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Stochastic Resonance and Nonlinear Response by NMR Spectroscopy L. Viola1†, E. M. Fortunato2, S. Lloyd1, C.-H. Tseng2, and D. G. Cory2 1 d’Arbeloff Laboratory for Information Systems and Technology, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 2 Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Let us consider a two-state quantum system whose We revisit the phenomenon of quantum stochastic reso- density operator ρ is represented in terms of the Bloch nance in the regime of validity of the Bloch equations. We vector~sasρ=(1+~s ~σ)/2i.e.,s (t)= σ (t) ,i=1,2,3, find that a stochastic resonance behavior in the steady-state · i h i i in the customary pseudo-spin formalism [7]. Within the responseof thesystemispresentwheneverthenoise-induced semigroup approach for open quantum systems [8], the relaxation dynamics can be characterized via a single relax- 0 most general (completely) positive relaxation dynamics 0 ationtimescale. Thepictureisvalidatedbyasimplenuclear 0 magnetic resonance experiment in water. inducedbythecouplingtosomeenvironmentisdescribed 2 by a quantum Markov master equation of the form 03.65.-w, 05.30.-d, 76.60.-k n 3 a i 1 ρ˙ = [H(t),ρ]+ a [σ ρ,σ ]+[σ ,ρσ ] . (1) J −¯h 2 X kl{ k l k l } The interplay between dissipation and coherent driv- k,l=1 6 ing in the presence of dynamical nonlinearities gives rise 2 The Hamiltonian H(t), which describes the interac- to a variety of intriguing behaviors. The most paradig- tion of the TLS with the (classical) driving field, can 1 matic and counterintuitive example is the phenomenon v of stochastic resonance (SR), whereby the response of be expressed as H(t) = h¯ω0σ3/2 + V(t), V(t) = 5 the system to the driving input signal attains a maxi- ¯h(2ω1)cos(Ωt)σ1/2, where the Larmor frequency ω0 = 9 mum at an optimum noise level [1]. By now, stochastic (E2 E1)/¯h associated with the TLS energy splitting 0 − 1 resonancehas been demonstratedin severaloverdamped andtheRabifrequencyω1proportionaltothealternating fieldamplitude havebeenintroduced. The aboveHamil- 0 bistable systems as diverse as lasers, semiconductor de- 0 vices, SQUID’s, and sensory neurons, the required noise tonian is identical to the one describing a driven tunnel- 0 ing process in a symmetric double-well system with “lo- tuning being accomplished by either controlling the in- h/ jection of external noise or by suitably varying the tem- calized” states provided by σ1-eigenstates: By rotating p perature of the noise-inducing environment. thespincoordinatebyπ/2abouttheyˆ-axis,oneformally - recovers the picture of tunneling in the zˆ-representation t Due to the broad typology of situations it can exem- n that is encountered in the literature [1–6,10]. The dissi- plifyanditsinherentsimplicity,apreferredcandidatefor a pative component of the TLS dynamics is fully charac- theoreticalanalysisis representedbyadrivendissipative u terized by the positive-definite 3 3 relaxation matrix q two-level system (TLS). The investigation has only re- × : cently been taken into the quantum world, where some A= akl , determining the equilibrium state of the sys- v { } tem and the relaxation time scales connected with the i prominentresults have been established for the so-called X spin-boson model. The latter schematizes the archety- equilibration process. The Bloch equations correspond r pal situation of a driven quantum-mechanical tunneling to an especially simple realization of A, the non-zero el- a ements being specified interms of3 independent param- systemincontactwith aharmonic heatbath,the result- ing dissipation being commonly addressed in the linear eters: a11 = a22 = (2T1)−1, a33 = (T2)−1 (2T1)−1, Ohmic regime [2–6]. Within this framework, a quantum a12 = a∗21 =i(√2T1)−1seq. T1,T2, and seq are−identified asthelongitudinalandtransverselifetimes,andtheequi- SRphenomenoninducedbyaresonantirradiationwitha librium value of the population difference respectively. continuous-wavefieldhasbeencharacterizedanalytically Thus, Eq. (1) takes the following familiar form [9]: [3,5] and verified through exact numerical path-integral calculations [3,6]. odnriIavnnecntehtewphop-ersnetosaemtneetqnwuoanonrktou,cmcwuerssysshftoeormwas,thmwahutochaseswrteoildcaehxraastctiloiacnssrdeyos--f  sss˙˙˙321 === −−2ωωω00ss112co−−s(TTΩ22−−t)11sss221−+, T2ω1−11(cso3s(−Ωts)esq3),. (2) namicscanbeaccountedforbyconventionalBlochequa- In microscopic derivations of (2), including the ones tions. We find that, irrespective of the details of the mi- based on the spin-boson model in the appropriate limit croscopic picture, the essential requirement is the emer- [10,5], relaxation is caused by elementary processes in- gence of a single relaxation time scale. The prediction volving noise-assisted transitions between the TLS en- is neatly demonstrated by a nuclear magnetic resonance ergylevelsorpurelydephasingeventswithnoenergyex- experiment on a water sample. change between the system and the environment. The 1 overall relaxation rates T−1 are obtained by integrat- ~µ= (u,v,w) [7]. The steady-state solution to the Bloch 1,2 ing such fluctuation and dissipation effects over the en- equations is well known [11,12]. In particular,χ′ and χ′′ vironmental modes, weighted by the appropriate noise are read from the dispersive and absorptive components spectral densities. Since the latter contain the coupling u,voftheBlochvectorrespectively,andthespectralam- strengthbetweenthesystemandtheenvironment,relax- plitudeη(Ω=ω0,ω1)=(u2+v2)1/2. Asimpleexpression ation rates are themselves directly proportional to the is found for the nonlinear response: underlying noise intensity. Note that the above treat- ω1T2 ment in terms of a constant matrix A is only valid for η(Ω=ω0,ω1)=seq 1+ω12T1T2 . (4) externalfieldsthatarerelativelyweakontheTLSenergy scale i.e., 2ω1 ω0. In spite of the many restrictions Supposenowthatwehavethecapabilityofmanipulat- | | ≪ ingthestrengthofthecouplingoftheTLStoitsenviron- involved, it is remarkable that the Bloch equations (2) are of such a wide applicability to cover the majority of ment,therebychangingtherelaxationtimesT1,T2. Fora magnetic or optical resonance experiments. fixed driving amplitude ω1, η displays purely monotonic For times long compared to the time scales T1,2 of the behaviors if T1,T2 are varied independently. However, transient dynamics, the motion of the system reaches if a single relaxation time is present, T1 = T2 = T12, η develops a local maximum characterizedby a steady-state behavior that is insensitive to the initial condition and acquires the periodicity of the driv- T1∗2 =ω1−1, η(Ω=ω0,ω1,T1∗2)= seq . (5) ing. In particular, the asymptotic TLS coherence prop- 2 erties are captured by the off-diagonal matrix element The occurrence of such a peak in the steady-state re- s1(t), s2(t). It is standard practice to formulate an in- sponseasafunctionofthenoisestrengthcanbepictured put/outputproblem,where the TLSisregardedasa dy- as a stochastic resonance effect in the TLS. Physically, namical system generating s1(t)= σ1(t) as the output the condition for the maximum in (5) can be thought h i signalinresponsetoagiveninputdriveV(t). Byletting of as a synchronization between the periodicity of the limt→∞ σ1(t) = s∞1 (t) denote the limiting steady-state rotating-frame vector in the absence of relaxation and h i value of s1(t), a figure of merit for the system response the additional time scale emerging when dissipation is is the so-called fundamental spectral amplitude [1], present. In semiclassical terms, a constraint of the form η(Ω,ω1)=|s∞1 (t)|=h¯(2ω1)|χ(Ω,ω1)|, (3) Tti1es=ofktTh2e,flkuc=tucaotninsgt.,eninvdiriocnatmesentthaaltfitehldessaploecntgradliffdeernesnit- where the connection to the complex susceptibility directions are not independent upon each other. Simple χ(Ω,ω1)=χ′(Ω,ω1) iχ′′(Ω,ω1) is made explicit. examplesincludenoiseprocessesthateffectivelyoriginate − Thecompetitionbetweendrivinganddissipativeforces from a single direction or that equally affect the system sets the boundary between the linear vs. nonlinear re- in the three directions. In fact, the existence of a single sponse regimes. In the limit where ω1 ≪T1−,21, only first- relaxation time is a feature shared with earlier investi- order contributions in ω1 are significant and the suscep- gations of quantum SR based on the driven spin-boson tibility χ(Ω,ω1) = χ(Ω) in (3) can be calculated within model [3,5], where it arises as a necessary consequence ordinary linear response theory. The linear regime im- of the initial assumption that environmental forces ex- plies that absorption of energy from the applied field clusively act along the tunneling axis. However, we em- occurs without disturbing populations from their equi- phasizethatourdiscussionisdonewithoutreferencetoa librium value s . Linear behavior breaks down when- specificmodel,encompassinginprinciplealargervariety eq ever ω1 >T1−,21. Strongly nonlinear-response regimes can of physical situations. be enter∼ed for arbitrarily weak fields as long as the cou- Apart from this conceptual difference, the SR phe- plingtotheenvironmentandtheinducednoiseeffectsare nomenon evidenced above is characterized by the same weak enough. For both linear and nonlinear driving, the distinctive features found for the spin-boson model on amplitude η(Ω,ω1) of the output signal also depends on resonance [3,5]. According to (5), the maximum steady- the various parameters characterizing the noise process. state response is independent of the driving amplitude, Quite generally,the phenomenonofstochasticresonance whereas the position of the peak shifts toward shorter canbe associatedwiththe appearanceofnon-monotonic relaxation times with increasing ω1. Thus, weaker noise dependencies upon noise parameters, leading to the op- strengths require weaker input fields to attain a large timization of the response at a finite noise level. response. This brings the nonlinear nature of the SR We focus on the Bloch equations (2) with resonant mechanism to light, for weaker dissipation more eas- driving, Ω = ω0. It is then legitimate to invoke ily pushes the system into a regime where ω1T12 > 1. the rotating-wave approximation and replace the al- No SR peak occurs in the limit ω1T12 1 where∼lin- ≪ ternating field V(t) with V(t) = h¯ω1cos(Ωt)σ1/2 ear response theory applies and η seq(ω1T12) seq. − → ≪ ¯hω1sin(Ωt)σ2/2. The rotating-frame description of the Thus, SR results in efficient noise-assisted signal ampli- Blochvector~µisintroducedviathetime-dependentrota- fication. Breakdown of linear behavior is more convinc- tionR=exp(iΩσ3t/2)i.e.,ρR =RρR−1 =(1+~µ ~σ)/2, ingly demonstrated by looking at the dependence of the · 2 response (4) upon the externalfield strength. It is easily after the applicationofa π pulse causingMz(0)= M0. − checked that the condition (5) simultaneously optimizes Values of T2 were inferred from the decay of the echo theresponseagainstω1,withω1∗ =(T1T2)−1/2. However, signals in a standard Carr-Purcell sequence where π ro- it is only when T1 = kT2 that the existence of such an tations wereusedto refocus dephasingdue to inhomoge- optimal field amplitude coexists with a SR effect. neous broadening [13]. For the 5 concentrations utilized, Nuclear Magnetic Resonance (NMR) provides a natu- T1 and T2 were found to be within 1 ms of the average ralcandidateforadirectexperimentalverificationofthe value T12 which is listed for each sample in Table I. predicted phenomenon. In NMR, the Bloch equations (2) describe the motion of the magnetization vector M~ CuSO4 (mM) T12 (ms) ± 1 ms 1 40 45.5 of spin 1/2 nuclei ( H) that are subjected to a static 50 36.5 magnetic field B0 along the zˆ-axis anda radio-frequency 60 28.5 signal with amplitude 2B1 B0 applied at frequency | | ≪ 75 25.0 Ω along the xˆ-axis. The mapping is established by iden- tifying ~s =M~, seq = M0, ω0 = γB0, ω1 = γB1, M0 and 100 18.0 γ denoting the equilibrium magnetization and the gyro- TABLE I.Relaxation time T12 =T1=T2 as a function of magnetic ratio respectively. Relaxation processes arise theCuSO4 paramagnetic impurity concentration for the 5 due to a multiplicity of microscopic mechanisms [12]. water samples used in theexperiment. For a liquid spin 1/2 sample, the leading contribution arises from fluctuations of the local dipolar field caused For each sample, the response to a long external r.f. by bodily motion of the nuclei. The longitudinal relax- pulse was measuredfor variousvalues of the driving am- ation time T1 is essentially determined by the xˆ- and yˆ- plitude. For a given driving amplitude, the duration of components of the local magnetic fields at the Larmor the pulse was increased up to about 200 ms and the frequency, while the transverse lifetime T2 takes extra reading was continued until a constant e.m.f. value was contributions from static components of the zˆ-field, im- reached, confirming that all transient responses had suf- plying that T2 2T1 ordinarily [12,13]. Let us assume ficiently decayed. Under these conditions, the observed ≤ as above that Ω=ω0. Once the steady state is reached, steady-state value is equivalent to the one produced by the magnetization vector M~ precesses about the zˆ-axis a cw-irradiation as assumed in Eqs. (2). A delay long with the periodicity of the r.f. field. The variation of with respect to T12 was waited between each pulse to the dipole momentinthe tranverseplaneinduces amea- allow the sample to return to equilibrium. The value of surable e.m.f. in a Faraday coil. This provides access ther.f. amplitudewasdeterminedbyextrapolatingmea- to the relevant quantity η of Eqs. (3)-(4), which repre- surementsofthenutationrateν1 =ω1/2π atahighfield sents the length of the transverse magnetization vector, setting down to the relevant lower-field domain in the (u2+v2)1/2 =(Mx2(t)+My2(t))1/2. neighbourhoodofω1 ≈T1−21. Whiletherelativeerrorbe- Our experiment consists in probing the steady-state tweentwor.f. settingisfoundtobesmall,thesystematic magnetizationresponseofwaterasafunctionofthenoise error associated with the extrapolation turns out to be strength inducing the natural relaxation processes. The significant. Alinearcorrectionofthefrequencyscalewas 1 HLarmorfrequencyatB0 =9.4Tisω0/2π=400MHz, included in the analysis to compensate for such error. with relaxationtimes T1 =3.6 s,T2 =2.5s. The sample The experimental results are shown in Figs. 1 and 2. canbebroughttoaregimewhereT1 T2 uponaddition Fig. 1evidencestheSRpeakforthreevaluesofthedriv- ≃ of the paramagnetic salt copper sulfate (CuSO4). With ing amplitude. A bell-shaped maximum in the response concentrationsintherangebetween40mMand100mM, profile is clearly visible, as well as the expected shifting collision events with the impurity dominate the nuclear of the peak location with increasing ω1. For each curve, relaxation dynamics. This effectively pushes the system the SR condition T∗ ω−1 of Eq. (5) is in fairly good 12 1 ≈ into a regime of rapidmotion where the correlationtime agreement with the observed behavior, existing discrep- ofthelocalmagneticfieldsseenbythenucleiisveryshort anciesbeing accountedforby the residualerroraffecting on the scale ω0−1, thereby ensuring that T1 = T2 = T12 the determination of ω1. In Fig. 2 the complementary [13]. Higher concentrationsof the CuSO4 additive result characterization of the SR effect in terms of nonlinear in a shorter relaxationtime T12 hence implying an effec- response to the driving field is displayed for three val- tivetuningofthenoisestrength. Allmeasurementswere ues of T12. In each case, ordinary linear response theory performed at room temperature with a Bruker AMX400 is valid for small ω1 to the left side of the maximum. spectrometeronfivewatersampleswithadditiveconcen- Thus, SR reveals itself as a signal optimization marking tration in the above range. the crossoverbetween linear and nonlinear response. IndependentmeasurementsofT1 andT2 weremade to Besidevalidatingthepredictionsfromthe Blochequa- confirmthattheamountofCuSO4 wassufficienttomake tions (2), our experimental results also support the con- them equal. T1 was measured via an inversion recovery clusionsindependentlyreachedinearliertheoreticalanal- technique [13], by lookingat the recoverycurve ofM (t) yses [3,5]. A few remarks are in order concerning the z 3 specific case of NMR. First, the present experiment is Level Systems (Dover,New York,1975). not a stochastic NMR experiment [14]. While the ob- [8] R.AlickiandK.Lendi,Quantum DynamicalSemigroups vious similarity is that both methods probe the nuclear and Applications (Springer-Verlag, Berlin, 1987). spin system by looking at the transverse magnetization [9] F. Bloch, Phys. Rev. 70, 460 (1946). [10] A. Leggett et al.,Rev.Mod. Phys.59, 1 (1987). response, in stochastic NMR the system is directly ex- [11] H. C. Torrey, Phys. Rev. 76, 1059 (1949). cited by noise, which is therefore always extrinsic (and [12] A. Abragam, Principles of Nuclear Magnetism (Oxford classical) in origin. More importantly, as mentioned al- University Press, New York,1961). ready, the solutions to the Bloch equation have a long [13] C. P.Slichter, Principles of Magnetic Resonance (Sprin- history as a tool to investigate magnetic resonance be- ger, Berlin, 1990). haviors. In particular, the existence of an optimum r.f. [14] B.BlumichandD.Ziessow,J.Magn.Res.52,42(1983); amplitude ω1∗ =(T1T2)−1/2 is a feature pointed out long B. Blumich and R. Kaiser, ibid. 54, 486 (1983). agobyBlochhimself[9]. However,onlytheSRparadigm provides the motivation to regard relaxation features as controllable output parameters and to look at the usual response behavior along different axes in the parameter space. Evenoncethisisdone,thisdoesnotautomatically 0.5 leadtoSR.Rather,itistherecognitionthatasingleaxis e T1 =T2 is effectively needed to bring out the fingerprint ns o of the phenomenon and the novel element added to the p s standard NMR analysis. Re In summary,we establishedboth theoretically andex- d e z perimentally the occurrence of stochastic resonance in ali0.4 m two-state quantum systems whose relaxation dynamics or are described by Bloch equations. In addition to sub- N stantiallybroadeningtheexistingparadigmforstochastic resonance in quantum systems, our results point to the possibility of characterizing intrinsic relaxation behavior 0.3 via resonance effects. By offering an optimized way for 20 30 40 Relaxation Time (ms) input/output transmission against noise, stochastic resonance carries a great potential for systems configured FIG. 1. Normalized steady-state response η/seq vs. re- to performspecific signalprocessingand communication laxation time T12 for resonant driving Ω/2π = 400 MHz at tasks [1]. In particular, full exploitation of stochastic differentdrivingamplitudes: ω1/2π= 6.3 Hz(circles), 5.5 Hz (crosses),4.8Hz(squares). Solidlines: theoreticalpredictions resonance phenomena could potentially disclose a useful fromEq. (4),aftersystematiccorrectionforthefrequency(no scenarioforreliabletransmissionofquantuminformation free parameters). in the presence of environmental noise and decoherence. This work was supported by the U.S. Army Research OfficeundergrantnumberDAAG55-97-1-0342fromthe DARPA Microsystems Technology Office. † Corresponding author: [email protected] 0.5 e s n o p0.4 s e R d e z [1] For a comprehensive review, see L. Gammaitoni et al., ali0.3 Rev.Mod. Phys. 70, 223 (1998). orm [2] R.LöfstedtandS.N.Coppersmith,Phys.Rev.Lett.72, N 1947 (1994); Phys. Rev.E 49, 4821 (1994). [3] D. E. Makarov and N. Makri, Phys. Rev. B 52, R2257 0.2 (1995); Phys.Rev. E52, 5863 (1995). [4] M. Grifoni and P. Hänggi, Phys. Rev. Lett. 76, 1611 5 10 15 (1996); Phys.Rev. E54, 1390 (1996). Rabi Frequency (Hz) [5] T. P. Pareek, M. C. Mahato, and A. M. Jayannavar, Phys.Rev.B 55, 9318 (1997). FIG.2. Normalized steady-state response η/seq vs. Rabi [6] M. Thorwart et al.,Chem. Phys. 235, 61 (1998). frequency ω1/2π for resonant driving Ω/2π = 400 MHz at different relaxation times: T12= 18.0 ms (circles), 28.5 ms [7] L. Allen and J. H. Eberly, Optical Resonance and Two- (squares), 45.5 ms (diamonds). Solid lines as above. 4