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10 JUNE 2016

PHYSICAL REVIEW LETTERS

PRL 116, 230503 (2016)

Speeding up Adiabatic Quantum State Transfer by Using Dressed States
Alexandre Baksic, Hugo Ribeiro, and Aashish A. Clerk
Department of Physics, McGill University, 3600 rue University, Montréal, Quebec H3A 2T8, Canada(Received 14 December 2015; published 9 June 2016)
We develop new pulse schemes to significantly speed up adiabatic state transfer protocols. Our general
strategy involves adding corrections to an initial control Hamiltonian that harness nonadiabatic transitions.
These corrections define a set of dressed states that the system follows exactly during the state transfer. We
apply this approach to stimulated Raman adiabatic passage protocols and show that a suitable choice ofdressed states allows one to design fast protocols that do not require additional couplings, while
simultaneously minimizing the occupancy of the “intermediate” level.
DOI: 10.1103/PhysRevLett.116.230503

Introduction.—The general goal of moving quantum
states between two different systems finds numerous
applications in quantum information processing [1,2]. It
has generated intense theoretical interest, with numerous
approaches developed to allow high fidelity state transfer
that are robust against dissipation and noise. Among themore powerful and interesting strategies are adiabatic
transfer protocols [3]. These generically involve adiabatically evolving an eigenstate of a composite quantum
system, such that the state is initially localized in the“source” system and ends up being localized in the “target”
system [see Fig. 1(a)]. The adiabatic evolution thus
corresponds to a state transfer, with the initial state of
the source system “riding” the adiabatic eigenstates, andending up in the target system. The most famous examples
of such approaches are the stimulated Raman adiabatic
passage (STIRAP) [4] and coherent tunnelling by adiabatic
passage [5] protocols, well known in atomic physics.
There are two main advantages in using transfer protocols based on adiabatic passage instead of resonant
techniques. First, adiabatic passage is inherently morerobust against pulse area and timing errors. Second, it isuseful in situations where the source and target only interact
via a lossy “intermediate” system, as it allows one to use themediated coupling without being harmed by the noise. This
is of particular relevance in optomechanical state transfer
schemes, where a dissipative mechanical resonator is theintermediate system [6–9].
Despite these advantages, adiabatic schemes are necessarily slow, and hence can suffer from dissipation and noise in
the target and/or source system. Therefore, several
approaches have been put forward to speed up adiabatic
passage [10,11]. Among the known methods, counterdiabatic control [12], also referred to as transitionless driving[13], or its higher-order variants [14,15] are analyticalmethods that allow one to construct a modification of an
original Hamiltonian to compensate for nonadiabatic errors.
While in principle transitionless driving would allow a
perfect state transfer, it suffers from two major flaws: it
sometimes requires either a direct coupling of the source and0031-9007=16=116(23)=230503(5)target systems [16–19] or a coupling not available in theoriginal Hamiltonian [20]. The higher-order variants overcome the first flaw of transitionless driving, but do not allow
one to control the population in the intermediate system[14,15]. A related approach based on constructing dynamical
invariants [21] has also been applied to STIRAP, but it leads
to pulse schemes that either need an infinite energy gap to be
perfect [22], or do not smoothly turn on or off [22,23] and arethus extremely challenging to implement experimentally.
Finally, one could use the general framework of optimalquantum control [24], but as we will show there is no need touse such a complex procedure.
In this Letter, rather than constructing perfect protocols
from scratch, we present an approach that corrects existing
efficient adiabatic protocols such that they allow for a
perfect state transfer even in the nonadiabatic regime.
Moreover, the high flexibility of this approach allows
one to engineer and reduce the population in the intermediate lossy level. The main idea of our approach is
sketched in Fig. 1(b). We work with a basis of dressedstates whose very definition incorporates the nonadiabatic
processes. Then, by introducing additional control fields,
we can ensure these dressed states coincide with the desired
adiabatic eigenstate at the initial and final time of the
(a)

(b)

FIG. 1. (a) Schematic of a composite quantum system where
the source and the target systems (qubits in this schematic) arecoupled via some intermediate system. (b) Schematic of thepossible evolutions: (red line) perfect adiabatic evolution; (blueline) speeding up the evolution results in nonadiabatic errors
leading to an imperfect state transfer; (green line) by dressing theadiabatic eigenstates it is possible to design an evolution that
leads to a perfect state transfer.

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PHYSICAL REVIEW LETTERS

protocol. It is thus possible to do a state transfer by having
the exact dynamics follow these new dressed states, even if
the protocol is too fast to allow a naive adiabatic evolution.
We illustrate this general idea by developing simple andeffective pulses for speeding up adiabatic state transfer ingeneric Λ-system setups.
General problem.—We consider a general composite
quantum system, composed of source, intermediate, andtarget subsystems, respectively labeled A, B, and C. The
goal is to transfer some initial quantum state jψi (e.g., aqubit state) from subsystem A to the target subsystem C.
Adiabatic transfer achieves this goal by constructing a timedependent Hamiltonian whose instantaneous eigenstates
evolve in a way that facilitates the transfer. We start by
assuming that one has constructed such a protocol. The
instantaneous eigenstates (hereafter referred to as adiabatic
eigenstates) and corresponding adiabatic energies (bothindexed by k) are defined via
ĤðtÞjφk ðtÞi ¼ Ek ðtÞjφk ðtÞi:ð1Þ
A subset of eigenstates has been engineered to form a basisof the A subsystem at the initial time ti and a basis of thetarget system at the final time tf . In other words the
eigenstates fjφmj ðtÞignj¼0 will serve as “medium” states andhave the following properties:
jφmj ðti Þi ¼ jβj iA ⊗ jχ i iB;C ;jφmj ðtf Þi ¼ jχ f iA;B ⊗ jγ j iC ;

ð2Þwhere fjβj ignj¼0 and fjγ j ignj¼0 span the subspaces A and C,
respectively. The states jχ i iB;C and jχ f iA;B are not necessarily equal.
It follows that if the evolution is perfectly adiabatic (i.e.,
happens on a time scale τ ≫ 1=ΔE, where ΔE is thesmallest instantaneous energy gap of the system), the initialsource state will be mapped on the final target state.
However for τ ≲ 1=ΔE, the evolution will not be perfectly
adiabatic. It is convenient to move to the adiabatic frame
where the adiabatic eigenstates are time independent. The
P
relevant unitary is ÛðtÞ ¼ k jφk ihφk ðtÞj. At each instantin time, ÛðtÞ maps the adiabatic eigenstate jφk ðtÞi onto thetime-independent state jφk i. In the adiabatic frame, theHamiltonian becomesĤad ðtÞ ¼ Ĥ 0 ðtÞ þ ŴðtÞX
dÛðtÞ †Û ðtÞ:Ek ðtÞjφk ihφk j þ i
¼
dt
k

ð3Þ

The operator ŴðtÞ generically has off-diagonal matrix
elements connecting the various adiabatic eigenstates.
The magnitude of these matrix elements increases as τ
decreases, leading to an imperfect state transfer.
Correcting nonadiabatic errors.—In order to correct the
nonadiabatic errors, we look for a correction HamiltonianĤc ðtÞ such that the modified Hamiltonian Ĥmod ðtÞ¼ ĤðtÞþĤc ðtÞ leads to a perfect state transfer. For this scheme to beweek ending
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reasonable, we require that Ĥ mod ðtÞ has no unattainably
large coupling strengths and that Ĥc ðtÞ does not involve
couplings that cannot be experimentally implemented.
Our strategy is based on the observation that thecorrected dynamics only needs to evolve the system fromthe correct state at ti to the correct state at tf [cf. Fig. 1(b)].
This suggests a strategy whose crucial ingredients are: (I) a
new basis of dressed states jφ~ k ðtÞi formally defined by atime-dependent unitary transformation VðtÞ asjφ~ k ðtÞi ≡ V̂ † ðtÞjφk i;ð4Þand (II) a control field Ĥc ðtÞ that is added to the original
Hamiltonian.
The additional control Hamiltonian Ĥ c ðtÞ and dressedstate basis [i.e., V̂ðtÞ] must be chosen as to satisfy thefollowing constraints. (i) The dressed medium states
coincide with the medium states at time ti and tf
V̂ † ðtf Þjφmj i ¼ V̂ † ðti Þjφmj i ¼ jφmj i:ð5Þ(ii) For all j, the evolution of jφ~ mj ðtÞi is trivial in the basis
defined by V̂ðtÞ (i.e. jφ~ mj ðtÞihφ~ mj ðtÞj is a conserved
quantity).
If both these conditions are satisfied, then the perfectdesired state transfer will occur. A sketch of the generalidea is shown in Fig. 1(b). Condition (ii) is better defined bymoving in the frame defined by V̂ in which the Hamiltoniantakes the formdV̂ †Ĥnew ðtÞ ¼ V̂ Ĥ ad ðtÞV̂ † þ V̂ Û Ĥc ðtÞÛ† V̂ † þ i
V̂ : ð6Þ
dt
We have omitted the explicit time dependence of Û and V̂ forclarity. In this new frame the dressed states of Eq. (4) have no
intrinsic time dependence, and condition (ii) becomeshφ~ mj jĤnew jφ~ k i ¼ 0 for 1 ≤ k ≤ n;k ≠ mj :ð7Þ

In other words, Ĥc ðtÞ has to be designed such that it cancels
the unwanted off-diagonal elements in Ĥnew ðtÞ.
To summarize, the general method involves picking anappropriate pair of operators ½V̂ðtÞ; Ĥc ðtÞ: the unitary V̂ðtÞ
selects a (time-dependent) basis of dressed states, while theadditional control Hamiltonian Ĥc ðtÞ ensures the correct
dynamics. The net result is that the desired state transfer
dynamics occurs perfectly despite not being in the adiabatic limit.
Transitionless driving [12–14] is a special case of thisapproach and is retrieved by choosing V̂ðtÞ ¼ 1̂ andĤc ¼ −Û† Ŵ Û. The alternative schemes described inRefs. [14,15] are also recovered from our approach, bychoosing the dressed states as the superadiabatic states[25–27] (instantaneous eigenstates of Ĥ ad ) or its higher
order counterparts. In what follows, we use our method toderive truly new protocols.
Application: STIRAP.—We apply our general approachto the problem of STIRAP [3,4] in a three-level Λ-type

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system. For concreteness, each of the subsystems A, B, andC are qubits such that A and C only interact with B via theso-called pump and Stokes pulses (Ωp and ΩS , respectively). The Hamiltonian reads
ĤðtÞ ¼ Ωp ðtÞjBihAj þ ΩS ðtÞjBihCj þ H:c:

ΩS ðtÞ ¼ ΩðtÞ cos θðtÞ:

ð9Þ

The adiabatic eigenstates (see the Supplemental Material[28]) consist of two “bright” states jφ ðtÞi with energyE ðtÞ ¼ ΩðtÞ, a “dark” state jφD ðtÞi with ED ðtÞ ¼ 0, andjφ0 ðtÞi ¼ j000i with E0 ðtÞ ¼ 0. A general adiabatic statetransfer from qubit A to C can be performed using the
medium statesjφD ðtÞi ¼ cos θðtÞjAi þ sin θðtÞjCi
ð10Þ
and jφ0 ðtÞi, which operates a state transfer from jAi to jCi
by using the counterintuitive pulse sequence θðti Þ ¼ 0 andθðtf Þ ¼ π=2. As mentioned before, as the protocol time is
reduced, the perfect adiabatic transfer will be more andmore corrupted. This is described by going in the adiabatic
basis where the Hamiltonian (8) becomes_ M̂ y ;Ĥ ad ðtÞ ¼ ΩðtÞM̂z þ θðtÞ

ð11Þ

where
i−jφ
p
ﬃﬃﬃ M̂ z ¼jφþ ihφþ j−jφ− ihφ− j, M̂x ¼ðjφ−p
ﬃﬃﬃ þ iÞhφD j=
2 þH:c:, and M̂y ¼ iðjφþ i þ jφ− iÞhφD j= 2 þ H:c: arespin 1 operators, obeying the commutation relation½Mp ; Mq  ¼ iεpqr M r . The second term of the adiabatic
Hamiltonian (11) corresponds to the nonadiabatic couplings coming from the inertial term in Eq. (3).
Thanks to the analogy between the adiabatic
Hamiltonian (11) and a spin 1 in a magnetic field,
ingredient (I) (i.e., the construction of dressed states) of
our approach can be parametrized as a rotation of the spinwith Euler angles ξðtÞ, μðtÞ, and ηðtÞV̂ g ¼ exp ½iηðtÞM̂z  exp ½iμðtÞM̂ x  exp ½iξðtÞM̂z :

ð12ÞIn order to satisfy condition (i), the angle μðtÞ has to satisfyμðti Þ ¼ μðtf Þ ¼ 0ð2πÞ and the two other angles can havearbitrary values. It can be shown that by choosing theingredient (II) of our method to have the general formĤc ðtÞ ¼ Û †ad ðtÞðgx ðtÞM̂ x þ gz ðtÞM̂z ÞÛad ðtÞ

Moreover, in order to satisfy Eq. (7), the control parameters
have to be chosen as
gx ðtÞ ¼

ð8Þ

with jAi ¼ j100i, jBi ¼ j010i, jCi ¼ j001i. The pulses areparametrized by the frequency ΩðtÞ and the angle θðtÞΩp ðtÞ ¼ −ΩðtÞ sin θðtÞ;week ending
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PHYSICAL REVIEW LETTERS

ð13Þ

we find a control Hamiltonian Ĥc that does not directly
couple the states jAi and jCi. The corrected protocol willconsist in a simple modification of the original STIRAP
angle and amplitude
gx ðtÞ
~ ¼ θðtÞ − arctanθðtÞ → θðtÞ; ð14Þ
ΩðtÞ þ gz ðtÞ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ~ΩðtÞ → ΩðtÞ
¼ ðΩðtÞ þ gz ðtÞÞ2 þ g2x ðtÞ: ð15Þμ_− θ_ tan ξ;cos ξ

μ_ sin ξ − θ_gz ðtÞ ¼ −Ω þ ξ_ þ;
tan μ cos ξ

ð16Þ
ð17Þ

and are independent of ηðtÞ. Within our framework, it canbe shown that the population in the intermediate level jBi isgiven byjhψðtÞjBij2 ¼ sin2 μðtÞcos2 ξðtÞ:

ð18Þ

From now on, in order to keep the discussion simple, we
focus on the ξðtÞ ¼ 0 case.
Application to Vitanov-style pulses.—We apply thesedressed-state protocols to the optimal STIRAP pulses
discussed by Vitanov et al. in Ref. [29] and defined byΩðtÞ ¼ Ω0 ;θðtÞ ¼π
1;2 1 þ e−t=τð19Þ

where the time scale τ controls the effective duration of the
protocol. The simplest nontrivial choice of the dressedstates basis is the superadiabatic basis, for which
_ 
θðtÞ
μ ¼ − arctan_gz ðtÞ ¼ 0:
ð20Þ;
gx ðtÞ ¼ μ;ΩðtÞThis choice will be referred to as superadiabatic transitionless driving (SATD). With this choice the only way to
reduce the population in the intermediate level [cf. Eq. (18)]_is to decrease the magnitude of θðtÞ,and hence slow down
the protocol (i.e., longer τ). Interestingly, SATD represents
a nonperturbative version of the derivative removal by
adiabatic gate approach to leakage errors [30,31] applied tothis problem (see the Supplemental Material [28]).
Our approach allows one to construct alternatives to
SATD (based on alternate dressed states) that reduce theintermediate-level occupancy. This can be extremely beneficial in systems where the intermediate state is lossy, butwhere adiabatic evolution is impossible, as the protocol
must be fast to avoid dissipation of the source and/or target
system, or because of slow drifts of system parameters. A
concrete example with all these features is optomechanicalstate transfer [6–9]. By generalizing Eq. (20) to
 _

θðtÞ
μ ¼ − arctan;
gx ðtÞ ¼ μ_ ;fðtÞΩðtÞ
_θðtÞgz ðtÞ ¼ −Ω −
ð21Þ
tan μ
we can chose the auxiliary function fðtÞ to reduce μ(and hence the amount of state dressing) to avoid unnecessary B-state population. Here, we choose to consider thesimple class of functions fðtÞ¼1þAexpð−t2 =T 2 Þ[fðtÞ ≥ 1 ∀ t] with A > 0 and T > 0 two parameters

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that can be optimized for each τ to minimize the populationin B. As we show below, this intuitive and physicallymotivated choice allows for a sizeable reduction of theoccupancy of the intermediate level without having to rely
on more complex methods (e.g., control theory).
To compare protocols, we look at the relevant case where
the fidelity is limited both by a nonzero τ in Eq. (19) and by
the protocol starting and ending at a finite time. In theory,the protocol should start at ti ¼ −∞ and end at tf ¼ þ∞ inorder to achieve the requirement θðti Þ ¼ 0, θðtf Þ ¼ π=2,
and μðti Þ ¼ μðtf Þ ¼ 0ð2πÞ. To simulate pulses with a finite
duration, we have chosen tf ¼ −ti ¼ 15τ such thatΩp ðti Þ ¼ ΩS ðtf Þ < 10−6 Ω0 . With our choices of correction, the shorter the protocol time is, the bigger the~ τÞ is. We consider the case where each
amplitude Ωðt;corrected pulse cannot exceed its original maximal~~~~
amplitude
Ω0 fmaxt ½Ωðt;τÞsinθðt;τÞ;Ωðt;τÞcos
θðt;τÞ≤Ω0 ; ∀tg. This constraint implies that we can only correct
protocols with an effective protocol time τ > τmin ≃1=2.63Ω0 .
ε ¼ 1 − F ¼ 1 − jC hψðtf Þjψðti ÞiA j2 :

ð22ÞSince we are interested in a qubit state transfer and j000i
has a trivial dynamics, only the transfer of state jAi to jCi
gives rise to errors. Thus, we plot the fidelity for transferring the jAi state only, which sets an upper bound for theerror when transferring a superposition of an arbitrary qubit
state (see the Supplemental Material [28]). In Fig. 2(a), weplot the residual error ε as a function of τ for SATD
[Eq. (20)] and the modified SATD [Eq. (21)] with optimized parameters. Both choices reduce the residual error by
the same amount and lead to a several orders of magnitude
reduction as compared to the protocol defined by Eq. (19).
The oscillatory behavior is a direct consequence of having
finite-time pulses (see the Supplemental Material [28]).
To illustrate the additional advantage of our choice ofcorrection, we consider the time integral over the full
protocol duration of the population in jBi. In Fig. 2(b), weplot this quantity for both SATD and the modified SATD:
the integrated population is reduced between ≈21% −25.5% with the modified SATD [Eq. (21)] as comparedto SATD [Eq. (20)]. In Figs. 2(c) and (d), we plot thecorrected pump pulse for SATD and the modified SATD fordifferent values of τ. The Stokes pulse is the symmetricreflection of the pump pulse with respect to ðtf − ti Þ=2. The
SATD pulses rapidly converge to the Vitanov style pulses[Eq. (19)] when τ increases, while the modified SATD
pulses converge more slowly. This is due to the fact that themodified SATD pulses have been designed not only to
reduce the residual error, but also to reduce the populationin the mechanics, which slowly converges to 0 as τ → ∞.
Application to Gaussian pulses.—An additional advantage of our approach is that it allows one to correct
protocols for which the correction (20) does not work.

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FIG. 2. (a) Comparison of the residual error between STIRAP[Eq. (19)], SATD [Eq. (20)], and the modified SATD [Eq. (21)] as
a function of the effective protocol duration τ in units of τmin .
(b) Comparison of the integrated population in jBi over the whole
protocol time between SATD [Eq. (20)] and our new dressed state
approach [Eq. (21)] as a function of τ in units of τmin . Inset: ratioof those two quantities. The integrated population is reduced by atleast 21% and at most 26% with our new protocol. Plot of thecorrected pump pulse for SATD (c) and the modified SATD(d) for different values of τ as a function of time ðt − ti Þ in units ofthe total protocol time ðtf − ti Þ.

In particular, the most common approach to STIRAP uses
Gaussian pulses [3,4] Ωp ðtÞ ¼ Ω0 exp½−ðt − t0 =2Þ2 =τ2 and ΩS ðtÞ ¼ Ω0 exp½−ðt þ t0 =2Þ2 =τ2  with t0 the delay
time between the two pulses. Using the parametrization
defined in Eq. (9), we haveθðtÞ ¼ arctan ½expð2tt0 =τ2 Þ;qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 2t þ t20 =4
2 cosh ðtt0 =τ2 Þ:ΩðtÞ ¼ Ω0 exp −
τ2

ð23Þ

For this particular case, we cannot use the SATD prescription to construct a control Hamiltonian as the condition μðti Þ ¼ μðtf Þ ¼ 0ð2πÞ is not satisfied (for this choice_of pulse θðtÞ=ΩðtÞ→ þ∞ as t → ∞). However, ourdressed state approach allows one to find a control
Hamiltonian using Eq. (17) (ξ ¼ 0) and


_
θðtÞ
μðtÞ ¼ − arctan:
ð24Þ
gðtÞ=τ þ ΩðtÞ
Here, gðtÞ=τ is used to regularize μðtÞ: it has to be chosen_ In Fig. 3,
such that it tends to zero at ti and tf slower than θ.
we have plotted the residual error for STIRAP with
Gaussian densities [Eq. (23)] and for the modified
SATD [Eq. (24)]. We have chosen t0 ¼ 6=5τ and gðtÞ ¼
A= cosh ζt with A ¼ 1=40 and ζ ¼ 9=10τ, which givesτmin ≈ 1=1.27Ω0 . Under the condition Ωp ðti Þ ¼ ΩS ðtf Þ <230503-4

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FIG. 3. (a) Comparison of the residual error for STIRAP with
Gaussian densities [Eq. (23)] and the modified SATD [Eq. (24)]as a function of the effective protocol duration τ in units of τmin .
The residual error is reduced by several orders of magnitude in
the nonadiabatic regime. (b) Corrected pump pulse for different
values of τ as a function of time ðt − ti Þ in units of the total
protocol time ðtf − ti Þ.

10−6 Ω0 , we have tf ¼ −ti ¼ 6τ. This new pulse scheme
leads to a reduction of the residual error by several orders ofmagnitude [see Fig. 3(a)] in the nonadiabatic regime while
SATD [Eq. (20)] fails. In Fig. 3(b), we plot the corrected
pump pulse for different values of τ. The Stokes pulse is thesymmetric reflection of the pump pulse with respectto ðΩp ; tÞ ¼ ½0; ðtf − ti Þ=2.
Conclusion.—We have developed a general method toachieve a perfect state transfer between two quantum
systems coupled via an intermediate lossy system. In
contrast to previous schemes, our approach is both physically transparent and extremely flexible, allowing application to a wide variety of realistic experimental situations.
In future work, it could be interesting to investigate theresilience of the generated pulse sequences with respect toexperimental imperfections of the system and of the control
fields as in Refs. [32,33]. It would also be interesting toinvestigate the implementation of our method in more
complicated systems, where analytical diagonalization is
not possible. In particular one could study perturbative
variants of our approach as well as numerical diagonalization allowing one to look for transitionless driving andhigher order variants corrections [13–15].
We acknowledge funding from the University of
Chicago Quantum Engineering program and the AFOSR
MURI program. H. R. also acknowledges funding from the
Swiss SNF.

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013417 (2009).
[30] F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K.
Wilhelm, Phys. Rev. Lett. 103, 110501 (2009).
[31] J. M. Gambetta, F. Motzoi, S. T. Merkel, and F. K. Wilhelm,
Phys. Rev. A 83, 012308 (2011).
[32] A. Ruschhaupt, X. Chen, D. Alonso, and J. G. Muga, New J.
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Rev. Lett. 111, 050404 (2013).

230503-5

Supplementary information for “Speeding up adiabatic quantum state transfer byusing dressed states”
Alexandre Baksic, Hugo Ribeiro, and Aashish A. Clerk
Department of Physics, McGill University, 3600 rue University, Montréal, Quebec H3A 2T8, Canada

I.

INSTANTANEOUS EIGENSTATES OF THE STIRAP HAMILTONIAN

The Hamiltonian describing the interaction of the qubits A and C via the intermediate system B is given by
H = Ω(t) cos θ(t)|CihB| − sin θ(t)|AihB| + h.c. .
(S1)and its instantaneous eigenstates are described by the lines of the unitary

√
√ 
√
sin θ(t)/ 2 −1/ 2 − cos θ(t)/ 20√
sin θ(t) √  ,Uad =  cos θ(t)√sin θ(t)/ 2 1/ 2 − cos θ(t)/ 2(S2)

where the first and last rows describe the so called “bright” states (|ϕ+ i and |ϕ− i) with instantaneous energy E+ (t) =+Ω(t) and E− (t) = −Ω(t), respectively. The middle row describes the so called “dark” state |ϕD i (it does not involvethe intermediate state |Bi) with a zero instantaneous energy, ED = 0. Since our goal is to transfer the state of qubit
A to qubit C, we need to add to those eigenstate the trivial state |000i which has zero energy and a trivial dynamics.
II.

OPTIMAL A AND T PARAMETERS FOR VITANOV STYLE PROTOCOL WITH MODIFIED
SA-TD CORRECTION

To obtain the optimal reduction of the population in the intermediate state we chose!
θ̇(t)µ(t) = − arctan,
f (t)Ω(t)

(S3)

with
f (t) = 1 + A exp(−t2 /T 2 ).

(S4)

We numerically optimize the values of A and T to obtain the largest reduction of the “B” state population for eachvalue of τ . The results are plotted in Fig. (S1).
III.

EXACT DYNAMICS OF THE CORRECTED PROTOCOLS

Once the protocols are corrected, we are able to write a simple expression for the exact system dynamics, as thecorrection makes the Hamiltonian diagonal in the dressed-state basis. With the choiceµ̇− θ̇ tan ξ,
cos ξ

(S5)

µ̇ sin ξ − θ̇gz (t) = −Ω + ξ˙ +,tan µ cos ξ(S6)θ̇ + ξ˙ + µ̇ sin ξM̂z
sin µ cos ξ(S7)gx (t) =we can show that
Hnew = −= Ẽ(t)M̂z .
The evolution of an arbitrary initial state is thus given by
 Z t

†
|ψ(t)i = Uad(t)Vg† (t) exp −i
dt0 Ẽ(t0 )M̂z Vg (ti )Uad (ti )|ψ(ti )i.
ti

(S8)

(S9)

2

(b)

(a)

Figure S1: Optimized values of the parameters A and T appearing in the function f (t) Eq. (S4) as a function of τ /τmin forti = −tf .
A.

Fidelity and residual error of the state transfer

The adiabatic protocols that we have chosen (Vitanov style and Gaussian) should start at ti = −∞ and end at
tf = +∞ in order to satisfy θ(ti ) = 0 and θ(tf ) = π/2. In addition, their corrected versions (SA-TD and modified
SA-TD) should satisfy µ(ti ) = µ(tf ) = 0(2π) which is also done by choosing ti = −∞ and tf = +∞. However, ina realistic setup one has a finite duration for the protocol (both ti and tf are finite), which leads to a fidelity equal
or smaller than 1 for both the STIRAP and corrected protocols. The fidelity of the state transfer can be calculatedwith the help of Eq. (S9) and assuming an initial arbitrary qubit state |ψ(ti )i = cos(α/2)|Ai + eiβ sin(α/2)|000i . The|000i state has a trivial dynamics, whereas the evolution of state |Ai is described by Eq. (S9). The fidelity is thencalculated (here for ξ(t) = 0 as in the main text) via
F =|hψ(t)|Ci + sin2 (α/2)|2hn= cos2 (α/2) cos θi sin θf cos µi cos µf − cos Q(sin θi cos θf − sin θf cos θi sin µi sin µf )oi2
+ sin Q(sin θi sin θf sin µf + cos θi cos θf sin µi ) + sin2 (α/2)

(S10)

Rt
where µk = µ(tk ), θk = θ(tk ), and Q = tif dt0 Ẽ(t0 ). If we further consider symmetric protocols such that θf = π/2 − θiand µf = µi (in the Vitanov style and Gaussian pulses cases this corresponds to ti = −tf ) we obtainhnoi2
F = cos2 (α/2) cos2 θi cos2 µi + cos Q(sin2 µi cos2 θi − sin2 θi ) + sin Q sin µi sin 2θi ) + sin2 (α/2) .
(S11)
We thus see that two oscillating terms are present (cos Q and sin Q) which explains the oscillating behaviour of theerror ε in the main text (see Figs. 2(a) and 3(a) in the main text). One can calculate rigorous lower and upper bounds
on the fidelity, yielding:
Fmin (α) ≤ F ≤ 1,

(S12)

h
i2
Fmin (α) = cos2 (α/2)(cos2 θi cos 2µi − sin2 θi ) + sin2 (α/2) .

(S13)

with

Since |000i is shared by the three subsystems (source, intermediate, and target) and its dynamics is trivial, it can beperfectly transferred (α = π(2π) ⇒ F = 1 ). The errors due to a finite protocol time are thus due to the amplitude of|Ai only. Hence, the minimal fidelity is achieved for α = 0(2π), which corresponds to the plots in the main text. The
error ε and its maximal value εmax = 1 − Fmin (α = 0) have been plotted as a function of tf /τ and τ /τmin by choosing
symmetric protocols (ti = −tf ) for the Vitanov style pulses in Fig. S2 and for the Gaussian pulses pulses in Fig. S3
with t0 = 6τ /5.

3

Figure S2: Error ε = 1 − F Eq. (S11) as a function of the normalized effective protocol time τ /τmin and normalized final time
tf /τ (we chose tf = −ti ) for the Vitanov style corrected pulses : (a) with SA-TD correction, and (b) with modified SA-TD
correction. Upper bound for the error εmax = 1 − Fmin Eq. (S12) : (c) with SA-TD correction, and (d) with modified SA-TD
correction.
IV.

LINK BETWEEN DRAG-LIKE PERTURBATIVE APPROACHES AND THE SA-TD APPROACH

It is possible to find the SA-TD correction perturbatively by using a DRAG approach [1, 2]. We start from theadiabatic HamiltonianĤad =

1M̂z + θ̇(t)M̂y(t)1=Ĥ (−1) + Ĥ (0)
(t)

(S14)
(S15)

where (t) = 1/Ω(t). The second term (Ĥ (0) ) induces leakage out of the adiabatic subspace of interest (the subspacespanned by {|ϕD i, |000i}) that we want to eliminate at a given order in (t) by using a DRAG approach.
The idea is to search for a unitary transformation
!
n
X
k
(k)
V̂ = exp −i (t)Ŝ
,
(S16)
k=1

as well as a correction
Ĥc,ad =nX
k=1

(t)n Ĥ (n) ,

(S17)

4

Figure S3: (a) Error ε = 1 − F Eq. (S11) as a function of the normalized effective protocol time τ /τmin and the normalized
final time tf /τ (we chose tf = −ti ) for the Gaussian corrected pulses. (b) Upper bound for the error εmax = 1 − Fmin Eq. (S12)for the Gaussian pulses.

that diagonalizes the corrected Hamiltonian in the new frame∞V̂ (Ĥad + Ĥc,ad )V̂ † + i

dV̂ † X kV̂ =
 (t)Ĥk
dt

(S18)

k=0

at a given order n in (t) (i.e. Ĥk ∝ M̂z , ∀k ≤ n). The equations that Ĥ (k) and Ŝ (k) have to fulfill are given in the
Appendix C of Ref. [2].
The zeroth order (in (t)) equation of Appendix C in Ref. [2] is fullfiled by choosingŜ (1) = −θ̇(t)M̂x ,(S19)Ŝ (2) = 0,

(S20)

= −θ̈(t)M̂x ,(S21)

the first order by choosing

Ĥ

(1)

and the second order by choosingθ̇3 (t),3
Ĥ (2) = 0.

Ŝ (3) =

(S22)
(S23)

The easiest way to fulfill the DRAG equations at each order is to choose S (2k) = 0, H (2k) = 0 and H (2k+1) =dS (2k+1) /dt. By doing so at each order, one recovers the SA-TD solution :qdV̂ †V̂ (Ĥad + Ĥc,ad )V̂ † + i
V̂ = Ω(t) 1 + 2 (t)θ̇2 (t)hn dt oiV̂ = exp −i arctan (t)θ̇ M̂x .
Ĥc,ad = −iV̂ †

d
V̂ .
dt

(S24)
(S25)
(S26)

This strategy is equivalent to first diagonalize V̂ Ĥad V̂ † perturbatively in terms of the small parameter θ̇/Ω(t) andthen eliminate the non-diagonal contributions coming from the inertial term i ddtV̂ V̂ † by choosing the correction as to5exactly cancel it.

[1] F. Motzoi, J. Gambetta, P. Rebentrost, and F. Wilhelm, Phys. Rev. Lett. 103, 110501 (2009).
[2] J. M. Gambetta, F. Motzoi, S. T. Merkel, and F. K. Wilhelm, Phys. Rev. A 83, 012308 (2011).