aa r X i v : . [ qu a n t - ph ] J a n Effective Hamiltonians for almost-periodically driven quantum systems
David Viennot
Institut UTINAM (CNRS UMR 6213, Universit´e de Bourgogne-Franche-Comt´e,Observatoire de Besan¸con), 41bis Avenue de l’Observatoire, BP1615, 25010 Besan¸con cedex, France.
We present an effective Hamiltonian theory avaible for quasi-periodically driven quantum systemswhich does not need the knowledge of the Fourier frequencies of the control signal. It is also avaiblefor chaotically driven quantum systems. It is based on the Koopman approach which generalizes theFloquet approach used with periodically driven systems. We show the properties of the quasi-energystates (eigenvectors of the effective Hamiltonian) as quasi-recurrent states of the quantum system.
PACS numbers: 03.65.-w, 05.45.Mt, 32.80.Qk
I. INTRODUCTION
Periodically driven quantum systems is a subject ofgreat interest in quantum physics. It is well known thatthe consistent framework to treat this subject is the Flo-quet theorem [1] which has been firstly considered inquantum dynamics in [2]. Since this pionner work, thesubject has been extensively studied [3–14]. A usefullmethod to study periodically driven quantum systems,which is directly induced by the Floquet theory, consiststo use a time-independent effectif Hamiltonian governingthe stroboscopic dynamics (evolution on the whole pe-riod) [15].Some attempts to generalize this approach has been pro-posed for quasi-periodic driven systems (time-dependentsystems characterized by several irrationally related fre-quencies) [11, 16–18]. These studies has been confrontedto the fact the Floquet theory cannot be applied for non-periodic systems. In this paper, we want generalize theeffective Hamiltonian approach to almost-periodicallydriven systems, i.e. systems such that ∀ t, ∀ ǫ > ∃ T ǫ,t > η ǫ,t such that k H ( t + T ǫ,t ) − H ( t ) k < ǫ , with η ǫ,t such that k H ( t + η ǫ,t ) − H ( t ) k > ǫ ; H ( t ) being the time-dependent Hamiltonian of the driven system. This situ-ation includes periodically and quasi-periodically drivensystems (but in contrast with the previous works, we donot need the decomposition of time-dependent Hamilto-nian into Fourier modes associated with each frequen-cies), but also systems driven by classical flows withPoincar´e recurrence [19] (including chaotic Hamiltonianflows) and systems driven by some stochastic flows asfor example Brownian motions onto a compact manifoldwithout boundary. These two situations can modelizea quantum system driven by a periodic control but af-fected by (chaotic or stochastic) noises [20]. Our ap-proach is based on the Koopman approach of the dynami-cal systems [19, 21, 22] and follows our previous work con-cerning the mathematical properties of the Schr¨odinger-Koopman quasienergy states [23]. In some sense, ourappoach generalizes to time-dependent systems the phe-nomenon of quantum recurrence/revival [24, 25] foundfor time-independent Hamiltonians.This paper is organized as follows. Section II introducesthe effective Hamiltonians governing almost-periodically driven quantum systems. Concrete formulae for theseeffective Hamiltonians are computed in section III andwe present the expected dynamical behaviours inducedby the almost-periodicity. Finally, some illustrations arepresented section IV. II. SK AND FIRST RECURRENCE EFFECTIVEHAMILTONIANSA. The generic model
We consider a quantum system described by a Hilbertspace H and governed for its free evolution by a Hamilto-nian ~ ω ˆ H where ~ ω is the characteristic transition en-ergy of the system ( ˆ H is the reduced free Hamiltonian).The quantum system is driven by a classical discrete flow ϕ : Γ → Γ onto a phase space Γ supposed to be a compactmanifold without boundary (in general Γ is a N -torus).Let µ : T → R + be an invariant measure onto Γ ( T isa σ -algebra of Γ (generally the Borelian σ -algebra), and ∀ O ∈ T (open set of Γ), µ ( ϕ ( O )) = µ ( O )), and such that µ (Γ) < ∞ . Let T be the sampling period on which thedynamics of the quantum system is discretized. We sup-pose that the evolution operator of the driven quantumsystem can be written in the following form ∀ n ∈ N : U n ≡ U (( n + 1) T , nT ) = e − ı ω ω ˆ H e − ıV ( ϕ n ( θ )) (1)where ω = πT is the sampling frequency and V ( θ )is the interaction operator for the value θ ∈ Γ of thecontrol parameters, θ ∈ Γ are the initial values ofthe control parameters. This form is very general.It can correspond to a time-dependent Hamiltonian H ( t ) = H + V ( ˜ ϕ t ( θ )) where ϕ t ( θ ) = θ ( t ) are continu-ous time-dependent parameters. With T ≪ π ~ ω we have U n = e − ı ω ω ˆ H e − ıV ( ˜ ϕ nT ( θ )) + O (cid:16) ω ω (cid:17) (with ˆ H = H ~ ω ). Itcan also correspond to the time-dependent Hamiltonian H ( t ) = H + P n ∈ N W ( ϕ n ( θ )) δ (cid:16) t − nT + ∆( ϕ n ( θ )) ω (cid:17) of a kicked quantum system, where W ( θ ) is the kickoperator for the values θ ∈ Γ of the control parametersand 0 ≤ ∆( θ ) < π is the “angular” delay of the kick forthe value θ (the quantum system is kicked once duringa period T but the kick can be delayed). In that case, U n = e − ı H ~ ω (2 π − ∆( ϕ n ( θ ))) e − ıW ( ϕ n ( θ )) e − ı H ~ ω ∆( ϕ n ( θ )) (see for example [13]), which can be rewritten as U n = e − ı ω ω ˆ H e − ıV ( ϕ n ( θ )) with ˆ H = π ~ ω H and V ( ϕ n ( θ )) = e ı H ~ ω ∆( ϕ n ( θ )) W ( ϕ n ( θ )) e − ı H ~ ω ∆( ϕ n ( θ )) .By the Poincar´e recurrence theorem [19], we have for µ -almost all θ ∈ Γ ∀ ǫ > , ∃ p ǫ,θ > , k ϕ p ǫ,θ ( θ ) − θ k < ǫ (2)whereas ∃ n < p ǫ,θ for which k ϕ n ( θ ) − θ k > ǫ (the normin Γ is the Euclidean norm of the control parameters k θ k = P i ( θ i ) ). p ǫ,θ being not unique, we set p ǫ,θ asbeing the smallest value satisfying the relation (2). If θ is p -cyclic ( ϕ p ( θ ) = θ ), then p ǫ,θ = p ǫ,ϕ n ( θ ) = p ( p ǫ,θ isindependent of ǫ and is the same for all point of the orbitof θ : Orb( θ ) = { ϕ n ( θ ) } n ∈ N . This case correspondsto a periodically driven quantum system. We recover thequasi-periodic case if Orb( θ ) is a torus (the overline de-notes the topological closure) and if p ǫ,ϕ n ( θ ) = p ǫ,θ (thealmost-period is the same on the whole of Orb( θ )). Forthe case of a chaotic flow ϕ , p ǫ,θ is “erratically” depen-dent on ǫ and θ . And finally, if ϕ is a stochastic flow, p ǫ,θ is a random variable. Since a flow can have sev-eral behaviors, it can be interesting to decompose thephase space into ergodic components: Γ = S e Γ e , with µ (Γ e ∩ Γ e ′ ) = 0 (for e ′ = e ) and with Γ e = Orb( θ ) for µ -almost all θ ∈ Γ e . ϕ modelizes a control applied on the quantum system(by electromagnetic fields, STM, ultra-fast kicks,...), or aclassical noise affecting the quantum system (when ϕ ischaotic or stochastic); or the both ones. B. Definition of the effective Hamiltonians
Let U ( θ ) = e − ı ω ω ˆ H e − ıV ( θ ) . The Schr¨odinger-Koopman (SK) quasienergy states are defined as solu-tions of the equation (see [23]): U ( θ ) | Zµ ie , θ i = e − ıχ ie | Zµ ie , ϕ ( θ ) i (3)for θ ∈ Γ e and where χ ie is called quasienergy ( χ ie de-pends only on the ergodic component Γ e on which θ be-longs). A quasienergy becomes another quasienergy un-der the gauge change: χ ie → χ ie − ıλ and | Zµ ie , θ i → f λ ( θ ) | Zµ ie , θ i , where λ and f λ are a Koopman value andthe associated Koopman mode (i.e. f λ ( ϕ ( θ )) = e λ f λ ( θ ), λ ∈ ı R ). χ ie does not depend on θ ∈ Γ e (due to thequasienergy orbital stability theorem [23]) and in general | Zµ ie , θ i = 0 for θ Γ e . We choose { χ ie } i =1 ,..., dim H ; e such that ( | Zµ ie , θ i ) i =1 ,..., dim H be a basis of H for all θ ∈ Γ e . { χ ie } i =1 ,..., dim H ; e is called fundamental quasienergyspectrum of the driven quantum system. Moreover for µ -almost all θ ∈ Γ e , h Zµ ie , θ | Zµ je , θ i = δ ij .From equation (3) we define the SK effective Hamiltonian as being: H eff ( θ ) = X e X i χ ie | Zµ ie , θ ih Zµ ie , θ | (4)(by assuming that | Zµ ie , θ i = 0 for θ Γ e ). To under-stand the role of H eff ( θ ) it needs to consider its relationwith the first recurrence Hamiltonian H effǫ ( θ ) defined by e − ıp ǫ,θ H effǫ ( θ ) = U ( ϕ p ǫ,θ − ( θ )) ...U ( ϕ ( θ )) U ( θ ) (5)Firstly, if θ is p -cyclic, then H effǫ ( θ ) is independentof ǫ , e − ıpH effǫ ( θ ) = U ( ϕ p − ( θ )) ...U ( θ ) and is equalto the SK effective Hamiltonian H effǫ ( θ ) = H eff ( θ )which is here the usual Floquet effective Hamil-tonian of the periodic driven quantum system( | Zµ ie , θ i is the Floquet quasienergy state definedby U ( ϕ p − ( θ )) ...U ( θ ) | Zµ ie , θ i = e − ıpχ ie | Zµ i , θ i , see[13]).If θ is not p -cyclic, we have only U ( ϕ p ǫ,θ − ( θ )) ...U ( θ ) | Zµ ie , θ i = e − ıp ǫ,θ χ ie | Zµ ie , ϕ p ǫ,θ ( θ ) i ,but because of eq. (2) we have | Zµ ie , ϕ p ǫ,θ ( θ ) i = | Zµ ie , θ i + ∂ ν | Zµ ie , θ i ˜ ǫ ν ( θ ) + O ( ǫ )(6)where ˜ ǫ ( θ ) ≡ ϕ p ǫ,θ ( θ ) − θ ( k ˜ ǫ ( θ ) k = O ( ǫ )). Finally,we have e − ıp ǫ,θ H effǫ ( θ ) | Zµ ie , θ i = e − ıp ǫ,θ χ ie | Zµ ie , θ i + e − ıp ǫ,θ χ ie ∂ ν | Zµ ie , θ i ˜ ǫ ν ( θ ) + O ( ǫ ) and then e − ıp ǫ,θ H effǫ ( θ ) = (cid:0) A ν ( θ )˜ ǫ ν ( θ ) + O ( ǫ ) (cid:1) e − ıp ǫ,θ H eff ( θ ) (7)with A ν ( θ ) = X e X ij h Zµ je , θ | ∂ ν | Zµ ie , θ i| Zµ je , θ ih Zµ ie , θ | (8)More precisely, consider a sequence ( ǫ n ) n ∈ N such that ǫ n +1 < ǫ n , lim n → + ∞ ǫ n = 0 and such that ∀ ǫ ∈ ] ǫ n +1 , ǫ n ], p ǫ,θ = p ǫ n ,θ . Since θ is not cyclic, lim n → + ∞ p ǫ n ,θ = + ∞ .We have clearly,lim n → + ∞ k e ıp ǫn,θ H eff ( θ ) e − ıp ǫn,θ H effǫn ( θ ) − k = 0 (9)The SK effective Hamiltonian H eff is the limit in senseof eq. (9) of the first recurrence Hamiltonian whenthe recurrence accuracy tends to zero. Moreover, since e − ıp ǫ,θ H effǫ ( θ ) = e A ν ( θ )˜ ǫ ν ( θ )+ O ( ǫ ) e − ıp ǫ,θ H eff ( θ ) , we have(see appendix A): H effǫ ( θ ) = H eff ( θ ) + ı A ν ( θ ) ˜ ǫ ν ( θ ) p ǫ,θ + O (cid:18) ǫ p ǫ,θ (cid:19) (10)where A ν ( θ ) eji = ( A ν ( θ ) eii if i = j ıp ǫ,θ ( χ ie − χ je )1 − e − ıpǫ,θ ( χie − χje ) A ν ( θ ) eji if i = j (11)(with B eji ≡ h Zµ je , θ | B | Zµ ie , θ i ). Note that H eff is the limit of H effǫ in the sense of eq. (9),but lim n → + ∞ k H effǫ n ( θ ) − H eff ( θ ) k 6 = 0 since (1 − e − ıp ( χ ie − χ je ) ) − has no limit at p → + ∞ . Note that p ǫ,θ can be very large, the mean Poincar´e recurrence time be-ing h p ǫ,θ i ∼ µ (Γ e ) ǫ dim Γ (by supposing that µ ( B ǫ ( θ )) ∝ ǫ dim Γ where B ǫ ( θ ) is the ball of radius ǫ and centered on θ inΓ). C. Perturbation of quasienergy states
It could be interesting to relate the eigensystems of H eff and H effǫ . We can compute the first recurrenceeigenstates, H effǫ ( θ ) | Zµ i , θ, ǫ i = χ ie,ǫ ( θ ) | Zµ i , θ, ǫ i , byusing a perturbative expansion from eq. (10): χ ie,ǫ ( θ ) = χ ie + ı h Zµ i , θ | ∂ ν | Zµ i , θ i ˜ ǫ ν ( θ ) p ǫ,θ + O (cid:18) ǫ p ǫ,θ (cid:19) (12) | Zµ ie , θ, ǫ i = | Zµ ie , θ i− X j = i h Zµ je , θ | ∂ ν | Zµ ie , θ i ˜ ǫ ν ( θ )1 − e ıp ǫ,θ ( χ je − χ ie ) | Zµ je , θ i + O (cid:18) ǫ p ǫ,θ (cid:19) (13)or conversely χ ie = χ ie,ǫ ( θ ) − ı h Zµ i , θ, ǫ | ∂ ν | Zµ i , θ, ǫ i ˜ ǫ ν ( θ ) p ǫ,θ + O (cid:18) ǫ p ǫ,θ (cid:19) (14) | Zµ ie , θ i = | Zµ ie , θ, ǫ i + X j = i h Zµ je , θ, ǫ | ∂ ν | Zµ ie , θ, ǫ i ˜ ǫ ν ( θ )1 − e ıp ǫ,θ ( χ je,ǫ ( θ ) − χ ie,ǫ ( θ )) | Zµ je , θ, ǫ i + O ( ǫ ) (15) III. PHYSICAL MEANINGS OF THEEFFECTIVE HAMILTONIANA. Approximate first recurrence Hamiltonian
In this section we want to exhibit concrete expressionsfor H effǫ ( θ ).
1. Low frequency case
Firstly we consider the low frequency regime where ω ≪ ω (there are a lot of Rabi oscillations during asampling period). This regime is consistent only with akicked quantum system where the sampling period is thekick period. We have U ( θ p ) ...U ( θ ) = e − ı ω ω ˆ H e − ıV p ...e − ı ω ω ˆ H e − ıV (16)= e − ı ( p +1) ω ω ˆ H e − ı ˜ V p ...e − ı ˜ V (17) with V n ≡ V ( θ n ) and ˜ V n ≡ e ın ω ω ˆ H V n e − ın ω ω ˆ H . V ( θ ) be-ing supposed bounded, we have k V ( θ ) k ≪ ω ω . It followsthat U ( θ p ) ...U ( θ )= e − ı ( p +1) ω ω ˆ H e − ı P pn =0 ˜ V n + O ( p k V k ) (18)= e − ı ( p +1) ω ω ˆ H − ı P pn =0 f − ı ( p +1) ω ω H [ ˜ V n ]+ O (cid:16) pf ( ıp ω ω δ ) k V k (cid:17) (19)where f X [ Y ] = f (ad X )[ Y ] with f ( x ) = x − e − x andad X [ Y ] = [ X, Y ], see appendix A concerning the Baker-Campbell-Hausdorff formula. δ is the gap between eigen-values of ˆ H which maximizes f ( ıp ω ω δ ). By applying thisresult to the definition of H effǫ eq. (5) we find H effǫ ( θ ) = ω ω ˆ H + 1 p ǫ,θ p ǫ,θ − X n =0 f − ıp ǫ,θ ω ω ˆ H [ ˜ V n ( θ )]+ O (cid:18) f ( ıp ǫ,θ ω ω δ ) k V k (cid:19) (20)where ˜ V n ( θ ) = e ın ω ω ˆ H V ( ϕ n ( θ )) e − ın ω ω ˆ H . To interpretethis formula, we can re-express it in the eigenbasis of ˆ H ( ˆ H | i i = λ i | i i ): H effǫ ( θ ) = X i (cid:18) ω ω λ i + h i | ¯ V θ | i i (cid:19) | i ih i | + X i,j = i f (cid:18) − ıp ǫ,θ ω ω ( λ i − λ j ) (cid:19) h i | ¯ V θ | j i| i ih j | + O (cid:18) f ( ıp ǫ,θ ω ω δ ) k V k (cid:19) (21)where ¯ V θ = p ǫ,θ P p ǫ,θ − n =0 ˜ V n ( θ ) is the average of theinteraction along Orb( θ ). We see that the effectiveHamiltonian corresponds to the free Hamiltonian ω ω ˆ H with its energies perturbed by the average interac-tion. This one induces also couplings between the freeenergy states which have magnitudes proportional to | f (cid:16) − ıp ǫ,θ ω ω ( λ i − λ j ) (cid:17) | . The function x
7→ | f ( − ıx ) | is plotted fig. 1. Since p ǫ,θ ω ω is large, excepted ifmax i,j = i | λ i − λ j | is realy very small, these couplings arestrong. Moreover we have resonances if p ǫ,θ ω ω | λ i − λ j | ∈ π N ∗ . Due to these resonances, the behavior of H effǫ ( θ )will be strongly sensitive to the value of ω ω .
2. High frequency case
Now we consider the high frequency regime where ω ≫ ω (there are a lot of samplings by the discretedescription during one Rabi oscillation). This is theonly one regime for a time discretization of a dynamics | f (- (cid:1) x )| x FIG. 1: Plot of the function x (cid:12)(cid:12)(cid:12) − ıx − e ıx (cid:12)(cid:12)(cid:12) appearing in thedevelopments by the Baker-Campbell-Hausdorff formula (seeappendix A). governed by a continuous time-dependent Hamiltonianwhere the sampling period is the discretization step. Thisregime can be also be consistent with a kicked quantumsystem. We consider several subcases depending on thebehaviour of V ( θ ). a. Case 1: V ( θ ) ∼ O ( ω /ω ) We can compute U ( θ p ) ...U ( θ ) = e − ı ω ω ˆ H e − ıV p ...e − ı ω ω ˆ H e − ıV by using theBaker-Campbell-Hausdorff formula at the first order. Wehave then H effǫ ( θ ) = ω ω ˆ H + 1 p ǫ,θ p ǫ,θ − X n =0 V ( ϕ n ( θ )) + O (cid:18) ω ω (cid:19) ! (22)In that case the effective Hamiltonian is just the sumof free Hamiltonian and the average interaction. b. Case 2: [ V ( θ ) , V ( θ ′ )] = 0 We suppose that[ V ( θ ) , V ( θ )] = 0, ∀ θ , θ ∈ Orb( θ ). We have then byusing the Baker-Campbell-Hausdorff formula explainedappendix A: U ( θ p ) ...U ( θ )= e − ı ω ω ˆ H e − ıV p ...e − ı ω ω ˆ H e − ıV (23)= e − ı P pn =0 V n e − ı ω ω P pn =0 ˜ H n + O (cid:18) p ω ω (cid:19) (24)= e − ı P pn =0 V n − ı ω ω P pn =0 f − ı P pq =0 Vq [ ˜ H n ]+ O (cid:18) pf ( ıpδ ) ω ω (cid:19) (25)where ˜ H n = e ı P nq =0 V q ˆ He − ı P nq =0 V q . δ is the gap betweeneigenvalues of the average interaction operator whichmaximizes f ( ıpδ ). By applying this result to the defi- nition of H effǫ eq. (5) we find H effǫ ( θ ) = 1 p ǫ,θ p ǫ,θ − X n =0 V ( ϕ n ( θ ))+ ω ω p ǫ,θ p ǫ,θ − X n =0 f − ı P pǫ,θ − q =0 V ( ϕ q ( θ )) [ ˜ H n ( θ )]+ O f ( ıp ǫ,θ δ ) (cid:18) ω ω (cid:19) ! (26)where ˜ H n ( θ ) = e ı P nq =0 V ( ϕ q ( θ )) ˆ He − ı P nq =0 V ( ϕ q ( θ )) . c. Case 3: V ( θ ) = v ( θ ) + W ( θ ) with [ v ( θ ) , v ( θ ′ )] = 0 and W ( θ ) ∼ O ( ω /ω ) This case is the combinationof the two previous ones, with V ( θ ) = v ( θ ) + W ( θ ),where W ( θ ) ∼ O ( ω /ω ) and [ v ( θ ) , v ( θ )] = 0, ∀ θ , θ ∈ Orb( θ ). By using the Baker-Campbell-Hausdorff formulaexplained appendix A, we have e − ı ω ω ˆ H e − ı ( v + W ) = e − ı ω ω ˆ H e − ıf − − ıv [ W ]+ O (cid:18) ω ω (cid:19) e − ıv (27)= e − ı ω ω ˆ H − ıf − − ıv [ W ]+ O (cid:18) ω ω (cid:19) e − ıv (28)We set K = ω ω ˆ H + f − − ıv [ W ]. U ( θ p ) ...U ( θ )= e − ıK p + O (cid:18) ω ω (cid:19) e − ıv p ...e − ıK + O (cid:18) ω ω (cid:19) e − ıv (29)= e − ı P pn =0 v n e − ı P pn =0 ˜ K n + O (cid:18) p ω ω (cid:19) (30)= e − ı P pn =0 v n − ı P pn =0 f − ı P pq =0 vq [ ˜ K n ]+ O (cid:18) pf ( ıpδ ) ω ω (cid:19) (31)with ˜ K n = e ı P nq =0 v q K n e − ı P nq =0 v q . δ is the gap betweeneigenvalues of the average interaction operator whichmaximizes f ( ıpδ ). By applying this result to the defi-nition of H effǫ eq. (5) we find H effǫ ( θ )= 1 p ǫ,θ p ǫ,θ − X n =0 v ( ϕ n ( θ ))+ 1 p ǫ,θ p ǫ,θ − X n =0 f − ı P pǫ,θ − q =0 v ( ϕ q ( θ )) [ ˜ K n ( θ )]+ O f ( ıp ǫ,θ δ ) (cid:18) ω ω (cid:19) ! (32)with ˜ K n ( θ ) = ω ω e ı P nq =0 v ( ϕ q ( θ )) ˆ He − ı P nq =0 v ( ϕ q ( θ )) + f − − ıv ( ϕ n ( θ )) h e ı P nq =0 v ( ϕ q ( θ )) W ( ϕ n ( θ )) e − ı P nq =0 v ( ϕ q ( θ )) i . d. Interpretation We consider here only the case 2,the case 1 being obvious and the case 3 being the su-perposition of the two first cases. Since we have a lotof sampling periods (a lot of short interactions) duringa Rabi oscillation, the dynamics is dominated by the in-teraction operator. Let ¯ V θ = p ǫ,θ P p ǫ,θ − n =0 V ( ϕ n ( θ )) bethe average of the interaction along Orb( θ ). Let ( | ¯ a i ) a be the eigenbasis of ¯ V θ ( ¯ V θ | ¯ a i = ν a | ¯ a i ). Onto this basis,the effective Hamiltonian can be expressed as: H effǫ ( θ ) = X a ( ν a + ω ω h ¯ a | ˆ H | ¯ a i ) | ¯ a ih ¯ a | + ω ω X a,b = a f ( − ip ǫ,θ ( ν a − ν b )) h ¯ a | ˆ H | ¯ b i| ¯ a ih ¯ b | + O f ( ıp ǫ,θ δ ) (cid:18) ω ω (cid:19) ! (33)To understand this formula, it is instructive to con-sider the case where V ( θ ) is a kick operator of theform V ( θ ) = λP ( θ ) where λ is the kick strengh and P ( θ ) = | w ( θ ) ih w ( θ ) | is a rank-1 projection. This meansthat the interaction consists to kick the quantum systemin the “direction” | w ( θ ) i . For example, with a two-levelsystem, | w ( θ ) i as a point onto the Bloch sphere defines adirection in the 3D-space. If the system is a spin kickedby ultra-short magnetic pulses, the direction defined ontothe Bloch sphere is identified with the polarization direc-tion of the magnetic field. If ǫ is sufficiently small, wehave ¯ V θ ≃ λ Z Γ e P ( θ ) dµ ( θ ) = λµ (Γ e ) ρ e (34)(with θ ∈ Γ e ), where ρ e is a mixed state (a density ma-trix) corresponding to the average of P ( θ ) onto Γ e =Orb( θ ) endowed with the probability measure dµ ( θ ) µ (Γ e ) . Wehave then ν a = λµ (Γ e ) p a where { p a } = Sp( ρ e ) arethe probabilities to find the direction | ¯ a i in the statis-tical mixture ρ e . ¯ V θ can be then viewed as a kick ofstrenght λµ (Γ e ) in a direction randomly chosen in {| ¯ a i} a with the probability law { p a } . ˆ H induces perturba-tive corrections onto this probability law but it inducesalso quantum coherences of magnitudes proportional to | f ( − ip ǫ,θ ( ν a − ν b )) | , which are strong since p ǫ,θ is large.Anew, resonances occur if p ǫ,θ | ν a − ν b | ∈ π N ∗ . Notethat in the case of a kick operator with kick delays asviewed in the introduction, V ( θ ) depends on ω ω (by rela-tive phases in its representation on the eigenbasis of ˆ H ).The behavior of H effǫ ( θ ) will be also in this case stronglysensitive to the value of ω ω because of these resonances.
3. About the accuracy of the approximations
The formulae for H effǫ ( θ ) found in this section arevery rough approximations. We can only consider themfor qualitative discussions or for physical interperta-tions. We cannot use them in qualitative discussions,especially for numerical computations. The reason ofthis bad accuracy are the error magnitude of the orderof f ( ıp ǫ,θ δ ) (cid:16) ω ω (cid:17) . This one is reasonable only if thealmost-period p ǫ,θ is small (and out of the resonances associated with the poles of f ). But, with an almost-periodic or a chaotic dynamics, it is small only if ǫ islarge. In that case, it is the almost-periodicity assump-tion which is very rough. We have a small p ǫ,θ onlyfor strictly periodic dynamics or for dynamics extremelyclose to a periodic dynamics.To make numerical computations of the effective Hamil-tonians, it is more usefull to use directly the definition ofthe first recurrence Hamiltonian eq. 5 or to solve eq. 3for example by the method explained in appendix B. B. Expected behaviours with almost-periodicallydriven systems
An orbit Orb( θ ) is characterized by three quantities.The first one is its almost-period p ǫ,θ . The secondone is its mean diameter ∠ Orb( θ ), the characteristicdistance onto Γ between two opposite points of Orb( θ ).It measures the mean dispersion of Orb( θ ) onto Γ.And finally λ θ the Lyapunov exponent ([6]) which mea-sures the “chaoticity” of the dynamics starting in theneighbourhood of θ . In this section we want to presentexpected dynamical behaviors of almot-periodicallydriven quantum systems with respect to the values ofthese quantities, when the interaction V ( θ ) is perturba-tive.By eq. 5 the dynamics generated by H effǫ ( θ ) during p ǫ,θ steps is the same than U ( ϕ p ǫ,θ − ( θ )) ...U ( θ ) and then ∀ ψ ( k ψ k = 1) F str ( θ ) = |h ψ | e ı ( p ǫ,θ +1) H effǫ ( θ ) U p ǫ,θ ( θ ) | ψ i| (35)= 1 + O ( ǫ ) (36)where U n ( θ ) = U ( ϕ n ( θ )) ...U ( θ ). For a p -cyclic orbit, H eff ( θ ) governs the global regime (on the time scale of p steps) where the transient regime (with time scale lowerthan p steps) is erased. The stroboscopic dynamics gov-erned by H eff ( θ ) is the exact true dynamics of the quan-tum system with the stroboscopic period p . For almost-periodic orbit, we can then be interested by the strobo-scopic fidelity of the dynamics governed by H effǫ ( θ ): F strn ( θ ) = |h ψ | e ı ( np ǫ,θ +1) H effǫ ( θ ) U np ǫ,θ ( θ ) | ψ i| (37)As previously said, we have exactly F strn ( θ ) = 1( ∀ n ∈ N ) for a p -cyclic orbit. Under what condi-tions is F strn ( θ ) ≃ λ ( θ ) = 0, because if the flowis chaotic, due to the sensitivity to initial conditions([6]) Orb( ϕ p ǫ,θ ( θ )) exponentially separates from Orb( θ )as e nλ θ ǫ (even if k ϕ p ǫ,θ ( θ ) − θ k < ǫ ). In particular,we have p ǫ,ϕ pǫ,θ ( θ ) = p ǫ,θ : the recurrence of the flowin the neighbourhood of θ is erratic, the almost-perioddrastically change at each recurrence. We can thinkthat F strn ( θ ) decreases if ω ω increases (or equivalentlythe approximation F strn ( θ ) ≃ n if ω ω is larger). Indeed, as we can see it inthe low frequency regime ( ω ω ≫
1, eq. 21), the largefactors | f ( − ıp ǫ,θ ω ω ( λ i − λ j )) | reinforce the θ -dependentcouplings h i | ¯ V θ | j i which become non-perturbative. So,the small difference between h i | ¯ V θ | j i and h i | ¯ V ϕ pǫ,θ ( θ ) | j i will be amplified by these factors (especially close tothe resonances). This problem does not occur in thehigh frequency regime. Finally we can think that theapproximation F strn ( θ ) ≃ ω ω is better with not too large diameters ∠ Orb( θ ). Inthe high frequency regime, ¯ V θ − ¯ V ϕ npǫ,θ ( θ ) ∼ O ( µ ( S n ))where S n ⊂ Γ is the region delimited by Orb( θ ) andOrb( ϕ np ǫ,θ ( θ )). It is more probable that µ ( S n ) quicklybecomes large with n if ∠ Orb( θ ) is large.By eq. 5 and 3 we have |h Zµ ie , ϕ n +1 ( θ ) | U n ( θ ) | Zµ ie , θ i| = 1 (38)(with θ ∈ Γ e ). U n ( θ ) describes the complete dynamicswhereas e − ınH effǫ ( θ ) describes the global dynamics with-out the transient fluctuations occuring at time scale lowerthan the almost-period. As eigenvector of e − ıH eff ( θ ) ,the quasi-energy state | Zµ ie , θ i is then the steady stateof the global dynamics. Its evolution could be almoststeady with the fluctuations associated with the transientregime. Let the survival probability of the quasi-energystate be: P survn ( θ ) = |h Zµ ie , θ | U n ( θ ) | Zµ ie , θ i| (39)We have P survp ǫ,θ ( θ ) = 1 + O ( ǫ ) and P survnp ǫ,θ ( θ ) ≃ | ψ i = | Zµ ie , θ i ). The quasi-energy states are thenstates of the quantum system which are almost recur-rent. They are then very important as the cyclic quantumstates associated with the Floquet theory (which theyare a generalization) and are associated with some quan-tum phenomena as the quantum revivals [24, 25]. Butmoreover we can hope that P survn ( θ ) ≈ V ( ϕ n ( θ )) − ¯ V θ on | Zµ ie , θ i are small( ¯ V θ being the average interaction operator along Orb( θ )).This needs that ∠ Orb( θ ) ≪
1, because the variations of V ( ϕ n ( θ )) will be large if the orbit is large. We can thinkthat the assumption P survn ( θ ) ≈ p ǫ,θ is not large, in order to the duration of the transientregime be short. Moreover, if V ( θ ) depends on ω ω asfor a kick operator with kick delay (as explained in theintroduction), we must have ω ω V ( ϕ n ( θ )) induces strongdifference between V ( ϕ n ( θ )) and ¯ V θ . And finally, to have P survn ( θ ) ≈ n > p ǫ,θ , we need λ θ = 0 because ofthe sensitivity to initial conditions of the chaotic flowswhich implies that ¯ V ϕ pǫ,θ ( θ ) ¯ V θ . IV. ILLUSTRATION
In order to illustrate the concepts presented in thispaper, we consider the following driven quantum sys-tem: a two-level quantum system defined by the canon-ical basis ( | i , | i ) and the free Hamiltonian H = ~ ω | ih | (for example a -spin system with Zeeman ef-fect), kicked with a frequency ω following the kick op-erator V ( θ ) = λ | w ( θ , θ ) ih w ( θ , θ ) | with | w ( θ , θ ) i =cos θ | i + e ı ω ω θ sin θ | i . λ = 0 . θ is the angular kick delay, and ( θ , ω ω θ )defines the kick direction in a spherical coordinates withthe z -axis corresponding to the direction of the Zeemanmagnetic field. We have then U ( θ ) = e − ı ω ω ˆ H e − ıV ( θ ) (with ˆ H = 2 π | ih | ). Since V is proportional to a rank 1projection, it is easy to compute the matrix exponentials:we have in the basis ( | i , | i ): e − ı ω ω ˆ H = (cid:18) e − ı π ω ω (cid:19) (40) e − ıV ( θ ) = 1 + ( e − ıλ − × cos θ e ı ω ω θ sin(2 θ ) e − ı ω ω θ sin(2 θ ) sin θ (41)Remark : V ( θ ) = λe ı H ~ ω θ | w ( θ , ih w ( θ , | e − ı H ~ ω θ = λ | w ( θ , ih w ( θ , | + ı λθ ~ ω [ H , | w ( θ , ih w ( θ , | ] + O ( ω /ω ). In the high frequency regime with constant θ the system belongs to the case 3 viewed in sectionIII A 2 c.The phase space is Γ = T (the 2-torus generated by θ = ( θ , θ ) ∈ [0 , π ] ). We consider the uniform mea-sure onto T : dµ ( θ ) = dθ dθ (2 π ) (with T the Borelian σ -algebra). The flow ϕ ∈ Aut( T ) is then an invariantautomorphism of the 2-torus. We choose the Chirikovstandard map defined by: ϕ ( θ ) = (cid:18) θ + K sin( θ ) mod 2 πθ + θ + K sin( θ ) mod 2 π (cid:19) (42)with K = 2. The phase portrait of this flow is plottedfig. 2. We can distinguish three different areas. The firstone is the chaotic sea (in blue fig. 2) which is an ergodiccomponent Γ = Orb( θ ) associated with a chaotic orbit.The fixed point (0 ,
0) is embedded in this chaotic sea. Abig island of stability centered on the fixed point (0 , π )is constituted by quasi-periodic orbits. The irrationallyrelated frequencies of these orbits are numerous. We con-sider five ergodic components { Γ e = Orb( θ e ) } e =1 ,..., inthis island for the numerical study. And finally, we have adouble small island of stability with two connected com-ponents centered on ( π,
0) and ( π, π ) (( π, ⇆ ( π, π ) isa 2-cyclic orbit). We consider these two components aspart of a same island because the inner orbit jump froma component to the other one. We consider three ergodiccomponents { Γ e = Orb( θ e ) } e =6 , , in this island. We FIG. 2: Phase portrait of the Chirikov standard map onto thetorus T , with 9 orbits considered in the simulations.TABLE I: Properties of the orbits used in the dynamics, with ǫ = 10 − . The almost-period p ǫ,θ e does not depend on thechoice of θ e ∈ Γ e except for Γ . In some simulations we con-sider also ǫ = 10 − for Γ , in that case p ǫ,θ = 734. Themean diameter ∠ Orb( θ e ) has been estimated as the averagebetween the maximum and the minimum diameters of the al-most closed orbits in the islands of stability. Since the chaoticsea covers a large part of T its mean diameter is 2 π .e p ǫ,θ e ∠ Orb( θ e ) λ θ e region0 25801 2 π .
415 chaotic sea1 108 4 . . . . .
96 0 big island center6 26 2 0 double small island border7 430 2 0 double small island8 42 0 . have choosen an initial point θ e in each ergodic compo-nent to start the dynamics. Except for Γ , this choicehas no influence onto the results. The properties of thenine considered orbits are reported table I.In the numerical simulations, we can compute U effǫ ( θ ) = e − ıH effǫ ( θ ) by two manners. The first oneconsists to use eq. 5: U effǫ ( θ e ) = pǫ,θe q U ( ϕ p ǫ,θe − ( θ e )) ...U ( θ e ) (43)The second one consists to use the method presented inappendix B. The two ones provide very similar numericalresults.
50 100 150 200 250 300 n0.20.40.60.81.0 |< | U n ( )| > |< | e - i(cid:0)(cid:2)e(cid:3)(cid:4) ( θ ) | > FIG. 3: Comparision of the true dynamics |h ψ | U n ( θ ) | ψ i| and the effective dynamics |h ψ | e − ınH effǫ ( θ ) | ψ i| for the orbit e = 6 during 12 almost-periods, with ω ω = 3 . | ψ i = √ ( | i + | i ). A. Stroboscopic fidelity of the dynamics
As previously explained, the dynamics governed by H effǫ ( θ ) is the global dynamics without the fluctuationsof the transient regime with small time scale. Fig. 3 givesan illutration of this. In this example, the global dynam-ics is the envelope of the complete dynamics, but it isnot always the case (it is necessary to tune ω ω to havethis simple behavior). It is more interesting to study thestroboscopic fidelity eq. 37. As illustration, we can seefig. 4. In order to enlighten the efficiency of the effec-tive description for the stroboscopic dynamics we havecompute the avergage stroboscopic fidelity: F str ( θ ) = 1 N + 1 N X n =0 F strn ( θ ) (44)at short term ( N = 12 almost-periods) and at long term( N = 120 almost-periods), for the three regimes (low,medium and high frequency). Since we have shown thatthe behaviours are very sensitive to the value of ω ω wehave considered for each regime three different values ofthe period ratio. The results are presented tables II andIII. These results are in accordance with the discussionof the section III B. Except for the case of the chaoticorbit Orb( θ ), the average stroboscopic fidelity is largefor the high frequency regime whereas it is good in thelow frequency regime only at short term. Moreover theresults seem better in the double small island and in thecenter of the big island, confirming that with a too largeorbit diameter ∠ Orb( θ ) the stroboscopic fidelity is lower.
40 60 80 100 120 n0.20.40.60.81.0 |< ψ | U (cid:5)(cid:6) θ ( θ )| ψ > |< ψ | e - i ( (cid:7)(cid:8) θ + ) H (cid:9)(cid:10)(cid:11) (θ) | ψ > FIG. 4: Comparision of the true stroboscopic survival proba-bility |h ψ | U np ǫ,θ ( θ ) | ψ i| and the effective stroboscopic sur-vival probability |h ψ | e − ı ( np ǫ,θ +1) H effǫ ( θ ) | ψ i| for the orbit e = 5, with ω ω = 4 . | ψ i = √ ( | i + | i ). We seea small dephasing occuring with a large number of almost-periods. The average stroboscopic fidelity during 12 almost-periods is 99 .
7% whereas due to the dephasing, during 120almost-periods, it is only 93 . | ψ i = √ ( | i + | i ), forthe different orbits and for different values of ω ω . We writte inbold the good results ( ≥ ≥ ≥ < e High freq. ω ω ≪ ω ω ∼ ω ω ≫ √ .
03 0 . √ . . √ . . . % . % 79 .
1% 77 . . % 75 . . % 78 . . %1 % % . % % . % . % . % 83 . . %2 % % % . % . % . % . % . % . %3 % % % . % . % . % . % . % . %4 % % % % % . % . % . % . %5 % % % % . % . % . % . % . %6 % % % % . % % . % . % . %7 % % % % % % . % . % . %8 % % % % % % . % . % . % B. Almost steady states
By construction, we know that |h Zµ ie , ϕ n +1 ( θ ) | U n ( θ ) | Zµ i , θ i| = 1. We are in-terested by the survival probability P survn ( θ ) = |h Zµ ie , θ | U n ( θ ) | Zµ ie , θ i| , as for example fig. 5.As expected, the quasi-energy states presents quasi-recurrences at each almost-period (in the same conditionsthan the previous discussion about the stroboscopicfidelity). But we can see also, that the fluctuations dur-ing the transient regime (between two almost-periods)seems not too large. With different parameters, thequasi-energy state is even an almost steady state as we TABLE III: Same as table II but with averaging during 120almost-periods. Moreover for the orbit e = 0 the presentedresults correspond to ǫ = 10 − (whereas ǫ = 10 − for theother orbits). e High freq. ω ω ≪ ω ω ∼ ω ω ≫ √ .
03 0 . √ . . √ . . . % . % . % . % . % . % . % . % . %1 % % . % . % . % . % . % . % . %2 . % . % . % 79 . . % . % . % . % . %3 . % . % . % 88 . . % . % . % . % 80 . % % % . % . % 76 . . % . % . %5 . % . % . % . % . % . % . % . % . %6 . % . % . % . % . % . % . % . % . %7 % % % % . % % . % 76 . . %8 . % . % . % . % . % . % . % . % . % |< Z μ (cid:12)(cid:13), θ | U n ( θ )| (cid:14) μ (cid:15)(cid:16)(cid:17) θ > |< (cid:18) μ (cid:19)(cid:20)(cid:21) θ | U mp θ ( θ )| (cid:22) μ ie (cid:23) θ >
50 100 150 200 250 300 n0.20.40.60.81.0
FIG. 5: Survival probability of a quasi-energy state |h Zµ i , θ | U n ( θ ) | Zµ i , θ i| for the orbit e = 6 during 12almost-periods with ω ω = 3 .
4. The circles show the survivalprobabilities at each almost-period { mp ǫ,θ } m =0 ,..., . can see it fig. 6. To enlight this behaviour, we havecompute the average survival probability: P surv ( θ ) = 1 N + 1 N X n =0 P survn ( θ ) (45) |< (cid:24) μ (cid:25)(cid:26)(cid:27) θ | U n ( θ )| (cid:28) μ (cid:29)(cid:30)(cid:31) θ > |< μ !" θ | U mp θ ( θ )| $ μ ie % θ >
50 100 150 200 250 300 n0.20.40.60.81.0
FIG. 6: Same as fig. 5 but with ω ω = 0 . TABLE IV: Average survival probability of a quasi-energystate during 120 almost-periods for different orbits and dif-ferent values of ω ω . For the orbit e = 0 the presented resultscorrespond to ǫ = 10 − (whereas ǫ = 10 − for the other or-bits). We writte in bold the good results ( ≥ ≥ ≥ < e High freq. ω ω ≪ ω ω ∼ ω ω ≫ √ .
03 0 . √ . . √ . . . % . % . % . % . % . % . % . % . %1 . % . % 80 . . % . % . % . % . % 89 . %2 . % . % . % . % 87 .
7% 82 .
5% 80 . . % . %3 . % . % . % 88 . . % . % . % . % . %4 . % . % . % . % . % 84 . . % . % . %5 . % % % . % . % . % 87 . . % 75 . . % . % . % . % 88 . . % . % . % . %7 . % . % . % . % . % . % . % 80 . . %8 % % % % % . % . % . % . % |< & μ ’() θ | U n ( θ )| * μ +-. θ >
100 200 300 n0.20.40.60.81.0
FIG. 7: Survival probability of a quasi-energy state |h Zµ i , θ | U n ( θ ) | Zµ i , θ i| for the chaotic orbit e = 0 dur-ing 1 almost-period with ω ω = 0 . during N = 120 almost-periods for the three regimes(low, medium and high frequency). The results are pre-sented table IV. The steadiness of the quasi-energy stateseems better in the center of the islands, confirming that ∠ Orb( θ ) must be small. Moreover it seems better for or-bits with small almost-periods. Due to the dependencyto ω ω of V ( θ ), the steadiness of the quasi-energy state isvalid only for high and medium frequency regimes. Notethat the results do not depend on the time scale, wefind very close survival probabilities by averaging during12 almost-periods and by averaging during 120 almost-periods.For the chaotic orbit e = 0, we can have an almoststeadiness of the quasi-energy states only during the firstalmost-period (because of the sensitivity to initial condi-tions), and for ǫ not too small (because p ǫ,θ must not betoo large). We can see this with fig. 7 and table V. Butsince the Orb( θ ) covers a large part of T , the steadinessis middling for the chaotic orbit. TABLE V: Same as table IV but with the averaging during 1almost-period and with ǫ = 10 − . e High freq. ω ω ≪ ω ω ∼ ω ω ≫ √ .
03 0 . √ . . √ . .
50 89 .
7% 89 . . . % . % . % . % 75 . . % . % C. True steady states
The steadiness viewed in the previous section is justan approximation in some conditions. But we can useeq. 3 to built true steady states by considering a setof copies of the driven quantum system and not only asingle one. For example, consider Orb( θ ) and 26 copiesof the driven quantum system. We suppose that the copylabeled by ( m ) ( m ≤ ϕ m ( θ ) as initial conditionfor its kick system. We choose as initial states for thedifferent copies, the quasi-energy states: | ψ ( m )0 i = | Zµ i , ϕ m ( θ ) i (46)After one quick, we have U ( ϕ m ( θ )) | ψ ( m )0 i = e − ıχ i | Zµ i , ϕ m +1 ( θ ) i (47)Everything happens as if the copy ( m ) takes the place ofthe copy ( m + 1) (for m <
25) and as if the copy (25)takes the place of the copy (0) (since ϕ ( θ ) = θ + O ( ǫ )).The set of 26 copies is then unchanged by the dynamics(even if individually each copy changes). We have thena true steady state (up to an error of magnitude ǫ ) byconsidering the mixed state of the set of copies: ρ n = 126 X m =0 U n ( ϕ m ( θ )) | ψ ( m )0 ih ψ ( m )0 | U n ( ϕ m ( θ )) † (48)If the initial mixed state is the quasi-energy state ofthe orbit, ρ m does not evolve up to an error of mag-nitude ǫ . This reasoning can be applied to all orbit, andto many orbits together. For example, we consider thesmall island of stability into the chaotic sea. We con-sider N = 1391 copies of the quantum system, with kickinitial conditions corresponding to the points of Orb( θ ),Orb( θ ), Orb( θ ) and the 893 points of Orb( θ ) corre-sponding to the precision ǫ = 2 × − ( ǫ = 10 − forthe orbits in the small island). We consider the followingdensity matrix ρ n = 1 N N X m =1 U n ( θ ( m ) ) | ψ ( m )0 ih ψ ( m )0 | U n ( θ ( m ) ) † (49)where θ ( m ) is the kick initial condition of the copy ( m );with the two initial states: | ψ ( m )0 i = | Zµ ie m , θ ( m ) i (with θ ( m ) ∈ Orb( θ e m )), and | ψ ( m )0 i = | i (for comparison).A population and the coherence of ρ n are drawn fig. 8.As expected, ρ Zµ, is a steady states whereas any statepresents oscillations due to the alsmot-periodicities of thefour orbits.0 < | ρ n | >< | ρ Z μ , n | >
500 1000 1500 n0.20.40.60.81.0 |< | ρ n | >||< | ρ Z μ , n | >|
500 1000 1500 n0.20.40.60.81.0
FIG. 8: Population of the state | i (up) and coherence (down)of the mixed states ρ ,n (with ρ , = | ih | ) and ρ Zµ,n (with ρ Zµ, = N P Nm =1 | Zµ ie ( m ) , θ ( m ) ih Zµ ie ( m ) , θ ( m ) | ) during2 almost-periods in the chaotic sea (with ǫ = 2 × − ) with ω ω = 3 . V. CONCLUSION
The effective Hamiltonian defined by the Schr¨odinger-Koopman approach permits to extend the approach ofthe Floquet effective Hamiltonian to quasi-periodicallydriven systems without knowledge of the different fre-quencies of these systems. It can be also applied to chaot-ically driven systems, whereas in this case it is diffult toexploit the dynamical behaviour (but this a direct con-sequence of the definition of chaos).
Acknowledgments
The author acknowledge support from ISITEBourgogne-Franche-Comt´e (contract ANR-15-IDEX-0003) under grants from I-QUINS and GNETWORKSprojects, and support from the R´egion Bourgogne-Franche-Comt´e under grants from APEX project.Numerical computations have been executed on com-puters of the Utinam Institute supported by the R´egionBourgogne-Franche-Comt´e and the Institut des Sciencesde l’Univers (INSU).The author thanks Professor Atushi Tanaka for usefulexchanges. [1] G. Floquet,
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J. Phys. A: Math. Gen. , R105 (2003). Appendix A: BCH formula
For some computations, we consider the following ver-sion of the Baker-Campbell-Hausdorff formula [26]: e X e Y = e X + f X [ Y ]+ O ( k Y k ) (A1)where f X = f (ad X ) with ad X [ Y ] = [ X, Y ] (ad X is anoperator onto the bounded operator space B ( H )) and f ( x ) = x − e − x ( f (ad X ) is defined by functional calculus[27]). We have also e Y e X = ( e − X e − Y ) − = e X + f − X [ Y ]+ O ( k Y k ) (A2)and conversely e X + Y = e f − − X [ Y ]+ O ( k Y k ) e X (A3)To compute f X we have two possibilities. The first oneconsists to use the Taylor serie of f : f (ad X ) = 1 − + ∞ X n =1 (1 − Ad( e X )) n n ( n + 1) (A4)where Ad( g ) Y = gY g − . The second one consists toconsider the Hilbert-Schmidt space of the operators of H ([27]). To simplify the discussion here, we supposethat H is finite dimensional (dim H = N ). The Hilbert-Schmidt space can be identified to HS = C N . TheHilbert-Schmidt representation of an operator Y ∈ L ( H )is then Y = Y ... Y N ... . . . ... Y N ... Y NN → | Y ii = Y ... Y N Y ... Y NN (A5)for a matrix representation in the choosen orthonormalbasis. The inner product of HS is hh Z | Y ii = tr( Z † Y ).We have moreover | XY ii = X ⊗ N | Y ii (A6) | Y X ii = 1 N ⊗ X T | Y ii (A7)It follows that ad X = X ⊗ N − N ⊗ X T (where T denotes the transposition). ad X can be then viewedas a N -order square matrix. Let { x n } n =1 ,...,N be the spectrum of ad X and P be such that P − ad X P = diag( x , ..., x N ) (the diagonal matrixhaving ( x , ..., x N ) on the diagonal). We have then f (ad X ) = P diag( f ( x ) , ..., f ( x N )) P − .For example to find eq. (10) we start from e A eν ˜ ǫ ν + O ( ǫ ) e − ıpH eff = e − ıpH eff + f ıpHeff [ A eν ]˜ ǫ ν + O ( ǫ ) . H eff = diag( χ e , ..., χ eN ) in the basis ( | Zµ i , θ i ) i =1 ,...,N ,and then ad H eff = diag(0 , χ e − χ e , ..., χ e − χ eN , χ e − χ e , , ..., χ e − χ eN , ..., χ eN − χ eN − , f (ad ıpH eff ) = diag ( j = i ıp ( χ ei − χ ej )1 − e − ıp ( χei − χej ) if j = i ! i,j (A8)(lim x → f ( x ) = 1). Appendix B: Direct computation of the effectiveHamiltonian