Generation of a Maximally Entangled State Using Collective Optical Pumping

Abstract

We propose and implement a novel scheme for dissipatively pumping two qubits into a singlet Bell state. The method relies on a process of collective optical pumping to an excited level, to which all states apart from the singlet are coupled. We apply the method to deterministically entangle two trapped ^{40}Ca^{+} ions. Within 16 pumping cycles, an initially separable state is transformed into one with 83(1)% singlet fidelity, and states with initial fidelity of ⪆70% converge onto a fidelity of 93(1)%. We theoretically analyze the performance and error susceptibility of the scheme and find it to be insensitive to a large class of experimentally relevant noise sources.

Full text

Generation of a Maximally Entangled State Using Collective Optical Pumping
M. Malinowski ,1,* C. Zhang ,1V. Negnevitsky,1I. Rojkov ,1F. Reiter ,1T.-L. Nguyen ,1
M. Stadler ,1D. Kienzler ,1K. K. Mehta,1and J. P. Home1,2,†
1Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland
2Quantum center, ETH Zürich, 8093 Zürich, Switzerland
(Received 10 August 2021; revised 10 December 2021; accepted 11 January 2022; published 22 February 2022)
We propose and implement a novel scheme for dissipatively pumping two qubits into a singlet Bell state.
The method relies on a process of collective optical pumping to an excited level, to which all states apart
from the singlet are coupled. We apply the method to deterministically entangle two trapped 40Caþions.
Within 16 pumping cycles, an initially separable state is transformed into one with 83(1)% singlet fidelity,
and states with initial fidelity of ⪆70% converge onto a fidelity of 93(1)%. We theoretically analyze the
performance and error susceptibility of the scheme and find it to be insensitive to a large class of
experimentally relevant noise sources.
DOI: 10.1103/PhysRevLett.128.080503
Quantum entanglement is a resource for quantum com-
putation [1], communication [2], cryptography [3], and
metrology [4]. Entangled states are typically prepared using
a two-step process, the first involving initialization of a
separable state by optical pumping, followed by a unitary
transformation which generates entanglement [5]. In such
an open-loop process the final state is sensitive to the
parameters of the drive used to create it and is not protected
from future errors. An alternative mode of operation is to
use a closed-loop process, where feedback from a low-
entropy reference system drives the system continuously
toward the desired state or subspace. This can be done using
measurement-conditioned classical control (e.g., quantum
error correction or outcome heralding) or through dissipative
engineering [6–10]. Dissipation engineering allows useful
quantum states to be created in the steady state, making the
process self-correcting with regard to transient errors
[11,12], and resulting in a resource state or subspace which
is continuously available. Entanglement of qubits using
dissipative engineering has previously been demonstrated
using trapped ions [13,14],atomicensembles[15],and
superconducting circuits [16–18]. Beyond qubit-based
approaches, reservoir engineering has been used to create
and stabilize nonclassical states of bosonic systems [6,19,20]
as well as to perform quantum error correction [20,21].
A widely used strategy for dissipation engineering is to
rely on engineered resonances, whereby pumping into the
desired entangled state is achieved by resonant drives,
while leakage processes out of the desired state are off
resonant [11,14,22,23]. This approach has proven to be
versatile, and has been theoretically extended to the
generation of multiqubit states [24–26], quantum error
correction [27,28], and quantum simulation [29]. However,
these protocols can be slow to converge. This is because, in
order to suppress leakage processes, the drives need to be
weak compared to the splittings of the resonances. The
resulting competition with additional uncontrolled dissipa-
tion channels limits the achievable fidelities. It has been
proposed that this issue could be overcome by dissipative
schemes based on symmetry [30–34].
In this Letter, we present a method for dissipatively
generating two-body entanglement using a deterministic
collective optical pumping process which does not couple
to the target entangled state: the singlet Bell state jΨ−i≡
ðj↑↓i−j↓↑iÞ=
ffiffiffi
2
p. Unlike previous demonstrations, our
method relies on symmetry, involving only global fields
which couple equally to each system. We thereby overcome
the speed limitations of previous schemes, achieving a
faster convergence. Our scheme is robust to global error
processes. We implement the protocol using two trapped
ions in a surface-electrode trap with integrated optical
control fields [35], achieving a 93(1)% fidelity with jΨ−i.
Compared to earlier trapped-ion approaches, our method
does not require ground-state cooling.
The scheme is illustrated in Fig. 1(a). We consider a spin
ground-state manifold consisting of the collective states
j↓↓i;j↑↑i;jΨþi≡ðj↑↓iþj↓↑iÞ=
ffiffiffi
2
p(spin triplet) and
jΨ−i(spin singlet), as well as excited states, of which
the most important for our purposes consists of both
systems in a particular excited state jei. Three elements
define the pumping process. The first is a collective
excitation (A) from j↓↓ito jeei. Its collective nature
Published by the American Physical Society under the terms of
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PHYSICAL REVIEW LETTERS 128, 080503 (2022)
0031-9007=22=128(8)=080503(7) 080503-1 Published by the American Physical Society

means that it does not couple to the other states in the
ground-state manifold. The state jeeiis quenched through a
decay channel (B), redistributing its population into all four
spin states. (A) and (B) together provide a collective optical
pumping process which moves the population from j↓↓ito
the other ground states. To prepare only the singlet, this is
supplemented by a symmetric drive (C) which resonantly
drives both spins with equal amplitude and phase. Because
of its symmetry, this drive cycles population within the
triplet subspace, while leaving the singlet untouched. Thus
the triplet states have a chance of being repumped through
the collective pumping, while population in the singlet is
dark to all drives. jΨ−ithen becomes the steady state of the
system. The protocol can be implemented continuously or
by sequentially applying each component. For our imple-
mentation, we expect the latter to be more robust to
experimental imperfections and analyze it below. The
continuous implementation is analyzed in detail in
Supplemental Material (SM), Sec. VI [36].
To identify optimal settings, we optimize a superoperator
which combines the three drives. For the collective exci-
tation this is derived from a unitary,
UAðΦÞ¼e−iΦS2
x;e ;ð1Þ
with Sx;e ¼σx;↓e⊗1þ1⊗σx;↓e,σx;↓e¼jeih↓jþj↓ihej,
and 1 is a 3×3identity operator. This provides a full
transfer from j↓↓ito jeeifor Φ¼π=4. Drive (B) repumps
the population from jeiwith branching ratios parametrized
by pe→↓=pe→↑¼tan2ðγÞ. Drive (C) is described by a
unitary UCðθÞ¼exp½iðθ=2Þσx⊗exp½iðθ=2Þσx, where
σx¼j↑ih↓jþj↓ih↑j. After N≫1cycles, the singlet
fidelity, defined as FðjΨ−iÞ¼hΨ−jρjΨ−ifor the system
density matrix ρ, approaches unity as FðjΨ−iÞ ∝
1−expð−N=N0Þ. Through eigenvalue analysis we find
the most rapid convergence for Φ¼π=4,θ≈0.72π,and
γ≈0.22π, where N0¼7.62 cycles (SM, Sec. I [36]). The
steady state is insensitive to the values of Φ,γ,andθ, i ndicating
that these parameters do not require precise calibration.
We implement the protocol on a pair of 40
Caþions
confined in the surface-electrode trap described in
Ref. [35]. The qubit is encoded into ground-state Zeeman
sublevels j↓i¼jS1=2;m
j¼−1=2iand j↑i¼jS1=2;m
j¼
þ1=2iwhich have a frequency splitting of 2π×16.5MHz
in the applied magnetic field of 0.59 mT. We use an
ancilliary state jei¼jD5=2;m
j¼−1=2i. Narrow-linewidth
laser light at 729 nm is delivered through trap-integrated
photonics, and coherently drives the S1=2↔D5=2transi-
tions. Free-space laser beams are used for cooling, repump-
ing, and readout. The j↓i↔j↑itransition is driven by
resonant radio-frequency magnetic fields.
The collective excitation step (A) is implemented using a
bichromatic 729 nm laser field with Rabi frequency Ωand
two frequency components detuned by δ¼2π×14.7kHz
from the red and blue motional sidebands of the j↓i↔jei
transition, using the axial stretch mode (where ions oscillate
out of phase) at ωm≈2π×2.4MHz for which the Lamb-
Dicke parameter η¼0.026. This results in a Hamiltonian
HA¼1
2ℏηΩSx;eðˆ
aeiδtþˆ
a†e−iδtÞwhich implements a force
on the oscillator whose phase depends on the eigenstate
of Sx;e.ThisiscommonlyreferredtoasaMølmer-Sørensen
drive, and is one of the primary methods for performing two-
qubit gates with trapped ions [37–39]. A pulse of duration t
then results in the unitary
UA¼e½αðtÞˆa†−αðtÞˆaSx;e eiΦðtÞS2
x;e ;ð2Þ
where αðtÞ¼−iðηΩ=δÞe−iδt=2sinðδt=2Þis an oscillator
phase-space displacement amplitude and ΦðtÞ¼
ðη2Ω2=4δ2Þ½δt−sinðδtÞ is a collective phase factor.
Equation (2) reduces to a pure S2
x;e coupling of the form of
Eq. (1) in two cases. The first, appropriate to a continuous
implementation, is when jδj≫ηΩand so the oscillator
excitation can be neglected [40]. The second, which is the
main focus of this Letter, is when t¼2nπ=δwith n∈Z,for
which Φ¼nπη2Ω2=ð2δ2Þ[41]. Repump (B) is implemented
using a laser at 854 nm, which couples all D5=2sublevels to
the short-lived P3=2states, which primarily decay into the
ground-state manifold. A second decay channel to the D3=2
states is repumped using a laser at 866 nm. After 5μs, we
measure a probability of leaving D5=2of >0.9999 with a
branching ratio of γ≃0.3π. The symmetric drive (C) is
implemented by passing a current through a track on a circuit
board at around 1 mm distance from the ions.
We employ a number of error mitigation techniques. The
collective excitation step (A) is implemented as a sequence
of two pulses (t¼2nπ=δwith n¼2) with ηΩ¼δ=2,
resulting in Φ¼π=4and a drive time of t¼150 μs.
(a) (b)
FIG. 1. (a) High-level description of the protocol. When drives
(A), (B), and (C) are switched on, the system is pumped into a
maximally entangled state jΨ−i. (b) Atomic transitions and drives
in 40Caþused in this work. (A) and (B) are driven by laser beams,
while (C) is implemented by an oscillating Bfield. Dashed lines
denote motional sidebands of the j↓i↔jeitransition.
PHYSICAL REVIEW LETTERS 128, 080503 (2022)
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The phase of the force acting on the oscillator is shifted by
πfor the second pulse, thus canceling any residual
displacement produced by a single pulse [42]. To our
surprise the ac Stark shift of the j↓i↔j↑itransition
produced by the collective drive was different by ≈2π×
2.5kHz on the two ions, causing a near-complete failure of
the protocol (SM, Sec. IV [36]). To mitigate this, we
replace the optimal value of θapplied in each cycle with
two values, θ1¼πapplied in odd cycles (drive time
tC¼6.4μs) and θ¼π=2(tC¼3.2μs) applied in even
cycles. This has the desired effect of a spin echo, but at the
cost that high-fidelity singlet states are produced only after
even cycles. One cycle of the protocol takes ≈165 μson
average. In the absence of other errors, the protocol
produces jΨ−iregardless of the ions’temperature.
However, finite temperature amplifies existing errors asso-
ciated with residual oscillator excitation [i.e., when
jαðtÞj >0]. For this reason, we cool the ion close to the
motional ground state.
We measure the Pð↓↓Þ,Pð↓↑ÞþPð↑↓Þ,andPð↑↑Þ
populations by shelving j↓iinto ancillary D5=2sublevels,
followed by state-dependent fluorescence [43]. This allows
us to extract the ground-state parity hσzσzi,whilehσxσxiand
hσyσyiare obtained by measuring the parity following radio-
frequency spin rotations exp½iðπ=2Þσx⊗exp½iðπ=2Þσx
and exp½iðπ=2Þσy⊗exp½iðπ=2Þσy, respectively. These
are combined to estimate the singlet state fidelity, using
FðjΨ−iÞ ¼ 1
4ð1−hσxσxi−hσyσyi−hσzσziÞ.
Figure 2(a) shows the measured fidelity as a function of
the number of cycles of the protocol, applied to a range of
initial states, showing the expected convergence toward the
singlet. Different starting states were created by initializing
the ions to j↓↓iand mapping it to a mixture of singlet and
triplet states (SM. Sec. V [36]). Figure 2(b) illustrates how,
after 16 cycles of the protocol, states with initial fidelities
⪆0.75 are mapped onto output states with the same final
fidelity. Averaged over all the data, we find a fidelity of
93(1)% at 16 cycles.
We analyze the noise robustness of the protocol by
considering the bichromatic drive (A) as the dominant
source of errors. Assuming j↑iis spectroscopically
decoupled from (A), we can describe all errors through
16 elementary error channels fIe;X
e;Ye;Z
eg⊗2. These act
as Pauli operators on the fj↓i;jeig subspace, and as
identity on j↑i(SM, Sec. II [36]). The effect of those
errors acting with probability pper cycle in a depolarizing
model is shown in Fig. 3(a). We find that the final fidelity is
independent of all global errors (such as XeXeor XeZe).
On the other hand, all local errors (such as XeIeor IeZe)
become amplified. A particularly experimentally relevant
class of errors is correlated local errors. These include
correlated bit-flip errors (corresponding to an application of
the operator IeXeþXeIe) which arise due to residual spin-
motion entanglement at the end of the collective excitation
step [i.e., when αðtÞ≠0] or off-resonant excitation of
“spectator”transitions (i.e., nearby undesired resonances).
Magnetic-field fluctuations common to both ions would
produce a correlated phase-flip error (IeZeþZeIe). We
find that such correlations increase the fidelity of the
collective optical pumping compared with uncorrelated
errors with similar constituent operators. Results of sim-
ulations showing this are displayed in Fig. 3(b).For
example, a bit-flip error with probability pper cycle
reduces the singlet fidelity by ≈5.2pwhen uncorrelated
and ≈3.2pwhen correlated. Correlated phase-flip errors
leave the fidelity unaffected since jΨ−iresides in a
decoherence-free subspace.
These insights are matched by simulations of the
dynamics of the collective optical pumping in the presence
of experimentally relevant error sources. These reveal that,
compared to either a single entangling gate or a two-
loop phase-modulated entangling gate [44] based on the
Hamiltonian HA, our protocol reduces the effect of qubit
frequency errors and Rabi frequency errors. For motional
frequency errors and fast (Markovian) optical qubit dephas-
ing, our protocol does not provide benefits. Details and
discussion of practical applicability of these results are
presented in SM, Sec. III [36].
It is challenging to exactly account for the measured
error from first principles due to a number of setup-specific
imperfections. We experience kilohertz-level drifts in
(a)
(b)
FIG. 2. Entanglement generation. (a) The effect of applying up
to 16 cycles of the protocol. Measurements for different initial
states are shown in different colors and connected by dashed
lines. All cases converge toward jΨ−i. (b) Comparison of the
singlet fidelity before and after the protocol is applied. The phase
ϕprep of the preparation pulse (SM, Sec. V [36]) sets the input
fidelity to FðjΨ−iÞ ≈0.88cos2ϕprep (green dashed line). After 16
cycles, states with fidelities ⪆0.75 converge onto the same state
of FðjΨ−iÞ ≈0.93 (blue dashed line). Statistical 1σerror bars
are smaller than data points, and the result spread is dominated by
experimental drifts.
PHYSICAL REVIEW LETTERS 128, 080503 (2022)
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motional mode frequencies due to charging of the trap
surface by light shining through the integrated waveguides
[45]. These occasionally lead to a mode spectrum where the
collective excitation step off-resonantly excites spectator
optical transitions. This error can be corrected by tuning
mode frequencies, but it is challenging to estimate its
magnitude between calibrations. Mode frequency drifts
associated with laser power changes were also the primary
reason we worked with a fixed number of protocol cycles
(N¼16, corresponding to ≈3ms of 729 nm light per
shot). High heating rates mean that each cycle of the
protocol starts with a higher occupancy of motional modes
(≈0.5quanta per cycle on a 1 MHz center-of-mass mode),
leading to an increase in a correlated bit-flip error during
the drive (A) as the protocol progresses. The combined
effect of all the error sources is a bit-flip error probability of
p≈0.01 for the first cycle, and p≈0.02 after 16 cycles.
The measured 16-cycle fidelity 93(1)% is consistent with
the value of 1−3.2ppredicted by the correlated bit-flip
error model. The obtained fidelity is significantly reduced
compared to the unitary gate based on the same
Hamiltonian HA, which produces ðj↓↓i−ijeeiÞ=
ffiffiffi
2
pwith
fidelity of ⪆99% [35], though it does improve our ability
to prepare jΨ−i, which is currently limited by errors in
single-ion addressing in our coherent implementation.
In order to verify that the correlated bit-flip error model
captures the essential performance limitations of the pro-
tocol, we measure the 16-cycle fidelity, as well as the
bit-flip probability in the collective excitation step, for a
range of experimental miscalibrations. For each parameter,
we experimentally approximate the steady-state fidelity
FðjΨ−iÞ by first preparing jΨ−iwith fidelity around 0.75
using unitary methods, and then applying 16 cycles of the
protocol. The bit-flip probability pfor the last cycle is
independently estimated by applying 16 cycles of the
protocol, followed by optical pumping and a single round
of drive (A) (SM, Sec. V [36]). The comparison between ;
FðjΨ−iÞ and the bit-flip model prediction is shown in
Fig. 3(c). We find qualitative agreement, suggesting that the
bit-flip model accurately captures the errors of the protocol.
Figure 3(d) illustrates the challenge associated with spectral
crowding. We modify the spectator mode spectrum by
adding an additional quadrupole potential with eigenaxes at
45° to the trap surface in the radial plane. This changes
the radial mode orientations, frequencies, and temperatures,
while keeping the (axial) gate mode frequency approx-
imately constant. We find that the spectator spectrum is
clear only for a narrow range of curvatures (here between
5.7×107and 6.1×107V=m2), which needs recalibrating
every few hours.
All of the limitations listed above are setup specific and
do not pose a fundamental limitation to the protocol.
Coupling to spectator transitions could be suppressed by
increasing the magnetic field. The heating rate observed in
this trap is particularly high, exceeding levels observed in
cryogenic traps with comparable ion-electrode distances
by a factor of ≈100 [46,47]. Reducing it to more typical
levels, combined with better shielding of nearby dielectrics
[48,49], would suppress drifts within each collective
pumping sequence and hence allow the protocol to reach
(a)
(b) (d)
(c)
FIG. 3. (a) Simulated steady-state error associated with individual error channels of probability p. (b) Comparison of simulated steady-
state errors associated with uncorrelated (solid lines) and correlated (dashed lines) errors. (c),(d) Experimentally measured values of
FðjΨ−iÞ (blue dots) compared to the prediction (1−3.2p) of a correlated bit-flip error model (gray lines), with pobtained from
independent experimental measurements. Error bars (smaller than most data points) show a 1σconfidence interval. In several data
points in (d) the measured values of pare large, and we can no longer apply the linear approximation. Inset in (d) highlights the typical
operation region
PHYSICAL REVIEW LETTERS 128, 080503 (2022)
080503-4

its true steady state. Alternatively, motional mode temper-
ature could be stabilized throughout the protocol by
sympathetic cooling [50].
We have presented and implemented a novel protocol
for collective optical pumping into a maximally entangled
two-qubit state. We measure a singlet fidelity of 93(1)%
after 16 cycles of the protocol (at which point a quasisteady
state has been achieved), to our knowledge exceeding
previously reported dissipative methods, and slightly below
a simultaneous work by Cole et al. [34]. The observed
infidelity is consistent with measured bit-flip errors of the
effective S2
xdrive which for our implementation is mediated
by a motional mode. The protocol can be practically
beneficial in experiments limited by global errors,
especially as a method of purifying lower-fidelity Bell
states. Dissipative generation of high-fidelity entangled
states could find application in a variety of quantum
information processing tasks, such as dissipative encoding
[51], error-corrected quantum sensing [52], and as a
supply of entangled resource states for quantum gate
teleportation [53].
While the analysis in this Letter focused on a specific
implementation in 40
Caþions, the protocol is general, and we
anticipate it might be applied in a wide range of platforms
where collective excitation may be engineered. These
include nitrogen-vacancy centers (via direct spin-spin inter-
actions [54]), neutral atom platforms (via Rydberg dressing
[55]) or superconductors (via parametric drives [32]).
We acknowledge funding from the Swiss National
Science Foundation (Grant No. 200020_165555), the
National Centre of Competence in Research for Quantum
Science and Technology (QSIT), ETH Zürich, and the
Intelligence Advanced Research Projects Activity
(IARPA) via the U.S. Army Research Office Grant
No. W911NF-16-1-0070. F. R. and I. R. acknowledge finan-
cial support from the Swiss National Science Foundation
(Ambizione Grant No. PZ00P2_186040). J. P. H. devised the
scheme, which was then simulated and analyzed by M. M.
and V. N. M .M., C. Z., and K. K. M. carried out the experi-
ments presented here in an apparatus with significant
contributions from M. M, C. Z., K. K. M., T.-L. N., and
M. S. M. M. analyzed the data and constructed the error
model. I. R. and F. R. developed an analytic approach to
describe the protocol. V. N. performed an initial experimen-
tal investigation. J. P. H., K. K. M., and D. K. supervised the
work. The Letter was written by M. M., F. R., and J. P. H.
with input from all authors.
*maciejm@phys.ethz.ch
†jhome@phys.ethz.ch
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