Quantum computing and simulation with arrays of Rydberg … · 2019. 6. 30. · Antoine Browaeys Laboratoire Charles Fabry, Institut d’Optique, CNRS, FRANCE CEA-LETI, Grenoble,
Upload
others
View
2
Download
0
Embed Size (px)
344 x 292
429 x 357
514 x 422
599 x 487
Citation preview
CEA-LETI, Grenoble, june 28th 2019
Quantum computing and simulation with arrays of Rydberg
atoms:
from proof-of-principle experiments to the startup PASQAL
Rydberg atoms for Quantum Information Processing
VOLUME 85, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 4
SEPTEMBER 2000
Fast Quantum Gates for Neutral Atoms
D. Jaksch, J. I. Cirac, and P. Zoller Institut für Theoretische
Physik, Universität Innsbruck, Technikerstrasse 25, A-6020
Innsbruck, Austria
S. L. Rolston National Institute of Standards and Technology,
Gaithersburg, Maryland 20899
R. Côté1 and M. D. Lukin2
1Physics Department, University of Connecticut, 2152 Hillside Road,
Storrs, Connecticut 06269-3046 2ITAMP, Harvard-Smithsonian Center
for Astrophysics, Cambridge, Massachusetts 02138
(Received 7 April 2000)
We propose several schemes for implementing a fast two-qubit
quantum gate for neutral atoms with the gate operation time much
faster than the time scales associated with the external motion of
the atoms in the trapping potential. In our example, the large
interaction energy required to perform fast gate operations is
provided by the dipole-dipole interaction of atoms excited to
low-lying Rydberg states in constant electric fields. A detailed
analysis of imperfections of the gate operation is given.
PACS numbers: 03.67.Lx, 32.80.Pj, 32.80.Rm
In recent years, numerous proposals to build quantum information
processors have been made [1]. Because of their exceptional ability
of quantum control and long coherence times, quantum optical
systems such as trapped ions [2] and atoms [3], and cavity QED [4],
have taken a leading role in implementing quantum logic in the
labora- tory. Quantum computing with neutral atoms [5] seems
particularly attractive in view of very long coherence times of the
internal atomic states and well-developed techniques for cooling
and trapping atoms in optical lattices, far off-resonance light
traps, and magnetic microtraps [3]. Preparation and rotations of
single qubits associated with long-lived internal states can be
performed by addressing individual atoms with laser pulses. A
central issue is to design fast two-qubit gates.
First of all, it is difficult to identify a strong and con-
trollable two-body interaction for neutral atoms, which is required
to design a gate. Furthermore, the strength of two-body
interactions does not necessarily translate into a useful fast
quantum gate: large interactions are usually associated with strong
mechanical forces on the trapped atoms. Thus, internal states of
the trapped atoms (the qubits) may become entangled with the
motional degrees of freedom during the gate, resulting effectively
in an ad- ditional source of decoherence. This leads to the typical
requirement that the process is adiabatic on the time scale of the
oscillation period of the trapped atoms in order to avoid
entanglement with motional states. As a result, ex- tremely tight
traps and low temperatures are required.
In the present Letter, we propose a fast phase gate for neutral
trapped atoms, corresponding to a truth table je1! ≠ je2! !
eie1e2wje1! ≠ je2! for the logical states jei! with ei ! 0, 1,
which (i) exploits the very large interactions of permanent dipole
moments of laser excited Rydberg states in a constant electric
field to entangle atoms, while (ii) allowing gate operation times
set by
the time scale of the laser excitation or the two particle
interaction energy, which can be significantly shorter than the
trap period. Among the attractive features of the gate are the
insensitivity to the temperature of the atoms and to the variations
in atom-atom separation.
Rydberg states [6] of a hydrogen atom within a given manifold of a
fixed principal quantum number n are de- generate. This degeneracy
is removed by applying a con- stant electric field E along the z
axis (linear Stark effect). For electric fields below the
Ingris-Teller limit the mix- ing of adjacent n manifolds can be
neglected, and the en- ergy levels are split according to DEnqm !
3nqea0E"2 with parabolic and magnetic quantum numbers q ! n 2 1 2
jmj, n 2 3 2 jmj, . . . , 2#n 2 1 2 jmj$ and m, re- spectively, e
the electron charge, and a0 the Bohr ra- dius. These Stark states
have permanent dipole moments m % mzez ! 3nqea0ez"2. In alkali
atoms the s and p states are shifted relative to the higher angular
momentum states due to their quantum defects, and the Stark maps of
the m ! 0 and m ! 1 manifolds are correspondingly modified
[6].
Consider two atoms 1 and 2 at fixed positions (see Fig. 1a), and
initially prepared in Stark eigenstates, with a dipole moment along
z and a given m, as selected by the polarization of the laser
exciting the Rydberg states from the ground state. They interact
and evolve according to the dipole-dipole potential
Vdip#r$ ! 1
jrj5
(1)
with r the distance between the atoms. We are interested in the
limit where the electric field is sufficiently large so that the
energy splitting between two adjacent Stark states is much larger
than the dipole-dipole interaction.
2208 0031-9007"00"85(10)"2208(4)$15.00 © 2000 The American Physical
Society
VOLUME 87, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 16
JULY 2001
Dipole Blockade and Quantum Information Processing in Mesoscopic
Atomic Ensembles
M. D. Lukin,1 M. Fleischhauer,1,2 and R. Cote3
1ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge,
Massachusetts 02138 2Fachbereich Physik, Universität
Kaiserslautern, D-67663 Kaiserslautern, Germany
3Physics Department, University of Connecticut, Storrs, Connecticut
06269
L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller Institut für
Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck,
Austria
(Received 7 November 2000; published 26 June 2001)
We describe a technique for manipulating quantum information stored
in collective states of meso- scopic ensembles. Quantum processing
is accomplished by optical excitation into states with strong
dipole-dipole interactions. The resulting “dipole blockade” can be
used to inhibit transitions into all but singly excited collective
states. This can be employed for a controlled generation of
collective atomic spin states as well as nonclassical photonic
states and for scalable quantum logic gates. An example involving a
cold Rydberg gas is analyzed.
DOI: 10.1103/PhysRevLett.87.037901 PACS numbers: 03.67.Lx,
03.75.Fi, 42.50.Gy, 73.23.–b
Recent advances in quantum information science have opened a door
for a number of fascinating potential appli- cations ranging from
the factorization of large numbers and secure communication to
spectroscopic techniques with enhanced sensitivity. But the
practical implementation of quantum processing protocols such as
quantum computa- tion requires coherent manipulation of a large
number of coupled quantum systems which is an extremely difficult
task [1]. Challenges ranging from a long-time storage of quantum
information to scalable quantum logic gates are by now well known.
It is generally believed that precise manipulation of microscopic
quantum objects is essential to implement quantum protocols. For
example, in most of the potentially viable candidates for quantum
comput- ers an exceptional degree of control over submicron sys-
tems is essential for performing single-bit operations and the
two-bit coupling is accomplished by interactions be- tween nearest
neighbors [2]. Related techniques are also being explored that
involve photons to connect qubits [3], and to construct potentially
scalable quantum networks [4]. However, since the single-atom
absorption cross section is very small, reliable coupling to light
requires high-finesse microcavities [5].
In the present Letter we describe a technique for the coherent
manipulation of quantum information stored in collective
excitations of mesoscopic many-atom en- sembles. This is
accomplished by optically exciting the ensemble into states with a
strong atom-atom interaction. Specifically, we consider the case
involving dipole-dipole interactions in an ensemble of cold atoms
excited into Rydberg states. Under certain conditions the level
shifts associated with these interactions can be used to block the
transitions into states with more than a single ex- citation. The
resulting “dipole blockade” phenomenon closely resembles similar
mesoscopic effects in nanoscale solid-state devices [6]. In the
present context it can take place in an ensemble with a size that
can exceed many
optical wavelengths. Combined with the exceptional degree of
control that is typical for quantum optical systems and long
coherence times, this allows one to considerably alleviate many
stringent requirements for the experimental implementation of
various quantum processing protocols. In particular, we show that
this technique can be used to (i) generate superpositions of
collective spin states (or Dicke states [7]) in an ensemble; (ii)
coherently convert these states into corre- sponding states of
photon wave packets of prescribed di- rection, duration, and pulse
shapes and vice versa using the collectively enhanced coupling to
light [8]; and (iii) per- form quantum gate operations between
distant qubits. Corresponding applications including (i) subshot
noise spectroscopy and atom interferometry [9], (ii) secure
cryptography protocols [10], and (iii) scalable quantum logic
devices can be foreseen. In general, no strongly coupling
microcavities and no single particle control are required to
implement computation and communication protocols. We further
anticipate that the approach can be applied to a variety of
interacting many-body systems ranging from trapped ions to
specifically designed semi- conductor structures.
The basic element of the present scheme is an ensemble of N
identical multistate atoms (Fig. 1) contained in a volume V . Using
well-developed techniques all atoms can be trapped and prepared in
a specific sublevel (gi , i ! 1, . . . , N) of the ground state
manifold. Relevant states of each atom include a pair of metastable
sublevels of the ground state manifold qi that are used for
long-time storage of qubits (storage states) and long-lived Rydberg
states ri , p
0 i , p
00 i . Additional Rydberg sublevels as well as
lower electronic excited states can be used for specific
applications. We assume modest atomic densities, such that
interactions between atoms can safely be neglected whenever they
are in the sublevels of the ground state. This also implies long
coherence lifetimes — up to a few
037901-1 0031-9007!01!87(3)!037901(4)$15.00 © 2001 The American
Physical Society 037901-1
Rydberg atoms and their van der Waals interaction
Lifetime > 100 μs Transition dipole: d n2ea0
<latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit>
Lifetime > 100 μs Transition dipole: d n2ea0
<latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit>
+ +
Lifetime > 100 μs Transition dipole: d n2ea0
<latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">AAAC1HicjVHLSsNAFD2Nr1ofrbp0EyyCq5IUwS4LblxWsA9oa0nSaR2aF5mJUKorcesPuNVvEv9A/8I7YwpqEZ2Q5My559yZe68b+1xIy3rNGUvLK6tr+fXCxubWdrG0s9sSUZp4rOlFfpR0XEcwn4esKbn0WSdOmBO4Pmu7k1MVb1+zRPAovJDTmPUDZxzyEfccSdSgVBz2BA/M8LJqMmdgmYNS2apYepmLwM5AGdlqRKUX9DBEBA8pAjCEkIR9OBD0dGHDQkxcHzPiEkJcxxluUSBvSipGCofYCX3HtOtmbEh7lVNot0en+PQm5DRxSJ6IdAlhdZqp46nOrNjfcs90TnW3Kf3dLFdArMQVsX/55sr/+lQtEiPUdA2caoo1o6rzsiyp7oq6ufmlKkkZYuIUHlI8Iexp57zPpvYIXbvqraPjb1qpWLX3Mm2Kd3VLGrD9c5yLoFWt2FbFPj8u12vZqPPYxwGOaJ4nqOMMDTT1zB/xhGejZdwYd8b9p9TIZZ49fFvGwwdSzJR0</latexit>
+ +
Rydberg atoms and their van der Waals interaction
Lifetime > 100 μs Transition dipole: d n2ea0
<latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit><latexit
sha1_base64="pTo/3/3LqD9o9gjA9H/PH0vWtGQ=">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</latexit>
+ +
R <latexit
sha1_base64="O/jBCx4552tN7ec8bncGvbo9p8A=">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</latexit><latexit
sha1_base64="O/jBCx4552tN7ec8bncGvbo9p8A=">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</latexit><latexit
sha1_base64="O/jBCx4552tN7ec8bncGvbo9p8A=">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</latexit><latexit
sha1_base64="O/jBCx4552tN7ec8bncGvbo9p8A=">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</latexit>
Interaction in Rydberg state = 1011 x ground state interaction!! ⇒
Switchable for ! R 1 10µm
<latexit
sha1_base64="b+ozCjZz8mqXbdoJEgWoqwfirQ8=">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</latexit><latexit
sha1_base64="b+ozCjZz8mqXbdoJEgWoqwfirQ8=">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</latexit><latexit
sha1_base64="b+ozCjZz8mqXbdoJEgWoqwfirQ8=">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</latexit><latexit
sha1_base64="b+ozCjZz8mqXbdoJEgWoqwfirQ8=">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</latexit>
A B
A B C6
A B C6
R6 <latexit
sha1_base64="wu4hp892Ln1bfVbOdvCvop/c4cU=">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</latexit><latexit
sha1_base64="wu4hp892Ln1bfVbOdvCvop/c4cU=">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</latexit><latexit
sha1_base64="wu4hp892Ln1bfVbOdvCvop/c4cU=">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</latexit><latexit
sha1_base64="wu4hp892Ln1bfVbOdvCvop/c4cU=">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</latexit>
C6
C6
C6
Dipole trap light 850 nm
Single atoms in optical tweezers
1 µm
1 mK
Dipole trap light 850 nm
Fluorescence 780nm
Single atoms in optical tweezers
1 µm
1 mK
Fluorescence 780nm
9.0 9.5 10.0 10.5 11.0 11.5 8.5 0
20
40
60
80
100
Single atoms in optical tweezers
A single Rb atom (20 μK)!
Non deterministic
1 µm
1 mK
'(x, y) 0
SLM pattern
Bergamini, JOSA B 21, 1889 (2004) Nogrette, PRX 4, 021034
(2014)
FT[ei'(x,y)] 2
Spatial Light Modulator
(liquid crystals) Reconfigurable
'(x, y) 0
FT[ei'(x,y)] 2
Bergamini, JOSA B 21, 1889 (2004) Nogrette, PRX 4, 021034 (2014) 10
μm
Atom-by-atom assembling of 2D arrays 10 μm
Problem: stochastic loading (p ~ 0.5)
Atom-by-atom assembling of 2D arrays 10 μm
Problem: stochastic loading (p ~ 0.5)
Solution: sort atoms in arrays
Barredo, Science 354, 1021 (2016) Also Harvard (1D) &
Korea
Atom-by-atom assembling of 2D arrays 10 μm
Problem: stochastic loading (p ~ 0.5)
Solution: sort atoms in arrays
Barredo, Science 354, 1021 (2016) Also Harvard (1D) &
Korea
In iti al
Fi na l
~ 100 atoms
474 nm
795 nm
E 87Rb
|ri <latexit
sha1_base64="EBJKlO8wz/yloVozRtr/CGHg/dE=">AAACzHicjVHLSsNAFD2Nr/quunQTLIKrkkhBlwU3rqSCfUhbJEmnNXTyYDIRSu3WH3Cr3yX+gf6Fd8YpqEV0QpIz595zZu69fsrDTDrOa8FaWFxaXimurq1vbG5tl3Z2m1mSi4A1goQnou17GeNhzBoylJy1U8G8yOes5Y/OVLx1x0QWJvGVHKesF3nDOByEgSeJur4XXeHFQ85uSmWn4uhlzwPXgDLMqielF3TRR4IAOSIwxJCEOTxk9HTgwkFKXA8T4gShUMcZplgjbU5ZjDI8Ykf0HdKuY9iY9soz0+qATuH0ClLaOCRNQnmCsDrN1vFcOyv2N++J9lR3G9PfN14RsRK3xP6lm2X+V6dqkRjgVNcQUk2pZlR1gXHJdVfUze0vVUlySIlTuE9xQTjQylmfba3JdO2qt56Ov+lMxap9YHJzvKtb0oDdn+OcB83jikv4slquVc2oi9jHAY5onieo4Rx1NMg7wiOe8GxdWNKaWNPPVKtgNHv4tqyHD3cvkwo=</latexit><latexit
sha1_base64="EBJKlO8wz/yloVozRtr/CGHg/dE=">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</latexit><latexit
sha1_base64="EBJKlO8wz/yloVozRtr/CGHg/dE=">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</latexit><latexit
sha1_base64="EBJKlO8wz/yloVozRtr/CGHg/dE=">AAACzHicjVHLSsNAFD2Nr/quunQTLIKrkkhBlwU3rqSCfUhbJEmnNXTyYDIRSu3WH3Cr3yX+gf6Fd8YpqEV0QpIz595zZu69fsrDTDrOa8FaWFxaXimurq1vbG5tl3Z2m1mSi4A1goQnou17GeNhzBoylJy1U8G8yOes5Y/OVLx1x0QWJvGVHKesF3nDOByEgSeJur4XXeHFQ85uSmWn4uhlzwPXgDLMqielF3TRR4IAOSIwxJCEOTxk9HTgwkFKXA8T4gShUMcZplgjbU5ZjDI8Ykf0HdKuY9iY9soz0+qATuH0ClLaOCRNQnmCsDrN1vFcOyv2N++J9lR3G9PfN14RsRK3xP6lm2X+V6dqkRjgVNcQUk2pZlR1gXHJdVfUze0vVUlySIlTuE9xQTjQylmfba3JdO2qt56Ov+lMxap9YHJzvKtb0oDdn+OcB83jikv4slquVc2oi9jHAY5onieo4Rx1NMg7wiOe8GxdWNKaWNPPVKtgNHv4tqyHD3cvkwo=</latexit>
5p <latexit
sha1_base64="/qAtFa8eXFmMittMBFOJ/09h120=">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</latexit><latexit
sha1_base64="/qAtFa8eXFmMittMBFOJ/09h120=">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</latexit><latexit
sha1_base64="/qAtFa8eXFmMittMBFOJ/09h120=">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</latexit><latexit
sha1_base64="/qAtFa8eXFmMittMBFOJ/09h120=">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</latexit>
474 nm
795 nm
E 87Rb
/(2) = 0.5 5MHz <latexit
sha1_base64="/DbtHOkr4nXAhjff9GDJQa6KBuI=">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</latexit><latexit
sha1_base64="/DbtHOkr4nXAhjff9GDJQa6KBuI=">AAAC5nicjVHLLgRBFD3a+z1Y2hQTCQmjWwgbicTGRpCYIdEi3T01o6Jfqa6WMLG2sxNbP2DLp4g/4C/cKi3BRKhOd586955Tde/101BkyrZfOqzOru6e3r7+gcGh4ZHR0th4LUtyGfBqkISJPPS9jIci5lUlVMgPU8m9yA/5gX+2qeMH51xmIon31UXKjyOvGYuGCDxF1Elpyt2JeNNbnF1yUzG3zuzKCltgK+58y5UR2966vDople2KbRZrB04ByijWblJ6hos6EgTIEYEjhiIcwkNGzxEc2EiJO0aLOElImDjHFQZIm1MWpwyP2DP6Nml3VLAx7bVnZtQBnRLSK0nJMEOahPIkYX0aM/HcOGv2N++W8dR3u6C/X3hFxCqcEvuX7jPzvzpdi0IDa6YGQTWlhtHVBYVLbrqib86+VKXIISVO4zrFJeHAKD/7zIwmM7Xr3nom/moyNav3QZGb403fkgbs/BxnO6gtVRy74uwtlzeWi1H3YRLTmKV5rmIDW9hFlbyv8YBHPFmn1o11a919pFodhWYC35Z1/w4jS5oQ</latexit><latexit
sha1_base64="/DbtHOkr4nXAhjff9GDJQa6KBuI=">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</latexit><latexit
sha1_base64="/DbtHOkr4nXAhjff9GDJQa6KBuI=">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</latexit>
Blockade + collective excitation √2 Gaétan, Nat. Phys. 5, 115
(2009)
Blockade Urban, Nat. Phys. 5, 110 (2009)
Early demonstrations of blockade and gate with 2 atoms
1.0
0.8
0.6
0.4
0.2
0.0
1p 2 (|rgi+ |gri)
Blockade + collective excitation √2 Gaétan, Nat. Phys. 5, 115
(2009)
Entanglement Wilk, PRL 104, 010502 (2010)
Blockade Urban, Nat. Phys. 5, 110 (2009)
C-NOT gate Isenhower, PRL 104, 010503 (2010)
Early demonstrations of blockade and gate with 2 atoms
F = 0.75 <latexit
sha1_base64="r97T6E0+y6c6EeYgo2I0tDGgkPg=">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</latexit><latexit
sha1_base64="r97T6E0+y6c6EeYgo2I0tDGgkPg=">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</latexit><latexit
sha1_base64="r97T6E0+y6c6EeYgo2I0tDGgkPg=">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</latexit><latexit
sha1_base64="r97T6E0+y6c6EeYgo2I0tDGgkPg=">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</latexit>
1p 2 (|rgi+ |gri)
State-of-the-art on gate, fidelity and entanglement (2019)
two-qubit gates, there can be large improvements in efficiency and
error tolerance by using native multi-qubit operations. These are
discussed in section 5.4. Although gate protocols have been
developed that promise high fidelity compatible with scalable
architectures there are a plethora of technical challenges that
remain to be solved. An overview of these issues is provided in
section 5.5.
5.1. One-qubit gates
Single qubit gates acting on qubits encoded in atomic hyperfine
states can be performed with microwaves [56, 129], with
two-frequency Raman light [55, 130], or with a com- bination of
microwaves and a gradient field for addressing of individual qubits
[59–61, 131] or groups of qubits [132]. The most recent experiments
have provided detailed character- ization of gate fidelity at Stark
shift selected sites in large multi-qubit arrays. Using randomized
benchmarking [133] average fidelities for Clifford gates of 0.992
[59] and 0.996 [61] have been demonstrated. Crosstalk errors to
nearby, non- targeted qubits were 0.014 [59] and 0.0046 [61].
Improved gate fidelity and reduced crosstalk were demonstrated in
[61] by implementing a sequence of pulses which make gate errors
sensitive to the fourth power of small beam pointing errors.
The error budgets in these experiments are associated with
inhomogeneities in the microwave field, variations in trap induced
qubit frequency shifts, and errors from the Stark addressing beams
due to imperfect spatial addressing, leakage to nontargeted sites,
and residual light scattering. Reduced sensitivity to pointing
errors together with reduced leakage to other sites can be achieved
by spatial shaping of the Stark beam [134]. While much work remains
to be done, it should be possible to reduce single qubit gate
errors to ~ -10 4 and below, a level of performance that has
already been demon- strated with trapped ion hyperfine qubits [3,
135].
5.2. Two-qubit gates
Two-qubit entanglement and logic gates using Rydberg state
interactions have been demonstrated in several experiments and are
listed in table 1. The first Rydberg blockade entan- glement
experiments were performed in 2010 [33–35]. These were followed by
improved results in 2015 [10, 11]. Exper- imental gate results are
shown in figure 7. Two-qubit entan- glement was also achieved using
local spin exchange with
atoms in movable tweezers in 2015 [31], but with lower fidelity
than the Rydberg experiments. While single qubit gate operations
with neutral atom qubits have already reached high fidelity, as
discussed in section 5.1, there is a large gap between the fidelity
results summarized in table 1 and the very high entanglement
fidelities that have been demonstrated with trapped ion [2–4] and
superconducting [5, 6, 136] qubits.
Scalable computation requires quantum error correction (QEC) with
the gate fidelity needed for QEC dependent on the size and
architecture of the code blocks. Roughly speak- ing larger codes
can tolerate gates with higher errors [7, 8], with large scale
surface codes that combine hundreds of physical qubits to create a
single logical qubit having threshold error rates ∼0.01. The
requirement of managing atom loss in a neutral atom qubit array,
see figure 4, suggests that smaller code sizes are preferable.
Concatenated codes with sizes of 25 qubits or less have thresholds
∼0.001 and for scalability gate error rates should be at least a
factor of 10 better. We conclude that scalable neutral atom quantum
computing will require a two-qubit gate fidelity of ~F 0.9999. This
is not a fundamental limit, and could be
relaxed with the invention of efficient codes that tolerate higher
errors, but serves as a placeholder with which to evaluate the
status of current approaches.
Comparing this target performance with table 1 it is apparent that
in order for Rydberg gates to be viable for scalable quantum
computation the fidelity needs to be greatly improved. It is
therefore important to identify the reasons for the relatively low
fidelity demonstrated to date. There are two aspects of gate
fidelity that can be considered separately. The
Table 1. Experimental Bell state fidelity achieved in two-qubit
neutral atom experiments. All fidelities were measured using parity
oscillations [137]. Values in parentheses are less than the
sufficient entanglement threshold of 0.5, but may still represent
entangled states as was explicitly verified in [31]. Post selected
values are corrected for atom loss during the experimental
sequence. Experiments were performed in the year indicated.
Atom Method Bell fidelity Post selected fidelity Year and
reference
87Rb Blockade, simultaneous addressing (0.46) 0.75 2009 [35] 87Rb
Blockade, separate addressing (0.48) 0.58 2009 [33] 87Rb Blockade,
separate addressing 0.58 0.71 2010 [34] Cs Blockade, separate
addressing 0.73 0.79 2015 [10] Cs Dressing, simultaneous addressing
0.60 0.81 2015 [11] 87Rb Local spin exchange (0.44) 0.63 2015
[31]
Figure 7. Rydberg gate experiments with Cs atoms. (a) Eye diagram
of CNOT gate with and without blockade as a function of the
relative phase f between p 2 pulses on the target qubit from [10].
(b) Parity oscillations of Bell states. Reprinted by permission
from Macmillan Publishers Ltd: Nature Physics [11], copyright
2015.
8
J. Phys. B: At. Mol. Opt. Phys. 49 (2016) 202001 Topical
Review
Review
State-of-the-art on gate, fidelity and entanglement (2019)
two-qubit gates, there can be large improvements in efficiency and
error tolerance by using native multi-qubit operations. These are
discussed in section 5.4. Although gate protocols have been
developed that promise high fidelity compatible with scalable
architectures there are a plethora of technical challenges that
remain to be solved. An overview of these issues is provided in
section 5.5.
5.1. One-qubit gates
Single qubit gates acting on qubits encoded in atomic hyperfine
states can be performed with microwaves [56, 129], with
two-frequency Raman light [55, 130], or with a com- bination of
microwaves and a gradient field for addressing of individual qubits
[59–61, 131] or groups of qubits [132]. The most recent experiments
have provided detailed character- ization of gate fidelity at Stark
shift selected sites in large multi-qubit arrays. Using randomized
benchmarking [133] average fidelities for Clifford gates of 0.992
[59] and 0.996 [61] have been demonstrated. Crosstalk errors to
nearby, non- targeted qubits were 0.014 [59] and 0.0046 [61].
Improved gate fidelity and reduced crosstalk were demonstrated in
[61] by implementing a sequence of pulses which make gate errors
sensitive to the fourth power of small beam pointing errors.
The error budgets in these experiments are associated with
inhomogeneities in the microwave field, variations in trap induced
qubit frequency shifts, and errors from the Stark addressing beams
due to imperfect spatial addressing, leakage to nontargeted sites,
and residual light scattering. Reduced sensitivity to pointing
errors together with reduced leakage to other sites can be achieved
by spatial shaping of the Stark beam [134]. While much work remains
to be done, it should be possible to reduce single qubit gate
errors to ~ -10 4 and below, a level of performance that has
already been demon- strated with trapped ion hyperfine qubits [3,
135].
5.2. Two-qubit gates
Two-qubit entanglement and logic gates using Rydberg state
interactions have been demonstrated in several experiments and are
listed in table 1. The first Rydberg blockade entan- glement
experiments were performed in 2010 [33–35]. These were followed by
improved results in 2015 [10, 11]. Exper- imental gate results are
shown in figure 7. Two-qubit entan- glement was also achieved using
local spin exchange with
atoms in movable tweezers in 2015 [31], but with lower fidelity
than the Rydberg experiments. While single qubit gate operations
with neutral atom qubits have already reached high fidelity, as
discussed in section 5.1, there is a large gap between the fidelity
results summarized in table 1 and the very high entanglement
fidelities that have been demonstrated with trapped ion [2–4] and
superconducting [5, 6, 136] qubits.
Scalable computation requires quantum error correction (QEC) with
the gate fidelity needed for QEC dependent on the size and
architecture of the code blocks. Roughly speak- ing larger codes
can tolerate gates with higher errors [7, 8], with large scale
surface codes that combine hundreds of physical qubits to create a
single logical qubit having threshold error rates ∼0.01. The
requirement of managing atom loss in a neutral atom qubit array,
see figure 4, suggests that smaller code sizes are preferable.
Concatenated codes with sizes of 25 qubits or less have thresholds
∼0.001 and for scalability gate error rates should be at least a
factor of 10 better. We conclude that scalable neutral atom quantum
computing will require a two-qubit gate fidelity of ~F 0.9999. This
is not a fundamental limit, and could be
relaxed with the invention of efficient codes that tolerate higher
errors, but serves as a placeholder with which to evaluate the
status of current approaches.
Comparing this target performance with table 1 it is apparent that
in order for Rydberg gates to be viable for scalable quantum
computation the fidelity needs to be greatly improved. It is
therefore important to identify the reasons for the relatively low
fidelity demonstrated to date. There are two aspects of gate
fidelity that can be considered separately. The
Table 1. Experimental Bell state fidelity achieved in two-qubit
neutral atom experiments. All fidelities were measured using parity
oscillations [137]. Values in parentheses are less than the
sufficient entanglement threshold of 0.5, but may still represent
entangled states as was explicitly verified in [31]. Post selected
values are corrected for atom loss during the experimental
sequence. Experiments were performed in the year indicated.
Atom Method Bell fidelity Post selected fidelity Year and
reference
87Rb Blockade, simultaneous addressing (0.46) 0.75 2009 [35] 87Rb
Blockade, separate addressing (0.48) 0.58 2009 [33] 87Rb Blockade,
separate addressing 0.58 0.71 2010 [34] Cs Blockade, separate
addressing 0.73 0.79 2015 [10] Cs Dressing, simultaneous addressing
0.60 0.81 2015 [11] 87Rb Local spin exchange (0.44) 0.63 2015
[31]
Figure 7. Rydberg gate experiments with Cs atoms. (a) Eye diagram
of CNOT gate with and without blockade as a function of the
relative phase f between p 2 pulses on the target qubit from [10].
(b) Parity oscillations of Bell states. Reprinted by permission
from Macmillan Publishers Ltd: Nature Physics [11], copyright
2015.
8
J. Phys. B: At. Mol. Opt. Phys. 49 (2016) 202001 Topical
Review
2016-17: importance of laser phase noise + new generation
expt
Lukin PRL (2018)
Saffman, JPhysB (2016)
Next, we characterize the coherence of single atoms and demonstrate
single-qubit control. To begin, we experimen- tally measure the
lifetime of the Rydberg state in Fig. 2(a). The measured T1 ¼ Tr→g
¼ 51ð6Þ μs is consistent with the 146 μs Rydberg state lifetime
[35] when combined with the ∼80 μs timescale for off-resonant
scattering of the 1013 nm laser from jei. A Ramsey experiment shows
Gaussian decay that is well explained by thermal Doppler shifts
[see Fig. 2(b)]. At 10 μK, the random atomic velocity in each shot
of the experiment appears as a random detuning δD
from a Gaussian distribution of width 2π × 43.5 kHz, resulting in
dephasing as jψi → ð1=
ffiffiffi 2
p Þðjgiþ eiδ
DtjriÞ. However, since the random Doppler shift is constant over
the duration of each pulse sequence, its effect can be
eliminated via a spin-echo sequence [orange in Fig. 2(b)]. Note
that the spin-echo measurements display some small deviations from
the numerical simulations, indicating the presence of an additional
dephasing channel. Assuming an exponential decay, we measure a
fitted T2 ¼ 32ð6Þ μs and extract a pure dephasing time T ¼ ½1=T2 −
1=ð2Tr→gÞ&−1 ¼ 47ð13Þ μs. We hypothesize that this dephasing
may result from residual laser phase noise. Finally, we demonstrate
a single-atom phase gate by
applying an independent focused laser that shifts the energy of the
ground state jgi [see Fig. 2(c)] [27]. By controlling the duration
of the applied laser pulse, we impart a controlled dynamical phase
on jgi relative to jri. The contrast of the resulting phase gate
(embedded in a spin- echo sequence) is close to the limit imposed
by detection and spin-echo fidelity. We next turn to two-atom
control. To this end, we
position two atoms at a separation of 5.7 μm, at which the
Rydberg-Rydberg interaction is U= ¼ 2π × 30 MHz Ω ¼ 2π × 2 MHz. In
this so-called Rydberg blockade regime, the laser field globally
couples both atoms from jggi to the symmetric state jWi ¼ ð1=
ffiffiffi 2
p Ω [see Fig. 3(a)] (here
the excited states jri are defined in the rotating frame to
incorporate the spatial phase factors eikx, as discussed in [27]).
The measured probabilities for the states jggi, jgri, jrgi, and
jrri (denoted by Pgg, Pgr, Prg, and Prr, respec- tively) show that
indeed no population enters the doubly excited state (Prr <
0.02, consistent with only detection error). Instead, there are
oscillations between the manifold of zero excitations and the
manifold of one excitation with a fitted frequency of 2π × 2.83
MHz≈
ffiffiffi 2
p Ω [see Fig. 3(b)].
These collective Rabi oscillations can be used to directly prepare
the maximally entangled Bell state jWi by applying a π pulse at the
enhanced Rabi frequency (denoted byXW
π ). To determine the fidelity of this experimentally prepared
entangled state, given by F ¼ hWjρjWi, we express it in terms of
diagonal and off diagonal matrix elements of the density operator
ρ:
F ¼ 1
1
2 ðρgr;rg þ ρrg;grÞ; ð1Þ
where ραβ;γδ ¼ hαβjρjγδi for α, β, γ, δ ∈fg; rg. The diagonal
elements can be directly measured by applying a π pulse and then
measuring the populations. The results closely match those of a
perfect jWi state after accounting for state detection errors, with
ρgr;gr þ ρrg;rg ¼ 0.94ð1Þ, relative to a maximum possible value of
0.95(1). To measure the off diagonal elements of the density
matrix, we make use of the single-atom phase gate Zð1Þ
demonstrated in Fig. 2(c), which introduces a variable phase on one
atom (as demonstrated in [36]). Specifically, a local beam adds a
light shift δ to jgri but not to jrgi,
(a) (b)
(c)
FIG. 2. Characterization of single-atom coherence and phase
control. (a) The lifetime of jri is measured by exciting from jgi
to jri with a π pulse, and then deexciting after a variable delay.
The probability to end in jgi (denoted Pg) decays with an extracted
lifetime of T1 ¼ 51ð6Þ μs (fitted to an exponential decay model
with no offset). (b) A Ramsey experiment (blue) shows Gaussian
decay with a 1=e lifetime of T '
2 ¼ 4.5ð1Þ μs, limited by thermal Doppler shifts. Inserting an
additional π pulse (orange) between the π=2 pulses cancels the
effect of the Doppler shifts and results in a substantially longer
coherence lifetime of T2 ¼ 32ð6Þ μs (fitted to an exponential decay
model with an offset of 0.5). (c) A single-atom phase gate is
implemented by applying an indepen- dent 809 nm laser which induces
a light shift δ ¼ 2π × 5 MHz on the ground state for time t,
resulting in an accumulated dynamical phase ¼ δt. The gate is
embedded in a spin-echo sequence to cancel Doppler shifts. In each
measurement shown here, the 1013 nm laser remains on for the entire
pulse sequence, while the 420 nm laser is pulsed according to the
sequence shown above each plot. Each data point is calculated from
200–500 repeated measurements with a single atom per
realization.
PHYSICAL REVIEW LETTERS 121, 123603 (2018)
123603-3
State-of-the-art on gate, fidelity and entanglement (2019)
two-qubit gates, there can be large improvements in efficiency and
error tolerance by using native multi-qubit operations. These are
discussed in section 5.4. Although gate protocols have been
developed that promise high fidelity compatible with scalable
architectures there are a plethora of technical challenges that
remain to be solved. An overview of these issues is provided in
section 5.5.
5.1. One-qubit gates
Single qubit gates acting on qubits encoded in atomic hyperfine
states can be performed with microwaves [56, 129], with
two-frequency Raman light [55, 130], or with a com- bination of
microwaves and a gradient field for addressing of individual qubits
[59–61, 131] or groups of qubits [132]. The most recent experiments
have provided detailed character- ization of gate fidelity at Stark
shift selected sites in large multi-qubit arrays. Using randomized
benchmarking [133] average fidelities for Clifford gates of 0.992
[59] and 0.996 [61] have been demonstrated. Crosstalk errors to
nearby, non- targeted qubits were 0.014 [59] and 0.0046 [61].
Improved gate fidelity and reduced crosstalk were demonstrated in
[61] by implementing a sequence of pulses which make gate errors
sensitive to the fourth power of small beam pointing errors.
The error budgets in these experiments are associated with
inhomogeneities in the microwave field, variations in trap induced
qubit frequency shifts, and errors from the Stark addressing beams
due to imperfect spatial addressing, leakage to nontargeted sites,
and residual light scattering. Reduced sensitivity to pointing
errors together with reduced leakage to other sites can be achieved
by spatial shaping of the Stark beam [134]. While much work remains
to be done, it should be possible to reduce single qubit gate
errors to ~ -10 4 and below, a level of performance that has
already been demon- strated with trapped ion hyperfine qubits [3,
135].
5.2. Two-qubit gates
Two-qubit entanglement and logic gates using Rydberg state
interactions have been demonstrated in several experiments and are
listed in table 1. The first Rydberg blockade entan- glement
experiments were performed in 2010 [33–35]. These were followed by
improved results in 2015 [10, 11]. Exper- imental gate results are
shown in figure 7. Two-qubit entan- glement was also achieved using
local spin exchange with
atoms in movable tweezers in 2015 [31], but with lower fidelity
than the Rydberg experiments. While single qubit gate operations
with neutral atom qubits have already reached high fidelity, as
discussed in section 5.1, there is a large gap between the fidelity
results summarized in table 1 and the very high entanglement
fidelities that have been demonstrated with trapped ion [2–4] and
superconducting [5, 6, 136] qubits.
Scalable computation requires quantum error correction (QEC) with
the gate fidelity needed for QEC dependent on the size and
architecture of the code blocks. Roughly speak- ing larger codes
can tolerate gates with higher errors [7, 8], with large scale
surface codes that combine hundreds of physical qubits to create a
single logical qubit having threshold error rates ∼0.01. The
requirement of managing atom loss in a neutral atom qubit array,
see figure 4, suggests that smaller code sizes are preferable.
Concatenated codes with sizes of 25 qubits or less have thresholds
∼0.001 and for scalability gate error rates should be at least a
factor of 10 better. We conclude that scalable neutral atom quantum
computing will require a two-qubit gate fidelity of ~F 0.9999. This
is not a fundamental limit, and could be
relaxed with the invention of efficient codes that tolerate higher
errors, but serves as a placeholder with which to evaluate the
status of current approaches.
Comparing this target performance with table 1 it is apparent that
in order for Rydberg gates to be viable for scalable quantum
computation the fidelity needs to be greatly improved. It is
therefore important to identify the reasons for the relatively low
fidelity demonstrated to date. There are two aspects of gate
fidelity that can be considered separately. The
Table 1. Experimental Bell state fidelity achieved in two-qubit
neutral atom experiments. All fidelities were measured using parity
oscillations [137]. Values in parentheses are less than the
sufficient entanglement threshold of 0.5, but may still represent
entangled states as was explicitly verified in [31]. Post selected
values are corrected for atom loss during the experimental
sequence. Experiments were performed in the year indicated.
Atom Method Bell fidelity Post selected fidelity Year and
reference
87Rb Blockade, simultaneous addressing (0.46) 0.75 2009 [35] 87Rb
Blockade, separate addressing (0.48) 0.58 2009 [33] 87Rb Blockade,
separate addressing 0.58 0.71 2010 [34] Cs Blockade, separate
addressing 0.73 0.79 2015 [10] Cs Dressing, simultaneous addressing
0.60 0.81 2015 [11] 87Rb Local spin exchange (0.44) 0.63 2015
[31]
Figure 7. Rydberg gate experiments with Cs atoms. (a) Eye diagram
of CNOT gate with and without blockade as a function of the
relative phase f between p 2 pulses on the target qubit from [10].
(b) Parity oscillations of Bell states. Reprinted by permission
from Macmillan Publishers Ltd: Nature Physics [11], copyright
2015.
8
J. Phys. B: At. Mol. Opt. Phys. 49 (2016) 202001 Topical
Review
2016-17: importance of laser phase noise + new generation
expt
Lukin PRL (2018)
|GHZi / |0101...i+ |1010...i <latexit
sha1_base64="9pt9m09OJGtqXRjpEg8XEBEMJ+o=">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</latexit><latexit
sha1_base64="9pt9m09OJGtqXRjpEg8XEBEMJ+o=">AAAC/HicjVHLSsNAFD2Nr/quunQTLIIghEQKuhRc2KWCtUUrksSxBpNMmEwEafVP3LkTt/6AW12Lf6B/4Z1xii9EJyQ599x7zsydG2RxlEvXfS5ZA4NDwyPl0bHxicmp6crM7G7OCxGyRshjLlqBn7M4SllDRjJmrUwwPwli1gxON1S+ecZEHvF0R55n7CDxO2l0HIW+JOqwstrrtkVib9b3LtrCTzsxa2eCZ5L3XM/1HMcx7HKPQvcjPqxUXcfVy/4JPAOqMGuLV57QxhE4QhRIwJBCEo7hI6dnHx5cZMQdoEucIBTpPMMFxkhbUBWjCp/YU/p2KNo3bEqx8sy1OqRdYnoFKW0skoZTnSCsdrN1vtDOiv3Nu6s91dnO6R8Yr4RYiRNi/9L1K/+rU71IHGNN9xBRT5lmVHehcSn0raiT25+6kuSQEafwEeUF4VAr+/dsa02ue1d36+v8i65UrIpDU1vgVZ2SBux9H+dPsLvieIS3a9X1mhl1GfNYwBLNcxXrqGMLDfK+wj0e8GhdWtfWjXX7XmqVjGYOX5Z19wb8XaQ2</latexit><latexit
sha1_base64="9pt9m09OJGtqXRjpEg8XEBEMJ+o=">AAAC/HicjVHLSsNAFD2Nr/quunQTLIIghEQKuhRc2KWCtUUrksSxBpNMmEwEafVP3LkTt/6AW12Lf6B/4Z1xii9EJyQ599x7zsydG2RxlEvXfS5ZA4NDwyPl0bHxicmp6crM7G7OCxGyRshjLlqBn7M4SllDRjJmrUwwPwli1gxON1S+ecZEHvF0R55n7CDxO2l0HIW+JOqwstrrtkVib9b3LtrCTzsxa2eCZ5L3XM/1HMcx7HKPQvcjPqxUXcfVy/4JPAOqMGuLV57QxhE4QhRIwJBCEo7hI6dnHx5cZMQdoEucIBTpPMMFxkhbUBWjCp/YU/p2KNo3bEqx8sy1OqRdYnoFKW0skoZTnSCsdrN1vtDOiv3Nu6s91dnO6R8Yr4RYiRNi/9L1K/+rU71IHGNN9xBRT5lmVHehcSn0raiT25+6kuSQEafwEeUF4VAr+/dsa02ue1d36+v8i65UrIpDU1vgVZ2SBux9H+dPsLvieIS3a9X1mhl1GfNYwBLNcxXrqGMLDfK+wj0e8GhdWtfWjXX7XmqVjGYOX5Z19wb8XaQ2</latexit><latexit
sha1_base64="9pt9m09OJGtqXRjpEg8XEBEMJ+o=">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</latexit>
Next, we characterize the coherence of single atoms and demonstrate
single-qubit control. To begin, we experimen- tally measure the
lifetime of the Rydberg state in Fig. 2(a). The measured T1 ¼ Tr→g
¼ 51ð6Þ μs is consistent with the 146 μs Rydberg state lifetime
[35] when combined with the ∼80 μs timescale for off-resonant
scattering of the 1013 nm laser from jei. A Ramsey experiment shows
Gaussian decay that is well explained by thermal Doppler shifts
[see Fig. 2(b)]. At 10 μK, the random atomic velocity in each shot
of the experiment appears as a random detuning δD
from a Gaussian distribution of width 2π × 43.5 kHz, resulting in
dephasing as jψi → ð1=
ffiffiffi 2
p Þðjgiþ eiδ
DtjriÞ. However, since the random Doppler shift is constant over
the duration of each pulse sequence, its effect can be
eliminated via a spin-echo sequence [orange in Fig. 2(b)]. Note
that the spin-echo measurements display some small deviations from
the numerical simulations, indicating the presence of an additional
dephasing channel. Assuming an exponential decay, we measure a
fitted T2 ¼ 32ð6Þ μs and extract a pure dephasing time T ¼ ½1=T2 −
1=ð2Tr→gÞ&−1 ¼ 47ð13Þ μs. We hypothesize that this dephasing
may result from residual laser phase noise. Finally, we demonstrate
a single-atom phase gate by
applying an independent focused laser that shifts the energy of the
ground state jgi [see Fig. 2(c)] [27]. By controlling the duration
of the applied laser pulse, we impart a controlled dynamical phase
on jgi relative to jri. The contrast of the resulting phase gate
(embedded in a spin- echo sequence) is close to the limit imposed
by detection and spin-echo fidelity. We next turn to two-atom
control. To this end, we
position two atoms at a separation of 5.7 μm, at which the
Rydberg-Rydberg interaction is U= ¼ 2π × 30 MHz Ω ¼ 2π × 2 MHz. In
this so-called Rydberg blockade regime, the laser field globally
couples both atoms from jggi to the symmetric state jWi ¼ ð1=
ffiffiffi 2
p Ω [see Fig. 3(a)] (here
the excited states jri are defined in the rotating frame to
incorporate the spatial phase factors eikx, as discussed in [27]).
The measured probabilities for the states jggi, jgri, jrgi, and
jrri (denoted by Pgg, Pgr, Prg, and Prr, respec- tively) show that
indeed no population enters the doubly excited state (Prr <
0.02, consistent with only detection error). Instead, there are
oscillations between the manifold of zero excitations and the
manifold of one excitation with a fitted frequency of 2π × 2.83
MHz≈
ffiffiffi 2
p Ω [see Fig. 3(b)].
These collective Rabi oscillations can be used to directly prepare
the maximally entangled Bell state jWi by applying a π pulse at the
enhanced Rabi frequency (denoted byXW
π ). To determine the fidelity of this experimentally prepared
entangled state, given by F ¼ hWjρjWi, we express it in terms of
diagonal and off diagonal matrix elements of the density operator
ρ:
F ¼ 1
1
2 ðρgr;rg þ ρrg;grÞ; ð1Þ
where ραβ;γδ ¼ hαβjρjγδi for α, β, γ, δ ∈fg; rg. The diagonal
elements can be directly measured by applying a π pulse and then
measuring the populations. The results closely match those of a
perfect jWi state after accounting for state detection errors, with
ρgr;gr þ ρrg;rg ¼ 0.94ð1Þ, relative to a maximum possible value of
0.95(1). To measure the off diagonal elements of the density
matrix, we make use of the single-atom phase gate Zð1Þ
demonstrated in Fig. 2(c), which introduces a variable phase on one
atom (as demonstrated in [36]). Specifically, a local beam adds a
light shift δ to jgri but not to jrgi,
(a) (b)
(c)
FIG. 2. Characterization of single-atom coherence and phase
control. (a) The lifetime of jri is measured by exciting from jgi
to jri with a π pulse, and then deexciting after a variable delay.
The probability to end in jgi (denoted Pg) decays with an extracted
lifetime of T1 ¼ 51ð6Þ μs (fitted to an exponential decay model
with no offset). (b) A Ramsey experiment (blue) shows Gaussian
decay with a 1=e lifetime of T '
2 ¼ 4.5ð1Þ μs, limited by thermal Doppler shifts. Inserting an
additional π pulse (orange) between the π=2 pulses cancels the
effect of the Doppler shifts and results in a substantially longer
coherence lifetime of T2 ¼ 32ð6Þ μs (fitted to an exponential decay
model with an offset of 0.5). (c) A single-atom phase gate is
implemented by applying an indepen- dent 809 nm laser which induces
a light shift δ ¼ 2π × 5 MHz on the ground state for time t,
resulting in an accumulated dynamical phase ¼ δt. The gate is
embedded in a spin-echo sequence to cancel Doppler shifts. In each
measurement shown here, the 1013 nm laser remains on for the entire
pulse sequence, while the 420 nm laser is pulsed according to the
sequence shown above each plot. Each data point is calculated from
200–500 repeated measurements with a single atom per
realization.
PHYSICAL REVIEW LETTERS 121, 123603 (2018)
123603-3
Transverse B Longitudinal B Spin-spin int.
H = ~ 2
Transverse B Longitudinal B Spin-spin int.
H = ~ 2
C6
Transverse B Longitudinal B Spin-spin int.
H = ~ 2
e.g. magnetism, topology… IO: Nature 2016, PRX 2018, Science
2019
regime where the blockade radius Rb, i.e., the distance over which
interatomic interactions prevent the excitation of two atoms, was
much larger than the lattice spacing a, rendering the underlying
lattice hardly relevant. In this case, the observed correlations
are liquidlike, and observing the crystal-like ground state of the
system [28] would require exponentially long ramps [29]. More
recently, experiments with arrays of optical tweezers allowed
exploring the regime Rb a, studying nonequilibrium dynamics follow-
ing quenches [30] or slow sweeps [31]. Here, we use a Rydberg-based
platform emulating an
Ising antiferromagnet to study the growth of correlations during
ramps of the experimental parameters in 1d and 2d arrays of up to
36 single atoms with different geometries. We operate in the regime
Rb a, where the interactions act to a good approximation only
between nearest neighbors. We dynamically tune the parameters of
the Hamiltonian and observe the buildup of antiferromagnetic order.
We also observe the influence of the finite ramp speed on the
extent
of the correlations, and we follow the development in space and
time of these correlations during a ramp. Numerical simulations of
the dynamics of the system without any adjustable parameters are in
very good agreement with the experimental data and show that
single-particle dephasing arising from technical imperfections
currently limits the range of the observed correlations. Finally,
we observe a characteristic spatial structure in the correlations,
which can be understood qualitatively by a short-time expansion of
the evolution operator for both square and triangular lattices.
This study is a benchmarking of a state-of-the- art quantum
simulator of spin models in nontrivial settings (two-dimensional
geometries, including frustrated ones). It shows that, although
single-particle dephasing is so far a limitation for the study of
ground-state properties, it does not prevent the observation of
interesting features in the dynamics of these systems, in
particular concerning the propagation of correlations during
dynamical tuning of the parameters.
(a)
(c)
(d)
(b)
FIG. 1. Studying the AF Ising model on 1d and 2d systems. (a)
Examples of single-shot fluorescence images of single-atom arrays
used in our experiments: a 24-atom 1d chain with periodic boundary
conditions, a 6 × 6 square array, and a 36-atom triangular array.
Each atom is used to encode a spin-1=2, whose internal states j↑i
and j↓i are coupled with Rabi frequency Ω and detuning δ. (b) Time
dependence of the Rabi frequency ΩðtÞ and detuning δðtÞ used to
probe the buildup of correlations. (c) Sketched ground-state phase
diagrams of the Ising model in Eq. (1), in the nearest-neighbor
interaction limit, for a 1d chain, a 2d square lattice, and a 2d
triangular lattice. In the figure, AFM stands for
antiferromagnetic, PM for paramagnetic, and OBD for order by
disorder. (d) Typical experimental correlation functions obtained
for these geometries (see text). For the 1d chain, the correlation
length ξ ¼ 1.5 sites (bottom left panel).
VINCENT LIENHARD et al. PHYS. REV. X 8, 021070 (2018)
021070-2
Quantum simulation with “large” arrays
Programmable “quantum simulator”
Transverse B Longitudinal B Spin-spin int.
H = ~ 2
Many-body physics
e.g. magnetism, topology… e.g. chemistry, optimization… IO: Nature
2016, PRX 2018, Science 2019
regime where the blockade radius Rb, i.e., the distance over which
interatomic interactions prevent the excitation of two atoms, was
much larger than the lattice spacing a, rendering the underlying
lattice hardly relevant. In this case, the observed correlations
are liquidlike, and observing the crystal-like ground state of the
system [28] would require exponentially long ramps [29]. More
recently, experiments with arrays of optical tweezers allowed
exploring the regime Rb a, studying nonequilibrium dynamics follow-
ing quenches [30] or slow sweeps [31]. Here, we use a Rydberg-based
platform emulating an
Ising antiferromagnet to study the growth of correlations during
ramps of the experimental parameters in 1d and 2d arrays of up to
36 single atoms with different geometries. We operate in the regime
Rb a, where the interactions act to a good approximation only
between nearest neighbors. We dynamically tune the parameters of
the Hamiltonian and observe the buildup of antiferromagnetic order.
We also observe the influence of the finite ramp speed on the
extent
of the correlations, and we follow the development in space and
time of these correlations during a ramp. Numerical simulations of
the dynamics of the system without any adjustable parameters are in
very good agreement with the experimental data and show that
single-particle dephasing arising from technical imperfections
currently limits the range of the observed correlations. Finally,
we observe a characteristic spatial structure in the correlations,
which can be understood qualitatively by a short-time expansion of
the evolution operator for both square and triangular lattices.
This study is a benchmarking of a state-of-the- art quantum
simulator of spin models in nontrivial settings (two-dimensional
geometries, including frustrated ones). It shows that, although
single-particle dephasing is so far a limitation for the study of
ground-state properties, it does not prevent the observation of
interesting features in the dynamics of these systems, in
particular concerning the propagation of correlations during
dynamical tuning of the parameters.
(a)
(c)
(d)
(b)
FIG. 1. Studying the AF Ising model on 1d and 2d systems. (a)
Examples of single-shot fluorescence images of single-atom arrays
used in our experiments: a 24-atom 1d chain with periodic boundary
conditions, a 6 × 6 square array, and a 36-atom triangular array.
Each atom is used to encode a spin-1=2, whose internal states j↑i
and j↓i are coupled with Rabi frequency Ω and detuning δ. (b) Time
dependence of the Rabi frequency ΩðtÞ and detuning δðtÞ used to
probe the buildup of correlations. (c) Sketched ground-state phase
diagrams of the Ising model in Eq. (1), in the nearest-neighbor
interaction limit, for a 1d chain, a 2d square lattice, and a 2d
triangular lattice. In the figure, AFM stands for
antiferromagnetic, PM for paramagnetic, and OBD for order by
disorder. (d) Typical experimental correlation functions obtained
for these geometries (see text). For the 1d chain, the correlation
length ξ ¼ 1.5 sites (bottom left panel).
VINCENT LIENHARD et al. PHYS. REV. X 8, 021070 (2018)
021070-2
Variationnal problems
Transverse B Longitudinal B Spin-spin int.
H = ~ 2
e.g. chemistry, optimization… IO: Nature 2016, PRX 2018, Science
2019
regime where the blockade radius Rb, i.e., the distance over which
interatomic interactions prevent the excitation of two atoms, was
much larger than the lattice spacing a, rendering the underlying
lattice hardly relevant. In this case, the observed correlations
are liquidlike, and observing the crystal-like ground state of the
system [28] would require exponentially long ramps [29]. More
recently, experiments with arrays of optical tweezers allowed
exploring the regime Rb a, studying nonequilibrium dynamics follow-
ing quenches [30] or slow sweeps [31]. Here, we use a Rydberg-based
platform emulating an
Ising antiferromagnet to study the growth of correlations during
ramps of the experimental parameters in 1d and 2d arrays of up to
36 single atoms with different geometries. We operate in the regime
Rb a, where the interactions act to a good approximation only
between nearest neighbors. We dynamically tune the parameters of
the Hamiltonian and observe the buildup of antiferromagnetic order.
We also observe the influence of the finite ramp speed on the
extent
of the correlations, and we follow the development in space and
time of these correlations during a ramp. Numerical simulations of
the dynamics of the system without any adjustable parameters are in
very good agreement with the experimental data and show that
single-particle dephasing arising from technical imperfections
currently limits the range of the observed correlations. Finally,
we observe a characteristic spatial structure in the correlations,
which can be understood qualitatively by a short-time expansion of
the evolution operator for both square and triangular lattices.
This study is a benchmarking of a state-of-the- art quantum
simulator of spin models in nontrivial settings (two-dimensional
geometries, including frustrated ones). It shows that, although
single-particle dephasing is so far a limitation for the study of
ground-state properties, it does not prevent the observation of
interesting features in the dynamics of these systems, in
particular concerning the propagation of correlations during
dynamical tuning of the parameters.
(a)
(c)
(d)
(b)
FIG. 1. Studying the AF Ising model on 1d and 2d systems. (a)
Examples of single-shot fluorescence images of single-atom arrays
used in our experiments: a 24-atom 1d chain with periodic boundary
conditions, a 6 × 6 square array, and a 36-atom triangular array.
Each atom is used to encode a spin-1=2, whose internal states j↑i
and j↓i are coupled with Rabi frequency Ω and detuning δ. (b) Time
dependence of the Rabi frequency ΩðtÞ and detuning δðtÞ used to
probe the buildup of correlations. (c) Sketched ground-state phase
diagrams of the Ising model in Eq. (1), in the nearest-neighbor
interaction limit, for a 1d chain, a 2d square lattice, and a 2d
triangular lattice. In the figure, AFM stands for
antiferromagnetic, PM for paramagnetic, and OBD for order by
disorder. (d) Typical experimental correlation functions obtained
for these geometries (see text). For the 1d chain, the correlation
length ξ ¼ 1.5 sites (bottom left panel).
VINCENT LIENHARD et al. PHYS. REV. X 8, 021070 (2018)
021070-2
Hybrid approach (class./Q)
Transverse B Longitudinal B Spin-spin int.
H = ~ 2
e.g. chemistry, optimization… IO: Nature 2016, PRX 2018, Science
2019
regime where the blockade radius Rb, i.e., the distance over which
interatomic interactions prevent the excitation of two atoms, was
much larger than the lattice spacing a, rendering the underlying
lattice hardly relevant. In this case, the observed correlations
are liquidlike, and observing the crystal-like ground state of the
system [28] would require exponentially long ramps [29]. More
recently, experiments with arrays of optical tweezers allowed
exploring the regime Rb a, studying nonequilibrium dynamics follow-
ing quenches [30] or slow sweeps [31]. Here, we use a Rydberg-based
platform emulating an
Ising antiferromagnet to study the growth of correlations during
ramps of the experimental parameters in 1d and 2d arrays of up to
36 single atoms with different geometries. We operate in the regime
Rb a, where the interactions act to a good approximation only
between nearest neighbors. We dynamically tune the parameters of
the Hamiltonian and observe the buildup of antiferromagnetic order.
We also observe the influence of the finite ramp speed on the
extent
of the correlations, and we follow the development in space and
time of these correlations during a ramp. Numerical simulations of
the dynamics of the system without any adjustable parameters are in
very good agreement with the experimental data and show that
single-particle dephasing arising from technical imperfections
currently limits the range of the observed correlations. Finally,
we observe a characteristic spatial structure in the correlations,
which can be understood qualitatively by a short-time expansion of
the evolution operator for both square and triangular lattices.
This study is a benchmarking of a state-of-the- art quantum
simulator of spin models in nontrivial settings (two-dimensional
geometries, including frustrated ones). It shows that, although
single-particle dephasing is so far a limitation for the study of
ground-state properties, it does not prevent the observation of
interesting features in the dynamics of these systems, in
particular concerning the propagation of correlations during
dynamical tuning of the parameters.
(a)
(c)
(d)
(b)
FIG. 1. Studying the AF Ising model on 1d and 2d systems. (a)
Examples of single-shot fluorescence images of single-atom arrays
used in our experiments: a 24-atom 1d chain with periodic boundary
conditions, a 6 × 6 square array, and a 36-atom triangular array.
Each atom is used to encode a spin-1=2, whose internal states j↑i
and j↓i are coupled with Rabi frequency Ω and detuning δ. (b) Time
dependence of the Rabi frequency ΩðtÞ and detuning δðtÞ used to
probe the buildup of correlations. (c) Sketched ground-state phase
diagrams of the Ising model in Eq. (1), in the nearest-neighbor
interaction limit, for a 1d chain, a 2d square lattice, and a 2d
triangular lattice. In the figure, AFM stands for
antiferromagnetic, PM for paramagnetic, and OBD for order by
disorder. (d) Typical experimental correlation functions obtained
for these geometries (see text). For the 1d chain, the correlation
length ξ ¼ 1.5 sites (bottom left panel).
VINCENT LIENHARD et al. PHYS. REV. X 8, 021070 (2018)
021070-2
Hybrid approach (class./Q)
C6
R6 <latexit
sha1_base64="zyNgjEEE+3h0Bn0WKCICpDgJvow=">AAAC0HicjVHLTsJAFD3UF+ILdemmkZi4Iq0xyJKEjUsk8kgASVsGbOjLdmokxBi3/oBb/SrjH+hfeGccEpUYnabtmXPvOTP3Xjvy3IQbxmtGW1hcWl7JrubW1jc2t/LbO80kTGOHNZzQC+O2bSXMcwPW4C73WDuKmeXbHmvZ46qIt65ZnLhhcM4nEev51ihwh65jcaJ61X6pG1Jcr1+Ucv18wSgacunzwFSgALVqYf4FXQwQwk
EKHwwBOGEPFhJ6OjBhICKuhylxMSFXxhlukSNtSlmMMixix/Qd0a6j2ID2wjORaodO8eiNSanjgDQh5cWExWm6jKfSWbC/eU+lp7jbhP628vKJ5bgk9i/dLPO/OlELxxBlWYNLNUWSEdU5yiWVXRE3179UxckhIk7gAcVjwo5UzvqsS00iaxe9tWT8TWYKVuwdlZviXdySBmz+HOc8aB4VTcJnx4VKWY06iz3s45DmeYIKTlFDg7yv8IgnPGt17Ua70+4/U7WM0uzi29IePgCreZNz</latexit><latexit
sha1_base64="zyNgjEEE+3h0Bn0WKCICpDgJvow=">AAAC0HicjVHLTsJAFD3UF+ILdemmkZi4Iq0xyJKEjUsk8kgASVsGbOjLdmokxBi3/oBb/SrjH+hfeGccEpUYnabtmXPvOTP3Xjvy3IQbxmtGW1hcWl7JrubW1jc2t/LbO80kTGOHNZzQC+O2bSXMcwPW4C73WDuKmeXbHmvZ46qIt65ZnLhhcM4nEev51ihwh65jcaJ61X6pG1Jcr1+Ucv18wSgacunzwFSgALVqYf4FXQwQwk
EKHwwBOGEPFhJ6OjBhICKuhylxMSFXxhlukSNtSlmMMixix/Qd0a6j2ID2wjORaodO8eiNSanjgDQh5cWExWm6jKfSWbC/eU+lp7jbhP628vKJ5bgk9i/dLPO/OlELxxBlWYNLNUWSEdU5yiWVXRE3179UxckhIk7gAcVjwo5UzvqsS00iaxe9tWT8TWYKVuwdlZviXdySBmz+HOc8aB4VTcJnx4VKWY06iz3s45DmeYIKTlFDg7yv8IgnPGt17Ua70+4/U7WM0uzi29IePgCreZNz</latexit><latexit
sha1_base64="zyNgjEEE+3h0Bn0WKCICpDgJvow=">AAAC0HicjVHLTsJAFD3UF+ILdemmkZi4Iq0xyJKEjUsk8kgASVsGbOjLdmokxBi3/oBb/SrjH+hfeGccEpUYnabtmXPvOTP3Xjvy3IQbxmtGW1hcWl7JrubW1jc2t/LbO80kTGOHNZzQC+O2bSXMcwPW4C73WDuKmeXbHmvZ46qIt65ZnLhhcM4nEev51ihwh65jcaJ61X6pG1Jcr1+Ucv18wSgacunzwFSgALVqYf4FXQwQwk
EKHwwBOGEPFhJ6OjBhICKuhylxMSFXxhlukSNtSlmMMixix/Qd0a6j2ID2wjORaodO8eiNSanjgDQh5cWExWm6jKfSWbC/eU+lp7jbhP628vKJ5bgk9i/dLPO/OlELxxBlWYNLNUWSEdU5yiWVXRE3179UxckhIk7gAcVjwo5UzvqsS00iaxe9tWT8TWYKVuwdlZviXdySBmz+HOc8aB4VTcJnx4VKWY06iz3s45DmeYIKTlFDg7yv8IgnPGt17Ua70+4/U7WM0uzi29IePgCreZNz</latexit><latexit
sha1_base64="zyNgjEEE+3h0Bn0WKCICpDgJvow=">AAAC0HicjVHLTsJAFD3UF+ILdemmkZi4Iq0xyJKEjUsk8kgASVsGbOjLdmokxBi3/oBb/SrjH+hfeGccEpUYnabtmXPvOTP3Xjvy3IQbxmtGW1hcWl7JrubW1jc2t/LbO80kTGOHNZzQC+O2bSXMcwPW4C73WDuKmeXbHmvZ46qIt65ZnLhhcM4nEev51ihwh65jcaJ61X6pG1Jcr1+Ucv18wSgacunzwFSgALVqYf4FXQwQwk
EKHwwBOGEPFhJ6OjBhICKuhylxMSFXxhlukSNtSlmMMixix/Qd0a6j2ID2wjORaodO8eiNSanjgDQh5cWExWm6jKfSWbC/eU+lp7jbhP628vKJ5bgk9i/dLPO/OlELxxBlWYNLNUWSEdU5yiWVXRE3179UxckhIk7gAcVjwo5UzvqsS00iaxe9tWT8TWYKVuwdlZviXdySBmz+HOc8aB4VTcJnx4VKWY06iz3s45DmeYIKTlFDg7yv8IgnPGt17Ua70+4/U7WM0uzi29IePgCreZNz</latexit>
Platform mature enough to envision startups… PHYSICAL REVIEW A 97,
053803 (2018)
Editors’ Suggestion
Analysis of imperfections in the coherent optical excitation of
single atoms to Rydberg states
Sylvain de Léséleuc, Daniel Barredo, Vincent Lienhard, Antoine
Browaeys, and Thierry Lahaye*
Laboratoire Charles Fabry, Institut d’Optique Graduate School,
CNRS, Université Paris-Saclay, 91127 Palaiseau Cedex, France
(Received 28 February 2018; published 3 May 2018)
We study experimentally various physical limitations and technical
imperfections that lead to damping and finite contrast of optically
driven Rabi oscillations between ground and Rydberg states of a
single atom. Finite contrast is due to preparation and detection
errors, and we show how to model and measure them accurately. Part
of these errors originates from the finite lifetime of Rydberg
states, and we observe its n3 scaling with the principal quantum
number n. To explain the damping of Rabi oscillations, we use
simple numerical models taking into account independently measured
experimental imperfections and show that the observed damping
actually results from the accumulation of several small effects,
each at the level of a few percent. We discuss prospects for
improving the coherence of ground-Rydberg Rabi oscillations in view
of applications in quantum simulation and quantum information
processing with arrays of single Rydberg atoms.
DOI: 10.1103/PhysRevA.97.053803
Arrays of single atoms trapped in optical tweezers and excited to
Rydberg states are a promising platform for quantum simulation [1–
5] and quantum information processing [6]. They combine a hyperfine
qubit with demonstrated individual control and one-qubit gates with
high fidelities [7– 9], the possibility to scale the system to
large numbers of qubits [10– 12] and strong interactions. Coherent
ground-Rydberg Rabi oscillations have been observed in dilute gases
[13,14], in sin- gle atoms [15– 18], and in blockaded ensemble
“superatoms” [19– 21]. Long coherence times of ground-Rydberg Rabi
os- cillations are a crucial element in the context of both quantum
simulation, to accurately emulate interacting systems
LOAD MORE