LETTERS
PUBLISHED ONLINE: 28 NOVEMBER 2016 | DOI: 10.1038/NPHYS3967
Accelerated quantum control using superadiabatic
dynamics in a solid-state lambda system
Brian B. Zhou
1
, Alexandre Baksic
2
, Hugo Ribeiro
2
, Christopher G. Yale
1
, F. Joseph Heremans
1,3
,
Paul C. Jerger
1
, Adrian Auer
4
, Guido Burkard
4
, Aashish A. Clerk
2
and David D. Awschalom
1,3
*
Adiabatic processesare useful for quantum technologies
1–3
but,
despite their robustness to experimental imperfections, they
remain susceptible to decoherence due to their long evolution
time. A general strategy termed shortcuts to adiabaticity
4–9
(STA) aims to remedy this vulnerability by designing fast dyn-
amics to reproduce the results of a slow, adiabatic evolution.
Here, we implement an STA technique known as superadiabatic
transitionless driving
10
(SATD) to speed up stimulated Raman
adiabatic passage
1,11–14
in a solid-state lambda system. Using
the optical transitions to a dissipative excited state in the
nitrogen-vacancy centre in diamond, we demonstrate the
accelerated performance of dierent shortcut trajectories
for population transfer and for the initialization and transfer
of coherent superpositions. We reveal that SATD protocols
exhibitrobustness todissipationandexperimental uncertainty,
and can be optimized when these eects are present. These
results suggest that STA could be eective for controlling a
variety of solid-state open quantum systems
11–16
.
Coherent control of quantum states is a common building block
behind quantum technologies for sensing, information processing,
and simulation. A powerful class of such techniques is based on
the adiabatic theorem, which ensures that a system will remain
in the same instantaneous eigenstate if changes to the system are
sufficiently slow. Although adiabatic techniques are attractive for
their robustness to experimental fluctuations, their effec tiveness
is limited when decoherence occurs on timescales comparable to
that required by the adiabatic theorem. To mitigate this drawback,
exact dynamics resulting from specially designed control fields were
proposed to realize the same purpose as adiabatic evolutions do,
but without condition on the evolution time
8
. These approaches for
accelerating adiabatic protocols are collectively known as ‘shortcuts
to adiabaticity’ (STA)
4–10
. Beyond providing practical benefits, they
address quantum mechanical limits on the spe ed of dynamical
evolution and the efficiency of thermodynamics
8
.
Among t he strategies for STA is counterdiabatic (or transition-
less) driving, which introduces, in its simplest formulation, an
auxiliary control field that precisely cancels non-adiabatic transi-
tions between the adiabatic (instantaneous) eigenstates of an initial
Hamiltonian. Offering broad applicability, counterdiabatic driving
has been demonstrated to speed up state transfer in two-level
quantum systems
17,18
, as well as the expansion
19
and transport
20
of
trapped atoms. Theoretically, it has also been proposed to facilitate
the preparation of many-body states for quantum simulation
21
.
However, implementat ion of the counterdiabatic field, par ticularly
in hig her-dimensional systems, c an be challenging as it may require
complex experimental resources to realize interactions absent in
the original Hamiltonian. Moreover, as STA protocols generally
assume ideal (unitary) e volution and perfect implementation, their
robustness to dissipation and experimental uncertainty remains an
open question.
To explore these issues, we demonstrate a generalization
10,22
of
the counterdiabatic strategy to expedite coherent manipulations in
a three-level 3 system. Our starting point is stimulated Raman
adiabatic passage (STIRAP), whereby population t ransfer between
two levels is mediated by their coupling to a third intermediate
level (Fig. 1a). An overlapping sequence of two driving fields, w ith
the Stokes pulse Ω
S
(t) preceding the pump pulse Ω
P
(t), guides the
system along a dark state that evolves from the initial to target state
without occupying the intermedi ate level
1
. For STIRAP, however,
achieving transitionless evolution in the adiabatic basis requires
a counterdiabatic field that directly couples the initial and target
levels
6
, a previously unnecessary interaction. To maintain the full
utility of STIRAP by introducing modifications only to the original
Stokes and pump fields, we instead enforce transitionless evolution
in a dressed st ate basis that reproduces the desired initial and final
conditions, but does not track the adiabatic evolution (s ee ref. 10
and Supplementary Section 1). This novel approach, which we
term ‘superadiabatic’ transitionless driving (SATD), is illustrated in
Fig. 1b. An example superadiabatic shortcut (solid, red line) drives
the same transfer as the adiabatic e volution (dashed, red line), but
via an alternate trajectory; this defines the ‘dressed dark state’.
In contrast to the adiabatic evolution for STIRAP, our shortcut
trajectories deliberately occupy the intermediate state, and hence are
sensitive to its dissipation. However, the degree of occupation can be
tailored by choice of the dressed dark state
10
. For the case of resonant
STIRAP (one- and two-photon detunings ∆ =0,δ =0, respectively,
as labelled in Fig. 1a), we start with a pulse (Vitanov shape) known to
be adiabat ically optimal
23
and show how it evolves to ensure finite-
time, transitionless driving with respect to two distinct choices
for the dressed basis: the ‘superadiabatic’ basis
22
(SATD protocol)
and a modified basis (MOD-SATD protocol). The latter is derived
from the former by reducing the intermediate level occupation
(Supplementary Section 1). The form of the superadiabatic Ω
S
(t)
(shown in Fig. 1c) and Ω
P
(t) pulses are determined by the shape
parameter A
shape
=Ω
shape
/Ω
min
, where Ω
shape
denotes the maximum
Rabi coupling assumed in the theoretical pulse calculation and
Ω
min
is a reference value proportional to the inverse of the pulse
duration L (see Methods). Under unitary evolution, per fect state
transfer is achieved by employing pulses with A
shape
matching the
experimental adiabaticity A =Ω/Ω
min
, where Ω denotes the actual
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
1
Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA.
2
Department of Physics, McGill University, Montreal,
Quebec H3A 2T8, Canada.
3
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA.
4
Department of Physics, University of
Konstanz, D-78457 Konstanz, Germany.
*
e-mail: awsch@uchicago.edu
330 NATURE PHYSICS | VOL 13 | APRIL 2017 | www.nature.com/naturephysics
NATURE PHYSICS DOI: 10.1038/NPHYS3967
LETTERS
1
2
3
4
∞
1.0
Time (norm.)
0.5
0.0
1.0
0.5
0.0
c e
b
Non-adiabatic
Adiabatic
Superadiabatic shortcut
Dissipation
d
To APD
AEOM
PEOM
Interact
Readout
Initialize
Cryostat/
sample (5.5 K)
AWG
SG
IQ
DC
DC
Freq.
Time
a
SATD MOD-SATD
Time (ns)
020406080100
PL (kcts)
0.0
0.5
1.0
1.5
2.0
2.5
Model
Data (|−1〉 → |A
2
〉)
100 ns
STIRAP
Δ
(= 0)
δ
(= 0)
Ω
P
(t)
Ω
= 2π × 171 MHz
Ω
S
(norm.)
|+1〉
|0〉
|−1〉
|A
2
〉
Transfer
S
(t)e
i
S
Ω
φ
| 〉
ϕ
|
F
〉
ψ
|
I
〉
ψ
A
shape
(
shape
/
min
)
Ω
Ω
A
shape
(
shape
/
min
)
Ω
Ω
1
2
3
4
∞
1.0
Time (norm.)
0.5
0.0
1.0
0.5
0.0
Ω
S
(norm.)
Hilbert space
Figure 1 | Concept and implementation of three-level superadiabatic transitionless driving. a, State transfer in a NV centre 3 system by STIRAP. The
|+1i/|−1i ground-state spin levels are coupled by resonant Stokes Ω
S
(t) and pump Ω
P
(t) optical fields to the |A
2
iexcited state, which acts as the
intermediate state for STIRAP. b, Schematic of possible dynamics. An adiabatic protocol transfers the initial state |ψ
I
ito the final state |ψ
F
ialong the
dashed, red trajectory, which is followed exactly only in the infinite time limit. For finite-time realizations, |ψ
I
imay be transferred to a dierent state |ϕi
due to non-adiabatic transitions (blue). Our superadiabatic shortcut (solid red) implements modified driving pulses to reproduce the same final transfer of
the adiabatic protocol, but for arbitrary evolution time and along a dierent path determined by the choice of dressed basis. Dissipation leads to errors for
all evolutions. c, Example of the modified Ω
S
(t) pulses for SATD and MOD-SATD, corresponding to two dierent basis choices. The shape parameter
A
shape
specifies the appropriate driving pulse under unitary evolution for a particular experimental coupling strength Ω and pulse duration L. The modified
Ω
P
(t) pulses (not shown) mirror the Ω
S
(t) pulses about the midpoint of the protocol. d, Experimental set-up utilizing EOMs to shape the Stokes and pump
pulses from a single laser on sub-nanosecond timescales. AWG, arbitrary waveform generator; IQ, quadrature modulation; SG, signal generator; P/AEOM,
phase/amplitude electro-optic modulator; DC, dichroic mirror; APD, avalanche photodiode. e, Optically driven Rabi oscillations between the |−1iand |A
2
i
levels. The oscillations damp due to excited state dissipation (lifetime and dephasing) and spectral diusion. The solid line is an example of a fit to a master
equation model using the rates given in the main text.
value of the experimental Rabi coupling. As A →∞, conditions
are fully adiabatic and the superadiabatic correction vanishes to
reproduce the original Vitanov shape (A
shape
→∞). Alternatively,
A =1 corresponds to the most non-adiabatic condition (Ω =Ω
min
)
whose corrected pulse (A
shape
= 1) does not exceed the original
maximum Rabi coupling.
We realize our protocols using optical driving in a solid-state 3
system hosted by a single nitrogen-vacancy (NV) centre in diamond
at low temperature (T =5.5 K). With its rich energy level structure,
spin–photon interface, and natural coupling to proximal nuclear
spins, this defect spin presents a dynamic arena for techniques
in quantum information
12,24–28
and metrology
29
. Passing a single
tunable l aser (637.2 nm) through a phase electro-optic modulator
(PEOM) produces frequency harmonics to resonantly excite both
the |−1i and |+1i ground-state spin levels, Zeeman-split by
1.414 GHz, to the |A
2
i spin–orbit excited state, which serves as the
intermediate state for STIRAP (Fig. 1a,d). The intensities of the
harmonics are subse quently modulated by an amplitude electro-
optic modulator (AEOM), such that co ordinated control of the
PEOM and AEOM with a 10 GHz arbitrary waveform generator
produces the temporal profiles for Ω
S
(t) and Ω
P
(t) used in
superadiabatic driving (Fig. 1d and Methods).
As we incorporate the excited state |A
2
i into our shortcut
dynamics, its lifetime T
1
and orbital dephasing rate 0
orb
provide
us unique insight on the effect of dissipation on STA. Moreover,
spectral diffusion of the excited level, a ubiquitous feature of
solid-state systems, probes the robustness of our protocol to
fluctuations from one-photon resonance. We first illustrate these
effects through measurement of the photolumines cence (PL) during
constant excitation of the | − 1i to |A
2
i transition (Fig. 1e).
Proportional to the occ upation of |A
2
i, the PL reveals coherent
Rabi osci llations between the g round and excited states
30
. These
oscillations damp due to a combination of spectral diffusion
(estimated as Gaussian-distributed with standard deviation σ ∼
2π ×31 MHz), lifetime T
1
=11.1 ns (1/T
1
=2π ×14.3 MHz), and
dephasing 0
orb
=1/(18 ns)=2π ×8.8 MHz for the NV centre used
(parameter determination is described in Supplementary Sections 2
and 3). Moreover, an overall decay in PL stems predominantly from
trapping into |+1i, the dark state defined by our driving field
24,31,32
.
We begin by examining the effectiveness of our superadiabatic
proto cols as a function of the maximum optical Rabi strength
Ω of the Stokes and pump pulses. For a constant pulse duration
L =16.8 ns (with an additional 2 ns buffer at each end for switching
on and off the optical fields), t he weakest Rabi coupling that can
be corrected without exceeding the maximum amplitude of the
adiabatic pulse is Ω
min
=2π ×72.6 MHz (∝L
−1
, see Methods).
After initializing into |−1i, we transfer the population into |+1i
using STIRAP pulses of varying Ω to explore different regimes of the
experimental adiabaticity A =Ω/Ω
min
. In Fig. 2a, we demonstrate
that SATD and MOD-SATD pulses with shape parameter A
shape
=
A, as prescribed by theory for unitary evolution, significantly
outperform the Vitanov (adiabatic) shape in transfer efficiency.
Despite the presence of dissipation and spectral instability, which
preclude the perfect efficiencies predicted in their absence, the
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© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
331
LETTERS
NATURE PHYSICS DOI: 10.1038/NPHYS3967
Time (ns)
01020304050
PL (kcts)
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.2
0.4
0.6
0.8
75 100 125 150 175 200
Transfer eciency
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.0
(L = 16.8 ns)
1.5 2.0 2.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0 1.5 2.0 2.5 3.0 3.5 4.0
Transfer eciency
0.5
0.6
0.7
0.8
0.9
1.0
Microwave reference
Adiabatic (Vitanov shape)
MOD-SATD
SATD
Errorbar
a
Final |+1〉 population
b
Errorbar
d
|A
2
〉 population (norm.)
Adiabatic
SATD
MOD-SATD
c
Trans. eff.
0.95
0.45
SATD:
MOD-SATD
SATD
Adiabatic
Microwave ref.
A
A
shape
opt
A
shape
opt
Experiment Model
1.0 1.5 2.0 2.5
1.0
1.5
2.0
2.5
Ω
min
= 2π × 73 MHz
Ω
= 2π × 115 MHz, L = 16.8 ns
Ω
= 2π × 113 MHz
Ω
P
Ω
S
A = 1.58
Adiabaticity, A (/
min
)
ΩΩ
Rabi frequency /2π (MHz)
Ω
A
shape
(
shape
/
min
)
ΩΩ
Shape param. A
shape
(
shape
/
min
)
ΩΩ
A (/
min
)
ΩΩ
1.0 1.5 2.0 2.5
A (/
min
)
ΩΩ
Figure 2 | Performance and robustness of superadiabatic pulses. a, STIRAP transfer eciency of MOD-SATD, SATD, and adiabatic (Vitanov) pulses as a
function of the maximum optical Rabi strength Ω. The superadiabatic protocols utilize pulses prescribed for unitary evolution: the shape parameter A
shape
is equal to the experimental adiabaticity A, determined by Ω and the constant pulse duration L =16.8 ns. The right y-axis indicates the absolute population
in |+1iat the end of the protocol. The left y-axis estimates a transfer eciency that accounts for imperfect initialization by using direct microwave transfer
from |0iinto |+1ito establish a reference transfer eciency of 1. b, Robustness of the transfer eciency as a function of the pulse shape A
shape
for
Ω =2π ×115 MHz. The maximum transfer eciency for the superadiabatic protocols occurs for a shape parameter A
opt
shape
< A, reflecting the presence of
dissipation and spectral diusion. Typical errorbars in a and b correspond to 95% confidence. c, False colour plot of the experimental (lef t) and simulation
(right) transfer eciency for SATD as a function of A and A
shape
. The dashed black lines represent A
shape
=A. The data points and fitted cyan line on the
experimental plot delineate the extracted A
opt
shape
, while the interval corresponds to ±1% in transfer eciency. The deviation A
opt
shape
< A is consistent
with the dissipative model (cyan trace denotes A
opt
shape
in model results). d, Photoluminescence (PL) (left y-axis) and converted |A
2
ipopulation (right
y-axis) measured during the adiabatic, SATD, and MOD-SATD pulses for Ω =2π ×113 MHz, highlighting the designed occupation of |A
2
i(less for
MOD-SATD) by the superadiabatic pulses.
superadiabatic protocols realize enhancements of >40% in absolute
efficiency over the adiabatic protocol as conditions become
increasingly non-adiabatic (A → 1). This indicates the relative
importance of minimizing transitions out of each protocol’s dark
state. Furthermore, the design of MOD-SATD to reduce the excited
state occupation in the evolution of the dressed dark state decreases
its exposure to dissipation and al lows it to surpass SATD in
efficiency (Fig. 2a).
To investigate the robustness of these protocols, we implement
pulse shapes deviating from A
shape
=A, anticipating applications
where errors in the determination of A or pulse shape may occur.
In Fig . 2b, we fix both Ω and L, resulting in A =1.58, and then
apply pulses with A
shape
ranging from 1 to 4. We find that both
SATD and MOD-SATD achieve better transfer efficiency than the
adiabatic protocol (magenta bar) for a wide range of pulse shapes.
Moreover, within each family of modified shapes, the pulse shape
that maximizes the transfer efficiency does not correspond to
A
shape
=A, as expected in the absence of dissipation, but to a value
A
opt
shape
< A. In Fig. 2c (left), we confirm this trend as A varies via
the optical power. A linear fit (cyan line) to the extracted transfer
efficiency maxima (black points) yields A
opt
shape
=0.81(2)A +0.09(3)
for SATD (see Supplementary Section 4.2 for MOD-SATD).
Although part of the deviation from A
opt
shape
= A probably
results from attenuation of the pulse shape in the experimental
hardware, our master equation model produces a similar deviation
simply by incorporating the measured lifetime (T
1
), dephasing
(0
orb
), and spectral diffusion of the |A
2
i excited state (Fig . 2c,
right). Physically, t he presence of the dissipative mechanisms and
fluctuations from one-photon resonance damp transitions to and
from the intermediate level, requiring more accentuated drive pulses
(A
opt
shape
< A) to mimic the optimal trajec tory found in the unitary
and zero-detuning (∆ = 0) limit. In Supplementary Section 4.3,
we present data using deliberate off-resonant driving that support
a shift towards more accentuated optimal pulses for nonzero
detuning, as similarly induced by spectral diffusion. Taking a wider
perspective, the broad funnel of enhanced transfer efficiency in
Fig. 2c demonstrates that these protocols are resilient to moderate
dissipation and to potential imperfections in real applications,
such as in the pulse shape (A
shape
), laser intensit y (Ω), or laser
frequency (∆).
In Fig. 2d, we confirm the dynamics of our superadiabatic
shortcuts by measuring the time-resolved PL during the adiabatic
pulse and during the optimal SATD and MOD-SATD pulses for
Ω =2π ×113 MHz. Strikingly, the converted |A
2
ipopulations peak
near the centre of the pulse sequence for the shortcut protocols,
prior to when they peak for the adiabatic protocol. This offset is
a signature of the shortcut’s aim to preemptively place population
into the intermediate state during the first half of the sequence
and to coherently retrieve that population during the second half
(see simulations in Supplementary Section 4.4). In contrast, any
population in |A
2
i during the adiabatic pulse is unintentional and
detrimental to fidelity. Moreover, we verif y that the maximal |A
2
i
population for MOD-SATD is ∼20% lower than SATD, consistent
with its theoretical design
10
. Some parasitic |A
2
ipopulation during
the shortcuts, such as the weak s econd bump in the SATD trace,
is apparent due to the imperfect initialization and fidelity of
our implementat ion.
To characterize the speed-up of our superadiabat ic shortcuts,
we turn to measurements of the transfer efficiency by varying the
adiabaticity A through the pulse length L (that is, Ω
min
∝L
−1
). As
shown in Fig. 3 for constant Ω =2π ×122 MHz, the optimal SATD
and MOD-SATD pulses maintain much higher transfer efficiencies
as L is reduced. Interpolating bet ween the data points, we infer
that the pulse length L for MOD-SATD (SATD) required to reach
a transfer efficiency of 90% is ∼2.7 (2.0) times shorter than that
for the adiabatic pulse (Fig. 3 inset). For the coupling strength
shown, our shortest superadiabatic protocol length of 12.6 ns, which
maintains efficiencies >85%, is just over twice the minimal time
set by quantum mechanics for transfer between two levels through
an intermediate state: L
QSL
=
√
2π/Ω =5.8 ns (see Supplementary
332
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
NATURE PHYSICS | VOL 13 | APRIL 2017 | www.nature.com/naturephysics
NATURE PHYSICS DOI: 10.1038/NPHYS3967
LETTERS
Pulse length, L (ns)
10 20 30 40 50 60
123456
Transfer eciency
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Errorbar
Transfer eciency
0.85 0.90 0.95
Speed-up (L
A
/L
SA
)
1.5
2.0
2.5
3.0
3.5
4.0
MOD-SATD
SATD
Interpolated:
MOD-SATD
SATD
Adiabatic
Interpolants
Quantum speed limit
Microwave ref.
Adiabaticity, A (/
min
)
ΩΩ
Ω
= 2π × 122 MHz
Figure 3 | Speed-up of superadiabatic protocols. STIRAP transfer eciency
for the optimal MOD-SATD and SATD pulses versus the adiabatic pulse as
a function of the pulse duration L ∝Ω
−1
min
for a constant Rabi strength
Ω =2π ×122 MHz. The vertical grey bar at 5.8 ns represents the quantum
speed limit for state transfer via an intermediate state for this coupling
strength Ω. The solid grey lines represent interpolating functions used to
invert the plot and estimate the pulse length L
A
(L
SA
) of the adiabatic
(superadiabatic) protocol needed to attain a given transfer eciency. The
inset displays the speed-up factor, given by the ratio L
A
/L
SA
, as a function
of the desired transfer eciency. Dashed lines in the inset represent
extrapolations outside the range of experimentally attained
transfer eciencies.
Section 4.5 for full discussion). This fastest transfer, occurring at the
so-called ‘quantum speed limit’ (QSL), utilizes a ‘hybrid’ rectangular
pulse scheme that significantly occupies the dissipative intermediate
level and would likewise not realize perfect efficiency.
To emphasize that our protocols retain phase coherence,
we utilize them to expedite the transfer and initialization of
superposition states (Fig. 4a). Starting with an initial superposition
|ψ
I
i=1/
√
2(|0i + e
iφ
I
| − 1i) and applying STIRAP on the
|−1i component, we propagate the initialized phase to the
ideal transferred state |ψ
F
i=1/
√
2(|0i+e
iφ
F
|+1i). Incoherent
effects, such as the spontaneous emission, dephasing, and energy
uncertainty of |A
2
i, will decohere the transferred phase, but can
nevertheless result in p opulation transfer. In Fig. 4b, we show the
quadrature amplitudes X and Y of |ψ
F
ion a polar plot to visualize
φ
F
tracking the increment of φ
I
for Ω =2π ×133 MHz. The MOD-
SATD and SATD pulses achie ve higher phase visibilities
√
X
2
+Y
2
than the adiabatic pulse does, affirming their superiority for
coherent manipulations. Moreover, comparing the phase visibility
to the square root of the protocol’s population transfer efficiency
(delineated by the corresponding solid arc in Fig. 4b) reveals that
the superadiabatic population transfers are predominantly coherent,
whereas incoherent contributions account for a larger fraction
of the adiabatic transfer (Supplementary Section 4.6). Finally, as
detailed in Supplementary Section 1, our analytical framework
can be extended to derive pulse shapes that accelerate fractional
STIRAP (f-STIRAP)
1
. In normal f-STIRAP, the Stokes and pump
pulses adiabatically turn off with a fixed amplitude and phase
relation to initialize arbitrary superpositions of the initial and target
states. In Fig. 4c, we show t hat the preparing the superposition
|ψ
F
i=1/
√
2(|−1i±|+1i) by f-STIRAP achieves an average
fidelity of F =0.93 ±0.01 for the SATD protocol, an improvement
over F =0.83 ±0.01 for the adi abatic pulse at Ω =2π ×135 MHz.
Our work establishes SATD as a fast and robust technique for
coherent quantum control, with applications to other adiabatic pro-
tocols and physical systems. The extension of adiabatic techniques
to more open quantum systems highlights the importance of STA as
a means to outpace decoherence, without sacrificing robustness. For
STIRAP in engineered, solid-state systems involving ladder energy
structures
13,14
or cavity-qubit states
16
, dissipation is unavoidable as
it affects multiple levels, rather than only the intermediate level.
AmplitudeAmplitude
SATD:
c
Adiabatic:
ab
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
F = 0.82 ± .01
F = 0.93 ± .01
F = 0.84 ± .01
F = 0.94 ± .01
X (norm. ampl.)
0.0 0.2 0.4 0.6 0.8 1.0
Y (norm. ampl.)
0.0
0.2
0.4
0.6
0.8
1.0
|0〉
|−1〉
|0〉
|+1〉
STIRAP
|+1〉
|−1〉
Fractional
STIRAP
|
I
〉 phase
increment
STIRAP
MOD-SATD
SATD
Adiabatic
|
F
〉
ψ
|
F
〉:
ψ
|
F
〉 =
ψ
|
I
〉
ψ
ψ
Ω
= 2π × 133 MHz, A = 1.83
Fractional STIRAP
Ω
= 2π × 135 MHz, A = 1.86
σ
I
σ
X
σ
Y
σ
Z
σ
I
σ
X
σ
Y
σ
Z
(|−1〉 + |+1〉)/√2 (|−1〉 − |+1〉)/√2
Figure 4 | Accelerating the transfer and initialization of superposition states. a, Bloch sphere schematic for phase-coherent STIRAP processes. (Top)
Transfer of superpositions: the phase relation within an initial superposition |ψ
I
iof the |0i/|−1istates is transferred by STIRAP to a target superposition
|ψ
F
iof the |0i/|+1istates. (Bottom) Initialization of superpositions: fractional STIRAP enables the creation of arbitrary superpositions of the |−1i/|+1i
states by maintaining a particular phase and amplitude relation between Ω
S
(t) and Ω
P
(t) as both fields are simultaneously ramped to zero. b, Visualization
of the phase of the transferred superposition |ψ
F
ion a polar plot for MOD-SATD, SATD, and adiabatic protocols as the phase of |ψ
I
iis incremented. X and
Y are the components of the projections of |ψ
F
ionto 1/
√
2(|0i+|+1i) and 1/
√
2(|0i+i|+1i), respectively, that vary with the initialized phase. The
phase visibility
√
X
2
+Y
2
can be compared to the square root of the population transfer eciency (delineated by the solid arcs) to gauge the coherent
fraction of the population transfer for each protocol. c, State tomography and fidelity F for the initialization of two dierent final superposition states
|ψ
F
i=1/
√
2(|−1i±|+1i) by fractional STIRAP via a shortcut SATD protocol (top) and an adiabatic protocol (bottom).
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333
LETTERS
NATURE PHYSICS DOI: 10.1038/NPHYS3967
In t hese cases, our ‘speed above all’ approach with SATD, together
with its flexibility to design transitionless evolutions tailored to
specific criteria, offers unique advantages. Promisingly, we show that
SATD protocols are robust against small experimental imperfections
and uncertainties, though for optimal performance, they should be
adjusted to reflect any dissipative dynamics. Looking for ward, the
dissipative 3 configuration here is exemplified in a future quan-
tum transducer, where a lossy mechanical mode connects qubits to
photons in a quantum network.
Methods
Methods, including statements of data availability and any
associated accession codes and references, are available in the
online version of this pap er.
Received 11 July 2016; accepted 27 October 2016;
published online 28 November 2016
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Acknowledgements
We thank C. F. de las Casas and D. J. Christle for experimental suggestions and
P. V. Klimov and G. Wolfowicz for thoughtful readings of the manuscript. C.G.Y., F.J.H.
and D.D.A. were supported by the US Department of Energy, Office of Science, Office of
Basic Energy Sciences, Materials Sciences and Engineering Division. B.B.Z. and P.C.J.
were supported by the Air Force Office of Scientific Research and the National Science
Foundation DMR-1306300. In addition, A.A. and G.B. acknowledge support from the
German Research Foundation (SFB 767). A.B., H.R. and A.A.C. acknowledge support
from the Air Force Office of Scientific Research.
Author contributions
H.R. and B.B.Z. engaged in preliminary discussions. A.B., H.R. and A.A.C. developed the
superadiabatic theory. B.B.Z., C.G.Y., F.J.H. and P.C.J. performed the experiments. A.B.,
A.A., H.R. and G.B. completed the master equation modelling. D.D.A. advised all efforts.
All authors contributed to the data analysis and writing of the manuscr ipt.
Additional information
Supplementary information is available in t he online version of the paper. Reprints and
permissions information is available online at www.nature.com/reprints.
Correspondence and requests for materials should b e addressed to D.D.A.
Competing financial interests
The authors declare no competing financial interests.
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