# Enhancing the Robustness of Dynamical Decoupling Sequences with Correlated Random Phases

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. The Effect of Controlled Imperfection

#### 2.1.1. Standard Protocol

#### 2.1.2. Randomisation Protocol

#### 2.1.3. Correlated Randomization Protocol

#### 2.2. Comparison of Different Protocol Performances

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NV | nitrogen-vacancy |

DD | dynamical decoupling |

CP | Carr–Purcell |

## References

- Viola, L.; Lloyd, S. Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A
**1998**, 58, 2733. [Google Scholar] [CrossRef] [Green Version] - Yang, W.; Wang, Z.-Y.; Liu, R.-B. Preserving qubit coherence by dynamical decoupling. Front. Phys.
**2010**, 6, 2–14. [Google Scholar] [CrossRef] [Green Version] - Suter, D.; Álvarez, G.A. Colloquium: Protecting quantum information against environmental noise. Rev. Mod. Phys.
**2016**, 88, 041001. [Google Scholar] [CrossRef] - Cai, J.M.; Retzker, A.; Jelezko, F.; Plenio, M.B. A large-scale quantum simulator on a diamond surface at room temperature. Nat. Phys.
**2013**, 9, 168–173. [Google Scholar] [CrossRef] [Green Version] - Rondin, L.; Tetienne, J.P.; Hingant, T.; Roch, J.F.; Maletinsky, P.; Jacques, V. Magnetometry with nitrogen-vacancy defects in diamond. Rep. Prog. Phys.
**2014**, 77, 056503. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wu, Y.; Jelezko, F.; Plenio, M.B.; Weil, T. Diamond Quantum Devices in Biology. Angew. Chem. Int. Ed.
**2016**, 55, 6586–6598. [Google Scholar] [CrossRef] [PubMed] - Suter, D.; Jelezko, F. Single-spin magnetic resonance in the nitrogen-vacancy center of diamond. Prog. Nucl. Magn. Reson. Spectrosc.
**2017**, 98–99, 50–62. [Google Scholar] [CrossRef] [Green Version] - Degen, C.L.; Reinhard, F.; Cappellaro, P. Quantum sensing. Rev. Mod. Phys.
**2017**, 89, 035002. [Google Scholar] [CrossRef] [Green Version] - Doherty, M.W.; Manson, N.B.; Delaney, P.; Jelezko, F.; Wrachtrup, J.; Hollenberg, L.C.L. The nitrogen-vacancy colour centre in diamond. Phys. Rep.
**2013**, 528, 1–45. [Google Scholar] [CrossRef] [Green Version] - Ryan, C.A.; Hodges, J.S.; Cory, D.G. Robust Decoupling Techniques to Extend Quantum Coherence in Diamond. Phys. Rev. Lett.
**2010**, 105, 200402. [Google Scholar] [CrossRef] [Green Version] - de Lange, G.; Wang, Z.; Riste, D.; Dobrovitski, V.; Hanson, R. Universal Dynamical Decoupling of a Single Solid-State Spin from a Spin Bath. Science
**2010**, 330, 60–63. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bar-Gill, N.; Pham, L.M.; Jarmola, A.; Budker, D.; Walsworth, R.L. Solid-state electronic spin coherence time approaching one second. Nat. Commun.
**2013**, 4, 1743. [Google Scholar] [CrossRef] [PubMed] - Abobeih, M.H.; Cramer, J.; Bakker, M.A.; Kalb, N.; Markham, M.; Twitchen, D.J.; Taminiau, T.H. One-second coherence for a single electron spin coupled to a multi-qubit nuclear-spin environment. Nat. Commun.
**2018**, 9, 2552. [Google Scholar] [CrossRef] [PubMed] [Green Version] - de Lange, G.; Riste, D.; Dobrovitski, V.V.; Hanson, R. Single-Spin Magnetometry with Multipulse Sensing Sequences. Phys. Rev. Lett.
**2011**, 106, 080802. [Google Scholar] [CrossRef] [Green Version] - Taminiau, T.H.; Wagenaar, J.J.T.; van der Sar, T.; Jelezko, F.; Dobrovitski, V.V.; Hanson, R. Detection and Control of Individual Nuclear Spins Using a Weakly Coupled Electron Spin. Phys. Rev. Lett.
**2012**, 109, 137602. [Google Scholar] [CrossRef] - Kolkowitz, S.; Unterreithmeier, Q.P.; Bennett, S.D.; Lukin, M.D. Sensing Distant Nuclear Spins with a Single Electron Spin. Phys. Rev. Lett.
**2012**, 109, 137601. [Google Scholar] [CrossRef] - Zhao, N.; Honert, J.; Schmid, B.; Klas, M.; Isoya, J.; Markham, M.; Twitchen, D.; Jelezko, F.; Liu, R.B.; Fedder, H.; et al. Sensing single remote nuclear spins. Nat. Nanotechnol.
**2012**, 7, 657. [Google Scholar] [CrossRef] [Green Version] - Müller, C.; Kong, X.; Cai, J.-M.; Melentijevic, K.; Stacey, A.; Markham, M.; Isoya, J.; Pezzagna, S.; Meijer, J.; Du, J.; et al. Nuclear magnetic resonance spectroscopy with single spin sensitivity. Nat. Commun.
**2014**, 5, 4703. [Google Scholar] [CrossRef] [Green Version] - Casanova, J.; Wang, Z.-Y.; Plenio, M.B. Noise-Resilient Quantum Computing with a Nitrogen-Vacancy Center and Nuclear Spins. Phys. Rev. Lett.
**2016**, 117, 130502. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.-Y.; Casanova, J.; Plenio, M.B. Delayed entanglement echo for individual control of a large number of nuclear spins. Nat. Commun.
**2017**, 8, 14660. [Google Scholar] [CrossRef] [Green Version] - Haase, J.F.; Wang, Z.-Y.; Casanova, J.; Plenio, M.B. Soft Quantum Control for Highly Selective Interactions among Joint Quantum Systems. Phys. Rev. Lett.
**2018**, 121, 050402. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lang, J.E.; Broadway, D.A.; Hall, G.A.L.W.L.T.; Stacey, A.; Hollenberg, L.L.; Monteiro, T.S.; Tetienne, J.-P. Quantum Bath Control with Nuclear Spin State Selectivity via Pulse-Adjusted Dynamical Decoupling. Phys. Rev. Lett.
**2019**, 123, 210401. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bradley, C.E.; Randall, J.; Abobeih, M.H.; Berrevoets, R.C.; Degen, M.J.; Bakker, M.A.; Markham, M.; Twitchen, D.J.; Taminiau, T.H. A ten-qubit solid-state spin register with quantum memory up to one minute. Phys. Rev. X
**2019**, 9, 031045. [Google Scholar] [CrossRef] [Green Version] - Zhao, N.; Hu, J.-L.; Ho, S.-W.; Wan, J.T.K.; Liu, R.B. Atomic-scale magnetometry of distant nuclear spin clusters via nitrogen-vacancy spin in diamond. Nat. Nanotechnol.
**2011**, 6, 242. [Google Scholar] [CrossRef] - Shi, F.; Kong, X.; P, P.W.; Kong, F.; Zhao, N.; Liu, R.B.; Du, J. Sensing and atomic-scale structure analysis of single nuclear-spin clusters in diamond. Nat. Phys.
**2014**, 10, 21–25. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.-Y.; Haase, J.F.; Casanova, J.; Plenio, M.B. Positioning nuclear spins in interacting clusters for quantum technologies and bioimaging. Phys. Rev. B
**2016**, 93, 174104. [Google Scholar] [CrossRef] [Green Version] - Abobeih, M.H.; Randall, J.; Bradley, C.E.; Bartling, H.P.; Bakker, M.A.; Degen, M.J.; Markham, M.; Twitchen, D.J.; Taminiau, T.H. Atomic-scale imaging of a 27-nuclear-spin cluster using a quantum sensor. Nature
**2019**, 576, 411–415. [Google Scholar] [CrossRef] [Green Version] - Gullion, T.; Barker, D.B.; Conradi, M.S. New, compensated Carr-Purcell sequences. J. Magn. Reson.
**1990**, 89, 479–484. [Google Scholar] [CrossRef] - Casanova, J.; Wang, Z.-Y.; Haase, J.F.; Plenio, M.B. Robust dynamical decoupling sequences forindividual-nuclear-spin addressing. Phys. Rev. A
**2015**, 92, 042304. [Google Scholar] [CrossRef] [Green Version] - Genov, G.T.; Schraft, D.; Vitanov, N.V.; Halfmann, T. Arbitrarily Accurate Pulse Sequences for Robust Dynamical Decoupling. Phys. Rev. Lett.
**2017**, 118, 133202. [Google Scholar] [CrossRef] [Green Version] - Loretz, M.; Boss, J.M.; Rosskopf, T.; Mamin, H.J.; Rugar, D.; Degen, C.L. Spurious Harmonic Response of Multipulse Quantum Sensing Sequences. Phys. Rev. X
**2015**, 5, 021009. [Google Scholar] [CrossRef] - Haase, J.F.; Wang, Z.-Y.; Casanova, J.; Plenio, M.B. Pulse-phase control for spectral disambiguation in quantum sensing protocols. Phys. Rev. A
**2016**, 94, 032322. [Google Scholar] [CrossRef] [Green Version] - Lang, J.E.; Casanova, J.; Wang, Z.-Y.; Plenio, M.B.; Monteiro, T.S. Enhanced Resolution in Nanoscale NMR via Quantum Sensing with Pulses of Finite Duration. Phys. Rev. Appl.
**2017**, 7, 054009. [Google Scholar] [CrossRef] [Green Version] - Shu, Z.; Zhang, Z.; Cao, Q.; Yang, P.; Plenio, M.B.; Müller, C.; Lang, J.; Tomek, N.; Naydenov, B.; McGuinness, L.P.; et al. Unambiguous nuclear spin detection using an engineered quantum sensing sequence. Phys. Rev. A
**2017**, 96, 051402. [Google Scholar] [CrossRef] [Green Version] - Wang, Z.-Y.; Lang, J.E.; Schmitt, S.; Lang, J.; Casanova, J.; McGuinness, L.; Monteiro, T.S.; Jelezko, F.; Plenio, M.B. Randomization of Pulse Phases for Unambiguous and Robust Quantum Sensing. Phys. Rev. Lett.
**2019**, 122, 200403. [Google Scholar] [CrossRef] - Cai, J.M.; Naydenov, B.; Pfeiffer, R.; McGuinness, L.P.; Jahnke, K.D.; Jelezko, F.; Plenio, M.B.; Retzker, A. Robust dynamical decoupling with concatenated continuous driving. New J. Phys.
**2012**, 14, 113023. [Google Scholar] [CrossRef] [Green Version] - Souza, A.M.; Álvarez, G.A.; Suter, D. Robust dynamical coupling. Philos. Trans. R. Soc. A
**2012**, 370, 4748. [Google Scholar] [CrossRef] - Carr, H.Y.; Purcell, E.M. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys. Rev.
**1954**, 94, 630. [Google Scholar] [CrossRef]

**Figure 1.**Repetition of a basic dynamical decoupling (DD) pulse unit. (

**a**) A basic unit of DD pulse sequence, which is defined by the positions and phases of the $\pi $ pulses. The lower panel is the example of an XY8 sequence. (

**b**) The standard protocol to construct a longer DD sequence is to repeat the same basic pulse unit illustrated in (

**a**) M times. Because the error contribution has the same phase factor for each DD pulse unit, the errors coherently add up (see the lower panel for the case of $M=6$). (

**c**) The randomization protocol shifts all the pulses within each unit by a common, independent, random phase ${\Phi}_{r,m}$. Because of the random phases, the error terms of the basic DD pulse unit add up incoherently, suppressing the growing of error contribution (see the lower panel for an example). (

**d**) The correlated randomization protocol imposes constraint on the phases on the random phases ${\Phi}_{r,m}$, such that the sum of their random phase factors vanishes. The lower panel illustrates an example in which the sum of random phase factors of every three successive DD pulse units is zero.

**Figure 2.**Robustness of the protocols for a small M. The fidelity (the value of ${P}_{\psi}$ in the absence of an external signal) of Carr-Purcell sequences as a function of detuning and amplitude (Rabi frequency) errors for standard protocol (

**a**), randomization protocol in [35] (

**b**), and correlated randomization protocol with the elimination size $G=2$ (

**c**). (

**d**) difference between the fidelity of the correlated protocol in (

**c**) and the uncorrelated protocol in (

**b**). (

**e**), as (

**d**) but for the elimination size $G=3$ for the correlated randomization protocol. (

**f**–

**j**) (

**k**–

**o**), as (

**a**–

**e**), but for the XY8 [YY8 [34] sequences. In all figures, the regions in white have values out of the ranges shown in the color bars. All the sequences consists of 48 $\pi $ pulses (that is, $M=6$ for XY8 and YY8 sequences). The time duration of each $\pi $ pulse is 15 ns, and the inter pulse spacing is 200 ns. All the plots for the protocols with random phases are average results of 100 random sequences.

**Figure 3.**Robustness of the XY8 and YY8 [34] protocols for $M=24$. (

**a**–

**e**) Results for XY8 sequences. (

**a**–

**c**) show the fidelity of XY8 sequences as a function of detuning and amplitude (Rabi frequency) errors for standard protocol (

**a**), randomization protocol [35], and correlated randomization protocol (

**c**), respectively. (

**d**,

**e**) are the fidelity enhancement over the randomization protocol by using the correlated randomization protocol for elimination sizes $G=2$ and $G=3$, respectively. (

**f**–

**j**), as (

**a**–

**e**), but for YY8 sequences. The control parameters are the same as those used in Figure 2 but with a larger $M=24$. In all figures, the regions in white have values out of the ranges shown in the color bars. All the plots for the protocols with random phases are average results of 100 random sequences.

**Figure 4.**Quantum spectroscopy with DD. (

**a**) Simulated survival probability ${P}_{\psi}$ (red solid line) as a function of the DD frequency [$1/\left(2\tau \right)$ for pulse spacing $\tau $] for the standard XY8 protocol with a total number of 200 $\pi $ pulses. The DD $\pi $ pulses have a non-zero time duration of 100 ns. The frequency detuning and amplitude errors of the $\pi $ pulses have a static value of 10% of the ideal Rabi frequency. The ${}^{1}\mathrm{H}$ spin to be sensed is coupled to the NV center via the hyperfine-field components [29] $({A}_{\perp},{A}_{\Vert})=2\pi \times (2,4)$ kHz. A ${}^{13}\mathrm{C}$ spin representing a noise source is coupled to the NV center via the hyperfine-field components $({A}_{\perp},{A}_{\Vert})=2\pi \times (10,200)$ kHz. The presence of ${}^{13}\mathrm{C}$ spin and imperfect control perturbs sensing signal and generates a spurious peak around 1740 kHz (compare it with the green dash-dotted line obtained by an ideal error-free DD sequence). (

**b**) The use of randomization protocol suppresses the effect of pulse imperfection. The effect of errors is further suppressed in (

**c**,

**d**) by the use of correlated random phases elimination sizes $G=2$ and $G=3$, respectively. A magnetic field 400 G is applied along the symmetry axis of the NV center in the simulation. The hyperfine fields used for the H and C atoms correspond to relative separation of 3.1 and 0.52 nm respectively away from the NV center in our simulation. All the plots for the protocols with random phases are average results of 100 random sequences.

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**MDPI and ACS Style**

Wang, Z.; Casanova, J.; Plenio, M.B.
Enhancing the Robustness of Dynamical Decoupling Sequences with Correlated Random Phases. *Symmetry* **2020**, *12*, 730.
https://doi.org/10.3390/sym12050730

**AMA Style**

Wang Z, Casanova J, Plenio MB.
Enhancing the Robustness of Dynamical Decoupling Sequences with Correlated Random Phases. *Symmetry*. 2020; 12(5):730.
https://doi.org/10.3390/sym12050730

**Chicago/Turabian Style**

Wang, Zhenyu, Jorge Casanova, and Martin B. Plenio.
2020. "Enhancing the Robustness of Dynamical Decoupling Sequences with Correlated Random Phases" *Symmetry* 12, no. 5: 730.
https://doi.org/10.3390/sym12050730