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 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.09.003 188 G.P. Berman et al. / Physics Letters A 331 (2004) 187–192 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 aver- age 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.