Coherence protection and decay mechanism in qubit ensembles under concatenated continuous driving
CCoherence protection and decay mechanism in qubitensembles under concatenated continuous driving
Guoqing Wang ( 王王王 国国国 庆庆庆 ) Research Laboratory of Electronics and Department of Nuclear Science andEngineering, Massachusetts Institute of Technology, Cambridge, MA 02139
Yi-Xiang Liu ( 刘刘刘 仪仪仪 襄襄襄 ) Research Laboratory of Electronics and Department of Nuclear Science andEngineering, Massachusetts Institute of Technology, Cambridge, MA 02139
Paola Cappellaro
Research Laboratory of Electronics and Department of Nuclear Science andEngineering, Massachusetts Institute of Technology, Cambridge, MA 02139Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139E-mail: [email protected]
Abstract.
Dense ensembles of spin qubits are valuable for quantum applications, even thoughtheir coherence protection remains challenging. Continuous dynamical decouplingcan protect ensemble qubits from noise while allowing gate operations, but it ishindered by the additional noise introduced by the driving. Concatenated continuousdriving (CCD) techniques can, in principle, mitigate this problem. Here we providedeeper insights into the dynamics under CCD, based on Floquet theory, that leadto optimized state protection by adjusting driving parameters in the CCD schemeto induce mode evolution control. We experimentally demonstrate the improvedcontrol by simultaneously addressing a dense Nitrogen-vacancy (NV) ensemble with10 spins. We achieve an experimental 15-fold improvement in coherence time foran arbitrary, unknown state, and a 500-fold improvement for an arbitrary, knownstate, corresponding to driving the sidebands and the center band of the resultingMollow triplet, respectively. We can achieve such coherence time gains by optimizingthe driving parameters to take into account the noise affecting our system. Byextending the generalized Bloch equation approach to the CCD scenario, we identifythe noise sources that dominate the decay mechanisms in NV ensembles, confirm ourmodel by experimental results, and identify the driving strengths yielding optimalcoherence. Our results can be directly used to optimize qubit coherence protectionunder continuous driving and bath driving, and enable applications in robust pulsedesign and quantum sensing. a r X i v : . [ qu a n t - ph ] A ug oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving
1. Introduction
Scaling up the size of quantum systems is desirable in many quantum technologies,ranging from quantum simulators to quantum sensors. However, manipulating a largequantum system while simultaneously protecting the coherence remains challenging,even when the quantum application only requires collective control, such as some specialensemble-based quantum sensors or simulators. In particular, frequency and drivinginhomogeneities typically increase when the system size increases. Various techniquessuch as pulsed and continuous dynamical decoupling [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,13, 14, 15, 16, 17], as well as spin-locking [18] have been used to protect the coherenceof quantum systems.Beyond achieving robust quantum memories, manipulating the quantum devicewhile maintaining its coherence remains a non-trivial task [19], but could be helped byusing continuous decoupling schemes. Unfortunately, these often introduce additionalsources of noise linked to the added driving fields. A technique termed concatenatedcontinuous driving (CCD), which consists of adding multiple resonant modulated fields,can combat external noise and fluctuations in the control fields [20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30]. A modulation field on resonance with the main driving term cansuppress decoherence, provided that its amplitude is much larger than the fluctuationsof the main driving.Here, we use the CCD scheme to protect the coherence of an ensemble of qubits andto achieve their collective manipulation in the presence of frequency and driving fieldinhomogeneities. Experimentally, we achieve a 15-fold improvement in the coherencetime for an arbitrary, unknown state (corresponding to the transverse coherence time)and we also show how to tune the CCD control to protect an arbitrary, known statewith a 500-fold improvement in its coherence. These results are achieved through amore comprehensive understanding of the modulated dynamics, which can be describedby Floquet theory as giving rise to a Mollow triplet [31]. A strong modulation has beendemonstrated to have a broad feature in the synchronization by evaluating the powerand detuning dependence of the evolution amplitude.The long coherence times we achieve are also predicated on selecting the optimalcontrol parameters given the characteristics of the noise. We thus carefully characterizethe experimental noise sources by evaluating the power and detuning dependences of theRabi signal coherence, and analyze the noise effects under the CCD scheme by extendingthe theoretical framework of the generalized Bloch equation to this scenario. Thisanalysis, confirmed by experimental results, allows not only to optimize the coherencetime by adjusting the drive parameters, but it could be also used to reconstruct thepower spectral density (PSD) of various noise sources. Finally, we briefly discuss thepotential applications in the protection of nuclear spin coherence, perfect pulse designand AC magnetic field sensing. oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving FluorescenceMW Input1Photodetector Magnet 2Magnet 1Input2
Setup | ۧ0cos𝜋 𝑖𝜋4 sin𝜋 |1ۧ| ۧ0𝜖 𝑚 𝜖 𝑚 𝜖 𝑚 Lab frame 1 st rotating frame MW y zx 𝜔 Ωcos 𝜔 𝑡 +𝜙 −2𝜖 𝑚 sin 𝜔 𝑡 +𝜙 cos(𝜔 𝑚 𝑡 +𝜙 𝑚 ) nd rotating frame MW y zx Ω2 𝜙 y z x 𝜙 𝑚 𝜙 𝜖 𝑚 Protected state
CCD
Figure 1: CCD for coherence protection. (a) Schematic of the experimental setup.(b) Principle of the amplitude-modulated CCD scheme. Upon applying a modulatedwaveform, we can enter into two rotating frames where direction and strength ofthe driving field can be set by the modulation parameters (cid:15) m , φ , φ m . (c) Coherenceprotection of the Mollow triplet sidebands with the phase-modulated CCD scheme.Parameters Ω = ω m = (2 π )7 . , φ m = 0 are chosen such that the effective drivingfield in the second rotating frame is perpendicular to the initial state | (cid:105) . Upper panel isa normal Rabi oscillation with (cid:15) m = 0. Lower panel is with (cid:15) m = (2 π )1MHz. Coherencetimes τ i are fitted with c + (cid:80) i c i e − tτi cos( ω i t + φ i ). (d) Mollow triplet in the CCD scheme.Parameters Ω = ω m = (2 π )7 . | (cid:105) . Sidebands aremeasured with φ m = 0 and center band is measured with φ m = π/
2. Frequency valuesare fitted from the Rabi oscillations with the same function as in (c). Solid lines are thetheoretical predictions of the frequency values ω m , ω m ± (cid:15) m . (e) Coherence protectionof the center band. Modulation strength (cid:15) m = (2 π )2 . | (cid:105) and the driving parameters are φ = 0 , φ m = π/
2. In lower panel, the initial state iscos( π ) | (cid:105) + e i π sin( π ) | (cid:105) and the driving parameters are φ = − π , φ m = π . Coherencetime τ and index α are fitted with c + c e − ( tτ ) α cos( ω t + φ ). oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving
2. Coherence protection with the CCD scheme
NV centers in diamond have emerged as a promising platform for quantum informationprocessing [32], thanks in part to good control techniques that have pushed theircoherence times nearly up to the relaxation limit [33, 34, 35, 36, 37, 20, 24, 28, 30].Our device is based on an ensemble of NV centers in diamond as previously reported inRef. [38] [see Fig. 1(a) for a schematic of the setup]. A pair of permanent magnets applya static magnetic field along the NV axis, B ≈ | m S = ± (cid:105) states. The energy gap between the | m S = 0 (cid:105) and | m S = − (cid:105) states thatwe address in experiments is 2.207GHz when the N nuclear spin is in state | m I = 1 (cid:105) .Laser illumination not only initializes the NV electronic spin in the | m S = 0 (cid:105) state, butalso polarizes the N nuclear spin states to 73% in | m I = 1 (cid:105) . Microwave is deliveredthrough a 0.7mm loop structure on a PCB board. Three photodiodes are attached tothe surface of the diamond, and glued on the same PCB to measure the fluorescence. Byfocusing a 0.4mW green laser beam to a 30 µm spot, we simultaneously address ∼ spins. An arbitrary waveform generator mixes a ∼ | m S = 0 (cid:105) and | m S = − (cid:105) as the logical | (cid:105) and | (cid:105) states of an effective qubit. Due to field and driving inhomogeneities across the sample, the coherence time undernormal Rabi driving is about 1 µ s [see Fig. 1(c) upper panel]. To overcome theselimitations, we use a CCD scheme, whose basic principles are shown in Fig. 1(b).Consider a two-level system with a static splitting ω along z, coupled to an amplitude-modulated microwave along the x axis Ω cos( ωt + φ ) − (cid:15) m sin( ωt + φ ) cos( ω m t + φ m ).When the rotating wave approximation (RWA) condition Ω , (cid:15) m (cid:28) ω is satisfied and φ = 0, going into the first rotating frame defined by H = ω σ z and neglecting thecounter-rotating term, the Hamiltonian becomes H I = − δ σ z + Ω2 σ x + (cid:15) m cos( ω m t + φ m ) σ y (1)Phase modulation can also engineer a similar Hamiltonian through a phase-modulatedwaveform Ω cos (cid:2) ωt + (cid:15) m Ω cos( ω m t + φ m ) (cid:3) . In the first rotating frame defined by H ( t ) = ω σ z − (cid:15) m ω m Ω sin( ω m t + φ m ) σ z , the Hamiltonian becomes H I = − δ σ z + Ω2 σ x + (cid:15) m ω m Ω sin( ω m t + φ m ) σ z . (2)When the second RWA condition (cid:15) m (cid:28) Ω is satisfied, the Rabi oscillations displaycontributions from a center band ω m and two sidebands ω m ± (cid:112) (cid:15) m + ( ω m − Ω R ) ,forming the Mollow triplet [Fig. 1(d)]. The intensity of the center and sidebands can oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving φ , φ m of the driving ( mode control ). When the initial stateis in the direction of the driving field in the second rotating frame, only the centerband appears; when the initial state is perpendicular to the field, only the sidebandsexist. For a generic initial state, all three bands contribute to the signal. Beyond theRWA, higher order frequency components as well as frequency and amplitude shiftscomplicate the dynamics, but nevertheless their effects can be precisely predicted byFloquet theory [31]. Here we focus on the dynamics within the RWA.The two sidebands are affected by fluctuations of the second driving field, whereasthe center band frequency is robust against noise, as it depends only on the modulationfrequency which is set with high precision. Thus, while a generic, unknown statecoherence is limited by the shorter, sideband coherence, we can use our knowledge ofthe central band dynamics to better protect a known , arbitrary state, by synchronizingthe mode of the center band to the qubit state. To demonstrate these improvements, inexperiments we evaluate the coherence improvement of the center band and sidebandsseparately, by setting different modulation phases and initial states. First, we show inFig. 1(d) that the coherence of the sidebands displays a large improvement by more thanan order of magnitude, when compared to a normal Rabi oscillation. By further tuningthe parameters φ , φ m in the CCD scheme, we can orient the driving field in the secondrotating frame along the direction of the initial state to be protected. This synchronizesthe state evolution to the Mollow center band, and achieves a 500-fold improvementin the coherence time, compared to the conventional Rabi oscillations. Fig. 1(e) showsthe coherence of two different initial states synchronized to the center band. In theupper panel, the initial state is | (cid:105) and the driving phases are φ = 0 , φ m = π/
2; inthe lower panel, the initial state is cos( π ) | (cid:105) + e i π sin( π ) | (cid:105) and the driving phases are φ = − π , φ m = π . Note that the coherence times of both states are similar, indicatingthat an arbitrary known state can be protected.We further study the robustness of the mode-synchronized driving protocol againstinhomogeneities in the driving and static fields that occur when manipulating largeensemble of spins. We measure the Rabi oscillations from t = 50 µ s to t = 50 . µ s toensure that only the center band survives, and extract the oscillation contrast c byfitting the signal to c + c cos( ω t + φ ). In Fig. 2, we compare the results with a (a)strong and (b) weak modulation strength (cid:15) m . In the first case, the center band has alarge amplitude in a broader region beyond the resonance condition √ Ω + δ = ω m ,showing that more spins are driven even if their detuning and Rabi frequency deviatesfrom the nominal ones due to inhomogeneities. Another evidence of robustness is thatthe measured oscillation contrast (intensity in the color map) at a nominal δ = 0 , Ω = ω m under strong modulation is larger than that under weak modulation, indicating that thestrong modulation improves the protection of the center band coherence.We note that similar order-of-magnitude improvements in the qubit coherence hadonly been observed for single NV centers [20], or for small ensembles of NVs [24, 30],while here we are able to engineer robust control over a large volume consisting ofan ensemble of 10 NV spins. In addition, we identified the mechanism for robust oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving Strong modulation /(2 ) (MHz) -20-1001020 / ( ) ( M H z ) -3 Weak modulation /(2 ) (MHz) -20-1001020 / ( ) ( M H z ) -3 m =0.5 m =0.04 ( + )= m ( + )= m (a) (b) Figure 2: Synchronization of a single mode evolution. (a) Power and detuningdependence of the center band oscillation amplitude. Initial state is prepared to | (cid:105) and modulation frequency is ω m = (2 π )7 . (cid:15) m = Ω kept unchanged. φ m = π/ ω m , δ = 0. Rabi oscillations are measured from 50 µ s to 50 . µ s to ensure that only thecenter band is alive. The intensity represents the value of contrast c fitted from Rabioscillations with c + c cos( ω t + φ ). (b) Similar experiment with a weak modulation (cid:15) m = Ω , φ m = π/
2. Note that the signal contrast of a normal Rabi oscillation measuredin our sample is around 1%-2%.protection of known quantum states via mode control, that was only previously achievedwith mechanical driving [28].
3. Coherence time analysis
To further understand the protection afforded by the CCD scheme, as well as selectthe optimal driving parameters, it is critical to develop a theoretical framework for thecoherence time of qubit ensembles under this scenario, and implement experiments toverify the theoretical predictions.In the regime of a qubit weakly coupled to the bath, its decay rate under a singletransverse driving field can be predicted by the generalized Bloch equation (GBE) wherethe relaxation rates are given by the spectral components of the noise on resonancewith the corresponding transition energies of the qubit [39]. The coherence time ofthe qubit is thus determined by the power spectral density (PSD) of the noise [40, 41].Here we generalize the GBE model to an ensemble of spins, modeling the ensembleas a single spin qubit, where field inhomogeneities are included as an additional zero-frequency component in the noise spectrum. With a semi-classical treatment, the fieldfluctuations can be included as a stochastic component in the amplitude-modulated oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving H = ω σ z + (Ω + ξ Ω ) cos( ωt ) σ x − (cid:15) m + ξ (cid:15) m ) sin( ωt ) cos( ω m t + φ m )) σ x + ξ x σ x + ξ z σ z (3)where ξ Ω , ξ (cid:15) m are the fluctuations of the driving fields, comprising both drivingfluctuations and inhomogeneities. ξ x , ξ z are the fluctuations of the transverse andlongitudinal field giving rise to T and T decay in the absence of driving, withcontributions from both the bath and the static field inhomogeneities. Assumingstationary processes, the time correlation of these fluctuations is the Fouriertransformation of their noise PSD, (cid:104) ξ j ( t ) ξ j ( t ) (cid:105) = π (cid:82) ∞−∞ dνS j ( ν ) e − iν ( t − t ) where S j ( ν )( j = x, z, Ω , (cid:15) m ) is the PSD of the corresponding noise in the lab frame. In thesame way we can better understand the unitary dynamics by applying rotating frametransformations to the Hamiltonian, here we can analyze the noise effects and derivethe expected decay rates by expressing the PSDs in the rotating frame as a functionof the PSDs in the lab frame [39, 41, 40]. This is important as only some frequencycomponents of the PSD contribute mostly to the decay in any given frame: the transverseon-resonance noise component contributes to the qubit random bit flips, whereas thelongitudinal noise components at zero frequency contribute to random phase flips.For a single driving field ( (cid:15) m = ξ (cid:15) m = 0) and under the resonance condition ω = ω ,the longitudinal and transverse relaxation times in the first rotating frame, T ρ , T ρ , are1 T ρ = 12 S x ( ω ) + S z (Ω) = 12 T + S z (Ω) (4)1 T ρ = 12 T ρ + 12 S x ( ω ) + 14 S Ω (0) . (5)Given long T relaxation times, the longitudinal relaxation T ρ , corresponding to thespin-locking condition, is dominated by S z (Ω), the longitudinal field fluctuations. Fora zero-frequency centered noise spectrum, larger driving strengths Ω result in bettercoherence as S z (Ω) picks the noise at a frequency farther away from zero. The transverserelaxation time T ρ describes the decay of a conventional Rabi oscillation. The dominantterms are typically S z (Ω) + S Ω (0), leading to competing effects as a function ofΩ. When Ω is increased, S z (Ω) decreases but S Ω (0) increases. In Fig. 3(a), westudy the driving strength dependence of the Rabi coherence. Since the coherence timemonotonically decreases in the measured range, S Ω (0) is the dominant source. Whenthe Rabi driving is off-resonance, the Hamiltonian in the rotating frame has componentsin the x − z plane. Then, the transverse decay rate includes a term ∝ δ Ω + δ S z (0)[see details in Appendix D] that soon dominates since it probes the spectrum at zerofrequency. Thus, the Rabi coherence dependence on the detuning δ provides informationabout the static field fluctuation S z (0), and locally optimal coherence is obtained underthree resonance frequencies corresponding to three nuclear spin sub-levels. To extractthe values of inhomogeneities from the experimental data, we simulate the decay ratewith a simple model by directly integrating the Rabi oscillation over a static Gaussiandistribution of the driving field inhomogeneities ξ Ω and static field inhomogeneities oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving ξ z . The optimal fit to experiments is obtained with parameters σ Ω ∼ . σ ω = 4 S z (0) = (2 π )0 . σ ω ≈ . ω . /(2 ) (MHz) R ab i de c a y t i m e ( s ) SimulationMeasurement /(2 ) (MHz) / ( M H z ) /(2 ) (GHz) R ab i de c a y t i m e ( s ) SimulationMeasurement (a) (b) m I =0 m I =+1m I =-1 Figure 3: Inhomogeneity characterization. (a) Power dependence of the Rabi coherencetime T ρ . Microwave frequency ω = ω = (2 π )2 . | m I = +1 (cid:105) sublevel of nuclear spin of N. (b) Detuning dependence of the Rabicoherence time T ρ . Microwave power is chosen such that Rabi frequency underresonance condition is 0 . f (Ω+ ξ Ω , ω + ξ z ) = πσ Ω σ ω exp( − ξ σ − ξ ω σ ω ) and summing up the three species of nuclear spin sublevelswith the population of each sublevel obtained from the ESR measurement. Values σ Ω = 0 . σ ω = (2 π )0 . c + c exp( − tτ ) cos( ω t + φ ) is used in the fitting to extract the coherence time τ .We can extend this analysis to the CCD protocol by entering a second rotatingframe. On resonance ω m = Ω, we obtain the longitudinal and transverse relaxationtimes T ρρ , T ρρ in the second rotating frame,1 T ρρ = 14 S Ω ( (cid:15) m ) + 34 S x ( ω ) + 14 [ S z (Ω − (cid:15) m ) + S z (Ω + (cid:15) m )] (6)1 T ρρ = 12 T ρρ + 14 S (cid:15) m (0) + 12 S z (Ω) + 14 S x ( ω ) (7)= 14 S (cid:15) m (0) + 18 S Ω ( (cid:15) m ) + 12 S z (Ω) + 18 [ S z (Ω − (cid:15) m ) + S z (Ω + (cid:15) m )] + 58 S x ( ω ) T ρρ is the coherence time under the spin-locking condition in the second rotating frame,which corresponds to the center band in the Mollow triplet. T ρρ is the coherence timefor the sidebands. By analyzing the dominant noise sources, we can explain the decayrates observed in experiments, as shown in Fig. 4, and propose good control strategies.The coherence time T ρρ shows a strong dependence on the driving powers, (cid:15) m . Thefast initial increase in coherence time is due to the fast decrease of S Ω ( (cid:15) m ) as (cid:15) m grows,followed by a broad plateau. When (cid:15) m approaches Ω, a fast decrease happens in the oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving m /(2 ) (MHz) ( s ) Decay time ( /(2 )=7.5MHz) m /(2 ) (MHz) Decay time ( /(2 )=3MHz) m /(2 ) (MHz) / ( M H z ) (b)(a) m - mm + mm Phase mod Amp. mod
Figure 4: (a) Coherence times T ρρ , T ρρ as a function of (cid:15) m . Parameters ω m = Ω =(2 π )7 . , δ = 0, and initial state is prepared to | (cid:105) . Sidebands coherence times areplotted in red points for ω m + (cid:15) m and blue points for ω m − (cid:15) m . Center band coherencetimes are plotted in light blue points. Solid points and curves are for phase modulationwhereas hollow points and dashed lines are for amplitude modulation. Two y axesare used for sidebands (left) and center band (right). (b) Coherence times T ρρ , T ρρ dependence on (cid:15) m . Parameters ω m = Ω = (2 π )3MHz. The inset is the decay rate 1 /τ of the center band under the phase modulation condition.coherence time as observed in Fig. 4(b) due to the increase of the noise term S z (Ω − (cid:15) m )around zero frequency. The values of the optimal coherence time T ρρ in both (a) and (b)approach the spin-locking coherence T ρ under the same driving strength Ω [see detailsin Fig. D1 in Appendix D], which verifies that the coherence of both the spin-lockingcondition and the center band in the CCD scheme is dominated by S z . This result pointsto a strategy to improve the spin-locking coherence time under CDD, by increasing thesecond drive strength past the first, (cid:15) m > Ω. We note that in this regime, high-orderFloquet effects need to be taken into account [31]. Since no S (cid:15) m term is involved in thecenter band coherence, phase modulation and amplitude modulation do not display asignificant difference.The coherence time for the sidebands T ρρ also shows a maximum as a function of (cid:15) m . This is due to the competing effects of S Ω ( (cid:15) m ) + S z (Ω + (cid:15) m ), which decreases withincreasing (cid:15) m and S (cid:15) m (0) + S z (Ω − (cid:15) m ), which instead increases. Since S (cid:15) m (0) alwayspicks up the DC noise components, it soon dominates when (cid:15) m keeps increasing, so thatthe coherence degradation happens earlier than for the center band. Characterizing thevarious noise spectrum components can inform the best driving parameters for optimalcoherence.For an ideal situation of phase modulation, we should have ξ (cid:15) m = 0, S (cid:15) m (0) = 0,and the decrease of coherence time T ρρ should only be induced by the S z (Ω − (cid:15) m ) oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving T ρρ . However, both previous experiments[24] and this work do not find significant difference between phase modulation andamplitude modulation, although indeed the maximum is shifted towards higher (cid:15) m . Inboth strong power and weak power cases in Figs. 4(a) and 4(b) respectively, the phasemodulation only shows a slight improvement for the sidebands and the coherence timeof the sidebands is 1-2 orders of magnitude smaller than the center band. The noise ξ (cid:15) m under the phase modulation may come from the phase noise of the microwave field.
4. Conclusion
In this work, we explore optimal coherence protection by the CCD technique in denseNV ensembles. We show that any arbitrary states can be protected by aligning thedriving field with the state to be protected, thus engineering a single mode evolutionthat corresponds to the center band in the Mollow triplet. Our experiments show thatsuch a technique can be used to synchronize the dynamics of qubit ensembles even in thepresence of large inhomogeneity. We generalize the GBE to include driving fluctuationsand to analyze the coherence under the CCD protocol. By experimentally measuring thedependence of the coherence time on the second driving strength (cid:15) m , we can validate ourtheoretical analysis and analyze the interplay of competing noise sources. In addition toproviding a useful tool for protecting known and unknown quantum states, the insightsinto the CCD dynamics have found applications in high-frequency AC magnetic fieldsensing [42, 43]. The robust driving of the NV center ensemble could further enablethe indirect protection of the N nuclear spin associated with the NV center, whosecoherence is limited by the random telegraph noise caused by the T relaxation of theNV electronic spin. The nuclear spin protection requires rapid flips of the NV electronspin [44], which can be accomplished by the CCD scheme. Similarly, robust driving ofthe electronic spin bath [34], could enhance the NV coherence time. Finally, the schemedemonstrated in this work can be used to design robust quantum control pulses [21]. Acknowledgments
This work was supported in part by DARPA DRINQS and NSF PHY1915218. Wethank Pai Peng for fruitful discussions and Thanh Nguyen for manuscript revision.
Appendix A. Dynamics of the CCD scheme
To predict the precise dynamics of the CCD scheme, we utilize Floquet theory tosimulate the evolution. The eigenvectors of a time-periodic Hamiltonian are given by e − iλ a t Φ a ( t ) where { λ a } are the eigen-energies, and Φ a ( t ) = Φ a ( t + T ) are periodic intime, with T = πω m . The evolution of an arbitrary qubit state can then be writtenas Ψ( t ) = c + e − iλ − t Φ + ( t ) + c − e − iλ + t Φ − ( t ) with the coefficients c ± set by the initialconditions. If the initial state is one of the two eigenstates Φ ± (0), then the spin-locking oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving λ ± . The Rabi oscillations will only include frequency componentsthat are integer multiples of ω m . Otherwise, the evolution will be a superposition ofthese two modes, and the Rabi oscillations will involve three sets of frequencies, nω m and nω m ± ( λ + − λ − ). By tuning the driving parameters, the evolution mode can bewell controlled. Strong modulation /(2 ) (MHz) -20-1001020 / ( ) ( M H z ) Weak modulation /(2 ) (MHz) -20-1001020 / ( ) ( M H z ) Simulation, m =0.5 Simulation, m =0.04 mm ( + )= m ( + )= m (a) (b) Figure A1: Floquet simulation for the experiment in Fig. 2. The oscillation contrastof the center band is calculated assuming a single NV under the same driving conditionas in the experiment seen in Fig. 2. The decay effects caused by the noise are takeninto account by only keeping the initial contrast of the center band while neglectingthe contrast contributed from the sidebands in the simulation. The intensity of thecolormap represents the contrast value. Under the optimum driving condition, themaximum contrast (= 1) is achieved. Please see Ref. [31] for details on the Floquetsimulation.In the presence of inhomogeneities of the first drive, a stronger modulation isneeded to synchronize more ensemble spins to the center band evolution. In the maintext (Fig. 2), we experimentally measured the power and detuning dependence of thecenter band oscillation contrast (twice the oscillation amplitude) under strong and weakmodulation strengths (cid:15) m . In Fig. A1, we use Floquet theory to calculate the center bandoscillation contrast of a single NV under the experimental conditions, and find a goodmatch with the experiment results. On resonance ω m = Ω , δ = 0, the simulation predictssimilar contrast for both strong and weak modulation. However, in the experiments inFig. 2 we found that the oscillation contrast was larger with the strong modulation.This can be easily understood by considering that even under a nominal resonancecondition, many spins in the ensemble have an offset, due to inhomogeneities, and onlya strong drive is enough to achieve a good control. In Fig. A2, we plot 1D cuts ofboth the experimental data (symbols) and results obtained from simulation (curves).Under strong modulation, the oscillation amplitude as a function of detuning has a fullwidth at half maximum much larger than the hyperfine coupling constant A = 2 . oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving Detuning dependence -20 -10 0 10 20 /(2 ) (MHz)
Power dependence /(2 ) (MHz) (b)(a)
Simulation m =0.5 m =0.04 m =0.5 m =0.0450 s, m =0.5250 s, m =0.550 s, m =0.04 50 s, m =0.5250 s, m =0.550 s, m =0.04Simulation Figure A2: 1D cuts of robustness experiments in Fig. 2 and simulations in Fig. A1.(a) Detuning δ dependence of center band amplitude, which is a cut of the intensityplot in Fig. 2 (a) along Ω = (2 π )7 . µ s (from 50 µ s to50 . µ s) and 250 µ s (from 250 µ s to 250 . µ s) correspondingly under strong modulation (cid:15) m = Ω , φ = π/
2. Yellow squares are data at time 50 µ s with weak modulation (cid:15) m = Ω , φ = π/
2. Red and yellow curves are theoretical prediction of Floquet theory.(b) Power dependence of the oscillation coefficients which is a cut of the intensity plotsalong δ = 0. The colors correspond to those in (a). Note that the plots in (a) and(b) are manually normalized by their maximum value for easier comparison of the peakwidth, thus the y axis has an arbitrary scale.between NV electronic spin and N nuclear spin. This indicates that we can effectivelysynchronize all nuclear spin sublevels to the center band, and thus protect a known NVstate irrespective of the nuclear state.
Appendix B. Comparison to previous works
Table B1 makes a comparison of our work to previous ones. An order of magnitudecoherence improvement was achieved in both single NV [20] and sparse NV ensembles[24] by applying the CCD scheme with resonant microwave, and the CCD scheme wasalso able to improve the coherence of NVs in nano diamonds [30]. The CCD schemehas also been explored by combining high quality mechanical driving, serving as themodulation field, and microwave driving [28]; the coherence of a single NV was improvedby one order of magnitude. In comparison, we achieve a 15-fold improvement for thecoherence of the two sidebands in a large volume of NV ensembles with 10 spins. Inaddition, we also observe a 500-fold improvement for the central band at ω m , whoselong coherence was only previously identified in the mechanical driving experiment.While the latter achieved a similar coherence enhancement, using microwaves onlyfurther allows the implementation of mode control of the evolution, and the phase- oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving Parameters Ref. [20](2012) Ref. [24](2017) Ref. [28](2017) Ref. [30](2020) This work
Sample Single NV NVensemble(10 spins) Single NV Nanodiamond NVensemble(10 spins) T / (2 π ) , T ∗ ρ µ s 9MHz,0.81 µ s 5.83MHz,5.3 µ s 8.06MHz 7.5MHz,1 µ s σ Ω / (2 π ) ∼ . ∼ ∼ . ∼ . ∼ (cid:15) m / (2 π ) ∼ ∼ ∼ . ∼ . T ρρ µ s ∼ µ s ∼ ∼ µ s ∼ . T ρρ ∼ µ s ∼ µ s Appendix C. Inhomogeneity characterization
To characterize the inhomogeneity in our sample, we study the power and detuningdependence of the Rabi coherence. Assuming that inhomogeneities in the drivingpower and static field are the two main sources of decay for the Rabi oscillations,we simulate the coherence time. We assume a Gaussian distribution of their values, f (Ω + ξ Ω , ω + ξ ω ) = πσ Ω σ ξz exp( − ξ σ − ξ ω σ ω ), where ξ Ω describes the driving strengthinhomogeneity and ξ ω the static field along the z axis. We take the inhomogeneity of thedrive to be proportional to the driving amplitude, ξ Ω = r Ω · Ω, where ξ ω is fixed. Rabioscillations are simulated by a two dimensional integration over the power and detuninginhomogeneity P | (cid:105) ( t ) = (cid:88) i =1 c i (cid:90) ∞−∞ (cid:90) ∞−∞ dξ Ω dξ ω f (Ω + ξ Ω , ω + ξ ω )Ω (cid:112) Ω + ( ω − ω i ) cos( (cid:112) Ω + ( ω − ω i ) t ) e − t/τ In Fig. 3 of the main text, we plot the simulation results (blue points) when varyingthe Rabi amplitude and resonance frequency. By comparing the dependence of the oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving σ Ω = 0 . σ ω = (2 π )0 . σ Ω , while the peak width as a function of ω (Fig. 3b) ismore sensitive to the static field variance, σ ω , which can be explained with the theoryin Appendix D. To make the simulation best fit the experiment, we choose an intrinsiccoherence time τ = 13 µ s, which gives a constant offset to the decay rates and maycome from other noise sources, such as the spin bath. Appendix D. Coherence limit
To analyze the effect of noise on the spin coherence, we consider a semiclassicalmodel, where the noise is taken to be a fluctuating field originating from a classicalbath. Introducing these stochastic components and assuming φ = 0, the amplitude-modulated CCD Hamiltonian reads H = ω σ z +(Ω+ ξ Ω ) cos( ωt ) σ x − (cid:15) m + ξ (cid:15) m ) sin( ωt ) cos( ω m t + φ m )) σ x + ξ x σ x + ξ z σ z (D.1)where ξ x , ξ z are the fluctuations of the effective transverse and longitudinal fields and ξ Ω , ξ (cid:15) m are the fluctuations of the driving field. Within the RWA, Ω (cid:28) ω , the Hamiltonianin the first rotating frame is H (1) I = (cid:20) − δ ξ z (cid:21) σ z + (cid:20) Ω + ξ Ω ξ x cos( ω t ) (cid:21) σ x + (cid:20) ( (cid:15) m + ξ (cid:15) m ) cos( ω m t + φ m ) − ξ x sin( ω t ) (cid:21) σ y (D.2)where δ = ω − ω is the resonance offset. In the following, we will analyze fourcases based on this model: on-resonance single driving, off-resonance single driving,amplitude-modulated CCD, and phase-modulated CCD. Appendix D.1. Single driving with δ = 0We first analyze the case without the second driving with (cid:15) m = 0 and compare withprevious work in Ref. [39]. The PSDs in the first rotating frame S (1) j can be expressedas a function of the PSDs in the lab frame S (1) x ( ν ) = 14 S Ω ( ν ) + 14 (cid:20) S x ( ν + ω ) + S x ( ν − ω ) (cid:21) S (1) y ( ν ) = 14 (cid:20) S x ( ν + ω ) + S x ( ν − ω ) (cid:21) (D.3) S (1) z ( ν ) = S z ( ν ) oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving α of the α = { x, y, z } components of the qubit Paulimatrix as Γ x = 14 (cid:20) S x ( ω + Ω) + S x ( ω − Ω) (cid:21) + S z (Ω)Γ y = 12 S x ( ω ) + S z (Ω) + 14 S Ω (0) (D.4)Γ z = 12 S x ( ω ) + 14 (cid:20) S x ( ω + Ω) + S x ( ω − Ω) (cid:21) + 14 S Ω (0)where we used the fact that the decay along one axis is determined by the sum of therotating frame spectra along the two other axes, Γ α = S (1) β + S (1) γ . In turn, these ratescan be used to write the longitudinal and transverse relaxation time in the first rotatingframe T ρ , T ρ . With the approximation S x ( ω ± Ω) ≈ S x ( ω ), we obtain1 T ρ = Γ x = 12 S x ( ω ) + S z (Ω) (D.5)1 T ρ = 12 (Γ y + Γ z ) = 34 S x ( ω ) + 12 S z (Ω) + 14 S Ω (0) = 12 T ρ + 1 T (cid:48) ρ where we defined the pure dephasing time T ρ (cid:48) with T (cid:48) ρ = S x ( ω ) + S Ω (0) = T + S Ω (0). Our analysis up to here is consistent with previous work in Ref. [39]except for an additional microwave fluctuation term. Fig. D1 is a measurement of spin-locking coherence as a function of Ω. The coherence time increases with Ω due to thedecreasing of S z (Ω). The inset (b) plots the decay rate and is a direct measurement of S z (Ω) [45, 41]. Appendix D.2. Single driving with δ (cid:54) = 0When there is a frequency offset, δ (cid:54) = 0, we can diagonalize the non-stochasticHamiltonian (in the σ x basis) by defining a new set of axes in the first rotating framewith σ z = ΩΩ R σ z (cid:48) − δ Ω R σ x (cid:48) , σ x = ΩΩ R σ x (cid:48) + δ Ω R σ z (cid:48) , σ y = σ y (cid:48) where Ω R = √ δ + Ω is theeffective driving field. Then, the Hamiltonian in the first rotating frame becomes H (1) I = − δ σ z + Ω2 σ x − ξ x sin( ω t ) σ y + (cid:20) ξ Ω ξ x cos( ω t ) (cid:21) σ x + ξ z σ z (D.6)= Ω R σ x (cid:48) + ξ z (cid:20) ΩΩ R σ z (cid:48) − δ Ω R σ x (cid:48) (cid:21) + (cid:20) ξ Ω ξ x cos( ω t ) (cid:21) (cid:20) ΩΩ R σ x (cid:48) + δ Ω R σ z (cid:48) (cid:21) − ξ x sin( ω t ) σ y (cid:48) Accordingly, the PSDs in this modified first rotating frame S (1) α (cid:48) can be expressed as afunction of the PSDs in the lab frame as S (1) x (cid:48) ( ν ) = δ Ω R S z ( ν ) + Ω Ω R (cid:20) S Ω ( ν ) + 14 ( S x ( ν + ω ) + S x ( ν − ω ) (cid:21) S (1) y (cid:48) ( ν ) = 14 (cid:20) S x ( ν + ω ) + S x ( ν − ω ) (cid:21) (D.7) S (1) z (cid:48) ( ν ) = Ω Ω R S z ( ν ) + δ Ω R (cid:20) S Ω ( ν ) + 14 ( S x ( ν + ω ) + S x ( ν − ω ) (cid:21) oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving (a) (b) Figure D1: Spin-locking coherence time τ as a function of the driving strength Ω / (2 π ).After initializing the qubit to | (cid:105) state, a π/ √ ( | (cid:105) + | (cid:105) ), then an on-resonance transverse driving field is applied continuously for atime interval, finally a π/ | (cid:105) . By measuring the population in | (cid:105) whilevarying the time intervals between two π/ π )7 . τ as a dependence of the driving strength Ω / (2 π ).Similar to what was done above, we can obtain the decay rates of the σ α (cid:48) componentsand then combine them to obtain the longitudinal and transverse relaxation rates in therotating frame. With the approximation S x ( ω ± Ω R ) ≈ S x ( ω ), we obtain1 T ρ = Γ x (cid:48) = 12 S x ( ω ) + Ω Ω R S z (Ω R ) + δ Ω R (cid:20) S Ω (Ω R ) + 12 S x ( ω ) (cid:21) (D.8)1 T ρ = 12 (Γ y (cid:48) + Γ z (cid:48) ) (D.9)= δ Ω R S z (0) + Ω R (cid:20) S Ω (0) + 2 S z (Ω R ) (cid:21) + 18 δ Ω R S Ω (Ω R ) + (cid:20)
34 + δ Ω R (cid:21) S x ( ω )This analysis helps explain the experimental results in Fig. 3(b) where the coherence ofRabi oscillation becomes worse when the detuning increases, due to the δ Ω R S z (0) termin the transverse decay rate T ρ . For a spin-locking experiment with detuning, instead,the decay rate T ρ ≈ S x ( ω ) = T approaches the T relaxation time when δ → ∞ . Appendix D.3. Amplitude-modulated CCD
To simplify the calculation, we assume ω = ω, ω m = Ω , φ = φ m = 0 for all thefollowing discussions. We enter into the second rotating frame defined by ω m σ x and oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving H (2) I = (cid:15) m σ y + (cid:20) ξ Ω ξ x cos( ω t ) (cid:21) σ x + (cid:20) ξ (cid:15) m ω m t )) − ξ x sin( ω t ) cos( ω m t ) + ξ z sin( ω m t ) (cid:21) σ y (D.10)+ (cid:20) ξ (cid:15) m ω m t ) + ξ x sin( ω t ) sin( ω m t ) + ξ z cos( ω m t ) (cid:21) σ z The PSDs in the second rotating frame S (2) j become S (2) x ( ν ) = 14 S Ω ( ν ) + 14 (cid:20) S x ( ν + ω ) + S x ( ν − ω ) (cid:21) S (2) y ( ν ) = 14 S (cid:15) m ( ν )+ 116 (cid:20) S (cid:15) m ( ν + 2 ω m ) + S (cid:15) m ( ν − ω m ) (cid:21) + 14 (cid:20) S z ( ν + ω m ) + S z ( ν − ω m ) (cid:21) (D.11)+ 116 (cid:20) S x ( ν + ω + ω m ) + S x ( ν + ω − ω m ) + S x ( ν − ω + ω m ) + S x ( ν − ω − ω m ) (cid:21) S (2) z ( ν ) = 116 (cid:20) S (cid:15) m ( ν + 2 ω m ) + S (cid:15) m ( ν − ω m ) (cid:21) + 14 (cid:20) S z ( ν + ω m ) + S z ( ν − ω m ) (cid:21) + 116 (cid:20) S x ( ν + ω + ω m ) + S x ( ν + ω − ω m ) + S x ( ν − ω + ω m ) + S x ( ν − ω − ω m ) (cid:21) In the second rotating frame, the static field is along the y axis, and the decay rates canbe analyzed in a similar way Γ x = S (2) y (0) + S (2) z ( (cid:15) m )Γ y = S (2) x ( (cid:15) m ) + S (2) z ( (cid:15) m ) (D.12)Γ z = S (2) y (0) + S (2) x ( (cid:15) m )Define the longitudinal and transverse relaxation times in the second rotating frame as T ρρ , T ρρ . Assume that S x ( ω ± Ω ± (cid:15) m ) ≈ S x ( ω ) with Ω , (cid:15) m (cid:28) ω , then1 T ρρ = Γ y = 14 S Ω ( (cid:15) m ) + 34 S x ( ω )+ 116 (cid:20) S (cid:15) m (2Ω − (cid:15) m ) + S (cid:15) m (2Ω + (cid:15) m ) (cid:21) + 14 (cid:20) S z (Ω − (cid:15) m ) + S z (Ω + (cid:15) m ) (cid:21) (D.13)1 T ρρ = 12 (Γ x + Γ z ) = 12 T ρρ + 14 S (cid:15) m (0) + 18 S (cid:15) m (2Ω) + 12 S z (Ω) + 14 S x ( ω ) = 1 T (cid:48) ρρ + 12 T ρρ (D.14)where T (cid:48) ρρ = S (cid:15) m (0) + S (cid:15) m (2Ω) + S z (Ω) + S x ( ω ) is defined as the pure dephasingrate in the second rotating frame. oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving (cid:15) m ≈ Ω and S Ω (Ω ± (cid:15) m ) ≈ S Ω (Ω), the coherence times in the second rotatingframe simplifies to1 T ρρ ≈ S Ω ( (cid:15) m ) + 34 S x ( ω ) + 18 S (cid:15) m (2Ω) + 12 S z (Ω)= 12 T ρ + 14 S Ω ( (cid:15) m ) + 12 S x ( ω ) + 18 S (cid:15) m (2Ω) (D.15)1 T ρρ ≈ S (cid:15) m (0) + 18 S Ω ( (cid:15) m ) + 316 S (cid:15) m (2Ω) + 34 S z (Ω) + 58 S x ( ω ) (D.16)When (cid:15) m ≈ Ω, the approximation here is no longer valid and the coherence is dominatedby S z (Ω − (cid:15) m ). Appendix D.4. Phase-modulated CCD
There are two-fold differences in the phase-modulated CCD. First, the modulationamplitude could be assumed in principle to be noise-free, since it arises from the phasemodulation that should be very precise, as it depends mostly on the signal source,and not on how it is delivered to the spins. Second, the modulated drive is along thez-direction (instead of the y-direction as it is the case for the amplitude-modulatedCCD). In the lab frame, we assume φ = 0 and add fluctuation parameters to thephase-modulated CCD Hamiltonian H = ω σ z + (Ω + ξ Ω ) cos( ωt + 2 (cid:15) m Ω cos( ω m t + φ m )) σ x + ξ x σ x + ξ z σ z (D.17)where ξ x , ξ z are the fluctuations of the transverse and longitudinal fields and ξ Ω is thefluctuation of the driving field. With the RWA and resonance condition Ω (cid:28) ω = ω ,we can enter into the first rotating frame where H (1) I = Ω2 σ x + (cid:15) m sin( ω m t + φ m ) σ z + (cid:20) ξ Ω ξ x cos( ω t + 2 (cid:15) m Ω cos( ω m t + φ m )) (cid:21) σ x (D.18) − ξ x sin( ω t + 2 (cid:15) m Ω cos( ω m t + φ m )) σ y + ξ z σ z Assume φ m = 0 , ω m = Ω and then the Hamiltonian in the second rotating frame is H (2) I = (cid:15) m σ y + (cid:20) ξ Ω ξ x cos( ω t + 2 (cid:15) m Ω cos( ω m t )) (cid:21) σ x + (cid:104) ξ z sin( ω m t ) − ξ x sin( ω t + 2 (cid:15) m Ω cos( ω m t )) cos( ω m t ) (cid:105) σ y (D.19)+ (cid:104) xi z cos( ω m t ) + ξ x sin( ω t + 2 (cid:15) m Ω cos( ω m t )) sin( ω m t ) (cid:105) σ z The term cos( ω t + 2 (cid:15) m Ω cos( ω m t )) or sin( ω t + 2 (cid:15) m Ω cos( ω m t )) can be approximated bycalculating the expansion of cos(2 (cid:15) m Ω cos( ω m t )) or sin(2 (cid:15) m Ω cos( ω m t )) to first order when oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving (cid:15) m / Ω is small. For example,cos( ω t + 2 (cid:15) m Ω cos( ω m t )) = cos( ω t ) cos(2 (cid:15) m Ω cos( ω m t )) − sin( ω t ) sin(2 (cid:15) m Ω cos( ω m t )) ≈ cos( ω t ) − sin( ω t )2 (cid:15) m Ω cos( ω m t ) (D.20)The PSDs in the second rotating frame S (2) j become S (2) x ( ν ) ≈ S Ω ( ν ) + 14 (cid:20) S x ( ν + ω ) + S x ( ν − ω ) (cid:21) + 14 ( (cid:15) m Ω ) (cid:20) S x ( ν + ω + ω m ) + S x ( ν + ω − ω m ) + S x ( ν − ω + ω m ) + S x ( ν − ω − ω m ) (cid:21) S (2) y ( ν ) ≈ (cid:20) S z ( ν − ω m ) + S z ( ν + ω m ) (cid:21) + 116 (cid:20) S x ( ν + ω + ω m ) + S x ( ν + ω − ω m ) + S x ( ν − ω + ω m ) + S x ( ν − ω − ω m ) (cid:21) + 14 ( (cid:15) m Ω ) (cid:20) S x ( ν + ω ) + S x ( ν − ω ) (cid:21) (D.21)+ 116 ( (cid:15) m Ω ) (cid:20) S x ( ν + ω + 2 ω m ) + S x ( ν + ω − ω m ) + S x ( ν − ω + 2 ω m ) + S x ( ν − ω − ω m ) (cid:21) S (2) z ( ν ) ≈ (cid:20) S z ( ν − ω m ) + S z ( ν + ω m ) (cid:21) + 116 (cid:20) S x ( ν + ω + ω m ) + S x ( ν + ω − ω m ) + S x ( ν − ω + ω m ) + S x ( ν − ω − ω m ) (cid:21) + 116 ( (cid:15) m Ω ) (cid:20) S x ( ν + ω + 2 ω m ) + S x ( ν + ω − ω m ) + S x ( ν − ω + 2 ω m ) + S x ( ν − ω − ω m ) (cid:21) In the second rotating frame, the static field is along the y axis, the decay rates can beanalyzed in a similar way Γ x = S (2) y (0) + S (2) z ( (cid:15) m )Γ y = S (2) x ( (cid:15) m ) + S (2) z ( (cid:15) m ) (D.22)Γ z = S (2) y (0) + S (2) x ( (cid:15) m )The longitudinal and transverse relaxation time T ρρ , T ρρ , under resonance conditionΩ = ω m and assuming S x ( ω ± Ω ± (cid:15) m ) ≈ S x ( ω ) with Ω , (cid:15) m (cid:28) ω , are given by1 T ρρ = Γ y = (cid:20)
34 + 54 ( (cid:15) m Ω ) (cid:21) S x ( ω ) + 14 S Ω ( (cid:15) m ) + 14 (cid:20) S z (Ω − (cid:15) m ) + S z (Ω + (cid:15) m ) (cid:21) (D.23)1 T ρρ = 12 (Γ x + Γ z ) = 12 T ρρ + 12 S z (Ω) + (cid:20)
14 + 34 ( (cid:15) m Ω ) (cid:21) S x ( ω ) = 1 T (cid:48) ρρ + 12 T ρρ (D.24)where T (cid:48) ρρ = S z (Ω) + ( + ( (cid:15) m Ω ) ) S x ( ω ) is defined as the pure dephasing rate in thesecond rotating frame. oherence protection and decay mechanism in qubit ensembles under concatenated continuous driving (cid:15) m (cid:28) Ω and S Ω (Ω ± (cid:15) m ) ≈ S Ω (Ω), the longitudinal coherence time in thesecond rotating frame becomes1 T ρρ ≈ S Ω ( (cid:15) m ) + 34 S x ( ω ) + 12 S z (Ω) = 12 T ρ + 14 S Ω ( (cid:15) m ) + 12 S x ( ω ) (D.25)The transverse coherence time in the second rotating frame1 T ρρ ≈ S Ω ( (cid:15) m ) + 34 S z (Ω) + 58 S x ( ω ) (D.26)Expressions here can also be obtained by simply setting ξ (cid:15) m = 0 in the amplitude-modulated situation. References [1] E. L. Hahn. Spin Echoes.
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