Abstract
We present a novel continuous dynamical decoupling scheme for the construction of a robust qubit in a threelevel system. By means of a clock transition adjustment, we first show how robustness to environmental noise is achieved, while eliminating drivenoise, to firstorder. We demonstrate this scheme with the spin sublevels of the NVcentre’s electronic ground state. By applying drive fields with moderate Rabi frequencies, the drive noise is eliminated and an improvement of 2 orders of magnitude in the coherence time is obtained compared to the pure dephasing time. We then show how the clock transition adjustment can be tuned to eliminate also the secondorder effect of the environmental noise with moderate drive fields. A further detailed theoretical investigation suggests an additional improvement of more than 1 order of magnitude in the coherence time which is supported by simulations. Hence, our scheme predicts that the coherence time may be prolonged towards the lifetimelimit using a relatively simple experimental setup.
Introduction
The reliable and efficient construction and manipulation of qubits is necessary for the implementation of quantum technological applications and quantum information processing. In solidstate and atomic systems, ambient magnetic field fluctuations constitute a serious impediment, which usually limits the coherence time to a small fraction of the inherent lifetime. Pulsed dynamical decoupling^{1,2,3} has proven to be very useful in prolonging the coherence time^{4,5,6,7,8,9,10,11,12}. However, in order to mitigate both environmental and controller noise, composite highfrequency pulse sequences must usually be applied^{13,14,15,16,17} which require large field strengths^{18}. Continuous dynamical decoupling^{18,19,20,21,22,23,24,25,26,27} offers another possibility of suppressing environmental noise, where diminishing the effect of the controller noise can be achieved by different approaches. In this context, a rotary echo scheme^{28,29} can be viewed as analogous to pulsed dynamical decoupling. The concatenation of several onresonance driving fields^{30,31,32,33,34} is another concept, but inherently connected to a reduction of the dressed energy gap, eventually limiting the performance of the scheme, and in particular reducing the qubit gate operation time.
Multistate systems allow for yet a different approach. By applying continuous driving fields on a multilevel structure, a fully robust qubit  a qubit that is robust to both external and controller noise  can be obtained^{35,36}. However, with these multistate schemes, which utilise onresonance driving fields, it is not possible to achieve robustness against drive noise in a threelevel system^{36}. A protected qubit subspace within a threelevel configuration can be realised by the application of offresonant strong driving fields^{37} making the experimental realisation challenging.
In this report we show how a fully robust qubit can be simply constructed by means of a clock transition adjustment^{27} using a threelevel system. We start with a basic version of our scheme where both continuous onresonant and offresonant driving fields are utilised. The onresonant driving fields result in robustness to environmental noise, whereas the offresonant driving fields facilitate robustness against driving noise, which typically limits continuous dynamical decoupling schemes. Similar to clock states, which possess a transition that is insensitive to firstorder magnetic shifts for a given magnetic field value, the offresonant driving fields generate a transition that is insensitive to firstorder shifts in the drivefield amplitudes. We demonstrate this scheme by utilising the ground state spin level of the nitrogenvacancy centre (NV) in diamond. The states are addressed by a combination of four microwave fields, adjusted to the same Rabi frequency, Ω, at different transition frequencies, ω_{1} and ω_{2}, and with a detuning, Δ, respectively. All experiments presented here were realized using Qudi Software Suite^{38}. The spin states of the NV centre are initialised and read out by a 532 nm laser identifying spin dependent fluorescence^{39,40,41,42}. We are operating in the vicinity of the excited state level anticrossing^{43,44} at a magnetic field of 35.8 mT. At this bias field the intrinsic nitrogen nuclear spin becomes polarised through optical pumping and does not contribute to the level structure (more information concerning the setup configuration and the system parameters can be found in Supplementary Information S1 and S2). Our implementation demonstrates that, to first order, drivenoise is eliminated, and compared to the pure dephasing time an improvement of 2 orders of magnitude in the coherence time is obtained even for a moderate drive field strength. Finally, we present an improved version of the scheme, where the clock transition adjustment is extended to also eliminate the secondorder effect of the environmental noise. Supported by simulations, our analysis shows that with a moderate driving field strength a further improvement of more than 1 order of magnitude in the coherence time can be obtained. Hence, our scheme allows prolonging the coherence time towards the lifetime limit in a simple experimental setup and without requiring exceptionally strong drive fields. The protocol is applicable to both the optical and microwave domain, and hence to a variety of atomic and solid state systems, such as trapped ions, rareearth ions, and defect centres.
The Basic Scheme
We consider a threelevel system with states 0〉 and ±1〉, where the ±1〉 states are dipole coupled to the 0〉 state, as illustrated in Fig. 1a. The energy gaps (\(\hslash =1\)) between the 0〉 state and the ±1〉 states are ω_{1} and ω_{2}, respectively. Our scheme utilises four driving fields which can be expressed by the driving Hamiltonian
Moving to the interaction picture (IP) with respect to H_{01} = ω_{1}−1〉〈−1 + ω_{2}1〉〈1, changing the basis to {0〉, B〉, D〉}, with \(B\rangle =(1\rangle +1\rangle )/\sqrt{2}\) and \(D\rangle =(1\rangle 1\rangle )/\sqrt{2}\), and applying the rotatingwave approximation, we obtain the Hamiltonian
which is illustrated by a level scheme depicted in Fig. 2a. The states 0〉 and D〉 are onresonantly coupled by a single lambda drive with a strength \({{\rm{\Omega }}}_{D}=\sqrt{2}{{\rm{\Omega }}}_{1}\). Under the assumption of Δ > Ω_{2}, an offresonant coupling between 0〉 and B〉 by \({{\rm{\Omega }}}_{B}=\sqrt{2}{{\rm{\Omega }}}_{2}\) is obtained.
The drive, Ω_{D}, in Fig. 2a transforms into the dressed states {u〉, B〉, d〉}, as schematically illustrated in Fig. 2b, with the corresponding eigenvalues {+Ω_{D}, 0, −Ω_{D}}, respectively. Here, we introduced the states \(u\rangle =(0\rangle +D\rangle )/\sqrt{2}\) and \(d\rangle =(0\rangle D\rangle )/\sqrt{2}\).
For a strong enough drive, Ω_{1}, robustness to magnetic noise is obtained. As the magnetic noise, δB, couples between the B〉 state and the u〉 and d〉 states, we can set Ω_{D} such that the power spectrum of the noise, S_{BB}(Ω_{D}), is much smaller than 1/T_{1}, where T_{1} is the system lifetime. This condition ensures that the first order effect of the magnetic noise is negligible.
The coherence time of the dressed states is then mainly limited by driving amplitude fluctuations, Ω → Ω(1 + δ(t)), where δ(t) represents a random noise contribution. To additionally obtain robustness to drivefield fluctuations, we consider the effect of the second detuned drive, Ω_{B}, on the dressed states. We therefore move to the basis of the dressed states and to the IP with respect to H_{02} = ΔB〉〈B, and obtain
The eigenstates of H_{II}, denoted by \(\{\tilde{u}\rangle ,\tilde{B}\rangle ,\tilde{d}\rangle \}\), are termed as the doublydressed states and their relative level scheme is illustrated in Fig. 2c.
The effect of drive fluctuations can be introduced in H_{II} by replacing Ω_{i} with Ω_{i}(1 + δ_{i}). As both driving fields originate from the same source, we can assume that the noise is mostly correlated. Thus, for each set of eigenstates, k〉 and j〉, we define the driving coherence time as \({T}_{2}^{{{\rm{\Omega }}}_{k,j}}=\sqrt{2}/{e}_{\delta }^{k}{e}_{\delta }^{j}\), where \({e}_{\delta }^{k}\) is the first order term in δ_{1} and δ_{2} of the eigenvalue expansion of k〉. For a given driving noise configuration, i.e., for a given relation between δ_{1} and δ_{2}, the driving parameters Ω_{1}, Ω_{2}, and Δ can be chosen such that the driving coherence time of the two negative eigenvalues of H_{II} is \({T}_{2}^{{{\rm{\Omega }}}_{\tilde{B}\rangle ,\tilde{d}\rangle }}\gg {T}_{1}\). In this case \({e}_{\delta }^{\tilde{B}}\approx {e}_{\delta }^{\tilde{d}}\) (see Supplementary Mathematica notebook), which means that the transition frequency of the robust qubit is insensitive to firstorder driving fluctuations. Hence, with moderate driving fields, the coherence time of this doublydressed qubit is mainly limited by the secondorder effect of the magnetic noise \(\sim \delta {B}^{2}/{{\rm{\Omega }}}_{D}\). Increasing the strength of the driving fields reduces the secondorder effect of the magnetic noise but introduces an increased secondorder effect of the drive noise, \(\sim \delta {{\rm{\Omega }}}_{D}^{2}\). For more details and the explicit forms of the leading terms of the magnetic and Rabi frequency noise see the Supplementary Mathematica notebook.
We consider a NV centre electron spin with a pure dephasing time of \({T}_{2}^{\ast }\approx 2\) μs resulting from magnetic noise described by an OrnsteinUhlenbeck random process^{45,46} with a correlation time of τ_{c} ≈ 15 μs. In this case, with Ω_{1} = Ω_{2} = 2π · 6 MHz, a coherence time of ≈1 ms is obtained (see Fig. 3), which is limited by the secondorder effect of both magnetic and drive noise. Further increase of the drive strength, Ω, would allow for a further improvement in the coherence time up to the point where the secondorder drive noise is too strong and begins to dominate, which in our case is at Ω_{1} = Ω_{2} ≈ 2π · 10 MHz (see inset of Fig. 3).
Experimental Results
The experimental implementation of the proposed scheme follows the protocol illustrated in Fig. 1b. Four drive fields are applied on the bare basis states {−1〉, 0〉, 1〉} of the NV centre (see Fig. 1a). The field amplitudes of the onresonant drives are adjusted to yield an identical Rabi frequency, Ω, for both transitions, ω_{1} and ω_{2} (see Suppl. B). The offresonant drives are obtained by adding a detuning, Δ, to the resonant drives, resulting in the total field
synthesised by a signal generator (Ω_{1} = Ω_{2} = Ω = 2π · 2.27 MHz) and thereby implementing Eq. (1).
The interaction with the field couples the drive, Ω, to the bare spin states (see Fig. 1a). Onresonant drives induce Rabi oscillations at a rate Ω_{D}, and a positive (negative) detuning results in ‘red’ (‘blue’) detuned ACStark shifted energy levels of the d〉 and B〉 (u〉 and B〉) states, as shown in Fig. 2b. In combination, these drives create the doublydressed states, which are depicted in Fig. 2c. The appearing energy levels of the doublydressed states, \(\tilde{u}\rangle ,\tilde{d}\rangle \), and \(\tilde{B}\rangle \), are eventually all coupled to the drive fields (see Fig. 2).
By adjusting the detuning, Δ, a configuration can be obtained, in which two states (either \(\tilde{d}\rangle \leftrightarrow \tilde{B}\rangle \) or \(\tilde{u}\rangle \leftrightarrow \tilde{B}\rangle \)) experience the same drive noise, δΩ. This eliminates the energy gap fluctuations (due to δΩ) between the two considered states and reflects the robustness of the qubit against drive strength fluctuations, δΩ. The large energy gap in the dressed states, Ω_{D}, which originates from the onresonant drives, ensures a sufficient decoupling from external magnetic noise contributions, δB, as it also increases the energy gap of the robust qubit, Ω_{RQ} (see Fig. 2c).
To determine the performance of the scheme the detuningdependent coherence times of the protected states have to be recorded, yielding the optimal ACStark shifted energy levels that are least sensitive to drive fluctuations. The measurement is performed in the dressedstate basis and is analogous to a free induction decay (FID) or Ramsey measurement (see Fig. 1b). Here, by the application of an onresonant π/2 pulse with Ω_{D}, a superposition is created between the B〉 and 0〉 states (which is also a superposition of the D〉 and 0〉 states, but with a different initial phase factor in the −1〉 state). In the next step, the double lambda drive (depicted in Fig. 1a) is applied on the superposition states as a function of interaction time τ, revealing the present energy gaps (in Fig. 2c) as a coherent evolution. By mapping the coherences to populations with a consecutive π/2 pulse (with Ω_{D}), spinstate dependent fluorescence is observed upon the application of a laser pulse. Finally, the recorded readout signal, S(τ), contains frequency components proportional to the energy gaps of the doublydressed states (between the states in Fig. 2c). The robust state is identified by the longest measured coherence time in Fig. 4 as energy fluctuations in the robust state are significantly suppressed while the oscillations induced by the other energy gaps decay quickly.
The measured coherence times, \({T}_{2}^{{{\rm{\Omega }}}_{\tilde{B}\rangle ,\tilde{d}\rangle }}\), are extracted by fitting a sinusoidal exponential decay to S(τ) and plotted as a function of detuning, Δ, in Fig. 4 (for a detailed explanation of the procedure see Supplementary Information S3). The asymmetric shape of the curve provides insights about the appearing dynamics. Starting from the limit of a very large detuning, Δ, only the dressed states u〉, d〉 and B〉 are present in the scheme and the ACStark shifts are negligible (see Fig. 2b). Thus, the coherence time of the robust qubit is then predominantly determined by the noise, δΩ, as δB has a smaller impact at this drive field strength, Ω (see Suppl. S2B). Decreasing the detuning, Δ, introduces Stark shifts on the u〉, d〉 and B〉 states, which effectively reduces the fluctuations of the doublydressed state energy gap, thereby prolonging the coherence time of the robust qubit. Hence, the height of the peak is now mainly limited by secondorder noise contributions from δB (see the obtained peak in Fig. 4). Further decrease of the detuning starts to drive 0〉 ↔ B〉 (see Fig. 2a) and introduces thereby again more drive noise, which impacts strongly on the doublydressed states. Approaching zero detuning will eventually eliminate the doublydressed states, \(\tilde{u}\rangle \), \(\tilde{d}\rangle \) and \(\tilde{B}\rangle \), and at Δ = 0 only the twolevel system transition, ω_{1}, remains to be addressed by 2Ω (see Eq. (4) and Fig. 1). The same behaviour as described above is expected for a negative detuning, Δ.
The theoretical model describing the coherence time as a function of detuning allows us to introduce three parameters in order to mimic the experimental situation. These are the drive noise, δΩ_{B} and δΩ_{D}, on the dressed states (see Fig. 1b) and the magnetic noise, δB. It is important to note that these noise parameters do not change the asymmetric shape of the curve as the shape is fully determined by the model. However, an imbalance between the drive noise strengths impacts the optimal detuning, whereas the magnetic noise solely impacts the attainable coherence time.
The measured free induction decay time, \({T}_{2}^{\ast }\mathrm{=(1.78}\pm \mathrm{24)}\) μs, of the bare states sets the theoretical limited for the coherence time of ~190 μs when the drive noise is eliminated. By selecting a four times higher noise, \(\delta {{\rm{\Omega }}}_{B}\approx 4\,\delta {{\rm{\Omega }}}_{D}\), the theoretical dependence of the coherence time on the detuning plotted in Fig. 4 is obtained. The analytical function of the coherence time is given by \({T}_{2}^{{{\rm{\Omega }}}_{\tilde{B}\rangle ,\tilde{d}\rangle }}=\sqrt{2}/((\sqrt{2}\mathrm{/190)}+{e}_{\delta }^{\tilde{B}}{e}_{\delta }^{\tilde{d}})\) μs. In addition, simulations with magnetic and driving noise models were preformed for several detuning values which reproduce the experimental results very well. These simulation results are presented in Fig. 4 (for more details on the simulations see Supplementary Information S6). In the following, we clarify how quantitatively the noise parameters are grasped in the experiment.
It appears that the DC and AC components of the magnetic noise, δB, have equal contributions to both transitions, ω_{1} and ω_{2}, as we obtain (within the error bar) the same values for \({T}_{2}^{\ast }\) and T_{2} = (215 ± 31) μs (see Supplementary Information S2C and D). However, by comparing the coherence time of the Rabi drives, we obtain a drastic difference, \({T}_{2}^{{\rm{\Omega }}({\omega }_{1})}=\mathrm{(62}\pm \mathrm{12)}\) μs and \({T}_{2}^{{\rm{\Omega }}({\omega }_{2})}=\mathrm{(159}\pm \mathrm{24)}\) μs, which hints at a drive frequency dependent noise spectrum. As all the fields are produced by the same signal generator, it is valid to consider correlations in the drive noise. The combination of both effects can truly cause the noise imbalance between δΩ_{B} and δΩ_{D}, which directly affects the position of the peak with respect to the detuning, Δ. As this is a setup specific setting, the ACStark shifts have to be adjusted to compensate for this value. However, the coherence time improving effect, as theoretically predicted, is expected to be within the range of 50 MHz at the utilised drive field, Ω, as larger detunings have a negligible energy shift on the states. A further and more detailed investigation of the drive noise is presented in Supplementary Informations S4 and S5.
Improved Scheme
So far, our scheme shows how to eliminate the firstorder effect of the drive fluctuations, δΩ, where for moderate drive fields the coherence time is mainly limited by the secondorder effect of the magnetic noise ~δB^{2}/Ω. However, the secondorder effect of the magnetic noise can be suppressed in a similar way as demonstrated for the elimination of the firstorder drive fluctuations, δΩ.
To see this, we consider the onresonant drive (Ω_{1} in Fig. 2). For the dressed states, the secondorder effect of the magnetic noise is given by \(\sim \delta {B}^{2}/{{\rm{\Omega }}}_{1}(u\rangle \langle ud\rangle \langle d)\), which describes the fluctuation of the robust energy gap (between B〉 and d〉) with \( \sim \delta {B}^{2}/{{\rm{\Omega }}}_{1}\). By introducing a onephoton detuning, Δ_{0}, which denotes a detuning of the coupling between D〉 and 0〉, the symmetry is broken. In this case the secondorder effect of the magnetic noise is given by ~δB^{2}/Ω_{1}(au〉〈u − bd〉〈d − cB〉〈B). By adjusting the onephoton detuning, Δ_{0}, we can set b = c, and achieve a clock transition that is insensitive to magnetic field fluctuations, δB (up to secondorder).
We now combine this idea with the presented elimination of the firstorder drive fluctuations, δΩ, to obtain a true clock transition. Including the onephoton detuning, Δ_{0}, in the driving fields of both Λ systems and magnetic noise, which is given by δBS_{z} = δB(+1〉〈+1 − −1〉〈−1), Eq. (3) results in
and the IP is now obtained with respect to \({H^{\prime} }_{02}={\rm{\Delta }}B\rangle \langle B+{{\rm{\Delta }}}_{0}0\rangle \langle 0\). We continue by moving to the basis of the eigenstates of the drives (the double dressed states)
where E_{i} are the eigenvalues, and α and β are real coefficients. The drive noise is treated as before, where we require \({e}_{\delta }^{\tilde{B}}\approx {e}_{\delta }^{\tilde{d}}\). This gives us one constraint on Δ and Δ_{0}. The second constraint comes from the elimination of the secondorder effect of the magnetic noise. Moving to the IP with respect to \({H^{\prime} }_{03}=\,{\rm{\Delta }}\tilde{B}\rangle \langle \tilde{B}\) we obtain the time independent Hamiltonian
This enables the calculation of the secondorder contribution of the magnetic noise to the eigenvalues, which are the b and c coefficients, as a function of Δ and Δ_{0}. The two constraints, \({e}_{\delta }^{\tilde{B}}\approx {e}_{\delta }^{\tilde{d}}\) and b ≈ c, allow us to determine the optimal values of Δ and Δ_{0}. For more details and the explicit forms of the leading terms of the magnetic and Rabi frequency noise, see Supplementary Mathematica notebook.
For the considered drive noise, δ_{1} = δ_{2} = 0.005, in Fig. 3, and for a moderate drive of Ω_{1} = Ω_{2} = 2π · 2 MHz, the optimal detunings would be obtained by Δ = 2π · 8.9956 MHz and Δ_{0} = 2π · 1.7386 MHz. We simulated the present driving configuration under the effect of the same magnetic noise model considered for the simulations of the experiment in Fig. 4 (with \({T}_{2}^{\ast }\approx 2\) μs). The results of the simulation are shown in the inset of Fig. 5 indicating an improvement of more than 1 order of magnitude in the coherence time compared to the original scheme. There are two limiting factors on the coherence time. The first factor are the higherorder terms of the noise; the secondorder term of the driving noise and the fourth order term of the magnetic noise. With the parameters considered in the simulation these terms result in a limit of ~10 ms on the coherence time. The second factor is the amplitude mixing between the eigenstates due to fast rotating terms, which introduces first order driving noise. In our case the mixing is ~0.2%, which means that the coherence time is limited to 500 times the driving noise limited coherence time(~20 μs here), and hence the limit on the coherence time is ~10 ms. Therefore, taking both factors into account we conclude that the coherence time is limited to ~5.3 ms, which is in agreement with the simulation results. For low drive fields, the drive noise contributions (secondorder terms and amplitude mixing) is small and the coherence time is mainly limited by the fourthorder magnetic noise. For higher drive fields the fourth order magnetic noise becomes negligible and the coherence time is mainly limited by the drive noise contributions. The estimated maximal coherence time for different drive field strengths is shown in Fig. 5. The amplitude mixing can also be decreased by increasing the Zeeman splitting. Given the noise parameters, one can optimise the driving parameters with respect to these factors and obtain the optimal coherence time.
Conclusion
In this work we presented and experimentally demonstrated a new scheme for the creation of a robust qubit in a threelevel system by means of a clock transition adjustment. The basic scheme is based on the application of continuous resonant and offresonant drive fields. The resonant drive fields provide robustness to environmental noise, whereas the offresonant drive fields eliminate the firstorder effect of the drive noise by tuning a clock like transition that is insensitive to firstorder shifts of the drivefield amplitudes. For the case of the NV centre in diamond, we achieved an improvement of ~2 orders of magnitude in the coherence time compared to the pure dephasing time while utilising moderate drive fields. In the optimal version of the scheme, the clock transition adjustment is extended to also eliminate the secondorder effect of the environmental noise without necessitating strong drive fields. Hence, our scheme proposes the possibility to prolong the coherence time towards the lifetime limit using a relatively simple experimental setup and without requiring extremely strong drive fields. However, further measurements are encouraged to verify the performance of the improved scheme.
This scheme facilitates the sensing of AC magnetic fields, and in particular, high frequency fields in the GHz regime, where the sensitivity would be solely limited by the coherence time of the robust qubit. While this work has focused solely on the NV centre, we believe that this scheme is applicable to a variety of atomic and solidstate systems with optical or microwave transitions, such as trapped ions, rareearth ions, and other defect centres. We therefore believe that the scheme has potential applications in a wide range of tasks in the fields of quantum information science and technology, and in particular quantum sensing.
Data Availability
The authors declare that all relevant data supporting the findings of this study are available within the paper (and its Supplementary information file). Any raw data can be obtained from the corresponding authors on reasonable request.
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Acknowledgements
We would like to thank Christian Osterkamp and Kristian Hagsted Rasmussen for the sample preparation. A.S., A.H., H.A.R.E.E. and U.L.A. acknowledge funding from the Innovation Foundation Denmark through the project EXMAD and the Qubiz centre, the Danish National Research Foundation, and the Danish Research Council via the Sapere Aude project (DIMS). A.R. acknowledges the support of the Israel Science Foundation (grant no. 1500/13) and ERC grant QRES under grant agreement no. 770929.
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N.A. and A.R. conceived the idea and developed the theory. A.H., A.R., F.J. and U.L.A. designed and supervised the project. A.S. performed, planned and developed the concept of the experiment with support from H.A.R.E.E. and A.H. A.S. and N.A. analysed the data. N.A. planned and carried out the simulations. A.S. and N.A. took the lead in writing the manuscript with support from A.H.. All authors contributed to the interpretation of the results, provided critical feedback and helped to shape the research, analysis and manuscript.
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Stark, A., Aharon, N., Huck, A. et al. Clock transition by continuous dynamical decoupling of a threelevel system. Sci Rep 8, 14807 (2018). https://doi.org/10.1038/s41598018319844
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Keywords
 Clock Transition
 Threelevel System
 Driven Field
 Driver Noise
 Coherence Time
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