Berry-phase gates for fast and robust control of atomic clock states

We propose and experimentally demonstrate a fast Berry-phase gate, which is implemented by picosecondtimescale optical pulses to make the qubit system of atomic clock states adiabatically evolve on a closed loop. The characteristic features of the proposed gate are gate speed and robustness against control fluctuations, which can potentially resolve the decoherence and reliability issues in quantum information processing, at the same time. The experiment is conducted with two linearly polarized, chirped optical pulses, interacting with five single rubidium atoms simultaneously in an array of optical tweezer dipole traps, to demonstrate the proposed picosecond-timescale clock-state gates. The robustness of the qubit rotation angle δ /δA = 1.5% is achieved with respect to the laser intensity (of pulse area A) fluctuation.


I. INTRODUCTION
The Berry phase is one of the hallmarks of quantum mechanics, dealing with the geometric phase, gained by a quantum wave function subjected to an adiabatic process, which can remain nonzero even after a cyclic evolution in which the more familiar dynamic phase disappears [1]. It appears ubiquitously in numerous physical phenomena including the Aharonov-Bohm effect, the quantum Hall effect, and neutron interferometry, to list a few [1][2][3]. The Berry phase written as a unitary operator U for a cyclic evolution is holonomy, which depends only on the evolution path but not on other dynamic details during the evolution. Thus, a geometrical manipulation of two-state systems utilizing the Berry phase is expected for robust quantum information processing against environment and parameter noises (characteristically of local nature) due to their independence from local phase changes (dynamic phases).
One way to implement these holonomic quantum gates [4,5] is adiabatic time evolution [1,6]. The time evolution of a qubit system |ψ (t i ) = α|0 + β|1 driven by the time-varying field of the Hamiltonian H I (t ) from t i to t f is written in the bare basis as where φ d is the dynamic phase that is only global, thus ignorable, and is the geometric phase. The adiabatic time evolution of a qubit system allows no leakage from the initial adiabatic state of degenerate eigenenergy, so if an * jwahn@kaist.ac.kr appropriate interaction picture removes this eigenenergy, a parallel transport condition ψ (t )|H I (t )|ψ (t ) = 0 can be imposed for the holonomy. Original proposals for holonomic quantum gates are based on this adiabatic evolution [4,[7][8][9]. However, in many physical systems of limited coherence time, it is difficult to satisfy the adiabatic condition and experimental implementation has been limited to longlived transitions [10] or the shortcut to adiabaticity [11,12].
In the present paper we propose a method implementing adiabatic holonomic gates, thus Berry-phase gates, of atomic clock states in picosecond timescales. Experimental demonstration is performed on the hyperfine states of an atomic system, interacting with chirped optical pulses that allow adiabatic time evolution in the qubit system, to achieve the robustness of the Rabi oscillation angle δ /δA = 1.5% with respect to the laser intensity (pulse area) fluctuation. We further demonstrate a scheme for rotation operators about arbitrary axes, with which a set of universal one-qubit quantum gates can be constructed.

II. THEORETICAL CONSIDERATION
Let us consider an atomic system [see Fig. 1(a)], of an excited level |e = |P 1/2 and a ground level |g = |S 1/2 , in which |g consists of two ground hyperfine states (qubit states) |0 = |S 1/2 , F = I + 1/2, m F = 0 and |1 = |S 1/2 , F = I − 1/2, m F = 0 . When the hyperfine energy splittinghω hf is negligible compared to the inverse of the gate-operation time, these qubit states can be considered as energy degenerate states.
We utilize a cyclic time evolution of the qubit system, by a pair of chirped pulses [see Fig. 1(b)]. As each chirped pulse implements a rapid adiabatic passage (RAP) [19] that provides robust adiabatic population transfer between |g and |e , Energy-level diagram of a ground level |g , an excited level |e , and two degenerate qubit eigenstates |0 and |1 within the ground level |g . A cyclic transition between |g and |e results in the evolution from the initial qubit state |ψ to U |ψ . (b) Pulse sequence of Berry-phase gates (top) and the corresponding time evolutions of the two-level system (bottom). When the system undergoes successive adiabatic passages by two linearly polarized, chirped laser pulses with relative polarization angle θ [(i) θ = −π/2, (ii) θ = 0, and (iii) θ = π/4], the first RAP excites the system from |g to |e along the path shown by the blue line and then the second RAP deexcites the system back to |g along the two different paths labeled by σ + and σ − , shown by red lines. The geometric phase gained through the cyclic transitions is proportional to the shaded area enclosed by the two red lines in the Bloch sphere. a pair of RAP applications adiabatically drives the transition from the ground initial state |ψ (t i ) = α|0 + β|1 to |e by the first pulse and then back to the ground state by the second. Suppose the pulses propagate along the quantization axis (+ẑ axis) of the qubit states |0 and |1 , being linearly polarized with a relative polarization angle θ between them. Here we set the coordinate system to let the polarization unit vector of the first pulse be thex-axis unit vectorx = (R +L)/ √ 2, whereR andL are the right and left circular polarization unit vectors, respectively. Then the polarization unit vector of the second pulse is expressed as (e −iθR + e iθL )/ √ 2. Correspondingly, the qubit system is considered in the Cartesian basis, as |ψ (t i ) = α−β √ 2 |− + α+β √ 2 | +, where |± ≡ (|0 ± |1 )/ √ 2 are the fine-structure states |S 1/2 , m J = ±1/2 in our case. By the dipole selection rule, the right and left circular polarizations drive σ ± transitions between |∓ and |e , respectively.
The Bloch sphere representation in the lower figures of Fig. 1(b) shows the time evolution pathways of |∓ driven by respective polarization components. After the cyclic evolution by the two pulses, |∓ states get geometric phases ±θ − π , respectively, corresponding to − 1 2 of the solid angle enclosed by the evolution pathways [5], while the dynamic phase φ d is due to the intensity-and detuning-dependent eigenenergy [19] and dynamic Stark shift from neighboring transitions [20]. Thus we get |ψ (t f ) = α−β √ 2 e iφ − |− + α+β √ 2 e iφ + | +, in which φ ∓ = ±θ − π + φ d are the phases gained during the time evolution (see the Appendix for more details). The final state |ψ (t f ) is then expressed in the qubit basis as where Ux( ) is the X -rotation operator of the qubit states by the rotation angle = 2θ . Note that the dynamic phase φ d is always global because linear polarization guarantees an equal magnitude of the σ ± transitions and thus the same dynamic phase for each transition. Therefore, this scheme implements the holonomic transition determined by only the geometric phase , robust against laser parameters such as intensity and detuning. Qubit rotations about an arbitrary axisn = n xx + n yŷ can be implemented with an additional pair of time-delayed pulses. Since our scheme works in the regime where the hyperfine splitting is neglected, we adopt a method utilizing the hyperfine interaction in a longer timescale [20]. In the interaction picture where the qubit basis is |0 = |0 and |1 = e −iω hf t |1 , the Cartesian basis is given by |± = |0 ± e iω hf t |1 . Then, with the second Berry-phase gate applied after time delay T , the time evolution of the qubit system from t i + is the rotation operator of the qubit states about the axisn, with n x = cos(ω hf T ) and n y = sin(ω hf T ) controlled by the time delay T .

III. EXPERIMENTAL PROCEDURE
An experimental demonstration of the Berry-phase gates of atomic clock states was performed with an array of single rubidium atoms driven by pulse-shaped ultrafast optical pulses [see Fig. 2(a)]. Fast optical control of atomic and ion systems has been studied before through direct or Raman excitations [21][22][23], which we borrowed in the present work to implement adiabatic operations for geometric phases. Laser pulses were produced by a femtosecond Ti:sapphire amplifier system operated at a 1-kHz repetition rate (carrier frequency 377.1 THz, bandwidth 3.8 THz), which were resonant to the D 1 transition. The pulses were linearly chirped by an acousto-optic pulse shaper to stretch the pulse length to 1.5 ps with a chirp rate of 2.6 ps −2 , to satisfy the adiabatic condition for the RAP (see the Appendix for more details). Each pulse was split into two pairs of double pulses, with the interpair (intrapair) delay T = 70-370 ps (τ = 6.7 ps). The relative polarization angle θ was varied by a combination of a half waveplate and a polarizer, realizing Ux with the first pair and Un with the second. These pulses were delivered along the counterpropagating directions (±ẑ), respectively, to the atom array in a magneto-optical trap.
Five single atoms ( 87 Rb) were prepared by optical tweezers [24,25] at fixed positions of 26.5-μm spacing along the transverse direction of the laser beam propagation so that they experienced different intensities of the same laser pulses [see Fig. 2(b)]. The optical tweezers were tightly focused 852-nm laser beams (2-μm 1/e 2 diameter) with a trap depth of 1.6 mK. The atoms were first optically pumped to the |0 = |5S 1/2 , F = 2, m F = 0 qubit state using π -polarized continuous light resonant with the F = 2 → F = 2 transition of the D 1 line and the F = 1 → F = 2 transition of the D 2 line, in the presence of an applied magnetic field of 2.4 G which defined the quantization axis along the laser propagation axis.
Then the laser pulse sequence, each pair of which constituted one Berry-phase gate operation, was focused to the single-atom array with a beam waist of 60 μm (90 μm) for pulses 1 and 3 (pulses 2 and 4) which was smaller than the array size of 106 μm. Thus each atom in the array experienced largely different laser intensities. Finally, a push-out measurement [26] was applied to record the probability of the |1 = |5S 1/2 , F = 1, m F = 0 state of each atom with an electron multiplying charge-coupled device camera.

IV. RESULTS AND DISCUSSION
With the experimental apparatus, we first demonstrate the robustness of Ux( ) in Eq. (2) against laser power fluctuation. We used the first pair of pulses [pulses 1 and 2 in Fig. 2(a)], while blocking the second pair (pulses 3 and 4), and measured the state |1 probability of each atom as a function of the relative polarization angle θ . In Fig. 3(a) the measured probabilities P(θ ) = | 1|ψ (t f ) | 2 of the five atoms exposed to different position-dependent pulse areas [A max ≈ 5 × A min ; see Fig. 3(b)] are plotted and numerically fitted to the function with fitting parameters γ , θ , and η. The ideal case is γ = 1 and θ = η = 0, while experimental imperfection results in degraded fringe visibility (γ < 1 and η > 0) and a fringe shift ( θ = 0). Imperfections in γ and η that result in gate infidelities are due to errors in state preparation and measurement (SPAM). In our experiment, there exist optical pumping infidelity (∼4%), push-out measurement infidelity (∼3%), and polarization mismatch between the pulses and the quantization axis (∼1%), in addition to the effect of weak pre-and postpulses [27]. However, we note that the SPAM errors are not directly related to the robustness of the Berry-phase gate.
On the other hand, a nonzero fringe shift θ = 0 could imply failure of the intensity robustness of the proposed Berryphase gate. This error mainly came from the birefringence of the vacuum window [28], which affected the polarization of the laser pulses and the imbalance between the σ ± transitions causing a dynamic phase error. This polarization imperfection was verified by measuring the gradient of the fringe shift δθ/δA vs polarization ellipticity, as shown in Fig. 3(c). In our present demonstration of the Ux which was limited by the remaining polarization ellipticity of 1/40, a robustness against the laser intensity is achieved up to δ /δA = 2δθ/δA = 1.5%, and the ultralow birefringence technique [29,30] of 1/3000 ellipticity is expected to further improve this below δ /δA = 0.01%.
In the second experiment, we tested the robustness of the rotational axisn, using two Berry-phase gates Ux (pulses 1 and 2) and Un (pulses 3 and 4). The relative polarization angle θ of both gates was fixed to π/4 for maximum visibility and the Ramsey fringe of the F = 1 state probability P(θ, T ) = | 1|Un (T ) (2θ ) Ux(2θ )|0 | 2 was measured, with respect to the time delay T between the pulse pairs, and numerically fitted to the function where γ R is the fringe visibility, ϕn = ω R T /2 is the angle of the rotational axis, ϕ 0 is the Ramsey phase shift, and η R is the offset. At each atom, the measured frequency resulted in ω R = (6.79 ± 0.08) × 2π GHz, agreeing well with the 87 Rb hyperfine frequency within the 95% confidence interval, in which the equivalent time-domain error is as small as 0.9 ± 1.8 ps.
In order to estimate how robust the rotation axis of the Berry-phase gate is, we consider the possibility of an intensity-dependent axis shift ϕ 0 (A), i.e.,n(T ) →n (T, A)  Fig. 4, with the Ramsey fringe of each atom in the inset. The Ramsey fringes in the insets exhibit the same phase shift for all the atom positions regardless of the pulse areas, demonstrating the robustness inn. In other words, all the values of ϕ 0 are zero within 95% confidence intervals among the atom positions. The mean value of the confidence interval radius for all atom positions is 0.016π , while the standard deviation of the pulse area among all the gates is 1.45π . So the robustness of the rotation axis against the laser intensity is given within their ratio, i.e., δϕ 0 /δA < 1.1%. Now we turn our attention to the numerical estimation of the fidelity and robustness of the given Berryphase gates (for an ideal case without SPAM errors). The Lindblad master equation is used to calculate the amplitude and phase of the transition between the ground hyperfine states |5S 1/2 , F = 2, m F and |5S 1/2 , F = 1, m F , via |5P 1/2 , m J = ±1/2 |I = 3/2, m I = ∓1/2 and |5P 1/2 , m J = ±1/2 |I = 3/2, m I = ±1/2 , in the presence of the offresonant coupling to |5P 3/2 and spontaneous decay. The gate fidelity [31] of U gate is defined by where U ideal = Ux(π ) and U gate = U (θ 2 − θ 1 = π/2), averaged over the set of input states, i.e., |ψ in ∈ {|0 , |1 , (|0 + |1 )/ √ 2, (|0 + i|1 )/ √ 2}. The contributing experimental parameters are the spectral width (FWHM) of the pulses ω, Ramsey phase difference ϕ 0 , which is defined as the deviation of the Ramsey phase at each atom from the mean value of it. As in the inset graphs, the Ramsey fringe at each atom position (and each corresponding total pulse area) is measured by the transition probability from |0 to |1 after applying Ux (π/2) (pulse 1 and pulse 2) and Un(π/2) (pulse 3 and pulse 4) with various interpair time delays T , consecutively. Fitted values of ϕ 0 are plotted with the error bars indicating the 95% confidence intervals. the chirp parameter c p , the pulse area A = A 1 + A 2 , the time delay τ , the amplitude imbalance of the two pulses α = (A 2 − A 1 )/(A 2 + A 1 ), the frequency detuning = ω 1 − ω 0 = ω 2 − ω 0 , and the relative phase between the laser pulses δφ L = φ L 2 − φ L 1 , i.e., F = f ( ω, c p , A, τ, α, , δφ L ). Figure 5 shows the numerical calculation. The contour plot of the fidelity F (c p , ω) in Fig. 5(a) shows the high-fidelity region around c p ≈ 0.072 ps −2 and ω ≈ 2π × 4 THz. The spectral width is upper bounded by the leakage D 2 transition to 5P 3/2 and lower bounded by the insufficient spectral width (smaller than required by the chirp) that corresponds to the spectral width ranging from about 2π × 3 THz to 2π × 4 THz in our experiment. To investigate the dependence of the fidelity on the pulse area, amplitude imbalance, and frequency detuning, respectively, in the rest of Fig. 5, we choose ω = 2π × 4 THz and c p = 0.072 ps −2 . First, in Fig. 5(b), the fidelity F (A) is calculated as a function of the pulse area A for various time delays τ . The result exhibits a nearly flat high-fidelity region within a wide range of pulse area, e.g., between 3π and 8π for a 2.36-ps short pulse. Thus, the Berry-phase gates are robust against the pulse area (or the laser power fluctuation). This A-robust region (δF/δA ≈ 0) is lower bounded by nonadiabaticity and upper bounded by the interference between temporally close two pulses. Next the robustness against the amplitude imbalance (α robustness δF/δα ≈ 0) is shown in Fig. 5(c), where a sufficiently large pulse area ensures the α robustness as the adiabatic condition of the chirped RAP breaks down for a weaker pulse of insufficient Rabi frequency. Finally, the robustness against the detuning and relative laser phase δφ L is shown in Fig. 5(d). The robustness (δF/δ ≈ 0) is achieved around zero detuning, regardless of the relative phase (δφ L robustness), while the asymmetry between positive and negative detunings stems from the dynamic Stark shift (due to the D 2 transition) of the 5S 1/2 level. Thus, we expect that the fidelity above 0.999 can be achieved for about half of the laser spectral width in the current experiment. It is worthwhile to compare the robustness of the proposed scheme with a state-of-the-art result. Wang et al. [32] considered the robustness of their single-qubit Pauli Z gate, which is a composite-pulse dynamic phase gate. Their robustness, obtained as δ = 21 mrad × (δ f / f ) 2 for a laser-induced Stark shift f , estimates the fidelity robustness of δF = 10 −3 over a 7% change in laser intensity. In comparison, our Berry-phase gate is robust over a 40% change in laser intensity, around A = 6π , within δF = δ 2 /4 = 10 −3 , estimated based on measurements in Fig. 3(c).

V. CONCLUSION
We have implemented qubit rotations of atomic clock states using Berry phases induced by two linearly polarized chirped pulses in picosecond timescales. The results show, as a characteristic of geometric phases of adiabatic passages, gate-operation robustness against laser parameter errors, which has been hard to achieve in previous nonadiabatic holonomic schemes. Berry-phase gates can offer a fast and robust qubit control not only for atomic systems, but also for solid-state systems of relatively short coherence time.

ACKNOWLEDGMENTS
The authors thank the anonymous referees for valuable comments and suggestions. This research was supported by Samsung Science and Technology Foundation through Grant No. SSTF-BA1301-12.

APPENDIX: DETAILED DESCRIPTION OF THE FAST BERRY-PHASE GATES
The proposed Berry-phase gates for the atomic clock states are achieved with two successive chirped optical pulses that are linearly polarized or in equal magnitudes of left and right circular polarizations. Each circular polarization component of the chirped pulses adiabatically drives population transfer between the fine-structure ground and excited levels according to the transition selection rules. The Berry-phase difference between the two driven evolution paths is determined by the relative polarization angle between the two pulses, resulting in robust qubit rotation insensitive to other laser parameters except the polarization. After we briefly review the chirped rapid adiabatic passage, we describe the Berry-phase gates for atomic systems.

Chirped rapid adiabatic passages
The robust population transfer is implemented by using the chirped rapid adiabatic passage [19]. Let us consider a twolevel system, of the fine-structure states |g and |e with energy separationhω 0 , interacting with a Gaussian chirped pulse of an electric field written in the frequency domain as where E 0 is the peak amplitude, ω L is the laser center frequency, ω is the bandwidth, and c p is the chirp parameter [33]. The corresponding time-domain electric field is given by with

The Hamiltonian of this interaction is given by
h is the Rabi frequency, and μ is the transition dipole moment. The eigenstates of Eq. (A3) are given by with The corresponding eigenenergies are given by Here, since the detuning (t ) is linearly dependent on time, the eigenstate | − (t ) [| + (t ) ] evolves from |g (|e ) to |e (|g ) as time changes from t = −∞ to ∞, along the meridian of the Bloch sphere. Thus, the complete population transfer between |g and |e is achieved as when the adiabatic condition is satisfied. The rapid adiabatic passage ensures the robustness against the fluctuation of the laser parameters E 0 , ω L , and φ L (amplitude, frequency, and phase).

Description of the fast Berry-phase gates in atomic systems
In our consideration, the qubit states are the hyperfine states |0 = |S 1/2 , F = I + 1/2, m F and |1 = |S 1/2 , F = I − 1/2, m F of the ground state |g = |S 1/2 , m J = ±1/2 of an alkali-metal atom, while the excited level is |e = |P 1/2 , m J = ±1/2 . Berry-phase gates are implemented by successive optical transitions between |g and |e , which induce the phase gates for the qubit system of |0 and |1 (atomic clock states for m F = 0).