Decay of Rabi Oscillations by Dipolar-Coupled Dynamical Spin Environments

V.V. Dobrovitski,1 A. E. Feiguin,2,3 R. Hanson,4 and D.D. Awschalom5

1Ames Laboratory U.S. DOE, Iowa State University, Ames, Iowa 50011, USA2Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

3Microsoft Station Q, University of California, Santa Barbara, California 93106, USA4Kavli Institute of Nanoscience Delft, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands5Center for Spintronics and Quantum Computation, University of California, Santa Barbara, California 93106, USA

(Received 1 April 2009; published 9 June 2009)

We study the Rabi oscillations decay of a spin decohered by a spin bath whose internal dynamics is

caused by dipolar coupling between the bath spins. The form and rate of decay as a function of the

intrabath coupling is obtained analytically, and confirmed numerically. The complex form of decay

smoothly varies from power law to exponential, and the rate changes nonmonotonically with the intrabath

coupling, decreasing for both slow and fast baths. The form and rate of Rabi oscillations decay can be used

to experimentally determine the intrabath coupling strength for a broad class of solid-state systems.

DOI: 10.1103/PhysRevLett.102.237601 PACS numbers: 76.20.+q, 03.65.Yz, 76.30.�v

Measurement of the Rabi oscillations decay is an im-portant step in studying decoherence of quantum systems.E.g., studies of Rabi oscillations in superconducting qubitsdamped by bosonic baths and by 1=f noise [1,2] helped toextend the decoherence time to �s range [3], placingsuperconducting qubits among most promising solid-statecandidates. Recently, much progress has been achieved inexperimental implementation of long-living coherent Rabioscillations in various spin systems: magnetic molecules[4], NV impurity centers in diamond [5–7], rare-earth ionsin host crystals [8], quantum dots [9–11]. Many of thesesystems are attractive for basic studies of quantum spincoherence effects, and show much promise for quantuminformation processing, coherent spintronics, or high-precision magnetometry, provided that detailed under-standing of the decoherence processes will be achieved.

A major decoherence source in these spin systems is thecoupling of the central spin (e.g., the electron spin of theNV center) to other spins in the sample (environmentalbath spins, e.g., the spins of nitrogen atoms in diamond).Moreover, for many relevant spin systems, the direct cou-pling between the environmental spins is essential, produc-ing internal dynamics of the bath. Here we exploretheoretically the influence of such a dynamical spin bathon the Rabi oscillations of the central spin. Avoiding thecommonly used framework of generalized Bloch equations[12–15], we are able to investigate the form and rate ofdecay as a function of the intrabath coupling strength. Wefind interesting behavior of the Rabi oscillations, whichcontradicts the expectations based on standard Redfield-type analysis: e.g., the slow bath leads to pronounceddecay, while for the fast bath the decay rate vanishes butthe Rabi frequency becomes renormalized. We demon-strate how the form and rate of the Rabi oscillations decaycan be used to experimentally characterize the intrabathdynamics, and provide a rather simple recipe for analysisof data.

We focus on dilute dipolar-coupled bath: in a nonmag-netic crystal, a single central spin of species S (e.g., aparamagnetic impurity) is coupled to a bath of dilute spinsof species I (e.g., other kind of paramagnetic impurities, ornuclear spins). The coupling between the bath spins is non-negligible, and is caused by dipolar interactions. Thissituation encompasses a wide range of interesting solid-state spin systems, from Er ions in CaWO4 studied in1960s [16] to the NV centers in diamond [5–7] whichgained much attention recently. The NV centers may beparticularly suitable for detailed studies of Rabi oscilla-tions and verification of the model due to possibility ofoptical readout of a single central spin [17], and tunablespin bath made of surrounding nitrogen atoms [7].We assume that a very large magnetic field Bz is applied

along the z axis (as in standard NMR/ESR settings), lead-ing to Zeeman splittings !S ¼ �SBz for the central spin Sand !I ¼ �IBz for the bath spins Ik (�S and �I are thegyromagnetic ratios of the S and I spins, respectively).Also, strong Rabi driving field HR is applied at the fre-quency !S. The difference j!S �!Ij is much larger thanany other energy scale, so the mutual flips of the centralspin and the bath spins involve huge Zeeman energy mis-match and are strongly suppressed. Hence, the terms likeSþI�k become irrelevant on a time scale of Rabi oscilla-

tions decay (see below); omitting these terms and trans-forming into rotating frame we obtain a secularHamiltonian [12,13]:

H ¼ hxSx þXk

AkSzIzk þ

Xk;l

Cklð3IzkIzl � IkIlÞ; (1)

where Sx;y;z and Ix;y;zk are the spin operators in the rotating

frame, and hx ¼ HR=2 is the rotating-frame Rabi drivingfield. The coupling constants Ak ¼ �S�I½1� 3ðnzkÞ2�=r3kare determined by the positions rk of the bath spins Ik (k ¼1; . . .N), where rk ¼ jrkj and nk ¼ rk=rk (the origin of thecoordinate frame coincides with the central spin). Simi-

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larly, the intrabath couplingsCkl ¼ �2I ½1� 3ðnzklÞ2�=r3kl are

determined by the vectors rkl ¼ rk � rl. Note that thesame Hamiltonian (1) can be obtained without externalfield Bz if the transition frequencies !S and !I are deter-mined by the zero-field splitting (e.g., due to anisotropicinteractions). Similarly, the assumption (used below) that Sand Ik are spins 1=2 is not essential: for larger spins, we canconsider each pair of levels as a pseudospin 1=2 [12,13].

Initially the central spin is in the state ‘‘up,’’ and the bathis in a maximally mixed state (unpolarized bath at hightemperatures); i.e., the initial density matrix of the wholesystem is �ð0Þ ¼ 2�Nj"ih"j � 1B, where 1B is the 2N � 2N

identity matrix. This is appropriate for most experiments(for nuclear spins at temperatures above a few nK, forelectron spins—above tens of K). We calculate the time-dependent z component of the central spin hSzðtÞi ¼Tr�ðtÞSz, where �ðtÞ ¼ expð�iHtÞ�ð0Þ expðiHtÞ. In orderto see well-pronounced long-living Rabi oscillations, thedriving field should be large, so we assume that hx is muchlarger than all other energy scales [18]. We calculate theoscillations damping in the lowest order in 1=hx, treatingthe second (spin-bath coupling) and the third (bath internalHamiltonian) terms in Eq. (1) perturbatively, and excludingthe bath internal Hamiltonian HB ¼ P

k;lBklð3IzkIzl � IkIlÞby the interaction representation transformation. The re-sulting second-order Hamiltonian

H0 ¼ hxSx þ SxB2ðtÞ=ð2hxÞ; (2)

where BðtÞ ¼ expðiHBtÞB expð�iHBtÞ, and the operator

B ¼ PkAkI

zk.

The evolution of BðtÞ is complex, involving intricatecorrelations between bath spins. However, exact dynamicsof every single bath spin is not important, since hSzðtÞiinvolves tracing over all bath spins. This is typical formany spin-bath decoherence problems, from magneticresonance to quantum information processing, and manyapproaches have been developed from the early days ofNMR/ESR theory [19,20] till very recently [21–28].Below, following the works [19,20], we approximate theeffect of the bath by a random field BðtÞ, which is modeledas an Ornstein-Uhlenbek process with the correlation func-tion hBðtÞBð0Þi ¼ b2 expð�RtÞ, where the dispersion b ¼ffiffiffiffiffiffiffiffiffiffiffiffiP

kA2k

q, while the correlation decay rate R is determined

by HB. This model may be oversimplified for complicatedsituations, e.g., the description of advanced control proto-cols requires more sophisticated treatment [23,24,28].However, our direct numerical simulations evidence thatthis model is quantitatively adequate for description of theRabi oscillations decay, while providing a clear descriptionof the physics underlying the Rabi oscillations decay, andallowing access to the regimes outside of the Bloch equa-tions framework.

The Hamiltonian (2) reflects a simple physical picture.The zero-order eigenstates of the Hamiltonian (1) corre-

spond to the central spin states jþi ¼ 1=ffiffiffi2

p ½j"i þ j#i� and

j�i ¼ 1=ffiffiffi2

p ½j"i � j#i�, separated by a large Rabi fre-quency hx. Since hx � b and hx � R, the field BðtÞ hasno spectral components of noticeable magnitude at theRabi frequency, and does not lead to transition betweenthe states jþi and j�i. The only relevant process is the puredephasing, when the field B destroys the initial phaserelation between the states jþi and j�i, leading to decayof hSzðtÞi, so that

hSzðtÞi ¼ 1=2hcos�i ¼ 1=2Rehexpði�Þi; (3)

where h�i denotes average over all possible realizations ofBðtÞ, and the phase

� ¼ hxtþ 1

2hx

Z t

0B2ðsÞds ¼ hxtþ� (4)

is the total phase difference between the states jþi and j�iaccumulated during time t, cf. the Hamiltonian (2). Thekey quantity MðtÞ ¼ hexpði�Þi is an analytically comput-able Gaussian path integral over the Ornstein-Uhlenbeckprocess, which gives the answer

hSzðtÞi ¼ 1=2Re½MðtÞ expðihxtÞ�½MðtÞ��2 ¼ expð�RtÞ½coshPtþ ðR=PÞ sinhPt�

� ib2

hxPexpð�RtÞ sinhPt;

(5)

where P ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � 2ib2R=hx

p.

Analysis of Rabi oscillations is often based on Bloch-type equations [12] or its generalizations [14], derived forvarious systems from quantum optics [15] to solid state[14]. It is based on the Redfield-type approach [12,13],taking into account only the terms which are secular withrespect to the Rabi driving hxSx. In addition to dephasing,these terms describe the actual transitions between thestates jþi and j�i, which lead to a longitudinal relaxation(along the x axis) of the central spin with the rate �l �b2R=h2x. This rate is of second order in 1=hx, and isdetermined by the spectral density of BðtÞ at the Rabifrequency hx. Also, the generalized Bloch equations, hav-ing constant coefficients, always predict the decay to havea (multi)exponential form. In contrast, our results are notlimited to the terms secular with respect to Rabi driving,and give the decay rate of the first order in 1=hx. Thesolution (5) predicts the decay which has, in general, nosimple form (multiexponential, Gaussian, power law, etc.)The familiar exponential decay occurs only at specialvalues of b, hx, and R, see below.The effect of the bath internal dynamics may be impor-

tant even for slow baths, with R � b2=hx. In this limit,

P ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2iR=hx

p, see Eq. (5). At short times t � 1=jPj,

the bath behaves as static, and the Rabi oscillations en-velope exhibits nonexponential slow decay of the form

½1þ ðb2t=hxÞ2��1=4, in accordance with the exact resultsobtained earlier [9,25–27]. At long times t � 1=jPj, thedecay has exponential form expð�bt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiR=4hx

p Þ, with therate which decreases very slowly with decreasing R. As aresult, even for very slow bath the effect of the bath

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dynamics remains noticeable. In experiments, the above-described behavior of the Rabi oscillations decay can bedetected by varying hx (since the ratio of R to b2=hxdetermines how fast or slow the bath is), and makes itpossible to estimate b and R.

Another interesting feature of our results is that thedecay of Rabi oscillations changes nonmonotonicallywith R: it is fastest for R� b2=hx, slowing down forboth slow and fast baths. This is confirmed by directnumerical simulations, see Fig. 1 and discussion below.

The regime of fast bath, with R � b2=hx, is even moreinteresting and unexpected. In this case

MðtÞ ¼ exp

�ib2t

2hx� b4t

4h2xR

�; (6)

and the Rabi oscillations exhibit exponential decay withthe rate �t ¼ b4=ð4h2xRÞ, which vanishes quickly for largeR. However, it does not mean that the effect of the fast bathdisappears. The bath still noticeably affects the centralspin, shifting its Rabi frequency by b2=2hx, and only thedecaying part of the bath effect vanishes [29]. This slowingof the Rabi oscillations decay is confirmed by direct nu-merical simulations (Fig. 2).

On the other hand, for fast baths, the transitions betweenthe states jþi and j�i become important. While the de-phasing between these states just shifts the Rabi frequency,the transitions lead to longitudinal relaxation along thex axis with the rate �l � b2R=h2x. This implies [12] ex-ponential decay of Rabi oscillations with the rate �l=2,which is comparable with the decay rate �t caused by thepure dephasing. Above, for simplicity, we omitted thelongitudinal relaxation (since it is of order 1=h2x), but wecan include it using the Redfield-type approach: the answer(5) for SzðtÞ just has to be multiplied by expð��lt=2Þ. Thisexplains how the system enters the generalized Blochequation regime at R � b: the dephasing effect becomesnegligible and the longitudinal relaxation becomes domi-nant. In experiment, this regime can be identified by com-paring �l and �t: they would be of the same order,changing as 1=h2x for all sufficiently large hx.To ensure that the physical picture above is adequate for

real dipolar-coupled bath, we performed direct numericalsimulations of the Rabi oscillations decay. We place thecentral spin and N bath spins randomly, with uniform den-

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FIG. 1 (color online). Exact simulations for N ¼ 15 bathspins. (a) Correlation function CðtÞ ¼ ð1=b2ÞhBð0ÞBðtÞi (normal-ized by b2) obtained from direct simulations (black), see also[31]. Its fitting (magenta or gray) determines the bath parametersb1, b2, and R for each value of the intrabath coupling scale EB

(here EB ¼ 1, b1 ¼ 0:62, b2 ¼ 0:58, R ¼ 0:095). (b) Rabi os-cillations decay for hx ¼ 14:14, b ¼ 0:85, and EB ¼ 1; individ-ual oscillations are not resolved. Numerics (black) agrees wellwith analytics (magenta, only the envelope shown).(c) Envelopes of simulated Rabi decay for EB ¼ 1 (blue ordark gray), 0.1 (green or light gray), and 0 (static bath, red orgray); corresponding individual oscillations near t ¼ 300 areshown on panel (d) by the same colors. Analytical results practi-cally coincide with simulations, and are not shown. The decayrate changes nonmonotonically with EB: the decay is slower forEB ¼ 1 (blue) and 0 (red) in comparison with EB ¼ 0:1 (green).At t ¼ 300, on panel (d), the oscillation amplitude for EB ¼ 0:1(green) is twice smaller than that for EB ¼ 0.

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FIG. 2 (color online). Numerical simulations for N ¼ 159 bathspins. (a) Correlation function CðtÞ ¼ ð1=b2ÞhBð0ÞBðtÞi (normal-ized by b2) obtained from P-representation sampling simulations(black); its fitting (magenta or gray) determines the parameter Rfor each value of the intrabath coupling scale EB (here EB ¼ 1and R ¼ 0:097). In spite of noticeable statistical fluctuations[e.g., Cð0Þ is slightly larger than 1 due to them], the parameter Ris determined precisely enough to be used in Eq. (5). (b) Rabioscillations decay for hx ¼ 15:0, b ¼ 1:39, and EB ¼ 0:1; indi-vidual oscillations are not resolved. Numerics (black) agreeswell with analytics (magenta, only the envelope shown).(c) Envelopes of simulated Rabi decay for EB ¼ 0:1 (blue ordark gray), 1 (green or light gray), and 10 (red or gray); thelongitudinal decay for EB ¼ 10 is taken into account. Analyticalresults practically coincide with simulations, and are not shown.The decay rate decreases for faster baths: the decay is slowest forEB ¼ 10 (red). Individual oscillations for EB ¼ 10 near t ¼ 20are shown on panel (d) in red: their frequency is shifted byb2=2hx ¼ 0:064 from the value hx ¼ 15:0 (oscillations withfrequency hx ¼ 15:0 are shown by dotted black line to demon-strate the phase difference accumulated since t ¼ 0).

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sity n ¼ 1, inside a cube with the side ðN þ 1Þ1=3, centralspin being in the center of the cube. All interaction coef-ficients are calculated according to Eq. (1), using the actualcoordinates of the spins. The value ESB ¼ �S�I, whichdetermines the strength of coupling between the centralspin and the bath, is set to 1, thus defining the energy andtime scales for all quantities below. The value EB ¼ �2

I ,which governs the energy scale of intrabath couplings, isvaried, making the bath slower or faster. We simulate thedynamics of the system using two numerical approaches.For smallN (e.g.,N ¼ 15 in Fig. 1) we exactly solve of theSchrodinger equation with the Hamiltonian (1) viaChebyshev polynomial expansion [30]. The dipolar-coupled systems with N > 15 are difficult to model thisway (due to exponentially increasing resources require-ments), so we use the P-representation sampling [30] formodeling larger baths (N ¼ 159 in Fig. 2). To comparenumerical solutions with the analytics, we calculate the

total coupling to a bath b ¼ ½PNk¼1 A

2k�1=2 directly from the

positions of the spins [see Eq. (1) and below]. The corre-lation decay rate R requires a separate simulation: wecalculate hBð0ÞBðtÞi and find R by fitting it to a decayingexponent, see also [31]. Varying the system parameters in awide range, we simulated baths with N ¼ 15, 59, and 159spins, and found good agreement between numerics andanalytics; typical results are given in Figs. 1 and 2.

The resulting experimental recipe is rather simple. If thedecay has a power law form at shorter times, changing toexponent later, with the decay constants changing as 1=hxand the duration of the two regimes varying with hx, thenthe bath is slow. If the Rabi oscillations decay is exponen-tial, with the rate changing as 1=h2x and the frequency shiftvarying as 1=hx, then the bath is fast. The fast bath can bemade slow by strong increase in hx.

Summarizing, we studied the decay of the Rabi oscil-lations of the central spin interacting with a dipolar-coupled dynamical spin bath. Approximating the effectof the bath as a random field (Ornstein-Uhlenbeck pro-cess), we find analytically the form of the decay. Validity ofthe approximation is confirmed by direct numerical simu-lations. The oscillations decay has interesting features,such as nonmonotonic variation of the decay rate withincreasing the intrabath coupling, and slowing down ofthe decay for fast baths. Studying the Rabi oscillationsdecay may help in experimental characterization of thedynamical spin bath, and is well within experimentalreach, e.g., for NV centers in diamond.

Work at Ames Laboratory was supported by the Depart-ment of Energy—Basic Energy Sciences under ContractNo. DE-AC02-07CH11358. We acknowledge support fromAFOSR (D.D.A.), FOM and NWO (R.H.). A. E. F. ac-knowledges support from the Microsoft Corporation.

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Saikin et al., Phys. Rev. B 75, 125314 (2007).[29] This is in clear contrast with the motional narrowing

phenomenon, when the free oscillations decay rate van-ishes for fast bath [12,13,19,20]. In the motional narrow-ing case, extremely fast bath does not affect the centralspin at all.

[30] W. Zhang et al., J. Phys. Condens. Matter 19, 083202(2007). P-representation sampling, although not exact, isexpected to give accurate results for our problem at largeN.

[31] For N ¼ 15, due to finite-N effects, the correlations decayto a finite value, so that the field BðtÞ, in addition to theOrnstein-Uhlenbeck component, has a static component(of the order of 1=

ffiffiffiffiN

p), and the correlation function has

the form hBð0ÞBðtÞi ¼ b21 expð�RtÞ þ b22, with restrictionb21 þ b22 ¼ b2.

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