Quantum-Classical Correspondence of Shortcuts to Adiabaticity”

Home , Adiabatic theorem

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The commutator in the quantum system is replaced by and the potential satisfies the dispersionless KdV equa- the Poisson bracket in the classical limit. We note that tion ξ is not uniquely determined from Eq. (3) [6]. This arbi- ∂U(x, α) ∂U(x, α) + U(x, α) =0. (9) trariness is discussed in the formulation discussed below. ∂α ∂x It is a simple task to show that the adiabatic theorem This equation can be obtained by removing the third holds in this counterdiabatic driving. Taking the deriva- derivative term, the dispersion term, in the KdV equa- tive of J in Eq. (1) with respect to α and using Eq. (3), tion [18]. The form of the potential is different between we obtain the quantum and classical STA for the same form of the dJ counterdiabatic term in Eq. (8). This property is con- = dxdp ξ,H δ (E H ) . (4) dα − { 0} 0 − 0 trasted with that in the scale-invariant system where the Z quantum and classical STA give the same result. In the This integral is evaluated by the surface contributions same way, we can find the correspondence between the and goes to zero in systems with a smooth trajectory as KdV and dispersionless KdV hierarchies at each odd or- we see in the following examples (Section A of Ref. [10]). der in p. Although it is a difficult problem to implement This result means that J is determined by the initial con- the higher order terms in an actual experiment, some dition and is independent of t. The proof clearly indicates deformation of the counterdiabatic term is possible to that the counterdiabatic term is introduced so that the represent the term by a potential function [12]. adiabatic theorem holds exactly. We note that the time Before studying the solutions of the dispersionless KdV variable t does not appear in Eq. (3) explicitly, which equation, we reformulate the classical STA by using the allows us to handle the adiabatic invariant defined geo- Hamilton–Jacobi theory. The standard classical adia- metrically in phase space. This result is the same as that batic theorem is described in periodic systems since the in Ref. [6]. validity of the approximation is written in terms of the The solution of Eq. (3) can be studied systematically period T as T α/α˙ 1. Although the adiabatic in- as was done in Ref. [12] for the quantum system. As an variant is treated| in|ergodic ≪ systems [19–21], its gener- example, we set the reference Hamiltonian in a standard alization is a delicate and difficult problem. To find the form quantum-classical correspondence of the adiabatic systems, we need to extend the formulation to general sys- 2 H0(x, p, α(t)) = p + U(x, α(t)). (5) tems. This can be done by the Hamilton–Jacobi theory. The adiabatic invariant is related to the action Then we show that the solution of Eq. (3) is given by the t t S = dt′ L = dt′ (xp ˙ H). This is a function of x(t), dispersionless Korteweg–de Vries (KdV) hierarchy [13, 0 0 t, and the whole history− of α(t): S = S(x(t), t, α(t) ). 14] (Section B of Ref. [10]). The corresponding method TheR property thatR S is independent of the history{ of x}(t) in the quantum STA was developed in Ref. [12] and the is shown by using the equation of motion. In the coun- KdV hierarchy was found. The reference Hamiltonian terdiabatic driving, trajectories in phase space are deter- and the counterdiabatic term represent the Lax pair in mined from Eq. (2) and the Hamiltonian satisfies Eq. (3) the corresponding nonlinear integrable system [15]. The that has no explicit time dependence. These properties dispersionless KdV hierarchy is known as the “classical” imply that the dynamics is characterized at each t, ir- limit of the KdV hierarchy [16]. respective of past history. By considering the variation When ξ(x, p, α) is linear in p, the potential is of the α(t′) α(t′)+ δα(t′) of S at an arbitrary t′ between 0 form and t,→ and using the equation of motion, we obtain the 1 x x0(t) deviation of the action as (Section C of Ref. [10]) U(x, α(t)) = 2 u − , (6) γ (t) γ(t) t ∂H δS = dt′ δα(t′) 0 + ξ,H [δα(t′)ξ]t . − ∂α { 0} − 0 where u is an arbitrary function, and α represents both Z0 x0 and γ, the former represents a translation and the (10) latter a dilation. The counterdiabatic term is given by We use Eq. (3) to find that the function defined as t γ˙ ′ ′ HCD =x ˙ 0p + (x x0)p. (7) Ω= S(x(t), t, α(t) )+ dt E (α(t )) (11) γ − { } 0 Z0 This is known as the scale-invariant driving and was is independent of the history of α(t). This function is found in previous works [6, 7, 17]. A new result is ob- a simple generalization of the Hamilton’s characteristic tained when we set that ξ is third order in p. We find function, or the abbreviated action, which is usually de- that the counterdiabatic term is given by fined for constant E0 by the Legendre transformation. It satisfies 2 H =αξ ˙ =α ˙ pU(x, α)+ p3 , (8) ∂Ω ∂Ω CD 3 = p(x, α), = ξ(x, p(x, α), α). (12) ∂x ∂α − 3

The momentum p is represented as a function of x and its definition shows that the derivative of Ω with respect α as we see from Eq. (2). These derivatives have no to ǫ0 gives the relation explicit t dependence. This implies that Ω is a function ∂Ω t dt′ of x and α, and not of t, just like the property of the = = τ(t), (19) ∂ǫ γ2(t′) Legendre transformation. The explicit t dependence of Ω 0 Z0 can be removed by adding a time-dependent term, which where the last equality is the definition of the rescaled does not change the trajectory, to the Hamiltonian. As time τ(t). We conclude that Ω as a function of t and α a result we can set Ω = Ω(x(t), α(t)). in the scale-invariant system satisfies the relation ∂S The Hamilton–Jacobi equation is given by ∂t + H =0 ∂S Ω(x(t; α(t)), α(t)) = Ω(x(τ(t); α(0)), α(0)). (20) with p = ∂x . We substitute Ω to this equation. Noting that the time derivative is replaced in the present system The left-hand side represents Ω at t obtained in the pro- with ∂ ∂ +α∂ ˙ , we find that the Hamilton–Jacobi t → t α tocol α(t) and the right-hand side represents Ω at τ(t) equation for the counterdiabatic driving is decomposed in the fixed protocol α(0). When the latter system gives as a closed trajectory, Ω at the period is equal to the adi- ∂Ω(x, α) abatic invariant in Eq. (1) and is written as Ω = pdx H0 x, p = , α = E0(α), (13) (Section D of Ref. [10] for an example of the harmonic os- ∂x cillator). This relation shows that the adiabatic invariantH ∂Ω(x, α) ∂Ω(x, α) ξ x, p = , α = . (14) is directly obtained from the corresponding nonperiodic ∂x − ∂α trajectory. For nonscale-invariant systems, Eq. (20) is not satis- These equations are solved as a function of x and α, fied. We treat the dispersionless KdV system as an ex- and have no explicit time dependence. We note that ample. H is given by Eq. (5) and the potential U satis- the counterdiabatic term is given by H =αξ ˙ (x, p, α). 0 CD fies the dispersionless KdV equation (9). We can rederive Thus the counterdiabatic driving is characterized by Ω. Eq. (8) in the present formalism by assuming that E0 is The definition of the action shows that Ω is written as 2 constant. Substituting U = E0 (∂xΩ) to Eq. (9), we t find (Section C of Ref. [10]) − Ω(x(t), α(t)) = dt′ (xp ˙ (x, α) αξ˙ (x, p(x, α), α)) − 3 Z0 ∂Ω ∂Ω 2 ∂Ω t + U + =0. (21) ′ ∂H0 ∂ξ ∂α ∂x 3 ∂x = dt p +α ˙ p ξ , (15) 0 ∂p ∂p − Z This equation shows that the counterdiabatic term ob- where we use the equation of motion in the second line. tained from Eq. (14) is given by Eq. (8). This expression shows that Ω, as a function of t and α(t), Equation (9) is solved by the hodograph method as satisfies the following equations: U(x, α)= f (x αU(x, α)) , (22) − ∂Ω ∂H ∂Ω ∂ξ = 0 p, = p ξ. (16) where f is an arbitrary function [22, 23]. As a simple ex- ∂t ∂p ∂α ∂p − 2 α t ample we consider the case f(x)= x . Then, by solving Second equation states that Ω is independent of α(t) the quadratic equation, we obtain when the counterdiabatic term is linear in p. This is 2αx +1 √4αx +1 U(x, α)= − . (23) the case of the scale-invariant driving where H0 is of the 2α2 form (5) with (6). The Hamilton–Jacobi equation (13) We take the negative branch of the equation so that the reads trajectories are bound. This potential is well-defined for 1 ∂Ω 2 1 x x x> and we set the parameters so that this relation + u − 0 = E (α), (17) − 4α ∂x γ2 γ 0 is satisfied throughout the time evolution. By taking the limit α 0, we have the harmonic oscillator U(x, 0) = 2 → and we find that E0 and Ω take the form E0(α(t)) = x . ǫ /γ2(t) and Ω = Ω((x x )/γ), respectively. By setting 0 − 0 γ(0) = 1, we can regard ǫ0 as the initial energy at t = 0. The counterdiabatic term is calculated as The equation of motion is solved numerically ∂Ω ∂Ω γ˙ and we show the trajectories in phase space and HCD = x˙ 0 γ˙ =x ˙ 0p + (x x0)p, (18) Ω(x(t; α(t)), α(t)) in Figs. 1 and 2, respectively. We use − ∂x0 − ∂γ γ − the protocol α(t) = α(0) 1 sin2(st) . We see from where we use the property that the derivatives of Ω with Fig. 2 that Eq. (20) is not satisfied− since Ω is not necessar- respect to x0 and γ are translated to that with x in the ily a monotone increasing function and the time rescal- present system. Ω is also a function of E0(α(0)) = ǫ0 and ing cannot give the result at s = 0. The counterdiabatic 4

the proof. See Sec. E of Ref. [10]. The period T can also be calculated there and we find T = π in the present case. Thus the adiabatic invariant can be calculated directly from the real nonperiodic trajectory. The Hamilton–Jacobi theory of the classical STA makes a link between the classical and quantum systems. In scale-invariant Hamiltonian in Eq. (5) with (6), the counterdiabatic term in classical system becomes the same as that in quantum system if we use the sym- metrization as px (ˆpxˆ +x ˆpˆ)/2. In the KdV systems, the form of the Hamiltonian→ is unchanged but the potential function U satisfies an equation that is different from Eq. (9). In the quantum system the state is described by the wavefunction which satisfies the Schrödinger equation

FIG. 1. Trajectories in phase space for the dispersionless KdV ∂ i¯h ψ(x, t) = (Hˆ0 + HˆCD)ψ(x, t). (25) system in Eq. (5) with (23). We take the initial condition as ∂t α(0) = 1/4, E0(α(0)) = 1, and (x(0),p(0)) = (−3/4, 0). The solid line represents s = 0.0, the dashed line s = 4.5, and the For example, we consider the Hamiltonian (5) and (8) dotted line s = 9.5. with the replacement pU (ˆpUˆ + Uˆpˆ)/2. This setting gives a counterdiabatic driving→ when the potential satis- ﬁes the KdV equation [18]

∂U(x, α) ∂U(x, α) ∂3U(x, α) + U(x, α) + =0. (26) ∂α ∂x ∂x3 By substituting the wavefunction

ψ(x, t)=e−iE0t/h¯ A(x, α(t))eiΩ(x,α(t))/h¯ , (27)

where A and Ω are real functions, to the Schrödinger equation and taking the limit ¯h 0, we can obtain Eq. (21). Thus the classical STA→ is obtained from the quantum version by taking the classical limit. To find this relation, it is crucial to develop the classical STA by the Hamilton–Jacobi theory as we discuss in this letter. In the quantum STA, the wavefunction is given by the adiabatic state of H0(ˆx, p,ˆ α(t)). By using the instanta- FIG. 2. The characteristic function Ω(x(t; α(t)), α(t)) in the neous eigenstate of H , n(α(t)) , and the instantaneous case of Fig. 1. The circle denotes the point (T, Ω(T ))=(π,π) 0 energy of H , E (α(t)),| we cani write the wave function where T denotes the period of the closed trajectory for a con- 0 n stant α. Inset: The potential function U(x, α) in Eq. (23). as i t ψ(ad)(t) = exp dt′ E (α(t′)) ψ˜ (α(t)) , | n i −¯h n | n i driving determines Ω instantaneously as we showed in Z0 the above analysis. This can be confirmed numerically (28) in the present system. In Figs. 1 and 2, we consider an where oscillating α(t) with the parameter s and we see that Ω with s is equal to Ω with s = 0 when t = 2πn/s with α(t) ˜ ′ ′ ∂ ′ integer n. We also find that Ω is equal to the adiabatic ψn(α(t)) = exp dα n(α ) n(α ) | i − h |∂α′ | i invariant J when t is equal to the period of the closed Zα(0) ! trajectory which is defined for a fixed α: n(α(t)) . (29) ×| i

Ω(x(T ; α(T )), α(T )) = J. (24) The point is that ψ˜n(α(t)) , the adiabatic state without the dynamical phase,| is writteni in terms of α, not of t. This relation holds for an arbitrary choice of α(t) and can We deﬁne the unitary operator Vˆ as be proved by assuming that T is independent of the initial energy. We use the theory of action-angle variables for ψ˜ (α(t)) = Vˆ (α(t)) ψ˜ (α(0)) . (30) | n i | n i 5

This operator was introduced to develop the path integral atom cooling in harmonic traps: Shortcut to adiabaticity, formulation of the adiabatic theorem [24]. Then we can Phys. Rev. Lett. 104, 063002 (2010). show that the counterdiabatic term is written as [5] E. Torrontegui, S. Ibáñez, S. Mart´ınez-Garaot, M. Mod- ugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, ∂Vˆ (α) X. Chen, and J. G. Muga, Shortcuts to adiabaticity, Adv. Hˆ (t)= i¯hα˙ (t) Vˆ †(α). (31) CD ∂α At. Mol. Opt. Phys. 62, 117 (2013). [6] C. Jarzynski, Generating shortcuts to adiabaticity in We note that, in this expression, the unitary operator Vˆ quantum and classical dynamics, Phys. Rev. A 88, can be replaced by the total time evolution operator to 040101(R) (2013). find the formula by Demirplak and Rice [1]. By using Vˆ , [7] S. Deffner, C. Jarzynski, and A. del Campo, Classical we can write the formula of Hˆ in a more suggestive and quantum shortcuts to adiabaticity for scale-invariant CD driving, Phys. Rev. X 4, 021013 (2014). form. We write Vˆ as [8] A. Patra and C. Jarzynski, Classical and quantum short- i cuts to adiabaticity in a tilted piston, J. Phys. Chem. B Vˆ (α) = exp Ω(ˆ α) , (32) ¯h 10.1021/acs.jpcb.6b08769 (2016). [9] C. Jarzynski, S. Deffner, A. Patra, and Y. Suba¸si, and this operator Ωˆ is the quantum analogue of the char- Fast forward to the classical adiabatic invariant, arXiv:1611.06437 (2016). acteristic function Ω(x, α) in the classical system. For [10] See Supplemental Material for the detailed calculations. example, in the scale-invariant Hamiltonian (5) with (6), [11] H. R. Lewis and W. B. Riesenfeld, An exact quantum Ωˆ is given by theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic Ω(ˆ α(t)) = pˆ(x (t) x (0)) 10 − 0 − 0 field, J. Math. Phys. , 1458 (1969). 1 γ(t) [12] M. Okuyama and K. Takahashi, From classical nonlinear (ˆp(ˆx x (t))+(ˆx x (t))ˆp) ln . (33) −2 − 0 − 0 γ(0) integrable systems to quantum shortcuts to adiabaticity, Phys. Rev. Lett. 117, 070401 (2016). The counterdiabatic term in this case is written as [13] D. R. Lebedev, Conservation laws and Lax representation of Benney’s long wave equations, Phys. Lett. A 74, 154 ∂Ω(ˆ α) (1979). Hˆ (t)= α˙ (t) . (34) CD − ∂α [14] V. E. Zakharov, Benney equations and quasiclassical approximation in the method of the inverse problem, Func- This expression formally coincides with Eq. (14). tional Analysis and Its Applications 14, 89 (1980). To summarize, we have developed the classical STA by [15] P. D. Lax, Integrals of nonlinear equations of evolution using the Hamilton–Jacobi theory. The system is charac- and solitary waves, Commun. Pure Appl. Math. 21, 467 terized by the generalized characteristic function Ω(x, α) (1968). and the counterdiabatic term is obtained from this func- [16] K. Takasaki and T. Takabe, Quasi-classical limit of KP hierarchy, W-symmetries and free fermions tion. The equation for the counterdiabatic term can be , Zap. Nauchn. Sem. POMI 235, 295 (1996) [J. Math. Sci. 94, studied systematically and is solvable when the system 1635 (1999)]. falls in the dispersionless KdV hierarchy. Our formula- [17] A. del Campo, Shortcuts to adiabaticity by counterdia- tion also gives a relation to the adiabatic theorem. We batic driving, Phys. Rev. Lett. 111, 100502 (2013). can also show that the classical STA is reduced from the [18] D. J. Korteweg and G. de Vries, On the change of form quantum STA by taking the standard semiclassical ap- of long waves advancing in a rectangular canal, and on a 39 proximation. new type of long stationary waves, Philos. Mag. , 422 We acknowledge financial support from the ImPACT (1895). [19] P. Hertz, Uber¨ die mechanischen Grundlagen der Ther- Program of the Council for Science, Technology, and In- modynamik, Ann. Phys. (Leipzig) 338, 225 (1910). novation, Cabinet Office, Government of Japan. K.T. [20] P. Hertz, Uber¨ die mechanischen Grundlagen der Ther- was supported by JSPS KAKENHI Grant No. 26400385. modynamik, Ann. Phys. (Leipzig) 338, 537 (1910). [21] E. Ott, Goodness of ergodic adiabatic invariants Phys. Rev. Lett. 42, 1628 (1979). [22] Y. Kodama, A method for solving the dispersionless KP equation and its exact solutions, Phys. Lett. A 129, 223 [1] M. Demirplak and S. A. Rice, Adiabatic population trans- (1988). fer with control fields, J. Phys. Chem. A 107, 9937 [23] Y. Kodama and J. Gibbons, A method for solving the (2003). dispersionless KP hierarchy and its exact solutions II, [2] M. Demirplak and S. A. Rice, Assisted adiabatic passage Phys. Lett. A 135, 167 (1989). revisited, J. Phys. Chem. B 109, 6838 (2005). [24] T. Kashiwa, S. Nima, and S. Sakoda, Berry’s phase and [3] M. V. Berry, Transitionless quantum driving, J. Phys. A euclidean path integral, Ann. Phys. 220, 248 (1992). 42, 365303 (2009). [4] X. Chen, A. Ruschhaupt, S. Schmidt, A. del Campo, D. Guéry-Odelin, and J. G. Muga, Fast optimal frictionless 6

Supplemental Material for “Quantum-Classical Correspondence of Shortcuts to Adiabaticity”

Manaka Okuyama and Kazutaka Takahashi

A. CLASSICAL STA This equation shows that ξ is odd order in p. We note that the additional relation p2 = E (α) U(x, α) is used 0 − We start from Eq. (2) to derive the formula of the to solve this equation. classical STA. Taking the time derivative, we have As an example, we ﬁrst assume that ξ is linear in p: dH (x, p, α) dE (α) ξ(x, p, α)= pξ˜ (x, α). (B.2) 0 =α ˙ (t) 0 . (A.1) 0 dt dα We also assume The left-hand side is rewritten by using the equations of ∂H ∂H k motionx ˙ = andp ˙ = as E0(α)= ǫ0α , (B.3) ∂p − ∂x

∂H dE where ǫ0 is constant. Equation (B.1) reads α˙ 0 H,H =α ˙ 0 . (A.2) ∂α −{ 0} dα ∂ ∂U ∂U k 2(E U) + ξ˜ = + E . (B.4) This gives Eq. (3). − 0 − ∂x ∂x 0 − ∂α α 0 The adiabatic theorem is proved as follows. We take the derivative with respect to α of Eq. (1) to find This equation holds for an arbitrary choice of initial energy and we obtain dJ dE0 ∂H0 = dxdp δ (E0 H0) . (A.3) k dα dα − ∂α − ξ˜ (x, α)= x + c (α), (B.5) Z 0 −2α 0 Using Eq. (3), we can write where c0 is an arbitrary function of α. The potential dJ function satisfies = dxdp ξ,H δ (E H ) dα − { 0} 0 − 0 Z ∂U k ∂U ∂ξ ∂ ∂ξ ∂ α + x + αc0(α) kU =0. (B.6) = dxdp θ (E H ) . ∂α − 2 ∂x − ∂x ∂p − ∂p ∂x 0 − 0 Z k ˜ (A.4) Substituting U(x, α) = α U(x, α) to this equation, we obtain This integration is written by the surface contributions ∂U˜ k ∂U˜ as α + x + αc (α) =0. (B.7) ∂α − 2 0 ∂x dJ ∂ξ p+ = dx θ (E0 H0) This equation shows that the potential has the scale- dα ∂x − p− Z invariant form ∂ξ x+ dp θ (E0 H0) . (A.5) k k x x0(α) − ∂p − x− U = α U˜ = α U − , (B.8) Z 0 αk/2 When x is equal to its maximum or minimum value, x+ where U0 is an arbitrary function and x0 is determined or x−, p takes a fixed value for a smooth trajectory. In this case, this derivative goes to zero and we can prove from the equation the adiabatic theorem. dx (α) k αk 0 x (α)= αc (α). (B.9) dα − 2 0 0 B. DISPERSIONLESS KDV HIERARCHY By using a proper reparametrization of the parameters, we obtain the potential in Eq. (6).

For the reference Hamiltonian H0 in Eq. (5), we study We next consider the case where ξ is third order in p: possible forms of the potential U and the corresponding 3 ξ(x, p, α)= pξ˜0(x, α)+ p ξ˜2(x, α). (B.10) counterdiabatic term HCD =αξ ˙ (x, p, α). ξ is obtained by solving Eq. (3). In the present case, it is written as We also assume that E0 is constant: p ∂ ∂U ∂ dE ∂U + ξ = 0 . (B.1) dE0(α) −m ∂x ∂x ∂p dα − ∂α =0. (B.11) dα 7

Then Eq. (3) is written as x(t′) x(t′)+ δx(t′) as → ′ ′ ∂L ′ ∂L 2 ∂ ∂U ˜ 4 ∂ 2 ∂U ˜ ∂U δS = dt δx(t ) + δx˙(t ) 2p + ξ0 + 2p +3p ξ2 = , ∂x ∂x˙ − ∂x ∂x − ∂x ∂x − ∂α Z t (B.12) ∂L d ∂L ∂L = dt′ δx(t′) + δx(t′) . ∂x − dt′ ∂x˙ ∂x˙ and we obtain the conditions Z 0 (C.5) ∂ξ˜ 2 =0, (B.13) The first term goes to zero if we use the equation of ∂x motion. This means that the action is independent of 1 ∂ξ˜ ∂U 0 +3 ξ˜ =0, (B.14) the history of x and is determined as a function of x(t). −m ∂x ∂x 2 The derivative of the action with respect to x(t) gives the ∂U ∂U second equality in Eq. (C.4). ξ˜ = . (B.15) ∂x 0 − ∂α In a similar way we can consider the variation α(t′) α(t′)+ δα(t′): → The first equation shows that ξ˜2 is independent of x: t ∂p ∂H ∂H0 ∂ξ ˜ δS = dt′ δα x˙ α˙ ξ2(x, α)= c2(α), (B.16) ∂α − ∂p − ∂α − ∂α Z0 ˜ ∂ξ ∂ξ ∂ξ the second equation shows that ξ2 is written in terms of +x ˙ +p ˙ +α ˙ [δαξ]t . (C.6) ∂x ∂p ∂α − 0 U: Using the equation of motion and the condition for the ξ˜ =3mc (α)U + c (α), (B.17) 0 2 0 counterdiabatic term in Eq. (3), we obtain and the third equation gives t ∂H δS = dt′ δα 0 + ξ,H [δαξ]t − ∂α { 0} − 0 1 ∂U c0(α) ∂U Z0 + U + =0. (B.18) t 3mc (α) ∂α 3mc (α) ∂x dE0(α) t 2 2 = dt′ δα [δαξ] . (C.7) − dα − 0 This is essentially equivalent to the dispersionless KdV Z0 equation in Eq. (9). This shows that Ω defined as Eq. (11) is independent of We can go further to derive the higher order dispersion- the history of α(t). less KdV equations. They are fifth order in p, seventh The Hamilton-Jacobi equation is derived by equating order, and so on. Eqs. (C.2) and (C.3). Using the definition of the Hamil- tonian H =xp ˙ L, we obtain − ∂S(x, t, α(t) ) ∂S C. HAMILTON-JACOBI THEORY { } + H x, p = ; α(t) =0. (C.8) ∂t ∂x We note that the action is dependent on x, t, and α(t), We briefly review the Hamilton-Jacobi theory to for- and the time derivative acts on α. By replacing ∂ with mulate the classical STA. The system is characterized by t ∂ +α∂ ˙ , we write the action defined from the Lagrangian: t α ∂S(x, t, α(t) ) ∂S t { } + H0 x, p = , α(t) ′ ′ ′ ∂t ∂x S = dt L(x(t ), x˙(t ); α(t)). (C.1) 0 ∂S(x, t, α(t) ) ∂S Z +α ˙ { } + ξ x, p = , α(t) =0. It satisfies, by definition, ∂α ∂x (C.9) dS = L. (C.2) dt This equation and the definition of Ω show that the Hamilton-Jacobi equation is decomposed as Eqs. (13) On the other hand we can write and (14). dS ∂S ∂S ∂L ∂S The derivation of the counterdiabatic term of the scale- = x˙ + = x˙ + , (C.3) dt ∂x ∂t ∂x˙ ∂t invariant system was done in Eq. (18). Here we consider the case of the dispersionless KdV equation. Substituting where we use the conjugate momentum 2 U = E0 (∂xΩ) to Eq. (9) and assuming E0 is constant, ∂L ∂S we have− p = = . (C.4) ∂x˙ ∂x ∂Ω ∂ ∂Ω ∂Ω ∂2Ω ∂Ω 2 + U +2 =0. − ∂x ∂x ∂α ∂x ∂x2 ∂x The first equality is the definition of the momentum and the second is derived from the variation of the action, (C.10) 8

1 with ω(t)=1+ 2 sin(st). This system is described as a scale-invariant system and the parameter ω represents the dilation eﬀect as γ = 1/√ω. At s = 0, we have a static system and the trajectory in phase space becomes a closed curve. This is not the case for systems with s = 0 as we see in Fig. D.1. Ω(t) for several values of s are6 plotted in Fig. D.2. By changing the horizontal axis t ′ ′ as τ(t) = 0 dt ω(t ), we can conﬁrm that all curves fall on the curve at s = 0. In the present example, Ω can be R calculated analytically and is given by E (ω(t)) Ω(x(t; ω(t)),ω(t)) = 0 (τ(t) sin τ(t)cos τ(t)) . ω(t) − (D.2)

E0(ω(t)) We can show Eq. (20) by using the relation ω(t) = E0(ω(0)) FIG. D.1. Trajectories in phase space for the harmonic oscilla- . 1 ω(0) tor in Eq. (D.1). We use the protocol ω(t)=1+ 2 sin(st), and E. ACTION-ANGLE VARIABLES take the initial condition as E0(ω(0)) = 1 and (x(0),p(0)) = (−2, 0). The adiabatic invariant J is independent of the parameter α(t). It is a function of the initial energy E0(α(0)) = ǫ0: J = J(ǫ0). Then the characteristic function is written as a function of x, α, and J: Ω=Ω(x, α, J). In the periodic systems, we know that the action-angle variables are useful to characterize the system. We deﬁne the angle variable ∂Ω w = . (E.1) ∂J In the theory of canonical transformation, w and the action variable J are interpreted as the canonical variables. J represents the conjugate momentum of w. We assume, for simplicity, that the instantaneous energy is constant: E0(α(t)) = ǫ0. In this case, the angle variable is represented as

∂ǫ0 ∂Ω ∂ǫ0 FIG. D.2. The characteristic function Ω(x(t; ω(t)),ω(t) in the w = = t. (E.2) ∂J ∂ǫ0 ∂J case of Fig. D.1. We note that J(ǫ0) is independent of α(t). This means that w is proportional to t: w = t/T (J). Taking the This gives derivative of w with respect to x, we obtain 3 ∂ ∂Ω ∂Ω 2 ∂Ω ∂w ∂2Ω ∂p + U + =0. (C.11) = = . (E.3) ∂x ∂α ∂x 3 ∂x " # ∂x ∂x∂J ∂J By gauging out an irrelevant constant term, we obtain For a fixed α, we consider the integration of w over the Eq. (21). closed trajectory. Then we obtain ∂w ∂p ∂ w = dx = dx = dxp =1. (E.4) D. HARMONIC OSCILLATOR ∂x ∂J ∂J I I I This result shows that T is equal to the period of the trajectory. Integrating w with respect to J, we write We consider an example of the harmonic oscillator 2 J t(x, α, J ′) 2 ω (t) 2 Ω(x, α, J)= dJ ′ , (E.5) H0 = p + x , (D.1) T (J ′) 4 Z0 9 where we set the boundary condition Ω = 0 at J = 0. where x± represent the end points of the trajectory. Since We note that this expression does not mean that Ω is the potential has the form U(x, α) = u(αx)/α2, we can proportional to t. t in the right hand side is defined as write t = ∂Ω and is a function of x, α, and J. Thus Eq. (E.5) ∂ǫ0 has no explicit time dependence and the real dynamics z+ 2 2 is not implemented in this expression. J = 2 dz ǫ0α u(z), (E.9) α z− − Now we substitute x and α at t = T (J) to Eq. (E.5). Z p Then where z± = αx±. We know from the adiabatic theorem Ω(x(T (J),J), α(T (J)),J) that this function is independent of α. Then we can J t(x(T (J),J), α(T (J)),J ′) evaluate this integral by taking the limit α 0. u(z) is = dJ ′ . (E.6) 2 → T (J ′) replaced with z and we find Z0 We note that x is a function of t and J. If T is indepen- 2 πǫ α2 dent of J, the function t in the right-hand side of this 0 J = 2 = πǫ0. (E.10) equation must be equal to T . We obtain in that case α 2

Ω(x(T,J), α(T ),J)= J. (E.7) This shows that the period of the trajectory is given by As an example, we consider the dispersionless KdV ∂J system with the potential in Eq. (23). The adiabatic T = = π. (E.11) invariant is represented as ∂ǫ0

x+ J =2 dx ǫ0 U(x, α), (E.8) This is independent of the initial energy ǫ0. x− − Z p