Abstract
Realizing a long coherence time quantum memory is a major challenge of current quantum technology. Until now, the longest coherencetime of a single qubit was reported as 660 s in a single ^{171}Yb^{+} ionqubit through the technical developments of sympathetic cooling and dynamical decoupling pulses, which addressed heatinginduced detection inefficiency and magnetic field fluctuations. However, it was not clear what prohibited further enhancement. Here, we identify and suppress the limiting factors, which are the remaining magneticfield fluctuations, frequency instability and leakage of the microwave referenceoscillator. Then, we observe the coherence time of around 5500 s for the ^{171}Yb^{+} ionqubit, which is the time constant of the exponential decay fit from the measurements up to 960 s. We also systematically study the decoherence process of the quantum memory by using quantum process tomography and analyze the results by applying recently developed resource theories of quantum memory and coherence. Our experimental demonstration will accelerate practical applications of quantum memories for various quantum information processing, especially in the noisyintermediatescale quantum regime.
Introduction
Quantum coherence is a vital component for scalable quantum computation^{1,2,3}, quantum metrology^{4,5}, and quantum communication^{6,7,8,9,10}. In practice, decoherence, loss of coherence in the computational basis, in the quantum system comes from the coupling with the surrounding environment and fluctuations of control parameters in quantum operations, which can lead to the infidelity of quantuminformation processing, the low sensitivity of quantum sensors, and the inefficiency of quantum repeater based protocols in quantum communication networks. Limited coherence time may also undermine quantuminformation applications such as quantum money^{11,12}. It is thus of practical importance to have a stable quantum memory with a longcoherence time.
Numerous experimental attempts have been made to enhance the coherence time of quantum memory in a variety of quantum systems. With ensembles of trapped ions and nuclear spins in a solid, coherence time of 10 min^{13,14}, and 40 min at room temperature^{15,16} and a few hours at 4 K^{17} have been reported, respectively. For a single qubit quantum memory, which is the essential buildingblock for quantum computers^{18,19} and quantum repeaters^{20,21}, records of coherence time have been reported to the time scale of a minute in trapped ion qubit^{22,23,24,25}. For the coherence time of a minute, the limitation mainly came from the qubitdetection inefficiency^{25,26,27} due to the motional heating of qubitions without Doppler lasercooling. The problem was addressed by sympathetic cooling by other species of ion, which allowed further improvements of coherence time to over 10 min with the support of dynamical decoupling^{28}. While the fundamental limit is far beyond 10 min; however, it remains a major technological challenge to further enhance the quality of a trappedion quantum memory.
Here we address this challenge by improving the coherence time of a ^{171}Yb^{+} ionqubit memory from 10 min to over one hour. This is achieved by identifying and suppressing the three dominant error sources: magneticfield fluctuation, the phase noise of the local oscillator, and microwave leakage for qubit operation. Furthermore, with the capability of full control on a single qubit, we systematically study the decoherence process of the quantum memory by quantum process tomography. Typically, the decoherence process has been characterized by the coherence time T_{2} at which the Ramsey contrast, corresponding to the size of the offdiagonal entry in the qubit densitymatrix, decays to 1/e^{13,14,15,16,17,28}. We experimentally study the decoherence dynamics by relevant quantum channels of depolarization and dephasing, which allows us to use recently developed coherence quantifiers^{29,30,31}. We also use our data to study recently developed resource theories of quantum memory and coherence, such as the robustness of quantum memory (RQM) that quantifies how well a memory preserves quantum information^{32} and relative entropy of coherence (REC) that quantifies how much coherence is maintained in the state.
Results
Two species of atomic ions
In our experiment, we load one ^{171}Yb^{+} ion and one ^{138}Ba^{+}ion in a fourrod Paul trap as shown in Fig. 1a. Two hyperfine levels of the ^{171}Yb^{+} ion in the S_{1/2} manifold are used to encode the qubit with \(\{\left0\right\rangle \equiv \leftF=0,{m}_{F}=0\right\rangle ,\left1\right\rangle \equiv \leftF=1,{m}_{F}=0\right\rangle \}\) and a frequency difference of 12642812118 + 310.8B^{2} Hz, where B is the magnetic field in Gauss. As a sympathetic cooling ion, ^{138}Ba^{+} is used since it has a similar atomic mass with ^{171}Yb^{+}, which can be used for efficient cooling. We apply Dopplercooling laser beams on the ^{138}Ba^{+} ion all the time, which provides continuous cooling for the whole system. In this way, we can measure the final state of the ^{171}Yb^{+}qubit by standard fluorescence detection technique without losing any detection fidelity^{25,26,27}.
Suppression of ambient magnetic field
We suppress the ambient noise of the magnetic field by installing a magneticfield shielding with a permanent magnet^{33}. We enclose our main vacuum chamber that contains the Paul trap with a twolayer of μmetal shielding as shown in Fig. 1a. By using a fluxgate meter, we observe more than 40 dB attenuation at 50 Hz inside the shielding, which is the main frequency of noise in the lab due to the AC powerline. To generate stable magnetic field of 5.8 G, we replace coils with a Sm_{2}Co_{17} permanent magnet, which has a temperature dependence of −0.03%/K^{33}. The magnetic field strength can be adjusted by changing the position of the magnet from the location of ions. After these modifications, we observe the coherence time of the fieldsensitive Zeeman qubit is increased to more than 30 ms. We study the noise spectrum by dynamical decoupling sequences^{34,35} and observe that noise of 50 Hz and 150 Hz are below 16 μG and 32 μG, respectively.
Improvements of microwave frequency stability
We perform coherent manipulation of the qubit by applying a resonant microwave. Qubit coherence is typically measured by the contrast of Ramsey fringe, which requires control and interrogation of the system by a local oscillator that can bring in phase noise^{36,37}. In our case, this part of the noise is determined by the microwave signal generator and its reference. For microwave signal, phase noise in the lowfrequency regime is mainly determined by those of the reference signal. We use a crystal oscillator as the reference, which has an orderofmagnitude smaller Allan variance at 1 s observation time than our previous Rb clock oscillator^{28}.
Suppression of microwave powerleakage
We also find that leakage of the microwave can introduce relaxation of the qubit memory. We include a microwave switch after amplifier as shown in Fig. 1b, which reduces the leakage by over 70 dB. In total, we suppress the microwave output by 164 dB after turning off all the switches. With π pulse duration of 175 μs, the effect of leakage is negligible for 0.4 s pulse interval time, which would be further suppressed by dynamical decoupling pulses. At the same time, we also use AOMs, EOPP and a mechanical shutter to suppress the leakage of ^{171}Yb^{+} ion resonant laser beams as same as Ref. ^{28}.
Dynamical decoupling pulse sequence
We measure the coherence time of the ^{171}Yb^{+} ion qubit by observing the dependence of Ramsey contrasts on the storage time. The experimental sequence is shown in Fig. 2. As discussed above, cooling laser beams for ^{138}Ba^{+}are applied during the whole sequence. We initialize the state of the ^{171}Yb^{+} ion qubit to \(\left0\right\rangle\) by the standard optical pumping technique, apply the π/2Ramsey pulses, and detect the probability in \(\left1\right\rangle\) state by the standard statedependent fluorescence method. In the Ramsey measurement, we observe the coherence time of 1.6 s (see “Methods” for the details). We note that we have a detection efficiency of 98.6%, which is corrected by the calibrated error magnitude with the uncorrelated error assumption as shown in Ref. ^{38}.
To enhance coherence time, we first apply a spinecho pulse that uses a single π pulse to compensate lowfrequency noise. We observe the coherence time is improved to 11.1 s with the single spinecho pulse (see “Methods” for the details). We then apply the dynamical decoupling scheme^{16,17,28,34,35,39,40}, which contains multiple of spinecho pulses. Performance of dynamic decoupling pulses is described by the filter function \(\widetilde{y}(\omega ,T)=\frac{1}{\omega }\mathop{\sum }\nolimits_{j = 0}^{T/\tau }{(1)}^{j}\left({{\rm{e}}}^{{\rm{i}}\omega {t}_{j}}{{\rm{e}}}^{{\rm{i}}\omega {t}_{j+1}}\right)\), with t_{0} = 0, t_{(T/τ)+1} = T, t_{j} = (j − 0.5)τ when 1≤j≤T/τ, and τ is the interval of pulses. Then Ramsey fringe contrast^{34} is \(W(T)={{\rm{e}}}^{\frac{2}{\pi }\mathop{\int}\nolimits_{o}^{\infty }S(\omega ){\left\widetilde{y}(\omega ,T)\right}^{2}{\rm{d}}\omega }\) with S(ω) being the noise spectrum density. In our experiment, we use KDD_{xy}(Knill dynamical decoupling)^{17,28,40} pulses, where all the pulses are equally spaced and have periodic phases as shown in Fig. 2. The filter function of the KDD_{xy} pulses has a peak at the frequency of \(\omega =\frac{\pi }{\tau }\). Most of the noise is suppressed except the part with frequencies around the peak, which is instead amplified. When the total time T is fixed, the position of the peak is determined by the pulse interval, which can be optimized depending on the noise spectrum. After comparing different parameters, we choose 0.4 s as the pulse interval, which leads to the peak of the filter function at 2π × 1.25 Hz.
Resulting coherence time
With different initial states, we show the time dependence of the Ramsey contrast up to 960 s in Fig. 3. By assuming exponential decay of the Ramsey contrast, we find the coherence time of states \(\left0\right\rangle\) and \(\left1\right\rangle\) to be 16000 ± 3200 s. Other four superposition states (ϕ = 0, \(\frac{\pi }{2}\), π, and \(\frac{3\pi }{2}\) shown in the legends of Fig. 3) have a coherence time of 5500 ± 670 s. Both of the uncertainties are from fitting errors. As shown in the inset of Fig. 3, the coherence time is increased by an orderofmagnitude compared to the previous stateoftheart result^{28}.
Experimental study of decoherence process
We further analyze the decoherence process by performing quantum process tomography, which completely characterizes unknown dynamics of a quantum system, at different storage time following Refs. ^{41,42}. The procedure of quantum process tomography is as follows. For a quantum process ε, we consider its process χ matrix, which is defined by \(\varepsilon (\rho )={\sum }_{mn}{\chi }_{mn}{\hat{E}}_{m}\rho {\hat{E}}_{n}^{\dagger }\) with \({\hat{E}}_{m}\in \{\hat{I},\hat{X},\hat{Y},\hat{Z}\}\)^{42}. We measure the χ matrix of our single ionqubit memory by preparing four different input states \(\left0\right\rangle\), \(\left1\right\rangle\), \((\left0\right\rangle +\left1\right\rangle )/\sqrt{2}\), \((\left0\right\rangle +i\left1\right\rangle )/\sqrt{2}\), applying the memory, and finally measuring the output states with four measurements I, X, Y and Z. We use the maximum likelihood method to reconstruct the process matrix^{41}. We observe the time dependence of the process matrix as shown in Fig. 4a. The ideal quantum memory process is described by \({\chi }_{mn}^{{\rm{id}}}={\delta }_{m,1}{\delta }_{n,1}\). With the experimentally measured process matrix \({\chi }^{\exp }\), we can obtain the process fidelity \({F}_{{\rm{p}}}={\rm{Tr}}({\chi }^{{\rm{id}}}{\chi }^{\exp })={\chi }_{11}^{\exp }\). The infidelity mainly comes from the dephasing and depolarization effects. The process with these two noises can be described by the following matrix as
where T_{1} and T_{2} are depolarizing and total dephasing time, respectively^{43,44}. The process matrix describes a quantum memory with full coherence at T = 0 but which has transitioned to a fully mixed state for \(T\gg \min ({T}_{1},{T}_{2})\). By fitting the experimental process tomography results with the above process matrix of Eq. (1), we obtain T_{1} = 12000 ± 2200 s and T_{2} = 4200 ± 580 s (see Methods for the details). We also plot the model of Eq. (1) and the experimental data in Fig. 4a.
From experimental quantum process tomography, the performance of the quantum memory on arbitrary quantum states can be accurately estimated, which can be simplified as the mean fidelity, \({F}_{{\rm{mean}}}={\langle {\rm{Tr}}(\rho \varepsilon (\rho ))\rangle }_{\rho }\), which is the averaged output fidelity with all possible input states ρ^{45,46,47}. The mean fidelity is a function of wait time T since the process matrix of quantum memory is different depending on wait time T. We use the Monte Carlo method to get the mean fidelity with 10^{5} different input states, generated by uniformly sampled random unitary operations according to the Haar measure^{48}. As shown in Fig. 4a, we obtain the coherence time, the time constant of fitted exponential decay function, 5200 ± 500 s for the mean fidelities. We note that within the error bar, this coherence time is consistent with that of a simple estimation of the mean fidelity from the formula of F_{mean} = (2F_{p} + 1)/3^{45}, where it provides 5600 ± 650 s.
Benchmark of quantum memory and quantum coherence
Recently due to the fundamental importance of quantum coherence, there have been serious developments of rigorous theories of quantum coherence and quantum memory as a physical resource. In our manuscript, we relate our experimental results with uptodate resource theories of quantum coherence and quantum memory such as REC and RQM, respectively.
The REC is a distancebased coherence quantifier, which is suggested as a gold standard measure^{45}. The REC can be interpreted as the minimal amount of noise required for making a quantum state fully decohere^{31}. The REC has the same formula with distillable coherence, which has an analogy to the distillable entanglement, a standard widely using entanglement quantifier. The distillable coherence is the optimal number of maximally coherent singlequbit states that can be obtained in a given qubit state through incoherent operations and fulfills all the requirements as a proper coherence quantifier^{31}. The formal definition of the REC^{30} is written as C(ρ) = S(Δ(ρ)) − S(ρ), with \(\Delta (\rho )={\sum }_{i}\left\langle i\right\rho \lefti\right\rangle \lefti\right\rangle \left\langle i\right\), \(\{\lefti\right\rangle \}\) being the computational basis, and \(S(\rho )={\rm{Tr}}(\rho \, {\mathrm{log}}_{2} \, \rho )\) being the Von Neumann entropy.
In our analysis, we use the ratio of the REC between the output state and the input state instead of directly using the REC because each input state has a different value of the REC. Based on the process matrix \({\chi }^{\exp }\), we numerically calculate the ratio of the REC. We study the time dependence of the mean ratio of the REC \({C}_{{\rm{mean}}}^{\prime}={\langle C(\varepsilon (\rho ))/C(\rho )\rangle }_{\rho }\), where we average over 10^{5} random input states. Note that we only consider states with REC larger than 0.01. As shown in Fig. 5, the mean REC ratio decays to 1/e after 3500 ± 1100 s by the exponential fitting. The relatively short duration and the large fluctuation of the results mainly stem from stringent condition and sensitivity of the REC to small errors in the process matrix.
The RQM, which was introduced by Ref. ^{32}, quantifies how well the memory preserves quantum information that includes coherence. Here the quantum memory, which stores a quantum state for later retrieval, is considered as a channel that maps an input state to an output state. Ideally, it should be an identity channel. The quantifier of RQM is developed based on the approach that considers the quantum memories as a resource and provides a means to benchmark quantum memories. Basically, the higher the RQM is, the more noise the quantum memory can sustain before it is unable to preserve quantum information. In contrast, a classical memory that cannot preserve quantum information is characterized as a measureandprepare (MP) memory that destroys the input state by measurement, and stores only the classical measurement result.
The RQM is defined as the least portion of the classical memory that needs to be mixed with the quantum memory so that the resultant mixture belongs to MP memory, which is formally written as \(R({\mathcal{N}})={\min }_{{\mathcal{M}}\in {\mathcal{F}}}\left\{s \, \ge \, 0\left\frac{{\mathcal{N}}+s{\mathcal{M}}}{s+1}\right.\in {\mathcal{F}}\right\}\), where \({\mathcal{N}}\) is the quantum memory of interest, \({\mathcal{M}}\) is a classical memory that is in the set of MP memories \({\mathcal{F}}\), and s is the amount of mixture of the quantum memory \({\mathcal{N}}\) with the classical memory \({\mathcal{M}}\). The RQM is the minimum value of s to make the mixed memory in \({\mathcal{F}}\). We note that the RQMs of all classical memories are zero since the MP memories cannot maintain quantum information. We obtain the RQM from the experimental process matrix. In general, the \(R({\mathcal{N}})\) can be found by a numerical search of the minimum s. Assuming offdiagonal elements in the process matrix are negligible, the RQM can be simplified to \(\max \{2{F}_{{\rm{p}}}1,0\}\) for qubit quantum memories. In our experimental process tomography, no noticeable difference is observed between the numerical search and the simplified formula. As shown in Fig. 5, the RQM of our system lasts 6300 s before it decays to zero by the exponential fitting.
Discussion
In conclusion, we report a trappedion based single qubit quantum memory with over one hour coherence time, an orderofmagnitude enhancement compared to the stateoftheart record^{28}. The quantum memory with the longcoherence time will accelerate the development of scalable quantum computation^{3,49,50}, longdistance quantum communication^{9,51}, highprecision quantum metrology^{4,5}, and quantum money^{11,12}, in particular, in the nearterm noiseintermediatescale quantum regime where there will be no quantum error correction. Our research can be also extended to realize a generalpurpose quantum memory that contains multiple qubits capable of individual storage and retrieval of quantum information at any required time with further enhancement of coherence time and increase of the number of individually controllable qubits.
Further enhancement of the coherence time to day level (≈10^{5} s) may be achievable by improving the stability of the classical oscillator and magneticfield fluctuation as shown in Fig. 6 (see also “Methods”). To reach the ultimate coherence time limited by the lifetime of the excited hyperfine state that is expected to be thousands of years to our estimation (see Methods), we need to suppress the hopping of ions, decoherence from scatting of ^{138}Ba^{+} lasers, leakage of the microwave, and collision of the background gas. Microwave leakage can be simply addressed by adding switches. The other sources of decoherence are related to the background gas collisions. The collisions cause hopping of ions, which introduces frequency shift from different magneticfield strengths between two positions and collision frequency shift due to change of motional distribution and phase of atomic superposition^{52}. The background gas collisions can be significantly suppressed by locating the ion trap system in a cryostat environment^{53}, which naturally suppresses the hopping rates and collisioninduced shift. No hopping allows us to shed the cooling laser beams only on the ^{138}Ba^{+} ion, which eliminates the scattinginduced decoherence of the ion qubit (see “Methods”).
Our work can be extended to the general purpose of quantum memory, quantum money for example, that requires a large number of qubits by using a long ionchain in a trap with an individual addressing system. The necessary technical improvement for such quantum memory is to eliminate the hopping problem because hopping ruins the individual tracking of the quantum memory. The hopping problem in the longlinear chain can be also suppressed by a cryostat ion trap as discussed above. We also notice that in the long ionchain, the micromotion induces inefficiency of statedetection^{54}. Individual compensation of the micromotions can be achieved by a sophisticated trap with the capability of localfield control.
Methods
Expected limitations of coherence time
The expected limitations of coherence time caused by different decoherence sources are summarized in Fig. 6. We note that in the analysis, we do not consider the imperfection of the KDD_{xy}pulses because we find the KDD_{xy}sequence is robust against the typical errors as flipangle error and frequencyoffset errors even at the levels of errors in our system^{40}. For the flipangle error of 10^{−2} and the frequencyoffset error of 100 Hz, around 2 × 10^{10} pulses and 3 × 10^{10} can be applied before the output results decay to 1/e, respectively, which correspond to 0.8 × 10^{10} s and 1.2 × 10^{10} s, respectively, for our choice of the gaptime, 0.4 s.
(i) Phase noise of local oscillator: the new frequency reference for local oscillator has an orderofmagnitude smaller Allan variance \(\sigma {({\tau }_{0})}^{2}\) at τ_{0} = 1 s than that of previous one in Ref. ^{28}, which indicates an orderofmagnitude smaller phasenoises spectrum density S_{LO}(ω), assuming the shape of S_{LO}(ω) is the same for both references. It is because of the relation between Allan variance \(\sigma {({\tau }_{0})}^{2}\) and phasenoise spectrum density S_{LO}(ω) is \(\sigma {({\tau }_{0})}^{2}=\frac{1}{\pi }\mathop{\int}\nolimits_{0}^{\infty }{S}_{{\rm{LO}}}(\omega ){\sin }^{4}(\frac{{\tau }_{0}}{2}\omega ){\rm{d}}\omega\)^{55}. With the orderofmagnitude smaller S_{LO}(ω), the Ramsey fringe contrast^{34}\(W(T)={{\rm{e}}}^{\frac{2}{\pi }\mathop{\int}\nolimits_{o}^{\infty }{S}_{{\rm{LO}}}(\omega ){\left\widetilde{y}(\omega ,T)\right}^{2}{\rm{d}}\omega }\) will also takes an orderofmagnitude longer time to reach 1/e. Therefore, the current an order of magnitude enhancement of coherence time is mainly limited by the phase noise of local oscillator.
(ii) Magneticfield fluctuation: magneticfield noise is suppressed by shielding and permanent magnet. The comparison of magneticfield fluctuation before and after the suppression is shown in Fig. 7. The coherence time of the Zeeman state is improved by around 30 times improvement after magneticfield noise suppression similar to that in Ref. ^{33}. Therefore, we expect the limitation of the coherence time of the clockstate qubit due to the magneticfield fluctuation is increased by 30 times, which is around 2 × 10^{4} s.
(iii) Ion hopping: hopping of the ions between two positions that have the qubitfrequency difference of 0.22 Hz (60 μG difference) occurs about every 10 min. The estimated infidelity of a superposition state due to the alternating frequency changes from the ion hopping is around 2.7 × 10^{−3} per hopping. Assuming the infidelity increases exponentially with the number of hopping, the limitation of coherence time due to the ion hopping is expected to be around 2 × 10^{5} s \((=10\,\min /(2.7\times 1{0}^{3}))\). We estimate the infidelity per hopping as follows. Since a small amount of constant frequency shift almost does not introduce infidelity due to the KDD_{xy}sequences, we ignore the nohopping period. When hopping occurs, the effect of frequency shift cannot be compensated by the dynamical decoupling pulses, which introduces the infidelity of the state. We assume in one KDD_{xy}unit, hopping occurs at most once with a uniform distribution of time, which is reasonable since the duration of one KDD_{xy}unit (4 s) is much shorter than the period of the hopping (10 min). Finally, we average out the infidelities at different occurring time of hopping.
(iv) Scatting of ^{138}Ba^{+}lasers: we estimate the spontaneous emission rate of the ^{171}Yb^{+} ion assuming the cooling laser beams (493 nm and 650 nm) of the ^{138}Ba^{+} ion are entirely applied to the ^{171}Yb^{+} ion. The spontaneous emission rate of the dipole transition of the ^{171}Yb^{+} ion is written as^{28,56,57,58}
where γ ≈ 2π × 20 MHz is the spontaneous emission rate from the ^{2}P states, \(g=\frac{\gamma }{2}\sqrt{I/(2{I}_{{\rm{sat}}})}\), Δ_{HF} = 2π × 12.6 GHz, Δ_{FS} = 2π × 100 THz. For 493 nm laser, power P = 35 μW, beam waist ω = 31.4 μm, I_{493} = 21.8I_{sat}, Δ_{D1} = 2π × 203.8 THz, then we get a scattering rate of 1.09 × 10^{−6} Hz. For the 650 nm laser, power P = 66 μW, beam waist ω = 22.9 μm, I_{650} = 75.5I_{sat}, Δ_{D1} = 2π × 349.9 THz, scattering rate 1.29 × 10^{−6} Hz. Therefore, both 493 nm and 650 nm laser beams provide the limitation of the coherence time around 4 × 10^{5} s.
(v) Leakage of microwave: after improving the frequency stability of the local oscillator and suppressing magneticfield fluctuations, the coherence time was improved to only twice, 1200 s, which was limited by the microwave leakage. We suppress the leakage by adding the microwave switch with 70 dB isolation at the final stage before the horn. We observe the enhancement of coherence time to 5400 s, which now is mainly limited by the frequency stability of the local oscillator as discussed in section (i). We estimate that the 70 dB isolation suppresses the carrier Rabifrequency by microwave leakage around 3000 times, which improves the coherencetime limitation to around 4 × 10^{6} (≈1200 s × 3000).
(vi) Collision of background gas: backgroundgas collisions cause decoherence by collision frequency shift. The model in Ref. ^{52} estimates that ^{27}Al^{+} optical transition clock has a frequency shift of order 10^{−16} after 0.15 s probe from the background gas collision of H_{2} in the pressure of 38 nPa at room temperature. The model estimates that a microwave transition has a larger shift as the level of 10^{−14} with 4 s probe, due to no suppression introduced by the Debye–Waller factor. This shift will be an upper bound of our collision frequency shift because the model does not include the suppression by the sympathetic cooling. We numerically simulate the collision frequency shift with KDD_{xy} sequences. The infidelity of a superposition state is estimated by around 1.7 × 10^{−9} for each KDD_{xy} unit, which leads the coherencetime limitation to 4 s × 1/(1.7 × 10^{−9}) ~ 2 × 10^{9} s, where we assume the infidelity increase exponentially with the number of KDD_{xy} gate numbers^{59}.
(vii) Lifetime of hyperfine state: The spontaneous emission rate of magnetic dipole transitions is written as \(\gamma =\frac{{\rm{\alpha }}{{\Delta }_{{\rm{HF}}}}^{3} M{ }^{2}}{3{{m}_{{\rm{e}}}}^{2}{{\rm{c}}}^{4}}\), where M is the magnetic dipole matrix element that is expected to be of order ℏ, α is the finestructure constant, and Δ_{HF} the energy splitting of hyperfine qubit^{60}. For the ground hyperfine level of ^{171}Yb^{+} ions, we estimate it as \({\tau }_{{\rm{HF}}}=\frac{1}{\gamma } \sim 5\times 1{0}^{11}\) s, where we assume M ~ ℏ.
Process matrix evolution
We obtain the T_{1} and T_{2} in the diagonal elements of χ of Eq. (1) by fitting χ_{XY} ≡ 0.5(χ_{22} + χ_{33}) and χ_{IZ} ≡ 1 − (χ_{11} − χ_{44}) to the functions of \(\frac{1{{\rm{e}}}^{t/{T}_{1}}}{4}\) and \(1{{\rm{e}}}^{t/{T}_{2}}\), respectively. As shown in Fig. 8, we obtain T_{1} = 11900 ± 2200 s and T_{2} = 4200 ± 580 s by fitting χ_{XY} and χ_{IZ}, respectively. We note that ideally the total dephasing time T_{2} = 4200 ± 570 s in the process tomography should be matched to the coherence time of 5500 ± 670 s. The discrepancy originates from the quantum fluctuation noise in the other bases measurements of the process tomography. The process tomography requires measurements of four different bases for different input states. For example, a superposition input state, \((\left0\right\rangle +\left1\right\rangle )/\sqrt{2}\) (an eigenstate of σ_{x}), we need to measure the expectation values of identity, σ_{x}, σ_{y}, and σ_{z}. In principle, both 〈σ_{y}〉 and 〈σ_{z}〉 should be zero (even there exists serious decoherence). However, due to the quantum fluctuation noise, they deviated from zero, which introduced the reduction of the T_{2} in the process tomography in our measurement. If these results are zero, the Ramsey coherence time and the total dephasing time of the process tomography will be perfectly matched. We believe if the number of measurements for the process tomography approaches infinity, the difference should converge to zero.
Simple coherence time measurement
Many experiments of interest can take advantage of dynamical decoupling pulses, but some of them cannot or can only apply a single spinecho pulse. This makes the enhancement of these specialcases coherence time more attractive for some applications. Figure 9 shows the measurement results for direct Ramsey measurement and one spinecho pause case.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors thank Roee Ozeri, Rene Gerritsma, Jianwei Zhang, and Jize Han for helpful discussions. This work was supported by the National Key Research and Development Program of China under Grant Nos. 2016YFA0301900 and 2016YFA0301901, the National Natural Science Foundation of China Grant Noa. 11574002, 11974200, and 11504197, Singapore Ministry of Education through Tier 1 Grant No. RG190/17, the Singapore National Research Foundation through Fellowship No. NRFNRFF201602, and NRFANR Grant No. NRF2017NRFANR004 VanQuTe. X.Y. acknowledges the support from Simons Foundation.
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P.W., C.Y.L., M.Q., M.U., J.Z., and Y.W. contributed to constructing the experimental system. P.W. with the assistance of C.Y.L. performed the data taking and analysis. X.Y., M.G., and J.N.Z provided theoretical support. K.K. supervised the experiment. All authors discussed the results and contributed to the writing of the manuscript.
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Wang, P., Luan, CY., Qiao, M. et al. Single ion qubit with estimated coherence time exceeding one hour. Nat Commun 12, 233 (2021). https://doi.org/10.1038/s4146702020330w
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