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Performance of superadiabatic quantum machines

Obinna Abah and Eric Lutz

Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, D-91058 Erlangen, Germany

We investigate the performance of a quantum thermal machine operating in ﬁnite time based on

shortcut-to-adiabaticity techniques. We compute eﬃciency and power for a quantum harmonic Otto

engine by taking the energetic cost of the superadiabatic driving explicitly into account. We further

derive generic upper bounds on both quantities, valid for any heat engine cycle, using the notion of

quantum speed limits for driven systems. We demonstrate that these quantum bounds are tighter

than those stemming from the second law of thermodynamics.

PACS numbers:

Superadiabatic (SA) techniques allow the engineer-

ing of adiabatic dynamics in ﬁnite time. While truly

adiabatic transformations require inﬁnitely slow driving,

transitionless protocols may be implemented at ﬁnite

speed by adding properly designed time-dependent terms

HSA(t) to the Hamiltonian of a system [1, 2]. By sup-

pressing nonadiabatic excitations, these fast processes re-

produce the same ﬁnal state as that of adiabatic driving.

In that sense, they provide a shortcut to adiabaticity. In

the last few years, there has been remarkable progress,

both theoretical [1–7] and experimental [8–17], in devel-

oping superadiabatic methods for quantum and classical

systems (see Ref. [18] for a review). Successful appli-

cations include high-ﬁdelity driving of a BEC [11], fast

transport of trapped ions [12, 13], fast adiabatic passage

using a single spin in diamond [14] and cold atoms [15],

as well as swift equilibration of a Brownian particle [16].

Superadiabatic protocols have recently been extended

to thermal machines as a means to enhance their per-

formance. Classical [19, 20] and quantum [21, 22] sin-

gle particle heat engines, as well as multiparticle quan-

tum motors [22–24] have been theoretically investigated.

Nonadiabatic transitions are well-known sources of en-

tropy production that reduce the eﬃciency of thermal

machines [25–27]. Successfully suppressing them using

superadiabatic methods thus appears a promising strat-

egy to boost their work and power output.

However, a crucial point that needs to be addressed

in order to assess the usefulness of shortcut techniques

in thermodynamics is the proper computation of the ef-

ﬁciency of a superadiabatic engine. Since the excitation

suppressing term HSA(t) in the Hamiltonian is often as-

sumed to be zero at the begin and at the end of a trans-

formation [18], its work contribution vanishes. The en-

ergetic cost of the additional superadiabatic driving is

therefore commonly not included in the calculation of

the eﬃciency [19–24]. As a result, the latter quantity

reduces to the adiabatic eﬃciency, even for fast nonadi-

abatic driving of the machine: the superadiabatic driv-

ing thus appears to be for free. This situation is some-

what reminiscent of the power of a periodic signal which

is zero at the beginning and at the end of one period.

While the instantaneous power vanishes at the end of

the interval, the actual power of the signal is given by

the non-zero time-averaged power [28]. As a matter of

fact, the energetic cost of superadiabatic protocols was

lately deﬁned in universal quantum computation and adi-

abatic gate teleportation models as the time-averaged

norm of the superadiabatic Hamiltonian HSA(t) [29, 30]

(see also Refs. [31–34]). However, the chosen Hilbert-

Schmidt norm is related to the variance of the energy

and not to its mean [35]. It is hence of limited relevance

to the investigation of the energetics of a heat engine.

In this paper, we evaluate the performance of a su-

peradiabatic thermal machine by properly taking the en-

ergetic cost of the transitionless driving into account.

We consider commonly employed local counterdiabatic

(LCD) control techniques [5–7] which have latterly been

successfully implemented experimentally in Refs. [9–11].

We evaluate both eﬃciency and power for a paradig-

matic harmonic quantum Otto engine [36–41]. We ex-

plicitly compute the cost of the superadiabatic protocol

as the time-averaged expectation value of the Hamilto-

nian HSA(t) for compression and expansion phases of the

engine cycle. We ﬁnd that the energetic cost of the su-

peradiabatic driving exceeds the potential work gain for

moderately rapid protocols. Superadiabatic engines may

therefore only outperform traditional quantum motors

for very fast cycles, albeit with an eﬃciency much smaller

than the corresponding adiabatic eﬃciency. We addi-

tionally derive generic upper bounds on both superadia-

batic eﬃciency and power, valid for any heat engine cycle,

based on the concept of quantum speed limit times for

driven unitary dynamics [42]. We demonstrate that these

quantum bounds are tighter than conventional bounds

that follow from the second law of thermodynamics.

Quantum Otto engine. We consider a quantum en-

gine whose working medium is a harmonic oscillator with

time-dependent frequency ωt. The corresponding Hamil-

tonian is of the usual form, H0(t) = p2/(2m) + mω2

tx2/2,

where xand pare the position and momentum opera-

tors of an oscillator of mass m. The Otto cycle consists

of four consecutive steps as shown in Fig. 1 [36–41]: (1)

Isentropic compression A→B: the frequency is varied

from ω1to ω2during time τ1while the system is isolated.

The evolution is unitary and the von Neumann entropy

is constant. (2) Hot isochore B→C: the oscillator is

weakly coupled to a bath at inverse temperature β2at

ﬁxed frequency and thermalizes to state C during time

τ2. (3) Isentropic expansion C→D: the frequency is

arXiv:1611.09045v1 [quant-ph] 28 Nov 2016

2

(2) Hot isochore

Heat added

D

B

(3) Isentropic expansion

Work done

3

W

(4) Cold isochore

Heat removed

A

C

(1) Isentropic compression

Work done

1

W

2

Q

4

Q

FIG. 1: Quantum Otto engine for a harmonic trap with time-

dependent frequency. The cycle consists of four consecutive

steps: (1) isentropic compression, (2) isochoric heating, (3)

isentropic expansion and (4) isochoric cooling. Work is pro-

duced during the ﬁrst and third unitary strokes, while heat is

absorbed from the hot reservoir during the heating phase (2).

changed back to its initial value during time τ3at con-

stant von Neumann entropy. (4) Cold isochore D→A:

the system is weakly coupled to a bath at inverse temper-

ature β1> β2and relaxes to state A during τ4at ﬁxed

frequency. We will assume, as commonly done [36–41],

that the thermalization times τ2,4are much shorter than

the compression/expansion times τ1,3. The total cycle

time is then τcycle =τ1+τ3= 2τfor equal step duration.

In order to evaluate the performance of the Otto en-

gine, we need to compute work and heat for each of the

above steps. Work is performed during the ﬁrst and third

unitary strokes, whereas heat is exchanged with the baths

during the isochoric thermalization phases two and four.

The mean work may be calculated by using the exact

solution of the Schr¨odinger equation for the parametric

oscillator for any given frequency modulation [43, 44].

For the compression/expansion steps, it is given by [41],

hW1i=~

2(ω2Q∗

1−ω1) coth β1~ω1

2,(1)

hW3i=~

2(ω1Q∗

3−ω2) coth β2~ω2

2,(2)

where we have introduced the dimensionless adiabaticity

parameter Q∗

i(i= 1,3) [45]. It is deﬁned as the ratio of

the mean energy and the corresponding adiabatic mean

energy and is thus equal to one for adiabatic processes

[44]. Its explicit expression for any frequency modulation

ωtmay be found in Refs. [43, 44]. Furthermore, the mean

5 10 15 20

0

2

4

6

8

10

Time τ

Energetic cost

FIG. 2: Energetic cost of the superadiabatic driving

H1

SAτ+H3

SAτ, each deﬁned as the time average of Eq. (8),

for the compression and expansion steps (1) and (3) (red

dotted-dashed) as a function of the driving time τ. The corre-

sponding nonadiabatic work hW1iNA +hW3iNA, deﬁned as the

diﬀerence between the actual and the adiabatic work, is shown

for comparison (grey dotted). Parameters are ω1= 0.32,

ω2= 1, β1= 0.5 and β2= 0.05.

heat absorbed from the hot bath reads [41],

hQ2i=~ω2

2coth β2~ω2

2−Q∗

1coth β1~ω1

2.

(3)

For an engine, the produced work is negative, hW1i+

hW3i<0, and the absorbed heat is positive, hQ2i>0.

Superadiabatic driving. The compression and expan-

sion phases (1) and (3) may be sped up, while suppress-

ing unwanted nonadiabatic transitions, by adding a local

harmonic potential HSA to the system Hamiltonian H0.

The local counterdiabatic Hamiltonian may then written

in the form HLCD(t) = H0(t) + HSA (t) with [5–7],

HSA =m

2Ω2

t−ω2

tx2=m

2−3 ˙ω2

t

4ω2

t

+¨ωt

2ωtx2.(4)

Boundary conditions ensuring that HSA(0, τ ) = 0 at the

beginning and at the end of the driving are given by,

ω(0) = ωi,˙ω(0) = 0,¨ω(0) = 0,

ω(τ) = ωf,˙ω(τ)=0,¨ω(τ)=0,(5)

where ωi,f =ω1,2denote the respective initial and ﬁ-

nal frequencies of the compression/expansion steps. The

conditions (5) are, for instance, satisﬁed by [5–7],

ωt=ωi+10(ωf−ωi)s3−15(ωf−ωi)s4+6(ωf−ωi)s5,(6)

with s=t/τ. Note that Ω2

t>0 to avoid trap inversion.

Implementing the superadiabatic driving (4) leads to a

unit adiabaticity parameter, Q∗

i(τ) = 1 (i= 1,3). As a

consequence, the work performed in ﬁnite time during the

two compression/expansion phases is equal to the adia-

batic work, hW1iSA =hW1iAD and hW3iSA =hW3iAD.

3

NASA QSL

0 5 10 15 20 25

0.50

0.55

0.60

0.65

0.70

Time τ

Efficiency

16 20 24

0.67

0.68

FIG. 3: Superadiabatic eﬃciency ηSA (red dotted-dashed),

Eq. (7), together with the nonadiabatic eﬃciency ηNA (blue

dashed) and the adiabatic eﬃciency ηAD (black dotted) as a

function of the time τ. The green solid line shows the quan-

tum speed limit bound (11). Same parameters as in Fig. 2.

Eﬃciency of the superadiabatic engine. We deﬁne the

eﬃciency of the superadiabatic motor as,

ηSA =energy output

energy input =−(hW1iSA +hW3iSA )

hQ2i+hH1

SAiτ+hH3

SAiτ

.(7)

In the above expression, the energetic cost of the tran-

sitionless driving is taken into account by including the

time-average, Hi

SAτ= (1/τ )Rτ

0dt Hi

SA(t)(i= 1,3),

of the local potential (4) for the compression/expansion

steps. Equation (7) reduces to the adiabatic eﬃciency

ηAD in the absence of these two contributions. For fur-

ther reference, we also introduce the usual nonadiabatic

eﬃciency of the engine, ηNA =−(hW1i+hW3i)/hQ2i,

based on the formulas (1)-(3) without any shortcut.

The expectation value of the local counterdiabatic po-

tential (4) may be calculated explicitly for an initial ther-

mal state in terms of the initial energy of the system

hH0(0)i. We ﬁnd (see Supplemental Material [46]),

hHSA(t)i=ωt

ωihH0(0)i−˙ω2

t

4ω4

t

+¨ωt

4ω3

t.(8)

We use Eq. (8) to numerically compute the time averages

Hi

SAτ(i= 1,3) for compression/expansion that are

needed to evaluate the superadiabatic eﬃciency (7).

Figure 2 shows, as an illustration, the energetic cost

of the superadiabatic driving H1

SAτ+H3

SAτ, for the

compression and expansion steps (1) and (3) as a function

of the driving time τ. We also display, for comparison,

the corresponding nonadiabatic work hW1iNA +hW3iNA,

deﬁned as the diﬀerence between the actual work and the

adiabatic work, hWiiNA =hWii−hWiiAD (i= 1,3); this

quantity measures the importance of nonadiabatic exci-

tations induced by fast protocols and is often referred to

as internal friction [21, 27, 31, 39]. We observe that the

time-averaged superadiabatic energy (red dotted-dashed)

increases signiﬁcantly with decreasing process time as

expected. This increase is much faster than that of

the nonadiabatic work hW1iNA +hW3iNA (grey dotted).

Eventually, for very rapid driving, the energetic price of

the shortcut will dominate nonadiabatic energy losses.

Figure 3 exhibits the superadiabatic eﬃciency ηSA (red

dotted-dashed), Eq. (7), as a function of the driving time

τ, together with the adiabatic eﬃciency ηAD (black dot-

ted) and the nonadiabatic eﬃciency ηNA (blue dashed).

Three points are worth emphasizing: i) if the energetic

cost of the shortcut is not included, the superadiabatic

eﬃciency is equal to the maximum possible value given

by the constant adiabatic eﬃciency ηAD, as noted in

Refs. [19–24], ii) by contrast, if that energetic cost is

properly taken into account, the superadiabatic eﬃciency

ηSA drops for decreasing τ, reﬂecting the sharp augmen-

tation of the time-averaged superadiabatic energy seen

in Fig. 2, iii) we further observe that ηSA < ηNA for large

time τ, while ηSA > ηNA only for small enough τ(inset).

We can thus conclude that the superadiabatic driving is

only of advantage for suﬃciently short cycle durations.

For large cycle times, the energetic cost of the shortcut

outweighs the work gained by emulating adiabaticity.

Another beneﬁt of the superadiabatic driving ap-

pears for very small time τ. An examination of

Eq. (3) reveals that the heat hQ2ibecomes negative

for very strongly nonadiabatic processes, when Q∗

1(τ)>

coth(β2~ω2/2)/coth(β1~ω1/2). In this regime, heat is

pumped into the hot reservoir, instead of being absorbed

from it, and the machine stops working as an engine [47].

Since Q∗

1(τ) = 1 for all τfor the local counterdiabatic

driving, this problem never occurs for the superadiabatic

motor, even in the limit of very short cycles τ→0.

Power of the superadiabatic engine. The power of the

superadiabatic machine is given by,

PSA =−hW1iSA +hW3iSA

τcycle

.(9)

Since the superadiabatic protocol ensures adiabatic work

output, hWiiSA =hWiiAD (i= 1,3), in a shorter cycle

duration τcycle, the superadiabatic power PSA is always

greater than the nonadiabatic power PNA =−(hW1i+

hW3i)/τcycle (see Fig. 4). This ability to considerably

enhance the power of a thermal machine is one of the

true advantages of the shortcut to adiabaticity approach.

However, in view of the discussion above, it is not pos-

sible to reach arbitrarily large power at maximum eﬃ-

ciency, as sometimes claimed [19–22]. This observation

is in complete agreement with recent general proofs that

forbid the simultaneous attainability of maximum power

and maximum eﬃciency [48].

Universal quantum speed limit bounds. We ﬁnally de-

rive generic upper bounds for both the superadiabatic

eﬃciency (7) and the superadiabatic power (9), based on

the concept of quantum speed limits (see Refs. [49–51]

and references therein). Contrary to classical physics,

quantum theory limits the speed of evolution of a system

4

SA

NA

QSL

5 10 15 20 25

0.0

0.5

1.0

1.5

2.0

Time τ

Power

FIG. 4: Superadiabatic power PSA (red dotted-dashed),

Eq. (8), together with the nonadiabatic power PNA (blue

dashed) as a function of the driving time τ. The green solid

line shows the quantum speed limit bound (12). Same pa-

rameters as in Fig. 2.

between given initial and ﬁnal states. In particular, there

exists a lower bound, called the quantum speed limit time

τQSL ≤τ, on the time a system needs to evolve between

these two states. An important restriction of the supera-

diabatic technique is the time required to successfully im-

plement the counterdiabatic driving (4), which depends

on the ﬁrst two time derivatives of the frequency ωt[52].

For this unitary driven dynamics, a Margolus-Levitin-

type bound on the evolution time reads [42],

τ≥τQSL =~L(ρi, ρf)

hHSAiτ

,(10)

where L(ρi, ρf) denotes the Bures angle between the ini-

tial and ﬁnal density operators of the system [46, 53]

and hHSAiτthe time-averaged superadiabatic energy (8).

We expect Eq. (10) to be a proper bound for the com-

pression/expansion phases, when the engine dynamics is

dominated by the superadiabatic driving for small τ.

To derive an upper bound on the superdiabatic eﬃ-

ciency (7), we use inequality (10) to obtain,

ηSA ≤ηQSL

SA =−hW1iAD +hW3iAD

hQ2i+~(L1+L3)/τ ,(11)

where Li(i= 1,3) are the respective Bures angles for

the compression/expansion steps. On the other hand, an

upper bound on the superadiabatic power (9) is,

PSA ≤PQSL

SA =−hW1iAD +hW3iAD

τ1

QSL +τ3

QSL

,(12)

where τi

QSL (i= 1,3) are the respective speed-limit

bounds (10) for the compression/expansion phases.

The two quantum speed limit bounds (11) and (12) are

shown in Figs. (3)-(4) (green solid). We ﬁrst notice that

the quantum bound (11) on the eﬃciency is sharper than

the thermodynamic bound given by the constant adia-

batic eﬃciency ηAD. Remarkably, quantum theory fur-

ther imposes an upper bound on the power, whereas the

second law of thermodynamics does not [48]. Quantum

thermodynamics hence establishes tighter bounds than

classical thermodynamics. The latter result may be un-

derstood by noting that thermodynamics does not have

the notion of time scale, contrary to quantum mechanics.

Finally, we stress that the two speed limit bounds directly

follow from the deﬁnitions of eﬃciency and power. As a

result, they are independent of the thermodynamic cycle

considered and generically apply to any quantum heat

engine, not just to the quantum Otto motor.

Conclusions. We have performed a detailed study of

both eﬃciency and power of a superadiabatic quantum

heat engine. We have explicitly accounted for the ener-

getic cost of the superadiabatic driving, deﬁned as the

time average of the local counterdiabatic potential. We

have found that the eﬃciency of the engine markedly

drops with decreasing cycle time. However, this drop

is much slower than that of the nonadiabatic eﬃciency

without the shortcut. As result, superadiabatic ma-

chines outperforms their conventional counterparts for

very short cycles, when the work gain generated by the

counterdiabatic driving outweighs its energetic cost. We

have additionally derived generic upper bound on supera-

diabatic eﬃciency and power based on the idea of quan-

tum speed limits. These quantum bounds, valid for gen-

eral thermal motors, are tighter than the usual bounds

based on the second law of thermodynamics. We there-

fore expect them to be useful for future investigations of

thermal machines in the quantum regime.

Acknowledgments This work was partially supported

by the EU Collaborative Project TherMiQ (Grant Agree-

ment 618074) and the COST Action MP1209.

Supplemental Material

Appendix A: Local counterdiabatic energy

We here present a derivation of the mean energy of the

local counterdiabatic Hamiltonian HLCD and of the cor-

responding adiabaticity parameter Q∗

LCD used during the

compression/expansion protocols. We consider a time-

dependent harmonic oscillator with Hamiltonian,

H0(t) = p2

2m+mω2

tx2

2,(A1)

where ωtis the time-dependent angular frequency, mthe

mass, and (p,x) the respective momentum and position

operators. The initial energy eigenstates at t= 0 with

ω(0) = ω0in coordinate representation are given by

ψn(x, 0) = 1

√2nn!mω0

π~1/4

exp −mω0

2~x2Hnrmω0

~x,

(A2)

5

where Hnare Hermite polynomials and E0

n=~ω0(n+

1/2) the corresponding energy eigenvalues. The instan-

taneous eigenstates and their corresponding eigenvalues

are obtained by replacing ω0with ωt.

The shortcut to adiabaticity may be implemented by

adding a time-dependent counterdiabatic (CD) term to

the system Hamiltonian (A1) [3–6]:

HCD

SA (t) = −˙ωt

4ωt

(xp +px) = i~˙ωt

4ωt

(a2−a†2).(A3)

The last equality is obtained by expressing x=

p~/2mωt(a†+a) and p=ip~mωt/2 (a†−a), in terms

of the annihilation and creation operators aand a†. The

total Hamiltonian HCD(t) = H0(t) + HCD

SA (t) is still

quadratic in xand pand may thus be considered that

of a generalized harmonic oscillator [4, 54]. However,

since the Hamiltonian (A3) is a nonlocal operator, it is

often convenient to look for a unitarily equivalent Hamil-

tonian with a local potential [5, 6]. Applying the canon-

ical transformation, Ux= exp im ˙ωx2/4~ω, which can-

cels the cross terms xp and px, to the Hamiltonian (A3)

leads to a new local counterdiabatic (LCD) Hamiltonian

of the form [5, 6],

HLCD(t) = U†

x(HCD(t)−i~˙

UxU†

x)Ux

=p2

2m+mΩ2

tx2

2,(A4)

with the modiﬁed time-dependent (squared) frequency,

Ω2(t) = ω2

t−3 ˙ω2

t

4ω2

t

+¨ωt

2ωt

.(A5)

This resulting Hamiltonian is local and still drives the

evolution along the adiabatic trajectory of the system of

interest. By demanding that HLCD =H0at t= 0, τ and

imposing ˙ω(τ) = ¨ω(τ) = 0, the ﬁnal state is equal for

both dynamics, even in phase, and the ﬁnal vibrational

state populations coincide with those of a slow adiabatic

process [5]. It can be readily shown that Ω2(t) approaches

ω2(t) for very slow expansion/compression process [6].

Exact solutions of the Schr¨odinger equation for a time-

dependent harmonic oscillator have been extensively in-

vestigated [55–57]. Following Lohe [57], a solution based

on the invariants of motion is of the form,

I(t) = b2

2mp2+m˙

b2

2x2−b˙

b

2(px +xp) + mω2

0

2b2x2,(A6)

where ω0is an arbitrary constant—a convenient choice

is to set ω0=ω(0),(ω2

0>0). The scaling factor b=b(t)

is a solution of the Ermakov diﬀerential equation,

¨

b+ω2

tb=ω2

0/b3.(A7)

In the adiabatic limit, ¨

b'0 and

b(t)→bad =rω0

ωt

.(A8)

Equation (A7) is valid for any given ωtand its general

solution can be constructed from the solutions f(t) of the

linear equation of motion for the classical time-dependent

harmonic oscillator [58],

¨

f+ω2

tf= 0,(A9)

according to b2/ω0=f2

1+W−2f2

2, where f1,f2are

independent solutions of Eq. (A9) and the Wronskian

W[f1, f2] = f1˙

f2−˙

f1f2is a nonzero constant. We note

that the Wronskian properties of Eq. (A9) can be used

to show the equivalence of the adiabaticity parameter

derived here and that of Husimi [44, 45] (see Ref. [23]).

The general solution of the time-dependent Schr¨odinger

equation for the Hamiltonian (A4) is hence,

Ψn(x, t) = 1

√2nn!mω0

π~b21/4

exp "im˙

b

2~bx2−iZt

0

ω0(n+ 1/2)

b(t0)2dt0#exp −mω0

2~b2x2Hnrmω0

~

x

b.(A10)

The time-dependent energy eigenstates,

HLCD|Ψn(x, t)i=E|Ψn(x, t)i, are explicitly given

by,

E=hΨn(x, t)|HLCD|Ψn(x, t)i

=E0

n

b2−m

2b¨

b−˙

b2hΨn(x, 0)|x2|Ψn(x, 0)i

+˙

b

2bhΨn(x, 0)|xp +px|Ψn(x, 0)i.(A11)

We next consider a quantum oscillator initially pre-

pared in thermal equilibrium state with density operator,

ρeq =

∞

X

n=0

p0

n|Ψn(x, 0)ihΨn(x, 0)|,(A12)

where p0

n= exp(−βE0

n)/Z0is the probability that the

oscillator is in state |Ψn(x, 0)iand Z0is the partition

function. The initial thermal mean energy at t= 0 is

accordingly,

hH(0)i=mω2

0x2(0)=~ω0

2coth β~ω0

2.(A13)

6

0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

2.5

3.0

3.5

Time t/τ

Parameter Q*

FIG. 5: Adiabaticity parameter Q∗

LCD(t), Eq. (18), (green

dashed) and Eq. (51) of Ref. [22], ¯

Q∗

LCD(t) = 1 + ˙ω2

t/(8ω4

t),

(blue dotted) as a function of t/τ for ω0/ω1= 0.15.

The expectation value of the local counterdiabatic Hamil-

tonian HLCD(t) at time tfollows from Eqs. (11)-(13) as

hHLCD(t)i=

∞

X

n=0

p0

nhΨn(x, t)|HLCD|Ψn(x, t)i

=hH(0)i

b2+(˙

b2−b¨

b)

2ω2

0hH(0)i,(A14)

where we have used the fact that h{x, p}(0)i= 0 for

thermal equilibrium state. Since the squared frequency

(A5) can be rewritten in the adiabatic limit as,

Ω2

t=ω2

t−¨

bad

bad

,(A15)

we obtain, using Eqs. (A5), (A8) and (A15), the expres-

sion,

b¨

b−˙

b2=b2

ad (−¨ωt

2ωt

+1

2˙ωt

ωt2).(A16)

Finally, substituting Eq. (A16) into Eq. (A14), the mean

energy of the local counterdiabatic driving is found to be,

hHLCD(t)i=Q∗

LCD(t)

b2

ad hH(0)i,(A17)

where the adiabaticity parameter Q∗

LCD(t) is given by,

Q∗

LCD(t) = 1 −˙ω2

t

4ω4

t

+¨ωt

4ω3

t

.(A18)

Note that Eq. (A18) corrects Eq. (51) in Ref. [22].

Appendix B: Superadiabatic energy

The expectation value of the superadiabatic potential

HSA may be evaluated from Eqs. (4) and (17). We have,

HSA(t) = HLCD (t)−H0(t)

=m

2−3 ˙ω2

t

4ω2

t

+¨ωt

2ωtx2.(B1)

As a consequence, we obtain,

hHSA(t)i=ωt

ω0hH(0)i−˙ω2

t

4ω4

t

+¨ωt

4ω3

t.(B2)

The properties of the shortcut imply that hHSA(0, τ )i= 0

at the beginning and at the end of the protocol.

Appendix C: Bures length

The Bures length between initial and ﬁnal density op-

erators of the system is L(ρτ, ρ0) = arccos pF(ρτ, ρ0),

where the F(ρτ, ρ0) is the ﬁdelity between the two states

[59]. For the considered driven harmonic oscillator, initial

and ﬁnal states are Gaussian and the ﬁdelity is explicitly

given by [53]:

F(ρτ, ρ0) = 2

pct2(β0/2) + ct2(β1/2) + 2Q∗ct(β0/2)ct(β1/2) + c2(β0/2)c2(β1/2) −c(β0/2)c(β1/2) .(C1)

where i=~ωi,ct(x) = coth(x) and c(x) = csch(x).

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