pair of complex equations: the mdure to and tme to propose av a:=-iwoai-iv(t)(a;+ai)-7(ai-ai) is useful to consider the realistic case of dissipative pen +i号(ag-4+a-a4-a). (6 dulums driven by an external fo orce,which is a p-2

G. Salerno et al. S (t) ex i (t) 1 (t)i 2 S N N S F Figure 1: Sketch of the physical system under consideration. Each pendulum contains a magnet and the modulation of its natural oscillation frequency is controlled by the timedependent current i1,2(t) flowing in the corresponding coil. The first pendulum may be externally driven by a time-dependent force Fex(t). the small oscillation regime where the displacements are much smaller than the length L of the pendulums: in this regime, the natural oscillation frequency of each isolated pendulum is the usual ω0 = p g/L. The friction constants of the two pendulums have the same value ξ1 = ξ2 = ξ. The coupling between the pendulums occurs via a spring of constant k. The key element to achieve the dynamical decoupling and isolation effects is the temporal modulation of the system, which is included in Eq. (2) and Eq. (4) as a temporal modulation of the restoring force strengths of relative amplitude ν1,2(t). One of the possible concrete realizations of this model is sketched in Fig. 1: each pendulum contains a magnet which feels the magnetic field generated by a coil located below its axis. In this way, the gravitational restoring force felt by each pendulum is supplemented by a contribution of magnetic origin, which can be controlled via the (time-dependent) current i1,2(t) flowing in the corresponding coil. The effective modulation of the natural oscillator frequencies then has the form: ω 2 1,2 (t) = ω 2 0 [1 + ν1,2(t)] , (5) where the ν1,2(t) are proportional to the currents i1,2(t). Introducing, for notational simplicity, the rescaled quantities: v1,2(t) = ν1,2(t) ω0 2 , Ω = k mω0 , γ = ξ 2m , the four equations of motion Eqs. (1)-(4) can be summarized in a pair of complex equations: α˙ i = −i ω0 αi − i vi(t)(αi + α ∗ i ) − γ (αi − α ∗ i ) + i Ω 2 (α3−i + α ∗ 3−i − αi − α ∗ i ). (6) for the αi=1,2 complex variables defined as: αi = r mω0 2 xi + i r 1 2mω0 pi . (7) Complex conjugate equations hold for the α ∗ 1,2 . Physically the square modulus |αi | 2 is the instantaneous energy of the i-th pendulum and the argument of αi is the oscillation phase. In experiments, both can be extracted by measuring the instantaneous position and velocity of the pendulums. In order for the coupling between the oscillators to be effective, the strong coupling condition, Ω ≫ γ, will be assumed. Among the many forms of the modulation considered in the literature [16, 17], in the following we shall concentrate our attention on sinusoidal ones: vi(t) = (−1)i I0 sin(wt). (8) As we shall see better in the following of this Letter, the most convenient regime where to obtain and observe the dynamical localization physics is characterized by the inequality chain: ω0 ≫ w ≫ Ω. (9) The latter inequality is essential to describe the system in terms of an effective, time-averaged coupling. The former helps to reduce those parametric instabilities that might otherwise occur for large values of the modulation amplitude I0 [28, 29]. Dynamical decoupling of the oscillators. – As a preliminary step, we have numerically integrated Eq. (6) in the vanishing friction case, γ = 0. The integration method is a standard fourth order Runge-Kutta. In Fig. 2 we have plotted the evolution of the |α1,2| sampled at the frequency w of the modulation. At these stroboscopic times tj = 2πj/w, the system is described by a timeindependent discrete evolution law. The different panels of Fig. 2 correspond to the same modulation frequency w but different values of the amplitude I0. In panel (a), there is no modulation, I0 = 0: the amplitudes of the two oscillators exhibit the usual beating effect, i.e. a periodic exchange of energy between the oscillators at the coupling frequency Ω. As a result of the modulation vi(t), for increasing values of its amplitude I0, the frequency of the beat is modified: in particular, for the parameters of panels (b) and (c), the beat frequency is more and more reduced. For even larger amplitudes I0, a non-monotonic behaviour of the effective beat frequency is observed (not shown). Experimental proposal and dynamical isolation. – In order to perform a quantitative study of the effect of the modulation and, at the same time, to propose a viable procedure to experimentally observe these phenomena, it is useful to consider the realistic case of dissipative pendulums driven by an external force, which is assumed to be monochromatic, Fex(t) = 2fex cos ωext and to act on p-2

Dynamical decoupling and isolation in modulated coupled pendulums 0.7 000 0.25 -0.01-005 001 0.005 e,%00o6 0.01 0.025 0.5 025 00050 1000 200, 3000 4000 lA)as a function of the fr Figure 2:Numerical integration of the equations of motion ond)pendulum,no )d normalized to the peak amplit tudef(2)of a the line are guides nof the two pendu olatedpe imit. The solid (see text).The different p e for diffo 0a 0=1.21b 231().The in 2/w0=0.68X1 (c)shows an er gm=4×10-2 transfer to the second tions for2 then take the form: main comparable. as=-i(0-)a;-iv(t)(ai+ai)+i6i..Fex(t) Panel (b)show a case where the c of the twe pendulums is dramatically suppr sed this effect isap -i(侵+n）a+号ag-4+a-) (10) peaks merge into a sir sign nt exc 1 where we have defined the short-hand o=wo+/2 and at rest is the usual Kronecker delta. a negligible oscillation amplitude.This behaviour is the or each the equations of motion Eq.(10)have driven-dissipative manifestation of the dynamical decou inadyseen in the lowest panel o 1/.For the stroboscopic sampling at t 2rj/w,the Panel sho th s a novel regime while some efe steady oscillations have the form:(t)A)e lobalexcitation hy the exter nal force is su Ai are ained via a Fourier sed.The cal solutions.The moduli as a function of the fre- amplitude of both pendulums.The pre esence of a signit 好 erent regimes re The case of no modulation is shown in panel (a):the ks,split by amplitude lo.For each of these values,we plot the max- The imum of the amp Ai(wex)over the correspo se at the peaks p-3

Dynamical decoupling and isolation in modulated coupled pendulums 0 1000 2000 3000 4000 5000 0 0.5 1 ω0 t |α | / |α 1 (t=0) | 0 1000 2000 3000 4000 5000 0 0.5 1 ω0 t |α | / |α 1 (t=0) | 0 1000 2000 3000 4000 5000 0 0.5 1 ω0 t |α | / |α 1 ( t=0) | (c) (b) (a) Figure 2: Numerical integration of the equations of motion Eq. (6). The blue (green) dots indicate the modulus |α1| (|α2|) of the oscillation amplitude of the first (second) pendulum, normalized to the initial amplitude, while the lines are guides to the eyes. At the initial time t = 0, only the first oscillator is excited, while α2(t = 0) = 0. The evolution of the two pendulums is stroboscopically followed at the modulation frequency w. The three panels are for different values of the modulation amplitude, I0/w = 0 (a), I0/w = 0.632 (b), and I0/w = 1.053 (c). System parameters are w/ω0 = 7.6×10−2 , Ω/ω0 = 0.68×10−2 , γ/ω0 = fex/ω0 = 0. the first pendulum only. Energy transfer to the second pendulum is made possible by the spring that couples the two pendulums. The driven-dissipative motion equations for αi=1,2 then take the form: α˙ i = −i (¯ω0 − iγ) αi − i vi(t)(αi + α ∗ i ) + i δi,1Fex(t) − i Ω 2 + iγ α ∗ i + i Ω 2 (α3−i + α ∗ 3−i ) (10) where we have defined the short-hand ω¯0 = ω0+Ω/2 and δi,j is the usual Kronecker delta. For each ωex, the equations of motion Eq. (10) have been integrated until a steady-state regime, showing regular periodic oscillations, is achieved at long times t ≫ 1/γ. For the stroboscopic sampling at tj = 2πj/w, the steady oscillations have the form: αi(t) ≈ Ai(ωex) e−iωext . The complex amplitudes Ai are obtained via a Fourier transformation of the stroboscopically sampled numerical solutions. The moduli |Ai |, as a function of the frequency of the external force, give the response spectra shown in Fig. 3: each panel corresponds to different value of the modulation amplitude I0 and illustrates a different regime. The case of no modulation is shown in panel (a): the spectra are characterized by a pair of peaks, split by Ω and of equal width γ. The lower (upper) frequency peak corresponds to the eigenmode where the two pendulums −0.01 −0.005 0 0.005 0.01 0 0.25 0.5 0.75 (ωex −ω0 )/ω0 2 |A(ωex ) | γ / fex −0.01 −0.005 0 0.005 0.01 0 0.25 0.5 0.75 (ωex −ω0 )/ω0 2 |A(ωex ) | γ / fex −0.01 −0.005 0 0.005 0.01 0 0.25 0.5 0.75 (ωex −ω0 )/ω0 2 |A(ωex ) | γ / fex −0.01 −0.005 0 0.005 0.01 0 0.025 (ωex −ω0 )/ω0 (a) (c) (b) Figure 3: Response spectra |Ai(ωex)| as a function of the frequency ωex of the external force. The stroboscopically sampled results are shown for the first (second) pendulum as blue (green) dots, normalized to the peak amplitude fex/(2γ) of a single isolated pendulum in the small γ limit. The solid lines show the result of an analytical calculation based on the rotating wave approximation (see text). The different panels corresponds to different values of the modulation amplitude, I0/w = 0 (a), I0/w = 1.21 (b), I0/w = 2.31 (c). The inset in (c) shows an enlargement of the main plot. System parameters: w/ω0 = 7.6 × 10−2 , Ω/ω0 = 0.68 × 10−2 , γ/ω0 = 0.1 × 10−2 , fex/ω0 = 4 × 10−2 . oscillate with the same (opposite) phase. At all frequencies, the oscillation amplitudes of the two pendulums remain comparable. Panel (b) shows a case where the coupling of the two pendulums is dramatically suppressed: this effect is apparent in the figure as the two peaks merge into a single peak and no significant excitation is transferred to the second pendulum, which remains basically at rest with a negligible oscillation amplitude. This behaviour is the driven-dissipative manifestation of the dynamical decoupling effect, already seen in the lowest panel of Fig. 2 for vanishing friction. Panel (c) shows a novel regime: while some effective coupling of the two pendulums is still present, their global excitation by the external force is suppressed. The suppressed excitation is visible as a very small oscillation amplitude of both pendulums. The presence of a signifi- cant coupling is apparent in the inset where the response of the first pendulum is still showing a doublet. These different regimes are illustrated in more detail in Fig. 4. Response spectra have been numerically calculated for a number of different values of the modulation amplitude I0. For each of these values, we plot the maximum of the amplitude |Ai(ωex)| over the external drive frequency ωex, that is the resonant response at the peaks. p-3

G.Salerno et al. 0.7 0.5 0.25 0.5 22.5 0.2 0。 05 6 25 0.5 225 Figure 5 e sold lines numerical calculations as dis wave approximation (see text).System parameters as in Fig3 of this effect will be given in the next section /w1.2.The dynamical isolation seen in Fig.3(c) modulated system can be obtained within the so-called max(A)that 2.4.In the fig rotating wave solid yamical (RW)approximation,well-known from e point aroun is much larg of the rotating wave approximation that will be presented than all other internal frequencies in the problem,that is afterwards. Further insight on the effective coupling of the two rdtaeat pendulums,resulting from the temporal modulation,is given in Fig.5.In panel (a)we present the num by neglecting the aterms in the motion Equations (10) for the a variables the pendulums:this quantity is observable as the beat espectra of eh福 ing efre to at )e phase factor involving the modulation is exactly equal to Iat the stroboscopic that are considered we are in th function no od tion er panel Fig.5 of the equations(10)over a modulation period T=2/w: of-phase oscillations,the modulation allows to tune the =-i(@0-)品1-+i2港/2)32+if(11) relative phase of the oscillation of the two 2=-i(-we)-1+i(/2) (12 any valu 2T. In math ical t the The effective couplings in Eqs.(11)-(12)are defined by nlin and they are complex conjugate to each other,= p-4

G. Salerno et al. 0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 I 0 / w max ( 2 |A(ωex)| ) γ / fex Numerical 1st osc. Numerical 2nd osc. Theory 1st osc. Theory 2nd osc. | f ex eff / (2 f ex )| Figure 4: Blue (green) dots show the normalized maximum of the oscillation amplitude for the first (second) oscillator as a function of the modulation amplitude I0/w. The solid lines show the result of an analytical calculation based on the rotating wave approximation (see text). System parameters as in Fig. 3. The dynamical decoupling seen in Fig. 3(b) corresponds here to the minimum of max(|A2|) that is visible around I0/w ≃ 1.2. The dynamical isolation seen in Fig. 3(c) lies in the vicinity of the simultaneous minima of both max(|A1,2|) that are visible around I0/w ≃ 2.4. In the figure, note the further dynamical decoupling point around I0/w ≃ 2.7. The solid lines are the analytical predictions of the rotating wave approximation that will be presented afterwards. Further insight on the effective coupling of the two pendulums, resulting from the temporal modulation, is given in Fig. 5. In panel (a) we present the numerically estimated magnitude of the effective coupling between the pendulums: this quantity is observable as the beat frequency in Fig. 2, or as the separation between peaks in the response spectra of Fig. 3. Most remarkable features of Fig. 5(a) are the vanishing effective coupling at I0/w ≃ 1.2 and I0/w ≃ 2.7, corresponding to the zeros of only max(|A2|) in Fig. 4. In the following, we will see that these numerical values are related to the zeros of the zeroth-order Bessel function. Another crucial consequence of the temporal modulation is shown in the lower panel Fig. 5(b). While the eigenmodes of the bare system correspond to in-phase and outof-phase oscillations, the modulation allows to tune the relative phase of the oscillation of the two pendulums to any value from 0 to 2π. In mathematical terms, the effective coupling develops a non-trivial phase. Note how this phase displays π jumps whenever the effective coupling goes through zero. Before proceeding, it is worth noting in Fig. 3 a sizeable global shift of the resonance curves towards lower frequencies for growing I0. An explanation 0 0.5 1 1.5 2 2.5 3 0 0.25 0.5 0.75 1 I 0 /w | Ωeff /Ω | 0 0.5 1 1.5 2 2.5 3 0 2 4 6 I 0 /w arg( Ωeff /Ω ) Numerical Theory (a) (b) Figure 5: Modulus of the effective coupling frequency |Ωeff|/Ω [upper (a) panel] and its phase arg(Ωeff) (modulo 2π) [lower (b) panel]. Dots are the results of the numerical calculations as discussed in the text, while the solid lines are analytical predictions of the RW approximation. System parameters as in Fig. 3. of this effect will be given in the next section. Analytical expressions within the rotating wave approximation. – Analytical insight in the physics of the modulated system can be obtained within the so-called rotating wave (RW) approximation, well-known from quantum optics. This approximation relies on assuming that the natural oscillation frequency ω0 is much larger than all other internal frequencies in the problem, that is ω0 ≫ max(Ω, γ, w, |ωex −ω0|). Since the α variables rotate at ≈ ω0 and their conjugate variables α ∗ rotate at ≈ −ω0, the RW approximation is straightforwardly implemented by neglecting the α ∗ terms in the motion Equations (10) for the α variables. To obtain an effective, temporally averaged form of the stroboscopic dynamics of the system, we introduce the new variables: βi(t) = αi(t) exp[i R t 0 vi(t ′ ) dt ′ ] exp[i ωext]. For the chosen sinusoidal form of the modulation, the phase factor involving the modulation is exactly equal to 1 at the stroboscopic times tj = 2πj/w that are considered in the figures. Since we are in the fast modulation regime w ≫ Ω, effective equations, which no longer depend on the time t, can be obtained [30] by averaging the RW form of the equations (10) over a modulation period T = 2π/w: β˙ 1 = −i (¯ω0 − ωex) β1 − γ β1 + i (Ωeff 12/2) β2 + if eff ex (11) β˙ 2 = −i (¯ω0 − ωex) β2 − γ β2 + i (Ωeff 21/2) β1. (12) The effective couplings in Eqs. (11)-(12) are defined by: Ω eff ij ≡ Ω T Z T 0 dt e i R t 0 [vi(t ′ )−vj (t ′ )] dt ′ and they are complex conjugate to each other, Ω eff 12 = Ωeff∗ 21 . p-4

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 modulus and the phase of this quantity are plotted as a solid line in Fig. 5. In particular, the modulus |Ωeff| shows a series 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 complete 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 predictions 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 correction to the RW approximation. We allow for the αi to also have a small counter-rotating part evolving at frequency −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 straightforward 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 averaging 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 estimate of the effective coupling Ωeff from the numerical results. An explicit expression of it in terms of the oscillation 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 numerically 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 magnitude 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 numerator 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 compared to RW prediction of Eq. (14). The agreement is again very good, in particular for what concerns the position 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 pendulums to a driven-dissipative version of the Bose-Hubbard (BH) model of quantum condensed-matter. In the presence of an external gauge field [32], the BH model is described 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 hopping amplitude is J and the phase φij describes a nontrivial 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 correspond to processes where the total numbers of particles is not conserved. If particles are instead injected from a coherent source and lost from the system, the theoretical 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

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