Quantum computing and simulation with arrays of Rydberg … · 2019. 6. 30. · Antoine Browaeys...

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 additional 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, extremely 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 constant 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 energy 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 radius. 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 mesoscopic 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 applications 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 computation 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 systems is essential for performing single-bit operations and the two-bit coupling is accomplished by interactions between 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 ensembles. 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 excitation. 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 corresponding 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) perform 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
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Rydberg atoms and their van der Waals interaction
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Interaction in Rydberg state = 1011 x ground state interaction!! ⇒ Switchable for ! R 1 10µm
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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
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
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)
Solution: sort atoms in arrays
Barredo, Science 354, 1021 (2016) Also Harvard (1D) & Korea
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=">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</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=">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</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=">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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>
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 characterization 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, nontargeted 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 demonstrated 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 entanglement 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 entanglement 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
8
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 experimentally 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 independent 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
8
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=">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</latexit><latexit sha1_base64="9pt9m09OJGtqXRjpEg8XEBEMJ+o=">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</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 experimentally 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 independent 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
H = ~ 2
C6
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”
H = ~ 2
Many-body physics
e.g. magnetism, topology… e.g. chemistry, optimization… IO: Nature 2016, PRX 2018, Science 2019
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(c)
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021070-2
Variationnal problems
H = ~ 2
e.g. chemistry, optimization… IO: Nature 2016, PRX 2018, Science 2019
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(c)
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021070-2
Hybrid approach (class./Q)
H = ~ 2
e.g. chemistry, optimization… IO: Nature 2016, PRX 2018, Science 2019
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021070-2
Hybrid approach (class./Q)
C6
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Platform mature enough to envision startups… PHYSICAL REVIEW A 97, 053803 (2018)
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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 single atoms [15– 18], and in blockaded ensemble “superatoms” [19– 21]. Long coherence times of ground-Rydberg Rabi oscillations are a crucial element in the context of both quantum simulation, to accurately emulate interacting systems

Quantum computing and simulation with arrays of Rydberg … · 2019. 6. 30. · Antoine Browaeys...

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