Optomechanical entanglement between a movable mirror and a cavity ...

Optomechanical entanglement between a movable mirror and a cavity field D. Vitali,1 S. Gigan,2 A. Ferreira,2, 3 H. R. Böhm,2 P. Tombesi,1 A. Guerreiro,3 V. Vedral,2, 4 A. Zeilinger,2, 5 and M. Aspelmeyer2, 5 1 Dipartimento di Fisica, Universit` a di Camerino, I-62032 Camerino (MC), Italy Institut f¨ ur Experimentalphysik, Universit¨ at Wien, Boltzmanngasse 5, 1090 Wien, Austria 3 Faculdade Ciências Universidade do Porto, Rua do Campo Alegre 687 4169-007, Porto, Portugal 4 The School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK 5 Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Wien, Austria (Dated: February 9, 2008)

arXiv:quant-ph/0609197v2 7 Dec 2006

2

We show how stationary entanglement between an optical cavity field mode and a macroscopic vibrating mirror can be generated by means of radiation pressure. We also show how the generated optomechanical entanglement can be quantified and we suggest an experimental readout-scheme to fully characterize the entangled state. Surprisingly, such optomechanical entanglement is shown to persist for environment temperatures above 20K using state-of-the-art experimental parameters. PACS numbers: 03.67.Mn, 42.50.Lc, 05.40.Jc

Entanglement, “the characteristic trait of quantum mechanics” [1], has raised widespread interest in different branches of physics. It provides insight into the fundamental structure of physical reality [2] and it has become a basic resource for many quantum information processing schemes [3]. So far entanglement has been experimentally prepared and manipulated using microscopic quantum systems such as photons, atoms and ions [3, 4]. Nothing in the principles of quantum mechanics prevents macroscopic systems to attain entanglement. However, the answer to the question as to what extent entanglement should hold when going towards “classical” systems is yet unknown [5]. Therefore it is of crucial importance to investigate the possibilities to obtain entangled states of macroscopic systems [6] and to study the robustness of entanglement against temperature [7]. Experiments in this direction include single-particle interference of macro-molecules [8], the demonstration of entanglement between collective spins of atomic ensembles [9], and of entanglement in Josephson-junction qubits [10]. Mechanical oscillators are of particular interest since they resemble a prototype of “classical” systems. Thanks to the fast-developing field of microfabrication, micro- or nano-mechanical oscillators can now be prepared and controlled to a very high precision [11]. In addition, several theoretical proposals exist that suggest how to reach the quantum regime for such systems [12]. Experimentally, quantum limited measurements have been developed that could allow ground state detection [13]. However, quantum effects in mechanical oscillators have not been demonstrated to date. Optomechanical coupling via radiation pressure [14] is a promising approach to prepare and manipulate quantum states of mechanical oscillators. Proposals range from the quantum state transfer from light to a mechanical oscillator to entangling two such oscillators [15, 16, 17, 18]. In this paper we propose an experimen-

tal scheme to create and probe optomechanical entanglement between a light field and a mechanical oscillator. This is achieved using a bright laser field that resonates inside a cavity and couples to the position and momentum of a moving (micro)mirror. The proposal is based on feasible experimental parameters in accordance with current state of the art optics and microfabrication. In contrast to other proposals [15, 18] it neither requires non-classical states of light nor temperatures close to the oscillator’s ground state. Entanglement is shown to persist above a temperature of 20K. We begin by modelling the system and its coupling to the environment by using the standard Langevin formalism. Then we solve the dynamics and quantify the entanglement generated in the stationary state. Finally we discuss a suitable experimental apparatus capable of measuring the entanglement. We consider an optical Fabry-Perot cavity in which one of the mirrors is much lighter than the other, so that it can move under the effect of the radiation pressure force. The motion of the mirror is described by the excitation of several degrees of freedom which have different resonant frequencies. However, a single frequency mode can be considered when a bandpass filter in the detection scheme is used [19] and mode-mode coupling is negligible. Therefore we will consider a single mechanical mode of the mirror only, which can be modeled as an harmonic oscillator with frequency wm . The Hamiltonian of the system reads [20] ¯hwm 2 (p + q 2 ) − ¯hG0 a† aq 2 +ı¯hE(e−ıw0 t a† − eıw0 t a),

H=h ¯ wc a† a +

(1)

where q and p ([q, p] = i) are the dimensionless position and momentum operators of the mirror, a and a† ([a, a† ] = 1) are the annihilation and creation operators of the cavity modepwith frequency wc and decay rate κ, and G0 = (wc /L) ¯h/mwm is the coupling coefficient,

2 with L the cavity length in the absence of the intracavity field and m the effective mass of the mechanical mode [19]. The last two terms in Eq. (1) describe the driving laser with frequency w0pand E is related to the input hw0 . laser power P by |E| = 2P κ/¯ A proper analysis of the system must include photon losses in the cavity and the Brownian noise acting on the mirror. This can be accomplished by considering the following set of nonlinear Langevin equations, written in the interaction picture with respect to h ¯ w0 a† a q˙ = wm p,

(2a) †

p˙ = −wm q − γm p + G0 a a + ξ,

√ a˙ = −(κ + i∆0 )a + iG0 aq + E + 2κain ,

(2b) (2c)

where ∆0 = wc − w0 and γm is the mechanical damping rate. We have introduced the vacuum radiation input noise ain , whose only nonzero correlation function is [21] hain (t)ain,† (t′ )i = δ(t − t′ ),

(3)

and the Hermitian Brownian noise operator ξ, with correlation function [22] Z dω −iω(t−t′ ) hω ¯ γm ′ +1 e ω coth hξ(t)ξ(t )i = wm 2π 2kB T (4) (kB is the Boltzmann constant and T is the mirror temperature). We can always rewrite each Heisenberg operator as a c-number steady state value plus an additional fluctuation operator with zero mean value, a = αs + δa, q = qs + δq, p = ps + δp. Inserting these expressions into the Langevin equations of Eqs. (2), these latter decouple into a set of nonlinear algebraic equations for the steady state values and a set of quantum Langevin equations for the fluctuation operators [23]. The steady state values are given by ps = 0, qs = G0 |αs |2 /wm , αs = E/(κ + i∆), where the latter equation is in fact a nonlinear equation determining the stationary intracavity field amplitude αs , since the effective cavity detuning ∆, including radiation pressure effects, is given by ∆ = ∆0 − G20 |αs |2 /wm . The parameter regime relevant for generating optomechanical entanglement is that with a very large input power P , i.e., when |αs | ≫ 1. In this case, one can safely neglect the nonlinear terms δa† δa and δaδq and gets the linearized Langevin equations δ q˙ = wm δp, δ p˙ = −wm δq − γm δp + GδX + ξ, √ δ X˙ = −κδX + ∆δY + 2κX in , √ δ Y˙ = −κδY − ∆δX + Gδq + 2κY in ,

(5a) (5b) (5c) (5d)

where we have chosen the phase reference of the cavthe cavity field so that αs is real, we have defined √ † ity field quadratures δX ≡ (δa + δa )/ 2 and δY ≡ √ (δa − δa† )/i 2, and the corresponding √Hermitian input noise operators X in ≡ (ain + ain,† )/ 2 and Y in ≡

√ (ain − ain,† )/i 2. What is relevant is that the quantum fluctuations of the field and the oscillator are now coupled by the much √ larger effective optomechanical coupling G ≡ G0 αs 2, so that the generation of significant optomechanical entanglement becomes possible. When the system is stable it reaches a unique steady state, independently of the initial condition. Since the quantum noises ξ and ain are zero-mean quantum Gaussian noises and the dynamics is linearized, the quantum steady state for the fluctuations is a zero-mean bipartite Gaussian state, fully characterized by its 4 × 4 correlation matrix Vij = (hui (∞)uj (∞) + uj (∞)ui (∞)i) /2, where uT (∞) = (δq(∞), δp(∞), δX(∞), δY (∞)) is the vector of continuous variables (CV) fluctuation operators at the steady state √ (t → ∞). Defining the vector of √ noises nT (t) = (0, ξ(t), 2κX in (t), 2κY in (t)) and the matrix   0 wm 0 0  −wm −γm G 0  , (6) A=  0 0 −κ ∆  G 0 −∆ −κ Eqs. (5) can be written in compact form as u(t) ˙ = Au(t)+ Rt n(t), whose solution is u(t) = M (t)u(0)+ 0 dsM (s)n(t− s), where M (t) = exp{At}. The system is stable and reaches its steady state when all the eigenvalues of A have negative real parts so that M (∞) = 0. The stability conditions can be derived by applying the Routh-Hurwitz criterion [24], yielding the following two nontrivial conditions on the system parameters, 2 2 2γm κ ∆4 + ∆2 (γm + 2γm κ + 2κ2 − 2wm ) (7a) i 2 2 + γm κ + κ2 + wm + wm G2 ∆(γm + 2κ)2 > 0, 2 (7b) ∆2 + κ2 − wm G2 ∆ > 0, wm which will be considered to be satisfied from now on. When the system is stable one gets XZ ∞ Z ∞ Vij = ds ds′ Mik (s)Mjl (s′ )Φkl (s − s′ ), (8) k,l

0

0

where Φkl (s−s′ ) = (hnk (s)nl (s′ ) + nl (s′ )nk (s)i) /2 is the matrix of the stationary noise correlation functions. Due to Eq. (4), the mirror Brownian noise ξ(t) is not deltacorrelated and therefore does not describe a Markovian process [22]. However, quantum effects are achievable only using oscillators with a large mechanical quality factor Q = wm /γm ≫ 1. In this limit, ξ(t) becomes deltacorrelated [25], hξ(t)ξ(t′ ) + ξ(t′ )ξ(t)i /2 ≃ γm (2¯ n + 1) δ(t − t′ ), −1

(9)

where n ¯ = (exp{¯hwm /kB T } − 1) is the mean thermal excitation number, and one recovers a Markovian process. As a consequence, and using the fact that

3 the three components of n(t) are uncorrelated, we get Φkl (s − s′ ) = Dkl δ(s − s′ ), where D = Diag[0, γm (2¯ n+ 1), κ, κ] is a diagonal matrix, and Eq. (8) becomes V = R∞ T dsM (s)DM (s) . When the stability conditions are 0 satisfied, M (∞) = 0 and one gets the following equation for the steady-state CM, T

AV + V A = −D.

(11)

h 1/2 i1/2 , where η − ≡ 2−1/2 Σ(V ) − Σ(V )2 − 4 det V with Σ(V ) ≡ det A + det B − 2 det C, and we have used the 2 × 2 block form of the CM A C V ≡ . (12) CT B Therefore, a Gaussian state is entangled if and only if η − < 1/2, which is equivalent to Simon’s necessary and sufficient entanglement non-positive partial transpose criterion for Gaussian states [28], which can be written as 4 det V < Σ − 1/4. We have made a careful analysis in a wide parameter range and found a parameter region very close to that of recently performed optomechanical experiments [29], for which a significative amount of entanglement is achievable. Fig. 1 shows EN versus the normalized detuning ∆/wm for two different masses, 5 and 50 ng: optomechanical entanglement is present only within a finite interval of values of ∆ around ∆ ≃ wm . The robustness of such an entanglement with respect to the mirror’s environmental temperature is shown in Fig. 2. The relevant result is that for the 5 ng mirror optomechanical entanglement persists for temperatures above 20K, which is several orders of magnitude larger than the ground state temperature of the mechanical oscillator. For the 50 ng mirror entanglement vanishes at lower temperatures (Fig. 2). Figs. 1-2 refer to Q = 105 , but we found that entanglement persists even for Q ≃ 104 , although it becomes much less robust against temperature. In this case, entanglement persists up to 3(1) K for a 5(50) ng mirror. We finally discuss the experimental detection of the generated optomechanical entanglement. In order to measure EN at the steady state, one has to measure all the ten independent entries of the correlation matrix V. This has been recently experimentally realized [30] for

0.3

0.25 0.2 0.15 0.1

(10)

Eq. (10) is a linear equation for V and can be straightforwardly solved, but the general exact expression is too cumbersome and will not be reported here. In order to establish the conditions under which the optical mode and the mirror vibrational mode are entangled we consider the logarithmic negativity EN , a quantity which has been already proposed as a measure of entanglement [26]. In the CV case EN can be defined as [27] EN = max[0, − ln 2η − ],

EN

0.05 1

2

3

4

5

∆/wm

FIG. 1: Plot of the logarithmic negativity EN as a function of the normalized detuning ∆/wm in the case of an optical cavity of length L = 1 mm, driven by a laser with wavelength 810 nm and power P = 50 mW. The mechanical oscillator has a frequency wm /2π = 10 MHz, a damping rate γm /2π = 100 Hz, and its temperature is T = 400 mK. Full line refers to a mass m = 5 ng, and finesse F = 1.07 × 104 , while the dashed line refers to a mass m = 50 ng and finesse F = 3.4 × 104 .

EN

0.3

0.25 0.2 0.15 0.1 0.05 5

10

15

20

T(K)

25

FIG. 2: Plot of the logarithmic negativity EN versus the mirror temperature. The full line refers to a mass m = 5 ng, detuning ∆ = wm and finesse F = 1.07 × 104 ; the dashed line refers to a mass m = 50 ng, ∆ = wm /2, and finesse F = 3.4 × 104 . The other parameters are those of Fig. 1.

the case of two entangled optical modes at the output of a parametric oscillator. In our case, the measurement of the field quadratures of the cavity mode can be straightforwardly performed by homodyning the cavity output using a local oscillator with an appropriate phase. Measuring the mechanical mode is less straightforward. However, if we consider a second Fabry-Perot cavity C2 , adjacent to the first one and formed by the movable mirror and a third fixed mirror (see Fig. 3), it is possible to adjust the parameters of C2 so that both position and momentum of the mirror can be measured by homodyning the C2 output. In fact, assuming that the movable mirror has unit reflectivity at both sides so that there is no light coupling the two cavities, the annihilation operator of the second cavity, a2 , obeys an equation analogous to the linearized version of Eq. (2c), √ δ a˙ 2 = −(κ2 + i∆2 )δa2 + iG2 α2 δq + 2κ2 ain 2 (t), (13) where κ2 , ∆2 , α2 , ain 2 (t) are the bandwidth, the ef-

4 fective detuning, the intracavity field amplitude, and the inputpnoise of C2 , respectively. Moreover, G2 = (wc2 /L2 ) ¯h/mwm , where wc2 and L2 are the frequency and the length of C2 . The presence of the second cavity affects the mirror dynamics, which is no more exactly described by Eqs. (5). However, if C2 is driven by a much weaker intracavity field so that |α2 | ≪ |αs |, its backaction on the mechanical mode can be neglected and the relevant dynamics is still well described by Eqs. (5).

FIG. 3: Schematic description of the proposed experiment, including the second Fabry-Perot cavity on the right for the detection of the mechanical motion.

If we now choose parameters so that ∆2 = wm ≫ k2 , G2 |α2 |, we can rewrite Eq. (13) in the frame rotating at ∆2 = wm for the slow variables δ˜ o(t) ≡ δo(t) exp{−iwm t} and neglect the terms fastly oscillating at the frequency 2wm , so to get [15, 18] √ G2 α2 δa ˜˙ 2 = −κ2 δ˜ a2 + i √ δ˜b + 2κ2 a ˜in (14) 2 (t), 2 √ √ where δb = (iδp+δq)/ 2. If κ2 ≫ G2 |α2 |/ 2, the cavity mode adiabatically follows thepmirror dynamics and one √ ˜ has δ˜ a ≃ i(G α /κ 2)δ b + 2/κ2 a ˜in õut = 2 2 2 2 2 (t). Using a 2 √ in 2κ2 δ˜ a2 − a ˜2 [21], we finally get G2 α2 ˜in a õut = i √ δ˜b + a 2 (t), 2 κ2

(15)

showing that, in the chosen parameter regime, the output light of C2 gives a direct measurement of the mirror dynamics. By changing the phases of the two local oscillators and by measuring the correlations between the two cavity outputs one can determine all the entries of the CM V and from them numerically extract the logarithmic negativity EN by means of Eq. (11). In conclusion, we have shown that a Fabry-Perot cavity with an oscillating micro-mirror and driven by coherent light can produce robust and stationary entanglement between the optical intracavity mode and the mechanical mode of the mirror. The amount of entanglement is quantified by the logarithmic negativity and surprisingly robust against increasing temperature: for experimental parameters close to those of recently performed experiments [29] entanglement may persist above 20K in the case of a 5 ng mechanical oscillator. Finally, we suggest a readout scheme that allows a full experimental characterization of the CV Gaussian steady state of the system and hence a measurement of the generated entanglement.

We acknowledge helpful discussions with C. Brukner, J. Eisert, J. Kofler, M. Paternostro. D. V. thanks the IQOQI in Vienna for hospitality. This work was supported by the Austrian Science Fund FWF (SFB15), by the European Commission under the programs RamboQ and SECOQC and by the City of Vienna.

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