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Hybrid quantum logic and a test of Bell’s inequality
using two different atomic isotopes
C. J. Ballance, V. M. Sch¨afer, J. P. Home, D. J. Szwer, S. C. Webster, D. T. C. Allcock,
N. M. Linke, T. P. Harty, D. P. L. Aude Craik, D. N. Stacey, A. M. Steane and D. M. Lucas
Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, U.K.
(Dated: 27 November 2015, v40.43)
Entanglement is one of the most fundamental
properties of quantum mechanics [1–3], and is the
key resource for quantum information process-
ing [4, 5]. Bipartite entangled states of identical
particles have been generated and studied in sev-
eral experiments, and post-selected or heralded
entangled states involving pairs of photons, sin-
gle photons and single atoms, or different nuclei
in the solid state, have also been produced [6–
12]. Here, we use a deterministic quantum logic
gate to generate a “hybrid” entangled state of
two trapped-ion qubits held in different isotopes
of calcium, perform full tomography of the state
produced, and make a test of Bell’s inequality
with non-identical atoms. We use a laser-driven
two-qubit gate [13], whose mechanism is insensi-
tive to the qubits’ energy splittings, to produce
a maximally-entangled state of one 40Ca+qubit
and one 43Ca+qubit, held 3.5 µm apart in the
same ion trap, with 99.8(6)% fidelity. We test the
Clauser-Horne-Shimony-Holt [14] (CHSH) ver-
sion of Bell’s inequality for this novel entangled
state and find that it is violated by 15 stan-
dard deviations; in this test, we close the detec-
tion loophole [8] but not the locality loophole [7].
Mixed-species quantum logic is a powerful tech-
nique for the construction of a quantum computer
based on trapped ions, as it allows protection of
memory qubits while other qubits undergo logic
operations, or are used as photonic interfaces to
other processing units [15, 16]. The entangling
gate mechanism used here can also be applied to
qubits stored in different atomic elements; this
would allow both memory and logic gate errors
due to photon scattering to be reduced below the
levels required for fault-tolerant quantum error
correction, which is an essential pre-requisite for
general-purpose quantum computing.
For Schr¨odinger, entanglement was “the characteris-
tic trait of quantum mechanics” [1] and it has been at
the heart of debates about the foundations of quantum
mechanics since the framing of the Einstein-Podolsky-
Rosen paradox [2]. The theoretical work of Bell [3], and
of Clauser et al. [14], established an experimental test
which could be used to rule out local hidden-variable
theories on the basis of correlations between measured
properties of entangled particles, and numerous experi-
ments, starting with that of Freedman and Clauser, have
confirmed the predictions of quantum mechanics [6–10].
Tests of Bell’s inequality with trapped ions were the first
to close the so-called “detection loophole”; hitherto these
trapped-ion tests have exclusively been carried out with
identical atoms [8, 17, 18]. The entanglement explored
in tests of Bell’s inequality is typically an entanglement
between distinguishable particles, in the strict quantum
mechanical sense, but when the particles are identical in
their internal structure and state, they are distinguish-
able only through their spatial localization. By employ-
ing different isotopes, our experiments involve entities
that are also distinguishable by many internal properties,
such as baryon number, mass, spin, resonant frequencies,
and so on.
Apart from its intrinsic interest, entanglement is a
central resource for quantum information applications,
such as quantum cryptography [5] and quantum comput-
ing [4]. Trapped atomic ions are one of the most promis-
ing technologies for the implementation of quantum com-
putation; several demonstrations of simple multi-qubit
algorithms have been made [19] and the elementary set
of quantum logic operations has recently been demon-
strated with the precision required for the implemen-
tation of fault-tolerant techniques [20, 21]. Scaling up
trapped-ion systems to the large numbers of qubits re-
quired for useful quantum information processing and
quantum simulation will almost certainly require the use
of more than one species of ion, both for the purpose
of sympathetic laser-cooling (which allows independent
control of the external and internal atomic degrees of
freedom) [15, 22, 23] and for providing robust mem-
ory qubits. The best memory qubits reside in hyperfine
ground states [20, 24], which have essentially infinite life-
times against spontaneous decay, but are vulnerable to
the scattering of a single photon of resonant laser light.
In a complex, multi-zone, ion trap processor it will be dif-
ficult to shield the memory qubits sufficiently well from
resonant laser beams, hence it will be useful to employ
different species of ion, for example as memory and logic
qubits, and a high-fidelity entangling gate operation be-
tween the two species will be invaluable. A significant
initial step was the demonstration of coherent state trans-
fer between different species in the context of precision
metrology [25, 26]. The relative merits of using different
isotopes versus different elements are discussed below.
In the present work, we entangle qubits stored in
two different isotopes of calcium. The 40Ca+qubit
is stored in the Zeeman-split ground level, (|↓i,|↑i) =
(4S−1/2
1/2,4S+1/2
1/2), and the 43Ca+qubit is stored in the hy-
perfine ground states (|⇓i,|⇑i) = (4S4,+4
1/2,4S3,+3
1/2), see fig-
arXiv:1505.04014v2 [quant-ph] 27 Nov 2015
2
FIG. 1: Calcium ion energy levels and experimental geome-
try. (a) Qubit states and Raman transitions in 43Ca+and
40Ca+. The two Raman beams have a mean detuning of
∆ = −1.04 THz from the 4S1/2↔4P1/2(397 nm) transition,
and a difference frequency of δ=fz+δg≈2 MHz. (b) Ra-
man gate beam geometry. The two perpendicular beams are
aligned to set the lattice k-vector parallel to the trap axis ˆz.
The beams have waist radii w= 27 µm, a power of ≈5 mW
each, and orthogonal linear polarizations as indicated. A
third, π-polarized, Raman beam (not shown) co-propagates
with the σ∓beam and is used for sub-Doppler sideband cool-
ing and single-qubit operations on 40Ca+. The quantization
axis is set by a magnetic field B≈0.2 mT. The diagram is
not to scale: the ions are separated by 3.5 µm, which is 12 1
2
periods of the standing wave, and around 20,000 times the
atomic radius of calcium.
ure 1. The qubit energy splittings differ by some three
orders of magnitude (fl≈5.4 MHz, fm≈3.2 GHz), but
they may nevertheless be efficiently coupled via the two-
qubit gate mechanism of Leibfried et al. [13], in which
the “travelling standing wave” from a pair of far-detuned
laser beams exerts a qubit-state-dependent force on the
ions whose magnitude Fis largely independent of the
qubit frequency. The force originates from a spatially-
varying light shift, oscillates at the difference frequency
δbetween the two beams and, when δ=fz+δgis set
close to the resonant frequency fzof a normal mode of
motion of the two-ion crystal, a two-qubit phase gate may
be implemented by applying the force for a time (1/δg).
An advantage of this type of gate is that the phase of the
optical field does not need to be referenced to either of
the qubit phases (see Methods); this makes scaling the
system easier because the relative optical phase does not
need to be controlled between different trap zones, or
during time delays between gates.
An important difference in the gate mechanism com-
pared with the case of identical ions is that the forces on
corresponding qubit states differ (F↑6=F⇑and F↓6=F⇓)
so that, in general, the four possible qubit states (↑⇑,
↑⇓,↓⇑,↓⇓) each acquire different phases. We choose to
implement the gate operation in two halves, each of du-
ration tg/2 = 1/δg, separated by spin-flip operations (π-
−200 −100 0 100 200
0
0.2
0.4
0.6
0.8
1
Probability
Analysis phase (degrees)
P(↑)
P(⇑)
P(↑⇓)+P(↓⇑)
Probability
Analysis phase (degrees)
P(↑)
P(⇑)
P(↑⇓)+P(↓⇑)
a
b
Analysis phase (degrees)
Probability
tg/2
FIG. 2: Entangling gate sequence and results. (a) Gate se-
quence, showing the operations applied to the 40Ca+(upper
line) and 43Ca+(lower line) qubits. The final state analy-
sis (tomography) π/2-pulses shown in green are optional; by
scanning their phase φwe can diagnose the state produced
by the gate. (b) Qubit populations and parity signal after
correcting for readout errors (see Methods). The individual
qubit populations are consistent with 1
2, as expected for the
Bell state (|↓⇓i+|↑⇑i)/√2. The parity signal P(↑⇓)+P(↓⇑),
i.e. the probability of the two qubits being in opposite states,
should oscillate between 0 and 1 as sin(2φ) for a perfect Bell
state. From the contrast of the parity signal and a measure-
ment of the populations without the analysis pulses, we infer
a Bell state fidelity of 99.8(6)%. The error bars show 1σsta-
tistical errors.
pulses) on the qubits (figure 2a). This symmetrizes the
gate operation such that the relative phases acquired by
the four states are (0,Φ,Φ,0). By setting the laser power
(i.e., effective Rabi frequency) and gate detuning δgap-
propriately, such that Φ = π/2, and enclosing the gate
operation in a Ramsey interferometer (two pairs of π/2-
pulses), we can generate the maximally-entangled Bell
state (|↓⇓i +|↑⇑i)/√2 from the initial state |↓⇓i. The
π-pulses also protect the qubits against dephasing due to
slow (tg) variations in magnetic fields.
In our experiment, we implement the gate using the
in-phase axial motional mode (at fz= 2.00 MHz) of a
linear Paul trap [27], with the ion separation (3.5 µm)
equal to a half-integer number of standing wavelengths,
thus exciting the motion maximally for the |↑⇓i and |↓⇑i
states. The Lamb-Dicke parameters for the two different
isotopes are η40 = 0.121 and η43 = 0.126. After initial
Doppler cooling, both axial modes are cooled close to
their ground states (mean occupation number ¯n < 0.1)
by Raman sideband cooling applied to the 40Ca+ion,
which sympathetically cools the 43Ca+ion [28]. Both
qubits are initialized by optical pumping, after which we
3
0
0.1
0.2
0.3
0.4
0.5
Re(ρ)
0
0.1
0.2
0.3
0.4
0.5
Im(ρ)
FIG. 3: Density matrix of the mixed-isotope Bell state. The
plots show the real (left) and imaginary (right) parts of the
density matrix, after correcting for qubit readout errors (see
Methods). The measurements were made by rotating each
qubit independently to perform full quantum state tomog-
raphy. We used a maximum likelihood method to find the
density matrix that best represents the experimental data.
This gives a separate estimate of the gate fidelity, 99(1)%.
apply the gate sequence shown in figure 2a, using a gate
duration tg= 27.4µs. Single-qubit π/2- and π-pulses,
for the spin-echo and tomography operations, are applied
using co-propagating Raman beams (for 40 Ca+) and mi-
crowaves (for 43Ca+). The ordering of the ion pair in the
trap was kept constant over the time taken to acquire
the full data set, to guard against systematic effects as-
sociated with ion position (see Methods). We implement
individual single-shot qubit readout by state-selectively
shelving both ions to the 3D5/2level simultaneously, then
detecting the ions’ fluorescence sequentially in two pho-
tomultiplier counting periods (see Methods).
From the contrast of the parity fringes shown in fig-
ure 2, and a measurement of the qubit populations be-
fore the analysis pulses [13], we estimate the fidelity of
the Bell state produced by the gate to be F= 99.8(6)%,
where the error is dominated by statistical uncertainty.
Known contributions to the gate error are significantly
smaller [27] than the statistical uncertainty; for example
the photon scattering error at the ∆ = −1.04 THz Ra-
man detuning used is estimated to be ≈0.1%. Since the
two qubits may be rotated independently by addressing
them in frequency space, we can also perform full tomog-
raphy of the entangled state and extract the density ma-
trix (figure 3); the density matrix is consistent with that
for the desired Bell state, to within the systematic errors
from the imperfect tomography pulses, and gives a sepa-
rate estimate of the fidelity F= 99(1)%. In both cases,
Frepresents the fidelity of the entangling gate operation;
it excludes errors due to state preparation and readout,
which we characterize in independent experiments (see
Methods).
To perform a test of the CHSH version of Bell’s in-
equality, we follow the gate sequence with further inde-
pendent single-qubit rotations and measurements. The
single qubit rotations have constant phase φbut varying
rotation angle θ. From these measurements we deter-
mine the two-particle correlation functions with results
θa(40Ca+)π/4 3π/4π/4 3π/4
θb(43Ca+)π/2π/2 0 0
E(θa, θb) 0.565(7) 0.530(7) 0.560(7) −0.573(8)
TABLE I: Bell/CHSH inequality test results, using the mixed-
isotope entangled state. The qubits aand bare independently
rotated through angles (θa, θ0
a) = (π/4,3π/4) and (θb, θ 0
b) =
(π/2,0), and for each combination of angles the correlation
function E(θa, θb) is measured, with results shown. (Eis
defined as in ref.[17].) The CHSH parameter is given by S=
|E(θa, θb)+E(θ0
a, θb)|+|E(θa, θ0
b)−E(θ0
a, θ0
b)|= 2.228(15) >2,
thus violating Bell’s inequality for this system of non-identical
atoms. The state detection errors are sufficiently small (≈6%,
see Methods), that it is not necessary to make a fair-sampling
assumption. For each angle setting 4,000 measurements were
made.
shown in table I. As is well known, the maximal CHSH
parameter Sallowed by local hidden-variable theories is
2, whereas quantum mechanics allows S≤2√2. In order
to avoid having to make a fair-sampling assumption, we
do not correct for qubit readout errors in these experi-
ments. The finite detection error then limits the CHSH
parameter to a detectable maximum Smax = 2.236(7) for
a perfect Bell state; our results give S= 2.228(15), con-
sistent with Smax to within the stated uncertainties, and
violating the CHSH inequality by ≈15σ.
The mixed-species quantum logic gate that we have
demonstrated has allowed us to create a novel entangled
state, leading to the first test of a Bell inequality violation
between isolated non-identical atoms. As an application,
the two isotopes used here could be employed for scal-
able quantum computing architectures based on trapped
ions; hyperfine qubits in 43Ca+at present constitute the
best single-qubit memories (T∗
2∼1 min) [20], whereas the
simpler atomic structure of 40Ca+is well suited for use as
a “photonic interconnect” qubit [16]. There are technical
advantages to using ions of similar mass for sympathetic
cooling and ion transport in multi-zone traps. However,
while the relatively small isotope shifts (∼1 GHz) allow
the convenient use of the same laser systems for manipu-
lation of both species, they may provide insufficient pro-
tection of qubits from stray resonant light unless tightly
focussed beams are used [18, 28]. Therefore in the long
term it may be necessary to use different atomic ele-
ments [22]. The gate mechanism employed here is inde-
pendent of the qubit frequency and thus can also be used
to couple qubits stored in different elements, provided
that the Raman laser fields exert sufficient force on both
qubits. We note that Ca+and Sr+ions are an attractive
choice in this respect: the 4S1/2↔4P1/2transition in
Ca+is separated from the 4S1/2↔4P3/2transition in
Sr+by 20 THz. A Raman laser detuning ∆ = −8 THz
(comparable to that used in our recent 43Ca+–43Ca+
two-qubit gate experiments [21]) would enable the imple-
mentation of a mixed-species logic gate with a photon-
scattering error of ∼10−4, significantly below the error
threshold for fault-tolerant operations [29].
4
Similar experiments using trapped-ion qubits stored in
two different elements (9Be+and 25Mg+) have recently
been carried out in the NIST Ion Storage Group [30].
Subsequent to the submission of our manuscript, a
CHSH-Bell test which closes both detection and locality
loopholes, using heralded entanglement of remote elec-
tron spins, has been reported [31].
METHODS
Ion crystal order. The 40 Ca+–43Ca+ion crystal
ordering is kept constant during the experiments to
control systematic errors. The principal error which
would arise if the ion order were not controlled is due to
an (undesired) axial magnetic field gradient that causes
the magnetic field between the two ions to differ by
0.18 µT. This means that the qubit frequencies for the
two possible ion orders differ by ≈5 kHz, which would
lead to errors in single-qubit rotations. We measure the
frequency of each qubit using slow (typically 100 µs)
carrier π-pulses, interleaved with the main experimental
pulse sequence, which allows us to detect and to correct
for both common-mode qubit frequency changes (due
to drift in the global magnetic field B) and differential
changes (due to incorrect ion crystal ordering). If the
ion order is wrong, we randomly reorder the crystal
until the order is correct with a short period of Doppler
heating to melt the crystal, followed by a short period
of Doppler cooling.
Single-qubit phases and light shifts. Despite the
qubits having very different frequencies, no special phase
control is needed to implement the entangling gate. The
43Ca+qubit phase is tracked by the microwave local os-
cillator, and the 40 Ca+qubit phase is tracked by the
difference phase of the co-propagating Raman beams,
in turn referenced to a radio-frequency local oscillator.
The phases of the Raman beams that implement the en-
tangling gate have no relationship to either of the qubit
phases. However, the travelling standing wave result-
ing from the interference of the Raman gate beams also
generates an isotope-dependent differential light shift on
each qubit with an amplitude that oscillates at the Ra-
man difference frequency δ. Over the course of the gate
operation this light shift adds phase shifts to the qubits
that depend on the (uncontrolled) optical phase differ-
ence of the Raman beams. These uncontrolled phase
shifts reduce the fidelity of the gate operation. We greatly
reduce this light shift error by shaping the turn-on and
turn-off of the Raman laser intensities with a character-
istic time of 1 µs; we estimate that without this pulse-
shaping the light shift would lead to an average gate error
of up to 5% (see ref.[27]).
We adjust the polarisation of each Raman beam
individually to null the differential light shift from each
single beam on the 40 Ca+qubit. (The interference of the
two gate beams nevertheless gives rise to a polarization
modulation which provides the state-dependent force.)
Due to the difference in atomic structure there is a
residual light shift on the 43Ca+qubit of ≈0.2% of
the light shift for a purely circularly-polarised beam of
the same intensity and frequency. This small light shift
does not cause any significant issues in the experiments
reported here; if necessary it could be suppressed further
by increasing the Raman detuning at the expense of
requiring more Raman beam power.
State preparation and measurement errors. To
perform individual single-shot qubit readout, we selec-
tively shelve one qubit state of each ion to the 3D5/2
level, then apply the Doppler cooling lasers sequentially
in time first for one isotope, then for the other. If an
ion was not shelved it fluoresces, and this is detected
with a photomultiplier. We simultaneously shelve the
two isotopes using a weak 393nm beam resonant with
the 43Ca+4S4,+4
1/2↔4P5,+5
3/2transition, with a 1.94 GHz
EOM sideband which drives the 40Ca+4S+1/2
1/2↔4P+3/2
3/2
transition. An intense 850nm beam resonant with the
40Ca+3D3/2↔4P3/2transition makes the shelving for
this isotope state-selective, through an EIT process in-
volving the two transitions [32]. The 43Ca+shelving is
state-selective due to the 3.2 GHz splitting between the
two qubit states [33]. Both these shelving processes have
a maximum theoretical efficiency of ≈90% due to leak-
age to 3D3/2(which for 43Ca+could be eliminated using a
further 850 nm beam if required [33]), leading to readout
errors of ¯≈5% when averaged over both qubit states.
From independent experiments (similar to those we de-
scribe in ref. [20]), we estimate the state-preparation er-
ror to be ≈0.1%, which is negligible compared with the
readout error.
We measure the readout errors for each qubit state of
each isotope, by preparing and measuring each state typ-
ically 10,000 times. Since the qubits are measured indi-
vidually, it is then straightforward to calculate the linear
mapping that corrects for the readout errors, provided
that they remain constant. The readout errors relevant
to the entangling gate experiment (figure 2) were mea-
sured to be ¯40 = 7.7(2)% for 40 Ca+and ¯43 = 4.4(2)%
for 43Ca+(averaged over both qubit states). Measure-
ments of the readout errors were interleaved with the gate
experimental runs, to check for systematic drifts, and
were made using the mixed-isotope crystal, to avoid sys-
tematic effects associated with ion position. We estimate
the systematic uncertainty in determining the readout er-
rors to be ≈0.1%, less than the statistical error in these
measurements. If we did not correct for readout errors,
the apparent infidelity in the Bell state would increase
by ≈3
2(¯40 + ¯43)≈18%. For the CHSH test, we do
not correct for readout errors, but we nevertheless mea-
sure them in order to calculate the maximum attainable
CHSH parameter (Smax).
5
Acknowledgements This work was supported by
the U.K. EPSRC “Networked Quantum Information
Technology” Hub and the U.S. Army Research Office
(contract W911NF-14-1-0217). D.M.L. would like to
thank Aida and Esther Andrade Castillo for their
hospitality while revising the manuscript.
Author contributions All authors contributed to the
development of the apparatus and/or the design of the
experiments. J.P.H. and D.M.L. conceived the experi-
ments and took preliminary data. C.J.B. and V.M.S.
designed and performed the experiments described
here, analysed data and produced the figures. C.J.B.
and D.M.L. wrote the manuscript, which all authors
discussed.
Author information Correspondence should be ad-
dressed to C.J.B. (c.ballance@physics.ox.ac.uk) or
D.M.L. (d.lucas@physics.ox.ac.uk).
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