Force sensing based on coherent quantum noise cancellation in a hybrid optomechanical cavity with squeezed-vacuum injection
Ali Motazedifard, F. Bemani, M. H. Naderi, R. Roknizadeh, D. Vitali
FForce sensing based on coherent quantum noise cancellation in a hybridoptomechanical cavity with squeezed-vacuum injection
Ali Motazedifard, ∗ F. Bemani, † M. H. Naderi, ‡ R. Roknizadeh, § and D. Vitali ¶ Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran Quantum Optics Group, Department of Physics, Faculty of Science,University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran Physics Division, School of Science and Technology, University of Camerino,I-62032 Camerino, Italy, and INFN, Sezione di Perugia, Perugia, Italy (Dated: November 6, 2018)We propose and analyse a feasible experimental scheme for a quantum force sensor based on theelimination of back-action noise through coherent quantum noise cancellation (CQNC) in a hybridatom-cavity optomechanical setup assisted with squeezed vacuum injection. The force detector,which allows for a continuous, broad-band detection of weak forces well below the standard quantumlimit (SQL), is formed by a single optical cavity simultaneously coupled to a mechanical oscillator andto an ensemble of ultracold atoms. The latter acts as a negative-mass oscillator so that atomic noisecancels exactly the back-action noise from the mechanical oscillator due to destructive quantuminterference. Squeezed vacuum injection enforces this cancellation and allows to reach sub-SQLsensitivity in a very wide frequency band, and at much lower input laser powers.
PACS numbers: 42.50.Dv, 03.65.Ta, 42.50.WkKeywords: Force sensing, Standard quantum limit, Hybrid optomechanics, Squeezed vacuum state
I. INTRODUCTION
Every measurement is affected by noise, degrading thesignal and consequently reducing the accuracy of themeasurement. However, noise cancellation techniquescan be applied if the noise can be identified and mea-sured separately, as, for example, in the acoustic domain[1]. The application of noise cancellation to quantumsystems has recently been introduced [2, 3] by using theso-called coherent quantum noise cancellation (CQNC)scheme, which relies on quantum interference. The ba-sic idea is that under certain conditions, it is possible tointroduce an “anti-noise” path in the dynamics of the sys-tem which can be employed to cancel the original noisepath via destructive interference.The measurement of weak forces at the quantum limit[4] and the search for quantum behavior in macroscopicdegrees of freedom have been some of the motivations atthe basis of the development of cavity optomechanics [5–7]. In a force measurement based on an optomechanicalscheme [8, 9], the competition between shot noise andradiation pressure back-action noise leads to the notionof SQL [4]. Shot noise is a known effect limiting high-precision interferometry at high frequencies [10], whileradiation pressure noise, recently observed for the firsttime [11, 12], becomes relevant only at large enough pow-ers and will be limiting in the low-frequency regime next-generation gravitational-wave detectors [13]. These two ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] noise sources have opposite scaling with the input fieldpower: increasing the input power in order to enhance themeasurement strength and decrease the shot noise leadsto increase the measurement back-action noise. There-fore, in order to improve the force detection sensitivityone has to eliminate the backaction noise.There are various proposals for reducing quantumnoise and overcoming the SQL in force measurements,including frequency-dependent squeezing of the inputfield [14], variational measurements [15, 16], the use ofKerr medium in a cavity [17], a dual mechanical oscil-lator setup [18], the optical spring effect [19], and two-tone measurements [20–23]. Preliminary experimentaldemonstrations of these ideas have been already carriedout [24–29], and recent clear demonstrations of quantum-nondemolition measurements have been given in [22, 30].A different approach for sub-SQL measurements hasrecently been introduced [2, 3], based on the CQNC ofback action noise via quantum interference. The idea isbased on introducing an“anti-noise” path in the dynam-ics of the optomechanical system via the addition of anancillary oscillator which manifests an equal and oppo-site response to the light field, i.e, an oscillator with aneffective negative mass. In the context of atomic spinmeasurements an analogous idea for coherent backactioncancellation was proposed independently [31, 32], andhas been applied for magnetometry below the SQL [33],demonstrating that Einstein-Podolski-Rosen (EPR)-likeentanglement of atoms generated by a measurement en-hances the sensitivity to pulsed magnetic fields. The orig-inal proposal [2] focused on the use of an ancillary cavitythat is red-detuned from the optomechanical cavity. Aquantum non-demolition coupling of the electromagneticfields within the two cavities yields the necessary anti-noise path, so that the backaction noise is coherently a r X i v : . [ qu a n t - ph ] J un cancelled. Ref. [34] considered in more detail the all-optical realization of the CQNC proposal put forwardedin [2, 3], and found that the requirements for its exper-imental implementation appear to be very challenging,especially for the experimentally relevant case of low me-chanical frequencies and high-quality mechanical oscilla-tors (MO) such as gravitational wave detectors. Other se-tups, which provide effective negative masses of ancillarysystems for CQNC, have been suggested based on em-ploying Bose-Einstein condensates [35], or the combina-tion of a two-tone drive technique and positive-negativemass oscillators [36].In recent years, hybrid optomechanical systems as-sisted by the additional coupling of the cavity mode withan atomic gas have attracted considerable attention. Ithas been found that the additional atomic ensemble maylead to the improvement of optomechanical cooling [37–41], thereby providing the possibility of ground statecooling outside the resolved sideband regime [42, 43].Moreover, the coupling of the mechanical oscillator to anatomic ensemble can be used to generate a squeezed stateof the mechanical mode [44], or robust EPR-type entan-glement between collective spin variables of the atomicmedium and the mechanical oscillator [31, 45].Inspired by the above considerations, more recentlya theoretical scheme for CQNC based on a dual cavityatom-based optomechanical system has been proposed[46]. In this scheme, a MO used for force sensing is cou-pled to an ultracold atomic ensemble trapped in a sepa-rate optical cavity which behaves effectively as an effec-tive negative mass oscillator (NMO). The two cavities arecoupled via an optical fiber. This system is a modificationof the setup suggested for hybrid cooling and electromag-netically induced transparency [47] and the interactionbetween the optomechanical cavity and the atomic en-semble leads to the CQNC. The atomic ensemble acts asa more flexible NMO, for which the “impedance match-ing” condition of a decay rate identical to the mechanicaldamping rate is easier to satisfy with respect to the full-optical implementation of Ref. [34].Here we propose to simplify and improve the atomicensemble implementation of CQNC of Ref. [46] by con-sidering a different setup, involving only a single optome-chanical cavity and a single cavity mode, coupled also toan atomic ensemble, which is also injected by squeezedvacuum (see Fig.1 (a)). The atomic ensemble is coupledto the radiation pressure and the coupling strength of theatom-field interaction is modulated. We show that theinteraction between the optomechanical cavity and theatomic ensemble leads to an effective NMO that can pro-vide CQNC conditions able to eliminate the backactionnoise of the MO. In fact, destructive quantum interfer-ence between the collective atomic noise and the back-action noise of the MO realizes an ‘anti-noise’ path, sothat the backaction noise can be cancelled (Fig.1 (b)).CQNC conditions are realized when the optomechanicalcoupling strength and the mechanical frequency are equalto the coupling strength of the atom-field interaction and to the effective atomic transition rate, respectively. Fur-thermore, the dissipation rate of the MO needs to bematched to the decoherence rate of the atomic ensemble.Here we exploit the injection of appropriately squeezedvacuum light in order to control and improve the noisereduction for force detection, applying within this newscenario, the properties of squeezing. In fact, it is wellknown that the injection of a squeezed state in the unusedport of a Michelson interferometer can improve interfer-ometric measurements [48–54], as recently demonstratedin the case of gravitational wave interferometers [55].The improvement of the performance of measurement viasqueezing injection has also been demonstrated in otherinterferometers, such as the Mach-Zehnder [56], Sagnac[57], and polarization interferometers [58]. Squeezing-enhanced measurement have been realized also withinoptomechanical setups: an experimental demonstrationof squeezed-light enhanced mechanical transduction sen-sitivity in microcavity optomechanics has been reportedin [59]. Moreover, by utilizing optical phase trackingand quantum smoothing techniques, improvement in thedetection of optomechanical motion and force measure-ments with phase-squeezed state injection has also beenverified experimentally [60]. Finally the improvementin position detection by the injection of squeezed lighthas been recently demonstrated also in the microwavedomain in Ref. [61]. We also notice that it has beenrecently theoretically shown that even the intracavitysqueezing generated by parametric down conversion canenhance quantum-limited optomechanical position detec-tion through de-amplification [62]. More recently Ref.[63] has investigated the response of a mechanical oscil-lator in an optomechanical cavity driven by a squeezedvacuum and has shown when it can be used as a highsensitive nonclassical light sensor.In the present paper we show that if the cavity mode isinjected with squeezed light with an appropriate phase,back-action noise cancellation provided by CQNC ismuch more effective because squeezing allows to sup-press the shot noise contribution at a much smaller inputpower, and one has a significant reduction of the forcenoise spectrum even with moderate values of squeezingand input laser power.The paper is organized as follows. Section II is devotedto the description of the model. The linear quantumLangevin equations of motion for the dynamical variablesinvolved in the sensing process are derived in Section III.The main results for force sensing and the increased sen-sitivity achieved in the case of back-action cancellationprovided by CQNC are given in Section IV. Finally, theconclusions are summarized in Section V. II. THE SYSTEM
The optomechanical setup considered in this paper isschematically described in Fig. 1(a). The system consistsof a single Fabry-P´erot cavity in which a MO, serving as a sq c ω ω= L ω c ∆ C a v i t y r e s p o n s e Γ m σ ω ω= effective two-level atom e (a) pumping field g cos( ) G G t ω κ L sq ω R Ω , L L P ω ˆ out a ext F ˆ x ( , , ) m m m γ ω ˆ( , ) c a ω MO R Ω e e eg σ δ σ δ R Ω (C) e γ (b) ˆ P ˆ X MO ˆ a X field ˆ a P ˆ d X ˆ d P signal ext F backaction noise anti backaction noise C QN C signal modified shot noise output NMO (atom) † ˆ ˆ( , ) a a FIG. 1. (Color online) (a) Schematic description of the sys-tem under consideration. A mechanical oscillator with fre-quency ω m is placed within a single-mode Fabry-P´erot cavitycontaining an atomic ensemble that can be controlled by aclassical pumping field with Rabi frequency Ω R with effectivetransition rate ω σ = ω m . An external force F ext is exertedon the mechanical oscillator acting as a sensor. The cavity isdriven by a classical laser field with power P L and frequency ω L , and also a squeezed light field, resonant with the cavitymode, ω sq = ω c , is injected into the cavity. (b) Flow chartrepresentation of the backaction noise cancellation caused bythe anti-noise path associated with the interaction of the cav-ity mode with the atomic ensemble acting as a negative massoscillator (NMO). (c) Atomic scheme leading to the effectiveFaraday interaction, with a double Λ atomic system coupledto the intracavity mode ˆ a (thin blue line) and driven by aclassical control field (thick blue line) of frequency ω G = ω c resonant with the cavity mode. test mass for force sensing, is directly coupled to the radi-ation pressure of an optical cavity field. Furthermore, thecavity contains an ensemble of effective two-level atomsthat is coupled to the intracavity mode. As is shown inFig. 1(c), and will be detailed further below, the two-levelatomic ensemble with time-modulated coupling constantconsidered in this scheme is achievable by considering adouble Λ-type atomic ensemble driven by the intracavitylight field and by a classical control field.We consider a standard optomechanical setup with asingle cavity mode driven by a classical laser field withfrequency ω L , input power P L , interacting with a sin-gle mechanical mode treated as a quantum mechanicalharmonic oscillator with effective mass m , frequency ω m ,and canonical coordinates ˆ x and ˆ p , with [ˆ x, ˆ p ] = i (cid:126) . Thissingle mode description can be applied whenever scatter-ing of photons from the driven mode into other cavitymodes is negligible [64], and if the detection bandwidthis chosen such that it includes only a single, isolated,mechanical resonance and mode-mode coupling is negli-gible [65]. Moreover, the cavity is injected by a squeezedvacuum field with central frequency ω sq which is assumed to be resonant with the cavity mode ω sq = ω c .The total Hamiltonian describing the system is givenby ˆ H = ˆ H c + ˆ H m + ˆ H om + ˆ H d + ˆ H at + ˆ H F , (1)where ˆ H c describes the cavity field, ˆ H m represents theMO in the absence of the external force F ext , ˆ H om de-notes the optomechanical coupling, ˆ H d accounts for thedriving field, ˆ H at contains the atomic dynamics, and ˆ H F denotes the contribution of the external force. The firstfour terms in the Hamiltonian of Eq. (1) are given byˆ H c = (cid:126) ω c ˆ a † ˆ a, (2a)ˆ H m = (cid:126) ω m ˆ b † ˆ b = ˆ p m + 12 mω m ˆ x , (2b)ˆ H om = (cid:126) g ˆ a † ˆ a (ˆ b + ˆ b † ) , (2c)ˆ H d = i (cid:126) E L (ˆ a † e − iω L t − ˆ ae iω L t ) , (2d)where ˆ a and ˆ b are the annihilation operators of the cav-ity field and the MO, respectively, whose only nonzerocommutators are [ˆ a, ˆ a † ] = [ˆ b, ˆ b † ] = 1. Furthermore,ˆ x = x ZP F (ˆ b + ˆ b † ) and ˆ p = ip ZP F (ˆ b † − ˆ b ), with x ZP F = (cid:112) (cid:126) / mω m and p ZP F = (cid:126) / x ZP F the zero-point position and momentum fluctuations of the MO. g = ( dω c /dx ) x ZP F is the single-photon optomechanicalstrength, while E L = (cid:112) P L κ in / (cid:126) ω L , with κ in the cou-pling rate of the input port of the cavity.For the atomic part, we consider an ensemble of N ul-tracold four-level atoms interacting non-resonantly withthe intracavity field and with a classical control field withRabi frequency Ω R and frequency ω G (see Fig. 1(c)).Considering the far off-resonant interaction, the two ex-cited states | e (cid:105) and | e (cid:105) will be only very weakly pop-ulated. In this limit, these off-resonant excited statescan be adiabatically eliminated so that the light-atominteraction reduces the coupled double - Λ system to aneffective two-level system, with upper level | e (cid:105) and lowerlevel | g (cid:105) , (Fig. 1(c)), driven by the so-called Faraday orquantum non-demolition interaction [66]. Apart from thelight-matter interface, the Faraday interaction has im-portant applications also in continuous non-demolitionmeasurement of atomic spin ensembles [67], quantum-state control/tomography [68] and magnetometry [69].In the system under consideration, we also assume thata static external magnetic field tunes the Zeeman split-ting between the states | e (cid:105) and | g (cid:105) into resonance withthe frequency ω m of the MO.Considering the effective two-level model for theatomic ensemble, we introduce the collective spin opera-tors ˆ S + = N (cid:88) i =1 (cid:12)(cid:12)(cid:12) e ( i ) (cid:69) (cid:68) g ( i ) (cid:12)(cid:12)(cid:12) = ( ˆ S − ) † , (3a)ˆ S z = 12 N (cid:88) i =1 (cid:12)(cid:12)(cid:12) e ( i ) (cid:69) (cid:68) e ( i ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) g ( i ) (cid:69) (cid:68) g ( i ) (cid:12)(cid:12)(cid:12) , (3b)where i labels the different atoms. The collective spin op-erators obey the commutation relations (cid:104) ˆ S + , ˆ S − (cid:105) = 2 ˆ S z and (cid:104) ˆ S ∓ , ˆ S z (cid:105) = ± ˆ S z , so that the effective Hamiltonianof the atomic ensemble can be written asˆ H at = (cid:126) ω m ˆ S z + (cid:126) G cos( ω G t )(ˆ a + ˆ a † )( ˆ S + + ˆ S − ) , (4)where G = E (Ω R /δ σ ) is the atom-field coupling, with E and δ σ denoting the cavity-mode Rabi frequency andthe detuning of the control beam from the excited atomicstates, respectively. Now we assume that the atoms areinitially pumped in the hyperfine level of higher energy, | e (cid:105) , which results in an inverted ensemble that can beapproximated for large N by a harmonic oscillator ofnegative effective mass. This fact can be seen formallyusing the Holstein-Primakoff mapping of angular momen-tum operators onto bosonic operators [70]. In our casewe have a total spin equal to N/ d such that ˆ S z = N/ − ˆ d † ˆ d , ˆ S + = √ N (cid:104) − ˆ d † ˆ d/N (cid:105) / ˆ d ,ˆ S − = √ N ˆ d † (cid:104) − ˆ d † ˆ d/N (cid:105) / , so that the commutationrules are preserved. As long as the ensemble remainsclose to its fully inverted state, we can take ˆ d † ˆ d/N (cid:28) S − (cid:39) √ N ˆ d † , ˆ S + (cid:39) √ N ˆ d . Therefore,under the bosonization approximation, we can rewriteEq. (4) asˆ H at = − (cid:126) ω m ˆ d † ˆ d + (cid:126) G cos( ω G t )(ˆ a + ˆ a † )( ˆ d + ˆ d † ) , (5)which shows that the atomic ensemble can be effectivelytreated as a NMO, coupled with the collective coupling G = G √ N with the cavity mode. Moving to theframe rotating at laser frequency ω L , where ˆ a → ˆ ae − iω L t ,choosing the resonance condition ω G = ω L , and apply-ing the rotating wave approximation in order to neglectthe fast rotating terms, i.e., the terms proportional to e ± i ( ω G + ω L ) t , one getsˆ H at = − (cid:126) ω m ˆ d † ˆ d + (cid:126) G a + ˆ a † )( ˆ d + ˆ d † ) . (6)Therefore, the total Hamiltonian of the system in theframe rotating at laser frequency ω L is time-independentand can be written asˆ H = (cid:126) ∆ c ˆ a † ˆ a + (cid:126) ω m ˆ b † ˆ b − (cid:126) ω m ˆ d † ˆ d + (cid:126) g ˆ a † ˆ a (ˆ b + ˆ b † )+ (cid:126) G a + ˆ a † )( ˆ d + ˆ d † ) + i (cid:126) E L (ˆ a † − ˆ a ) , (7)where ∆ c = ω c − ω L . III. DYNAMICS OF THE SYSTEM
The dynamics of the system is determined by the quan-tum Langevin equations obtained by adding damping and noise terms to the Heisenberg equations associatedwith the Hamiltonian of Eq. (7) [71],˙ˆ x = ˆ p/m (8a)˙ˆ p = − mω m ˆ x − p ZP F g ˆ a † ˆ a − γ m ˆ p + η + ˜ F ext , (8b)˙ˆ a = − i ∆ c ˆ a − ig ˆ a ˆ xx ZP F − i G d + ˆ d † ) + E L − κ a + √ κ ˆ a in , (8c)˙ˆ d = iω m ˆ d − i G a + ˆ a † ) − Γ2 ˆ d + √ Γ ˆ d in , (8d)where γ m is the mechanical damping rate, Γ is the collec-tive atomic dephasing rate, κ denotes the cavity photondecay rate. We have also considered an external clas-sical force ˜ F ext which has to be detected by the MO.The system is also affected by three noise operators:the thermal noise acting on the MO, η ( t ), the opticalinput vacuum noise, ˆ a in , and the bosonic operator de-scribing the optical vacuum fluctuations affecting theatomic transition, ˆ d in [72]. These noises are uncorre-lated, and their only nonvanishing correlation functionsare (cid:104) ˆ a in ( t )ˆ a in ( t ) † (cid:105) = (cid:104) ˆ d in ( t ) ˆ d in ( t ) † (cid:105) = δ ( t − t (cid:48) ) [72]. Here,we have assumed that the external classical force has noquantum noise. The Brownian thermal noise operator η ( t ) obeys the following correlation function [71] (cid:104) η ( t ) η ( t (cid:48) ) (cid:105) = mγ m (cid:126) (cid:90) dω π ωe − iω ( t − t (cid:48) ) (cid:20) coth (cid:18) (cid:126) ω k B T (cid:19) +1 (cid:21) , (9)where T is the temperature of the thermal bath of theMO. The mechanical quality factor Q m = ω m /γ m is typi-cally very large, justifying the weak damping limit wherethe Brownian noise can be treated as a Markovian noise,with correlation function [71] (cid:104) η ( t ) η ( t (cid:48) ) (cid:105) (cid:39) (cid:126) mγ m [ ω m (2¯ n m +1) δ ( t − t (cid:48) )+ iδ (cid:48) ( t − t (cid:48) )] , (10)where ¯ n m = (exp( (cid:126) ω m /k B T ) − − is the mean ther-mal phonon number and δ (cid:48) ( t − t (cid:48) ) is the time derivativeof the Dirac delta. The term proportional to the deriva-tive of the Dirac delta is the antisymmetric part of thecorrelation function, associated with the commutator of η ( t ) [71], but it does not contribute to the subsequent ex-pressions where we have always calculated symmetrized correlation functions.We define the optical and atomic quadrature operatorsˆ X a = (ˆ a † + ˆ a ) / √
2, ˆ P a = i (ˆ a † − ˆ a ) / √
2, ˆ X d = ( ˆ d + ˆ d † ) / √ P d = i ( ˆ d † − ˆ d ) / √ X ina = (ˆ a in, † + ˆ a in ) / √
2, ˆ P ina = i (ˆ a in, † − ˆ a in ) / √ X ind = ( ˆ d in, † + ˆ d in ) / √ P ind = i ( ˆ d in, † − ˆ d in ) / √ X = ˆ x/ √ x ZP F and ˆ P = ˆ p/ √ p ZP F ,so that [ ˆ X, ˆ P ] = i . We then consider the usual regimewhere the cavity field and the atoms are strongly drivenand the weak coupling optomechanical limit, so that wecan linearize the dynamics of the quantum fluctuationsaround the semiclassical steady state. After straight-forward calculations, the linearized quantum Langevinequations for the quadratures’ fluctuations are obtainedas δ ˙ˆ X = ω m δ ˆ P , (11a) δ ˙ˆ X d = − ω m δ ˆ P d − Γ2 δ ˆ X d + √ Γ ˆ X ind , (11b) δ ˙ˆ X a = ∆ c δ ˆ P a − κ δ ˆ X a + √ κ ˆ X ina , (11c) δ ˙ˆ P = − ω m δ ˆ X − γ m δ ˆ P − gδ ˆ X a + √ γ m ( ˆ f + F ext ) , (11d) δ ˙ˆ P a = − ∆ c δ ˆ X a − gδ ˆ X − Gδ ˆ X d − κ δ ˆ P a + √ κ ˆ P ina , (11e) δ ˙ˆ P d = ω m δ ˆ X d − Gδ ˆ X a − Γ2 δ ˆ P d + √ Γ ˆ P ind , (11f)where the effective linearized optomechanical couplingconstant is g = 2 g α s , ∆ c = ∆ c − g | α s | /ω m isthe effective cavity detuning, and α s is the intracavityfield amplitude, solution of the nonlinear algebraic equa-tion ( κ/ i ∆ c ) α s = E L − iG ω m Re α s / (Γ / ω m ),which is always possible to take as a real number byan appropriate redefinition of phases. Finally we haverescaled the thermal and external force by defining f ( t ) = η ( t ) / √ (cid:126) mω m γ m and F ext = ˜ F ext ( t ) / √ (cid:126) mω m γ m . Theseequations are analogous to those describing the CQNCscheme proposed in [3] and then adapted to the case whenthe NMO is realized by a blue detuned cavity mode inRef. [34], and by an inverted atomic ensemble in Ref.[46]. Compared to the latter paper, the cavity mode tun-nel splitting 2 J is replaced by the effective cavity modedetuning ∆ c .As suggested by the successful example of the injec-tion of squeezing in the LIGO detector [55] and more re-cently in an electro-mechanical system [61], we now showthat the force detection sensitivity of the present schemecan be further improved and can surpass the SQL whenthe cavity is driven by a squeezed vacuum field, witha spectrum centered at the cavity resonance frequency ω sq = ω c .The squeezed field driving is provided by the finitebandwidth output of an optical parametric oscillator(OPO), shined on the input of our cavity system, imply-ing that the cavity mode is subject to a non-Markoviansqueezed vacuum noise, with two-time correlation func-tions given by [73] (cid:10) ˆ a in ( t )ˆ a in ( t (cid:48) ) (cid:11) = M b x b y b x + b y (cid:0) b y e − b x τ + b x e − b y τ (cid:1) , (12a) (cid:10) ˆ a in, † ( t )ˆ a in ( t (cid:48) ) (cid:11) = N b x b y b y − b x (cid:0) b y e − b x τ − b x e − b y τ (cid:1) , (12b)where τ = | t − t (cid:48) | , while b x and b y define the bandwidthproperties of the OPO driven below threshold [74] forthe generation of squeezed light. The squeezing band-widths and the parameters M and N are related to theeffective second-order nonlinearity ε and the cavity decayrate γ of the OPO by b x = γ/ − | ε | , b y = γ/ | ε | and M = ( εγ/ /b x + 1 /b y ), N = ( | ε | γ/ /b x − /b y ).It is clear that N ≥ b x ≥
0. The chosen parametrization satisfies the well known condition | M | ≤ N ( N + 1) for squeezednoise. In the case of pure squeezing, there are onlytwo independent parameters, one can parametrize M =(1 /
2) sinh(2 r ) exp( iφ ) and N = sinh r , with r and φ be-ing, respectively, the strength and the phase of squeezing,so that | M | = N ( N + 1) and b y = b x (cid:112) N + | M | + 1).In the white noise limit, i.e., when b x,y → ∞ , whilekeeping M and N constant, the correlation functionscan be written in Markovian form, i.e., (cid:104) ˆ a in ( t )ˆ a in ( t (cid:48) ) (cid:105) = M δ ( t − t (cid:48) ) and (cid:10) ˆ a in, † ( t )ˆ a in ( t (cid:48) ) (cid:11) = N δ ( t − t (cid:48) ). We will re-strict to this white noise limit from now on, which is jus-tified whenever the two bandwidths b x,y are larger thanthe mechanical frequency ω m and the cavity line-width κ . In the next section, we will study how CQNC eliminat-ing the effect of backaction noise and squeezed-vacuuminjection can jointly act in order to improve significantlythe detection of a weak force acting on the MO. IV. FORCE SENSING AND CQNC
An external force acting on the MO shifts its position,which in turn is responsible for a change of the effectivelength of the cavity and therefore of the phase of theoptical cavity output. As a consequence, the signal as-sociated to the force can be extracted by measuring theoptical output phase quadrature, ˆ P outa , with heterodyneor homodyne detection. The expression for the outputfiled can be obtained from the standard input-output re-lation [72, 75], i.e., ˆ a out = √ κδ ˆ a − ˆ a in , so that the outputquadrature is given byˆ P outa = √ κδ ˆ P a − ˆ P ina , (13)and solving Eqs. (11) for δ ˆ P a . Typically stationary spec-tral measurements of forces are carried out and thereforewe are interested in the solution for P outa in the frequencydomain. After straightforward calculations, we getˆ P outa = √ κχ (cid:48) a (cid:110) − gχ m √ γ m (cid:16) ˆ f + F ext (cid:17) + √ κ (cid:20)(cid:18) − χ (cid:48) a κ (cid:19) ˆ P ina − ∆ c χ a ˆ X ina (cid:21) − Gχ d √ Γ (cid:20) ˆ P ind − ˆ X ind (cid:18) Γ / iωω m (cid:19)(cid:21) + √ κχ a (cid:0) g χ m + G χ d (cid:1) ˆ X ina (cid:111) , (14)where we have defined the susceptibilities of the cavityfield, the MO, and of the atomic ensemble, respectivelyas χ a ( ω ) = 1 κ/ iω ,χ m ( ω ) = ω m ( ω m − ω ) + iωγ m ,χ d ( ω ) = − ω m ( ω m − ω + Γ /
4) + iω Γ , (15)and we have introduced the modified cavity mode sus-ceptibility1 χ (cid:48) a = 1 χ a − χ a ∆ c (cid:0) g χ m + G χ d − ∆ c (cid:1) . (16)Eq. (14) is the experimental signal which, after cali-bration, is used for estimating the external force F ext .Appropriately rescaling Eq. (14) we can rewrite F est ext ≡ − gχ (cid:48) a χ m √ κγ m ˆ P outa ≡ F ext + ˆ F N , (17)where the added force noise is defined asˆ F N = ˆ f − (cid:114) κγ m gχ m (cid:20)(cid:18) − χ (cid:48) a κ (cid:19) ˆ P ina − ∆ c χ a ˆ X ina (cid:21) + Gχ d gχ m (cid:115) Γ γ m (cid:20) ˆ P ind − Γ / iωω m ˆ X ind (cid:21) − g χ m + G χ d gχ m (cid:114) κγ m χ a ˆ X ina . (18)Eq. (18) shows that in the present scheme for force de-tection we have four different contributions to the forcenoise spectrum. The first term corresponds to the ther-mal noise of the MO, the second term corresponds tothe shot noise associated with the output optical field,which is the one eventually modified by the squeezed in-put field. The third term is the contribution of the atomicnoise due to its interaction with the cavity mode, whilethe last term describes the backaction noise due to thecoupling of the intracavity radiation pressure with theMO and with the atomic ensemble. A. CQNC conditions
The CQNC effect amounts to the perfect backactioncancellation at all frequencies, obtaining in this way sig-nificantly lower noise in force detection. From the lastterm in Eq. (18), it is evident that for g = G and χ m = − χ d the contributions of the backaction from themechanics and from the atoms cancel each other for allfrequencies. As shown in Fig. 1(b), they can be thoughtof as ‘noise’ and ‘anti-noise’ path contributions to the sig-nal force F ext . Therefore an effective NMO, in this caserealized by the inverted atomic ensemble, is necessary forrealizing χ m = − χ d . More in detail, CQNC is realizedwhenever:(i) the coupling constant of the optical field with theMO and with the atomic ensemble are perfectlymatched, g = G , which is achievable by adjustingthe intensity of the fields driving the cavity and theatoms;(ii) the atomic dephasing rate between the two loweratomic levels Γ must be perfectly matched with themechanical dissipation rate γ m (we have assumedthe atomic Zeeman splitting perfectly matchedwith the MO frequency ω m from the beginning); (iii) the MO has a high mechanical quality factor, orequivalently, Γ (cid:28) ω m so that the term Γ / χ d (see Eq. (15)).Mechanical damping rates of high quality factor MO arequite small, not larger than 1 kHz. As already pointedout in Sec. III of Ref. [46], the matching of the twodecay rates is easier in the case of atoms because groundstate dephasing rates can also be quite small [76, 77]. Onthe contrary, matching the dissipative rates in the casewhen the NMO is a second cavity mode, as in the fullyoptical model of Ref. [34], is more difficult because itrequires having a cavity mode with an extremely smallbandwidth which can be obtained only assuming largefinesse and long cavities.Note that under CQNC conditions the effective suscep-tibility of Eq. (16) becomes χ (cid:48) CQNCa = (1 /χ a + χ a ∆ c ) − .It is clear that under the CQNC conditions the last termin the noise force of Eq. (18) is identically zero and wecan rewriteˆ F N = ˆ f − (cid:114) κγ m gχ m (cid:20)(cid:18) − χ (cid:48) a κ (cid:19) ˆ P ina − ∆ c χ a ˆ X ina (cid:21) − (cid:20) ˆ P ind − Γ / iωω m ˆ X ind (cid:21) . (19)In order to quantify the sensitivity of the force mea-surement, we consider the spectral density of added noisewhich is defined by [34] S F ( ω ) δ ( ω − ω (cid:48) ) = 12 (cid:16)(cid:68) ˆ F N ( ω ) ˆ F N ( − ω (cid:48) ) (cid:69) + c.c (cid:17) . (20)Under perfect CQNC conditions one gets the force noisespectrum in the presence of squeezed-vacuum injectionwhich, in the experimentally relevant case κ (cid:29) ω , reads(see Appendix A for the explicit derivation) S F ( ω ) = k B T (cid:126) ω m + 12 (cid:18) ω + γ m / ω m (cid:19) + κg γ m | χ m | (cid:34) (cid:18)
12 + 2∆ c κ (cid:19) +Σ( M, N, ∆ c /κ ) (cid:35) , (21)whereΣ( M, N, ∆ c /κ ) = N (cid:18)
12 + 2∆ c κ (cid:19) + 2 ∆ c κ Im M (cid:18) c κ − (cid:19) +Re M (cid:34) c κ − (cid:18)
12 + 2∆ c κ (cid:19) (cid:35) (22)is the contribution of the injected squeezing to the op-tomechanical shot noise. Eq. (21) shows that whenCQNC is realized, the noise spectrum consists of threecontributions: the first term denotes the thermal Brown-ian noise of the MO, the second term describes the atomicnoise, and the last one represents the optomechanicalshot noise modified by squeezed-vacuum injection.We recall that with the chosen units, the noise spec-tral density is dimensionless and in order to convert itto N Hz − units we have to multiply by the scale factor (cid:126) mω m γ m . This noise spectrum has to be compared withthe noise spectrum of a standard optomechanical setupformed by a single cavity coupled to a MO at resonancefrequency (∆ c = 0) [5, 7], S stF ( ω ) = k B T (cid:126) ω m + 12 (cid:34) κg γ m | χ m | + 4 g κγ m (cid:35) . (23)As it is well known, the standard quantum limit for sta-tionary force detection comes from the minimization ofthe noise spectrum at a given frequency over the driv-ing power, i.e., over the linearized coupling squared g ,yielding S stF ( ω ) ≥ S SQL = 1 γ m | χ m ( ω ) | . (24)In the present case, the complete cancellation of the back-action noise term proportional to g has the consequencethat force detection is limited only by shot noise and thattherefore the optimal performance is achieved at verylarge power. In this limit force detection is limited onlyby the the additional shot-noise-type term that is inde-pendent of the measurement strength g correspondingto atomic noise (see Eq. (21)), and which is the price topay for the realization of CQNC, S CQNC = 12 (cid:18) ω + γ m / ω m (cid:19) , (25)(here we neglect thermal noise and other technical noisesources which are avoidable in principle). As alreadydiscussed in Ref. [46], in the limit of sufficiently largedriving powers when shot noise (and also thermal noise)is negligible, CQNC has the advantage of significantly in-creasing the bandwidth of quantum-limited detection offorces, well out of the mechanical resonance. This analy-sis can be applied also for the present scheme employinga single cavity mode, and it is valid also in the presence ofinjected squeezing, which modifies and can further sup-press the shot noise contribution. This is relevant be-cause it implies that one can achieve the CQNC limit ofEq. (25), by making the shot noise contribution negli-gible, much easily, already at significantly lower drivingpowers. In this respect one profits from the ability of in-jected squeezing to achieve the minimum noise at lowerpower values, as first pointed out by Caves [48].Let us now see in more detail the effect of the in-jected squeezing by optimizing the parameters under per-fect CQNC conditions. To be more specific, the injectedsqueezed light has to suppress as much as possible theshot noise contribution to the detected force spectrum,and therefore we have to minimize the function withinthe square brackets of Eq. (21), over the squeezing pa-rameters N , M and the detuning ∆ c . Defining y = ∆ c /κ the normalized detuning, one can rewrite this function as h ( M, N, y ) = (cid:18) N + 12 (cid:19) (cid:18)
12 + 2 y (cid:19) −| M | [ a ( y ) sin φ + b ( y ) cos φ ] , (26)where M = | M | e iφ , and we have introduced thedetuning-dependent functions a ( y ) = 2 y (1 − y ) b ( y ) = (cid:18)
12 + 2 y (cid:19) − y . (27) h ( M, N, y ) can be further rewritten as h ( M, N, y ) = (cid:18) N + 12 (cid:19) (cid:18)
12 + 2 y (cid:19) −| M | (cid:112) a ( y ) + b ( y ) cos [ φ − φ opt ( y )] , (28)where tan φ opt ( y ) = a ( y ) /b ( y ) and it is straightforwardto verify that (cid:112) a ( y ) + b ( y ) = (cid:0) / y (cid:1) . From thislatter expression is evident that, for a given detuning y ,and whatever value of N and | M | , the optimal value ofthe squeezing phase minimizing the shot noise contribu-tion is just φ = φ opt ( y ), for which one gets h ( | M | , N, y ) = (cid:18) N + 12 − | M | (cid:19) (cid:18)
12 + 2 y (cid:19) . (29)This latter expression can be easily further minimized byobserving that its minimal value is obtained by assumingpure squeezed light | M | = (cid:112) N ( N + 1) and also takingzero detuning y = 0, i.e., driving the cavity mode atresonance, so that for a given value of (pure) squeezing N , one has that h min ( N ) = 14 (cid:104) N + 1 / − (cid:112) N ( N + 1) (cid:105) , (30)which tends to zero quickly for large squeezing N , i.e., h min ( N (cid:29) → / (32 N ). As a consequence, the shotnoise contribution can be rewritten after optimizationover the squeezing and detuning as, S F shot , opt ( ω ) = κ g γ m | χ m | (cid:104) N + 1 / − (cid:112) N ( N + 1) (cid:105) . (31)We notice that the optimal value of the detuning, ∆ c = 0can be taken only in the present model with a singlecavity mode and not in the dual-cavity model of Ref. [46]where the parameter ∆ c is replaced by the coupling ratebetween the two cavities 2 J , which cannot be reduced tozero. This is an important advantage of the single cavitymode case considered here. Eq. (31) shows that injectedsqueezing greatly facilitates achieving the ultimate limitprovided by CQNC of Eq. (25) because in the optimalcase and at large squeezing N , the shot noise term issuppressed by a factor 1 / (4 N ) with respect to the case ω / ω m S F ( ω ) ( a ) ω / ω m S F ( ω ) ( b ) FIG. 2. (Color online) Force noise spectral density versus ω/ω m at the optimal value for the detuning ∆ c /κ = 0 andwith an optimized squeezed injected light with phase ϕ = ϕ opt (0) = 0 and | M | = √ N ( N + 1), for different values of thesqueezing parameter, N = 0 (dashed, red line), N = 10 (dot-dashed, purple line), and N = 100 (full, blue line). The dottedblack line corresponds to the SQL. The other parameters are ω m / π = 300 KHz, γ m / π = 30 mHz, g / π = 300 Hz, λ L ≃
780 nm, P L = 24 µ W and κ/ π = 1 MHz. In Fig. (2), the force noise spectral density S F ( ω ) op-timized over the detuning and the value of the squeezingparameter M , of Eq. (31), is plotted versus frequency atdifferent values of the injected squeezing parameter N .The corresponding SQL spectrum (dotted black line) issignificantly higher in a broadband around the mechan-ical frequency and force noise suppression increases forincreasing injected squeezing N .It should be noted that in the case of no CQNC andwithout atomic ensemble with squeezed injection theSQL (see Appendix B Eq.(B8)) is as the same as stan-dard SQL (Eq.(24)) since the effect of squeezing in lowfrequency is vanished by the oposit effect of squeezing dueto the back action noise in Eq.(B6). The suppression ofnoise in low frequency with squeezed injection is one ofthe advantage of our scheme in comparison to others al-though there was the proposal based on the squeezingcorrelation injection for suppressing the noise in the lowfrequency but here we can suppress the noise without us-ing the squeezing correlation. Also we could suppress the ω ω ω ω ω ω - - - ( g / g ) S F ( ω ) ( a ) ω = ω m - - ( g / g ) S F ( ω ) ( b ) ω - ω m = γ m FIG. 3. (Color online) Force noise spectral density versus( g/g ) (proportional to the input laser power) at the op-timal value for the detuning ∆ c /κ = 0 and with an opti-mized squeezed injected light with phase ϕ = ϕ opt (0) = 0 and | M | = √ N ( N + 1). (a) refers to the value at the mechanicalresonance, ω = ω m , while (b) refers to the off-resonant case ω = ω m + 4 γ m . The full red line refers to standard optome-chanical case without the atomic ensemble of Eq. (23); thelong-dashed line and double-dot dashed line refer to standardoptomechanical case without the atomic ensemble of Eq. (B6)with inject squeezing with N = 1 and N = 10, respectively;the dashed black line refers to the case with the atomic en-semble and CQNC but no injected squeezing, N = 0; thedotted purple line refers to the case with injected squeezingwith N = 1; the dot-dashed blue line refers to the case withinjected squeezing with N = 10. The other parameter valuesare as in Fig. 2. low frequency by squeezed injection and direct detectionof quadrature and this scheme is better than synodyne de-tection which is based on the squeezing correlation in lowfrequency. In other word we could improve low frequencynoise with squeezed injection due to the backaction noisecancellation by atomic ensemble as a NMO.In Fig. (3) the optimized force noise spectral densityis instead plotted versus ( g/g ) , which is proportionalto the input laser power P L , both at the mechanical res-onance (Fig. 3a, ω = ω m ) and off-resonance (Fig. 3b, ω = ω m +4 γ m ). Due to the back-action noise cancellationcaused by CQNC, force noise spectrum is always signifi-cantly suppressed at large power with respect to the case FIG. 2. (Color online) Force noise spectral density versus ω/ω m in the presence of perfect CQNC, with an optimizedsqueezed injected light with phase φ = φ opt (0) = 0 and | M | = (cid:112) N ( N + 1). (a) refers to the case with fixed squeezing N =10 and different detunings: ∆ c /κ = 0 (dot-dashed purpleline), ∆ c /κ = 1 / c /κ = 1 (full brownline). (b) refers to the optimal case ∆ c /κ = 0 and differentvalues of the squeezing parameter, N = 0 (dashed, red line), N = 10 (dot-dashed, purple line), and N = 100 (full, blueline). The dotted black line corresponds to the SQL. Theother parameters are ω m / π = 300 KHz, γ m / π = 30 mHz, g / π = 300 Hz, λ L (cid:39)
780 nm, P L = 24 µ W and κ/ π = 1MHz. without injected squeezing (compare Eq. (29) in the case M = N = y = 0 with Eq. (30) in the case when N (cid:29) g , and thereforemuch less optical driving power in order to reach thesame suppression of the shot noise contribution.Let us now illustrate how the combination of backac-tion cancellation by the atomic ensemble under CQNCand of the squeezing injected in the cavity may signif-icantly improve force detection sensitivity. We consideran experimentally feasible scheme based on a membrane-in-the-middle setup [78], coupled to an ultracold atomicgas confined in the cavity and in a magnetic field, likethe one demonstrated in Ref. [77] for light storage. A system of this kind has not been demonstrated yet, butthe coupling of an atomic ensemble with a membrane hasbeen already demonstrated in Refs. [41, 43]. We assumetypical mechanical parameter values for SiN membranes, ω m / π = 300 KHz, γ m / π = 30 mHz, g / π = 300 Hz, λ L (cid:39)
780 nm, P L = 24 µ W and κ/ π = 1 MHz (see alsothe caption of Fig. 2). The ground state sub-levels of theultracold atomic gas of Ref. [77] could be prepared in or-der to satisfy the CQNC condition, i.e., the Zeeman split-ting tuned in order that the effective atomic transitionrate coincides with ω m , the driving of the laser fields ad-justed so that the two linearized couplings with the cavitymode, G and g , coincide. Matching the dephasing rateΓ with the damping rate γ m is less straightforward butone can decrease and partially tune the atomic dephas-ing rate using the magic-value magnetic field techniqueand applying dynamical decoupling pulse sequences, asdemonstrated in Ref. [77].In Fig. (2), the force noise spectral density S F ( ω )optimized over the squeezing parameters, i.e., | M | = (cid:112) N ( N + 1), φ = 0, is plotted versus frequency. In Fig.2(a) we fix the squeezing parameter N = 10 and con-sider different values of the detuning: as shown above,force noise is minimum at the optimal case of resonantcavity driving ∆ c = 0. This plot clearly shows the ad-vantage of the present single cavity scheme compared tothe double cavity setup of Ref. [46], where the role of ∆ c is played by the mode splitting 2 J associated with theoptical coupling J between the cavity that cannot be putto zero. In Fig. 2(b) we fix the detuning at this optimalzero value, and we consider different values of the injectedsqueezing parameter N . At resonance ( ω = ω m ), CQNCand injected squeezing does not improve with respect tothe SQL spectrum (dotted black line), but force noisesuppression is remarkable in a broadband around the res-onance peak, and becomes more relevant for increasinginjected squeezing N . Notice that injected squeezing al-lows a further reduction of the off-resonance ( ω (cid:54) = ω m )force noise with respect to what can be achieved withCQNC alone (see in Fig. 2(b) the full blue line and thedot-dashed purple line compared to the dashed red linewhich refers to the case of no-injected squeezing, N = 0.)In Fig. (3) the force noise spectral density is insteadplotted versus ( g/g ) , which is proportional to the in-put laser power P L , both at the mechanical resonance(Fig. 3(a), ω = ω m ) and off-resonance (Fig. 3(b), ω = ω m + 4 γ m ). In both subfigures we compare the forcenoise spectrum with perfect CQNC and for a given op-timized squeezing N , with the corresponding spectrumwith the same injected squeezing but without atomicensemble and CQNC, for three different values of N , N = 0 , ,
10 (see also Appendix B where we evaluatethe general expressions of the force noise spectrum with-out imposing the CQNC condition). Back-action noisecancellation manifests itself with a significant noise sup-pression at large power, where minimum force noise isachieved. Without atoms and CQNC, force noise di-verges at large power due to backaction, and one has the ω / ω m S F ( ω ) FIG. 2. (Color online) Force noise spectral density versus ω/ω m at the optimal value for the detuning ∆ c /κ = 0 andwith an optimized squeezed injected light with phase ϕ = ϕ opt (0) = 0 and | M | = √ N ( N + 1), for different values of thesqueezing parameter, N = 0 (dashed, red line), N = 10 (dot-dashed, purple line), and N = 100 (full, blue line). The dottedblack line corresponds to the SQL. The other parameters are ω m / π = 300 KHz, γ m / π = 30 mHz, g / π = 300 Hz, λ L ≃
780 nm, P L = 24 µ W and κ/ π = 1 MHz. In Fig. (2), the force noise spectral density S F ( ω ) op-timized over the detuning and the value of the squeezingparameter M , of Eq. (31), is plotted versus frequency atdifferent values of the injected squeezing parameter N .The corresponding SQL spectrum (dotted black line) issignificantly higher in a broadband around the mechan-ical frequency and force noise suppression increases forincreasing injected squeezing N .It should be noted that in the case of no CQNC andwithout atomic ensemble with squeezed injection theSQL (see Appendix B Eq.(B7)) is as the same as stan-dard SQL (Eq.(24)) since the effect of squeezing in lowfrequency is vanished by the oposit effect of squeezing dueto the back action noise in Eq.(B5). The suppression ofnoise in low frequency with squeezed injection is one ofthe advantage of our scheme in comparison to others al-though there was the proposal based on the squeezingcorrelation injection for suppressing the noise in the lowfrequency but here we can suppress the noise without us-ing the squeezing correlation. Also we could suppress thelow frequency by squeezed injection and direct detectionof quadrature and this scheme is better than synodyne de-tection which is based on the squeezing correlation in lowfrequency. In other word we could improve low frequencynoise with squeezed injection due to the backaction noisecancellation by atomic ensemble as a NMO.In Fig. (3) the optimized force noise spectral densityis instead plotted versus ( g/g ) , which is proportionalto the input laser power P L , both at the mechanical res-onance (Fig. 3a, ω = ω m ) and off-resonance (Fig. 3b, ω = ω m + 4 γ m ). Due to the back-action noise cancel-lation caused by CQNC, force noise spectrum is alwayssignificantly suppressed at large power with respect tothe case without atoms and CQNC, but when injectedsqueezing is added, the force noise spectral density is sig- ω ω ω - - - ( g / g ) S F ( ω ) ( a ) ω = ω m - - ( g / g ) S F ( ω ) ( b ) ω - ω m = γ m FIG. 3. (Color online) Force noise spectral density versus( g/g ) (proportional to the input laser power) at the op-timal value for the detuning ∆ c /κ = 0 and with an opti-mized squeezed injected light with phase ϕ = ϕ opt (0) = 0 and | M | = √ N ( N + 1). (a) refers to the value at the mechan-ical resonance, ω = ω m , while (b) refers to the off-resonantcase ω = ω m + 4 γ m . The full red line refers to standard op-tomechanical case without the atomic ensemble of Eq. (23);the dashed black line refers to the case with the atomic en-semble and CQNC but no injected squeezing, N = 0; thedotted purple line refers to the case with injected squeezingwith N = 1; the dot-dashed blue line refers to the case withinjected squeezing with N = 10. The other parameter valuesare as in Fig. 2. nificantly suppressed already at much lower power andthe effect is stronger for increasing values of N .Fig. (4) shows the effect of detuning. It is clear thatthe optimum case is zero detuning. In the case of nosqueezing ( N = 0) the noise is suppressed by factor 1 / / (16 N ) (see Appendix C). As shown the zerodetuning is advantage of this scheme in comparison toothers.It is important to discussion how much sensitivityof matching is necessary, specially we would considerthe mismatch of atomic dephasing rate and mechanicaldamping rate which is more important than other CQNCcondition. FIG. 3. (Color online) Force noise spectral density versus( g/g ) (proportional to the input laser power) at the op-timal value for the detuning ∆ c /κ = 0 and with an opti-mized squeezed injected light with phase φ = φ opt (0) = 0 and | M | = (cid:112) N ( N + 1). (a) refers to the value at the mechanicalresonance, ω = ω m , while (b) refers to the off-resonant case ω = ω m + 4 γ m . In both subfigures we compare the force noisespectrum with perfect CQNC and for a given (optimized)squeezing N with the corresponding spectrum with the sameinjected squeezing but without atomic ensemble and CQNC.The full red line refers to standard optomechanical case N = 0without the atomic ensemble of Eq. (23), while the dashedblack line refers again to N = 0 with the atomic ensemble andperfect CQNC. The case with injected squeezing with N = 1corresponds to the long-dashed green line (without atoms andCQNC, see also Eq. (B6)), and to the dotted purple line (withatoms and CQNC). Finally the case with injected squeezingwith N = 10 corresponds to the double-dot dashed brown line(no atoms and CQNC), and to the dot-dashed blue line (withatoms and perfect CQNC). The other parameter values areas in Fig. (2). usual situation where minimum force noise is achievedat the SQL, at a given optimal power. In both cases,either with or without CQNC, injected squeezing with φ = 0 and | M | = (cid:112) N ( N + 1) is not able to improveforce sensitivity and to lower the noise at resonance (seeFig. 3(a)), i.e., the SQL value remains unchanged, butone has the advantage that for increasing N , the SQL isreached at decreasing values of input powers [48]. As al- ready suggested by Fig. (2), instead one has a significantforce noise suppression off-resonance and at large powersdue to backaction cancellation (Fig. 3(b)). B. The case of imperfect CQNC conditions
Backaction cancellation requires the perfect matchingof atomic and mechanical parameters. As we discussedabove, one can tune the effective atomic transition rateby tuning the magnetic field, and make it identical tothe mechanical resonance frequency ω m . Here we stillassume such a perfect frequency matching which, eventhough not completely trivial, can always be achieveddue to the high tunability of Zeeman splitting. As wediscussed in the previous subsection, one can also makethe two couplings with the cavity mode G and g identical,by adjusting the cavity and atomic driving, and finallyeven the two decay rates, Γ and γ m . However, both cou-pling rates matching and decay rate matching are lessstraightforward, and therefore it is important to investi-gate the robustness of the CQNC scheme with respect toimperfect matching of these two latter parameters.We have restricted our analysis to the parameterregime corresponding to the optimal case under per-fect CQNC conditions, i.e., the resonant case ∆ c = 0,with an optimized pure squeezing, φ = φ opt (0) = 0 and | M | = (cid:112) N ( N + 1). We have also fixed the squeezingvalue, N = 10, and considered again the parameter val-ues of the previous subsection, but now considering thepossibility of nonzero mismatch of the couplings and/orof the decay rates. We have used the expression of thespectrum of Eq. (B5). We first consider in Fig. (4) theeffect of parameter mismatch on the force noise spectrumversus ω . Fig. (4) shows that CQNC is more sensitiveto the coupling mismatch than to the decay rate mis-match. In fact, the spectrum is appreciably modified al-ready when ( G − g ) /g = 10 − , and force noise increasessignificantly and in a broadband around resonance al-ready when ( G − g ) /g = 10 − . This modification is quiteindependent from the value of the decay rate mismatch,(Γ − γ m ) /γ m , whose effect moreover is always concen-trated in a narrow band around resonance and for largervalues, (Γ − γ m ) /γ m = 0 .
5. There is a weak dependenceupon the sign of the two mismatches, which however istypically very small and not visible in the plots.In Fig. (5) instead we fix the frequency and considerthe dependence of the force noise spectrum versus g , i.e.versus the laser input power, either at resonance (Fig.5(a)), and off-resonance (Fig. 5(b)), similarly to whatwe did under perfect CQNC in Fig. (3). Due to the im-perfect CQNC caused by parameter mismatch, at largepower force noise spectrum increases again due to the un-cancelled, residual backaction noise, and the increase atlarge power is larger for larger parameter mismatch. Atresonance (Fig. 5(a)) both coupling mismatch and decayrate mismatch have an effect, and force noise increase islarger when both mismatches are nonzero and opposite,0 ω / ω m S F ( ω ) FIG. 4. (Color online) Force noise spectral density versus ω/ω m at the optimal value for the detuning ∆ c /κ = 0, with anoptimized squeezed injected light with phase φ = φ opt (0) =0, | M | = (cid:112) N ( N + 1), and N = 10. We consider differentcoupling and decay rate mismatches. The dashed purple lineand double-dot dashed red line, respectively refer to the SQLand perfect CQNC. The other curves correspond to: ( G − g ) /g = ± − , Γ = γ m (green, dot-dashed line); ( G − g ) /g = ± − , Γ = γ m (blue, solid line); (Γ − γ m ) /γ m = ± . g = G (black, solid line); ( G − g ) /g = ± − , (Γ − γ m ) /γ m = ± . due to the effect of the negative mass, yielding suscepti-bilities with opposite signs. As already shown in Fig. (4),the effect of decay rate mismatch is instead hardly appre-ciable out of resonance, and noise increase is caused bythe mismatch between the two couplings, regardless thevalue of the decay rate mismatch. The analysis of Figs.(4) and (5) allows us to conclude that CQNC is robustwith respect to mismatch of the decay rates, up to 10%mismatch, and especially off-resonance, where the ad-vantage of backaction cancellation is more relevant. Onthe contrary, CQNC is very sensitive to the mismatchbetween the atomic and mechanical couplings with thecavity mode, which have to be controlled at 0 .
1% levelor better. This means that in order to suppress backac-tion noise the intensity of the cavity and atomic drivinghave to be carefully controlled in order to adjust the twocouplings.
V. SUMMARY AND CONCLUSION
We have proposed a scheme for the realization ofCQNC scheme for the high-sensitive detection of forcesbased on a single optomechanical cavity containing anatomic ensemble and driven by squeezed vacuum light.The interaction of the atomic ensemble with the cavitymode leads to a destructive interference that perfectlycancels backaction noise of the MO, provided that atomicensemble parameters are chosen so that it acts as a neg-ative mass oscillator whose susceptibility perfectly can- FIG. 4. (Color online) Force noise spectral density versus ω/ω m at the optimal value for the detuning ∆ c /κ = 0 andwith an optimized squeezed injected light with phase ϕ = ϕ opt (0) = 0 and | M | = √ N ( N + 1), and N = 10 for differentmismatches. In top panel: The black dotted line refers toSQL, blue dot-dashed line refers to ( G − g ) /g = +0 .
1, browndot line refers to ( G − g ) /g = +0 .
1, green solid line refers to(Γ − γ m ) /γ m = 0 . P L = 2 . nW and the other parameter valuesare as in Fig. (2). It should be mentioned that if we go tohigher power, the effett of mismatch in g is dominate whichis due to the non-cancellation of backaction noise term. Indown panel: P L = 24 µW and brown dashed line refers to(Γ − γ m ) /γ m = 0 . without atoms and CQNC, but when injected squeezingis added, the force noise spectral density is significantlysuppressed already at much lower power and the effectis stronger for increasing values of N . As shown in thestandard optomechanical cavity with squeezed injectionthe noise is suppressed at lower frequency but in highfrequency the backaction noise is dominated.It is important to discussion how much sensitivityof matching is necessary, specially we would considerthe mismatch of atomic dephasing rate and mechanicaldamping rate which is more important than other CQNCcondition. Fiq. (5) shows the effect of mismatch betweencoupling constants and decoherence rates versus g /g aton-resonance detection. As shown, 10% mismatch is goodapproximation for back-action noise reduction, howeverour numerical calculations shows that 1% mismatch isexcellent approximation for CQNC. Moreover, we findwhen the sign of mismatch of coupling constants and de-coherence rates is the same, noise reduction is more thanwhen the mismatch sign is opposite. Also, it is clear thatin the case of no optimization value the noise reductionis worse then the choosing of ∆ c = 0 is important fromexperimental point of view in order to reach better noisecancellation. Fig. 5(c) shows that the effect of mismatchin off-resonance case is smaller specially for mismatchingof damping rates. V. SUMMARY AND CONCLUSION
We have proposed a simplified scheme for the realiza-tion of CQNC scheme for the high-sensitive detection offorces based on a single optomechanical cavity contain-ing an atomic ensemble and driven by squeezed vacuumlight. The interaction of the atomic ensemble with thecavity mode leads to a destructive interference that per-fectly cancels backaction noise of the MO, provided thatatomic ensemble parameters are chosen so that it actsas a negative mass oscillator whose susceptibility per-fectly cancels the mechanical one. Perfect CQNC occurs
12 34 56789 - - - ( g / g ) S F ( ω ) ( a ) ω = ω m - - - ( g / g ) S F ( ω ) ( b ) ω = ω m + γ m FIG. 5. (Color online) Force noise spectral density versus( g/g ) . (a) for the case on-resonance ( ω = ω m ). Curves 1(double-dot-dashed dark green line) and 5 (brown solid line)refer to the CQNC and standard optomechanical spectrum(Eq. (23)), respectively. (Γ − γ m ) /γ m = +0 . − γ m ) /γ m = ( G − g ) /g = ± . − γ m ) /γ m = − . G − g ) /g = − . G − g ) /g = +0 . G − g ) /g = − (Γ − γ m ) /γ m = − . G − g ) /g = − (Γ − γ m ) /γ m = +0 . ω = ω m + 4 γ m ). Curves 1 (double-dot-dashed dark green line)and 3 (brown solid line) refer to the CQNC and standardoptomechanical spectrum, respectively. (Γ − γ m ) /γ m = ± . G − g ) /g = − .
1, (Γ − γ m ) /γ m = ( G − g ) /g = − .
1, and ( G − g ) /g = − (Γ − γ m ) /γ m = − . G − g ) /g = 0 .
1, (Γ − γ m ) /γ m = ( G − g ) /g = 0 . G − g ) /g = − (Γ − γ m ) /γ m = 0 . N = 10,∆ c = 0, ϕ = ϕ opt (0) = 0 and | M | = √ N ( N + 1). The otherparameter values are as in Fig. (2). when the optomechanical and the atom-field interactioncoupling constants, the mechanical frequency and the ef-fective atomic transition rate, and finally the dissipationrate of the mechanical resonator and the decoherence rateof the atomic ensemble respectively coincides. In thepresence of the injected squeezing CQNC is much moreeffective because broadband high sensitive force detec-tion is achieved at the cavity resonance and at much lessinput laser power. In fact, the residual shot noise contri-bution to the force noise spectrum is strongly suppressedby the injected squeezed light, even at moderate values FIG. 5. (Color online) Force noise spectral density versus( g/g ) . (a) Resonant case ( ω = ω m ). Curves 1 (double-dot-dashed dark green line) and 9 (brown solid line) refer to theCQNC and standard optomechanical spectrum (Eq. (B6)),respectively. The other curves correspond to: (Γ − γ m ) /γ m =+0 . G = g (curve 2, red dashed); (Γ − γ m ) /γ m = ( G − g ) /g = ± . − γ m ) /γ m = − . g = G (curve 4, green dashed); ( G − g ) /g = − .
1, Γ = γ m (curve 5, pink dashed); ( G − g ) /g = +0 .
1, Γ = γ m (curve6, blue dashed); ( G − g ) /g = − (Γ − γ m ) /γ m = − . G − g ) /g = − (Γ − γ m ) /γ m = +0 . ω = ω m + 4 γ m ). Curves 1 (double-dot-dashed darkgreen line) and 5 (brown solid line) refer to the CQNC andstandard optomechanical spectrum, respectively. The othercurves correspond to: (Γ − γ m ) /γ m = ± . g = G (curve2, blue dashed); ( G − g ) /g = − . γ m , (Γ − γ m ) /γ m =( G − g ) /g = − .
1, and ( G − g ) /g = − (Γ − γ m ) /γ m = − . G − g ) /g = 0 . γ m , (Γ − γ m ) /γ m = ( G − g ) /g = 0 . G − g ) /g = − (Γ − γ m ) /γ m =0 . N = 10, ∆ c = 0, φ = φ opt (0) = 0 and | M | = (cid:112) N ( N + 1).The other parameter values are as in Fig. (2). cels the mechanical one. Perfect CQNC occurs when theoptomechanical and the atom-field interaction couplingconstants, the mechanical frequency and the effectiveatomic transition rate, and finally the dissipation rateof the mechanical resonator and the decoherence rate ofthe atomic ensemble, respectively coincide. The presentscheme could be implemented by combining state-of-the-art membrane-in-the-middle setup [78], with ultracoldatomic ensemble systems used for long-lived light stor-1age [77] and improves in various directions the dual cavityproposal of Ref. [46]. The optical coupling rate betweencavities J in Ref. [46] is replaced by the cavity modedetuning ∆ c in our scheme, and due to this fact, thepresent scheme reaches a stronger force noise suppres-sion because such a suppression is optimal at resonance∆ c = 0, which can be set only in the present scheme. Afurther noise suppression is realized by injected squeezedvacuum in the cavity mode: in fact, shot noise is furthersuppressed for increased squeezing, and this occurs atmuch lower input laser power. We have also analyzed indetail the effect of imperfect CQNC conditions, i.e., whenthe mechanical and atomic parameters are not perfectlymatched, focusing on the case when the two couplingswith the cavity modes and/or the decay rates are differ-ent. We have seen that backaction cancellation is robustwithe respect to the decay rate mismatch and 10% mis-match can be tolerated, especially off-resonance. InsteadCQNC is very sensitive to the mismatch of the couplingrates, and one has to tune the two couplings, by adjust-ing the cavity and atomic driving, at the 0 .
1% level atleast.
Acknowledgments
This work is supported by the Euro-pean Commission through the Marie Curie ITN cQOMand FET-Open Project iQUOEMS. A. M. and F. B. wishto thank the Office of Graduate Studies of the Universityof Isfahan for their support.
Appendix A: derivation of CQNC force noisespectrum
Using the definitions provided in the main text, theforce noise spectrum is explicitly written as (cid:68) ˆ F N ( ω ) ˆ F N ( − ω (cid:48) ) (cid:69) = (cid:68) ˆ f ( ω ) ˆ f ( − ω ) (cid:69) + κg γ m χ m ( ω ) χ m ( − ω ) (cid:20) (1 − χ (cid:48) a ( ω ) κ )(1 − χ (cid:48) a ( − ω ) κ ) (cid:68) ˆ P ina ( ω ) ˆ P ina ( − ω ) (cid:69) − ∆ c χ a ( − ω )(1 − χ (cid:48) a ( ω ) κ ) (cid:68) ˆ P ina ( ω ) ˆ X ina ( − ω ) (cid:69) − ∆ c χ a ( ω )(1 − χ (cid:48) a ( − ω ) κ ) (cid:68) ˆ X ina ( ω ) ˆ P ina ( − ω ) (cid:69) +∆ c χ a ( ω ) χ a ( − ω ) (cid:68) ˆ X ina ( ω ) ˆ X ina ( − ω ) (cid:69)(cid:105) + (cid:68) ˆ P ind ( ω ) ˆ P ind ( − ω ) (cid:69) − Γ / − iωω m (cid:68) ˆ P ind ( ω ) ˆ X ind ( − ω ) (cid:69) − Γ / iωω m (cid:68) ˆ X ind ( ω ) ˆ P ind ( − ω ) (cid:69) + ω + Γ / ω m (cid:68) ˆ X ind ( ω ) ˆ X ind ( − ω ) (cid:69) , (A1) where the correlation functions in the Fourier space aregiven by (cid:68) ˆ f ( ω ) ˆ f ( − ω (cid:48) ) (cid:69) = (¯ n m + 12 ) δ ( ω − ω (cid:48) ) (cid:39) k B T (cid:126) ω m δ ( ω − ω (cid:48) ) , (cid:68) ˆ X ina ( ω ) ˆ X ina ( − ω (cid:48) ) (cid:69) = 12 ((2 N + 1) + 2Re M ) δ ( ω − ω (cid:48) ) , (cid:68) ˆ P ina ( ω ) ˆ P ina ( − ω (cid:48) ) (cid:69) = 12 ((2 N + 1) − M ) δ ( ω − ω (cid:48) ) , (cid:68) ˆ X ina ( ω ) ˆ P ina ( − ω (cid:48) ) (cid:69) = i − i Im M ) δ ( ω − ω (cid:48) ) , (cid:68) ˆ P ina ( ω ) ˆ X ina ( − ω (cid:48) ) (cid:69) = − i i Im M ) δ ( ω − ω (cid:48) ) , (cid:68) ˆ P ind ( ω ) ˆ X ind ( − ω (cid:48) ) (cid:69) = − (cid:68) ˆ X ind ( ω ) ˆ P ind ( − ω (cid:48) ) (cid:69) = i δ ( ω − ω (cid:48) ) , (cid:68) ˆ X ind ( ω ) ˆ X ind ( − ω (cid:48) ) (cid:69) = (cid:68) ˆ P ind ( ω ) ˆ P ind ( − ω (cid:48) ) (cid:69) = 12 δ ( ω − ω (cid:48) ) . (A2)Inserting these expressions one finally gets the generalresult of Eq. (21). Appendix B: Exact expression of force noisespectrum
Based on Eq.(18), the exact expression of the forcenoise spectrum in the general case without CQNC con-dition is given by S F ( ω ) = S th ( ω ) + S f ( ω ) + S b ( ω ) + S atom ( ω ) + S fb ( ω ) , (B1)where S th ( ω ) = k B T / (cid:126) ω m is the thermal noise contri-bution, S f ( ω ) corresponds to the field contribution, thethird term is associated with the contribution of back-action noise, the fourth term corresponds to the atomiccontribution, and the last term is an interference termasscoaited with the joint action of the cavity field and ofthe atoms. The explicit expressions are given by S f ( ω ) = κg γ m | χ m ( ω ) | { ∆ c Im [ Z ( ω ) (1 − i Im M )]+ (cid:20) κ | χ (cid:48) a ( ω ) | − χ (cid:48) a ( ω ) κ | χ (cid:48) a ( ω ) | (cid:21) (cid:18) N + 12 − Re M (cid:19) +∆ c | χ a ( ω ) | (cid:18) N + 12 + Re M (cid:19)(cid:27) , (B2a) S b ( ω ) = 4 g κγ m (cid:18) N + 12 + Re M (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) G g R ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) , (B2b) S atom ( ω ) = | A ( ω ) | (cid:18) ω + Γ / ω m (cid:19) , (B2c) S fb ( ω ) = κγ m Im (cid:20) (2 i Im M − Z ( ω ) χ ∗ m ( ω ) (cid:20) G g R ( ω ) (cid:21)(cid:21) − κγ m ∆ c | χ a ( ω ) | (cid:18) N + 12 + Re M (cid:19) Re (cid:34) G g R ( ω ) χ ∗ m ( ω ) (cid:35) , (B2d)2where Z ( ω ) = χ a ( ω ) (cid:18) − κχ (cid:48)∗ a ( ω ) (cid:19) , (B3a) R ( ω ) = χ d ( ω ) χ m ( ω ) = − (1 + r ( ω )) , (B3b) r ( ω ) = iω ( γ m − Γ)( ω m − ω ) + iω Γ , (B3c) A ( ω ) = Gg (cid:115) Γ γ m R ( ω ) . (B3d)Notice that under perfect CQNC conditions, 1 +( G /g ) R ( ω ) = 0, and both contributions S b and S fb become zero. In the Markov limit, κ (cid:29) ω , we keep onlythe zero order of ω/κ , therefore we have χ a ( ω ) (cid:39) /κ + O ( ω/κ ) ,χ (cid:48)− a ( ω ) (cid:39) κ (cid:20)(cid:18) c κ (cid:19) − c κ g χ m ( ω ) (cid:18) G g R ( ω ) (cid:19)(cid:21) ,Z ( ω ) (cid:39) κ − (cid:20) (1 − c κ ) + 4 ∆ c κ g χ ∗ m ( ω ) (cid:18) G g R ∗ ( ω ) (cid:19)(cid:21) , (B4)When we choose the optimal case of zero cavity detun-ing, ∆ c = 0, the total force noise spectrum considerablysimplifies and we get S ( ω ) | ∆ c =0 = k b T (cid:126) ω m + κ g γ m | χ m ( ω ) | (cid:18) N + 12 − Re M (cid:19) + 4 g κγ m (cid:18) N + 12 + Re M (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) G g R ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:18) Gg (cid:19) Γ γ m | R ( ω ) | (cid:18) ω + Γ / ω m (cid:19) +Im (cid:20) (2 i Im M − γ m χ ∗ m ( ω ) (cid:20) G g R ( ω ) (cid:21)(cid:21) . (B5) Eq. (B5) shows the effect of mismatch. Without theatomic ensemble ( G = 0) the force noise spectrum ofthe optomechanical cavity with squeezed injection canbe written as S stopt ( ω ) = k B T (cid:126) ω m + 2Im M Q m ω m − ω ω m + 14 κg γ m | χ m | (cid:18) N + 12 − Re M (cid:19) + 4 g κγ m (cid:18) N + 12 + Re M (cid:19) . 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