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2006 New J. Phys. 8 107
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New Journal of Physics
The open–access journal for physics
Reconstructing the dynamics of a movable mirror in
a detuned optical cavity
M Paternostro1,4 , S Gigan1,2 , M S Kim3 , F Blaser1,2 , H R Böhm1,2
and M Aspelmeyer1,2
1
Institute for Quantum Optics and Quantum Information (IQOQI),
Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
2
Physics Faculty, Institute for Experimental Physics, University of Vienna,
Boltzmanngasse 5, A-1909 Vienna, Austria
3
School of Mathematics and Physics, Queen’s University,
Belfast BT7 1NN, UK
E-mail: [email protected]
New Journal of Physics 8 (2006) 107
Received 1 May 2006
Published 23 June 2006
Online at http://www.njp.org/
doi:10.1088/1367-2630/8/6/107
We consider the dynamics of a movable mirror in a Fabry-Perot
cavity coupled through radiation pressure to the cavity field and in contact with a
thermal bath at finite temperature. In contrast to previous approaches, we consider
arbitrary values of the effective detuning between the cavity and an external input
field. We analyse the radiation-pressure effect on the Brownian motion of the
mirror and its significance in the density noise spectrum of the output cavity field.
Important properties of the mirror dynamics can be gathered directly from this
noise spectrum. The presented reconstruction provides an experimentally useful
tool in the characterization of the energy and rigidity of the mirror as modified
by the coupling with light. We also give a quantitative analysis of the recent
experimental observation of self-cooling of a micromechanical oscillator.
Abstract.
4
Permanent address: School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, UK.
New Journal of Physics 8 (2006) 107
1367-2630/06/010107+15$30.00
PII: S1367-2630(06)23636-X
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
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Contents
1. Introduction
2. The system
3. Reconstruction of the cantilever dynamics
4. Revealing radiation-pressure based self-cooling
5. Conclusions
Acknowledgments
References
2
3
7
10
13
14
14
1. Introduction
The interaction between a movable mirror and the radiation field of an optical cavity has
recently been the focus of extensive investigations both at the theoretical and experimental
level. Optomechanical couplings can be used to infer the effects of weak forces acting on the
mirror and play a critical role in the sensitivity of advanced gravitational-wave detectors [1].
The quantum optical properties of a mirror coupled via radiation pressure to a cavity field show
interesting similarities to an intracavity Kerr-like interaction [2]. Recently, in the context of
classical investigations of nonlinear regimes, the dynamical instability of a driven cavity having
a movable mirror has been investigated [3]. From a quantum information processing perspective,
various schemes have been proposed for radiation pressure-based quantum state engineering
[4, 5] and for the creation of entangled states of continuous-variable systems [6, 7]. The possibility
of inducing non-classical behaviour in mechanical systems composed of billions of atoms is
appealing as a test-bed for fundamental questions of quantum physics [8].
Despite the plethora of protocols proposed so far, it is experimentally very challenging to
study the behaviour of mechanical systems at the quantum level. Up to now, no experiment has
achieved working conditions that would put the dynamics of such a system into the quantum
realm, although exciting progresses have been made in this direction [9].5 One way to accomplish
this task is to cool the mechanical system, treated as an oscillator of frequency ωM , to a temperature
T ∗ such that thermal fluctuations are small compared to the ground state energy, i.e. kB T ∗ h̄ωM
(kB is the Boltzmann constant). Recently, various cooling strategies have been suggested based
on optomechanical coupling between light and a mechanical oscillator. Besides noise reduction
in interferometric applications, some of these techniques might also allow to reach the quantum
regime of mechanical systems. Feedback-based protocols, engineered so as to reduce the thermal
vibration of a mirror in contact with a bath at finite temperature have been suggested [10]. The
first experimental evidence along this line [11] was based on a ‘cold damping technique’. Other,
not yet realized, techniques include the coupling of the vibrational mode of the mechanical
oscillator to a single qubit, such as a superconducting quantum interference device in the charge
regime, a trapped ion or a quantum dot embedded in the oscillator [12].
Besides these active techniques, it is also possible to modify the dynamics of a mirror in a detuned cavity so as to induce intrinsically optical cooling (named, from now on, self-cooling) [13].
5
The very recently claimed first observation of quantized displacement in macroscopic oscillators (Gaidarzhy
et al [9]) has been questioned by Schwab et al [9].
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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This mechanism, which applies to any optical force inducing changes in the intracavity
field characterized by a finite response time, has so far been described in a fully classical
picture. Metzger and Karrai [14] have proven its validity for photothermal forces and, very
recently, self-cooling involving radiation pressure forces has been observed [15].
In this paper, we provide a self-consistent quantum mechanical approach to the study of the
dynamics of a movable mirror in an optical cavity for arbitrary values of the detuning between
the optical cavity and an input driving field. Our work, based on the use of linearized Langevin
equations, allows for the exact reconstruction of the quantum statistical properties of the system
at hand [2]. It both generalizes and completes previous investigations about optomechanical
couplings, which have been mainly focused on effectively non-detuned light-cavity systems [6].
In addition to analysing the rich dynamics of a movable mirror coupled to light, our study also
provides an operative procedure to attain important figures of merit related to the light-induced
modification of the mirror rigidity and mean energy. Under this point of view, our work provides
a quantum mechanical picture of the self-cooling mechanism. We thus extend the so far entirely
classical picture of this process [13, 14], setting the most appropriate scenario for the inference
and study of the limitations to self-cooling due to quantum noise. While we defer to later the
quantitative assessment of the problem, here we provide a very brief picture of the central point
of our discussion. The experimentally accessible field leaking out from the cavity is, ideally, a
blank paper on which the statistical properties of the mirror dynamics are being written. In the
limit of a sufficiently large cavity-mode bandwidth, the intracavity field adiabatically follows
the mirror’s dynamics and the phase quadrature of the intracavity field directly depends on the
mirror’s position quadrature. Thus, phase-sensitive measurements of the extracavity density noise
spectrum (DNS) result in the deduction of the noise properties of the mirror’s position quadrature.
This allows us to directly infer the modified rigidity of the mirror induced by radiation pressure.
On the other hand, a simple post-processing of the DNS data permits the determination of
the mean square displacement of the mirror motion, which provides direct information about
the mean energy. A single set of phase-sensitive measurements gives, in principle, a complete
picture of the effects of radiation pressure on the dynamics of a movable mirror.
The paper is organized as follows. In section 2, we introduce the prototype of the system
of interest. We consider the Langevin equations associated with the dynamics of the noise
operators relative to the mirror and the cavity field. These are solved in the frequency domain
for arbitrary values of the effective detuning between the cavity field and an input laser driving
the cavity system. In section 3, we obtain the DNS of the position quadrature of the mirror
and we study the connections between the outcoming field noise spectrum and the analogous
quantity related to the mirror’s motion. We then introduce a function allowing for the mutual
conversion of the two DNSs, i.e. the reconstruction of the mirror’s dynamics from experimentally
accessible quantities. Section 4 is devoted to the application of this strategy to radiation pressureinduced self-cooling. We provide examples of the behaviour of the resonance frequency of the
mirror, its natural damping rate and temperature against the externally tunable effective detuning.
Section 5 summarizes our results.
2. The system
We consider a Fabry-Perot cavity composed of a massive mirror supposed to be fixed (the mirror
is heavy enough that we neglect its vibrational motion) and a lighter mirror, from now on named
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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c
c
c
ξ
Figure 1. Sketch of principle of the physical system we consider. An optical
cavity, in a Fabry-Perot configuration, interacting with an external driving field.
The cavity is formed by a fixed concave coupling mirror, schematically depicted
in the figure above so as to emphasize its concavity, and a lighter planar movable
mirror (a cantilever). The cantilever is in contact with a thermal bath at finite
temperature and undergoes Brownian motion, quantum mechanically described
by the noise operator ξ̂.
the cantilever, which is free to move along the x-direction of a properly set reference frame.
The fixed mirror couples the cavity to an external laser field of frequency ωL and amplitude Ẽ .
This configuration includes the possibility that the cavity system we describe constitutes one
arm of an interferometer for the detection of gravitational waves (in a LIGO-like configuration,
for example [1]) or the basic element of an experiment devoted to the study of non-classical
properties of macroscopic systems (as in the proposal by Marshall et al [5] or in the scheme by
Pinard et al in [7]). All these situations require devices that have a sensitivity bounded only by
quantum fluctuations, which implies the modelization of the movable mirror in terms of a single
quantum harmonic oscillator (or bosonic mode) with frequency ωm , effective mass m and energy
decay rate γm . Moreover, we invoke the condition 2ωm L c (c is the speed of light and L the
cavity length), which implies that the frequency of the cantilever is much smaller than the cavity
free spectral range so that scattering of photons into modes other than the one being considered
can be neglected [16]. In what follows, bolometric effects induced by photon-absorption by the
mirror are neglected.
The intracavity field is described by the annihilation (creation) operator ĉ (ĉ† ) and the cavity
is locked to the frequency ωc ωL . The input laser field populates the intracavity mode which
couples to the cantilever motion through radiation pressure (proportional to the intensity of the
cavity field). The field, in turn, is phase-shifted by an amount proportional to the cantilever’s
displacement. The less the power of the input laser, the smaller the radiation pressure. If no laser
is used, the cantilever will be coupled just to the thermal bath (due to the finite temperature of
the setup), giving rise to Brownian motion. The total energy of the system in a frame rotating at
the frequency of the laser can be written as
1 p̂2
2 2
+ mωm q̂ + i h̄E (ĉ† − ĉ).
Ĥ = Ĥ c + Ĥ rp + Ĥ m + Ĥ Lc = h̄(ωc − ωL )n̂c − h̄χn̂c q̂ +
2 m
(1)
In this equation, Ĥ c (Ĥ m ) is the free Hamiltonian of the cavity field mode (cantilever), Ĥ rp
is the radiation-pressure interaction term and Ĥ Lc is the coupling between the laser and the
cavity, respectively. Note that, in the range of validity of Ĥ, any photon-generation process
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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due to Casimir effect can be √
safely ignored. In equation (1) χ = ωc /L is the optomechanical
†
coupling rate, n̂c = ĉ ĉ, E = 2κẼ with κ the cavity decay rate. The momentum and position
quadratures of the cantilever are indicated as p̂ and q̂. The effect of Ĥ rp is a displacement of
the cantilever conditioned by the intensity of the intracavity field. For detailed discussions about
the capabilities of this conditioned displacement in the context of quantum state engineering of
continuous variables, see for example Mancini et al and Bose et al in [4].
The system we are considering is intrinsically open as the cavity field is damped by the
photon-leakage through the massive coupling mirror and the cantilever is connected to a bath at
finite temperature represented by background internal and external modes. This latter point
implies that, in the absence of radiation-pressure coupling, the cantilever would undergo a
pure Brownian motion driven by its contact with the thermal environment. The presence of
the optomechanical coupling alters this purely dissipative dynamics.
In order to account for the complete open dynamics of the subsystems involved in
this problem, an adequate choice is to use the formalism of the Langevin equations ∂t Ô =
(i/ h̄)[Ĥ, Ô] + N̂ , with Ô a generic operator of the system and N̂ its corresponding noise operator.
The explicit form of the set of Langevin equations for the problem at hand reads
∂t q̂ = p̂/m,
∂t p̂ = −mωm2 q̂ + h̄χn̂c − γm p̂ + ξ̂,
√
∂t ĉ = i(ωL − ωc )ĉ + iχq̂ĉ − κĉ + 2κĉin .
(2)
In these equations, ĉin is the
√ operator associated to the external field, characterized by the steady
average amplitude |Ẽ | = P/( h̄ωL ) (with P the laser power), and a fluctuating part quantum
mechanically described by the operator δĉin . A phase factor can be associated to the complex
laser amplitude. We will comment later about a judicious choice of this phase which allows for a
simplification of our approach. The fluctuating part of the external field has zero mean amplitude
and is delta-correlated as
δĉin (t)δĉ†in (t ) = δ(t − t ),
δĉin (t)δĉin (t ) = δĉ†in (t)δĉin (t ) = 0,
(3)
where δ(t) is the Dirac delta function and all the other correlation functions are zero. The
operator ξ̂ accounts for the (zero-mean) Brownian noise affecting the mirror. A consistent
quantum mechanical analysis, valid at any temperature, reveals that the Brownian noise is not
simply delta-correlated but exhibits a non-Markovian correlation function at different times
reading [17]
h̄γm
h̄ω
−iω(t−t )
m ωe
+ 1 dω,
(4)
coth
ξ̂(t)ξ(t ) =
2π
2kB T
where the cut-off frequency of the bath power spectrum has been taken as infinite and T is the
bath temperature. The non-Markovian nature of the Brownian noise is particularly relevant at
low temperatures. The Langevin equations with the correlation function for the Brownian-noise
operator provide a general description of the dynamics of the system at any temperature.
It is important to notice the nonlinearity in equation (1) arising from the coupling between
the intracavity intensity and the position quadrature of the cantilever. This makes the problem
difficult and is at the basis of nonlinear effects which can be encountered in the dynamics of
a mirror interacting with light through radiation pressure. A considerable simplification to the
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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analytical approach to equations (2) comes from taking a bright input laser field. In this case,
the system is characterized by a semiclassical steady state defined by a stable amplitude of
the intracavity field and a new equilibrium position of the cantilever, in accordance with the
classical description of a driven harmonic oscillator. In the semiclassical approximation, one
can rely on linearization techniques which allow us to treat equations (2) as an effective linear
set. This can be done by focusing our attention on the fluctuations of the relevant operators
in the problem, expanded as Ô = Os + δÔ (Ô = q̂, p̂, ĉ, ĉ† ) with Os the mean value and δÔ
the zero-mean operator accounting for fluctuations. It is worth stressing that, other than being
a well-established tool, this linearization approach allows for the exact reconstruction of the
quantum statistical properties of the system at hand, as far as the fluctuations of the investigated
operators are small with respect to the steady-state values [2]. The first step is the research of the
steady-state values Os , obtained by setting the time-derivative in (2) to zero. This leads to the
equilibrium position and the steady-state amplitude of the cavity field
ps = 0,
qs =
h̄χ|cs |2
,
mωm2
cs =
E
,
κ + i
(5)
where we introduced the effective cavity-laser detuning = ωc − ωL − χqs . The equation for
cs is very important for the understanding of the physics behind this problem. Firstly, from it
we clearly see how the cantilever dynamics affects the steady state of the intracavity field. The
coupling to the mirror changes the field inside the cavity in a way to induce a new stationary
intensity. The change occurs after a transient time depending on the response of the cavity and the
strength of the coupling to the cantilever. Secondly, this equation makes evident the nonlinearity
in the system. The third-order nature of the expression for cs could lead to the appearance of
instability and turning points in the cantilever dynamics (for an experimental observation of the
associated hysteresis cycle see for example [18]). In order to find the conditions of stability of
the system, i.e. the conditions for a steady state to settle in the dynamics at hand, we consider
the linearized Langevin equations for the fluctuation operators
∂t δq̂ = δp̂/m,
∂t δp̂ = −mωm2 δq̂ + h̄χ(cs δĉ† + cs∗ δĉ) − γm δp̂ + ξ̂,
∂t δĉ = −iδĉ + iχcs δq̂ − κδĉ +
√
2κδĉin ,
∂t δĉ = iδĉ +
†
†
iχcs∗ δq̂
− κδĉ +
†
√
(6)
2κδĉ†in .
A useful simplification comes from taking cs to be real. This is possible by adjusting the relative
phase between the intracavity√field and the external laser. In particular, taking E = |E |e−iθ ,
cs is real for e−iθ = (κ + i)/ κ2 + 2 . By summing and subtracting the last two equations
at (6), it is straightforward to get the dynamical equation for the quadratures of the cavity
field δx̂ = δĉ† + δĉ and δŷ = i(δĉ† − δĉ). Under a formal point of view, the vector of the
system’s fluctuation quadratures f̂ = (δq̂ δp̂ δx̂ δŷ)τ (where τ denotes transposition) satisfies
the homogeneous differential equation ∂t f̂ = Kf̂ with the kernel matrix K, which can be easily
deduced from equations (6) (the inhomogeneous part of the quadratures dynamics arising from
the noise contributions). The stability of the solution depends entirely on the properties of the
eigenvalues of K and can be determined by standard tools of modern control theory. A widely
applied technique is provided by the Routh–Hurwitz test [19]. A first stability condition comes
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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from considering the characteristic polynomial of K
2 h̄χ2 cs2
m
2
2
2
2
2
+(2κωm + γm + κ γm ) + ωm (κ + 2 ) = 0,
4 + (2κ + γm )3 + (2 + κ2 + ωm2 + 2κγm )2 −
(7)
where labels the eigenvalues of K. In order for the roots of this equation to be stable, the sign
of the coefficients in front of the different powers of has to be the same, which leads to the first
stability condition
ωm2 (κ2 + 2 ) >
2 h̄χ2 cs2
.
m
(8)
The additional condition we need is found by applying the recipe in [20], which leads to
2 h̄
χ2 cs2
(2κ + γm )2 + κγm 4 + 22 2κ2 + γm2 − 2ωm2 + 2κγm
m
+(κ2 + ωm2 )(κ2 + ωm2 + 2κγm ) + κ2 γm2 > 0.
(9)
The study of these conditions reveals the point at which the system enters an instable regime.
Here, we will restrict ourselves to the stable region.
3. Reconstruction of the cantilever dynamics
The linearized Langevin equations can be easily solved in the frequency domain, where the
Fourier transform of equations (6) leads to the following expression for the fluctuations of the
position quadrature of the cantilever
δq̂(ω) = −
√
1 2
[ + (κ − iω)2 ]ξ̂(ω) − i h̄ 2κχcs
d(ω)
× [(ω + iκ − )δĉ†in (ω) + (ω + iκ + )δĉin (ω)] ,
(10)
where ω stands for frequency and d(ω) = 2 h̄χ2 cs2 + m[2 + (κ − iω)2 ](ω2 − ωm2 + iωγm ). The
expression for δq(ω) shows the thermal contribution to the cantilever’s motion, dependent on
ξ̂(ω), given by the coupling with the environment. It is present regardless of the detuning and is independent from the optomechanical coupling. If no radiation pressure is assumed, the
cantilever is just a harmonic oscillator in contact with a thermal bath and undergoing Brownian
motion described by ξ̂. Thus, from the first term in equation (10), the spectrum of the displacement
of the cantilever has to be characterized by a Lorentzian susceptibility function, peaked at ωm
with width γm . The second term, which accounts for radiation pressure, is proportional to χ and
modifies the response of the cantilever.
It is useful to introduce the DNS of the cantilever position quadrature Sq (ω) which is
deduced from the two-frequency auto-correlation function δq̂(ω)δq̂() = Sq (ω)δ(ω + ). In
general, given a process P(ω), its DNS is defined as the Fouriertransform of the corresponding
auto-correlation function, see [21]. In details SP (ω) = (1/2π) de−i(ω+)t P(ω)P(). This
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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definition is perfectly equivalent to the one used in the body of this work. The calculation of the
cantilever DNS requires the use of the noise correlation functions
ĉin (ω)ĉ†in () = 2πδ(ω + ),
h̄ω
ξ̂(ω)ξ̂() = 2π h̄γm mω 1 + coth
2KT
(11)
δ(ω + ).
After a little calculation, we arrive at the expression for the symmetrized cantilever position
quadrature DNS
h̄
2 h̄κχ2 cs2 (2 + κ2 + ω2 ) + mωγm
Sq (ω) =
|d(ω)|2
h̄ω
2
2
2
2
2
2 2
.
(12)
× [ ( + 2κ − 2ω ) + (κ + ω ) ] coth
2kB T
In this expression, the contribution due to radiation pressure and the one due to the thermal contact
with the bath can be singled out. Again, a consistency check reveals that without radiationpressure any dependence from the effective detuning vanishes and the resulting spectrum is
simply that of a harmonic oscillator undergoing Brownian motion at temperature T . Any
information about the cantilever’s modified motion is obtained from the study of Sq (ω). The
quantitative analysis of the features characterizing Sq (ω) is performed in section 4, where the
mirror dynamics is studied under the perspective of radiation-pressure induced self-cooling.
An immediate observation is that Sq (ω) is peaked at a frequency ω − eff which, in general, is
different from ωm for any finite = 0.6
Here, we focus on the way the modified state of motion of the mirror influences the statistical
properties of the noise associated to the cavity field. Apart from the interest related to the
evidence of mirror-field optomechanical coupling, this gives us an operative method to infer the
mirror motion. Let us consider the solution of equations (6) for the intracavity field quadratures,
expressed in terms of the mirror quadrature fluctuation δq
√ δx̂in − (κ − iω)δŷin
2χcs (κ − iω)
δ
q̂
−
2κ
δŷ = 2
+ (κ − iω)2
2 + (κ − iω)2
= α()δq̂ + β(ω)δx̂in + η(ω)δŷin ,
(13)
showing that δŷ is related not only to the mirror but also to the statistical features of the vacuum
noise entering the cavity. From equation (13) and considering the correlations
δx̂in (ω)δx̂in () = δŷin (ω)δŷin () = 2πδ(ω + ),
δx̂in (ω)δŷin () = −δŷin (ω)δx̂in () = 2πiδ(ω + ),
6
(14)
Finding out an analytical expression for ωeff is, in general, a difficult problem due to the non-algebraic
nature of the maximization of Sq (ω). However, in the large temperature limit of kB T h̄ω, the approximation
coth( h̄ω/2kB T) 2kB T/ h̄ω can be used, which reduces the maximization of the mirror’s quadrature DNS to the
research for the roots of an algebraic polinomial in ω. A further simplification comes from the assumption of large
cavity bandwidth, such that κ ω, which allows us to get an analytical (but cumbersome) expression for ωeff . The
advantage of an approach to the modifications in the cantilever’s susceptibility function based on the study of Sq (ω)
is, obviously, its validity beyond the limits of a classical picture.
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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we find that the DNS of the intracavity field is related to Sq (ω) via the simple relation
Sy (ω) = |α()|2 Sq (ω) + additional terms.
(15)
The explicit form of the additional terms appearing in the above relation is not relevant for the
following discussion and we will not consider it. Under a physical point of view, equation (15)
formalizes the back action of the cantilever on the intracavity field dynamics. The statistics of the
two different subsystems (the optical cavity field and the mechanical oscillator) are inter-related
due to the radiation-pressure coupling.
√
We consider Collett and Gardiner’s input-output relations [22] δr̂out + δr̂in = 2κδr̂ (with
r̂ = ĉ, ĉ† ) in order to infer the spectrum of the noise associated to the field leaking out from the
cavity. In what follows, we concentrate on the structure of the DNS without directly referring to
any specific detection scheme. It is evident that any phase-sensitive detection device is suitable
for our purposes. The choice of one specific scheme implies the definition of a proper signal to
be detected. As an example, a phase modulation detection analogous to the Pound–Drever–Hall
technique [23] requires the introduction of two noise sources in the expression for the signal,
arising from the modulation (and subsequent demodulation) procedure. None of them is present
in a standard homodyne detection. As a consequence, the two signal DNS obtained with these
strategies will differ by the form of the shot noise term [24, 25]. On the other hand, each detection
procedure also affects the signal amplitude according to its specific response and sensitivity on
cavity detuning. Our approach provides the basic toolbox to be used in the appropriate definition
of a signal and is therefore totally general.
The evaluation of the DNS of δŷout proceeds as sketched before for the case of the mirror
position quadrature and leads directly to the expression
8 h̄κχ2 cs2
h̄ω
2 2
2
2
2
2
.
2 h̄κχ cs + m[(κ − γm )ω − κωm ] + mωγm (κ + ω ) coth
Syout (ω) = 1 +
|d(ω)|2
2kB T
(16)
Written in this form, Syout is quite informative. It is an even function of the frequency ω, as implied
by the stationarity of the process under consideration. The first contribution to the spectrum is
the quantum mechanical shot noise of the light. The first and the second terms inside the bracket
describe the back action of the field on to the cantilever dynamics, modified by the nonzero
detuning . The last term, proportional to the temperature of the bath, shows the effects of
thermal fluctuations.
By assuming the parameters used in [15], where the stability of the solution of equations
(6) is guaranteed for > 0, the term in the bracket is always positive, thus ruling out squeezing
of the observed quadrature. More generally, this always holds for sufficiently large temperatures
such that the thermal contribution overcomes the detuning-dependent term (T 1 K). On the
other hand, starting with a precooled mirror, the thermal contribution can become negligible and,
at low frequencies (such that ω < ωm ), noise reduction below the shot noise limit can occur for
proper values of [2]. Within the validity of equation (1), Syout (ω) is the general DNS of the
field leaking out from a detuned cavity with a movable mirror. It is valid at any temperature,
provides a general description of the modified dynamics of the field and also allows for the study
of the effects of quantum noise. Equation (16) generalizes to non-zero detunings the spectrum
evaluated in [17].
We now go back to the expression for the intracavity field given by equation (13) and notice
that the DNS of the outgoing field can be calculated by exploiting directly the position
√ quadrature
of the mirror and the quadratures of the input vacuum noise through δŷout + δŷin = 2κδŷ. Even
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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if formally equivalent to the derivation of Syout conducted before, this allows us to explicitly
single out the contribution to Syout given by the DNS of δq̂(ω). In complete analogy to what has
been said in equation (15), it is possible to see that
Syout (ω) = 1 + 2κ|α()|2 Sq (ω) + spurious terms,
(17)
where the spurious terms contain not just the contribution from the input noise but also terms
related to the correlation functions between δq̂(ω) and the vacuum noise. Nevertheless, by
assuming a large initial temperature (T ∼ 100 K) and parameters within the experimental state
of the art [15], |α()|2 Sq (ω) can be many orders of magnitude larger than the additional terms
appearing in equation (15),7 hence justifying the approximation Sy (ω) |α()|2 Sq (ω). An
analogous comparison between the spurious terms present in equation (17) and 2κSy (ω) leads
to similar figures of merit and enables us to neglect the effects of correlations different from
the auto-correlations of δq̂(ω). This analysis makes clear that, in virtue of the correspondence
between the spectra, a measurement of the DNS of δŷout corresponds to a faithful measurement
of the cantilever dynamics. The quantitative correspondence depends on |α()|2 , which acts as a
transfer function in the transformation Syout (ω) ↔ Sq (ω). In particular, the behaviour of |α()|2
for an assigned value of is very flat within a range of frequencies around ω − eff equal to
several times the full width at half maximum (FWHM) of the corresponding resonance peak. The
transfer function is thus well approximated by considering its value for ω = ω − eff,8 making
the conversion between the spectra frequency-independent. In these conditions, equation (17) is
the main tool for the faithful reconstruction of the dynamics of a movable mirror in a detuned
cavity. As will be shown in section 4, our analysis represents the rigorous theoretical background
to those experimental observations involving radiation pressure effects in a detuned cavity (as in
[15]). On the other hand, as already stressed, the general nature of equation (17) paves the way
to the study of the onset of quantum behaviour and the limit imposed to the reconstruction of the
dynamics of the cantilever. This corresponds to the analysis of the cases in which the additional
noise terms in equations (15) and (17) become relevant to the determination of Syout (ω).
4. Revealing radiation-pressure based self-cooling
Having established the connection between the modified cantilever motion and the observable
DNS of the output field, we now turn our attention to a specific example: the characterization
of the self-cooling effect induced by radiation pressure and the identification of its markers in
Syout (ω). This provides a significative test for the reconstruction strategy described above. For a
better understanding of our analysis, we mention the effects that have be expected in association
to self-cooling. On detuning, one expects to observe the modification of the mechanical rigidity
of the cantilever (resulting in overdamping and blue shifts of the resonance frequency) and the
reduction of its mean energy.
Let us start by analysing the properties of the DNS Sq (ω) under the self-cooling perspective.
A good source of information is represented by its study as a function of ω and . In order to fix
the ideas, in what follows we assume values for the physical parameters very close to those used
in the experiment [15] (see caption of figure 2). We choose to work in the stability conditions
The signal-to-noise ratio between |α()|2 Sq (ω) and the additional terms is 107 for the parameters in [15].
For all practical purposes, it is possible to neglect the variations of |α()|2 with ωeff and use its value at ω = ωm
as a transfer function.
7
8
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(a)
s
∇
ω
(b)
sy
∇
ω
(c)
srec
q
ω
∇
Figure 2. (a) DNS Sq (in units of 10−26 m2 s−1 ) against ω (units of 106 s−1 ) and
(units of 108 s−1 ). We have chosen λL = 2πc/ωL = 1064 nm, L = 25 mm,
P = 4 mW, ωm /2π = 275 kHz, T = 300 K, m = 15 ng, mechanical quality
factor Q = ωm /γm = 2.1 × 103 and cavity finesse equal to 400. (b) DNS for
the output field quadrature (in units of 109 ) against ω (units of 106 s−1 ) and (units of 108 s−1 ). (c) Sqrec (ω) (in units of 10−26 m2 s−1 ) reconstructed from the
extracavity field via the function |α()|2 .
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Normalized area
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1
332
0.5
166
0
5×107
1×108
1.5×108
2×108
Width
12
2.5×108
∆
Figure 3. FWHM and normalized peak area against ∈ [0, 2.5κ] (units per
second). The points (red curve, rightmost vertical scale (units of Hz)) show
the increasing width of the peaks for up to ∼ 0.45κ, thus demonstrating the
overdamping of the cantilever. The points (blue curve, leftmost vertical scale
(arbitrary units)) show the area below the resonance peaks (normalized with
respect to the area at = 0). The reduction in area is in correspondence with
the decrease in temperature of the cantilever. Self-cooling for up to ∼ 0.45κ is
shown.
characterized by > 0, which corresponds to the regime of self-cooling of the cantilever motion.
At a given value of , the DNS is well approximated by a Lorentzian curve [14]. As already
anticipated, Sq (ω) experiences detuning-dependent shifts which, for our choice of parameters,
bring the resonance peak to ω − eff < ωm . As demonstrated in figure 2(a), the red-shift of the
resonance peak achieves a maximum at ∼ 0.45κ. By further increasing the detuning, making
it comparable to the cavity bandwidth κ, ω − eff → ωm . This is because less power enters inside
the cavity, which becomes far-off resonant with respect to the input laser field, thus reducing
the radiation pressure influences. A second feature visible from figure 2(a) is the reduction
of the height of the peaks achieved by increasing up to ∼ 0.45κ. Further increases of push
the system toward the conditions achieved at zero detuning.
This analysis accounts for only two of the effects associated with radiation pressure acting
on a movable mirror in a detuned resonator. Nevertheless, the physics behind the dynamics of
our system is much richer. In particular, following [14], we have to expect an increase in the
FWHM of each peak associated to a different value of and, most importantly, a decrease of
the overall area below a resonance peak. This latter feature is relevant: the area below the DNS
of the position quadrature is proportional to the mean square displacement of the cantilever,
i.e. to its mean potential energy. We have also checked that the dimensionless variances of the
position and momentum quadratures of the cantilever follow the same quantitative behaviour
against the detuning, which implies a corresponding reduction in the kinetic energy of the
cantilever. Thus, the sole inference of the energy of the oscillating mirror through the area
underneath the position quadrature’s DNS provides a faithful indication of diminishing energy.
By considering the equipartition principle, this implies the reduction of the temperature of the
cantilever, a manifestation of the self-cooling mechanism predicted in [13, 14]. The overdamping
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and temperature-reduction effects are both shown in figure 3 for ∈ [0, 2.5κ]. The above
considerations are valid regardless of the specific choice for the set of physical parameters
characterizing a given configuration. Quantitatively, the modifications to the cantilever dynamics
depend on the actual values of the set of parameters entering Sq (ω). Experimentally, one can
identify in the input power, the damping rates κ and γm and the cantilever mass m some key
parameters in the self-cooling mechanism, whose values can be decided so to optimize the desired
effects. A general analysis of the dependence of self-cooling from the mentioned parameters
is difficult, the achievement of the desired level of cooling being a trade-off between κ, the
mechanical quality of the vibrational mode under study and the mass of the cantilever. We just
mention that, in general, a better optical quality of the cantilever (i.e. a small κ) and small values
of γm and m magnify the overdamping mechanism and improve the normalized area reduction.
The proportionality between Sq (ω) and Sy (ω) analysed in section 3 shows that the maximum
of Syout (ω) occurs at a frequency identical to ωeff . Moreover, because the transfer function acts
as a detuning-dependent rescaling factor, an analogous argument holds for the frequencies at
which Syout is equal to half of its maximum. Therefore, the FWHM and the frequency shifts of
the resonance peaks in Syout correspond to the analogous quantities in Sq (ω). The observation of
Syout thus provides reliable information about the cantilever’s rigidity.
However, |α()|2 is a function which decreases with . This affects the behaviour of
the inferred mirror temperature against as the decaying values of the transfer function
monotonically reduce the amplitude of each resonance peak. As a result, while after an optimal
detuning (corresponding to the maximum self-cooling) the areas below the DNS of the cantilever
position quadrature start to increase, the areas underneath the peaks in the noise spectra of δŷout
simply decay. This is demonstrated in figure 2(b), where we show the behaviour of Syout against ω
and . Evidently, with the increase of the detuning the height of the peaks shrinks continuously.
Moreover, the heating of the cantilever back to the temperature corresponding to = 0 is masked
by the spoiling effects due to the decreasing power entering the cavity, as discussed above. In
figure 2(a), indeed, up to = κ there is no evidence for ‘rising up’ of the peak height. No
reliable information about temperature can thus be directly gathered from the integrated DNS
of δŷout . The right trend can be regained only through the conversion via the transfer function
|α()|2 . As an illustrative example, in figure 2(c) we show the reconstructed Sqrec (ω) obtained
by calculating (Syout (ω) − 1)/2κ|α()|2 . The spectra shown in panels (a) and (c) are practically
indistinguishable, with the right behaviour of the area against detuning being evidently brought
back. We have checked that the effectiveness of the DNS reconstruction can be extended to a
wide range of values of the relevant physical parameters, the only necessary constraint being the
negligibility of the spurious terms in equations (15) and (17).
5. Conclusions
We have considered a driven Fabry-Perot cavity with a movable mirror, which is the paradigmatic
setup for the study of radiation-pressure effects. Our study aimed at the analysis and
reconstruction of the mirror dynamics under general conditions of detuning between the cavity
and the driving field. We have provided a self-consistent approach, sufficient to account for the
main features of the mirror motion as modified by the interaction with the driven intracavity
field. As an experimentally motivated application, we have considered the example of selfcooling induced by radiation pressure and the possibility to single out its effects by looking at the
statistical properties of the noise associated to the mirror motion. This provides the full quantum
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mechanical counterpart for the pioneering studies performed in [13, 14]. We have generalized
the studies performed so far within the context of radiation pressure. In addition, our approach
serves as a toolbox for the interpretation of experiments directed toward the observation of selfcooling involving radiation pressure effects. The presented framework allows us to investigate
the ultimate quantum limits on the effects of cavity-induced optomechanical coupling and to
study the role of quantum noise in their reconstruction.
Acknowledgments
We are grateful to K Jacobs, D Vitali and A Zeilinger for fruitful discussions. We acknowledge
financial support by the Austrian Science Fund (FWF) under SFB15 (Coherent Control of
Quantum Systems) and by the City of Vienna. MSK acknowledges support from KRF (2003070-C00024) and the UK EPSRC.
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