Quantum-Classical Correspondence of Shortcuts to Adiabaticity

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by  Manaka Okuyama

Quantum-Classical Correspondence of Shortcuts to Adiabaticity
Manaka Okuyama and Kazutaka Takahashi
Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan
(Dated: March 16, 2017)
We formulate the theory of shortcuts to adiabaticity in classical mechanics. For a reference
Hamiltonian, the counterdiabatic term is constructed from the dispersionless Korteweg–de Vries
(KdV)hierarchy. Thentheadiabatictheoremholdsexactlyforanarbitrarychoiceoftime-dependent
parameters. We use the Hamilton–Jacobi theory to define the generalized action. The action is
7
independent of the history of the parameters and is directly related to the adiabatic invariant.
1
The dispersionless KdV hierarchy is obtained from the classical limit of the KdV hierarchy for
0
the quantum shortcuts to adiabaticity. This correspondence suggests some relation between the
2
quantumand classical adiabatic theorems.

r
a
M
Shortcuts to adiabaticity (STA) is a method control- formulations will allow us to make a link between two
ling dynamical systems. The implementation of the theorems. In this letter, we develop the theory of the
5
method results in dynamics that are free from nona- classical STA. First, we formulate the classical STA in a
1
diabatic transitions for an arbitrary choice of time- generalwayso that the counterdiabaticterm canbe cal-

] dependentparametersinareferenceHamiltonian. Itwas culated, in principle, from the derived formula. Second,
h
developed in quantum systems [1–4] and its applications we show that the adiabatic theorem holds exactly in the
p
- have been studied in various fields of physics and engi- classical STA and the adiabatic invariant is obtained di-
s neering [5]. It is important to notice that this method, rectly from the nonperiodic trajectory. Third, we show
s
a decomposingtheHamiltonianintothereferencetermand that the quantum STA reduces to the classical STA by
l the counterdiabatic term, is applied to any dynamical taking the limit ¯h 0, which suggests some relation
c
→
. systems and offers a novel insight into the systems. between quantum and classical adiabatic dynamics.
s
c It is an interesting problem to find the correspond- We consider classical systems with one degree of free-
i ing method in classical mechanics from several points of dom for simplicity. The system is characterized by the
s
y view as we discuss in the following. Jarzynski studied Hamiltonian H = H(x,p;α(t)) = H +H . The dy-
0 CD
h STA for the classical system by using the adiabatic in- namicalvariablesinphasespacearedenotedby xandp.
p
variant [6]. He found the form of the counterdiabatic To consider the counterdiabatic driving, we use a time-
[
terminacertainsystembasedonageneratoroftheadia- dependentparameterα(t). Itisastraightforwardtaskto

2 batictransport. Althoughseveralapplicationshavebeen generalize the present formulation to systems including
v
discussed [7–9], the complete formulationof the classical severalparameters. Inthe adiabatictime evolution,α(t)
7
STA is still under investigation. represents a slowly-varying function. Here, we do not
2
9 In the quantum system, the adiabatic theorem is de- impose any conditions on α(t). In the quantum counter-
7 scribed by the adiabatic state constructed from the in- diabaticdriving,thestateisdeterminedinstantaneously.
0 stantaneous eigenstate of a reference Hamiltonian H . The same must be implemented for classical dynamics
0
.
1 When the time-dependence of the parameters in the and we impose the condition that, when the solution of
0 Hamiltonian is weak, the solution of the Schr¨odinger the equation of motion (x,p) = (x(t;α(t)),p(t;α(t))) is
7 equation can be approximated by the adiabatic state. substituted, the reference Hamiltonian H (x,p,α(t)) is
0
1
On the other hand, the adiabatic theorem in the classi- equal to the instantaneous energy as
:
v cal system is described by the phase volume defined in
Xi periodic systems. The closed trajectory in phase space H0(x(t;α(t)),p(t;α(t)),α(t)) =E0(α(t)). (2)
for a fixed parameter gives the adiabatic invariant
r Then,byconsideringthetimederivative,wefindthatthe
a
counterdiabatic term H = H H = α˙(t)ξ(x,p,α),
CD 0
−
J = dxdpθ(E0 H0), (1) added to the Hamiltonian, satisfies
−
Z
∂H (x,p,α) dE (α)
where E0 denotes the instantaneous energy. J is defined 0 = ξ(x,p,α),H0(x,p,α) + 0 ,(3)
∂α { } dα
instantaneously and the adiabatic theorem states that J
is approximately conserved when the parameter change where , denotesthePoissonbracket. Thisisobtained
{· ·}
is slow. by using the equation of motion (Section A of Ref. [10]).
Thus the quantum and classical adiabatic theorems Ifwe find ξ thatsatisfies this equation,we canrealizean
look very different and the relation between them is not ideal time evolution that is characterized by E (α(t)) at
0
obvious. The quantum STA is introduced so that the each time t. This equation corresponds to the equation
quantum adiabatic theoremholds exactly and we expect derived in Ref. [6] and may be related to the equation
that the same holds for the classical case. These two for the dynamical invariant in the quantum STA [4, 11].
2
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 fromEq. (3) [6]. This arbi- ∂U(x,α) ∂U(x,α)
+U(x,α) =0. (9)
trarinessis discussedinthe formulationdiscussedbelow. ∂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)
0 0 0 trastedwiththatinthe scale-invariantsystemwherethe
dα − { } −
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-
wesee inthe followingexamples(SectionAofRef.[10]). der in p. Although it is a difficult problem to implement
ThisresultmeansthatJ isdeterminedbytheinitialcon- the higher order terms in an actual experiment, some
ditionandisindependentoft. Theproofclearlyindicates 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 BeforestudyingthesolutionsofthedispersionlessKdV
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-
metricallyinphasespace. Thisresultisthesameasthat 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 Hamiltonianin a standard alization is a delicate and difficult problem. To find the
form quantum-classical correspondence of the adiabatic sys-
tems, we need to extend the formulation to general sys-
H (x,p,α(t))=p2+U(x,α(t)). (5)
0 tems. This can be done by the Hamilton–Jacobi the-
ory. The adiabatic invariant is related to the action
ThenweshowthatthesolutionofEq.(3)isgivenbythe S = tdt′L= tdt′(x˙p H). Thisisafunctionofx(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 R R { }
ThepropertythatS isindependentofthe historyofx(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
terdiabaticdriving,trajectoriesinphasespacearedeter-
and the counterdiabatic term represent the Lax pair in
minedfromEq.(2)andthe HamiltoniansatisfiesEq.(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 x (t) deviation of the action as (Section C of Ref. [10])
0
U(x,α(t))= u − , (6)
γ2(t) γ(t) t ∂H
(cid:18) (cid:19) 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 (cid:18) (cid:19)
x and γ, the former represents a translation and the (10)
0
latter a dilation. The counterdiabatic term is given by
We use Eq. (3) to find that the function defined as
γ˙ t
HCD =x˙0p+ γ(x−x0)p. (7) Ω=S(x(t),t,{α(t)})+ dt′E0(α(t′)) (11)
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 finedforconstantE0 bytheLegendretransformation. It
satisfies
HCD =α˙ξ =α˙ pU(x,α)+ 2p3 , (8) ∂Ω =p(x,α), ∂Ω = ξ(x,p(x,α),α). (12)
3 ∂x ∂α −
(cid:18) (cid:19)
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 ǫ gives the relation
0
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′)
Legendretransformation. TheexplicittdependenceofΩ 0 Z0
canberemovedbyaddingatime-dependentterm,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-invariantsystem satisfies the relation
TheHamilton–Jacobiequationisgivenby ∂S+H =0
∂t
with p = ∂S. We substitute Ω to this equation. Noting Ω(x(t;α(t)),α(t)) =Ω(x(τ(t);α(0)),α(0)). (20)
∂x
thatthetimederivativeisreplacedinthepresentsystem The left-hand side represents Ω at t obtainedin the pro-
with ∂t ∂t +α˙∂α, we find that the Hamilton–Jacobi 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-
abatic invariant in Eq. (1) and is written as Ω = pdx
∂Ω(x,α)
H0 x,p= ,α =E0(α), (13) (SectionDofRef.[10]foranexampleoftheharmonicos-
∂x H
(cid:18) (cid:19) cillator). Thisrelationshowsthattheadiabaticinvariant
∂Ω(x,α) ∂Ω(x,α)
ξ x,p= ,α = . (14) is directly obtained from the corresponding nonperiodic
∂x − ∂α trajectory.
(cid:18) (cid:19)
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 = α˙ξ(x,p,α). 0
CD fiesthedispersionlessKdVequation(9). Wecanrederive
Thus the counterdiabatic driving is characterizedby Ω.
Eq. (8) in the present formalism by assuming that E is
The definitionof the actionshowsthat Ω is written as 0
constant. Substituting U = E (∂ Ω)2 to Eq. (9), we
0 x
−
t find (Section C of Ref. [10])
Ω(x(t),α(t)) = dt′ (x˙p(x,α) α˙ξ(x,p(x,α),α))
Z0 − ∂Ω ∂Ω 2 ∂Ω 3
t ∂H ∂ξ +U + =0. (21)
= dt′ 0p+α˙ p ξ , (15) ∂α ∂x 3 ∂x
(cid:18) (cid:19)
∂p ∂p −
Z0 (cid:20) (cid:18) (cid:19)(cid:21) 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).
ThisexpressionshowsthatΩ,asafunctionoftandα(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
= 0p, = p ξ. (16) wheref is anarbitraryfunction[22,23]. Asasimple ex-
(cid:18)∂t (cid:19)α ∂p (cid:18)∂α(cid:19)t ∂p − ample we consider the case f(x)=x2. Then, by solving
the quadratic equation, we obtain
Second equation states that Ω is independent of α(t)
when the counterdiabatic term is linear in p. This is 2αx+1 √4αx+1
the case of the scale-invariantdriving where H0 is of the U(x,α)= −2α2 . (23)
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
∂Ω 2 1 x x x> 1 and we set the parameters so that this relation
+ u − 0 =E (α), (17) −4α
∂x γ2 γ 0 is satisfiedthroughoutthe time evolution. By takingthe
(cid:18) (cid:19) (cid:18) (cid:19) limit α 0, we have the harmonic oscillator U(x,0) =
→
and we find that E and Ω take the form E (α(t)) = x2.
0 0
ǫ /γ2(t)andΩ=Ω((x x )/γ),respectively. Bysetting
0 0
−
γ(0)=1, we can regard ǫ as the initial energy at t=0.
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
− ∂x − ∂γ γ −
0 the protocol α(t) = α(0) 1 sin2(st) . We see from
−
wherewe use the propertythat the derivativesofΩ with Fig.2thatEq.(20)isnotsatisfiedsinceΩisnotnecessar-
(cid:0) (cid:1)
respect to x and γ are translated to that with x in the ily a monotone increasing function and the time rescal-
0
presentsystem. ΩisalsoafunctionofE (α(0))=ǫ and ing cannot give the result at s=0. The counterdiabatic
0 0
4
the proof. See Sec. E of Ref. [10]. The period T canalso
becalculatedthereandwefindT =πinthepresentcase.
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 sys-
tems. In scale-invariant Hamiltonian in Eq. (5) with
(6),thecounterdiabaticterminclassicalsystembecomes
the same as that in quantum system if we use the sym-
metrization as px (pˆxˆ+xˆpˆ)/2. In the KdV systems,
→
theformoftheHamiltonianisunchangedbutthepoten-
tialfunctionU satisfiesanequationthatisdifferentfrom
Eq. (9).
In the quantum system the state is described by the
wavefunction which satisfies the Schr¨odinger equation
∂
FIG.1. TrajectoriesinphasespaceforthedispersionlessKdV i¯h ψ(x,t)=(Hˆ +Hˆ )ψ(x,t). (25)
0 CD
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
For example, we consider the Hamiltonian (5) and (8)
solid line representss=0.0, thedashed line s=4.5, and the
dotted line s=9.5. with the replacement pU (pˆUˆ +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 Schr¨odinger
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 crucialto developthe classicalSTAby
the Hamilton–Jacobi theory as we discuss in this letter.
Inthe quantumSTA,the wavefunctionis givenby the
adiabatic state of H (xˆ,pˆ,α(t)). By using the instanta-
0
FIG. 2. The characteristic function Ω(x(t;α(t)),α(t)) in the
neous eigenstate of H , n(α(t)) , and the instantaneous
0
caseofFig.1. Thecircledenotesthepoint(T,Ω(T))=(π,π) | i
energy of H , E (α(t)), we can write the wave function
whereT denotestheperiodoftheclosedtrajectoryforacon- 0 n
as
stant α. Inset: The potential function U(x,α) in Eq.(23).
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 (cid:18) Z0 (cid:19)
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) ∂
iinntveagreiarnnt.JWwehaelnsotfiinsdeqtuhaalttΩo itsheeqpuearliotod tohfethadeiaclboasteidc |ψ˜n(α(t))i=exp −Zα(0) dα′hn(α′)|∂α′|n(α′)i!
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˜ (α(t)) , the adiabatic state without
n
| i
the dynamical phase, is written in terms of α, not of t.
Thisrelationholdsforanarbitrarychoiceofα(t)andcan We define the unitary operator Vˆ as
beprovedbyassumingthatT isindependentoftheinitial
energy. We use the theory of action-angle variables for ψ˜ (α(t)) =Vˆ(α(t))ψ˜ (α(0)) . (30)
n n
| i | i
5
Thisoperatorwasintroducedtodevelopthepathintegral atom coolinginharmonictraps: Shortcut toadiabaticity,
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. Iba´n˜ez, S. Mart´ınez-Garaot, M. Mod-
ugno, A. del Campo, D. Gu´ery-Odelin, A. Ruschhaupt,
Hˆ (t)=i¯hα˙(t)∂Vˆ(α)Vˆ†(α). (31) X.Chen,andJ.G.Muga,Shortcuts toadiabaticity,Adv.
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 operatorVˆ quantum and classical dynamics, Phys. Rev. A 88,
can be replaced by the total time evolution operator to 040101(R) (2013).
findthe formulaby DemirplakandRice [1]. By usingVˆ, [7] S. Deffner, C. Jarzynski, and A. del Campo, Classical
we can write the formula of Hˆ in a more suggestive and quantum shortcuts toadiabaticity forscale-invariant
form. We write Vˆ as CD driving, Phys.Rev.X 4, 021013 (2014).
[8] A.PatraandC.Jarzynski,Classical and quantum short-
i cuts to adiabaticity in a tilted piston, J. Phys. Chem. B
Vˆ(α)=exp Ωˆ(α) , (32)
10.1021/acs.jpcb.6b08769 (2016).
¯h
(cid:18) (cid:19) [9] C. Jarzynski, S. Deffner, A. Patra, and Y. Suba¸si,
andthisoperatorΩˆ isthequantumanalogueofthechar- Fast forward to the classical adiabatic invariant,
arXiv:1611.06437 (2016).
acteristic function Ω(x,α) in the classical system. For
[10] See SupplementalMaterial for thedetailed calculations.
example,inthe scale-invariantHamiltonian(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
Ωˆ(α(t))= pˆ(x0(t) x0(0)) field, J. Math. Phys. 10, 1458 (1969).
− −
1 γ(t) [12] M.OkuyamaandK.Takahashi,Fromclassicalnonlinear
(pˆ(xˆ x (t))+(xˆ x (t))pˆ)ln . (33)
0 0 integrable systems to quantum shortcuts to adiabaticity,
−2 − − γ(0)
Phys. Rev.Lett. 117, 070401 (2016).
The counterdiabatic term in this case is written as [13] D.R.Lebedev,ConservationlawsandLaxrepresentation
of Benney’s long wave equations, Phys. Lett. A 74, 154
Hˆ (t)= α˙(t)∂Ωˆ(α). (34) (1979).
CD − ∂α [14] V.E.Zakharov,Benney equations and quasiclassical ap-
proximation in the method of the inverse problem, Func-
This expression formally coincides with Eq. (14). tional Analysis and Its Applications 14, 89 (1980).
Tosummarize,wehavedevelopedtheclassicalSTAby [15] P. D. Lax, Integrals of nonlinear equations of evolution
usingtheHamilton–Jacobitheory. Thesystemischarac- and solitary waves, Commun. PureAppl.Math. 21, 467
terizedbythe generalizedcharacteristicfunction Ω(x,α) (1968).
and the counterdiabaticterm is obtained fromthis func- [16] K. Takasaki and T. Takabe, Quasi-classical limit of
KP hierarchy, W-symmetries and free fermions, Zap.
tion. The equation for the counterdiabatic term can be
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).
canalso show that the classicalSTA is reduced fromthe [18] D. J. Korteweg and G. de Vries, On the change of form
quantum STA by taking the standard semiclassical ap- of longwaves advancing inarectangular canal, and ona
proximation. new type of long stationary waves, Philos. Mag. 39, 422
(1895).
We acknowledge financial support from the ImPACT
[19] P. Hertz, U¨ber die mechanischen Grundlagen der Ther-
Programof 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, U¨ber die mechanischen Grundlagen der Ther-
wassupportedbyJSPSKAKENHIGrantNo. 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.DemirplakandS.A.Rice,Adiabaticpopulationtrans- (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.DemirplakandS.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.
Gu´ery-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
thatthe additionalrelationp2 =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 0 0
=α˙(t) . (A.1)
dt dα
We also assume
The left-hand side is rewritten by using the equations of
motion x˙ = ∂H and p˙ = ∂H as E (α)=ǫ αk, (B.3)
∂p −∂x 0 0
∂H0 dE0 where ǫ0 is constant. Equation (B.1) reads
α˙ H,H =α˙ . (A.2)
0
∂α −{ } dα
∂ ∂U ∂U k
2(E U) + ξ˜ = + E . (B.4)
This gives Eq. (3). − 0− ∂x ∂x 0 −∂α α 0
(cid:20) (cid:21)
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 en-
ergy and we obtain
dJ dE ∂H
0 0
dα = dxdp dα − ∂α δ(E0−H0). (A.3) ξ˜(x,α)= k x+c (α), (B.5)
Z (cid:18) (cid:19) 0 −2α 0
Using Eq. (3), we can write
where c is an arbitrary function of α. The potential
0
dJ function satisfies
= dxdp ξ,H δ(E H )
0 0 0
dα − { } −
∂U k ∂U
Z
α + x+αc (α) kU =0. (B.6)
∂ξ ∂ ∂ξ ∂ 0
∂α −2 ∂x −
= dxdp θ(E0 H0). (cid:18) (cid:19)
∂x∂p − ∂p∂x −
Z (cid:18) (cid:19) (A.4) Substituting U(x,α) = αkU˜(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)
0
∂α −2 ∂x
dJ ∂ξ p+ (cid:18) (cid:19)
= dx θ(E H )
dα ∂x 0− 0 This equation shows that the potential has the scale-
Z (cid:20) (cid:21)p−
invariant form
∂ξ x+
dp θ(E H ) . (A.5)
0 0 x x (α)
−Z (cid:20)∂p − (cid:21)x− U =αkU˜ =αkU0 −αk/02 , (B.8)
(cid:18) (cid:19)
When x is equal to its maximum or minimum value, x
+
where U is an arbitrary function and x is determined
or x , p takes a fixed value for a smooth trajectory. In 0 0
−
from the equation
this case, this derivative goes to zero and we can prove
the adiabatic theorem.
dx (α) k
αk 0 x (α)=αc (α). (B.9)
0 0
dα − 2
B. DISPERSIONLESS KDV HIERARCHY By using a proper reparametrization of the parameters,
we obtain the potential in Eq. (6).
For the reference HamiltonianH in Eq.(5), we study We next consider the case where ξ is third order in p:
0
possible forms of the potential U and the corresponding
ξ(x,p,α)=pξ˜(x,α)+p3ξ˜(x,α). (B.10)
counterdiabatic term H = α˙ξ(x,p,α). ξ is obtained 0 2
CD
by solving Eq. (3). In the present case, it is written as
We also assume that E is constant:
0
p ∂ + ∂U ∂ ξ = dE0 ∂U. (B.1) dE0(α) =0. (B.11)
−m∂x ∂x ∂p dα − ∂α dα
(cid:18) (cid:19)
7
Then Eq. (3) is written as x(t′) x(t′)+δx(t′) as
→
∂L ∂L
2p2 ∂ + ∂U ξ˜ + 2p4 ∂ +3p2∂U ξ˜ = ∂U, δS = dt′ δx(t′) +δx˙(t′)
0 2 ∂x ∂x˙
(cid:18)− ∂x ∂x(cid:19) (cid:18)− ∂x ∂x(cid:19) −∂α Z (cid:18) (cid:19)
(B.12) ∂L d ∂L ∂L t
= dt′δx(t′) + δx(t′) .
∂x − dt′ ∂x˙ ∂x˙
and we obtain the conditions Z (cid:18) (cid:19) (cid:20) (cid:21)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˜0 +3∂Uξ˜ =0, (B.14) the history of x and is determined as a function of x(t).
2
−m ∂x ∂x Thederivativeoftheactionwithrespecttox(t)givesthe
∂Uξ˜ = ∂U. (B.15) second equality in Eq. (C.4).
∂x 0 −∂α Ina similarwaywe canconsiderthe variationα(t′)
→
α(t′)+δα(t′):
The first equation shows that ξ˜ is independent of x:
2
t ∂p ∂H ∂H ∂ξ
ξ˜(x,α)=c (α), (B.16) δS = dt′δα x˙ 0 α˙
2 2 ∂α − ∂p − ∂α − ∂α
Z0 (cid:20) (cid:18) (cid:19)
the second equation shows that ξ˜2 is written in terms of +x˙∂ξ +p˙∂ξ +α˙ ∂ξ [δαξ]t . (C.6)
U: ∂x ∂p ∂α − 0
(cid:21)
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 (cid:18) (cid:19)
+ U + =0. (B.18)
3mc2(α) ∂α (cid:18) 3mc2(α)(cid:19) ∂x = tdt′δαdE0(α) [δαξ]t . (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
the history of α(t).
Wecangofurthertoderivethehigherorderdispersion-
The Hamilton-Jacobi equation is derived by equating
less KdV equations. They are fifth order in p, seventh
Eqs. (C.2) and (C.3). Using the definition of the Hamil-
order, and so on.
tonian H =x˙p L, we obtain
−
∂S(x,t, α(t) ) ∂S
C. HAMILTON-JACOBI THEORY { } +H x,p= ;α(t) =0. (C.8)
∂t ∂x
(cid:18) (cid:19)
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
mulatetheclassicalSTA.Thesystemischaracterizedby t
∂ +α˙∂ , we write
the action defined from the Lagrangian: t α
∂S(x,t, α(t) ) ∂S
t { } +H0 x,p= ,α(t)
S = dt′L(x(t′),x˙(t′);α(t)). (C.1) ∂t (cid:18) ∂x (cid:19)
Z0 ∂S(x,t, α(t) ) ∂S
+α˙ { } +ξ x,p= ,α(t) =0.
It satisfies, by definition, ∂α ∂x
(cid:18) (cid:18) (cid:19)(cid:19)
(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 Thederivationofthecounterdiabatictermofthescale-
= x˙ + = x˙ + , (C.3)
dt ∂x ∂t ∂x˙ ∂t invariantsystem was done in Eq. (18). Here we consider
thecaseofthedispersionlessKdVequation. Substituting
where we use the conjugate momentum
U =E (∂ Ω)2 toEq.(9)andassumingE isconstant,
0 x 0
−
∂L ∂S we have
p= = . (C.4)
∂x˙ ∂x ∂Ω ∂ ∂Ω ∂Ω ∂2Ω∂Ω
2 +U +2 =0.
The firstequality is the definition of the momentum and − ∂x ∂x ∂α ∂x ∂x2 ∂x
(cid:20) (cid:18) (cid:19) (cid:21)
the second is derived from the variation of the action, (C.10)
8
with ω(t) = 1+ 1sin(st). This system is described as
2
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
6
are plotted in Fig. D.2. By changing the horizontal axis
as τ(t) = tdt′ω(t′), we can confirm that all curves fall
0
on the curve at s=0. In the present example, Ω can be
R
calculated analytically and is given by
E (ω(t))
0
Ω(x(t;ω(t)),ω(t))= (τ(t) sinτ(t)cosτ(t)).
ω(t) −
(D.2)
We can show Eq. (20) by using the relation E0(ω(t)) =
ω(t)
FIG.D.1. Trajectoriesinphasespacefortheharmonicoscilla- E0(ω(0)).
ω(0)
torinEq.(D.1). Weusetheprotocolω(t)=1+1sin(st),and E. ACTION-ANGLE VARIABLES
2
take the initial condition as E0(ω(0))=1 and (x(0),p(0))=
(−2,0).
TheadiabaticinvariantJ isindependentoftheparam-
eterα(t). ItisafunctionoftheinitialenergyE (α(0))=
0
ǫ : J =J(ǫ ). Thenthecharacteristicfunctioniswritten
0 0
as a function of x, α, and J: Ω=Ω(x,α,J).
Intheperiodicsystems,weknowthattheaction-angle
variablesareusefultocharacterizethesystem. Wedefine
the angle variable
∂Ω
w = . (E.1)
∂J
In the theory of canonicaltransformation, w and the ac-
tionvariableJ areinterpretedasthecanonicalvariables.
J represents the conjugate momentum of w.
We assume, for simplicity, that the instantaneous en-
ergy is constant: E (α(t)) = ǫ . In this case, the angle
0 0
variable is represented as
∂ǫ ∂Ω ∂ǫ
0 0
w= = t. (E.2)
FIG.D.2. ThecharacteristicfunctionΩ(x(t;ω(t)),ω(t)inthe ∂J ∂ǫ ∂J
0
case of Fig. D.1.
We note that J(ǫ ) is independent of α(t). This means
0
that w is proportional to t: w = t/T(J). Taking the
This gives derivative of w with respect to x, we obtain
∂ ∂Ω ∂Ω 2 ∂Ω 3 ∂w ∂2Ω ∂p
+U + =0. (C.11) = = . (E.3)
∂x"∂α ∂x 3(cid:18)∂x(cid:19) # ∂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
H =p2+ ω2(t)x2, (D.1) Ω(x,α,J)= JdJ′ t(x,α,J′), (E.5)
0 4 Z0 T(J′)
9
where we set the boundary condition Ω = 0 at J = 0. wherex representtheendpointsofthetrajectory. 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 2 z+
J = dz ǫ α2 u(z), (E.9)
is not implemented in this expression. α2 0 −
Now we substitute x and α at t = T(J) to Eq. (E.5). Zz− 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) replaced with z2 and we find →
T(J′)
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 J = 0 =πǫ . (E.10)
α2 2 0
equation must be equal to T. We obtain in that case
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 ǫ U(x,α), (E.8) This is independent of the initial energy ǫ .
0 0
−
Zx−
p