Robust phase-controlled gates for scalable atomic quantum processors using optical standing waves

2025-05-03 0 0 935.87KB 12 页 10玖币

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Robust phase-controlled gates for scalable atomic

quantum processors using optical standing waves

Shannon Whitlock

European Center for Quantum Sciences and aQCess - Atom Quantum Computing as a Service,

Institut de Science et d’Ingénierie Supramoléculaire (UMR 7006), University of Strasbourg and CNRS

A simple scheme is presented for realizing robust optically controlled quan-

tum gates for scalable atomic quantum processors by driving the qubits with

optical standing waves. Atoms localized close to the antinodes of the standing

wave can realize phase-controlled quantum operations that are potentially more

than an order of magnitude less sensitive to the local optical phase and atomic

motion than corresponding travelling wave conﬁgurations. The scheme is com-

patible with robust optimal control techniques and spatial qubit addressing in

atomic arrays to realize phase controlled operations without the need for tight

focusing and precise positioning of the control lasers. This will be particu-

larly beneﬁcial for quantum gates involving Doppler sensitive optical frequency

transitions and provides an all optical route to scaling up atomic quantum

processors.

1 Context and motivation

Trapped atoms and atom-like systems are a leading candidate for numerous quantum

technologies, including scalable quantum computing [1,2,3], optical atomic clocks [4] and

quantum sensors [5]. A key requirement in these applications is to reliably control the full

quantum state of the system, typically by applying a sequence of phase-controlled optical

pulses to the atoms. However, a major issue arises when scaling up to many atoms and

large numbers of quantum operations, as errors associated to optical phase noise, atom

position ﬂuctuations and motion lead to rapidly accumulating errors [6,7,8,9].

A common way to realize phase-controlled quantum operations is by utilizing atomic

transitions with frequency diﬀerences in the radio frequency or microwave regimes. These

transitions can then be driven by stable microwave ﬁelds or a pair of Raman lasers with

a small eﬀective wavevector in a Doppler insensitive conﬁguration [10]. On the other

hand, protocols involving optical frequency transitions [11,12,13,14] can have signiﬁcant

advantages when it comes to isolating the states and controlling diﬀerential light shifts,

performing high ﬁdelity state-resolved readout and mediating strong interactions exploiting

highly-excited Rydberg states [3]. However optical frequency transitions are generally much

more sensitive to atom position ﬂuctuations and motional dephasing making it diﬃcult to

realize high-ﬁdelity phase-controlled operations.

Here a general method is proposed for realising robust and high ﬁdelity quantum op-

erations (gates) that is also compatible with spatial addressing of atomic qubits in large

atomic arrays. By using two driving lasers in standing wave conﬁguration it is possible

Shannon Whitlock: whitlock@unistra.fr

Accepted in Quantum 2023-03-03, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.00576v2 [quant-ph] 6 Mar 2023

qubit

drive

addressing

(a) (b)

(c)

Figure 1: Setup for realizing robust phase controlled gates using optical standing waves. (a) Exemplary

level diagram showing an optical transition for encoding an optical frequency qubit and laser ﬁelds for

trapping and qubit addressing. (b) An interfering pair of counter propagating lasers produce an optical

standing wave characterised by a periodically varying coupling strength |Ω(x)|and a phase ϕ(x)which

is spatially uniform between adjacent nodes. By controlling both the amplitude and phase of the lasers

it is possible to realize phase-controlled quantum operations. (c) Scheme for single atom addressing

involving a global standing wave drive ﬁeld (red stripes) and an auxiliary far oﬀ-resonant beam (purple

shaded region) targeting the central qubit.

to strongly suppress the sensitivity to local optical phase noise, including atom position

variations and motion which are dominant errors in current experiments [7,15,8]. Ro-

bust phase-compensated gates compatible with standing wave or travelling wave conﬁg-

urations can also be applied to individual qubits in large atomic arrays using targeted

light-shifts [16,17,18], where unwanted rotations on non-targeted qubits are cancelled by

exploiting a time-reversal symmetry and geometric constraints. The proposed gates are

robust, technically simple and parallelizable which will enable scaling up atomic quantum

processors and high ﬁdelity qubit control to large atomic registers.

2 Phase-controlled gates using optical standing wave (OSW) ﬁelds

Our goal is to realize phase-controlled qubit manipulations, which for a single qubit can

be described by the following transformation acting on two atomic states

|0i → cos(θ/2)|0i − isin(θ/2)eiϕ|1i

|1i → cos(θ/2)|1i − isin(θ/2)e−iϕ|0i(1)

Generally the rotation angle θand qubit phase ϕis determined by the speciﬁc protocol

used to drive the qubit. In the case of resonant driving θis determined by the strength

Accepted in Quantum 2023-03-03, click title to verify. Published under CC-BY 4.0. 2

and duration of the atom-light coupling, while ϕis determined by, and very sensitive to

the relative phase between the atomic dipole and the optical ﬁeld at the position of the

atom.

To realize the transformation (1)one can resonantly drive the atoms by two counter

propagating optical ﬁelds with the same amplitude and phase forming optical standing

wave (OSW). The drive Hamiltonian in the rotating wave approximation can be written

Hd=~

2Ω1(t)eikx + Ω2(t)e−ikx|1ih0|+h.c (2)

where Ωα(t) = |Ωα(t)|eiϕα(t)are the complex valued Rabi frequencies of the two drive lasers

with wavevectors ±k, at the position of the atom x. For OSW gates Ω1(t)=Ω2(t) = Ω(t)

is assumed.

This Hamiltonian allows to realize arbitrary phase controlled single qubit gates which

are mostly insensitive to local optical phase noise and atomic position ﬂuctuations. While

an optical travelling wave (OTW) has a phase that is a linearly varying function of position

∝kx, an OSW on the other hand interferes to produce a ﬁeld with a spatially uniform

phase which steps between ϕand ϕ+πbetween adjacent nodes (Fig. 1b). The atoms are

assumed to be localized close to the antinodes of the OSW, either by making the atom

array commensurate with the standing wave (Fig. 1c) or by shifting the relative phase of

the drive ﬁelds between gates to address speciﬁc atoms. In this way it is possible to realize

phase-controlled quantum operations that are mostly insensitive to the precise positions

or velocities of the atoms. Instead the atom experiences a spatially varying intensity

∝cos2(kx). But by tailoring the time-dependence of |Ω(t)|and ϕ(t)(e.g., using electro-

optic or acousto-optic modulators which can act on the timescale of several nanoseconds)

it is possible to realize diﬀerent quantum gates that also correct for associated intensity

noise (or Rabi frequency) errors.

To assess the advantage of OSW ﬁelds over OTW ﬁelds the achievable gate ﬁdelity

for three diﬀerent gate protocols will be compared. The OTW-1 gate consists of a drive

pulse Ω1(t) = Asin(πt/T )and Ω2(t)=0with A=π2/(4T)(Fig. 2a). The corresponding

OSW-1 gate has Ω1(t) = Ω2(t) = (A/2) sin(πt/T )(note that the total required intensity

for the OSW-1 gate is half that of OTW-1 for the same gate duration due to construc-

tive interference). These simple parametric pulse shapes are favorable for experimental

implementations, particularly for fast gates where bandwidth of the control systems might

be limited. Finally the OTW-1 and OSW-1 gates are compared with a gate based on a

four pulse BB1 sequence [19] (OSW-BB1, see Fig. 2b) which additionally suppresses sen-

sitivity to Rabi frequency errors. In the absence of noise all three gates perfectly realize

aRx(θ=π/2) gate, equivalent to a √Xgate within a global phase factor, which is a

basic operation for realising more complex multiqubit gates and quantum circuits. See

Appendix Afor generalizations to Uxy(θ, φ)rotation gates according to Eq. (1). Several

Uxy(θ, φ)gates can be concatenated to realize arbitrary single qubit control using OSW

ﬁelds.

Figure 2c shows the gate inﬁdelity = 1 −Fas a function of the local optical phase kx

assuming the atom is at rest during the gate time. Fis estimated by simulating the time-

evolution operator ˆ

Uas a sequence of 400 piecewise constant segments and calculating

F=|Tr(ˆ

U†

target ˆ

U)|2/4, where ˆ

Utarget = ((1,−i),(i, 1))/√2. The simulations show the

OTW-1 gate is most sensitive to the local optical phase with quadratic dependence on kx

with  > 0.1for |kx|= 0.5. In contrast the OSW-1 gate exhibits a less sensitive quartic

dependence on |kx|with = 0.01 for |kx|= 0.5. The residual inﬁdelity in the OSW

conﬁguration can be attributed to the spatial varying intensity, which can be corrected

Accepted in Quantum 2023-03-03, click title to verify. Published under CC-BY 4.0. 3

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Robustphase-controlledgatesforscalableatomicquantumprocessorsusingopticalstandingwavesShannonWhitlockEuropeanCenterforQuantumSciencesandaQCess-AtomQuantumComputingasaService,InstitutdeScienceetd'IngénierieSupramoléculaire(UMR7006),UniversityofStrasbourgandCNRSAsimpleschemeispresentedforrealizingrobu...

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Robust phase-controlled gates for scalable atomic quantum processors using optical standing waves

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