New Journal of Physics
The open–access journal for physics

Back-action evasion and squeezing of a mechanicalresonator using a cavity detector
A A Clerk1,4 , F Marquardt2 and K Jacobs3
1
Department of Physics, McGill University, Montréal, Québec, H3A 2T8,
Canada2
Department of Physics, Arnold-Sommerfeld-Center for Theoretical Physics
and Center for NanoScience, Ludwig-Maximilians-Universität München,
Theresienstrasse 37, 80333 Munich, Germany
3
Department of Physics, University of Massachussets at Boston, Boston,
MA 02125, USA
E-mail: aashish.clerk@mcgill.ca, florian.marquardt@physik.uni-muenchen.de
and kjacobs@cs.umb.edu
New Journal of Physics 10 (2008) 095010 (20pp)

Received 14 April 2008
Published 30 September 2008
Online at http://www.njp.org/
doi:10.1088/1367-2630/10/9/095010

Abstract. We study the quantum measurement of a cantilever using aparametrically coupled electromagnetic cavity which is driven at the two
sidebands corresponding to the mechanical motion. This scheme, originally due
to Braginsky et al (Braginsky V, Vorontsov Y I and Thorne K P 1980 Science 209547), allows a back-action free measurement of one quadrature of the cantilever’s
motion, and hence the possibility of generating a squeezed state. We present
a complete quantum theory of this system, and derive simple conditions onwhen the quantum limit on the added noise can be surpassed. We also study
the conditional dynamics of the measurement, and discuss how such a scheme(when coupled with feedback) can be used to generate and detect squeezed
states of the oscillator. Our results are relevant to experiments in optomechanics,
and to experiments in quantum electromechanics employing stripline resonators
coupled to mechanical resonators.

4

Author to whom any correspondence should be addressed.

New Journal of Physics 10 (2008) 095010
1367-2630/08/095010+20$30.00© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2Contents

1. Introduction
2. Model and main results
2.1. Basic idea behind single quadrature detection . . . . . . . . . . . . . . . . . .
2.2. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Back-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Output spectrum and beating the SQL . . . . . . . . . . . . . . . . . . . . . .
2.5. Conditional squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6. Feedback for true squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Experimental prospects and conclusions
Acknowledgments
Appendix. Details of calculations
References

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4
6
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11
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20

1. Introduction

Considerable effort has been devoted recently to attempts at seeing quantum effects in micron
to nanometre scale mechanical systems. Experiments coupling such oscillators to mesoscopic
electronic position detectors have seen evidence of quantum back-action and back-action
cooling [1], and have demonstrated continuous position detection at a level near the fundamental
limit placed by quantum back-action [2]–[4]. Complementary to this work, experiments using
optomechanical systems (e.g. a cantilever coupled to a optical cavity) have been able to cool
micromechanical resonators by several orders of magnitude, using either passive [5]–[8] oractive (i.e. feedback-based) approaches [9].
Despite these recent successes, seeing truly quantum behaviour in a mechanical resonatorremains a difficult challenge. If one is only doing linear position detection, the quantum
behaviour of an oscillator is almost perfectly masked. Nonlinear detector–oscillator couplings
allow one to probe quantum behaviour such as energy quantization [10]–[14]; however,generating such couplings is generally not an easy task. Quantum behaviour could also be
revealed by coupling the resonator to a qubit [15]–[17]; this too is challenging, as it requires
relatively large couplings and a highly phase coherent qubit. Here, we consider an alternate
route to seeing quantum behaviour in a mechanical oscillator, one that requires no qubit andonly a linear coupling to position. As was first suggested by Braginsky et al [18, 19], by using
an appropriately driven electromagnetic cavity which is parametrically coupled to a cantilever,
one can make a measurement of just a single quadrature of the cantilever’s motion. As a result,quantum mechanical back-action need not place a limit on the measurement accuracy, as the
back-action affects only the unmeasured quadrature. One can then make (in principle) a perfectmeasurement of one quadrature of the oscillator’s motion. This is in itself useful, as it allows
for the possibility of ultra-sensitive force detection [20, 21]. Perhaps even more interesting, one
expects that such a measurement can result in a quantum squeezed state of the oscillator, where
the uncertainty of the measured quadrature drops below its zero-point value.
While the original proposal by Braginsky is quite old, there nonetheless does not exist a
fully quantum theory of the noise and back-action of this scheme; moreover, there exists notreatment of the measurement-induced squeezing. In this paper, we remedy this situation, andNew Journal of Physics 10 (2008) 095010 (http://www.njp.org/)3
present a fully quantum theory of measurement in this system. We calculate the full noise in thehomodyned output signal from the cavity (an experimentally measurable quantity), and derive
simple but precise conditions that are needed to beat the conventional quantum limit on theadded noise of a position detector [19, 20, 22, 23]. Somewhat surprisingly, we show that beating
the standard quantum limit (SQL) does not require a mechanical frequency which is much larger
than the cavity damping (i.e. one does not need to be deep in the good-cavity limit). Usinga conditional measurement approach, we also discuss the conditions required to squeeze themechanical resonator, and demonstrate how feedback may be used to unambiguously detect thissqueezing. Our results are especially timely, given the recent experimental successes in realizing
cavity position detectors using both superconducting stripline resonators [24] as well as opticalcavities [5]–[8], [25]; our theory is applicable to both these classes of systems. Note that Ruskov
et al [26] recently analysed a somewhat related scheme involving stroboscopic measurement ofan oscillator with a quantum point contact. Unlike that scheme, the system analysed here should
be much easier to implement, being directly related to existing experimental setups; our scheme
also has the benefit of allowing significant squeezing without the need to generate extremely
fast pulses.
The remainder of this paper is organized as follows. In section 2, we give a heuristic
description of how one may realize back-action free single-quadrature detection, introduce theBraginsky two-sideband scheme, and give a synopsis of our main findings. We conclude insection 3, and present a brief discussion on possible experimental realizations. Details of thecalculations are relegated to the appendix.
2. Model and main results

2.1. Basic idea behind single quadrature detection
Consider a high-Q mechanical oscillator having frequency ωM and annihilation operator ĉ. We
will use X and Y to denote the cosine and sine quadratures of the oscillator’s motion. Using
Schrödinger operators ĉ and ĉ† , the operators associated with the quadratures are:
1X̂ = √ ĉeiωM t + ĉ† e−iωM t ,(1a)2iŶ = − √2ĉeiωM t − ĉ† e−iωM t .

(1b)

The Heisenberg-picture position operator x̂(t) is then given by the Heisenberg-picture operatorsX̂ (t) and Ŷ (t) as expected:
√x̂(t) ≡ 2xzpt X̂ (t) cos ωM t + Ŷ (t) sin ωM t .
(2)Here, xzpt denotes the ground state position uncertainty of the mechanical resonator. Note thatX̂ and Ŷ are canonically conjugate:ihX̂ , Ŷ = i.
(3)
Also note that the definition of the quadrature operators relies on having an external clock in thesystem which defines the zero of time.
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

4

Figure 1. Schematic picture of the setup studied in the text. A cavity is driven by

an input beam that is amplitude-modulated at the mechanical frequency ωM of
the movable end-mirror. The radiation pressure force, as well as the cavity andmechanical frequencies and decay rates are indicated.
In general, X̂ (t) and Ŷ (t) will vary slowly in time (in comparison to ωM ) due to the external
forces acting on the oscillator. Our goal will be to make a weak, continuous measurement of
only X̂ (t), using the usual kind of setup where the position of the oscillator is linearly coupled
to a detector. We will use a detector–oscillator coupling Hamiltonian of the form:Ĥ int = −h̄ A x̂ F̂,(4)

where F̂ is some detector operator. It represents the force exerted by the detector on theoscillator; in the cavity-position detector we will consider, F̂ will be the number of photons
in an electromagnetic cavity. Ideally, if we only measure X̂ , the back-action of the measurement
will only affect the unmeasured quadrature Ŷ , and will not affect the evolution of X̂ at latertimes. The hope thus exists of being able to make a back-action free measurement, one which
is not subject to the usual SQL [20, 22, 23].
As is discussed extensively in [20, 21], single quadrature detection with the interaction
Hamiltonian in equation (4) can be accomplished by simply modulating the coupling strength
A at the oscillator frequency. Setting A = A(t) = 2( Ã/xzpt ) cos(ωM t), Ĥ int becomes:hi√Ĥ int = − 2h̄ Ã F̂ X̂ (1 + cos(2ωM t)) + Ŷ sin(2ωM t) .
(5)
In a time-averaged sense, we see the detector is only coupled to the X quadrature; we thus might
expect that the (time-averaged) output of the detector will tell us only about X . In principle, thisin itself does not imply a lack of back-action: via the coupling to Ŷ , noise in F̂ could affectthe dynamics of X̂ . To prevent this, we need the further requirement that the detector force has
no frequency components near ±2ωM . In this case, the effective back-action force F̂ sin(2ωM t)
will have no Fourier weight in the narrow-bandwidth around zero frequency to which X̂ issensitive, and it will not affect X̂ . Note that Ruskov et al [26] recently considered a linear
position detection scheme where the effective coupling constant is harmonically modulated;
however, their scheme does not satisfy the second requirement above of having a narrow-bandback-action force.
2.2. Model
We now consider a specific and experimentally realizable system which can realize the above
ideas; this system was first proposed by Braginsky et al [18, 19]. As shown schematically infigure 1, the setup consists of a high-Q mechanical oscillator which is parametrically coupled
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)5with strength A to a driven electromagnetic cavity:Ĥ = h̄(ωR − A x̂) â † a − hâ † ai + Ĥ M + Ĥ drive + Ĥ κ + Ĥ γ ,(6)where ωR is the cavity resonance frequency, Ĥ M = h̄ωM ĉ† ĉ is the mechanical oscillator
Hamiltonian, Ĥ drive describes the cavity drive, Ĥ κ describes the cavity damping and Ĥ γ is themechanical damping. Note this same system was recently shown (in a similar parameter regime
to what we will require) to allow back-action cooling to the ground state [27, 28].
Assuming a one-sided cavity, standard input–output theory [29, 30] yields the Heisenberg
equation of motion:√κâ − κ b̂in (t),(7)
â˙ = −iωR −2where κ is the cavity damping and b̂in describes both the drive applied to the cavity as well as
the noise (quantum and thermal) entering the vacuum port.
To implement back-action evasion in the cavity system, we will consider the case where
ωR  ωM , and take the resolved-sideband or ‘good cavity’ limit, where ωM  κ. We will also
take an amplitude-modulated cavity drive of the form:b̄LOsin(ωM t)e−iωR t .
(8)2
The same resolved-sideband limit is required to achieve ground state cooling [27, 28]; all thatis different from the setup here is the nature of the drive. Here, one drives the cavity equallyat both sidebands associated with the oscillator motion, whereas in the cooling case, one only
drives the red-detuned sideband.
To proceed, we may write the cavity annihilation operator â as the sum of a classical pieceˆā(t) and a quantum piece d:hb̂in (t)i ≡ b̄in (t) =ˆâ(t) = ā(t) + d(t),(9)ā(t) is determined solely by the response of the cavity to the (classical) external drive b̄in (t). In
the long-time limit, equation (7) yields:ā(t) = āmax cos (ωM t + δ) e−iωR t

(10)

with
r
āmax = b̄LO

κ
2
4ωM
+ κ2

,

δ = arctan (κ/ωM ) .

(11a)
(11b)

The phase δ plays no role except to set the definitions of the two quadratures X and Y ; thus,
without loss of generality, we will set it to zero. We will also be interested in a drive large
enough that āmax  1.
ˆ the quantum part of the cavity annihilation operator, is influenced by
In contrast to ā, d,both the mechanical oscillator and quantum noise associated with the cavity dissipation. Making
use of the solution for ā and the conditions ωR  ωM  κ, and keeping only terms which are atleast order ā, the term in the total system Hamiltonian coupling the oscillator to the cavity takes

New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)6
an analogous form to equation (5) with:Ã = 12 Axzpt āmax ,(12a)F̂ = eiωR t dˆ + e−iωR t dˆ† .

(12b)Thus, the chosen cavity drive gives us the required harmonically modulated coupling constant:in a time-averaged sense, the cavity is only coupled to the X -oscillator quadrature. Further, thesecond condition outlined in section 2.1 is also satisfied: because κ  ωM , F̂ has no appreciablenoise power at frequencies near ±2ωM . As such, we expect no back-action heating of the X̂quadrature in the resolved-sideband limit κ/ωM → 0. We will of course consider the effect of anonzero but small κ/ωM in what follows.
2.3. Back-action
Working in an interaction picture, one can easily derive Heisenberg equations of motion for thesystem, and solve these in the Fourier domain (cf equations (A.7a) and (A.7b)). As expected, one
finds that in the ideal good-cavity limit (κ/ωM → 0), the measured X quadrature is completely
unaffected by the coupling to the cavity, whereas the unmeasured Y quadrature experiences anextra back-action force due to the cavity. For finite κ/ωM , there is some small additional backaction heating of the X quadrature. The noise spectral densities of the quadrature fluctuations
are given by
Z ∞
Dn
oE
1
S X (ω) ≡ 2
dteiωt X̂ (t), X̂ (0)
−∞

=

γ /2

ω2 + (γ /2)2

SY (ω) = 12=Z ∞dteiωt
1 + 2 n eq + n bad ,DnŶ (t), Ŷ (0)

(13a)

oE

−∞

γ /2

ω2 + (γ /2)2
1 + 2 n eq + n BA + n bad ,

(13b)

where

−1
h̄ωM
n eq = exp
−1
(14)
kB T
is the number of thermal quanta in the oscillator. n BA parameterizes the back-action heating of
the Y quadrature as an effective increase in n eq ; in the relevant limit γ  κ one has:2
8 Ã2 2 Axzptn BA ==(15)(āmax )2 .
κγ
κγ
We have assumed here that there is no thermal noise in the cavity drive: it is shot noise-limited.
Finally, n bad parameterizes the spurious back-action heating of X which occurs when one
deviates from the good-cavity limit; to leading order in κ/ωM , it is simply given by: 
n BA κ 2n bad =
.
(16)
32 ωM
Note that there is no back-action damping of either quadrature (see discussion following
equation (A.7b)).


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7
2.4. Output spectrum and beating the SQL
We assume that a homodyne measurement is made of the light leaving the cavity. Using thesolution to the Heisenberg equations of motion (cf equations (A.7a) and (A.7b)) and standard
input–output theory, one can easily find the noise spectral density of the homodyne current I (t).
The information about X̂ (t) will be contained in a bandwidth ∼γ  κ around zero frequency.
Thus, focusing on frequencies ω  κ, we have simply:


κ
2
S0 .
S I (ω) = G S X (ω) +(17)
32 Ã2
Here, G is a gain coefficient proportional to the homodyne local oscillator amplitude, andS0 represents added noise in the measurement coming from both the cavity drive and in thehomodyne detection. If both are shot noise limited, we simply have S0 = 1. We can refer thisnoise back to the oscillator by simply dividing out the factor G 2 : the result is the measured
X -quadrature fluctuations:
S I (ω)κ
S X,meas (ω) ≡
=
S
(ω)
+
(18)
S0 .
X
G2
32 Ã2
Now, note that in the good-cavity limit the spurious heating of X described by n bad vanishes.
Thus, in this limit, the added noise term (second term in equation (18)) can be made arbitrarily
small by increasing the intensity of the cavity drive beam (and hence Ã), without any resulting
back-action heating of the measured X quadrature. Thus, in the good-cavity limit, there is noback-action imposed limit on how small we can make the added noise of the measurement(referred back to the oscillator). In contrast, for small but nonzero κ/ωM , one needs to worry
about the small residual backaction described by n bad ; one can still, nonetheless, beat the SQL
in this case, as we now show.
To compare against the SQL, consider S X,meas (0):
2κS X,meas (0) =1 + 2n eq + 2n bad +S0γ
32 Ã22
1 + 2n eq + 2n add .
(19)≡γ
In the last line, we have represented both the residual back-action n bad and the added noise of
the measurement as an effective increase in the number of oscillator quanta by an amount n add .
The SQL (which applies when both quadratures are measured) yields the condition n add > 1/2[20, 22, 23]. Here, we find: κγn BA κ 2
1
n add = n bad +
S0 =
+
S0 .
(20)
32 ωM
16n BA
128 Ã2
Thus, if we are in the ideal good-cavity limit (κ/ωM → 0) and shot-noise limited, beating the
SQL on n add requires a coupling strong enough that n BA > 1/8: the Y quadrature fluctuations
must be heated up by at least an eighth of an oscillator quantum.
In the more general case where κ/ωM is finite, one cannot increase the coupling indefinitely,as there is back-action on X̂ . One finds that for an optimized coupling of:ωM pñ BA =
2S0 ,
(21)
κ
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)8κ / ωΜ = 0.0, S0 = 1
κ / ωΜ = 0.5, S0 = 1
κ / ωΜ = 0.0, S0 = 10
κ / ωΜ = 0.5, S0 = 10

Added noise ( add)

4
2

1
4
2

0.1
4
2

0.01
0

2

4

6

8

10

BA

Figure 2. Plot of added noise in the single quadrature measurement (measured

as a number of quanta, n add , cf equation (20)) versus the strength of themeasurement (measured in terms of the back-action heating of the Y quadrature,n BA , cf equation (15)). Different curves correspond to different values ofκ/ωM and S0 , the noise associated with the homodyne measurement; S0 = 1
corresponds to a shot-noise limited measurement. The SQL of n add = 0.5 isshown as a horizontal dashed line.
the minimum added noise at resonance is given by:
r
S0
κ
.
n add
(22)
=
min
8ωM 2
Thus, even for moderately small κ/ωM , one can make n add smaller than the SQL value (seefigure 2). In practice, this fact could be quite valuable, as the large detuning of the cavity drive
needed to be in the good-cavity limit can make it hard to get sufficient power in the cavity, andhence achieve a sufficiently strong coupling n BA (cf equation (12a)).
2.5. Conditional squeezing
Given that the double-sideband scheme described here can allow for a near perfect measurementof the oscillator X quadrature, one would expect it could lead to a squeezed oscillator state,
where the uncertainty in X̂ drops below the zero point value of 1/2. However, equation (13a)indicates that in the good-cavity limit, the fluctuations of X̂ are completely unaffected by the
coupling to the cavity detector. To resolve this seeming contradiction, one must consider theconditional aspects of the measurement: what is the state of the resonator in a particular run ofthe experiment? In any given run of the experiment, the oscillator will indeed be squeezed.
However, the mean value of X̂ will have some nonzero value which is correlated with thenoise in the output signal. Once one averages over many realizations of the experiment, thisrandom motion of h X̂ i appears as extra noise, and masks the squeezing, resulting in the resultof equation (13a). We make these statements precise in what follows.
A rigorous description of the conditional evolution of the oscillator in the setup consideredhere can be developed in analogy to [31], which considered ordinary linear position detection
using a cavity. For simplicity, we focus on the good-cavity limit, where κ/ωM → 0. We first
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

9
define the parameter k̃, a measure of the rate at which the measurement extracts information, as:
32 Ã2= η (4γ n BA ) ,(23)
κwhere n BA represents as before the back-action heating of the Y quadrature, and η = S10 6 1
represents the efficiency of the homodyne detection (η = 1 corresponds to being quantum
limited). One has k̃ = 1/τmeas , where τmeas is the minimum time required to resolve a differencein hX i equal to the zero point rms value from the output of the detector; as we are interested in
weak measurements, we expect k̃/ωM  1. Note that k̃ = 8ηk, where k is the usual definitionof the strength of the measurement [32]. The scaled homodyne output signal may then be
written [32]:p
I (t) = k̃h X̂ (t)i + ξ(t),(24)
k̃ = η

where ξ(t) is white Gaussian noise. In a given run of the experiment, ξ(t) will be correlatedwith the state of the oscillator at times later than t.
In exact analogy to [31], a simple description of the conditional oscillator density matrix
is possible in the limit where κ  Ã. In this limit, this density matrix is Gaussian, being fully
determined by its means X̄ = h X̂ i and Ȳ = hŶ i, and its second moments VX = hh X̂ 2 ii, VY =hhŶ 2 ii and C = hh{ X̂ , Ŷ }/2ii. Note that X̄ , Ȳ represent the observer’s best estimates for theoscillator’s two quadrature amplitudes, given the measurement output record. In the interaction
picture (i.e. rotating frame at the oscillator frequency), the equations for the means (theestimates) arepγ
˙X̄ = − X̄ + k̃VX ξ,
(25a)
2
p
γ
˙
Ȳ = − Ȳ + k̃Cξ(25b)
2
and for the covariances are
V̇X = −k̃VX2 − γ (VX − T̃ eq ),(25c)

V̇Y = −k̃C 2 + k̃/(4η) − γ (VY − T̃ eq ),(25d)Ċ = −γ C − k̃VX C,(25e)

T̃ eq = 12 + n eq .

(26)

where
We stress that these equations are almost identical to the standard equations for conditional
linear position detection [26, 31], with the important exception that terms corresponding to
the bare oscillator Hamiltonian are missing. In a sense, the scheme presented here effectivelytransforms away the oscillator Hamiltonian.
To find the amount of squeezing in a particular run of the experiment, we simply find thestationary variances for the oscillator’s Gaussian state. We have
VY = 21 + n eq + n BA ,
(27a)
q

2 1 + 2n eq (ηn BA ) + 1/4 − 1/2
VX =
,
(27b)
4 (ηn BA )C = 0.
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

(27c)

)

10

quadrature variance (

1

0.1

8
6
4
2

8
6
4

0

5

10

15

20

25

30

BA

Figure 3. Plot of the conditional X-quadrature variance VX (cf equation (27b))

as a function of the measurement strength (parameterized in terms of the backaction heating of the Y quadrature, n BA , cf equation (15)); one clearly seesthat the X quadrature can be squeezed. Different curves correspond to different
values of the bath temperature (parameterized by n eq , cf equation (14)) andmeasurement efficiency η. The solid-red curve corresponds to n eq = 0 and η = 1;
the dashed black curve to n eq = 1 and η = 1; the dashed-dotted blue curve ton eq = 0 and η = 0.1. The horizontal dashed line corresponds to the ground statevalue of the variance, VX = 0.5.

Note first that the result for VY is in complete agreement with the unconditional resultof equation (13b): the measurement back-action heats the Y quadrature by an amountcorresponding to n BA quanta. In contrast, we find that unlike the unconditional result of
equation (13a), the measurement causes VX to decrease below its zero-coupling value: it is amonotonically decreasing function of n BA (see figure 3). This is the expected measurementinduced squeezing. Of particular interest is the minimum coupling strength needed to reduce
VX to its zero-point value:n eq
n BA =.
(28)
ηIn other words, lowering the X quadrature uncertainty from a thermal value of (1/2 + n eq ) to
the ground state value of (1/2) requires that we at least increase the Y -quadrature uncertainty
by the same amount. This minimum amount is only achieved for a quantum-limited detector
η = 1.
The equation describing the fluctuations of the mean quadrature amplitudes X̄ , Y¯ can also
be easily solved. Assuming that X̄ = Y¯ = 0 at the initial time, one always has h X̄ (t)i = hY¯ (t)i =0, where the average here is over many runs of the experiment. In the stationary state (i.e. oncethe variances VX , VY and C have attained their stationary values), one finds Y¯ (t) = 0 with nofluctuations. X̄ (t) continues to fluctuate, with an autocorrelation function:!
k̃X̄ (t) X̄ (0) =VX2 e−γ |t|/2 .
(29)γAgain the average here is over many runs of the experiment.
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11
We may now combine the results of equation (29), and equation (25c) to find VX,tot , thetotal (unconditional) X variance. One finds the simple result (valid in the stationary regime):
VX,tot ≡ VX + h X̄ 2 ik̃ 2 1(30)
V = + n eq .
γ X 2This shows that, as expected, averaging the results of the conditional theory over manymeasurement runs reproduces the result of the unconditional theory (i.e. the fluctuations of
the measured X quadrature are completely unaffected by the measurement).
= VX +

2.6. Feedback for true squeezing
In the previous section, we saw how the state of the resonator, once conditioned on themeasurement record, is squeezed. We can use feedback control to turn this conditional squeezing
into ‘real’ squeezing of the resonator, where the full, unconditional oscillator variance VX,tot (cfequation (30)) drops below the zero-point value. This is accomplished by applying a timedependent force to the resonator which is proportional to X̄ (t), the observer’s best estimateof h X̂ (t)i inferred from the measurement output. Such a force can be used to suppress thefluctuations in the mean value of X , and in the limit of strong feedback, can remove themcompletely. The only fluctuations that remain are quantified by the conditional variances, which
are squeezed. Note that a similar approach was considered in [26].
More precisely, if one makes the measurement at rate k̃ described above (cf equation (23)),and applies the feedback force F(t) = αγ X̄ sin ωM t in the laboratory frame, the result is insome ways similar to an effective damping of the X quadrature at a rate αγ /2. Calculating thefluctuations of the X quadrature under this feedback (the details of which are given in the next
section), we find that the total unconditioned X -quadrature variance reaches a stationary state:h X̄ 2 i
1+α(n eq + 1/2) + αVX
=
.
(31)
1+α
Here, VX is the conditioned variance given in equation (27b). Note that when α → 0, we again
get the result of equation (30): the unconditioned X -quadrature variance is not affected by thefbmeasurement. In contrast, in the limit of large α, one has VX,tot→ VX . Thus, as claimed above,in the limit of strong feedback, the total fluctuations of the X quadrature are reduced to theconditional variance; they may thus be squeezed.
It is also important to ask how this squeezing will manifest itself in the output signal ofthe measurement. Calculating S Ifb (ω), the spectrum of the homodyne current in the presence of
feedback, and referring it back to the X quadrature, we find:fb
VX,tot= VX +fb
S X,meas(ω) ≡S Ifb (ω)
k̃

=

1
k̃

+

fb
(1 + α)γ VX,tot
2 .
ω2 + (γ /2)(1 + α)(32)

We see that the output spectrum consists of the white added noise of the measurement (1/k̃) plus
a Lorentzian term arising from the oscillator. In the absence of feedback (i.e. α → 0), this term
is simply the measurement-independent X -quadrature fluctuation spectrum S X (ω); this is incomplete agreement with the unconditional theory (cf equation (19)). With the feedback turned
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)12fb
on, the weight of the Lorentzian term is now the total unconditional X -quadrature variance VX,tot
given in equation (31):


Z
dω
1
fb
fb
S X,meas (ω) −= VX,tot
.
(33)
2π
k̃
Thus, using feedback, the output of the measurement can be used to read out the fullfb
oscillator X -quadrature variance VX,tot: one simply measures the integral of the Lorentzian in
the output spectrum. In the limit of strong feedback (α → ∞), this variance tends to Vx (theconditional variance), and can exhibit squeezing (cf equation (27b)). One thus has a direct way
to detect the squeezing generated by the double-sideband back-action evasion scheme.
In practice, there are two potential drawbacks to directly using the output spectrum todetect squeezing. Firstly, in the strong feedback limit α → ∞, the weight of the Lorentzianpeak in the spectrum tends to a constant value VX , but its width diverges as αγ . For strong
feedback, it could thus become very difficult to resolve the Lorentzian term in the output
spectrum of equation (32) from the constant background 1/k̃. Secondly, one has to worry aboutaccusations that one is not seeing squeezing, but simply noise squashing [33]–[35]. This isa phenomenon where feedback reduces fluctuations in the output signal without necessarily
reducing the fluctuations of the system being measured. To understand this effect, note thatin driving the resonator with a force proportional to X̄ , we are driving it with a signal thatis correlated with the noise in the output signal. Feedback may thus lead to new correlations
between the fluctuations of X and the output noise. Such correlations could conceivably reduce
the oscillator’s contribution to the output noise, and are the source of the squashing effectin other systems [35]. In our case, the output spectrum in equation (32) does indeed reflectcorrelations between the measurement noise and the feedback force. Nonetheless, there is nosquashing here, as the weight of the Lorentzian in the output spectrum always coincides withthe full variance of the X -quadrature fluctuations. It would still, however, be preferable to havea scheme where one could eliminate the possibility of squashing without any detailed analysis.
A solution to these concerns is to make a second measurement of the mechanicalresonator’s X quadrature (e.g. by using a second cavity coupled to the resonator). One could
then use the corresponding output spectrum to measure the full X -quadrature fluctuations andany potential squeezing. As the output signal from the second measurement, I2 (t), is not part of
the feedback loop, its measurement noise is completely uncorrelated with the feedback signal.
As a result, there is no unwanted noise correlation, and no possibility of squashing. Surprisingly,
the lack of correlations also prevents the apparent bandwidth associated with the oscillator noisefrom diverging in the strong feedback limit; this point is more fully explained in the appendix.
It is straightforward to calculate the output spectrum of the second measurement. Note thatsince the second measurement is also a QND measurement of the X quadrature, it does not
fb
fb
affect the results for VX,tot
or S X,meas(ω) derived above. If the rate of the second measurement is
λ̃, then the spectrum of its output (again, referred back to the oscillator) isfb
S X,meas,2(ω) ≡S Ifb2 (ω)
λ̃

=

1
λ̃

+

4  (n eq + 1/2) − A+γ (2ω/γ )2 + (1 + α)2A2  ,(2ω/γ )2 + 1 + 2k̃VX /γ

(34)

where
A=α

(2n eq + 1)(1 + 2k̃VX /γ + α) − αVXα(2 + α) − 2(2n eq + 1)k̃/γ

New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

.

(35)

13
The first term in the spectrum above represents the added noise of the measurement (e.g.
shot noise), whereas the terms in square brackets are a direct measure of the oscillator’s
X -quadrature fluctuations. Unlike the output spectrum of the first, in-loop measurement (cfequation (32)), there is no contribution here from correlations between the measurement noise
and feedback force. As a result, the oscillator’s contribution to the output spectrum is a sum offb
two Lorentzians, the integral of which directly yields VX,tot, the total X -quadrature variance inthe presence of feedback


Z
dω
1
fb
fb
S X,meas,2 (ω) −= VX,tot
.
(36)
2π
λ̃
It thus follows from equation (31) that in the strong feedback limit (α → ∞), the area under
the resonant peaks in the output spectrum directly yields VX , and hence a direct measure ofsqueezing. The same was of course true for the spectrum of I1 (the in-loop output signal) given
in equation (32). However, unlike that spectrum, the bandwidth associated with the oscillator
noise here does not become infinite as α → ∞. In this limit, one finds simply
1 4
Vx
fb
lim S X,meas,2(ω) = +(37),α→∞λ̃ γ2 (2ω/γ2 )2 + 1
where the effective noise bandwidth γ2 is given byγ2 = γ 1 + 2k̃VX /γ = γ 1 + 8n BA VX .
(38)
Thus, for strong feedback, the squeezing of the oscillator can now be unambiguously detected in
the output signal of the second measurement: one obtains a simple Lorentzian resonance whosearea is simply VX . Note that from equation (31), one requires αγ  γ2 in order that the total
oscillator variance be reduced to the (possibly squeezed) variance VX .
3. Experimental prospects and conclusions

We close with a brief discussion of possible experimental realizations of the back-action evasion
scheme presented here. One promising class of experiments involves microwave transmission
line resonators coupled to nanomechanical resonators. The first experimental realization of sucha system was reported by Regal et al [24]. These authors were able to achieve ωM /κ ≈ 5,
placing them firmly in the good-cavity limit needed for the double-sideband back-action evasion
scheme. Unfortunately, the coupling strength in this first experiment was far too weak to
pursue back-action evasion. Assuming one could couple a power of 600 pW to the cavity(as was deemed reasonable by the authors of [24]), the system of [24] yields a couplingstrength parameter n BA ' 10−4 (cf equation (15)). Recall that beating the SQL requires at leastn BA > 1/8. This being said, [24] suggests that it should be possible to increase the couplingstrength A by at least a factor of 20. If, in addition, one could use a higher quality mechanicalresonator (the current experiment had a mechanical Q = 1.2 × 105 ) or increase the incident
power, one would be in the regime where n BA is large enough to contemplate back-action
evasion.
Experiments in optomechanics are also progressing to the point where one can realisticallyconsider implementing the scheme discussed here. The resolved-sideband limit was recently
reached by Schliesser et al [25] for an optical mode propagating in a microtoroidal cavity that
supports mechanical vibrations, with ωM /κ ≈ 20. Using parameter values provided in [25], andNew Journal of Physics 10 (2008) 095010 (http://www.njp.org/)14
assuming that a power of 1 µW is coupled to the cavity (as was achieved in the experiment),
one finds n BA ' 0.05. Thus, the optomechanical setup in [25] is already very close to being
able to implement back-action evasion: a modest increase in applied power or oscillator Q
factor would allow one to beat the SQL. Note that the chief advantage of the microtoroidal
optomechanical system compared to the current version of the microwave cavity system is amuch larger intrinsic coupling: the dimensionless ratio Ax zpt /κ = (dωR /dx)xzpt /κ is ∼1000
times larger in the experiment of [25] versus that in [24].
We now turn to our conclusions. In this paper, we have provided a thorough and fully
quantum treatment of back-action evasion using a driven electromagnetic cavity, which isparametrically coupled to a mechanical oscillator. We have considered both the unconditionaland conditional aspects of the measurement. In particular, we have derived exactly how strong
the coupling must be to beat the SQL, and to achieve a conditionally squeezed state. We havealso shown how feedback can be used to generate true squeezing, and how this squeezing canbe detected using a second measurement.
Acknowledgments

We thank S Girvin and K Schwab for useful conversations. AC acknowledges the support of
NSERC, the Canadian Institute for Advanced Research, and the Alfred P Sloan Foundation.
FM acknowledges support by NIM, SFB 631, and the Emmy-Noether program of the DFG.
Appendix. Details of calculations

A.1. Spectrum of the detector output
A.1.1. Equations of motion. The Heisenberg equations of motion (in the rotating frame) follow
directly from H0 and the dissipative terms in the total Hamiltonian:



√
κ
˙
dˆ = − dˆ − κ ξˆ(t)eiωR t − i Ã ĉ 1 + e−2iωM t + h.c.
2
i
√ h
κ ˆ √
iωR t
ˆ
= − d − κ ξ (t)e − i 2 Ã X̂ (1 + cos(2ωM t)) + Ŷ sin(2ωM t) ,2
(A.1a)γ√c˙ˆ = − ĉ − γ η̂(t)eiωM t − i Ã 1 + e2iωM t
2




†
ˆ
ˆ
d +d .

(A.1b)

Here, ξˆ describes noise in the cavity input operator b̂in . In the limit where there is only quantum
noise (i.e. shot noise) in the cavity drive, we havehξˆ† (t) · ξˆ(t 0 )i = 0,
(A.2a)hξˆ(t) · ξˆ† (t 0 )i = δ(t − t 0 ).

(A.2b)

In contrast, η̂ describes equilibrium noise due to the intrinsic damping of the mechanicaloscillator. One hashη̂† (t) · η̂(t 0 )i = n eq δ(t − t 0 ),hη̂(t) · η̂† (t 0 )i = n eq + 1 δ(t − t 0 ),(A.3a)(A.3b)

where n eq is a Bose–Einstein occupation factor evaluated at energy h̄ωM and temperature Tbath .
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

15
The equations of motion are easily solved by first writing them in terms of the quadrature
operators X̂ and Ŷ , and then Fourier transforming. To present these solutions, we first introduce
the cavity and mechanical oscillator susceptibilities as1
χR (ω) =(A.4a),−iω + κ/2
χM (ω) =1−iω + γ /2

(A.4b)and define the back-action force fˆBA via√ˆf BA (ω) = − Ã 2κχR (ω) ξˆ(ω + ωR ) + ξˆ(ω − ωR ) .

(A.5)

Note that while ξˆ describes white noise, the cavity susceptibility χR (ω) ensures that fˆBA (ω) is
only significant around a narrow bandwidth centered about zero frequency. Note also that we
define Fourier transformed operators via
Z ∞(A.6a)
Â(ω) ≡
dt Â(t)eiωt ,−∞Â (ω) ≡†Z ∞h

dt Â(t)i†eiωt .

(A.6b)−∞hi†
As such, one has Â(ω) = Â† (−ω).
The solutions of the Fourier-transformed quadrature operators then read
#
" r
ˆ
ˆ

f
(ω
+
2ω
)
−
f
(ω
−
2ω
)
γ
BA
M
BA
M
η̂(ω + ωM ) + η̂† (ω − ωM ) +,X̂ (ω) = χM (ω) −22i(A.7a)
"r

#
ˆ
ˆ

γ
f
(ω
+
2ω
)
+
f
(ω
−
2ω
)
BA
M
BA
M
Ŷ (ω) = χM (ω) iη̂(ω + ωM ) − η̂† (ω − ωM ) − fˆBA (ω) −.
2
2(A.7b)
Note from equations (A.7a) and (A.7b) that there is no back-action damping of either
quadrature, even when one deviates from the good-cavity limit by having κ/ωM > 0. This iseasy to understand on a purely classical level. Note first that it is only the cosine quadrature(i.e. dˆ + dˆ† ) of the cavity which couples to the mechanical resonator. As the cavity is itself
a harmonic oscillator, this means that only the cavity sine quadrature (i.e. dˆ − dˆ† ) will beaffected by the resonator motion. As the cavity cosine quadrature provides the back-action force
on the resonator (cf equation (12b)), it thus follows that the back-action force is completely
independent of both quadratures of the mechanical resonator’s motion. There is thus no backaction damping, as such damping requires a back-action force which responds (with some timelag) to the motion of the oscillator.
Equations (13a) and (13b) for the noise spectra of X̂ and Ŷ at frequencies ω  κ
now follow directly from equations (A.7a) and (A.7b), whereas equations (A.2a) and (A.3a)
determine the noises ξˆ and η̂.
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

16
A.1.2. Output spectrum and beating the SQL. Standard input–output theory [29, 30] yields thefollowing relation between b̂out , the field leaving the cavity, and b̂in , the field entering the cavity:b̂out (t) = b̂in (t) +√κ â(t).

(A.8)

In our case of a one-sided cavity, this relation becomes in the lab (i.e. non-rotating) frame:√−i(ω − ωR ) − κ/2b̂out (ω) = b̄out (ω) +ξˆ(ω) − i Ã 2κχR (ω − ωR ) · X̂ (ω − ωR ).
(A.9)−i(ω − ωR ) + κ/2
The first term on the rhs simply represents the output field from the cavity in the absence ofthe mechanical oscillator and any fluctuations. It will yield sharp peaks at the two sidebands
associated with the drive, ω = ωR ± ωM . The second term on the rhs of equation (A.9) represents
the reflected noise of the incident cavity drive. This noise will play the role of the ‘intrinsic
output noise’ or ‘measurement imprecision’ of this detector.
Finally, the last term on the rhs of equation (A.9) is the amplified signal: it is simply the
amplified quadrature X of the oscillator. We see that the dynamics of X̂ will result in a signalof bandwidth ∼γ centered at the cavity resonance frequency. This can be detected by making
a homodyne measurement of the signal leaving the cavity. Using a local-oscillator amplitudebLO (t) = iBe−iωR t with B real, and defining the homodyne current as:
∗(A.10)
Iˆ(t) = bLO(t) + b̂†out (t) bLO (t) + b̂out (t) ,one finds that the fluctuating part of I is given in frequency-space byih √ √iω + κ/2 (A.11)
Iˆ(ω) = −B 2 2 Ã κχR (ω) X̂ (ω) + i
ξˆ(ωR + ω) − ξˆ† (−ωR + ω) .
iω − κ/2
The signal associated with the oscillator will be in a bandwidth ∼γ  κ: for these frequencies,
the above expression simplifies to
h 4 Ã

i(A.12)
Iˆ(ω) = −B √X̂ (ω) + −i ξˆ(ωR + ω) − ξˆ† (−ωR + ω) .
κ/2
Using this equation along with equations (A.7a), (A.2a) and (A.3a), it is straightforward
to obtain the result for the homodyne spectrum S I (ω) given in equation (17).
A.2. Conditional evolution
To derive the stochastic master equation (SME) describing the conditional evolution of theresonator under the double sideband measurement scheme, (that is, the evolution given thecontinuous stream of information obtained by the observer), one uses a procedure that isessentially identical to that given in [31]. Once we have moved into the interaction picture (inwhich the quadratures are QND observables), the displacement picture [36] (that is, separated âinto a¯ and dˆ as per equation (9)), and made the rotating-wave approximation, the Hamiltonian
for the combined cavity and resonator system is√
H = − 2 Ã(dˆ + dˆ† )X.
(A.13)
We now perform homodyne detection of output from the (one-sided) cavity, and as a result theevolution of the system is given by the quantum optical SME [36, 37]
 


i
√
dσ = − [H, σ ] dt + κ D dˆ σ dt + ηκ H − idˆ σ dW,(A.14)h̄
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

17
where σ is the joint density matrix of the two systems as before, η is the detection efficiency,and κ is the cavity decay rate. The superoperators D and H are given by
2D[ĉ]σ = 2ĉσc ĉ† − ĉ† ĉσ − σ ĉ† ĉ,(A.15)

H[ĉ]σ = ĉσ + σ ĉ† − Tr[ĉσ + σ ĉ† ]σ,(A.16)

for an arbitrary operator ĉ.
We now wish to obtain an equation for the evolution of the resonator alone. This is possible
so long as the cavity decay rate is fast compared to the timescale of the cavity–resonatorinteraction. That is,pÃ hX 2 i γ∼ ≡   1.
(A.17)
κκThis means that the light ouput from the cavity spends sufficiently little time in the cavity that it
continually provides up-to-the-minute information about the oscillator. With this large damping
rate, the fluctuations of the light in the cavity about the average value a¯ are small, and we canthus expand the cavity state described by the operator dˆ about the vacuumaσ = ρ00 |0ih0| + (ρ10 |1ih0| + h.c.) + ρ11|1ih1| + (ρ20 |2ih0| + h.c.) + O( 3 ).

(A.18)The density matrix for the resonator is then given byρ = Trc [σ ] = ρ00 + ρ11 + O( 3 ),(A.19)

where Trc denotes the trace over the cavity mode. From the master equation (equation (A.14))
we then derive the equations of motion for the ρi j . Adiabatic elimination of the off-diagonal
elements ρ01 and ρ02 (described in detail in [31]) allows us to write a closed set of equations forthe diagonal elements ρ00 and ρ11 . The result is a SME for ρ = ρ00 + ρ11, which ispdρ = k[X, [X, ρ]] dt + 2ηk H[X ]ρ dW,(A.20)
where the measurement strength k = 4 Ã2 /κ. Defining k̃ = 8ηk, and making a Gaussian ansatz
for the quantum state, we find equations (25a)–(25e) for the means and variances of thequadratures X and Y .
A.3. Squeezing via feedback control
There are three formulations that can be used to analyse the behaviour of an observed linear
quantum system: the Heisenberg picture (the input–ouput formalism), the Schrödinger picture(the SME) and the equivalent classical formulation, introduced in [31]. We have already used
the first two methods in our analysis above. To analyse the effect of feedback we now use thethird. The equivalent classical formulation is given by the equationsqγdx = − x dt + γ T̃ eq dWx ,(A.21)
2
p
p
q
γ
k̃
λ̃
dy = − y dt + γ T̃ eq dW y +
dV1 +
dV2 .
(A.22)
222Here, x and y are the classical dynamical variables which represent the true, fluctuating
quadrature amplitudes of the oscillator. The noise sources Wi and Vi are, as always, mutuallyuncorrelated Wiener processes. We have now included two measurements of the x quadrature,
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)

18
one with strength k̃ and the other with strength λ̃, for reasons that will be explained below. The
measurement records (i.e. the homodyned output signals) for these measurements are given byp
dI1 = k̃x dt + dU1 ,(A.23)
pdI2 = λ̃x dt + dU2 .
(A.24)
Once again the Ui are mutually uncorrelated Wiener processes. Of interest are the quantities X̄ 1
and X̄ 2 , which are (respectively) the two observers’ estimates of the X quadrature. Note thatthese are not the same as x above. When λ̃ = 0, so that there is no second measurement, theequation of motion for X̄ 1 is naturally that given by equations (25a)–(25e). With the second
measurement, the dynamics of the means and variances for the first observer becomepd X̄ 1 = −(γ /2) X̄ dt + k̃VX dŨ 1 ,(A.25a)p
dY¯ 1 = −(γ /2)Y¯ dt + k̃C dŨ 1 ,(A.25b)V̇X = −k̃VX2 − γ (VX − T̃ eq ),(A.25c)V̇Y = −k̃C 2 + 2k + 2λ − γ (VY − T̃ eq ),(A.25d)Ċ = −γ C − k̃VX C,(A.25e)where k = k̃/(8η1 ) and λ = λ̃/(8η2 ) are the strengths of the respective measurements (under
the usual definition of measurement strength [32]), and the ηi are the respective efficiencies ofthe measurements. We also introduce a fourth set of noises, Ũ i , where Ũ 1 appears in the above
equations for the first observer, and Ũ 2 would appear in the equations for the second observer,
although we will not need those here. The Ũ i are given by [31]pdŨ i = dIi − X̄ i dt = k̃(x − X̄ i ) dt + dUi .
(A.26)
While it is not obvious, it turns out that the dŨ i are also mutually uncorrelated, and uncorrelatedwith all the other noise sources.
Armed with the above equations, we now introduce feedback into the system. We apply
a continuous feedback force F(t) = αγ X̄ sin(ωM t) to the system in the lab frame. Discarding
rapidly oscillating terms (making a rotating-wave approximation), this results in the following
dynamics for the system
q
γ
αγ 
dx = −
x+X̄ 1 dt + γ T̃ eq dWx ,(A.27)
22qqqγdy = − y dt + γ T̃ eq dW y + k̃/4 dV1 + λ̃/4 dV2 .
(A.28)
2
Note that if X̄ 1 = x, then the feedback simply provides a damping force on the X quadrature
with a rate αγ /2. As we will see, as X̄ 1 will not in general be precisely equal to x, the dynamics
is in fact slightly more complex.
To proceed, we note that applying a known force to the system cannot change the observers’
uncertainty regarding the classical coordinates. Thus, the equations of motion for the variances

New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)19
for both observers are unaffected by the feedback. The equations of motion for the means,however, also pick up exactly the same damping terms. Thus for observer one we havep
(1 + α)γ˙X̄ 1 = −X̄ 1 + k̃VX Ũ˙ 1 ,(A.29)
2pγ(A.30)
Ȳ˙ 1 = − Y¯ 1 + k̃C Ũ˙ 1 .
2
We now want to calculate the variance of the X quadrature under this feedback protocol,
and also the spectrum of the output signal for both observers. Since the X and Y quadratures
are not coupled, we need merely solve the two coupled equationsqγαγẋ = − x −X̄ 1 + γ T̃ eq Ẇx ,(A.31)22p
 γ αγ(A.32)X̄˙ 1 = −++ k̃VX X̄ 1 + k̃VX x + k̃VX U̇1 ,2
2
where we have used equation (26) to write the equation for X̄ 1 in terms of U̇1 rather thanŨ˙ 1 . The unconditional variance of the X quadrature under feedback, which we will denote byfb
fb
VX,tot, is given by the variance of x. Solving for the steady-state value of VX,tot
using the usual2
techniques of Ito calculus, and using the fact that T̃ eq = VX + (k̃/γ )VX (cf equation (30)), we
obtain equation (31). We see that as the feedback strength α tends to infinity, VXfb → VX , asclaimed above.
To calculate the spectrum of the output signal for the first observer we first transform
equations (A.31) and (A.32) to the frequency domain and solve them. The solution is of the
form
q
!


x(ω)
γ T̃ eq Ẇx (ω),(A.33)= M(ω) p
X̄ 1 (ω)
k̃VX U̇1 (ω)
withhẆx (ω)Ẇx (ω0 )i = hU̇1 (ω)U̇1 (ω0 )i = δ(ω + ω0 ).
The output signal for the first measurement isp
I1 (ω) = k̃x(ω) + U̇1 (ω).

(A.34)(A.35)

Note crucially that the first and second terms here are indeed correlated, as the feedback-force
driving x depends on the noise U̇1 . The output spectrum corresponding to I1 is defined via
hI1 (ω)I1 (ω0 )i = S I1 (ω)δ(ω + ω0 ).

(A.36)

Using equations (A.33) and (A.34), and referring the resulting spectrum back to the oscillator
by dividing by k̃, we obtain equation (A.32).
Similarly, the output of the second measurement is given byp
I2 (ω) = λ̃x(ω) + U̇ 2 (ω).
(A.37)
Unlike I1 , the two terms in this equation are completely uncorrelated, as the feedback force
applied to x is independent of the added noise U̇2 . Using this definition and equations (A.33)
and (A.34), and making use of equation (30), we find the output spectrum given in equation (34).
New Journal of Physics 10 (2008) 095010 (http://www.njp.org/)20
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