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Performance of superadiabatic quantum machines

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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.
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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) + 2
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 AB: 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 BC: 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 CD: 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 DA:
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
1ω1) coth β1~ω1
2,(1)
hW3i=~
2(ω1Q
3ω2) 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
2Q
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
22
tω2
tx2=m
23 ˙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(ωfωi)s315(ωfωi)s4+6(ωfωi)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 )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 τ0.
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)cycle (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],
ττQSL =~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),(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+2
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!0
π~1/4
exp 0
2~x2Hnr0
~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
(a2a2).(A3)
The last equality is obtained by expressing x=
p~/2t(a+a) and p=ip~t/2 (aa), 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+m2
tx2
2,(A4)
with the modified time-dependent (squared) frequency,
2(t) = ω2
t3 ˙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
2x2b˙
b
2(px +xp) + 2
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 b20=f2
1+W2f2
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!0
π~b21/4
exp "im˙
b
2~bx2iZt
0
ω0(n+ 1/2)
b(t0)2dt0#exp 0
2~b2x2Hnr0
~
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
b2m
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=2
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 ω01= 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+(˙
b2b¨
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,
2
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
23 ˙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(β0/2) + ct2(β1/2) + 2Qct(β0/2)ct(β1/2) + c2(β0/2)c2(β1/2) c(β0/2)c(β1/2) .(C1)
where i=~ωi,ct(x) = coth(x) and c(x) = csch(x).
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