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Reconstructing the dynamics of a movable mirror in a detuned optical cavity
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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 ∗ h¯ ωM (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].
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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 ˆ by the noise operator ξ. 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 E˜. 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 cˆ (ˆc† ) 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 pˆ 2 2 2 ˆ + mωm qˆ + i h¯ E (cˆ† − c). Hˆ = Hˆ c + Hˆ rp + Hˆ m + Hˆ Lc = h¯ (ωc − ωL )nˆ c − h¯ χnˆ c qˆ + 2 m (1) In this equation, Hˆ c (Hˆ m ) is the free Hamiltonian of the cavity field mode (cantilever), Hˆ rp is the radiation-pressure interaction term and Hˆ Lc is the coupling between the laser and the ˆ any photon-generation process cavity, respectively. Note that, in the range of validity of H, 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 † ˆ E = 2κE˜ with κ the cavity decay rate. The momentum and position coupling rate, nˆ c = cˆ c, quadratures of the cantilever are indicated as pˆ and q. ˆ The effect of Hˆ 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 Oˆ = ˆ Oˆ ] + Nˆ , with Oˆ a generic operator of the system and Nˆ its corresponding noise operator. (i/ h¯ )[H, The explicit form of the set of Langevin equations for the problem at hand reads ˆ ∂t pˆ = −mωm2 qˆ + h¯ χnˆ c − γm pˆ + ξˆ , ∂t qˆ = p/m, √ ∂t cˆ = i(ωL − ωc )cˆ + iχqˆcˆ − κcˆ + 2κcˆin . (2) In these equations, cˆin is the √ operator associated to the external field, characterized by the steady ˜ average amplitude |E | = P/( h¯ ωL ) (with P the laser power), and a fluctuating part quantum mechanically described by the operator δcˆin . 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 δcˆin (t)δcˆ†in (t ) = δ(t − t ),
δcˆin (t)δcˆin (t ) = δcˆ†in (t)δcˆin (t ) = 0,
(3)
where δ(t) is the Dirac delta function and all the other correlation functions are zero. The operator ξˆ 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] h¯ γm h¯ ω −iω(t−t ) ˆ m ωe + 1 dω, (4) coth ξ(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
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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 ˆ p, ˆ c, ˆ cˆ† ) with Os the mean value and δOˆ in the problem, expanded as Oˆ = Os + δOˆ (Oˆ = q, 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 =
h¯ χ|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 ˆ ∂t δqˆ = δp/m,
∂t δpˆ = −mωm2 δqˆ + h¯ χ(cs δcˆ† + cs∗ δc) ˆ − γm δpˆ + ξˆ ,
∂t δcˆ = −i δcˆ + iχcs δqˆ − κδcˆ +
√
2κδcˆin ,
∂t δcˆ = i δcˆ + †
†
iχcs∗ δqˆ
− κδcˆ + †
√
(6) 2κδcˆ†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 ˆ Under a formal point of view, the vector of the field δˆx = δcˆ† + δcˆ and δˆy = i(δcˆ† − δc). ˆ system’s fluctuation quadratures f = (δqˆ δpˆ δˆx δˆy)τ (where τ denotes transposition) satisfies the homogeneous differential equation ∂t fˆ = Kfˆ 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
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from considering the characteristic polynomial of K 2 h¯ χ2 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 h¯ χ2 cs2 . m
(8)
The additional condition we need is found by applying the recipe in [20], which leads to 2 h¯
χ2 cs2 (2κ + γm )2 + κγm 4 + 2 2 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 δq(ω) ˆ = −
√ 1 2 [ + (κ − iω)2 ]ξˆ (ω) − i h¯ 2κχcs d(ω)
× [(ω + iκ − )δcˆ†in (ω) + (ω + iκ + )δcˆin (ω)] ,
(10)
where ω stands for frequency and d(ω) = 2 h¯ χ2 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 ξˆ (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 ξˆ . 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 δq(ω)δ ˆ q( ) ˆ = Sq (ω)δ(ω + ). In general, given a process P(ω), its DNS is defined as the Fourier transform of the corresponding auto-correlation function, see [21]. In details SP (ω) = (1/2π) d e−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 cˆin (ω)cˆ†in ( ) = 2πδ(ω + ),
h¯ ω ξˆ (ω)ξˆ ( ) = 2π h¯ γm mω 1 + coth 2KT
(11) δ(ω + ).
After a little calculation, we arrive at the expression for the symmetrized cantilever position quadrature DNS
h¯ 2 h¯ κχ2 cs2 ( 2 + κ2 + ω2 ) + mωγm Sq (ω) = |d(ω)|2 h¯ ω 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 √ δˆxin − (κ − iω)δˆyin 2χcs (κ − iω) δ q ˆ − 2κ δˆy = 2 + (κ − iω)2 2 + (κ − iω)2 = α( )δqˆ + β(ω)δˆxin + η(ω)δˆyin ,
(13)
showing that δˆy 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 δˆxin (ω)δˆxin ( ) = δˆyin (ω)δˆyin ( ) = 2πδ(ω + ), δˆxin (ω)δˆyin ( ) = −δˆyin (ω)δˆxin ( ) = 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 h¯ ω, the approximation coth( h¯ ω/2kB T) 2kB T/ h¯ ω 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.
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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] δˆrout + δˆrin = 2κδˆr (with rˆ = c, ˆ cˆ† ) 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 δˆyout proceeds as sketched before for the case of the mirror position quadrature and leads directly to the expression
8 h¯ κχ2 cs2 h¯ ω 2 2 2 2 2 2 . 2 h¯ κχ 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 δˆyout + δˆyin = 2κδˆy. 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 ˆ In complete analogy to what has single out the contribution to Syout given by the DNS of δq(ω). 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 δq(ω) ˆ 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 δq(ω). ˆ This analysis makes clear that, in virtue of the correspondence between the spectra, a measurement of the DNS of δˆyout 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 .
New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
Normalized area
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1
332
0.5
166
0
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1×108
1.5×108
2×108
Width
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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 New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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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 δˆyout 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 δˆyout . 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 New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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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. References [1] Caves C M 1980 Phys. Rev. Lett. 45 75 Loudon R 1981 Phys. Rev. Lett. 47 815 Buonanno A, Chen Y and Mavalvala N 2003 Phys. Rev. D 67 122005 Mertz J, Marti O and Mlynek J 1993 Appl. Phys. Lett. 62 2344 Stowe T D, Yasumura K, Kenny T W, Botkin D, Wago K and Rugar D 1997 Appl. Phys. Lett. 71 288 Vitali D, Mancini S and Tombesi P 2001 Phys. Rev. A 64 051401 (R) [2] Fabre C, Pinard M, Bourzeix S, Heidmann A, Giacobino E and Reynaud S 1994 Phys. Rev. A 49 1337 Mancini S and Tombesi P 1994 Phys. Rev. A 49 4055 Corbitt T, Chen Y, Khalili F, Ottaway D, Vyatchanin S, Whitcomb S and Mavalvala N 2006 Phys. Rev. A 73 023801 [3] Marquardt F, Harris J G E and Girvin S M 2006 Appl. Phys. Lett. 96 103901 [4] Mancini S, Manko V I and Tombesi P 1997 Phys. Rev. A 55 3042 Tombesi P 1996 Quantum Interferometry ed F De Martini, G Denardo and Y Shih (Weinheim: VCH) Bose S, Jacobs K and Knight P L 1997 Phys. Rev. A 56 4175 [5] Marshall W, Simon C, Penrose R and Bouwmeester D 2003 Appl. Phys. Lett. 91 130401 [6] Mancini S, Giovannetti V, Vitali D and Tombesi P 2002 Appl. Phys. Lett. 88 120401 Pirandola S, Mancini S, Vitali D and Tombesi P 2003 Phys. Rev. A 68 062317 Pirandola S, Mancini S, Vitali D and Tombesi P 2003 J. Opt. B Quantum Semiclass. Opt. 5 S523 Pirandola S, Mancini S, Vitali D and Tombesi P 2003 Phys. Rev. A 68 062317 [7] Pinard M, Dantan A, Vitali D, Arcizet O, Briant T and Heidmann A 2005 Europhys. Lett. 72 747 Ferreira A, Guerreiro A and Vedral V 2006 Appl. Phys. Lett. 96 060407 Pirandola S, Vitali D, Tombesi P and Lloyd S 2005 Preprint quant-ph/0509119 [8] Leggett A J and Shimony A 1998 Quantum Measurement: Beyond Paradox ed R A Healey and G Hellman (Minneapolis, MN: University of Minnesota Press) p 1 Zurek W H 2003 Rev. Mod. Phys. 75 715 Bose S 2006 Appl. Phys. Lett. 96 060402 [9] Knobel R G and Cleland A N 2003 Nature (London) 424 291 LaHaye M D, Buu O, Camarota B and Schwab K C 2004 Science 304 74 Arcizet O, Briant T, Heidmann A and Pinard M 2006 Phys. Rev. A 73 033819 Gaidarzhy A, Zolfagharkhani G, Badzey R L and Mohanty P 2005 Appl. Phys. Lett. 94 030402 Schwab K C, Blencowe M P, Roukes M L, Cleland A N, Girvin S M, Milburn G J and Ekinci K L 2005 Appl. Phys. Lett. 95 248901 New Journal of Physics 8 (2006) 107 (http://www.njp.org/)
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[10] Mancini S, Vitali D and Tombesi P 1998 Phys. Rev. Lett. 80 688 Vitali D, Mancini S, Ribichini L and Tombesi P 2002 Phys. Rev. A 65 063803 Vitali D, Mancini S, Ribichini L and Tombesi P 2003 J. Opt. Soc. Am. B 20 1054 [11] Cohadon P F, Heidmann A and Pinard M 1999 Phys. Rev. Lett. 83 3174 [12] Armour A D, Blencowe M P and Schwab K C 2002 Phys. Rev. Lett. 88 148301 Tian L and Zoller P 2004 Appl. Phys. Lett. 93 266403 Martin I, Shnirman A, Tian L and Zoller P 2004 Phys. Rev. B 69 125339 Wilson-Rae, Zoller P and Imamoglu A 2004 Appl. Phys. Lett. 92 075507 [13] Braginsky V and Vyatchanin S P 2002 Phys. Lett. A 293 228 [14] Metzger C H and Karrai K 2004 Nature (London) 432 1002 [15] Gigan S, Böhm H R, Paternostro M, Blaser F, Langer G, Hertzberg B, Schwab K, Bäuerle D, Aspelmeyer M and Zeilinger A 2006 submitted [16] Law C K 1994 Phys. Rev. A 49 433 Law C K 1994 Phys. Rev. A 51 2537 [17] Giovannetti V and Vitali D 2001 Phys. Rev. A 63 023812 [18] Dorsel A, McCullen J D, Meystre P, Vignes E and Walther H 1983 Appl. Phys. Lett. 51 1550 [19] Hurwitz A 1964 Selected Papers on Mathematical Trends in Control Theory ed R Bellman and R Kalaba (New York: Dover) [20] DeJesus E X and Kaufman C 1987 Phys. Rev. A 35 5288 [21] Gardiner C W 1985 Handbook of Stochastic Methods (Berlin: Springer) [22] Gardiner C W 1991 Quantum Noise (Berlin: Springer) Collett M J and Gardiner C W 1984 Phys. Rev. A 30 1386 Gardiner C W and Collett M J 1985 Phys. Rev. A 31 3761 Gardiner C W 1985 Handbook of Stochastic Methods (Berlin: Springer) [23] Bjorklund G C 1980 Opt. Lett. 5 15 Black E D 2001 Am. J. Phys. 69 79 [24] Jacobs K, Tittonen I, Wiseman H M and Schiller S 1999 Phys. Rev. A 60 538 [25] Vitali D, Punturo M, Mancini S, Amico P and Tombesi P 2004 J. Opt. B: Quantum Semiclass. Opt. 6 S691
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