Wave Function Collapses in a Single Spin Magnetic Resonance Force Microscopy

Physics Letters A 331 (2004) 187–192 www.elsevier.com/locate/pla

Wave function collapses in a single spin magnetic resonance force microscopy

G.P. Berman a, F. Borgonovi b,c,∗,V.I.Tsifrinovichd

a Theoretical Division, MS B213, Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Dipartimento di Matematica e Fisica, Università Cattolica, via Musei 41, 25121 Brescia, Italy c INFM, Unità di Brescia and INFN, Sezione di Pavia, Italy d IDS Department, Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201, USA Received 19 July 2004; received in revised form 1 September 2004; accepted 2 September 2004 Available online 11 September 2004 Communicated by P.R. Holland

Abstract We study the effects of wave function collapses in the oscillating cantilever driven adiabatic reversals (OSCAR) magnetic resonance force microscopy (MRFM) technique. The quantum dynamics of the cantilever tip (CT) and the spin is analyzed and simulated taking into account the magnetic noise on the spin. The deviation of the spin from the direction of the effective magnetic ﬁeld causes a measurable shift of the frequency of the CT oscillations. We show that the experimental study of this shift can reveal the information about the average time interval between the consecutive collapses of the wave function.  2004 Elsevier B.V. All rights reserved.

Recently invented [1,2], the oscillating cantilever in [4–7]. The quantum theory of a single-spin mea- driven adiabatic reversals (OSCAR) technique has surement in OSCAR MRFM was presented in [8,9]. been successfully used for a single spin detection [3]. It was shown in [8,9] that the CT frequency can In this technique the cantilever tip (CT) vibrations take two values corresponding to two possible direc- in combination with a radio-frequency (rf) resonance tions of the spin relative to the direction of the ef- field causes adiabatic reversals of the effective mag- fective magnetic field, Beff. If the spin points initially netic field. Spins of the sample follow the effective in (or opposite to) the direction of Beff, then the CT magnetic field causing a small shift of the CT fre- has a definite trajectory with the corresponding pos- quency, which is to be measured. The quasiclassical itive (negative) frequency shift. In the general case, theory of the OSCAR technique has been developed the Schrödinger dynamics describes a superposition of two possible trajectories—the Schrödinger cat state. It * Corresponding author. was also shown in [8,9] that the interaction between E-mail address: [email protected] (F. Borgonovi). the CT and the environment causes two effects. The

first one is a rapid decoherence: the Schrödinger cat wave function? This Letter discusses the second prob- state transforms into a statistical mixture of two possi- lem. ble trajectories with the definite directions of the spin The CT position and momentum have finite quan- relative to the effective field. Physically this means tum uncertainties. Thus, when the spin direction devi- that the CT quickly selects one of two possible trajec- ates from the direction of Beff, the collapse does not tories, even if initially the spin does not have a definite occur instantly. During a finite time interval, the spin direction relative to the direction of Beff. The second and the CT are entangled. The spin does not have a effect is an ordinary thermal diffusion of the CT tra- definite direction, and the CT does not have a definite jectory. trajectory. The dynamics of the CT–spin system on the What was not taken into account in [8,9] was the time scale less than the time interval between two con- effect of the direct interaction between the spin and secutive collapses can be described by the Schrödinger the environment. In general, for an MRFM system one equation. During this time, the average CT frequency should consider two environments: one for the CT and shift is expected to be smaller (in absolute value) than the other for the spin. If the initial spin wave function the frequency shift corresponding to the definite di- describes a superposition of two spin directions rela- rection of the spin relative to Beff. We show that the tive to Beff, then the spin generates two trajectories for experimental study of this effect can reveal the answer the CT. The cantilever is a quasiclassical device which to a fundamental problem of quantum dynamics: at measures the spin projection relative to the direction what time does the collapse of the wave function occur of Beff. The CT environment collapses the CT–spin if the quasiclassical trajectories are not well separated? wave function selecting only one CT trajectory and a Indeed, if the quasiclassical trajectories are initially definite direction of the spin relative to Beff. This sit- well separated (the Schrödinger cat state) then the uation is similar to the Stern–Gerlach effect, but for a characteristic time of the collapse is the decoherence time-dependent Beff. The direct interaction of the spin time, i.e., the time of vanishing of the non-diagonal with its environment is extremely weak in comparison peaks of the density matrix (the non-diagonal peaks to the interaction between the CT and its environment. describe the quantum correlation between two trajec- However, this weak interaction causes a deviation of tories when these trajectories coexist during the same the spin from the direction of Beff after a collapse of time interval [11,12]). However, the Schrödinger cat the wave function. In turn, this deviation generates two state is a specific bizarre phenomenon in the macro- CT trajectories. This scenario occurs again and again scopic world. Namely, in a typical situation the col- in the OSCAR MRFM. Normally, a collapse “forces” lapse of the wave function occurs long before a well the spin back to its initial (after the previous collapse) defined separation develops between the two quasi- direction relative to Beff. But sometimes this collapse classical trajectories. There are a few simple cases, pushes the spin to the opposite direction, revealing the for which the exact solutions of the master equation quantum jump—a sharp change of the spin direction have been obtained (see, for example, [12]). The ex- and CT trajectory. It was shown in [5–7] that the main act solution describes a generation of the two qua- source of the magnetic noise on the spin was asso- siclassical trajectories, their decoherence, and a ther- ciated with the cantilever modes whose frequencies mal diffusion. Before the two quasiclassical trajecto- were close to the Rabi frequency. ries are well separated, the exact solution describes There are two basic problems associated with the the complicated dynamics of the density matrix ele- single spin OSCAR MRFM. The first problem is ments. It is not clear if the master equation is capable the theoretical description of the statistical properties to describe the wave function collapse when the two of quantum jumps. Unfortunately, the direct simula- trajectories are not well separated. Even if the col- tion of quantum jumps consumes too much computer lapse time for this case is “hidden” in the solution of time to be implemented. Recently, we have consid- the master equation, we still do not know how to ex- ered a simplified model which describes the statistical tract it analytically and numerically. Only experiments properties of quantum jumps [10]. The second prob- could resolve this fundamental problem. We show that lem is more sophisticated: what is the characteristic OSCAR MRFM could become one of these experi- time interval between two consecutive collapses of the ments. G.P. Berman et al. / Physics Letters A 331 (2004) 187–192 189

the wave function can be represented by a product of the CT and spin wave functions. In this case the average spin S has a magnitude 1/2. In the general case, the spin is entangled with the CT, and the average spin is smaller than 1/2. In our estimates we will use the following parameters from experiment [1]:

ωc µN = 6.6kHz,kc = 600 ,B= 300 µT, 2π m 1 5 T G = 4.3 × 10 ,Xm = 10 nm,T= 200 mK, m (3) Fig. 1. Cantilever set-up. Bext is the external static magnetic field, where Xm is the amplitude of the CT. Using these val- B1 the rotating magnetic field, m the magnetic moment of the fer- ues, we obtain the following values of parameters for romagnetic particle glued on the cantilever tip, S the spin to be our model measured. −21 X0 = 85 fm,P0 = 1.2 × 10 Ns,η= 0.078, The OSCAR MRFM setup considered here is Xm 5 shown in Fig. 1. xm = = 1.2 × 10 . (4) X The dimensionless Hamiltonian for the OSCAR 0 =| | MRFM in the rotating system of coordinates is taken The relative frequency shift ξ0 ωc/ωc of the can- as tilever vibrations can be estimated to be [7,8]: 2 p2 + x2 2Sη − H = x + + ξ = = 4.7 × 10 7. (5) εSx 2ηxSz, (1) 0 2 2 + 2 1/2 2 (2η xm ε ) where we use the quantum units of√ the momentum (We insert the factor 2S for future discussion.) P0 = h/X¯ 0 and the coordinate X0 = hω¯ c/kc.Here Suppose that initially the spin points opposite to ωc is the cantilever frequency, kc is the cantilever ef- the direction of Beff. The frequency shift of the CT = fective spring constant, ε γB1/ωc, γ is the mag- vibrations is then −ξ0. The magnetic noise acting on nitude of the electron gyromagnetic ratio and η = the spin causes a deviation of the spin direction from (1/2)(γ X0/ωc)G,whereG = ∂Bz/∂x, is the mag- the direction of the effective magnetic field. Thus, it netic field gradient at the spin location. The first term produces two trajectories of the CT with the frequen- in (1) describes the unperturbed oscillations of the CT, cies ±ξ0. Because of the quantum uncertainty of the the second term describes the interaction between the CT position during the finite time (which we call the spin and the resonant rf field, and the last term de- collapse time τcoll), the wave function of the CT describes the interaction between the spin and the CT. To scribes a single peak with an absolute value of the take into consideration the magnetic noise on the spin frequency shift less than ξ0. The collapse of the wave caused by the thermal CT vibrations we add a random function changes the frequency shift to the value −ξ0 = term ∆(τ)Sz to the Hamiltonian (1),whereτ ωct is (or, sometimes, ξ0, in the case of a quantum jump). the dimensionless time. We simulate the quantum dynamics between two The wave function of the system is assumed to have consecutive collapses of the wave function of the sys- the form tem. In our simplified model the magnetic noise ∆(τ) takes consecutively two values, ±∆ . The time in- Ψ = u (x, τ)α + u (x, τ)β, (2) 0 α β terval between two consecutive “kicks” of the func- where α and β are the basis spin functions in the tion ∆(τ) was taken randomly from the interval Sz representation corresponding to the values Sz = (3τR/4, 5τR/4),whereτR = 2π/ε is the dimension- ±1/2. The Schrödinger equation splits into two cou- less Rabi period. (In a more advanced theory the char- pled equations for uα(x, τ) and uβ (x, τ).Ifthesetwo acteristics of the magnetic noise should be derived functions are identical (up to a constant factor) then from the parameters of the thermal CT vibrations.) 190 G.P. Berman et al. / Physics Letters A 331 (2004) 187–192

The initial wave function is assumed to be a direct product of the CT and spin wave functions. The initial state of the CT is a coherent state | = 1/4 n/2 n − 2 +| |2 α0 π 2 α0 Hn(x) exp x α0 /2 , n (6) where Hn(x) is a Hermite polynomial, and α0 = √1 (x0 +ip0);herex0 and p0 are the quantum mechan- 2 ical averages of x and p at τ = 0. The initial direction of the spin is taken to be opposite to the direction of the effective magnetic field Beff = iε + 2kηx0,where i and k are the unit vectors in the positive x-and z-directions. Fig. 2. Deviation of τj from the unperturbed half period of the In our numerical simulations we expand the func- CT oscillations π, as a function of the number of half periods j,for tions uα(x, τ) and uβ (x, τ) in (2) over 400 eigenfunc- different ∆0 as indicated in the legend. Solid lines are the standard =| − | tions of the unperturbed oscillator Hamiltonian. Dur- linear fits. On the vertical axis δτj τj π . ing the time interval between two consecutive “kicks” of the noise function ∆(τ) we have a time-independent adequate one, then from the experimentally measured Hamiltonian. Thus, we find the evolution of the wave quantity ξ we could determine the time interval be- × function by diagonalizing the 800 800 matrix and tween two collapses τcoll. (It is clear from Fig. 2 that taking into consideration the initial conditions after ξ is directly related to the value of τcoll for the as- each “kick”. The output of our simulations is the sumed magnetic noise parameters.) Certainly, in a real time interval τj = τj − τj−1 between two consec- situation this opportunity does exist if the average fre- utive returns to the origin for the average value x: quency shift ξ is significantly smaller than the ex- − = = x(τj 1) 0and x(τj ) 0. To save computational pected shift ξ0. We will call such situation “the case of time, we have used the values of parameters ε = 10, the strong noise”. η = 0.3, p0 = 0, x0 = 13. These values provide a large Note, that a decrease of the frequency shift may be relative frequency shift of the CT oscillations [8,9] interpreted as an effective decrease of the spin δS.Us- −3 ing Eq. (5) we obtain ξ0 ≈ 7.9 × 10 . (7) (ξ−ξ )(2η2x2 + ε2)1/2 The results of our simulations are shown in Fig. 2, δS = 0 m . (9) 2 which demonstrates the deviation of τj from the un- 2η perturbed half-period of the CT oscillations π.We Previously we have shown that in the case of strong introduce δτj =| τj − π|. With no magnetic noise noise, an experimentalist could determine the time in- = (∆0 0) the deviation δτj does not change with time terval τcoll between two consecutive collapses of the (ξ0 is dimensionless) wave function by measuring the decrease of the frequency shift of the CT vibrations. Here we propose a δτ = δτ = πξ ≈ 0.025. (8) j 0 0 special experiment which allows one to determine the If ∆0 = 0, the value of δτj decreases with time time τcoll for the case of weak noise in which the av- until the collapse of the wave function destroys the erage frequency shift between two collapses is close Schrödinger cat state. (Fig. 2 demonstrates the case for to ξ0. which the time interval between the collapses equals The problem is the following. Even a very weak six half-periods of the CT vibrations.) noise generates a second CT trajectory with a fre- We introduce the effective (relative) frequency shift quency shift opposite to that of the first trajectory. ξj = δτj /π.FromFig. 2 we can find the average ef- Thus, two trajectories tend to separate at the same fective frequency shift ξ between two collapses of rate for any magnetic noise. Correspondingly, the time the wave function. If our model of the noise were the interval between two collapses is expected to be ap- G.P. Berman et al. / Physics Letters A 331 (2004) 187–192 191 proximately the same for any level of magnetic noise. near the Rabi frequency can be estimated to be: However, in the case of weak noise, the probability of 1/2 ωc kBT the second trajectory is small, so its contribution to the aT = = 38 fm. (11) average frequency shift becomes negligible. ωR 2kc To overcome this obstacle, we propose an arti- The square of the characteristic spin deviation during ficial change of the frequency shift using the “in- a single reversal (half of the CT period) is terrupted OSCAR technique” recently implemented 2 in [1].In[1], the rf field is turned off for a time inter- 2 GaT −7 ( θ1) ≈ 3.4 = 9 × 10 . (12) val equal to half of the CT period, which is equivalent ωcXm to the application of the π-pulse which changes the (In the diffusion approximation the square of the direction of the spin relative to the effective magnetic spin deviation is proportional to the number of re- field. We propose to turn off the rf field for the duration versals [6].) We have no idea about the order of the of the quarter of the CT period, which is equivalent to collapse time τcoll. If we assume that the collapse oc- the application of the π/2-pulse. Suppose that initially curs when the separation between the two trajectories the spin is parallel to the effective magnetic field, Beff. with the frequencies 1 ± ξ0 is equal to 1/2 (the quan- If we apply a “π/2-pulse”, the spin will become per- tum uncertainty of the CT position in the coherent state 2 pendicular to Beff. Thus, we have two CT trajectories, is ( x) =1/2), then we obtain for τcoll each with the same probability. Before these two tra- 1 jectories are separated, the CT will oscillate with the τcoll sin τcoll ≈ ≈ 4.4. (13) 4xmξ0 unperturbed frequency ωc. After the collapse the frequency shift is ±ξ0, with It follows from Eq. (13) that the wave function col- equal probabilities. Thus, using a “π/2-pulse” we can lapses during the second period of the CT vibrations. If achieve a maximum possible reduction of the fre- we estimate the probabilities of the two CT trajectories 2 quency shift. If we apply a periodic sequence of “π/2- as P1 ∼ 1 − ( θ1) (for the trajectory with the initial ∼ 2 pulses” with the period τp (τp >τcoll), then the aver- frequency shift) and P2 ( θ1) (for the trajectory age frequency shift ξ is with the opposite frequency shift), then the average CT frequency shift can be estimated to be = · + − 2 ξ 0 τcoll ξ0(τp τcoll) /τp δξ=ξ0(P1 − P2) = ξ0 1 − 2( θ1) . (14)

= ξ0(1 − τcoll/τp). (10) This estimated reduction of the CT frequency shift is clearly negligible. Manipulating τp one can achieve a signiﬁcant de- Consider the doubtful extreme case. Suppose that crease of ξ in comparison with ξ0.UsingEq.(10) the collapse occurs when the separation between the one can determine the collapse time from the experi- two trajectories is of the order of the thermal CT ﬂuc- 1/2 mental value ξ. Thus, the collapse time can be mea- tuations (kBT/kc) ≈ 150 pm or 1760 in dimension- sured for the case of weak magnetic noise. less units (as before, we used the values of parameters One may argue that for application of the effective in (3)). In this case, the time interval between the two 4 “π/2-pulse” we must be sure that the spin is placed consecutive collapses τcoll is of the order of 10 peri- at the center of the resonant slice. We believe that the ods of the CT oscillations. During this time, the char- positioning of the spin can be achieved by moving the acteristic spin deviation is ( θ)2 ∼ 0.02. Then, we 2 −3 cantilever near the spin and measuring the frequency have for the probability P1 ≈ ( θ) /4 ≈ 5 × 10 . −2 shift after the “pulse”. If the values ±ξ0 will be ob- The average frequency shift is δξ≈ξ0(1 − 10 ). served with equal probability then the implemented One can see that even in this extreme case the reduc- “pulse” is a “π/2-pulse”. tion of the frequency shift is expected to be small. Finally, based on quasiclassical theory [6] we Thus, the experimental conditions in [1] probably will estimate the reduction of the average frequency correspond to the case of the weak noise. In such a sit- shift caused by the noise for the experimental condi- uation the collapse time could be measured using the tions [1]. The amplitude of the thermal CT vibrations periodic sequence of “π/2-pulses” described earlier. 192 G.P. Berman et al. / Physics Letters A 331 (2004) 187–192

Finally, we will note the two requirements for Acknowledgements the collapse time measurement. First, the characteristic thermal fluctuation of the CT frequency must be We thank D. Rugar for discussions. This work was smaller than the change of the CT frequency shift supported by the Department of Energy under the con- caused by the quantum collapse. (The signal averaging tract W-7405-ENG-36 and DOE Office of Basic En- may relax this requirement. As an example in experi- ergy Sciences, by the Defense Advanced Research ment [3] the reliable signal was detected for the signal- Projects Agency (DARPA), by the National Security to-noise ratio about 0.06.) Second, the same require- Agency (NSA) and Advanced Research and Develop- ment is valid for the frequency noise caused by the ment Activity (ARDA) under Army Research Office optical measurement of the cantilever position. (The (ARO) contract # 707003. noise caused by the optical measurement in MRFM was studied in [13].) It may happen that the decoherence caused by the CT–environment interaction will References cause a quantum collapse long before the instant when the separation between the two CT trajectories is close [1] H.J. Mamin, R. Budakian, B.W. Chui, D. Rugar, Phys. Rev. to the quantum uncertainty of the CT position. In this Lett. 91 (2003) 207604. [2] B.C. Stipe, H.J. Mamin, C.S. Yannoni, T.D. Stowe, T.W. case the requirements formulated above cannot be sat- Kenny, D. Rugar, Phys. Rev. Lett. 87 (2001) 277602. isfied. [3] D. Rugar, R. Budakian, H.J. Mamin, B.W. Chui, Nature 430 In summary, we suggested a procedure for mea- (2004) 329. suring the mysterious collapse time in the OSCAR [4] G.P. Berman, D.I. Kamenev, V.I. Tsifrinovich, Phys. Rev. A 66 MRFM technique. We simulated the quantum dynam- (2002) 023405. [5] G.P. Berman, V.N. Gorshkov, D. Rugar, V.I. Tsifrinovich, Phys. ics of the spin–CT system. Unlike the previous studies Rev. B 68 (2003) 094402. of the quantum dynamics we took into consideration [6] G.P. Berman, V.N. Gorshkov, V.I. Tsifrinovich, Phys. Lett. the direct interaction between the spin and the envi- A 318 (2003) 584. ronment (the magnetic noise). This noise causes (i) a [7] D. Mozyrsky, I. Martin, D. Pelekhov, P.C. Hammel, Appl. deviation of the spin from the direction of the effective Phys. Lett. 82 (2003) 1278. [8] G.P. Berman, F. Borgonovi, V.I. Tsifrinovich, Quantum In- magnetic field and (ii) entanglement between the spin form. Comput. 4 (2004) 102. and the CT. The spin–CT entanglement influences the [9] G.P. Berman, F. Borgonovi, Z. Rinkevicius, V.I. Tsifrinovich, frequency of the CT oscillations before the wave func- Superlattices, Microstruct. 34 (2003) 509. tion collapse takes place. This effect can be described [10] G.P. Berman, F. Borgonovi, V.I. Tsifrinovich, quant- as an effective decrease of the single spin magnitude. ph/0402063. [11] W.H. Zurek, Phys. Today 44 (1991) 36. We demonstrated that the experimental measurement [12] G.P. Berman, F. Borgonovi, G.V. Lopez, V.I. Tsifrinovich, of the OSCAR MRFM frequency shift could reveal Phys. Rev. A 68 (2003) 012102. information about the time interval between two con- [13] T.A. Brun, H.-S. Goan, Phys. Rev. A 68 (2003) 032301. secutive collapses of the wave function.