PHYSICAL REVIEW A 89, 063805 (2014) Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir M. J. Woolley1 and A. A. Clerk2 1School of Engineering and Information Technology, UNSW Canberra, ACT, 2600, Australia 2Department of Physics, McGill University, Montreal,´ QC, Canada H3A 2T8 (Received 9 April 2014; published 11 June 2014) We study theoretically a three-mode optomechanical system where two mechanical oscillators are indepen- dently coupled to a single cavity mode. By optimized two-tone or four-tone driving of the cavity, one can prepare the mechanical oscillators in an entangled two-mode squeezed state, even if they start in a thermal state. The highly pure, symmetric steady state achieved allows the optimal fidelity of standard continuous-variable teleportation protocols to be achieved. In contrast to other reservoir engineering approaches to generating mechanical entanglement, only a single reservoir is required to prepare the highly pure entangled steady state, greatly simplifying experimental implementation. The entanglement may be verified via a bound on the Duan inequality obtained from the cavity output spectrum. A similar technique may be used for the preparation of a highly pure two-mode squeezed state of two cavity modes, coupled to a common mechanical oscillator. DOI: 10.1103/PhysRevA.89.063805 PACS number(s): 42.50.Dv, 03.67.Bg, 42.50.Lc, 85.85.+j I. INTRODUCTION of Ref. [8], the steady state is a (highly pure) two-mode squeezed state rather than an entangled mixed state. Therefore, The generation and detection of entangled states of macro- using the steady state as an Einstein-Podolsky-Rosen (EPR) scopic mechanical oscillators is an outstanding task in the channel, the optimal teleportation fidelity for a given amount study of mechanical systems in the quantum regime [1]. of entanglement can be achieved via the standard continuous- There exist a number of proposals for the generation of variable teleportation protocol, without the need for additional such states [2–6]. Perhaps most promising amongst these are local operations [18,19]. In contrast to the proposal of Ref. [9] approaches based on reservoir engineering [7–10], whereby (also described in the supplement to Ref. [8]), the two-mode the dissipation is engineered such that the steady state of squeezed state is generated here using only one, rather than the dissipative dynamics is the desired target state. These two, auxiliary cavity modes, simplifying the experimental proposals are highly attractive from an experimental point implementation. of view, requiring relatively minor modifications of existing Reservoir engineering has earlier been the subject of experimental configurations [11–14]. significant theoretical study in the context of optical and Here, we propose an approach for generating highly pure, atomic systems [7,20–25], culminating in the experimental highly entangled two-mode squeezed states of two mechanical demonstration by Krauter and co-workers of the entangle- oscillators via coupling to a driven cavity mode. The two-mode ment of atomic ensembles [26]. The utility of reservoir squeezed state is the simultaneous vacuum of two nonlocal engineering has also been shown with two-level systems, bosonic operators (so-called Bogoliubov modes) [15]. Hence, with demonstrations of superconducting qubit state control it is possible to prepare a mechanical two-mode squeezed state [27], as well as entanglement in both trapped ion [28] and by cooling these two modes. This can be achieved using two superconducting [29] systems. In other work pertaining to independent reservoirs (see, e.g. Refs. [7–9]). Here, however, mechanical entanglement, the entanglement of mechanical we show that the same goal can be achieved using just a single motion with a microwave field has been demonstrated [30]. reservoir. By making the Bogoliubov modes nondegenerate, Earlier, entanglement of phonons at the single-quantum level they will couple to different frequency components of a single was demonstrated [31], as well as the entanglement of motional reservoir (that is, the damped cavity). Since the Bogoliubov degrees of freedom of trapped ions [32]. transformation preserves the difference in number operators, Here, in Sec. II we introduce the multimode optome- this simply corresponds to a frequency difference of the two chanical system that we shall study. Section III describes mechanical oscillators. Viewed differently, one can say that approaches that one may take to reservoir engineering in we are exploiting the coherent dynamics of the mechanical this system, while in Sec. IV we describe how these strate- oscillators to effectively cool both Bogoliubov modes. Ulti- gies could be implemented in our system. In Sec. V,we mately, our protocol simply involves appropriately weighted consider the adiabatic limit, in which the cavity responds and detuned two-tone or four-tone driving of the coupled cavity rapidly to the mechanical motion, and obtain analytical mode. expressions for the entanglement, purity, and teleportation The approach we take here may be regarded as the coherent fidelity possible with the steady state. Section VI gives an feedback [16] analog of our measurement-based approach to analysis of the full linearized system, including the effects of the same task [5], and is related to a recent proposal for the counter-rotating terms and the possibility of instability. In Sec. preparation of a quantum squeezed state of a single mechanical VII, we derive a bound on the entanglement based on the cavity oscillator [17]. As compared with the measurement-based output spectrum. Section VIII provides an analysis of the three- approach for entanglement generation in Ref. [5], the purity of mode optomechanical system composed of two cavity modes the steady state achieved here is greater and the implementation coupled to a single mechanical oscillator, and demonstrates is greatly simplified. Further, in contrast to the proposal that the same physics can be realized in this system. 1050-2947/2014/89(6)/063805(17) 063805-1 ©2014 American Physical Society M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014) precisely in Eqs. (26a) and (26b), are rotating in a nontrivial manner. In particular, 1 +i(ω −)t † −i(ω −)t Xˆ + = [aeˆ a + aˆ e a 2 + + † − + + beˆ i(ωb )t + bˆ e i(ωb )t ], (5a) i +i(ω −)t † −i(ω −)t Pˆ− =− [aeˆ a − aˆ e a 2 + + † − + + beˆ i(ωb )t − bˆ e i(ωb )t ]. (5b) In any case, the two-mode squeezed state is defined as |r = Sˆ r | , FIG. 1. (Color online) (a) The system studied consists of two 2 2( ) 0 0 , where mechanical oscillators, each coupled to a common cavity (or circuit) † † Sˆ2(r) ≡ exp[r(aˆbˆ − aˆ bˆ )] (6) mode. (b) Frequencies in this system defined with respect to the cavity resonance frequency ωc. The blue lines indicate the is the two-mode squeezing operator with squeezing param- standard mechanical sidebands, at ±ωa and ±ωb. If the single-photon eter r [34]. Starting from aˆ |0,0 = 0 and bˆ |0,0 = 0it † optomechanical coupling rates (ga and gb) are equal, the required ˆ ˆ | = is straightforward to show that [S2(r)aˆS2(r)] r 2 0 and Hamiltonian (8) can be realized using only two cavity drive detunings † † that [Sˆ (r)bˆSˆ (r)] |r = 0. However, βˆ = Sˆ (r)aˆSˆ (r) and (±ω ), indicated by the vertical red lines. 2 2 2 1 2 2 m ˆ = ˆ ˆ ˆ† ˆ β2 S2(r)bS2(r), and therefore, the ground state of β1 and βˆ2 is the two-mode squeezed state with squeezing parameter II. SYSTEM AND HAMILTONIAN r. Our goal then is to engineer the driving Hamiltonian of (1) The system [see Fig. 1(a)] is composed of two mechanical such that the steady state of (2) results in the βˆi modes being oscillators, with resonance frequencies ωa and ωb, each cooled to their ground state, implying two-mode squeezing of independently, dispersively coupled (with strengths ga and the mechanical oscillators. gb, respectively) to a common cavity mode having resonance One method to achieve this is to use two cavity modes frequency ωc. The Hamiltonian is to independently cool the Bogoliubov modes [7–9]. Both † † † † † Bogoliubov modes are independently coupled to a cavity Hˆ = ωaaˆ aˆ + ωbbˆ bˆ + ωccˆ cˆ + ga(aˆ + aˆ )cˆ cˆ † mode with a beam-splitter-like interaction, i.e., βˆ cˆi + H.c. † † i + gb(bˆ + bˆ )cˆ cˆ + Hˆdrive + Hˆdiss, (1) (i = 1,2), an interaction which can be used to cool the Bogoliubov modes [35]. While such an approach can be where aˆ and bˆ denote mechanical mode lowering operators, effective, from a practical point of view it would be highly cˆ denotes the electromagnetic mode lowering operator, and advantageous if this could be achieved using only a single Hˆ accounts for driving of the electromagnetic mode. The drive cavity mode. term Hˆ accounts for dissipation, with the modes subject diss A seemingly simple way of using only a single reservoir to damping at rates γ , γ , and κ, respectively. The system a b would be to couple the cavity to one of the Bogoliubov modes, dynamics, within the usual approximations [33], is described say βˆ , and then couple βˆ to the other Bogoliubov mode by the master equation 1 1 (βˆ2) via an all-mechanical contribution to the Hamiltonian of =− Hˆ + + D + D † ˆ† ˆ + ρ˙ i[ ,ρ] γa(n¯a 1) [aˆ]ρ γan¯a [aˆ ]ρ the form β1β2 H.c., again a beam-splitter-like interaction. † This interaction will allow βˆ to be cooled (by swapping + γb(n¯b + 1)D[bˆ]ρ + γbn¯bD[bˆ ]ρ + κD[cˆ]ρ, (2) 2 quanta into βˆ ) even though it is not directly coupled to the Hˆ = Hˆ − ˆ 1 with the Hamiltonian Hdiss and the dissipative cooling reservoir.