Quantum entanglement and teleportation in pulsed cavity-optomechanics
Sebastian G. Hofer, Witlef Wieczorek, Markus Aspelmeyer, Klemens Hammerer
QQuantum entanglement and teleportation in pulsed cavity-optomechanics
Sebastian G. Hofer,
1, 2, ∗ Witlef Wieczorek, Markus Aspelmeyer, and Klemens Hammerer Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics,University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria Institute for Theoretical Physics, Institute for Gravitational Physics,Leibniz University Hannover, Callinstraße 38, 30167 Hannover, Germany
Entangling a mechanical oscillator with an optical mode is an enticing and yet a very challenginggoal in cavity-optomechanics. Here we consider a pulsed scheme to create EPR-type entanglementbetween a travelling-wave light pulse and a mechanical oscillator. The entanglement can be verifiedunambiguously by a pump–probe sequence of pulses. In contrast to schemes that work in a steady-state regime under continuous wave drive this protocol is not subject to stability requirements thatnormally limit the strength of achievable entanglement. We investigate the protocol’s performanceunder realistic conditions, including mechanical decoherence, in full detail. We discuss the relevanceof a high mechanical Q · f -product for entanglement creation and provide a quantitative statementon which magnitude of the Q · f -product is necessary for a successful realization of the scheme. Wedetermine the optimal parameter regime for its operation, and show it to work in current state-of-the-art systems. I. INTRODUCTION
In optomechanical systems a cavity mode can bestrongly coupled to a high-quality mechanical oscilla-tor via radiation pressure or dipole gradient forces [1–3].Quantum effects [4–6] are starting to play an increasinglyimportant role: In the microwave regime ground-statecooling via laser-cooling techniques [7], strong coupling[8, 9] and coherent control of single-phonon excitations[9] have been successfully achieved. In the optical regimecooling to the quantum ground state [10] and effects ofstrong coupling [11] have been demonstrated in recentexperiments. It is as yet an outstanding goal to ob-serve genuine quantum effects such as entanglement [12]at macroscopic length and mass scales.Entanglement of a mechanical oscillator with light hasbeen predicted in a number of theoretical studies [13–23]and would be an intriguing demonstration of optome-chanics in the quantum regime. These studies, as wellas similar ones investigating entanglement among sev-eral mechanical oscillators [24–32], explore entanglementin the steady-state regime . In this regime the optome-chanical system is driven by one or more continuous-wavelight fields and settles into a stationary state, for whichthe interplay of optomechanical coupling, cavity decay,damping of the mechanical oscillator, and thermal noiseforces may remarkably give rise to persistent entangle-ment between the intracavity field and the mechanicaloscillator.Entanglement in the steady-state regime shows twomain characteristic features: Firstly, entanglementreaches a maximal value when the system is driven closeto a point of dynamical instability.
Vice versa , the lim-its on the strength of entanglement achievable in proto-cols working in the steady-state regime are set by the ∗ [email protected] very conditions guaranteeing a dynamically stable, sta-tionary state. Recent studies indicate that these limi-tations can even become rather restrictive when a finitelaser linewidth is taken into account [20, 22]. Secondly,the verification of entanglement between the intracav-ity field and the moving mirror has to be performed viameasurements on the outcoupled light leaving the op-tomechanical system. Ultimately only correlations be-tween modes of the light field are measured, from whichany entanglement involving the mechanical oscillator hasto be inferred. However, due to the curious feature ofquantum correlations that “no entanglement is necessaryto distribute entanglement” [33, 34], this sort of infer-ence is in general a delicate issue. It is unambiguouslyonly possible under additional assumptions regarding theparticular dynamics (i. e., the system’s Hamiltonian) andstructure of the steady state [13].An alternative approach to achieving optomechanicalentanglement is to work in the pulsed regime , where en-tanglement is created and verified with two subsequentpulses of light. This strategy has first been developedin the context of atomic ensembles [35] and was recentlyconsidered for systems employing levitated microspherestrapped in an optical cavity [36]. A pulsed scheme doesnot rely on the existence of a stable steady state, whichprovides us with the benefit that entanglement is not lim-ited by stability requirements. The temporal ordering ofthe pulses excludes the possibility of distributing entan-glement without using entanglement [33, 34] such that itprovides a direct (i. e., without additional assumptions)and unambiguous test of entanglement. Similar protocolshave also been discussed for micromirrors in free space(i. e., without the use of an optical cavity) [37, 38].In this article we provide a complete treatment of aprotocol for the generation and verification of optome-chanical entanglement using pulsed light. In addition toan idealized scenario, which was briefly discussed in [36],we include in our description the full dynamics of theoptomechanical system. Our derivation provides an ex- a r X i v : . [ qu a n t - ph ] N ov haustive discussion of imperfections, how they affect theperformance of the protocol and how these effects canbe minimised. Most prominently, we (perturbatively) in-clude thermal decoherence of the mechanical system andfind that a high Q · f -product (quality factor times the fre-quency of the mechanical oscillator) plays a crucial rolein the creation of optomechanical entanglement. Morespecifically, we find that the relation Q · f (cid:29) k B T /h (where T is the temperature of the mechanical environ-ment) has to hold, which is a general and often verystringent condition to observe quantum effects in optome-chanical systems [39]. A large Q · f -product is thus oneof the most important aspects to consider in the designof novel high-quality mechanical resonators.To further explore the effect of imperfections on theprotocol, we optimise the amount of created entangle-ment with respect to key experimental parameters andpresent specific values for two existing optomechanicalsystems. We find that creation of entanglement is possi-ble in a parameter regime which is realistic yet challeng-ing for current state-of-the-art setups. Very importantly,our treatment also provides a quantitative statement onwhat magnitude of the Q · f -product is necessary in or-der to successfully create entanglement for a given bathtemperature T . These findings provide a general under-standing of the requirements to observe quantum effectsin optomechanical systems and represent essential infor-mation for the material development and for the furtherdesign of future optomechanical structures.We finally note that the quantum state created inthis protocol exhibits a type of entanglement known asEinstein–Podolsky–Rosen (EPR) entanglement [40] be-tween the mechanical oscillator and the light pulse [36].It thus provides the canonical resource for quantum in-formation protocols involving continuous variable (CV)systems [41, 42]. We give a detailed description of howoptomechanical EPR entanglement can be used for theteleportation of the state of a propagating light pulseonto a mechanical oscillator.Note that other promising perspectives for pulsed op-tomechanics have recently been discussed also in [43–47], albeit in a very different parameter regime employ-ing light pulses which are short on the time scale of amechanical oscillation. The importance of temporal or-dering in the verification of optomechanical entanglementhas also been pointed out in [15].The paper is organized as follows: Section II A containsthe main results of this work. We first describe entangle-ment creation under idealized circumstances and outlinea way to verify it unambiguously. Additionally, we showhow it can be used as a resource for CV teleportation.In Sec. II B we analyze the influence of imperfections onthe protocol’s performance and find the optimal param-eter regime for maximal entanglement. Section III givesa detailed description of the full system dynamics. TheAppendix contains a short derivation of the effective sys-tem Hamiltonian in the pulsed regime. FIG. 1. (Color online) Schematic of the system and the tele-portation protocol: (a) A blue detuned light pulse (A) is en-tangled with the mirror (B). (b) A second light pulse (V) isprepared in the input state and interferes with A on a beamsplitter. Two homodyne detectors measure P outl + X v and X outl + P v , yielding outcomes m X and m P , respectively. Feed-back is applied by displacing the mirror state in phase spaceby a unitary transformation D X m ( m X ) D P m ( m P ) . (c) To ver-ify the success of the protocol, the mirror state is coherentlytransferred to a red detuned laser pulse and a generalisedquadrature X (cid:48) l ( θ ) = X (cid:48) l out cos θ + P (cid:48) l out sin θ is measured. Re-peating steps (a)–(c) for the same input state but for differentphases θ yields a reconstruction of the mirror’s quantum state. II. CENTRAL RESULTSA. Motivation for the pulsed scheme
1. Cavity optomechanical system
Let us consider an optomechanical cavity in a Fabry–Pérot-type setup (Fig. 1), with mechanical oscillation fre-quency ω m , mechanical dissipation rate γ , optical reso-nance frequency ω c and cavity decay rate κ . A light pulseof duration τ and carrier frequency ω l impinges on thecavity and interacts with the oscillatory mirror mode viaradiation pressure.In a frame rotating with the laser frequency, the systemis described by the (effective) Hamiltonian [48] H = ω m a † m a m + ∆ c a † c a c + g (cid:0) a m + a † m (cid:1) (cid:0) a c + a † c (cid:1) (1)where a m and a c are annihilation operators of the me-chanical and optical mode, respectively. The condi-tions under which (1) is valid are discussed in detailin Sec. III A. The first two terms give the energy ofthe mechanical oscillator and the cavity field, where ∆ c = ω c − ω l is the detuning of the laser drive with respectto the cavity resonance. The last term describes the lin-earised optomechanical coupling (with coupling constant g ) via radiation pressure x m x c ∝ (cid:0) a m + a † m (cid:1) (cid:0) a c + a † c (cid:1) = (cid:0) a m a † c + a † m a c (cid:1) + (cid:0) a m a c + a † m a † c (cid:1) , which can be decom-posed into two terms [4, 25, 37, 38]: a beam-splitter-likeinteraction (the first term) and a two-mode-squeezing in-teraction (the second term). The former can be used tocool the mirror as well as to generate a state swap be-tween the mechanical and the optical mode, while thelatter term describes the optomechanical analogue to theoptical down-conversion process in an optical paramet-ric amplifier and is known to create entanglement fromcoherent input states [4, 49].We assume the pulse to approximately be a flat-toppulse, which has a constant amplitude for the largestpart, but possesses a smooth head and tail (see the Ap-pendix). The coupling constant g is then given by g = g (cid:115) κ ∆ + κ N ph τ , (2)with N ph being the number of photons in the pulse (seeSec. III A). The single-photon coupling constant g is de-fined by g = ω c x /L , where x is the size of the zero-point-motion of the mechanical oscillator and L is thecavity length. It is possible to make a single one of theinteraction terms dominant by tuning the laser such, thatone of its motional sidebands ω l ± ω m is resonant with thecavity, where for blue detuning the resonant scattering tothe lower (Stokes) sideband ( ω c = ω l − ω m ) enhances thedown-conversion interaction, while for red detuning theresonant scattering to the upper (anti-Stokes) sideband( ω c = ω l + ω m ) enhances the beam-splitter interaction[4]. In this pump-probe scheme we make use of bothdynamics separately: Pulses tuned to the blue side ofthe cavity resonance are applied to create entanglement,while pulses on the red side are later used to read outthe final mirror state. A similar separation of Stokes andanti-Stokes sideband was suggested in [37, 38] by select-ing different angles of reflection of a light pulse scatteredfrom a vibrating mirror in free space.The full system dynamics, including the dissipativecoupling of the mirror and the cavity decay, are de-scribed by quantum Langevin equations [50], which de-termine the time evolution of the corresponding opera-tors x m = ( a m + a † m ) / √ , p m = − i( a m − a † m ) / √ and a c , a † c . They read ˙ x m = ω m p m , (3a) ˙ p m = − ω m x m − γ p m − √ g (cid:0) a c + a † c (cid:1) − (cid:112) γ f, (3b) ˙ a c = − (i∆ c + κ ) a c − i √ g x m − √ κ a in , (3c)where we introduced the (self-adjoint) Brownian stochas-tic force f , and quantum noise a in entering the cav-ity from the electromagnetic environment. Both a in and—in the high-temperature limit— f are assumed tobe Markovian. Their correlation functions are thus givenby (cid:104) a in ( t ) a † in ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) (in the optical vacuum state)and (cid:104) f ( t ) f ( t (cid:48) ) + f ( t (cid:48) ) f ( t ) (cid:105) = (2¯ n + 1) δ ( t − t (cid:48) ) (in a ther-mal state of the mechanics) [50].
2. Creation of optomechanical entanglement
In this section we impose the following conditions onthe system’s parameters. Firstly, we drive the cavity witha blue detuned laser pulse ( ∆ c = − ω m ) and assume towork in the resolved-sideband regime ( κ (cid:28) ω m ) to en-hance the down-conversion dynamics. Note that in thisregime a stable steady state only exists for very weak op-tomechanical coupling [51], which poses a fundamentallimit to the amount of entanglement that can be createdin a continuous-wave scheme [14]. In contrast, a pulsedscheme does not suffer from these instability issues. Infact, it is easy to check by integrating the full dynamics(see Sec. III B) up to time τ , that working in this particu-lar regime yields maximal entanglement, which increaseswith increasing sideband resolution ω m /κ . Secondly weassume a weak optomechanical coupling g (cid:28) κ , such thatonly first-order interactions of photons with the mechan-ics contribute. This minimises pulse distortion and sim-plifies the experimental realization of the protocol. Takentogether, the conditions g (cid:28) κ (cid:28) ω m allow us to invokethe rotating-wave approximation (RWA), which amountsto neglecting the beam-splitter term in (1). Also, we ne-glect mechanical decoherence effects in this section. Weemphasise that this approximation is justified as long asthe total duration of the protocol is short compared tothe effective mechanical decoherence time /γ ¯ n , where γ is the mechanical damping rate and ¯ n is the thermal oc-cupation of the corresponding bath. Corrections to thissimplified model—including the treatment of mechani-cal decoherence and dynamics beyond the RWA—will beaddressed in Sec. II B.Based on the assumptions above we can now simplifyequations (3). For convenience we go into a frame rotat-ing with ω m by substituting a c → a c e i ω m t , a in → a in e i ω m t and a m → a m e − i ω m t . Note that in this picture the cen-tral frequency of a in is located at ω l − ω m = ω c . In theRWA the Langevin equations then simplify to ˙ a c = − κa c − i g a † m − √ κ a in , (4a) ˙ a m = − i g a † c . (4b)In the limit g (cid:28) κ we can use an adiabatic solution forthe cavity mode and we therefore find a c ( t ) ≈ − i gκ a † m ( t ) − (cid:114) κ a in ( t ) , (5a) a m ( t ) ≈ e Gt a m (0) + i √ G e Gt (cid:90) t d s e − Gs a † in ( s ) , (5b)where we defined G = g /κ . Equation (5b) shows thatthe mirror motion gets correlated to a light mode of cen-tral frequency ω l − ω m (which coincides with the cavityresonance frequency ω c ) with an exponentially shapedenvelope α in ( t ) ∝ e − Gt . Using the standard cavity input-output relations a out = a in + √ κ a c allows us to definea set of normalised temporal light modes A in = (cid:114) G − e − Gτ (cid:90) τ d t e − Gt a in ( t ) , (6a) A out = (cid:114) G e Gτ − (cid:90) τ d t e Gt a out ( t ) , (6b)which obey the canonical commutation relations [ A i , A † i ] = 1 . Together with the definitions B in = a m (0) and B out = a m ( τ ) we arrive at the following expressions,which relate the mechanical and optical mode at the endof the pulse t = τA out = − e Gτ A in − i (cid:112) e Gτ − B † in , (7a) B out = e Gτ B in + i (cid:112) e Gτ − A † in . (7b)By expressing equations (7) in terms of quadratures X i m = ( B i + B † i ) / √ and X i l = ( A i + A † i ) / √ , where i ∈ { in , out } , and their corresponding conjugate vari-ables, we can calculate the so-called EPR variance ∆ EPR of the state after the interaction. For light initially invacuum (∆ X inl ) = (∆ P inl ) = and the mirror in athermal state (∆ X inm ) = (∆ P inm ) = n + , the state isentangled iff [52] ∆ EPR = (cid:2) ∆( X outm + P outl ) (cid:3) + (cid:2) ∆( P outm + X outl ) (cid:3) = 2( n + 1) (cid:16) e r − (cid:112) e r − (cid:17) < , (8)where r = Gτ is the squeezing parameter and n the ini-tial occupation number of the mechanical oscillator. Notethat in the limit of large squeezing r (cid:29) we find that thevariance ∆ EPR ≈ ( n + 1) e − r / is suppressed exponen-tially, which shows that the created state asymptoticallyapproximates an EPR state. Therefore, this state can bereadily used to conduct optomechanical teleportation asdescribed in Sec. II A 4.Rearranging (8), we find that the state is entangled aslong as r > r = 12 ln (cid:18) ( n + 2) n + 1) (cid:19) n →∞ ∼
12 ln n . (9)This illustrates that in our scheme the requirement onthe strength of the effective optomechanical interaction,as quantified by the parameter r = g τκ , scales logarith-mically with the initial occupation number n of the me-chanical oscillator. This tremendously eases the proto-col’s experimental realization, as neither g nor τ can bearbitrarily increased—both for fundamental and techni-cal reasons—, as we will show in Sec. II B. Note that n need not be equal to the mean bath occupation ¯ n , butmay be decreased by laser pre-cooling to improve theprotocol’s performance.
3. Entanglement verification
To verify the successful creation of entanglement a reddetuned laser pulse ( ∆ c = ω m ) is sent to the cavity where it resonantly drives the beam-splitter interaction,and hence generates a state swap between the mechanicaland the optical mode. It is straightforward to show thatchoosing ∆ c = ω m leads to a different set of Langevinequations which can be obtained from (4) by droppingthe Hermitian conjugation ( † ) on the right-hand side.By defining modified mode functions α (cid:48) in(out) = α out(in) and corresponding light modes A (cid:48) in(out) one obtains in-put/output expressions in analogy to (7) A (cid:48) out = − e − Gτ A (cid:48) in + i (cid:112) − e − Gτ B in , (10a) B out = e − Gτ B in − i (cid:112) − e − Gτ A (cid:48) in . (10b)The pulsed state-swapping operation therefore also fea-tures an exponential scaling with Gτ . For Gτ → ∞ theexpressions above reduce to A (cid:48) out = − i B in and B out =i A (cid:48) in , which shows that in this case the mechanical state—apart from a phase shift—is perfectly transferred to theoptical mode. In the Schrödinger-picture this amountsto the transformation | ϕ (cid:105) m | ψ (cid:105) l → | ψ (cid:105) m | ϕ (cid:105) l , where ϕ and ψ constitute the initial state of the mechanics and thelight pulse, respectively. The state-swap operation thusallows us to access mechanical quadratures by measur-ing quadratures of the light and therefore to reconstructthe state of the bipartite system via optical homodynetomography. For this the protocol is operated in twosteps: After the blue detuned pulse is reflected fromthe cavity, it is sent to a homodyne detection setup,where a quadrature X l ( φ ) = X outl cos φ + P outl sin φ ( φ being the local oscillator phase) is measured. The sameprocedure is subsequently carried out for a red detunedpulse, measuring X (cid:48) l ( θ ) (see Fig. 1), which, for the caseof a perfect state swap, yields the mechanical quadrature X (cid:48) l ( θ ) = X m ( θ + π ) . Here the rotation by π is due to thephase shift from the swap operation. By repeating thisprocess multiple times for different local-oscillator phases ( φ i , θ j ) , the quantum state of the bipartite optomechan-ical system can be reconstructed. Having obtained thefull quantum state, entanglement can be analyzed by var-ious means [53], e. g., by applying the EPR criterion fromabove.
4. Optomechanical teleportation protocol
As we have shown above, pulsed operation allows usto create EPR-type entanglement, which forms the cen-tral entanglement resource of many quantum informationprocessing protocols [41]. An immediate extension thisscheme is an optomechanical continuous variables quan-tum teleportation protocol. The main idea of quantumstate teleportation in this context is to transfer an arbi-trary quantum state | ψ in (cid:105) of a traveling wave light pulseonto the mechanical resonator, without any direct in-teraction between the two systems, but by making useof optomechanical entanglement. The scheme works infull analogy to the CV teleportation protocol for pho-tons [54, 55]. Due to its pulsed nature it closely resem-bles the scheme used in atomic ensembles [35, 56] andit was recently also suggested in the context of levitatedmicrospheres [36] (see [57] for an exhaustive descriptionof a similar system comprising a nuclear-spin ensembleentangled with light): A light pulse (A) is sent to the op-tomechanical cavity and is entangled with its mechanicalmode (B) via the dynamics described above. Meanwhilea second pulse (V) is prepared in the state | ψ in (cid:105) , which isto be teleported. This pulse then interferes with A on abeam splitter. In the output ports of the beam splitter,two homodyne detectors measure two joint quadratures P outl + X v and X outl + P v , yielding outcomes m X and m P , respectively. This constitutes the analogue to theBell measurement in the case of qubit teleportation andeffectively projects previously unrelated systems A andV onto an EPR state [58]. Note that both the secondpulse and the local oscillator for the homodyne measure-ments must be mode-matched to A after the interaction;i. e., they must possess the identical carrier frequency aswell as the same exponential envelope. The protocol isconcluded by displacing the mirror in position and mo-mentum by m X and m P according to the outcome ofthe Bell-measurement. This can be achieved by meansof short light pulses, applying the methods described in[43, 45]. After the feedback the mirror is then describedby [41] X finm = X outm + P outl + X v , = X v + (cid:16) e r − (cid:112) e r − (cid:17) ( X inm − P inl ) , (11a) P finm = P outm + X outl + P v , = P v + (cid:16) e r − (cid:112) e r − (cid:17) ( P inm − X inl ) , (11b)which shows that its final state corresponds to the in-put state plus quantum noise contributions. It is ob-vious from these expressions that the total noise addedto both quadratures [second term in (11a) and (11b), re-spectively] is equal to the EPR variance. Again, for largesqueezing r (cid:29) the noise terms are exponentially sup-pressed and in the limit r → ∞ , where the resource stateapproaches the EPR state, we obtain perfect teleporta-tion fidelity, i. e., X finm = X v and P finm = P v . In particularthis operator identity means, that all moments of X v , P v with respect to the input state | ψ in (cid:105) will be transferredto the mechanical oscillator, and hence its final state willbe identically given by | ψ in (cid:105) .To verify the success of the teleportation one has toread out the mirror state after completing the feedbackstep. This can be achieved by applying tomography onthe mechanical state as described in the previous section.The overlap of the reconstructed state ρ out and a pureinput state | ψ in (cid:105) then gives the teleportation fidelity F = (cid:104) ψ in | ρ out | ψ in (cid:105) . For coherent input states the fidelity isgiven by F = (cid:18) EPR (cid:19) − . (12) In order to beat the optimal classical strategy for trans-mission of quantum states (i. e., the optimal measure-and-prepare scheme), the achieved fidelity (averaged overall coherent states) must exceed F > / [59], which isequivalent to the condition for entanglement, ∆ EPR < . B. Optimised protocol including imperfections
1. Perturbations
In the previous section we found that in the ideal sce-nario the amount of entanglement essentially dependsonly on the coupling strength (or equivalently on the in-put laser power) and the duration of the laser pulse andthat it shows an encouraging scaling, growing exponen-tially with Gτ . This in turn means that the minimalamount of squeezing needed to generate entanglementonly grows logarithmically with the initial mechanicaloccupation n . In this section we will develop a morerealistic scenario including thermal noise effects and fullsystem dynamics, both of which will decrease the cre-ated entanglement. We will show, however, that underconditions already available in state-of-the-art optome-chanical experiments, one can find an optimal workingpoint such that the significance of these unwanted effectscan sufficiently be suppressed.To extend the validity of the previous, simplifiedmodel, we now include the following additional dynam-ics: contributions from the beam-splitter Hamiltonian,higher order interactions beyond the adiabatic approxi-mation, and decoherence effects due to mechanical cou-pling to a heat bath. In the following we will investigatetheir effect on our protocol and determine the parameterregime featuring maximal entanglement. The technicaldetails of how we include them in our calculations will beshown in Sec. III.Including the above-mentioned perturbations results ina final state which deviates from an EPR-entangled state.To minimise the extent of these deviations, the systemparameters must obey the following conditions:1. κ (cid:28) ω m results in a sharply peaked cavity responseand implies that the down-conversion dynamics isheavily enhanced with respect to the suppressedbeam-splitter interaction.2. g (cid:46) κ inhibits multiple interactions of a single pho-ton with the mechanical mode before it leaves thecavity. This suppresses spurious correlations to theintracavity field. It also minimises pulse distortionand simplifies the protocol with regard to modematching and detection.3. gτ (cid:29) is needed in order to create sufficientlystrong entanglement. This is due to the fact thatthe squeezing parameter r = ( g/κ ) gτ should belarge, while g/κ needs to be small.4. ¯ nγτ (cid:28) , where ¯ n is the thermal occupation of themechanical bath, assures coherent dynamics overthe full duration of the protocol, which is an es-sential requirement for observing quantum effects.As the thermal occupation of the mechanical bathmay be considerably large even at cryogenic tem-peratures, this poses (for fixed γ and ¯ n ) a very strictupper limit to the pulse duration τ .Note however that not all of these inequalities haveto be fulfilled equally strictly, but there rather exists anoptimum which arises from balancing all contributions.It turns out that fulfilling (4) is critical for successfulteleportation, whereas (1)–(3) only need to be weaklysatisfied. Taking the above considerations into account,we find a sequence of parameter inequalities ¯ nγ (cid:28) τ (cid:28) g (cid:28) κ (cid:28) ω m , (13)which defines the optimal parameter regime. In Sec. II Awe assumed the first two conditions to be well satisfiedand we neglected the existence of mechanical decoher-ence. If we now take into account that the mechanicaloscillator couples to a heat bath with an effective deco-herence rate ¯ nγ , we find that increasing the pulse dura-tion to values larger than the mechanical coherence timewill drastically decrease entanglement. This results in anupper bound for entanglement, as now both the interac-tion strength and the pulse duration, and therefore alsothe squeezing parameter r = ( g/κ ) gτ , are bounded fromabove.Dividing (13) by γ and taking a look at the outermostcondition ¯ n (cid:28) Q m , where Q m = ω m /γ is the mechanicalquality factor, we see that the ratio Q m / ¯ n defines therange which all the other parameters have to fit into. Itis intuitively clear, that a high quality factor and a lowbath occupation number, and consequently a low effec-tive mechanical decoherence rate, are favourable for thesuccess of the protocol. Equivalently, we can rewrite theoccupation number as ¯ n = k B T bath / (cid:126) ω m and thereforefind k B T bath / (cid:126) (cid:28) Q m · ω m , where now the Q · f -product( f = ω m / π ) has to be compared to the thermal fre-quency of the bath. Let us consider a numerical exam-ple: For a temperature T bath ≈
100 mK the left-hand sidegives k B T bath / (cid:126) ≈ π · Hz . The Q · f -product con-sequently has to be several orders of magnitude largerto successfully create entanglement. As current optome-chanical systems feature a Q · f -product of π · Hz andabove [60–63], this requirement seems feasible to meet.Note that in an experiment T bath will often depend onthe input laser power, as scattered light can heat up thecryogenic environment. Hence, for a given bath occupa-tion the coupling strength may be limited for technicalreasons.In order to find the optimal working point, it is con-venient to introduce the following dimensionless parame-ters: the sideband-resolution parameter η , the adiabatic-ity parameter ξ , and the ratio of pulse length to mechan- ical coherence time (cid:15) . They are given by η = κ/ω m , ξ = g/κ, (cid:15) = γτ. From (13) it follows that η (cid:28) , ξ (cid:28) and (cid:15) (cid:28) / ¯ n .Each of those small parameters can be used to realise aperturbative expansion of the additional dynamics listedabove. The perturbative solutions can then be used tocalculate the EPR variance and optimise the resultingexpressions.
2. Optimization
As illustrated above, we expect—for fixed values of ¯ n and Q m —to find optimal values for the remaining param-eters (cid:15), ξ and η . The maximal possible entanglement willultimately be set by ¯ n and Q m , which will also constitutehard boundaries in typical experiments.Figure 2 shows results of this optimization for different Q -factors, plotted against the thermal occupation num-ber of the mechanical bath. Figure (a) shows the min-imal value of ∆ EPR for a given Q m , and (b)–(d) showthe corresponding optimal values for η opt , ξ opt and (cid:15) opt .As expected the noise contribution from the mechanicalbath is found to be the most critical. As we show inIII B the EPR variance including thermal noise can beexpressed by ∆ EPR = (cid:2) ∆( X outm + P outl ) (cid:3) + (cid:2) ∆( P outm + X outl ) (cid:3) + (2¯ n + 1) (cid:15), (14)where X out i and P out i here denote the solutions for γ = 0 .The dashed curves in Fig. 2 (a) illustrate the noise contri-bution (2¯ n +1) (cid:15) opt of the thermal bath. As shown above,this quantity is added to the unperturbed EPR variance,and thus the system can only exhibit strong entanglementif its value is far below two. Note that working at the op-timal point keeps the fraction of thermal noise in ∆ EPR approximately constant over a wide range of ¯ n . This isshown in Fig. 2 (e), where we defined (cid:15) (cid:48) opt = (cid:15) opt / ∆ EPR .As we have seen in the previous section, the EPR vari-ance depends on the occupation number of the oscillatorat the initial time t = 0 . Due to this, the entanglementcan be drastically increased by pre-cooling the mechanicsby means of laser cooling before starting the actual pro-tocol. Fig. 2 shows that it is thus possible to create anentangled state even for a fairly large bath occupation.This works due to short pulse durations, during whichthe mechanical decoherence is small.Taking a look at figure (c) we note that the sideband-resolution shows rather large optimal values near unity,especially for increasing occupation numbers. This indi-cates that the beam-splitter dynamics only weakly dis-turbs the entangling interaction.Table I gives a list of the optimized key experimentalparameters for two existing optomechanical structures.The values in the first row correspond to the black dotsshown in Fig. 2. (a) . . . . . . . . . . . . . . ¯ nT bath [mK] at 3 . ∆ EP R (b) ¯ n ǫ o p t [ − ] (c) ¯ n η o p t [ − ] (d) ¯ n ξ o p t [ − ] (e) ¯ n ( n + ) ǫ ′ o p t . . . FIG. 2. (Color online) Optimised parameters for Q m = 10 and n = 50 [red (dark) line], and Q m = 10 and n = 0 [blue(light) line], where n is the initial mechanical occupation and ¯ n is the mean bath occupation. This corresponds to the twocases of large Q m with moderate pre-laser-cooling, and lower Q m with pre-cooling into the ground state. Clearly, entangle-ment creation is possible in both cases. (a) Minimal ∆ EPR as a function of ¯ n . The upper axis gives the correspondingbath temperature for a oscillator with a resonance frequencyof 3.8 MHz. To each point corresponds a triple (cid:15) opt , η opt , ξ opt [(b)–(d)] for which the minimal value is realised. The dotted,black line shows the upper bound up to which entanglementis present. The black dots marks the values for ¯ n = 1100 ( T = 200 mK ), which coincide with the values given in thefirst row of table I. The dashed lines show the respective ther-mal noise contribution (2¯ n +1) (cid:15) opt to ∆ EPR [see (14)]. (b)–(d)Values of (cid:15), η, ξ which optimise ∆ EPR for a given ¯ n . (e) Showsthe relative amount of noise induced by the coupling to themechanical environment ( (cid:15) (cid:48) opt = (cid:15) opt / ∆ EPR ). Over and above the fundamental imperfections, techni-cal losses, such as mode-mismatch and detector inefficien-cies, additionally decrease entanglement. They can allcollectively be described as (passive) beam-splitter lossesadding vacuum noise to the optical signal. They willhowever, never completely break entanglement, as long as the overall loss is smaller than unity. Noise contribu-tions of this type can easily be accounted for by addingappropriate noise terms to (7).Finally let us compare the amount of entanglementcreated in the pulsed and continuous-wave schemes [14]in terms of the logarithmic negativity E N [53] for theparameters used in the first row of table I ( Q m = 10 , ¯ n =1100 , n = 0 ). In the case of continuous driving one findsthe maximal negativity close to the instability region andfor a detuning of around ∆ c ≈ ω m , yielding E N ≈ . .For the pulsed protocol the optimisation yields a muchlarger value of E N ≈ . . Note that E N is a logarithmicquantity. III. DETAILED MODELA. Linearizing the dynamics
As was shown in [65], the radiation pressure interac-tion is inherently non-linear. In current micromechanicalsystems however, the single-photon coupling g is veryweak (the best values up to date are on the order of g /κ ≈ . [63]), and has to be enhanced by meansof a strong optical pump field. It is well known thatin the case of a strong continuous-wave light field, thesteady-state dynamics of the system is approximately lin-ear. We will show in the following, that this also holds ina (long) pulsed scheme. The linearization process followsthe same general idea as in the steady-state regime, it is,however, slightly more involved due to the explicit timedependence of the Hamiltonian.We consider a laser pulse with a fixed number of pho-tons N ph and an envelope function ε ( t ) , which is nor-malised in the sense that (cid:82) τ d t | ε ( t ) | = 1 . Its head andtail are assumed to be smooth and its amplitude shouldbe constant ε ( t ) ≈ / √ τ for the most part of τ . The fullHamiltonian for the system, including the laser drivingterm and the non-linear radiation pressure interaction[65] is then given by H ( t ) = ω m a † m a m + ∆ a † c a c + g a † c a c (cid:0) a m + a † m (cid:1) + i E ( t ) (cid:0) a c − a † c (cid:1) , (15)where ∆ = ω c , − ω l is the detuning for the case ofa cavity with fixed length and E ( t ) = (cid:112) κN ph ε ( t ) isthe driving strength. In the Appendix we show that wecan eliminate the driving term by going into a (time-dependent) displaced picture. The transformed Hamil-tonian then takes the form ¯ H ( t ) = ω m a † m a m + ∆ c ( t ) a † c a c + g a † c a c ( a m + a † m )+ g (cid:0) α ∗ ( t ) a c + α ( t ) a † c (cid:1) ( a m + a † m ) , (16)where the effective detuning ∆ c and the mean cavity field α now depend explicitly on time. The resulting expres-sions (see the Appendix) are essentially the same as found TABLE I. Specific optimal values for an Al x Ga − x As structure (3.8 MHz) [64] and an Si optomechanical crystal (3.7 GHz) [10].The whole set is fully determined by (cid:15) opt , η opt , ξ opt . The value for the mean power P = (cid:126) ω l N ph /τ is obtained using (2). (The x in the formula above is a number between 0 and 1 to indicate a ternary alloy between GaAs and AlAs.) ω m / π Q m T bath ¯ n n g / π κ opt / π τ opt P opt g opt / π ∆ EPR
200 mK 1100 0.0 . . . µ s 30 mW 0 .
97 MHz
200 mK 0.7 0.7 . .
26 GHz 0 . µ s 6 µ W 0 .
032 GHz . .
31 GHz 0 . µ s 8 µ W 0 .
040 GHz in the steady-state case (see for example [66]). The non-linear term in (16) can be neglected whenever | α | (cid:29) ,which is true for sufficiently strong driving | E | and willbe the case for the greatest part of the pulse duration.By assuming α ( t ) to be real and by introducing the ef-fective optomechanical coupling constant g ( t ) = g | α ( t ) | ,the procedure leaves us with a linear Hamiltonian in theform of (1).Note that for the case of ∆ c ≈ ω m the relative fre-quency shift induced by radiation pressure will be of theorder of O ( g/ω m ) , and therefore small. Consequently[together with the assumption that ε ( t ) ≈ / √ τ ] we willin the following drop the explicit time dependence of theeffective detuning and the effective coupling strength. B. Solving the full system
The full Langevin equations (3) resulting from the lin-earised Hamiltonian (including the beam-splitter inter-action and mechanical decoherence) can be rewritten inthe compact form dd t R ( t ) = ( S − D ) R ( t ) − √ DR in ( t ) , (17)where R = ( a m , a † m , a c , a † c ) and correspondingly R in de-notes the input noise. S and D are matrices compris-ing the respective coefficients. To solve this set of equa-tions we apply the Laplace transformation, introducing ¯ R ( s ) = L [ R ]( s ) , with L [ f ]( s ) = (cid:82) ∞ d t e − st f ( t ) . Solvingfor R we obtain ¯ R ( s ) = ¯ M ( s ) (cid:16) R (0) − √ D ¯ R in ( s ) (cid:17) , (18)and thus R ( τ ) = M ( τ ) R (0) − (cid:16) M ∗ √ DR in (cid:17) ( τ ) , (19)where ¯ M ( s ) = ( s − S + D ) − , M ( τ ) = L − [ ¯ M ]( τ ) and ∗ denotes the convolution integral ( f ∗ g )( t ) = (cid:82) t d s g ( s ) f ( t − s ) . In the case of a bipartite system it ispossible to find an exact expression for M for arbitraryparameters. The obtained solution, however, is very te-dious and will not be presented here. We proceed asfollows: We separate the mechanical decoherence in aperturbative approach (i. e., we expand M in powers of (cid:15) ), while the other dynamics will be treated exactly. This allows us to find input/output relations corresponding to(7) and to calculate the EPR variance for the full system.We established in Sec. II B 2 that for the protocol towork we require the effective mechanical decoherencetime to be much larger than the duration of the lightpulse, i. e., (cid:15) ¯ n (cid:28) . We emphasise that (cid:15) is the smallestof all parameters, and the coherent evolution will onlybe negligibly perturbed by the coupling to the mechani-cal bath. We will therefore only keep terms O ( (cid:15) ¯ n ) whileneglecting O ( (cid:15) ) . Based on this premise we simplify (19)twofold: Firstly we drop the mechanical damping fromthe first term, as it gives corrections on the order of O ( (cid:15) ) only; thus M ≈ M | γ =0 . This amounts to dropping theterm − γ p m in (3b). Secondly, we approximate the me-chanical noise contribution (second term) by only keep-ing the free, harmonic evolution, while neglecting theircoupling to the optical mode. This coupling is due to asecond-order process, and is therefore suppressed by anadditional factor of ξ . Note that by doing so we overes-timate the effect of mechanical noise, as it contributes tothe creation of optomechanical correlations when subjectto the coherent dynamics. The complete noise term en-tering in the evolution of the mechanical variables thentakes the form i √ γ (cid:90) τ d s e − i ω m s f ( τ − s ) =: (cid:114) γτ F s + i F c ) , (20)where we introduced (co)sine components F (c)s of theBrownian force. We can therefore write a m ( τ ) ≈ a m ( τ ) (cid:12)(cid:12) γ =0 + (cid:112) (cid:15)/ F s + i F c ) , and consequently X outm ≈ X outm (cid:12)(cid:12)(cid:12) γ =0 + √ (cid:15) F s , (21a) P outm ≈ P outm (cid:12)(cid:12)(cid:12) γ =0 + √ (cid:15) F c , (21b)while we neglect the mechanical noise contribution to theoptical mode, i. e., A out ≈ A out | γ =0 . Note that fromthe commutation relation of the Brownian noise term, [ f ( t + s ) , f ( t )] = i ω m δ (cid:48) ( s ) [50], it follows that [ F s , F c ] =i + O (1 /ω m τ ) and [ F i , F i ] = O (1 /ω m τ ) . The perturbedvariables (21) therefore approximately obey canonicalcommutation relations [ X outm , P outm ] ≈ i(1+ (cid:15) ) . Using (21)together with the correlation functions (cid:104) F i F i (cid:105) = ¯ n + leads to (14). As we have separated the mechanical noiseterms from the other dynamics, we will always assumethat M ≈ M | γ =0 and drop the γ dependence for the restof this section.We now use (19) together with the definitions of A out (6b) and B out to obtain input/output equations similarto (7), but for the full system dynamics. The resultingexpressions are of the form B out = c B in + c B † in + c a c (0) + c a † c (0)+ c (cid:90) τ d s α in , ( s ) a in ( s ) + c (cid:90) τ d s α ∗ in , ( s ) a † in ( s ) , with a similar expression for A out . The coefficients c i aswell as the light modes α in ,i are determined by the sys-tem dynamics [given by M ( t ) ] and the light mode α out selected from the output field. Note that these expres-sions are valid for γ = 0 only and thus have to be used inconjunction with (21) to account for mechanical noise.Following the treatment in Sec. II A 2 one easily findsthe corresponding EPR variance, which now includesnoise terms from the initial intracavity field and the extralight modes (both assumed to be in vacuum). The lat-ter contributions are given by the overlap of the differentlight modes (cid:82) τ d t α in ,i ( t ) α ∗ in ,j ( t ) .The resulting expression is an involved function of (cid:15), η and ξ and is not presented here. Numerical minimizationwith respect to those three variables for fixed values of ¯ n and Q m yields the results presented in Fig. 2. IV. CONCLUSIONS
We have developed a scheme to create and—due toits pump–probe operation—unambiguously verify EPRentanglement in optomechanical systems. Additionallyits application as an entanglement resource in quantumteleportation was discussed. Finally, by optimizing theexperimental parameters we showed that the suggestedprotocol is feasible with state-of-the-art optomechanicaldevices.
ACKNOWLEDGMENTS
We thank the European Commission (MINOS,QESSENCE), the European Research Council (ERC StGQOM), the Austrian Science Fund (FWF) (START, SFBFoQuS), and the Centre for Quantum Engineering andSpace-Time Research (QUEST) for support. W. W. ac-knowledges support from the Alexander von HumboldtFoundation. S. G. H. is a member of the FWF DoctoralProgramme CoQuS (W1210). We thank G. D. Cole, N.Kiesel, M. R. Vanner and A. Xuereb for useful discussion.
Appendix A: Transformed Hamiltonian
Starting from the Hamiltonian (15) we write down thestandard quantum-optical master equation for a dampedcavity mode ˙ ρ = − i [ H, ρ ] + κ (cid:0) a c ρa † c − a c a † c ρ − ρa c a † c (cid:1) , (A1) while we neglected the mechanical decoherence terms aswe are only interested in times far within the coherencetime of the oscillator. In order to eliminate the drivingfield E ( t ) we go into a displaced picture ¯ ρ = D c ( α ) D m ( β ) ρD † c ( α ) D † m ( β ) , (A2)with displacement operators D i ( α ) = exp( αa † i + α ∗ a i ) .The time-dependent, complex amplitudes α = α ( t ) and β = β ( t ) give the mean displacements due to the laserdrive and will be determined in the following.The transformed master equation can again be writtenin the form of (A1) by substituting ρ → ¯ ρ and H → ¯ H .The Hamiltonian ¯ H is then given by ¯ H = ω m a † m a m + [∆ + g ( β + β ∗ )] a † c a c + g ( α ∗ a c + αa † c )( a m + a † m ) + g a † c a c ( a m + a † m )+ (cid:110)(cid:2) i ˙ α + (i κ + ∆ ) α + g ( β + β ∗ ) α − i E (cid:3) a † c + h . c . (cid:111) + (cid:26) (cid:104) i ˙ β + ω m β + g | α | (cid:105) a † m + h . c . (cid:27) . (A3)The first two lines constitute the new Hamiltonian of thesystem, and the last two lines describe the mean (classi-cal) cavity and mirror amplitude, respectively. We canmake these terms disappear by choosing α and β suchthat they fulfill the following set of coupled, non-lineardifferential equations ˙ α = { i [∆ + g ( β + β ∗ )] − κ } α + E, (A4a) ˙ β = i ω m β + i g | α | . (A4b)We seek solutions to these equations for initial conditions α (0) = β (0) = 0 in terms of the driving field E ( t ) . Dueto their non-linear nature, no exact closed-form solutionwill exist in general and we will therefore look for ap-proximate solutions under the assumptions we made inSec. II B. We formally integrate equation (A4b) to find β ( t ) = i g (cid:90) t d s e i ω m s | α ( t − s ) | . (A5)Under the assumption that κ (cid:28) ω m and given that E ( t ) varies sufficiently slowly, we also expect | α ( t ) | to be aslowly varying function on the time scale of /ω m . Wewill check this for consistency at the end of this section.For this case we can use the adiabatic solution β ( t ) ≈ − g ω m | α ( t ) | . (A6)Plugging this into (A4a) and introducing the effectivedetuning ∆ c ( t ) = ∆ − g ( β ( t ) + β ∗ ( t ))= ∆ − g ω m | α ( t ) | (A7)we find the solution α ( t ) ≈ − e − (i∆ c + κ ) t i∆ c + κ E ( t ) (1 + δ ) ≈ E ( t )i∆ c + κ , (A8)0where δ is a correction, which is small if E ( t ) varies slowlyon a timescale of /κ . More precisely, one can show that acrude upper bound is given by | δ ( t ) | < sup s ∈ (0 ,t ) 1 κ | ˙ E ( s ) || E ( t ) | ,which must be much smaller than unity. Also, as we as-sume that κτ (cid:29) , we neglect the term e − κt , as this onlycontributes at the very beginning of the pulse. The ap-proximations made in (A8) amount to assuming that theslope of the pulse is small enough (with respect to κ ) thatit does not experience distortion due to the finite cavitylinewidth. 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