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

Abstract

A requirement for many quantum computation schemes is the ability to measure single spins. This paper examines one proposed scheme: magnetic resonance force microscopy, including the effects of thermal noise and back-action from monitoring. We derive a simplified equation using the adiabatic approximation, and produce a stochastic pure state unraveling which is useful for numerical simulations. Comment: 33 pages LaTeX, 9 figure files in EPS format. Submitted to Physical Review A

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arXiv:quant-ph/0302178v1 25 Feb 2003
Realistic simulations of
single-spin nondemolition measurement
by magnetic resonance force microscopy
Todd A. Brun
Institute for Advanced Study, Einstein Drive,
Princeton, NJ 08540 USA
Hsi-Sheng Goan
Center for Quantum Computer Technology,
University of New South Wales,
Sydney, NSW 2052 Australia
February 9, 2008
Abstract
A requirement for many quantum computation schemes is the abil-
ity to measure single spins. This paper examines one proposed scheme:
magnetic resonance force microscopy, including the effects of thermal
noise and back-action from monitoring. We derive a simplified equa-
tion using the adiabatic approximation, and produce a stochastic pure
state unraveling which is useful for numerical simulations.
1Introduction
Single-spin measurement is an extremely important challenge, and necessary
for the future successful development of several recent spin-based proposals
for quantum information processing. [1, 2, 3, 4, 5] There are both direct and
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indirect single-spin measurement proposals. The idea behind some indirect
proposals is to transform the problem of detecting a single spin into the
task of measuring charge transport [2, 6], since the ability to detect a single
charge is now available. For direct single-spin detection, magnetic resonance
force microscopy (MRFM) has been suggested [7, 8, 9] as one of the most
promising techniques. To date, the MRFM technique has been demonstrated
with sensitivity to a few hundred spins [10, 11].
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 (CAI).
[12, 10, 9] In this CAI MRFM technique, the frequency of the spin inversion
in the rotating frame is in resonance with the mechanical vibration of an
ultra thin cantilever, allowing it to amplify the otherwise extremely weak
force due to the spin. These amplified vibrations can then be detected by,
e.g., optical methods.
Previous studies [8, 9] of the dynamics of single-spin measurement by
MRFM considered only the unitary evolution of the spin and the cantilever
system, without including any effects of external environments or measure-
ment devices. Only recently, the effect of thermal noise environment on the
dynamics of the spin-cantilever system in the MRFM was studied [13] by us-
ing the Caldeira-Leggett master equation [14] in the high-temperature limit.
There is, however, a macroscopic device in the MRFM setup which mea-
sures the cantilever motion and hence provides 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 de-
vice (a fiber-optic interferometer) 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 non-demolition measurement of a single
spin by MRFM. Finally, we present some simulation results of the dynamics
of the single-spin measurement process.
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B
s
RF coil
fiber-optic
interferometer
cantilever
magnetic
tip
0
Figure 1: Schematic diagram of the MRFM setup.
2 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 which can oscillate only in the z-direction. A
ferromagnetic particle (or small magnetic material) mounted on the can-
tilever tip produces a non-uniform magnetic field or magnetic field gradient
of (∂Bz/∂Z)0on the single spin. As a result, a reactive force (or interaction)
acts back on the magnetic cantilever 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 frequency modulation of a rotating radio-frequency (RF)
magnetic field in the x-y plane. In this case, the rotating RF field can be
represented as B1x= B1cos{[ω + ∆ω(t)]t}, B1y = −B1sin{[ω + ∆ω(t)]t},
where the frequency modulation ∆ω(t) is a periodic function in time with
the resonant frequency ωmof the cantilever. In the reference frame rotating
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with the B1, the spin-cantilever Hamiltonian can be written as
ˆHSZ(t) =ˆHZ− ¯ h[ωL− ω − ∆ω(t)]ˆSz− ¯ hω1ˆSx− gµ
?∂Bz
∂Z
?
0
ˆZˆSz, (1)
where ωL= gµBz/¯ h and ω1= gµB1/¯ h are respectively the Larmor and Rabi
frequencies, Bz includes the uniform magnetic field B0 and the magnetic
field produced by the ferromagnetic particle, g and µ are the g-factor and
the electron or nuclear magneton, and
ˆHZ=
1
2mˆ p2+mω2
m
2
ˆZ2
(2)
is the Hamiltonian of the cantilever in isolation (i.e., with no external mag-
netic field coupling it to the spin). For ω = ωL, we arrive at an effective
cantilever-spin Hamiltonian of the form
ˆHSZ(t) =ˆHZ− 2ηˆZˆSz+ f(t)ˆSz− εˆSx,
where f(t) = ∆ω(t), η = (gµ/2)(∂Bz/∂Z)0and ε = ¯ hω1. We will discuss
in details 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 when the adiabatic approximation
is exact, the instantaneous eigenstates of the spin Hamiltonian in the rotating
frame of the B1field are the spin states parallel or antiparallel to the direction
of the effective magnetic field Beff(t) = (ε,0,−f(t)), denoted as |v±(t)?,
respectively. We define an operatorˆS′
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 (in the laboratory or rotating frame) of
(3)
zfor the component of spin along this
χ(0) = a| ↑? + b| ↓?(4)
in theˆSzrepresentation, we can rewrite this initial state in the basis of the
instantaneous eigenstates ofˆS′
zas
χ(0) = aeff|v+(0)? + beff|v−(0)?,(5)
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where
aeff
beff
= acos(Θ0/2) + bsin(Θ0/2),
= −asin(Θ0/2) + bcos(Θ0/2),
(6)
(7)
and Θ0≡ Θ(0) is the initial angle between the effective magnetic magnetic
field and the z-axis direction. This implies tan[Θ(t)] = Beff
−ε/f(t). It then follows from the adiabatic theorem that the spin state at
time t can be written as:
x(t)/Beff
z(t) =
χ(t) = aeff|v+(t)? exp(−i
+beff|v−(t)? exp(−i
¯ h
?t
0λ+(t′)dt′)
?t
¯ h
0λ−(t′)dt′), (8)
where λ±(t) are instantaneous eigenvalues. So the probabilities of finding the
spin to be in the instantaneous eigenstates |v±(t)? are respectively |aeff|2and
|beff|2. Since the coefficients aeffand beffare time independent, the probabil-
ities |aeff|2and |beff|2remain the same at all times. This provides us with an
opportunity to measure the initial spin state probabilities 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 eigen-
states of the spin Hamiltonian, the spin-cantilever interaction can be written
as 2ηˆZˆS′
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 interaction between them, the total state becomes entangled. Monitoring
the phase of the cantilever vibrations will give us the information about the
spin. Numerical simulations (see Fig. 4 with reasonable parameters for the
CAI approximations) indicate that as the amplitude of the cantilever vibra-
tions increases with time, the phase difference in the oscillations for the two
different initial spin eigenstates ofˆS′
surement 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 purpose of this paper is to present
a realistic and detailed analysis of the single-spin measurement scheme, in-
zcos[Θ(t)]. As a result, the phase of the driven cantilever vibrations
zapproaches π. In other words, the mea-
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cluding the effects of the measurement device and other relevant sources of
noise.
3 The rotating picture and adiabatic approx-
imation
We assume an effective cantilever-spin Hamiltonian of the form (3) where for
the moment we let f(t) and ε be arbitrary, andˆHZis the Hamiltonian given
by (2). It is useful to group this into three terms
ˆHSZ(t) =ˆHZ+ˆHI+ˆHS(t) , (9)
where
ˆHI
≡ −2ηˆZˆSz,
ˆHS(t) ≡ f(t)ˆSz− εˆSx. (10)
The state of the cantilever-spin system evolves according to the Schr¨ odinger
equation
d|ψ(t)?
dt¯ h
In realistic cases, the spin part of the Hamiltonian (representing preces-
sion under the magnetic field) gives an evolution which is very rapid compared
to the reaction time of the cantilever. It therefore makes sense to switch to an
interaction picture in which the state is rotating along with this precession.
We do this by introducing a (partial) time translation operator
= −i
ˆHSZ(t)|ψ(t)? .(11)
ˆUS(t) ≡: exp−i
¯ h
??t
0
ˆHS(t′)dt′
?
: ,(12)
where :: indicates that the integral is to be taken in a time-ordered sense;
this unitary operator obeys the differential equation
dˆUS(t)
dt
= −i
¯ h
ˆHS(t)ˆUS(t) . (13)
We then introduce the state |˜ψ(t)? in the rotating picture:
|˜ψ(t)? ≡ˆU†
S(t)|ψ(t)? ,(14)
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with |ψ(t)? the solution of the original Schr¨ odinger equation (11) at time t.
The evolution equation for |˜ψ(t)? is
d|˜ψ(t)?
dtdt
i
¯ h
= −i
¯ h¯ h
= −i
¯ h
We can define a locked spin operatorˆSL(t)
=
dˆU†
S(t)
|ψ(t)? +ˆU†
S(t)ˆHS(t)|ψ(t)? −i
ˆHZ|˜ψ(t)? −i
ˆHZ|˜ψ(t)? +2iη
S(t)d|ψ(t)?
dt
ˆU†
=
ˆU†
¯ h
S(t)ˆHSZ(t)|ψ(t)?
?
S(t)ˆSzˆUS(t)
?ˆU†
ˆZ
S(t)ˆHIˆUS(t)|˜ψ(t)?
?
¯ h
?ˆU†
|˜ψ(t)? . (15)
ˆSL(t) ≡
?ˆU†
S(t)ˆSzˆUS(t)
?
;(16)
in terms of this, the equation of motion for |˜ψ? becomes
d|˜ψ(t)?
dt
Unfortunately, it is difficult to get an exact solution forˆUS(t) for a general
function f(t). This means that it is also difficult to derive an exact expression
forˆSL(t), and the rotating picture (15), while formally correct, is not very
helpful.
However, while we cannot easily find an exact expression forˆUS(t) for
general f(t), we can easily find an approximate solution for a large class
of functions. Suppose that ε is large and f(t) is slowly varying, so that
|f(t)|,ε ≫ |f′(t)/f(t)| for typical values of f(t) and f′(t). ThenˆHS(t) is also
slowly varying, and if a spin begins in an instantaneous eigenstate ofˆHS(t),
it will remain close to an instantaneous eigenstate ofˆHS(t) for all times by
the adiabatic theorem.
The instantaneous eigenstates ofˆHS(t) are
ˆHS(t)|v±(t)? = λ±(t)|v±(t)? ≡ ±λ(t)|v±(t)? ,
where
= −i
¯ h
ˆHZ|˜ψ(t)? +2iη
¯ h
ˆZˆSL(t)|˜ψ(t)? .(17)
(18)
λ(t) =
?
f2(t) + ε2,
|v±(t)? =
ε
?
(f(t) ∓ λ(t))2+ ε2| ↓? −
f(t) ∓ λ(t)
(f(t) ∓ λ(t))2+ ε2| ↑? . (19)
?
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We use these instantaneous eigenvectors and eigenvalues to define an approx-
imation to the unitary operatorˆUS(t):
ˆU′
S(t) =ˆI ⊗ |v+(t)??v+(0)|e−iΦ(t)+ˆI ⊗ |v−(t)??v−(0)|eiΦ(t),
with the accumulated phase
(20)
Φ(t) ≡1
¯ h
?t
0λ(t′)dt′.(21)
Note that Φ(t) obeys dΦ(t)/dt = λ(t). This implies that
dˆU′
S(t)
dt
= −i
¯ hλ(t)ˆI ⊗ |v+(t)??v+(0)|e−iΦ(t)+i
+ˆI ⊗d|v+(t)?
dt
= −i
¯ h
+ˆI ⊗d|v−(t)?
dt
¯ hλ(t)ˆI ⊗ |v−(t)??v−(0)|eiΦ(t)
?v+(0)|e−iΦ(t)+ˆI ⊗d|v−(t)?
S(t) +ˆI ⊗d|v+(t)?
dt
dt
?v−(0)|eiΦ(t)
ˆHS(t)ˆU′
?v+(0)|e−iΦ(t)
?v−(0)|eiΦ(t), (22)
which has the form of (13) plus some additional terms. From the definition
(19) of |v±(t)?, we see
d|v±(t)?
dt2λ2(t)
= ±1ε df(t)
dt
|v∓(t)? . (23)
Provided that f(t) is slowly varying, the additional terms in (22) will be
small.
Just as before, we can define a rotating picture, now using the unitary
transformationˆU′
S(t),
|˘ψ(t)? ≡
?ˆU′
S(t)
?†|ψ(t)? .(24)
This gives us a new evolution equation for |˘ψ?:
d|˘ψ(t)?
dt
= −i
=
d(ˆU′
S(t))†
dt
ˆHZ|˘ψ(t)? +2iη
|ψ(t)? + (ˆU′
S(t))†d|ψ(t)?
dt
¯ h¯ h
ˆZ
?
(ˆU′
S(t))†ˆSzˆU′
S(t)
?
|˘ψ(t)?
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+ˆI ⊗
+|v−(0)?d?v−(t)|
?
|v+(0)?d?v+(t)|
dt
e−iΦ(t)
dt
eiΦ(t)
?
ˆU′
S(t)|˘ψ(t)? .(25)
At this point, it is helpful to introduce a new set of spin operators
ˆS′
x
=
¯ h
2
i¯ h
2
¯ h
2
ˆI ⊗ (|v+(0)??v−(0)| + |v−(0)??v+(0)|) ,
ˆI ⊗ (|v−(0)??v+(0)| − |v+(0)??v−(0)|) ,
ˆI ⊗ (|v+(0)??v+(0)| − |v−(0)??v−(0)|) .
ˆS′
y
=
ˆS′
z
=
(26)
Using the definition (20) forˆU′
S(t), we can solve for the various terms in (25):
(ˆU′
S(t))†ˆSzˆU′
S(t) = −f(t)
λ(t)
ˆS′
z−
ε
λ(t)
?ˆS′
xcos(2Φ(t)) −ˆS′
ysin(2Φ(t))
?
. (27)
ˆI ⊗
?
|v+(0)?d?v+(t)|
dt
e−iΦ(t)+ |v−(0)?d?v−(t)|
iε
¯ hλ2(t)dt
dt
eiΦ(t)
?
ˆU′
S(t) =
df(t)
?ˆS′
xsin(2Φ(t)) +ˆS′
ycos(2Φ(t))
?
.(28)
Substituting (26–28) into (25), we get
d|˘ψ(t)?
dt
= −i
¯ h
ˆHZ|˘ψ(t)? +2iη
+2iη
¯ hλ(t)
ε
¯ hλ2(t)
¯ h
ˆZf(t)
λ(t)
ˆS′
z|˘ψ(t)?
ˆZ
ε
?ˆS′
xcos(2Φ(t)) −ˆS′
ysin(2Φ(t))
?
|˘ψ(t)?
+i
df(t)
dt
?ˆS′
xsin(2Φ(t)) +ˆS′
ycos(2Φ(t))
?
|˘ψ(t)? . (29)
Note that this equation is still exact—it is equivalent to the original
Schr¨ odinger equation (11). However, we can see that if |f(t)|,ε are large, then
Φ(t) will be a rapidly growing function, and the last two terms of equation
(29) will oscillate very rapidly compared to the first two terms. Over a short
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period relative to the response time of the cantilever they will essentially
average away to nothing. In this limit, therefore, we can reasonably make a
rotating-wave approximation, to get the approximate evolution equation
d|˘ψ(t)?
dt
≈ −i
¯ h
?ˆHZ− 2η(f(t)/λ(t))ˆZˆS′
z
?
|˘ψ(t)? .(30)
This is equivalent to making an exact adiabatic approximation, as described
in section 2. We can see how this approximation compares to the complete
Hamiltonian for a reasonable set of parameter values in figures 2 and 3. This
set of parameters was chosen to match those of Berman et al. [9], as was the
set of times plotted in figure 3. Comparison shows that our results match
their unitary simulations to good precision. 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 |˘ψ?.
-50
-40
-30
-20
-10
0
10
20
30
40
50
050100
t
150200
<Z>
Full equation
Rotating wave
Figure 2: Mean cantilever position ?ˆZ? vs. t for the complete and rotating
wave Hamiltonians.
In this rotating-wave approximation, if the spin begins in an instanta-
neous eigenstate ofˆHS(t), it will remain in an instantaneous eigenstate at all
times. If it begins in a superposition of the two eigenstates, the spin and can-
tilever degrees of freedom will become entangled, with the two components of
the wavefunction corresponding to the two spin directions remaining undis-
turbed for all times. Monitoring the position of the cantilever then serves as
a nondemolition measurement of the spin, as we would wish.
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
z
Full equation
Rotating wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
Full equation
Rotating wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
z
Full equation
Rotating wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
z
Full equation
Rotating wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
z
Full equation
Rotating wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
z
Full equation
Rotating wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
z
Full equation
Rotating wave
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-40 -30 -20 -10 0 10 20 30 40
z
Full equation
Rotating wave
p(z)p(z)
p(z)
p(z)
p(z)
p(z) p(z)
t=20
t=104
t=126.4
t=160
t=182.4 =199.2
t
t=210.4
t=221.6
z
p(z)
Figure 3: The probability distribution, p(z), of finding the cantilever at posi-
tion z at a range of times for the complete and rotating wave Hamiltonians.
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Note that the corrections to the adiabatic approximation include terms
which can flip the spin. These terms must remain small for the system to
be a true nondemolition measurement. The result of the spin measurement
manifests itself as a π phase shift in the oscillation of the cantilever. We can
see this in figure 4.
-50
-40
-30
-20
-10
0
10
20
30
40
50
050 100
t
150200
<Z>
Spin up
Spin down
-50
-40
-30
-20
-10
0
10
20
30
40
50
150 155 160 165 170 175 180 185 190 195 200
t
<Z>
Spin up
Spin down
Figure 4: Mean cantilever position ?ˆZ? vs. t for initial spin up and down in
theˆS′
zdirection.
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4 The thermal environment
Unfortunately, in practice we cannot treat the cantilever as an isolated sys-
tem. It is coupled at least weakly to the vibrational modes of the bulk, and
is therefore subject to dissipation 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 [14] master equation
in the high-temperature limit:
˙ ρ = −i
¯ h[ˆHSZ(t),ρ] −iγm
¯ h
[ˆZ,{ˆ p,ρ}] −γm
2ℓ2[ˆZ,[ˆZ,ρ]] , (31)
where the parameters are
γm =
Γ
2m,
ℓ =
¯ h
2√mkT
,(32)
m is the cantilever mass, T is the temperature, k is Boltzmann’s constant (or
the equivalent for our system of units), and Γ is the strength of the coupling
to the thermal bath. We can interpret γm (with units of inverse time) as
the dissipation rate and ℓ (with units of length) as the thermal de Broglie
wavelength.
A feature of this equation is that it doesn’t necessarily preserve the posi-
tivity of ρ on short time scales (though at long times it is well-behaved) [17].
This arises because of the approximations which are made in the derivation,
which become invalid at very short times. While this may be physically
unimportant, it can be inconvenient; in particular, if we wish to unravel the
evolution into a stochastic Schr¨ odinger equation [23] (as we will in section
6), it is necessary to start with a master equation in Lindblad form, [16]
˙ ρ = −i
¯ h[ˆH,ρ] +
?
j
?
2ˆLjρˆL†
j− {ˆL†
jˆLj,ρ}
?
, (33)
for some HermitianˆH and set of general Lindblad operators {ˆLj}.
Caldeira-Leggett equation (31) is not of this form, which is why it can violate
positivity of ρ.
The exact quantum Brownian motion master equation was shown [17] not
to have the Lindblad form, but rather requires time-dependent coefficients
The
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to ensure the positivity of the density matrix at short times. However, by
keeping more terms from the high- or medium-temperature-limit expansion
in a consistent way, Di´ osi [18] has shown that the Caldeira-Leggett equation
can be replaced by another master equation which is of Lindblad form, and
which agrees with it except at very short times when the equation’s validity
is questionable in any case. This is done by adding a term to (31) of the form
−(γmℓ2/2¯ h2)[ˆ p,[ˆ p,ρ]]. The procedure is analogous to completing the square.
If we choose the ansatz
ˆL = AˆZ + iBˆ p(34)
with real A,B, plug it into the equation (33), and equate it to the Caldeira-
Leggett equation (31) plus the additional term, we get
˙ ρ = −(i/¯ h)[ˆH,ρ] − A2[ˆZ,[ˆZ,ρ]] − B2[ˆ p,[ˆ p,ρ]]
+iAB(−2ˆZρˆ p +ˆZˆ pρ + ρˆZˆ p + 2ˆ pρˆZ − ˆ pˆZρ − ρˆ pˆZ)
= −(i/¯ h)[ˆHSZ(t),ρ] −γm
+iγm
¯ h(ˆ pρˆZ −ˆZˆ pρ −ˆZρˆ p + ρˆ pˆZ) ,
2ℓ2[ˆZ,[ˆZ,ρ]] −γmℓ2
2¯ h2[ˆ p,[ˆ p,ρ]]
(35)
which implies that
A =
?
?
ˆHSZ(t) + (γm/2)(ˆZˆ p + ˆ pˆZ) ≡ˆH′
γm/2ℓ2,
B =
ˆH
γmℓ2/2¯ h2,
=
SZ(t) .(36)
So the Lindblad operator for this equation is
ˆL =
?
γm/2
?
(1/ℓ)ˆZ + i(ℓ/¯ h)ˆ p
?
,(37)
and the effective Hamiltonian, going to the rotating picture and making use
of the approximation derived in section 3, is
ˆH′
SZ(t) =
1
2mˆ p2+mω2
m
2
ˆZ2− 2η(f(t)/λ(t))ˆZˆS′
z+ (γm/2)(ˆZˆ p + ˆ pˆZ) . (38)
In order for the cantilever to be an effective measurement device, the loss
rate must be very low: ωm≫ γm.
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5The effects of monitoring
In order to serve as a measurement scheme, we must have some way of
monitoring the motion of the cantilever. Because 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 figure 1, the cantilever forms one side of an optical microcav-
ity and the cleaved end of the fiber forms the other side. As the cantilever
moves, the resonant frequency of the cavity changes. Because the timescale
of the cantilever’s motion is very long compared to the optical timescale, 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 [19] for the cantilever-spin-cavity system in the interaction
picture is
˙ ρ = −i
¯ h[ˆH′
−i[E(ˆ a†+ ˆ a) + ˆ a†ˆ a(∆ + κˆZ),ρ]
+(γc/2)(2ˆ aρˆ a†− ˆ a†ˆ aρ − ρˆ a†ˆ a) ,
SZ(t) andˆL are the Hamiltonian and Lindblad operator for the
cantilever and spin given by eqns. (37) and (38), E is the strength of the
laser driving, ∆ is the detuning from the “neutral” cavity frequency, κ is the
coupling strength of the cantilever to the cavity mode, and γcis the loss rate
of the cavity.
Suppose now that we perform homodyne measurement [15, 20] on the
light which escapes from the cavity. We would like to replace the equation
(39) above with an equation for the conditional evolution of ρ, conditioned
on the output photocurrent Ic(t). The conditional evolution equation for our
system then becomes [20, 21] (in Itˆ o calculus form)
SZ(t),ρ] + 2ˆLρˆL†−ˆL†ˆLρ − ρˆL†ˆL
(39)
whereˆH′
dρ = −i
¯ h[ˆH′
−i[E(ˆ a†+ ˆ a) + ˆ a†ˆ a(∆ + κˆZ),ρ]dt
+(γc/2) 2ˆ aρˆ a†− ˆ a†ˆ aρ − ρˆ a†ˆ a
+√γced
ˆ aρ + ρˆ a†− ?ˆ a + ˆ a†?ρ
SZ(t),ρ]dt +
?
2ˆLρˆL†−ˆL†ˆLρ − ρˆL†ˆL
?
dt
?
?
?
?
dt
dWt,(40)
15