Realistic simulations of single-spin nondemolition measurement by magnetic resonance force microscopy

Realistic simulations of single-spin nondemolition measurement by magnetic resonance force microscopy

Todd A. Brun*

Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540, USA Hsi-Sheng Goan^†

Center for Quantum Computer Technology, University of New South Wales, Sydney, New South Wales 2052, Australia 共Received 25 February 2003; published 5 September 2003兲

A requirement for many quantum computation schemes is the ability to measure single spins. This paper examines one proposed scheme: magnetic resonance force microscopy 共MRFM兲, including the effects of thermal noise and back action from monitoring. We derive a simplified equation using the adiabatic approxi- mation and produce a stochastic pure state unraveling which is useful for numerical simulations. We also calculate the signal-to-noise ratio for single-spin measurement by MRFM, using a quantum Langevin equation approach.

DOI: 10.1103/PhysRevA.68.032301 PACS number共s兲: 03.67.⫺a

I. INTRODUCTION

Single-spin measurement is an extremely important chal- lenge, and necessary for the future successful development of several recent spin-based proposals for quantum- information processing 关1–5兴. There are both direct and in- direct single-spin measurement proposals. The idea behind some indirect proposals is to transform the problem of de- tecting a single spin into the task of measuring charge trans- port 关2,6兴, since the ability to detect a single charge is now available. For direct single-spin detection, magnetic reso- nance force microscopy共MRFM兲 has been suggested 关7–9兴 as one of the most promising techniques. To date, the MRFM technique has been demonstrated with sensitivity to a few hundred spins关10,11兴.

In this paper we discuss how to read out the quantum state of a single spin using the MRFM technique based on cyclic adiabatic inversion 共CAI兲 关9,10,12兴. In this CAI MRFM technique, the frequency of the spin inversion in the rotating frame is in resonance with the mechanical vibration of an ultrathin cantilever, allowing it to amplify the otherwise ex- tremely weak force due to the spin. These amplified vibra- tions can then be detected by, e.g., optical methods.

Previous studies关8,9兴 of the dynamics of single-spin mea- surement by MRFM considered only the unitary evolution of the spin and the cantilever system, without including any effects of external environments or measurement devices.

Only recently, the effect of thermal noise environment on the dynamics of the spin-cantilever system in the MRFM was studied 关13兴 by using the Caldeira-Leggett master equation 关14兴 in the high-temperature limit.

There is, however, a macroscopic device in the MRFM

setup which measures the cantilever motion and hence pro- vides information about the spin state. To our knowledge, the back action of the measurement device and the effect of the thermal noise on the dynamics of the cantilever-spin system for the single-spin detection problem by MRFM have not yet been investigated systematically. In this paper, we include, in our analysis, a measurement device共a fiber-optic interferom- eter兲 to monitor the position of the cantilever. We consider various relevant sources of noise and calculate the signal-to- noise ratio of the output photocurrent of the measurement device. We also develop a realistic continuous measurement model and discuss the approximations and conditions to achieve a quantum nondemolition measurement of a single spin by MRFM. Finally, we present some simulation results of the dynamics of the single-spin measurement process.

II. THE MEASUREMENT SCHEME

A schematic illustration of the MRFM setup is shown in Fig. 1. A uniform magnetic field B₀ points in the positive z direction. A single spin is placed in front of the cantilever tip

*Email address: [email protected]

†Mailing address: Center for Quantum Computer Technology, C/-Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia.

Email address: [email protected] FIG. 1. Schematic diagram of the MRFM setup.

1050-2947/2003/68共3兲/032301共14兲/$20.00 68 032301-1 ©2003 The American Physical Society

which can oscillate only in the z direction. A ferromagnetic particle 共or small magnetic material兲 mounted on the canti- lever tip produces a nonuniform magnetic field or magnetic- field gradient of (⳵^Bz/⳵^Z)0 on the single spin. As a result, a reactive force共or interaction兲 acts back on the magnetic can- tilever tip in the z direction from the single spin. The origin is chosen to be the equilibrium position of the cantilever tip without the presence of the spin.

In CAI, the cantilever is driven at its resonance frequency to amplify the otherwise very small vibrational amplitude.

This is achieved by a modulation scheme using the fre- quency modulation of a rotating radio-frequency 共rf兲 mag- netic field in the x-y plane. In this case, the rotating RF field can be represented as B_1x⫽B1cos关␻^t⫹⌬␻^(t)兴, B1y

⫽⫺B1sin关␻^t⫹⌬␻^(t)兴, where the frequency modulation

⌬␻(t) is a periodic function in time with the resonant fre- quency ␻mof the cantilever. In the reference frame rotating with B₁, the spin-cantilever Hamiltonian can be written as

Hˆ_SZ共t兲⫽HˆZ⫺ប

冋

^{␻L⫺␻⫺dtd ⌬␻共t兲}

册

^{Sˆz⫺ប␻1Sˆx}

⫺g␮

冉

^⳵⳵^BZz

冊

0

ZˆSˆ_z, 共1兲

where␻L⫽g␮^Bz/ប and␻1⫽g␮^B1/ប are the Larmor and Rabi frequencies, respectively; B_zincludes the uniform mag- netic field B₀ and the magnetic field produced by the ferro- magnetic particle; g and␮ are the g factor and the electron or nuclear magneton, respectively; and

Hˆ

Z⫽ 1

2mpˆ²⫹m␻m 2

2 Zˆ² 共2兲

is the Hamiltonian of the cantilever in isolation共i.e., with no external magnetic field coupling it to the spin兲. For ␻

⫽␻L, we arrive at an effective cantilever-spin Hamiltonian of the form

Hˆ_SZ共t兲⫽HˆZ⫺2␩^ZˆSˆz⫹ f共t兲Sˆz⫺␧Sˆx, 共3兲 where f (t)⫽d关⌬␻^(t)兴/dt, ␩⫽(g␮^/2)(⳵^Bz/⳵^Z)0, and ␧

⫽ប␻1. We will discuss in detail the rotating picture and adiabatic approximation for the spin-cantilever system in the next section.

In the following, we briefly describe the basic principle of the single-spin measurement by CAI MRFM. In the case where the adiabatic approximation is exact, the instantaneous eigenstates of the spin Hamiltonian in the rotating frame of the B₁ field are the spin states parallel or antiparallel to the direction of the effective magnetic field B^eff(t)⫽„␧,0,

⫺ f (t)…, denoted as 兩v⫾(t)典, respectively. We define an op- erator Sˆ_z⬘for the component of spin along this axis. Note that the initial spin state in the laboratory frame has the same expression as the initial state in the rotating frame. Starting at a general initial spin state 共in the laboratory or rotating frame兲 of

␹共0兲⫽a兩↑典^⫹b兩↓典 ^共4兲

in the Sˆ_zrepresentation, we can rewrite this initial state in the basis of the instantaneous eigenstates of Sˆ_z⬘ ^as

␹共0兲⫽aeff兩v_⫹共0兲典^⫹beff兩v⫺共0兲典^{, 共5兲}

where

a_eff⫽a cos共⌰0/2兲⫹b sin共⌰0/2兲, 共6兲

b_eff⫽⫺a sin共⌰0/2兲⫹b cos共⌰0/2兲, 共7兲

and ⌰0⬅⌰(0) is the initial angle between the effective magnetic field and the z-axis direction. This implies tan关⌰(t)兴⫽Bx

eff(t)/B_z^eff(t)⫽⫺␧/ f (t). It then follows from the adiabatic theorem that the spin state at time t can be written as

␹共t兲⫽aeff兩v_⫹共t兲典^exp

冉

^⫺បi

^冕

^{0t␭⫹共t⬘兲dt⬘}

冊

⫹beff兩v_⫺共t兲典^exp

冉

^⫺បi

^冕

^{0t␭⫺共t⬘兲dt⬘}

冊

^{, 共8兲}

where␭_⫾(t) are instantaneous eigenvalues. So the probabili- ties of finding the spin to be in the instantaneous eigenstates 兩v_⫾(t)典 ^are 兩aeff兩² and兩beff兩², respectively. Since the coeffi- cients a_eff and b_eff are time independent, the probabilities 兩aeff兩²and兩beff兩² remain the same at all times. This provides us with an opportunity to measure the initial spin state prob- abilities at later times.

How do we measure these spin state probabilities? The idea is to transfer the information of the spin state to the state of the driven cantilever. In the interaction picture in which the state is rotating with the instantaneous eigenstates of the spin Hamiltonian, the spin-cantilever interaction can be writ- ten as 2␩^ZˆSˆz⬘^cos关⌰(t)兴. As a result, the phase of the driven cantilever vibrations depends on the orientation of the spin states. Suppose that the initial state is a product state of the cantilever and spin parts. At a later time, due to the interac- tion between them, the total state becomes entangled. Moni- toring the phase of the cantilever vibrations will give us in- formation about the spin. Numerical simulations共see Fig. 3兲 indicate that as the amplitude of the cantilever vibrations increases with time, the phase difference in the oscillations for the two different initial spin eigenstates of Sˆ_z⬘^approaches

␲. In other words, the measurement of the single-spin states can be achieved by monitoring the phases of the cantilever vibrations at some later time t. Phase-sensitive, optical ho- modyne measurements of the cantilever vibrations can be performed using a fiber-optic interferometer. The main pur- pose of this paper is to present a realistic and detailed analy- sis of the single-spin measurement scheme, including the ef- fects of the measurement device and other relevant sources of noise.

III. THE ROTATING PICTURE AND THE ADIABATIC APPROXIMATION

We assume an effective cantilever-spin Hamiltonian of form 共3兲 where for the moment we let f (t) and ␧ be arbi- trary, and Hˆ_Zis the Hamiltonian given by Eq.共2兲. It is useful to group this into three terms

Hˆ

SZ共t兲⫽HˆZ⫹HˆI⫹HˆS共t兲, 共9兲 where

Hˆ_I⬅⫺2␩^ZˆSˆz,

Hˆ_S共t兲⬅ f 共t兲Sˆz⫺␧Sˆx. 共10兲 The state of the cantilever-spin system evolves according to the Schro¨dinger equation

d兩␺共t兲典

dt ⫽⫺i

បHˆ

SZ共t兲兩␺共t兲典^. 共11兲 In realistic cases, the spin part of the Hamiltonian共repre- senting precession under the magnetic field兲 gives an evolu- tion which is very rapid compared to the reaction time of the cantilever. It therefore makes sense to switch to an interac- tion picture in which the state is rotating along with this precession. We do this by introducing a共partial兲 time trans- lation operator

Uˆ_S共t兲⬅:exp⫺i

ប

冋 ^冕

^{0tHˆS共t⬘兲dt⬘}

册

^{:, 共12兲}

where : : indicates that the integral is to be taken in a time- ordered sense; this unitary operator obeys the differential equation

dUˆ_S共t兲

dt ⫽⫺ i

បHˆ_S共t兲UˆS共t兲. 共13兲 We then introduce the state兩␺^{˜ (t)}典 in the rotating picture,

兩␺^˜共t兲典⬅UˆS

†共t兲兩␺共t兲典^, 共14兲

with兩␺^(t)典the solution of the original Schro¨dinger equation 共11兲 at time t. The evolution equation for 兩␺^{˜ (t)}典 ^is

d兩␺^˜共t兲典 dt ⫽dUˆ

S

†共t兲

dt 兩␺共t兲典⫹UˆS

†共t兲d兩␺共t兲典 dt

⫽ i បUˆ

S

†共t兲HˆS共t兲兩␺共t兲典^⫺_ប^iUˆS

†共t兲HˆSZ共t兲兩␺共t兲典

⫽⫺ i បHˆ

Z兩␺^˜共t兲典⫺ i ប 关Uˆ

S

†共t兲HˆIUˆ

S共t兲兴兩␺^˜共t兲典

⫽⫺ i

បHˆ_Z兩␺^˜共t兲典⫹2i␩ ប Zˆ关UˆS

†共t兲SˆzUˆ_S共t兲兴兩␺^˜共t兲典^. 共15兲

We can define a locked spin operator Sˆ_L(t)

Sˆ_L共t兲⬅关UˆS

†共t兲SˆzUˆ

S共t兲兴; 共16兲

in terms of this, the equation of motion for兩␺^˜典 ^becomes

d兩␺^˜共t兲典

dt ⫽⫺ i

បHˆ

Z兩␺^˜共t兲典⫹2i␩

ប ZˆSˆ_L共t兲兩␺^˜共t兲典^. 共17兲

Unfortunately, it is difficult to get an exact solution for Uˆ

S(t) for a general function f (t). This means that it is also difficult to derive an exact expression for Sˆ_L(t), and the rotating picture共15兲, while formally correct, is not very help- ful.

However, while we cannot easily find an exact expression for Uˆ

S(t) for general f (t), we can easily find an approximate solution for a large class of functions. Suppose that␧ is large and f (t) is slowly varying, so that兩 f (t)兩,␧Ⰷ兩f⬘^{(t)/ f (t)}兩 for typical values of f (t) and f⬘(t). Then, Hˆ_S(t) is also slowly varying, and if a spin begins in an instantaneous eigenstate of Hˆ_S(t), it will remain close to an instantaneous eigenstate of Hˆ_S(t) for all times by the adiabatic theorem.

The instantaneous eigenstates of Hˆ

S(t) are

Hˆ_S共t兲兩v⫾共t兲典⫽␭_⫾共t兲兩v⫾共t兲典⬅⫾␭共t兲兩v⫾共t兲典^, 共18兲

where

␭共t兲⫽

冑

^f2共t兲⫹␧², 兩v_⫾共t兲典⫽ ␧

冑

„f共t兲⫿␭共t兲…²⫹␧²兩↓典

⫺ f共t兲⫿␭共t兲

冑

„f共t兲⫿␭共t兲…²⫹␧²兩↑典^. 共19兲 We use these instantaneous eigenvectors and eigenvalues to define an approximation to the unitary operator Uˆ_S(t):

Uˆ

S⬘共t兲⫽Iˆ丢兩v_⫹共t兲典具^v⫹共0兲兩e^⫺i⌽(t)⫹Iˆ丢兩v_⫺共t兲典

⫻具^v⫺共0兲兩e^i⌽(t), 共20兲

with the accumulated phase

⌽共t兲⬅1

ប

冕

^{0t␭共t⬘兲dt⬘. 共21兲}

Note that⌽(t) obeys d⌽(t)/dt⫽␭(t). This implies that

dUˆ

S⬘共t兲

dt ⫽⫺ i

ប ␭共t兲Iˆ丢兩v_⫹共t兲典具^v⫹共0兲兩e^⫺i⌽(t)

⫹ i

ប ␭共t兲Iˆ丢兩v_⫺共t兲典具^v⫺共0兲兩e^i⌽(t)

⫹Iˆ丢d兩v_⫹共t兲典

dt 具^v⫹共0兲兩e^⫺i⌽(t)

⫹Iˆ丢d兩v_⫺(t)典

dt 具^v⫺共0兲兩e^i⌽(t)

⫽⫺ i បHˆ

S共t兲UˆS⬘共t兲

⫹Iˆ丢d兩v_⫹共t兲典

dt 具^v⫹共0兲兩e^⫺i⌽(t)

⫹Iˆ丢d兩v_⫺(t)典

dt 具^v⫺共0兲兩e^i⌽(t), 共22兲 which has the form of Eq. 共13兲 plus some additional terms.

From definition共19兲 of 兩v_⫾(t)典^{, we see} d兩v_⫾共t兲典

dt ⫽⫾1

2

␧

␭²共t兲 d f共t兲

dt 兩v_⫿共t兲典^. 共23兲 Provided that f (t) is slowly varying, the additional terms in Eq. 共22兲 will be small.

Just as before, we can define a rotating picture; now using the unitary transformation Uˆ

S⬘^(t),

兩␺^˘共t兲典^⬅„UˆS⬘共t兲…^†兩␺共t兲典^{. 共24兲} This gives us a new evolution equation for兩␺^˘典^:

d兩␺^˘共t兲典

dt ⫽d关UˆS⬘共t兲兴^†

dt 兩␺共t兲典⫹关UˆS⬘共t兲兴^†d兩␺共t兲典 dt

⫽⫺ i

បHˆ_Z兩␺^˘共t兲典⫹2i␩

ប Zˆ„关UˆS⬘共t兲兴^†Sˆ_zUˆ

S⬘共t兲…兩␺^˘共t兲典

⫹Iˆ丢

冉

^{兩v⫹共0兲典d具vdt⫹共t兲兩e⫺i⌽(t)}

⫹兩v_⫺共0兲典^d具^v⫺共t兲兩

dt e^i⌽(t)

冊

^{UˆS⬘共t兲兩␺˘共t兲. 共25兲}

At this point, it is helpful to introduce a new set of spin operators

Sˆ_x⬘⫽¹2Iˆ_丢„兩v⫹共0兲典具^v⫺共0兲兩⫹兩v_⫺共0兲典具^v⫹共0兲兩…, Sˆ_y⬘⫽i

2Iˆ_丢„兩v⫺共0兲典具^v⫹共0兲兩⫺兩v_⫹共0兲典具^v⫺共0兲兩…, Sˆ_z⬘⫽¹2Iˆ_丢„兩v⫹共0兲典具^v⫹共0兲兩⫺兩v_⫺共0兲典具^v⫺共0兲兩…. 共26兲 Using definition共20兲 for UˆS⬘(t), we can solve for the various terms in Eq. 共25兲:

关UˆS⬘共t兲兴^†Sˆ_zUˆ

S⬘共t兲⫽⫺ f共t兲

␭共t兲Sˆ_z⬘⫺ ␧

␭共t兲兵^Sˆx⬘^cos关2⌽共t兲兴

⫺Sˆy⬘^sin关2⌽共t兲兴其^{. 共27兲}

Iˆ_丢

冉

^{兩v⫹共0兲典d具vdt⫹共t兲兩e⫺i⌽(t)}

⫹兩v_⫺共0兲典^d具^v⫺共t兲兩

dt e^i⌽(t)

冊

^{UˆS⬘共t兲}

⫽ i␧ ប␭²共t兲

d f共t兲

dt 兵^Sˆx⬘^sin关2⌽共t兲兴⫹Sˆy⬘^cos关2⌽共t兲兴其^. 共28兲 Substituting Eqs.共26兲–共28兲 into Eq. 共25兲, we get

d兩␺^˘共t兲典

dt ⫽⫺i

បHˆ_Z兩␺^˘共t兲典⫹2i␩ ប Zˆf共t兲

␭共t兲Sˆ_z⬘兩␺^˘共t兲典

⫹2i␩ ប Zˆ ␧

␭共t兲兵^Sˆx⬘^cos关2⌽共t兲兴

⫺Sˆy⬘^sin关2⌽共t兲兴其兩␺^˘共t兲典

⫹i ␧

ប␭²共t兲 d f共t兲

dt 兵^Sˆx⬘^sin关2⌽共t兲兴

⫹Sˆy⬘^cos关2⌽共t兲兴其^兩␺^˘共t兲典^. 共29兲 Note that this equation is still exact—it is equivalent to the original Schro¨dinger equation共11兲. However, we can see that if兩 f (t)兩,␧ are large, then ⌽(t) will be a rapidly growing function, and the last two terms of Eq. 共29兲 will oscillate very rapidly compared to the first two terms. Over a short period relative to the response time of the cantilever they will essentially average away to nothing. In this limit, there- fore, we can reasonably make a rotating-wave approxima- tion, to get the approximate evolution equation

d兩␺^˘共t兲典

dt ⬇⫺ i

ប兵^HˆZ⫺2␩关 f 共t兲/␭共t兲兴ZˆSˆz⬘其^兩␺^˘共t兲典^. 共30兲 This is equivalent to making an exact adiabatic approxima- tion, as described in Sec. II. We can see how this approxi- mation compares to the complete Hamiltonian for a reason- able set of parameter values in Fig. 2. This set of parameters was chosen to match those of Berman et al. 关9兴—see Sec.

VII for further details on the simulation. A comparison shows that our results match their unitary simulations to a good precision. If the initial state is a Gaussian wave packet, it remains very close to a Gaussian at later times, just as in Ref. 关9兴; indeed, under the approximate Hamiltonian the state remains an exact Gaussian at all times. For the duration of our numerical simulations, the wave packets of the full and approximate equations remained virtually indistinguish- able.

We should point out, however, that while the parameters of the cantilever and driving force are plausible for near-term experiments, the initial condition shown is atypical. Gener- ally, thermal noise will cause the cantilever to begin with a rather higher amplitude than that shown. In this case, it will take longer for the phase difference between the two spin states to become fully evident. This might be important if spin-relaxation effects are taken into account.

For the rest of this paper we will be using the rotating- wave approximation and representing states in the rotating frame. For simplicity, we henceforth omit the accent from the state兩␺^˘典^.

In this rotating-wave approximation, if the spin begins in an instantaneous eigenstate of Hˆ

S(t), it will remain in an instantaneous eigenstate at all times. If it begins in a super- position of the two eigenstates, the spin and cantilever de- grees of freedom will become entangled, with the two com- ponents of the wave function corresponding to the two spin directions remaining undisturbed for all times. Monitoring the position of the cantilever then serves as a nondemolition measurement of the spin.

Note that the corrections to the adiabatic approximation include terms which can flip the spin. These terms must re- main small for the system to be a true nondemolition mea- surement. The result of the spin measurement manifests itself as a␲ phase shift in the oscillation of the cantilever. We can see this in Fig. 3.

IV. THE THERMAL ENVIRONMENT

Unfortunately, in practice we cannot treat the cantilever as an isolated system. It is coupled at least weakly to the vibra- tional modes of the bulk, and is therefore subject to dissipa- tion and thermal noise. Since the cantilever can be treated as a single harmonic oscillator, we can model the effects of this thermal bath by the well-known Caldeira-Leggett关14兴 mas- ter equation in the high-temperature limit:

␳^˙⫽⫺i

ប 关Hˆ_SZ共t兲,␳兴⫺i␥m

ប 关Zˆ,兵^pˆ,␳其^兴⫺ ␥m

2ᐉ²†Zˆ,关Zˆ,␳兴‡, 共31兲

where the parameters are

␥m⫽ ⌫ 2m,

ᐉ⫽ ប

2

冑

^{mkT, 共32兲}

m is the cantilever mass, T is the temperature, k is Boltz- mann’s constant 共or the equivalent for our system of units兲, and⌫ is the strength of the coupling to the thermal bath. We can interpret ␥m 共with units of inverse time兲 as the dissipa- tion rate andᐉ 共with units of length兲 as the thermal de Bro- glie wavelength.

A feature of this equation is that it does not necessarily preserve the positivity of ␳ on short-time scales 共though at long times it is well behaved兲 关15兴. This arises because of the approximations which are made in the derivation, which be- come invalid at very short times. While this may be physi- cally unimportant, it can be inconvenient; in particular, if we wish to unravel the evolution into a stochastic Schro¨dinger equation关16兴 共as we will show in Sec. VI兲, it is necessary to start with a master equation in the Lindblad form关17兴

␳^˙⫽⫺ i

ប 关Hˆ ,␳兴⫹

兺

_j ^{关2Lˆj␳Lˆ†j⫺兵Lˆ†jLˆj,␳其兴 共33兲}

FIG. 2. Mean cantilever position具^Zˆ典vs t for the complete and rotating-wave Hamiltonians.

FIG. 3. Mean cantilever position具^Zˆ典vs t for initial spin up and down in the Sˆ_z⬘^direction.

for some Hermitian Hˆ and a set of general Lindblad opera- tors 兵^Lˆj其. The Caldeira-Leggett equation共31兲 is not of this form, which is why it can violate the positivity of␳^.

The exact quantum Brownian-motion master equation was shown 关15兴 not to have the Lindblad form, but rather requires time-dependent coefficients to ensure the positivity of the density matrix at short times. However, by keeping more terms from the high- or medium-temperature-limit ex- pansion in a consistent way, Dio´si 关18兴 showed that the Caldeira-Leggett equation can be replaced by another master equation which is of the Lindblad form, and which agrees with it except at very short times when the equation’s valid- ity is questionable in any case. This is done by adding a term to Eq.共31兲 of the form ⫺(␥mᐉ²/2ប²)†pˆ,关pˆ,␳兴‡. The proce- dure is analogous to completing the square. If we choose the ansatz

Lˆ⫽AZˆ⫹iBpˆ 共34兲

with real A,B, plug it into Eq. 共33兲, and equate it to the Caldeira-Leggett equation 共31兲 plus the additional term, we get

␳^˙⫽⫺共i/ប兲关Hˆ,␳兴⫺A²†Zˆ,关Zˆ,␳兴‡⫺B²†pˆ,关pˆ,␳兴‡

⫹iAB共⫺2Zˆ␳^pˆ⫹Zˆpˆ␳⫹␳^{Zˆ pˆ}⫹2pˆ␳^Zˆ⫺pˆZˆ␳⫺␳^pˆZˆ兲

⫽⫺共i/ប兲关HˆSZ共t兲,␳兴⫺ ␥m

2ᐉ²†Zˆ,关Zˆ,␳兴‡⫺␥mᐉ²

2ប² †pˆ,关pˆ,␳兴‡

⫹i␥m

ប 共pˆ␳^Zˆ⫺Zˆpˆ␳⫺Zˆ␳^pˆ⫹␳^pˆZˆ兲, 共35兲 which implies that

A⫽

冑

␥m/2ᐉ², B⫽

冑

␥mᐉ²/2ប²,

Hˆ⫽HˆSZ共t兲⫹共␥m/2兲共Zˆpˆ⫹pˆZˆ兲⬅HˆSZ⬘ 共t兲. 共36兲 So the Lindblad operator for this equation is

Lˆ⫽

冑

␥m/2关共1/ᐉ兲Zˆ⫹i共ᐉ/ប兲pˆ兴, 共37兲 and the effective Hamiltonian, going to the rotating picture and making use of the approximation derived in Sec. III, is

Hˆ

SZ⬘ 共t兲⫽ 1

2mpˆ²⫹m␻m 2

2 Zˆ²⫺2␩关 f 共t兲/␭共t兲兴ZˆSˆz⬘

⫹共␥m/2兲共Zˆpˆ⫹pˆZˆ兲. 共38兲 In order for the cantilever to be an effective measurement device, the loss rate must be very low:␻mⰇ␥m.

V. THE EFFECTS OF MONITORING

In order to serve as a measurement scheme, we must have some way of monitoring the motion of the cantilever. Be-

cause of the microscopic scale of the motion, this is not so easily done. One approach is to use optical interferometry to measure the cantilever position.

As shown in Fig. 1, the cantilever forms one side of an optical microcavity and the cleaved end of the fiber forms the other side. As the cantilever moves, the resonant fre- quency of the cavity changes. Because the time scale of the cantilever’s motion is very long compared to the optical time scale, we can treat the effects of this in the adiabatic limit.

The cavity mode is also subject to driving by an external laser, and has a very high loss rate. The full master equation 关19兴 for the cantilever-spin-cavity system in the interaction picture is

␳^˙⫽⫺ i ប 关Hˆ

SZ⬘ 共t兲,␳兴⫹2Lˆ␳^Lˆ†⫺Lˆ^†Lˆ␳⫺␳^Lˆ†Lˆ⫺i关E共aˆ^†⫹aˆ兲

⫹aˆ^†aˆ共⌬⫹␬^Zˆ兲,␳兴⫹共␥c/2兲共2aˆ␳^aˆ†⫺aˆ^†aˆ␳⫺␳^aˆ†aˆ兲, 共39兲 where Hˆ

SZ⬘ (t) and Lˆ are the Hamiltonian and the Lindblad operator for the cantilever and spin given by Eqs. 共37兲 and 共38兲, E is the strength of the laser driving, ⌬ is the detuning from the ‘‘neutral’’ cavity frequency, ␬ is the coupling strength of the cantilever to the cavity mode, and ␥c is the loss rate of the cavity.

Suppose now that we perform a homodyne measurement 关20,21兴 on the light which escapes from the cavity. We would like to replace Eq. 共39兲 with an equation for the conditional evolution of␳, conditioned on the output photocurrent I_c(t).

The conditional evolution equation for our system then be- comes 关21,22兴 共in Itoˆ calculus form兲

d␳⫽⫺ i ប 关Hˆ

SZ⬘ 共t兲,␳兴dt⫹共2Lˆ␳^Lˆ†⫺Lˆ^†Lˆ␳⫺␳^Lˆ†Lˆ兲dt

⫺i关E共aˆ^†⫹aˆ兲⫹aˆ^†aˆ共⌬⫹␬^Zˆ兲,␳兴dt⫹共␥c/2兲共2aˆ␳^aˆ†

⫺aˆ^†aˆ␳⫺␳^aˆ†aˆ兲dt⫹

冑

␥ce_d共aˆ␳⫹␳^aˆ†⫺具^aˆ⫹aˆ^†典␳兲dWt, 共40兲 where 0⭐ed⭐1 is the detector efficiency and dWtis a real stochastic differential variable which obeys the statistics

M关dWt兴⫽0, M关dWtdW_s兴⫽␦共t⫺s兲dsdt, 共41兲 with M denoting an ensemble average. This noise is related to the output photocurrent关20–22兴

I_c共t兲⫽␤

冋

^{␥ced具aˆ⫹aˆ†典t⫹}

^冑

^{␥ced dWdtt}

册

^{, 共42兲}

where␤ is a constant giving the device’s range of response.

We want to operate in the ‘‘bad cavity’’ limit where ␥c

Ⰷ␻m. This means that the cavity mode will approach equi- librium on a time scale very short compared to that of the cantilever’s motion, so that the cavity mode can be adiabati- cally eliminated 关19,21–22兴 from this equation, leaving an equation in terms of the spin and cantilever position alone.

Let the detuning vanish,⌬→0, and the coupling␬ ^{to the} cantilever be very small. If we initially neglect this coupling altogether, we can solve for the steady state of the cavity mode in isolation from the cantilever:

⫺i关E共aˆ^†⫹aˆ兲,␳兴⫹共␥c/2兲共2aˆ␳^aˆ†⫺aˆ^†aˆ␳⫺␳^aˆ†aˆ兲⫽0, 共aˆ␳⫹␳^aˆ†⫺具^aˆ⫹aˆ^†典␳兲⫽0, 共43兲 which implies that ␳⫽兩␣0典具␣0兩, where aˆ兩␣0典⫽␣0兩␣0典 ^{is a} coherent state with

␣0⫽⫺2iE

␥c

. 共44兲

Now let us restore the coupling␬ between the cantilever and the cavity mode. If this coupling is very small, then the state of the cavity mode will remain very close to state兩␣0典^. In this case, it is very useful to switch to a displaced basis 关19,21,22兴 for the cavity mode. We switch from the operators aˆ,aˆ^† to displaced operators

bˆ⬅aˆ⫺␣0,

bˆ^†⬅aˆ^†⫺␣0*, 共45兲

and displaced number states

bˆ^†bˆ兩n典⫽n兩n典^. 共46兲 Obviously,兩0典⫽兩␣0典 ^and兩1典⫽aˆ^†兩␣0典⫺␣0*_兩␣0典^.

We now make the ansatz of keeping the two lowest dis- placed number states兩0,1典of the cavity mode and neglecting the rest关19,21,22兴. We then write the full density matrix for the spin-cantilever-cavity system as

␳共t兲⫽␳0共t兲丢兩0典具^0兩⫹␳1共t兲丢兩1典具^0兩⫹␳1

†共t兲丢兩0典具^1兩

⫹␳2共t兲丢兩1典具¹兩, 共47兲

where␳0,1,2 are operators which act on the Hilbert space of the cantilever and spin, and␳0,2are self-adjoint. The reduced density matrix of the spin-cantilever system alone is obtained by tracing out the cavity mode, yielding

␳SZ共t兲⫽␳0共t兲⫹␳2共t兲. 共48兲 If we substitute definitions 共45兲 and 共47兲 into the stochastic master equation 共40兲 and collect terms, we get a set of coupled equations in the operators ␳0,1,2:

d␳0⫽

冉

^{⫺ប 关i HˆSZ⬘ 共t兲,␳0兴⫹2Lˆ␳0Lˆ†⫺Lˆ†Lˆ␳0⫺␳0Lˆ†Lˆ}

冊

^dt

⫺4i␬^E2

␥c

2 关Zˆ,␳0兴dt⫹2␬^E

␥c 共Zˆ␳1⫹␳1

†Zˆ兲dt⫹␥c␳2dt

⫹

冑

␥ce_d共␳1⫹␳1

†⫺␳0Tr兵␳^1⫹␳1

†其兲dWt, 共49兲

d␳1⫽

冉

^{⫺ប 关i HˆSZ⬘ 共t兲,␳1兴⫹2Lˆ␳1Lˆ†⫺Lˆ†Lˆ␳1⫺␳1Lˆ†Lˆ}

冊

^dt

⫺i␬^Zˆ␳1dt⫺4i␬^E2

␥c

2 关Zˆ,␳1兴dt⫺2␬^E

␥c

共Zˆ␳0⫺␳2Zˆ兲dt

⫺共␥c/2兲␳1dt⫹

冑

␥ce_d共␳2⫺␳1Tr兵␳1⫹␳1

†其兲dWt, 共50兲

d␳2⫽

冉

^{⫺ប 关i HˆSZ⬘ 共t兲,␳2兴⫹2Lˆ␳2Lˆ†⫺Lˆ†Lˆ␳2⫺␳2Lˆ†Lˆ}

冊

^dt

⫺

冉

^{i␬⫹4i␥␬c2E2}

冊

^{关Zˆ,␳2兴dt⫺2␥␬cE共Zˆ␳1†⫹␳1Zˆ兲dt}

⫺␥c␳2dt⫺

冑

␥ce_d␳2Tr兵␳^1⫹␳1

†其^{dWt. 共51兲}

Both␳1 and␳2 contain damping terms, which imply that they will remain small at all times, provided ␬Zˆ is suffi- ciently small compared to ␥c. 共This also implies that our ansatz is reasonable for sufficiently small␬^.兲

By making use of the above equations, we can find the evolution equation for the reduced density matrix ␳SZ:

d␳SZ共t兲⫽d␳0共t兲⫹d␳2共t兲

⫽

冉

^{⫺ប 关i HˆSZ⬘ 共t兲,␳SZ兴⫹2Lˆ␳SZLˆ†⫺Lˆ†Lˆ␳SZ}

⫺␳SZLˆ^†Lˆ

冊

^dt⫺4i␥^␬c^E2

2 关Zˆ,␳SZ兴dt

⫹2␬^E

␥c 关Zˆ,␳1⫺␳1

†兴dt⫺i␬关Zˆ,␳2兴dt

⫹

冑

␥ce_d共␳1⫹␳1

†⫺␳SZTr兵␳^1⫹␳1

†其^兲dWt. 共52兲 If we keep only terms to second order in␬Zˆ we can neglect the ␳2 term. This leaves only the terms proportional to ␳1

⫾␳1

†, which we need know only to leading order in ␬^Zˆ.

Provided 共as we have already assumed兲 that the cantilever moves slowly compared to the time scale set by␥c and that

␬Zˆ can be treated as small, then to leading order d␳1 van- ishes;␳1 remains in an approximate equilibrium state. If we make use of this assumption we can共again to leading order兲 solve for␳1⫾␳1

†:

␳1⫹␳1

†⬇⫺4␬^E

␥c

2 兵^Zˆ,␳SZ其^,

␳1⫺␳1

†⬇⫺4␬^E

␥c

2 关Zˆ,␳SZ兴, 共53兲

which when inserted into Eq. 共52兲 gives us a closed evolu- tion equation for ␳SZ:

d␳SZ共t兲⫽

冉

^{⫺ប 关i HˆSZ⬘ 共t兲,␳SZ兴⫹2Lˆ␳SZLˆ†⫺Lˆ†Lˆ␳SZ}

⫺␳SZLˆ^†Lˆ

冊

^dt⫺4i␥^␬c^E2

2 关Zˆ,␳SZ兴dt

⫺8␬^2E2

␥c

3 †Zˆ,关Zˆ,␳SZ兴‡dt⫹

冑

␥ce_d4␬^E

␥c 2

⫻共Zˆ␳SZ⫹␳SZZˆ⫺2␳SZTr兵^Zˆ␳SZ其^{兲dWt. 共54兲} 共Note that we have absorbed a factor of ⫺1 into dWt.)

Examining the terms in Eq.共54兲, we see that by eliminat- ing the cavity mode we get another effective term in the Hamiltonian and another Lindblad operator. We can there- fore write this stochastic master equation in the form

d␳SZ共t兲⫽⫺ i ប 关Hˆ

eff共t兲,␳SZ兴dt⫹

兺

_j⫽1^{2 共2Lˆj␳SZLˆj†⫺Lˆ†jLˆj␳SZ}

⫺␳SZLˆ^†_jLˆ_j兲dt⫹

冑

^2ed关共Lˆ2⫺具^Lˆ2典兲␳SZ

⫹␳SZ共Lˆ2⫺具^Lˆ2典兲兴dWt, 共55兲 where we define

Lˆ₁⫽

冑

␥m/2关共1/ᐉ兲Zˆ⫹i共ᐉ/ប兲pˆ兴, Lˆ₂⫽

冑

⁸␬^2E2/␥c

3Zˆ,

Hˆ_eff共t兲⫽ 1

2mpˆ²⫹m␻m 2

2 Zˆ²⫺2␩关 f 共t兲/␭共t兲兴ZˆSˆz⬘⫹4␬^E2

␥c 2 Zˆ

⫹共␥m/2兲共Zˆpˆ⫹pˆZˆ兲. 共56兲

Note that the term 4␬^E2Zˆ/␥c

2 is a constant force, which just displaces the equilibrium position of the cantilever. It can be eliminated simply by changing the origin of Zˆ, and is in any case small for reasonable values of the parameters. The out- put from the homodyne measurement now corresponds to a measurement of the cantilever position具^Zˆ典^:

I_c共t兲⫽␤

冉

^{⫺8e␥d␬cE具Zˆ典⫹}

^冑

^{␥ced dWdtt}

冊

^{. 共57兲}

As we shall see in the following section, we can further unravel this stochastic master equation共55兲 into a stochastic Schro¨dinger equation for pure states. This further unraveling provides a considerable improvement in numerical effi- ciency, though it does not represent an actual measurement process.

VI. PURE STATE UNRAVELING

The stochastic master equation 共55兲 represents the evolu- tion of the cantilever-spin system, conditioned on the photo- current measurement record I_c(t). If we averaged over all possible measurement records, the dW_tterms would average

to zero, and we would be left with an ordinary deterministic master equation for the cantilever and spin. It is for this reason that the stochastic master equation is therefore often referred to as an unraveling of the average master equation.

For numerical purposes, it is often much easier to solve an equation for a pure state vector rather than a density matrix 关16,23兴. It is therefore useful to unravel Eq. 共55兲 still further to an equation which preserves pure states. We do this by introducing two additional stochastic processes to account for the thermal noise and the inefficiency of the detector.

We introduce the new master equation

d␳SZ共t兲⫽⫺ i

ប 关Hˆ_eff共t兲,␳SZ兴dt⫹

兺

j⫽1 2

共2Lˆj␳SZLˆ^†_j⫺Lˆj

†Lˆ_j␳SZ

⫺␳SZLˆ_j^†Lˆ_j兲dt⫹

冑

²关共Lˆ1⫺具^Lˆ1典兲␳SZ

⫹␳SZ共Lˆ1⫺具^Lˆ1典兲兴dW1t⫹

冑

^2ed关共Lˆ2⫺具^Lˆ2典兲␳SZ

⫹␳SZ共Lˆ2⫺具^Lˆ2典兲兴dW2t⫹

冑

²共1⫺ed兲

⫻关共Lˆ2⫺具^Lˆ2典兲␳SZ⫹␳SZ共Lˆ2⫺具^Lˆ2典兲兴dW3t, 共58兲

where the Hamiltonian and the Lindblad operators are the same as in Eq. 共56兲 and we now have three independent noise processes represented by stochastic differential vari- ables dW_1t, dW_2t, and dW_3t which satisfy

M关dWjt兴⫽0, M关dWitdW_js兴⫽␦共t⫺s兲␦i jdsdt. 共59兲 If we take the mean of Eq. 共58兲 over dW1t and dW_3t, we recover Eq. 共55兲. We can think of the additional stochastic processes as representing fictitious additional measurements, whose outcomes we average over to recover the state which is conditioned on the actual measurement.

However, Eq.共58兲 has a great advantage over Eq. 共55兲. If

␳SZis initially a pure state␳SZ⫽兩␺SZ典具␺SZ兩, it will remain a pure state at all times, the state, of course, depending on the stochastic processes W₁, W₂, and W₃. We can recover the solution of Eq.共55兲 by averaging

␳SZ共t兲⫽MW₁,W₃关兩␺SZ共t兲典具␺SZ共t兲兩兴. 共60兲 It would be useful to replace Eq. 共58兲 with an explicit evolution equation for兩␺SZ典^{instead of}␳SZ. This equation is the quantum state diffusion equation with real noise关24,25兴:

d兩␺SZ典⫽⫺ i បHˆ

eff共t兲兩␺SZ典^dt⫹

兺

_j⫽1^{2 共2具Lˆj†典Lˆj⫺Lˆ†jLˆj}

⫺兩具^Lˆj典兩²兲兩␺SZ典^dt⫹

冑

²共Lˆ1⫺具^Lˆ1典兲兩␺SZ典^dW1t

⫹

冑

^2ed共Lˆ2⫺具^Lˆ2典兲兩␺SZ典^dW2t

⫹

冑

²共1⫺ed兲共Lˆ2⫺具^Lˆ2典兲兩␺SZ典^dW3t. 共61兲 The nonlinearity of this equation arises to preserve the norm.

VII. NUMERICAL SIMULATION

We have simulated this system using the C⫹⫹ quantum state diffusion library关26兴 to numerically solve both the uni- tary evolution with Hamiltonian 共30兲 and the stochastic equation 共61兲. All the figures in this paper were generated using this software.

We chose our parameters based on those used by Berman et al.关9兴. These values are 共in arbitrary units兲

ប⫽␻m⫽m⫽1,

␩⫽0.3,

␧⫽400.0,

␥m⫽␻m/Q⫽10^⫺5,

k_BT⫽10⁵, 共62兲

where Q is the quality factor of the cantilever. The driving force f (t) takes the form

f共t兲⫽

再

^{⫺6000⫹300t1000 sin}共t⫺20兲 if t⬎20.^{if 0⭐t⭐20,} 共63兲 If we make contact with physical values for actual cantile- vers used in experiments, we have ␻m⬇10⁵ s^⫺1 and m

⬇10^⫺12kg. The value of k_BT above then corresponds to a temperature of around 0.1 K, which is within the bounds of experimental feasibility, though rather lower than the tem- peratures used in the current experiments共around 3 K兲 关11兴.

These are the physical values assumed in plotting the various figures. Since ␩⫽(g␮^/2)(⳵^Bz/⳵^Z)0, the value of ␩ ^corre- sponds to a field gradient of about 1.5⫻10⁷ T/m, which is higher than the current experiments by roughly two orders of magnitude 关11兴, but hopefully this too will improve with time. The cantilever would undergo displacements of about a nanometer.

Alternatively, rather than increasing the field gradient we could achieve similar numbers by lowering the spring con- stant of the cantilever, for instance, by shrinking the mass of the cantilever. Lowering the mass by a factor of 100 has the same relative effect on␩as increasing the field gradient by a factor of 10.

We then might ask about realistic parameters for the monitoring. A typical cavity size L is about a micrometer, with a laser frequency of␻c⬇1.4⫻10¹⁵s^⫺1. This cavity is generally quite lossy; reasonable quality factors might be in the range Q_c⬃10–100. The parameter E is a function of the laser power, E⫽

冑

^P␥c/ប␻c⫽

冑

^P/បQc. For P⬃1 ␮^{W and} Q_c⬃100 we have E⬃10¹³s^⫺1. The coupling between the cantilever and the cavity is given by a geometric factor ␬

⫽␻c/L⬃1.4⫻10²¹(m s)^⫺1. In arbitrary units, this gives coefficients

8␬^E

␥c

⫽1.9⫻10³, 共64兲

4␬^E2

␥c

2 ⫽7⫻10²,

冑

^8␬␥²c^E2 3 ⫽0.07.

The first value is the multiplier in Eq.共57兲; the second gives the equilibrium displacement of the cantilever; the third is the coefficient of the Lindblad operator Lˆ₂.

One question we can now easily address is how quickly the state of the spin collapses onto eigenstates of Sˆ_z⬘^{. In Fig.}

4 we plot具^Sˆz⬘典for ten different trajectories. We see that in all ten cases the spin converged to ⫾1/2 quite quickly, before t⫽0.8 ms.

If we compare this with the results of Fig. 3, we see that the spin state collapses rather more quickly than the cantile- ver oscillations can respond. We only get a clear output sig- nal when the two phases are well separated, which does not occur until nearly t⫽1.5 ms. Generically, the difficulty of collapsing the spin state is much less than the difficulty of obtaining an unequivocal readout.

The curves depicted in Fig. 3 are idealized, without the measurement noise which will always be present in the out- put current 共42兲 or 共57兲. In Fig. 5 we show what the actual output would look like for the set of parameters we are dis- cussing. Note that even with the noise, the two phases 共rep- resenting spin up and spin down兲 are clearly distinguishable.

In the following section, we derive an expression for the signal-to-noise ratio in more general situations.

VIII. SIGNAL-TO-NOISE RATIO

Since we have to detect the effect of a very weak force on the cantilever by the single spin, we need very high resolu- tion for the cantilever position measurements and a good control of the various noise sources in the MRFM device. As described in Sec. II, the small displacement of the cantilever FIG. 4. Expectation value具^Sˆz⬘典vs t for ten different trajectories, showing the rapid localization of the spin for an initial superposi- tion state关兩v⫹(0)典⫹兩v⫺(0)典^]/冑2. We have taken the convention that Sˆ_z⬘has eigenvalues⫾1/2.

is measured by a fiber-optic interferometer as a phase shift of the interference fringes. We shall analyze the quantum and thermal noise in this homodyne measurement scheme.

The Hamiltonian for the combined system of the spin, cantilever, and cavity mode, excluding coupling to the envi- ronments, in the spin-rotating frame is

Hˆ⫽HˆZ⫺2␩_␭共t兲^{f共t兲ZˆSˆ}z⬘⫹ប␻caˆ^†aˆ⫹បE共aˆ^†e^⫺i␻0t⫹aˆe^i␻0t兲

⫹ប␬^{aˆ†aˆZˆ.} 共65兲

Here, ␻c is the optical frequency of the cavity mode, ␻0

⬃␻cis the driving frequency of the external laser, and other terms and parameters have been described in Sec. V. The master equation approach in Sec. IV is valid in high- or medium-temperature case. Here, we analyze the noise in the Heisenberg picture, using the quantum Langevin equation approach that is valid at any temperature关27兴.

Using standard techniques关28,29兴, the reservoir 共environ- mental兲 variables may be eliminated, in the interaction pic-

ture with respect toប␻0aˆ^†aˆ, to give the following quantum Langevin equations describing the dynamics of the whole system:

dZˆ共t兲 dt ⫽ 1

mpˆ共t兲, 共66兲

d pˆ共t兲

dt ⫽⫺m␻m

2Zˆ共t兲⫺⌫

mpˆ共t兲⫺ប␬^aˆ†共t兲aˆ共t兲⫹Wˆ共t兲

⫹2␩_␭共t兲^f共t兲Sˆz⬘共t兲, 共67兲 daˆ共t兲

dt ⫽⫺

冉

^{i␻c⫺i␻0⫹␥2c}

冊

^{aˆ共t兲⫺i␬Zˆ}共t兲aˆ共t兲⫺iE

⫹

冑

␥caˆ_in共t兲, 共68兲

dSˆ_z⬘共t兲

dt ⫽0, 共69兲

dSˆ_x⬘共t兲

dt ⫽2␩_␭共t兲^f共t兲Zˆ共t兲Sˆy⬘共t兲, 共70兲 dSˆ_y⬘共t兲

dt ⫽⫺2␩_␭共t兲^f共t兲Zˆ共t兲Sˆx⬘共t兲. 共71兲 In the the equations, the usual optical input noise operator aˆ_in(t) is associated with the vacuum fluctuations of the con- tinuum of electromagnetic modes outside the cavity and its correlation function is given by

具^aˆin共t兲aˆin

†共t⬘兲典⫽␦共t⫺t⬘兲. 共72兲

The random force Wˆ (t) describes the thermal noise motion 共quantum Brownian motion兲 of the cantilever at temperature T. For the case of an Ohmic environment, the thermal ran- dom force correlation is given by 关27兴

具Wˆ共t兲Wˆ共t⬘兲典⫽ប⌫

␲ ^{关Fr共t⫺t}⬘兲⫹iFi共t⫺t⬘兲兴, 共73兲 where

Fr共t兲⫽

冕

0

⍀

d␻ ␻^cos共␻^t兲coth

冉

^2kប␻BT

冊

^{, 共74兲}

Fi共t兲⫽

冕

0

⍀

d␻ ␻^sin共␻^t兲, 共75兲

with⍀ the frequency cutoff of the reservoir spectrum. With- out the presence of the external driving force from the spin, the cantilever-cavity system can be characterized by a semi- classical steady state with a new equilibrium position for the cantilever, displaced by Z_st⫽⫺␬兩␣st兩²/(m␻m

2) with respect to that with no external driving laser field, and the cavity mode in a coherent state 兩␣st典 with the amplitude given by FIG. 5. Simulation of photocurrent output in arbitrary units,

including measurement noise, using the parameters of Sec. VII, with detector efficiency ed⫽0.85. We have chosen the scale␤ so that the vertical scale matches that of Fig. 3, and also plotted the expectation values具^Zˆ典without the noisy dW/dt components.