Optomechanical transducers for quantum information processing
K. Stannigel, P. Rabl, A. S. Sørensen, M. D. Lukin, P. Zoller
OOptomechanical transducers for quantum information processing
K. Stannigel , , P. Rabl , A. S. Sørensen , M. D. Lukin , and P. Zoller , Institute for Quantum Optics and Quantum Information,Austrian Academy of Sciences, 6020 Innsbruck, Austria Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria QUANTOP, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark and Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA (Dated: October 30, 2018)We discuss the implementation of optical quantum networks where the interface between sta-tionary and photonic qubits is realized by optomechanical transducers [K. Stannigel et al. , PRL , 220501 (2010)]. This approach does not rely on the optical properties of the qubit and therebyenables optical quantum communication applications for a wide range of solid-state spin- and charge-based systems. We present an effective description of such networks for many qubits and give aderivation of a state transfer protocol for long-distance quantum communication. We also describehow to mediate local on-chip interactions by means of the optomechanical transducers that canbe used for entangling gates. We finally discuss experimental systems for the realization of ourproposal.
PACS numbers: 03.67.Hk, 42.50.Wk, 07.10.Cm
I. INTRODUCTION
The distribution of quantum information between in-dividual nodes of larger quantum networks is a key re-quirement for many quantum information applications[1], in particular for long-distance quantum communi-cation and quantum key distribution protocols [2]. Incontrast to classical networks, quantum communicationchannels must allow the distribution of entanglement,which is the essential resource that can be harnessedby further local processing within the nodes. In build-ing high-fidelity quantum channels over long distances,qubits encoded in propagating photons play a uniquerole, since they provide the only way to transmit quan-tum information over kilometer distances [3, 4]. How-ever, also on a smaller scale rapid progress is made inthe design of nano-fabricated photonic circuits for classi-cal and quantum applications [5] and compared to elec-tric circuits [6, 7] or diverse qubit shuttling techniques[8, 9], such photonic circuits could provide a fast androbust alternative for ’on-chip’ distribution of entangle-ment. Therefore, the development of coherent interfacesbetween stationary and so-called ‘flying’ photonic qubitsis one of the essential steps in realizing quantum net-works and is commonly included in the list of key crite-ria for general purpose quantum information processingplatforms [10].The fundamental operation to be carried out in a quan-tum network is the transfer of an arbitrary quantum state | ψ (cid:105) between two nodes according to | ψ (cid:105) | (cid:105) → | (cid:105) | ψ (cid:105) ,which together with local operations allows the genera-tion of inter-node entanglement. Ideas for a physical im-plementation of this basic building block have first beendeveloped in the context of atomic cavity QED. Here,qubits are stored in spin or hyperfine states of trappedatoms or ions and can be selectively excited to other elec-tronic levels where they couple strongly to light. As was (a)(b) OMT
FIG. 1. (Color online) (a) Elementary quantum network forthe realization of state transfer protocols between two nodesvia the exchange of photons propagating along an opticalfiber. Tunable qubit decay rates Γ i allow for a controlledemission of the photon wave-packet and a perfect reabsorp-tion at the second node. (b) Schematic setup for implement-ing effective qubit-light interfaces based on optomechanicaltransducers (OMTs): The mechanical resonator mediates acoupling between a solid-state qubit and the driven opticalcavity mode. As a result, qubit excitations decay with an ef-fective rate Γ into the fiber via the cavity output (see text fordetails). shown by Cirac et al. [11], this level of control can be usedto design a deterministic state transfer protocol based onthe tailored emission of a photon, which is subsequentlyreabsorbed at a second node along an optical fiber (seeFig. 1(a)).In parallel to the developments in the field of atomicqubits, substantial progress has been made on nano-fabricated solid-state qubits, including for exampleimpurity-spins [12] in various host-materials (such as sil-icon [13] or diamond [14, 15]), quantum dots [16, 17] a r X i v : . [ qu a n t - ph ] J a n or superconducting devices [18–21]. In view of the re-markable level of coherence and control that has alreadybeen achieved in such systems, the challenge is now toidentify a suitable interconnection with optical quantumchannels also for this larger class of qubits. This is, how-ever, often hindered by lack of coherent optical transi-tions or the incompatibility with light. Early suggestionsto overcome this problem have been made in the contextof hybrid systems, where, e.g., superconducting devicesare coupled to close-by atomic [22–24], molecular [25, 26]or spin systems [27–29]. An optical interface could thenin principle be realized by using, for example, the atomsas a mediator. Recently, an alternative promising routehas become available which is fully solid-state based andmakes use of the quantized motion of macroscopic me-chanical resonators. In a cavity-optomechanical setting,these resonators can coherently interact with light [30]and their control near the single-quantum level has beendemonstrated [31–34]. On the other hand, they can alsointeract with solid-state qubits via magnetic [35–37] orelectric fields [38–41], and these abilities make them nat-ural candidates for mediating indirect qubit-photon in-teractions, thereby realizing the desired qubit-light in-terface.In a recent work [42] we proposed the use of suchoptomechanical transducers (OMTs) for the implemen-tation of optical communication protocols between twosolid-state qubits, while related ideas have also been dis-cussed in the context of traveling-wave phonon to photonconverters [43]. The basic idea of an OMT is illustratedin Fig. 1(b), where a spin- or charge-based qubit is cou-pled to the motion of a mechanical resonator via mag-netic field gradients or electrostatic interactions. Theresonator in turn interacts with the field of an optical cav-ity mode via radiation pressure or optical gradient forcesas is currently experimentally explored with various dif-ferent optomechanical (OM) settings [32–34, 44–46]. Asa result, this configuration induces effective interactionsbetween the qubit and photons which do not rely on theoptical properties of the qubit system. Therefore, thisapproach in principle enables quantum communicationapplications for a large range of solid-state qubits.The purpose of the present work is two-fold. Inthe first and main part of the paper we study long-distance optomechanical quantum networks as intro-duced in Ref. [42]. Here, we present a detailed analysisof the effective qubit-fiber interactions and derive a cas-caded master equation which describes the effective mul-tiqubit dynamics in such a network. In particular, wedescribe the implementation of tunable qubit-fiber cou-plings for quantum state transfer protocols and discussthe role of thermal noise and Stokes-scattering processes.These are the two main noise sources which are typi-cally absent in analogous atomic settings. In the secondpart of the paper we then investigate the case of a singlemode optical quantum channel relevant for optical on-chip communication schemes. In this setting, the qubitsare effectively connected by a finite number of mechani- cal and optical modes and we analyze the performance ofentangling operations mediated by this type of network.The remainder of this paper is structured as follows:In Section II we present a detailed description of a singleOMT as depicted in Fig. 1(b) and discuss the resultingeffective qubit-fiber interaction together with the rele-vant noise processes in this system. In Sec. III we extendthe model to an arbitrary number of nodes and derivea cascaded master equation for this multiqubit network.Further, we give a protocol for state-transfer betweentwo nodes and assess its performance. In Sec. IV wethen study an on-chip setting where the individual nodesare coupled by optical resonators with a discrete spec-trum and analyze the prospects for performing entanglinggates between the qubits. Finally, we discuss potentialexperimental realizations of the proposed OMT in Sec. Vand close with concluding remarks in Sec. VI. II. AN OPTOMECHANICAL INTERFACEBETWEEN STATIONARY QUBITS AND LIGHT
The aim of this work is to develop a theoreticalmodel for the description of optical quantum networkswhere the interconversion between stationary and pho-tonic qubits is realized by an OMT at each node. As afirst step we will investigate in this section the dynamicsof a single node of such a network and derive a simplifiedmodel for the effective qubit-light interactions mediatedby the OM device.
A. Model
We consider a single node of an optical network asschematically shown in Fig. 1(b), where the coupling be-tween a qubit and an optical fiber is mediated by anOMT. The total Hamiltonian for this system is H = H node + H fib + H cav-fib + H env , (1)where H node describes the coherent dynamics of the qubitand the OM device, H fib the free evolution of the opticalfiber modes, and H cav-fib the coupling between cavity andfiber. The last term in Eq. (1), H env , summarizes alladditional interactions with the environment leading todecoherence as specified below.For a general discussion we assume complete local con-trol over the qubit, which is taken to be encoded in twostates | (cid:105) and | (cid:105) of an isolated quantum system, andwhich interacts with the quantized motion of a mechan-ical resonator, e.g., via magnetic field gradients [36] orelectrostatic interactions [38–41]. The mechanical res-onator in turn is coupled to the field of an optical cavitymode via radiation pressure or optical gradient forces.In terms of the usual Pauli operators σ z = | (cid:105)(cid:104) | − | (cid:105)(cid:104) | and σ − = | (cid:105)(cid:104) | for the qubit, and the bosonic operator b for the mechanical mode, the Hamiltonian H node canbe written in the form ( (cid:126) = 1) H node = ω q σ z + λ σ − b † + σ + b ) + H om . (2)Here, ω q is the qubit’s energy splitting, λ the strength ofthe qubit-resonator coupling and H om is the Hamiltonianfor the coupled OM system. We point out that dependingon the physical implementation, the Jaynes-Cummingstype interaction assumed in Eq. (2) may emerge only asan effective description of an underlying driven two- ormultilevel system. In this case, also ω q is an effectiveparameter which can differ from the bare frequency scalesin the system. (This will be discussed in more detailin Sec. V where we consider the implementation of theHamiltonian (2) for the specific examples of spin andcharge qubits.)The OM system consisting of a mechanical resonatorcoupled to a single optical cavity mode is described bythe Hamiltonian H om = ω r b † b + ω c c † c + g c † c ( b + b † ) (3)+ i E ( t ) e − iω L t c † − i E ∗ ( t ) e iω L t c , where ω r is the mechanical vibration frequency and c isthe bosonic annihilation operator for the optical modeof frequency ω c . The quantity g = a ∂ω c ∂x is the singlephoton OM coupling and corresponds to the optical fre-quency shift per zero point motion a . In the last lineof Eq. (3) we have included an additional external laserfield of frequency ω L , which coherently excites the cav-ity field. As detailed below, this driving field will allowus to enhance and also to control the OM coupling byadjusting the slowly varying driving strength E ( t ). Wefinally stress that the optical cavity might support ad-ditional degenerate modes which we ignored in writingEq. (3). As will be discussed below, this is valid as longas the OM interaction does not cause scattering betweenthe degenerate modes. B. Quantum Langevin equations
The optical cavity of the OM system is coupled to themodes of an optical fiber, which in the multinode set-ting considered below will serve as our quantum commu-nication channel. The fiber supports many continua ofmodes described by bosonic operators a σω , normalized to[ a σω , a † σ (cid:48) ω (cid:48) ] = δ σ,σ (cid:48) δ ( ω − ω (cid:48) ), where σ labels, e.g., propa-gation direction or polarization. The free fiber Hamilto-nian is then simply given by H fib = (cid:80) σ (cid:82) d ω ω a † σω a σω .We assume that the OM system is located at position z = 0 along the fiber to which it is coupled according to H cav-fib = i (cid:88) σ (cid:90) ∞ d ω √ π (cid:16)(cid:112) κ σ ( ω ) c a † σω − h.c. (cid:17) , where the coupling constants (cid:112) κ σ ( ω ) are defined in thisway for later convenience. For definiteness, we consider the case where the cavity mode is side-coupled to a fiberwith left- and right-moving modes of a single relevantpolarization ( σ = R, L ). In this case, the cavity-fibercoupling is naively estimated to be proportional to theoverlap integral (cid:113) κ R,L ( ω ) ∼ (cid:90) ∆ z − ∆ z dz (cid:48) h ( z (cid:48) ) e ∓ iωz (cid:48) /c , (4)where h ( z ) is the mode function of the evanescent cav-ity field over the relevant interaction region of length2∆ z (see, e.g., Refs. [47, 48] for a detailed analysis ofthe coupling to a whispering-gallery mode cavity). For∆ z (cid:38) λ c , where λ c is the vacuum cavity wavelength,we can roughly distinguish between two cases. For astanding-wave cavity mode h ( z ) ∼ cos( ω c z/c ) and pho-tons from the cavity will be emitted into both direc-tions ( κ R ≈ κ L ), while for a running-wave mode suchas h ( z ) ∼ e iω c z/c , a preferred emission into a specific di-rection can be achieved (in this case κ R (cid:29) κ L ). Althoughin principle both configurations could be relevant for dif-ferent quantum communication applications, we will forconcreteness focus in the following on the case where c describes a circulating running-wave cavity mode thatcouples to the fiber according to H cav-fib = i (cid:112) κ f (cid:16) cf † R ( z = 0) − c † f R ( z = 0) (cid:17) , (5)where f R ( z ) = √ π (cid:82) ∞ d ω a Rω e iωz/c is the right-propagating fiber-field. The coupling of the cavity to thiscontinuum of modes will lead to an irreversible decay ofthe cavity field at the rate κ f ≡ κ R ( ω c ).To proceed, we eliminate the time-dependence inEq. (3) by transforming all photonic operators to a framerotating at the laser frequency. This amounts to replac-ing H om by¯ H om = ω r b † b + ∆ c c † c + g c † c ( b + b † ) (6)+ i (cid:0) E ( t ) c † − E ∗ ( t ) c (cid:1) , and H fib by ¯ H fib = (cid:80) σ (cid:82) d ω ∆ ω a † σω a σω , where ∆ c = ω c − ω L and ∆ ω = ω − ω L are the detunings of the cavityand fiber modes from the laser frequency, respectively.We then use a standard Born-Markov approximation toeliminate the fiber modes [49] and describe the resultingdissipative dynamics in terms of a quantum Langevinequation (QLE) for the cavity field,˙ c ≈ − i [ c, H node ] − κc − (cid:112) κ f f in ( t ) − √ κ f ( t ) . (7)Here, we have introduced a total decay rate κ = κ f + κ , where κ accounts for additional intrinsic losses ofthe optical cavity mode and f ( t ) is the associated noiseoperator. We have further defined operators f in ( t ) := f R (0 − , t ) and f out ( t ) := f R (0 + , t ) which obey the input-output relation f out ( t ) ≈ f in ( t ) + (cid:112) κ f c ( t ) , (8)and we specify vacuum white noise statistics for the fiberinput, i.e., [ f in ( t ) , f † in ( t (cid:48) )] = δ ( t − t (cid:48) ) and (cid:104) f in ( t ) f † in ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ), as well as for f ( t ). In addition, the mechanicalresonance generally has a finite intrinsic width and is con-nected via its support to a thermal bath of temperature T . This is commonly modeled by the QLE˙ b = − i [ b, H node ] − γ m b − √ γ m ξ ( t ) , (9)where γ m / ξ ( t ) with statistics (cid:104) ξ ( t ) ξ † ( t (cid:48) ) (cid:105) = ( N m + 1) δ ( t − t (cid:48) )and [ ξ ( t ) , ξ † ( t (cid:48) )] = δ ( t − t (cid:48) ). Here, N m is given by theBose occupation number for a mode of frequency ω r . Inthe high-temperature case, where k B T (cid:29) (cid:126) ω r and hence N m (cid:29)
1, the relevant thermal decoherence rate will turnout to be γ m N m ≈ k B T (cid:126) Q , where Q = ω r /γ m is the qualityfactor of the mechanical resonance and typically Q (cid:29) γ m → γ m N m constant. C. Linearized equations of motion
In typical experiments the OM coupling g is too smallto allow coherent interactions between the mechanicalsystem and individual cavity photons. Therefore, toachieve an appreciable and also tunable coupling we con-sider the case of a strongly driven OM system where theeffective coupling between mechanics and light is ampli-fied by the coherent field amplitude inside the cavity.Starting from the QLEs (7),(9), we perform a unitarytransformation c → c + α and b → b + β such that thec-numbers α , β describe the classical mean values of themodes and the new operators c and b represent quantumfluctuations around them. We require any classical (thatis c-number) contributions to the transformed QLEs tovanish, which yields the following system for the classicalresponse:˙ α = − ( i ∆ c + κ ) α − ig ( β + β ∗ ) α + E ( t ) , (10a)˙ β = − ( iω r + γ m / β − ig | α | . (10b)For E ( t ) = const and γ m → β = − g | α | /ω r with α determined from α = E i ∆ c + κ − i g | α | /ω r . (11)This relation is still approximately valid if E ( t ) is slowlyvarying compared to the characteristic response time.This means, e.g., ˙ E ( t ) / E ( t ) (cid:28) κ, ∆ c , ω r for the case | α | g (cid:28) ∆ c , ω r which is of relevance below. Within theselimits, a desired temporal profile α ( t ) can be directly re-lated to the applied laser power and phase by means ofEq. (11). For more rapid variations the system (10) hasto be integrated exactly to capture all retardation effects. In the strong driving regime where | α ( t ) | (cid:29) g relativeto those of order g | α | and g | α | . The result of thisprocedure is equivalent to replacing ¯ H om by the linearizedOM Hamiltonian H linom ( t ) = ω r b † b + ˜∆ c ( t ) c † c + ( G ( t ) c † + G ∗ ( t ) c )( b + b † ) . (12)Here, we have introduced the laser-enhanced OM cou-pling G ( t ) = g α ( t ), which describes interconversionbetween phonons and photons, as well as the renor-malized cavity detuning ˜∆ c ( t ) = ∆ c + g ( β + β ∗ ) =∆ c − | G ( t ) | /ω r . Although the latter shift is smallin the regime of interest identified below, one has totake it into account when varying G ( t ). We will dropthe tilde in what follows and also note that we have ig-nored a contribution λ/ β ( t ) σ + + h.c.) to H node , whichcan be compensated by local fields acting on the qubit.Finally, if the cavity supports many degenerate modes c µ that couple to the mechanical resonator according to g (cid:80) µ c † µ c µ ( b + b † ), then only the OM coupling to thedriven mode is enhanced. The others may then be ne-glected due to g (cid:28) | G | , which justifies our single-modetreatment. However, this argument breaks down if thereis phonon-assisted scattering between the modes, i.e., foran interaction of the form g (cid:80) µ,ν c † µ c ν ( b + b † ).In summary, the OM system can be described by anenhanced, linear coupling | G | (cid:29) g in H linom between theresonator and an undriven cavity mode. The qubit dy-namics is determined by the Heisenberg equations of mo-tion ˙ σ − = − iω q σ − + i λ σ z b , (13a)˙ σ z = − iλ ( σ + b − b † σ − ) , (13b)and the linearized OM dynamics (QLEs (7),(9) with H om → H linom ) can be written compactly by introducingthe vectors v = ( b, c, b † , c † ) T and S = ( σ − , , − σ + , T :˙ v ( t ) = − M v ( t ) − i λ S ( t ) − N ( t ) . (14)Here, the drift matrix M describes the response of theOM system and is explicitly given by M = i ω r − i γ m G ∗ ζGG ∆ c − iκ ζG − ζG ∗ − ω r − i γ m − G − ζG ∗ − G ∗ − ∆ c − iκ , (15)where ζ = 1 gives the full linearized OM couplingand ζ = 0 corresponds to the often-applied rotatingwave approximation (RWA), which is valid for κ, | G | (cid:28) ∆ c ≈ ω r and renders the coherent part of the dy-namics excitation-number- or energy-conserving. Thenoise sources driving the OM system are summarized in N ( t ) = (cid:112) κ f I ( t ) + R ( t ), where I ( t ) = (0 , f in , , f † in ) T represents the fiber input-field and the intrinsic noise iscontained in R ( t ) = ( √ γ m ξ, √ κ f , √ γ m ξ † , √ κ f † ) T .The statistics of these noise vectors follow directly fromthe definitions in the previous subsection and can alsobe found in App. B. We finally note that the system de-scribed by M generally exhibits self-oscillations for bluedetuning ∆ c < c > | G | > ( κ + ∆ c ) ω r / c (see, e.g., Ref. [50]). However, in this work we will onlyconsider parameters where these phenomena do not oc-cur. D. Effective qubit-light interface
The dynamics of a single node as described by Eqs. (13)and (14) allows for a coherent conversion of a qubit ex-citation into a photon propagating in the optical fiber,as is indicated in Fig. 2(a): First, the excitation is trans-ferred to the mechanical resonator at a rate ∼ λ , fromwhere it is then up-converted by the excitation-numberconserving terms of the OM interaction into a cavity pho-ton with a rate ∼ G , and subsequently emitted into thefiber at a rate ∼ κ f . At the same time, this transfer isaffected by decoherence in form of mechanical dissipa-tion and photon loss, thereby degrading the fidelity ofthe interface under realistic conditions. Our goal is nowto derive a simplified model for the interface which cap-tures all of these aspects in an effective description forthe qubit-fiber coupling only.To proceed, let us first look at the dynamics of thelinearized OMT, which is fully characterized by the driftmatrix M defined in Eq. (15) (see also Ref. [51] for anextensive discussion). It has two independent eigenval-ues which we identify with iω ± + γ ± , where ω ± > γ ± the decay rates of the associatedOM normal modes (the remaining eigenvalues are simplycomplex conjugates). We take G to be real and positivefor simplicity and focus on the case ∆ c ≈ ω r relevant forresonant excitation transfer, where we can distinguishbetween two regimes: For weak OM coupling, G < κ/ ω − ≈ ω r , ω + ≈ ∆ c . How-ever, due to a small admixing between the modes theoptical cavity provides an additional decay channel forthe mechanical system and can lead to significant mod-ifications of the mechanical damping rate. In the limit G (cid:28) κ it is given by γ − = γ m / G /κ (see left panel ofFig. 2(b). In the opposite regime of strong OM coupling, G > κ/
2, resonator and cavity hybridize and the OMsystem exhibits a normal mode splitting ( ω ± ≈ ω r ± G ).This hybridization also implies that both modes decaywith the same rate γ ± ≈ κ/ γ op , which for arbitrary parameters is definedas γ op = min { γ + , γ − } and reproduces γ op ≈ G /κ and γ op ≈ κ/ γ op is therate at which mechanical excitations are converted intotraveling photons, it sets a maximal transfer rate for theOMT. We expect γ op (cid:29) γ m / γ m → λ is small com-pared either to γ op or to the detuning from the near-est normal mode, that is, the regime where λ (cid:28) max( γ op , min( | ω ± − ω q | )). In this limit, the dynam-ics of the OM system is fast compared to the qubit-resonator coupling and can hence be adiabatically elim-inated, yielding the desired effective description of theOMT. To this end, we formally integrate Eq. (14) to ob-tain v ( t ) = v free ( t ) − i λ (cid:90) t −∞ d s e − M ( t − s ) S ( s ) , (16)where we have neglected transients and v free ( t ) = − (cid:82) t −∞ d s e − M ( t − s ) N ( s ) is the OM steady state solutionin the absence of the qubit (see Eq. (14) for λ → λ replace the qubit operators by their freeevolution, i.e., σ − ( s ) = σ − ( t ) e iω q ( t − s ) and re-insert theresult into Eq. (13). By keeping only the resonant terms,we obtain effective QLEs for the qubit operators,˙ σ − = − (cid:20) i ( ω q + ∆ ) + Γ2 (cid:21) σ − − √ Γ σ z F in ( t ) , (17a)˙ σ z = − Γ( + σ z ) + √ (cid:16) σ + F in ( t ) + F † in ( t ) σ − (cid:17) , (17b)with noise operators F in ( t ) = − i (cid:112) λ / b free ( t ). FromEq. (8) we further obtain an effective input-output rela-tion for the fiber field, f out ( t ) ≈ f in ( t ) + √ κ f c free ( t ) − (cid:112) η Γ e iφ σ − ( t ) . (18)In Eqs. (17)-(18) we have introduced the effective decayrate Γ and frequency shift ∆ by the relationsΓ = λ { A ( ω q ) } , ∆ = λ { A ( ω q ) } , (19)where A ( ω ) = ( M − iω ) − is the OM response matrix.Further, η = κ f /κ is the branching ratio resulting fromintrinsic cavity decay and φ = arg { iA ( ω q ) } . Note thatin writing Eq. (18) we have neglected small correctionsrelated to finite γ m , as well as non-RWA corrections andcounter-rotating terms ∝ σ + ( t ).The effective equations of motion (17) together withthe input-output relation (18) are familiar from atom-light interactions [49] and describe a two level systemwhich is directly coupled to an optical fiber with an ef-fective decay rate Γ. The OM system which mediates thisinteraction has disappeared from the dynamics, but still (a) (c) qubit resonator cavity fiberOMT (b) OMT OMT (d)
FIG. 2. (Color online) (a) Level scheme illustrating the decay of a qubit excitation into the fiber (only processes correspondingto ζ = 0 are show). The dashed box represents the OMT and wavy arrows indicate mechanical diffusion and intrinsic cavity loss.(b) Level structure of the OMT as seen by the qubit. Wavy arrows indicate decay into the optical fiber at γ op ≈ min { G /κ, κ/ } in the regimes of weak (left) and strong (right) OM coupling. (c) Effective qubit-decay Γ as a function of the linear OM coupling G and the qubit frequency ω q as given by Eq. (19). We have used κ = 0 . ω r and γ m = 0, and ∆ c = ω r is satisfied at G = 3 κ/ G = 3 κ/ c = ω r . The dashed lines in (c),(d) indicatepossible paths for tuning Γ between ∼ ∼ λ / κ (see text). determines Γ, ∆ , φ , which can thus be controlled by ap-propriately adjusting OM parameters. This can be seenin more detail from the expression for the decay rate,Γ ≈ λ G κ/ G + (∆ c − ω q )( ω q − ω r )) + κ ( ω q − ω r ) , (20)where we have assumed ζ = 0 and γ m →
0. A plot of thefull expression from Eq. (19) is given in Fig. 2(c) for thecase ω r ≈ ∆ c and one clearly observes the underlyingmode structure of the OM system as discussed above.The decay Γ is suppressed if the qubit is detuned fromthe OM eigenmodes, while under resonance conditions, ω q ≈ ω ± , it scales as Γ ∼ λ /γ op , and we obtain Γ ≈ λ κ/ G and Γ ≈ λ / κ in the weak and strong couplingregime, respectively.As we will discuss in more detail in Sec. III C below, itis essential for the implementation of deterministic quan-tum state transfer protocols within quantum networks tohave control over the shape of the emitted photon, whichcan be accomplished by considering a time-dependent de-cay rate Γ → Γ( t ). To achieve this, we may make use ofthe fact that in principle any of the OM parameters en-tering Γ could be varied in an experiment, that is, weconsider Γ( t ) = Γ( G ( t ) , ∆ c ( t ) , ω q ( t ) , ... ). The effectiveMarkovian description of the qubit dynamics and the al-gebraic relation in Eq. (19) remain valid as long as theOM parameters are tuned slowly relative to the time-scale of the effective resonator decay 1 /γ op , such that˙ G/G (cid:28) γ op , etc. In this sense we may adiabaticallyadjust the effective qubit decay rate and hence acquirecontrol over the generated signal photon. Examples . As a specific example we consider tuningthe OM coupling along a path as indicated by the dashedline in Fig. 2(c). Here, G ( t ) varies between 0 and ∼ κ/ ω q ≈ ω r − κ/ ω r ≈ ∆ c . This corresponds to tuning the lower nor-mal mode in and out of resonance with the qubit, suchthat Γ can be varied within a certain range [Γ min , Γ max ].For G → δ = | ω q − ω r | from the resonator mode to which it couples and there-fore, the effective decay rate is suppressed according toΓ ∼ λ γ op / δ ≈ λ G / κδ , which realizes Γ min → min ≈ λ γ m / δ , which, however, is negligible forthe parameters considered below. By increasing G up to ∼ κ/ max = λ / κ isachieved. Within these limits, a desired time-dependenceof the effective decay Γ( t ) can thus directly be translatedinto a profile G ( t ) via Eq. (19).Similarly, we can vary the cavity frequency along thedashed line in Fig. 2(d), where we chose ω q ≈ ω r − κ/ G varies according to Eq. (11),which has been included in the plot. We choose G = 3 κ/ c = ω r , such that Γ max for this path is the sameas for the previous one. Starting from this point, theeffective decay can be turned off by increasing the cavitydetuning as far as needed, with a residual decay due tothe resonator as before. E. Noise and imperfections
As already mentioned above, the OMT does not onlymediate the desired coherent interactions between thequbit and the optical channel, but under realistic con-ditions also adds noise. In the effective QLEs (17) thisnoise appears first of all in form of the operator F in ( t ),which generally represents an effective non-vacuum inputthat heats the qubit. Second, noise processes also affectthe out-field f out ( t ) given in Eq. (18) where on the onehand the term c free ( t ) represents excess photons whichare emitted into the fiber independently of the qubit stateand on the other hand η < F in ( t ) which ap-pears in the QLEs (17) for the qubit. From its definitionin terms of the OM drift matrix M we see that two-timecorrelations of F in ( t ) decay with γ op (cid:29) Γ. Therefore, onthe scale of the effective qubit dynamics, it can be con-sidered as nearly white noise and is characterized by aneffective occupation number N = 2 Re (cid:90) ∞ d t (cid:104) F † in ( t ) F in (0) (cid:105) e − iω q t . (21)This expression can be readily evaluated in Fourier spaceas shown in App. B and for γ m → N ≈ γ m N m κ κ + (∆ c − ω q ) G + κ + (∆ c − ω q ) c ω q . (22)The first contribution (given for ζ = 0) is due to thethermal noise of the resonator’s environment and roughlyscales as γ m N m /γ op . The second contribution to N de-scribes heating due to non-energy-conserving terms in theOM coupling, ∼ Gc † b † , and accounts for photons scat-tered from the strong driving beam into the cavity whilesimultaneously creating a mechanical excitation. We seethat for a qubit on resonance with an OM normal mode( ω q ≈ ∆ c ± G ) the conditions for small thermal noiseare γ m N m (cid:28) γ op , while to reduce the Stokes scatteringevents we need κ, G (cid:28) ∆ c , ω r . The latter conditions de-scribe the so-called resolved side-band regime and there-fore, we conclude that the conditions for a low noise OMTare equivalent to the requirements for OM ground statecooling [31].The second type of noise is the contamination of thecavity output with noise photons originating from up-converted thermal noise or Stokes scattering events, asdescribed by the term ∝ c free ( t ) in Eq. (18). To ana-lyze the effect of these excess photons let us in a firststep imagine that at time t = 0 the qubit is preparedin the excited state and that we record the number ofphotons emitted into the fiber during a time-interval T ∼ / Γ. The observed photon number is given by N ( T ) = (cid:82) T d t (cid:104) f † out f out (cid:105) ( t ) and for simplicity, we assumethat the qubit operator evolves according to σ − ( t ) = e − (Γ / iω q ) t σ − (0), while being uncorrelated with c free ( t ).We then obtain N ( T ) = 1 − e − Γ T + N ex ( T ) with the num-ber of excess photons given by N ex ( T ) = 2 κ f (cid:90) T d t (cid:104) c † free c free (cid:105) ( t ) , (23)which simply yields N ex ( T ) = 2 κ f T (cid:104) c † free c free (cid:105) (0) fortime-independent OM parameters. For T = Γ − , κ = 0, γ m →
0, and ω r = ∆ c this quantity can be estimated to be N ex (Γ − ) ≈ γ m N m / Γ + ( κ/ Γ)( G /ω r ), and for aqubit resonant with an OM mode we may use Γ ∼ λ /γ op to obtain the scaling N ex (Γ − ) ∼ γ λ (cid:18) γ m N m γ op + κγ op G ω r (cid:19) . (24)While the terms in parentheses are small under the lownoise conditions identified above, the prefactor is muchlarger than one due to the condition γ op (cid:29) λ underpin-ning the adiabatic elimination. The number of excessphotons may thus exceed one and make the signal pho-ton difficult to detect in general. However, this estimatedid not take into account the fact that the qubit emitsin a bandwidth Γ around ω q , while the OM noise flooris spread over a bandwidth γ op . Therefore, we expectthat the excess photons can be substantially reduced to N ex (Γ − ) Γ /γ op by appropriate filtering, which cancelsthe large factor γ /λ appearing in Eq. (24). Indeed,we find below that the OMT at a second node providesexactly this filtering. III. QUANTUM NETWORKS
With the concept of the OMT laid out in the previoussection we are now ready to consider a quantum network,where the OMTs in the various nodes serve to link thequbits to a common optical fiber, as depicted in Fig. 1.To obtain an effective description of the multiqubit dy-namics, we proceed as before by linearizing the OM cou-plings and eliminating the coupled OMTs. We will thendiscuss how to realize a state-transfer within a two-nodenetwork. The Hamiltonian for the complete network is astraight-forward generalization of Eq. (1) and reads: H = (cid:88) i (cid:0) H i node + H i cav-fib + H i env (cid:1) + H fib , (25)where H i node is given by Eq. (2) with appropriate nodeindices added and H i env summarize decoherence effects.We assume that the output of a given cavity is routedto the next, which is, e.g., realized by side-coupling allcavities to the right-moving field in the fiber as describedby H i cav-fib = i (cid:112) κ if (cid:16) c i f † R ( z i ) − c † i f R ( z i ) (cid:17) . (26)Here, f R ( z i ) is the right-moving field operator definedafter Eq. (5) and the nodes are located at positions z i < The steady state cavity occupation can be expressedas (cid:104) c † c (cid:105) free = (cid:82) d ω π C ( ω ), with the matrix C ( ω )defined in App. B. Neglecting non-RWA correctionsto the thermal contribution, we obtain (cid:104) c † c (cid:105) free ≈ (cid:82) d ω π (cid:0) γ m N m | A ( ω ) | + 2 κ | A ( ω ) | (cid:1) which we evaluateto leading order in G /ω r [52]. z i +1 along the fiber, with c i denoting the driven cavitymode at node i .In a first step, we eliminate the modes of the opticalfiber in a procedure similar to the one in Sec. II B. Foreach node, we obtain an input-output relation of the form(8) and in addition, we find a cascaded coupling wherethe output of a given cavity drives the subsequent one[49, 53]:˙ c i ( t ) ≈ i [ H i node , c i ] − κ f c i ( t ) − √ κ f f in ,i ( t ) ,f in ,i ( t ) ≈ f out ,i − ( t − τ i,i − ) e iω L τ i,i − . Here, H i om in H i node is replaced by the analogue of Eq. (6), τ ij = ( z i − z j ) /c is the propagation time between thenodes and f in , ( t ) = f in ( t ) is a vacuum white-noiseoperator. To simplify notation, we assume quantitieswithout index to be the same for all nodes and definerotated and retarded photonic operators, i.e., c i ( t ) → e iω L z i /c c i ( t − z i /c ) and similarly for f in ,i , f out ,i , E i . Allother quantities are redefined as, e.g., b i ( t ) → b i ( t − z i /c )and we can then shift the time in the QLEs for node i ac-cording to t → t + z i /c . Including local dissipative effectsas above, the full OM QLEs for λ → b i ( t ) = i [ H i node , b i ] − γ m b i − √ γ m ξ i ( t ) , (27)˙ c i ( t ) = i [ H i node , c i ] − κc i − √ κ f f in ,i ( t ) − √ κ f ,i ( t ) , where the cascaded coupling is encoded in the input-output relation f in ,i +1 ( t ) = f out ,i ( t ) = f in ,i ( t ) + √ κ f c i ( t ) . (28)Now, we can treat the classical laser drives by makingthe replacements c i → c i + α i and b i → b i + β i , andfrom the requirement of vanishing classical forces on theshifted operators we obtain the system˙ α i = − ( i ∆ c + κ ) α i − ig ( β i + β ∗ i ) α i + E eff i , (29a)˙ β i = − ( iω r + γ m / β i − ig | α i | . (29b)Here, E eff i = E i − κ f (cid:80) j
1, we may linearize the OM coupling as beforeand the resonators and cavities are then described by thelinear system˙ v i ( t ) = − M i v i ( t ) − R i ( t ) − √ κ f I i ( t ) . (30) where v i = ( b i , c i , b † i , c † i ) T and M i is defined asin Eq. (15). The vector I i ( t ) = (0 , f in ,i , , f † in ,i ) T contains the fiber-input of node i , while R i ( t ) =( √ γ m ξ i , √ κ f ,i , √ γ m ξ † i , √ κ f † ,i ) T summarizes localnoise inputs. A. Ideal effective qubit network
We derive an effective description of the qubits onthe basis of the assumption that their coupling to theresonators is slow compared to the OM dynamics (cf.Sec. II). This allows us to eliminate the OM degrees offreedom and most importantly, we expect the qubitsto inherit the cascaded nature of the coupling presentbetween the cavities. We have seen before that a sin-gle OMT mediates a qubit-fiber coupling ∼ √
Γ, andwe hence expect the effective cascaded coupling betweenqubits i and j due to emission and re-absorption of a pho-ton to scale as (cid:112) Γ i Γ j . The general elimination procedureis presented in App. A and B and for later conveniencein deriving the state transfer protocol and discussing im-perfections, we present the results in a master equation(ME) formulation for the reduced qubit density operator µ . To focus on the key points, we postpone the generalsituation to the next subsection and first discuss the ide-alized case in which we (i) take the OM coupling in RWA( ζ = 0) and (ii) assume all additional decay channels andnoise sources to be zero. The general effective ME (B1)then assumes a very simple form:˙ µ = L µ + L ideal µ . (31)Here, L µ = − i (cid:80) i ˜ ω iq [ σ iz , µ ] / ω iq = ω q + ∆ ,i , where the shifts∆ ,i are defined below. The second term describes theideal cascaded interaction [49] and reads L ideal µ = − iH eff µ + iµH † eff + S µ S † , (32) H eff = − i S † S − i (cid:88) i>j (cid:112) Γ i Γ j (cid:0) σ + i σ − j − σ − i σ + j (cid:1) , (33)where S = (cid:80) i √ Γ i σ − i is the collective jump operator andthe single-qubit decay rates Γ i are given below. The first,anti-hermitian term in the effective Hamiltonian H eff en-sures that L ideal is of Lindblad form with a single jumpoperator S and the second, hermitian term describes thecoherent part of the fiber-induced dynamics. In writingthe above equations, we have further absorbed phases e − iθ i σ − i → σ − i into the qubit operators to simplify nota-tion . Note that the two-qubit terms in the hermitian The two-qubit terms are determined by the cascaded coupling and non-hermitian parts of H eff have the same magnitudeand a specific phase-relation, such that they actually in-terfere to produce the cascaded coupling. This is moreevident when rewriting H eff as H eff = − i (cid:88) i Γ i σ + i σ − i − i (cid:88) i>j (cid:112) Γ i Γ j σ + i σ − j . (34)Apart from generating single-qubit decays, this Hamil-tonian may transfer excitations from qubit j to a sub-sequent qubit i > j , but not in the other direction. Asexpected, these cascaded interactions take place on thescale (cid:112) Γ i Γ j and we will exploit them to perform a statetransfer below.The effective quantities entering the above ME aregiven by the dynamical properties of the underlyingmultinode OM system. To determine the cascaded dy-namics we use the general relations from App. B,Γ i + 2 i ∆ ,i = λ X ii ( ω q ) , J ij = λ X ij ( ω q ) , (35)where the cascaded coupling J ij determines the two-qubitterms in H eff . The correlation function X ij ( ω ) is gener-ally defined as X ij ( ω ) = (cid:90) ∞ d τ (cid:104) [ b i ( τ ) , b † j (0)] (cid:105) free e iωτ , (36)where the subscript “free” denotes the expectation valuein the steady state of the OM dynamics in the absenceof the qubits as described by Eqs. (28) and (30). It canbe evaluated using the quantum regression theorem [49]with the initial condition (cid:104) [ b i (0) , b † j (0)] (cid:105) free = δ ij . Defin-ing the OM response matrix of node i as A i ( ω q ) =( M i − iω ) − , we find in general that X ii ( ω ) = A i ( ω )and also that J ij = 0 for i < j , which yields the uni-directionality of the coupling. In contrast, for i > j thecorrelation function describes the propagation of a signalfrom node j to node i including various filtering and ab-sorption effects taking place at intermediate nodes (seeApp. B for details). For the idealized case considered inthis section, these effects only amount to a phase and weobtain | J ij | = (cid:112) Γ i Γ j as expected. B. Full master equation
The ideal picture presented in the previous subsectionhas to be refined in order to discuss the impact of thevarious imperfections. As has already been discussed J ij introduced in Eq. (35). In OM RWA and for γ m = κ = 0 itcan be written as J ij = (cid:112) Γ i Γ j exp[ i ( θ i − θ j )] for i > j , where θ i = φ i + (cid:80) i − n =1 φ n with φ n = arg { iA n ( ω q ) } and G i taken tobe real for simplicity. This structure allows to absorb e − iθ i σ − i → σ − i . FIG. 3. (Color online) Accumulation of noise in a cas-caded network where. Thick, wavy arrows denote noise pho-tons (red, single-headed) and noise phonons (green, double-headed). Each node emits noise photons generated bynon-RWA scattering events or by up-conversion of thermalphonons. At successive nodes, these noise photons may bedown-converted and lead to decoherence of the qubits (thinwavy arrows). in Sec II, the effects of the thermal noise and non-RWAcorrections for a single node are two-fold: They lead toheating of the attached qubit as well as contamination ofthe OMT’s output with noise photons. In a multiqubitnetwork, these noise photons naturally affect successivenodes and when moving down the fiber we expect an ac-cumulation of noise, as illustrated in Fig. 3. The generalME for an N -node network capturing all these imperfec-tions is derived in App. B and reads:˙ µ = 12 (cid:88) i (cid:110) − i ˜ ω iq [ σ iz , µ ] + Γ i D [ σ − i ] µ (cid:111) (37) − (cid:88) i>j (cid:0) J ij (cid:2) σ + i , σ − j µ (cid:3) + J ∗ ij (cid:2) µσ + j , σ − i (cid:3)(cid:1) + 12 (cid:88) i Γ i N i (cid:0)(cid:2)(cid:2) σ + i , µ (cid:3) , σ − i (cid:3) + (cid:2) σ + i , (cid:2) µ, σ − i (cid:3)(cid:3)(cid:1) + (cid:88) i (cid:54) = j D ij (cid:2)(cid:2) σ + j , µ (cid:3) , σ − i (cid:3) , where D [ a ] µ = 2 aµa † − a † a µ − µ a † a is a Lindblad termwith jump operator a . The first two lines contain thefiber-induced decay and cascaded coupling of the qubitsas discussed in the previous subsection, with the associ-ated rates Γ i and J ij given in Eq. (35). The renormal-ized qubit frequencies are given by ˜ ω iq = ω q + ∆ i , with∆ i = ∆ ,i + ∆ th ,i , where the second, thermal contribu-tion ∆ th ,i is negligible for our purposes (see the discussionin App B). In contrast, the last two lines describe addi-tional decoherence processes, characterized by effectivebath occupation numbers N i and the rates of correlateddiffusion D ij . These quantities are given byΓ i N i = λ Y ii ( ω q ) , D ij = λ Y ij ( ω q ) , (38)where Y ij ( ω ) is determined by the OM dynamics accord-ing to Y ij ( ω ) = (cid:90) ∞−∞ d τ (cid:104) b † i ( τ ) b j (0) (cid:105) free e − iωτ , (39)0and the property Y ij ( ω ) = Y ∗ ji ( ω ) ensures that the N i arereal. Finally, we note that the explicit Lindblad form ofEq. (37) is in general not very illuminating and also te-dious to compute except for special cases as given belowor in the previous subsection. Nevertheless, it is ensuredby the property (cid:104) b † i ( τ ) b j (0) (cid:105) free = (cid:104) b † i (0) b j ( − τ ) (cid:105) free of theOM correlation functions that the qubit density opera-tor µ remains positive semi-definite under the evolutiondescribed by Eq. (37) (see Ref. [54]).In the previous subsection, we assumed the idealizedconditions of vanishing intrinsic decays κ = γ m = 0,and ζ = 0. In this limit, we obtain N i = 0, D ij = 0,and | J ij | = (cid:112) Γ i Γ j , which allows one to rewrite the resultEq. (37) in the simple form of Eq. (31). We will now relaxthese restrictions to discuss the influence of the variousimperfections and give the conditions under which theireffects are small.
1. Intrinsic cavity decays
For non-vanishing intrinsic cavity decay κ only a por-tion η = κ f /κ of the qubit excitation is actually emittedinto the fiber, while the rest gets lost to other channels.For the simple case of two nodes and ζ = 0, the MEcan be written exactly as the sum of a reduced cascadedinteraction plus additional on-site decays:˙ µ = L µ + η L ideal µ + (1 − η ) (cid:88) i =1 Γ i D [ σ − i ] µ , (40)with Γ i given in Eq. (20). For more than two nodes theME is less simple, since every cavity provides an ad-ditional decay channel with an associated jump oper-ator involving all previous qubits. Therefore, we onlynote that the cascaded couplings decay exponentiallywith the number of intermediate nodes, i.e., | J ij | = √ Γ i | t i − . . . t j +1 | (cid:112) Γ j where t i ( ω q ) = C i ( ω q ) are trans-fer amplitudes of the intermediate cavities with the ma-trices C i ( ω ) defined in App. B. For κ (cid:28) κ f and ∆ c = ω r we find 1 − | t i | (cid:46) κ /κ f .
2. Non-RWA corrections
The most important effect of the non-RWA terms inthe OM coupling is the appearance of additional con-tributions to the single- and multiqubit noise terms dueto Stokes scattering, as has already been discussed forthe single-qubit case in Sec. II (see Eq. (22)). Since ex-plicit expressions for these contributions are lengthy andnot very illuminating, we only mention that their effectscales as κ /ω r , | G | /ω r and they are therefore small inthe resolved-sideband limit. Note, however, that the non-RWA terms in the various nodes are in principle phase-coherent, which can lead to non-trivial interference ef-fects. Apart from the additional noise, one also obtains many rather quantitative corrections of the decay ratesand cascaded couplings as compared to the RWA case,such as the fact that the relation | J ij | ≈ (cid:112) Γ i Γ j becomesapproximate. However, the general picture of emissionand reabsorption of qubit excitations as conveyed by theideal cascaded Liouvillian L ideal remains valid.
3. Accumulation of noise
In Sec. II E we have discussed the photonic noisepresent in the output of a single OMT. Clearly, in thecurrent multinode setting these noise photons will prop-agate down the fiber and affect successive qubits (seeFig. 3). Since the local noise sources (encapsulated in the R i (t) in Eq. (30)) are uncorrelated, we expect the effec-tive bath occupation numbers to scale with the numberof preceding nodes, i.e., N i ∝ i . To be more specific, weapproximate them as N i ≈ N ,i + N c,i . (41)Here, N ,i is the local contribution as given before inEq. (22) and the effect of previous nodes is summarizedby the cascaded occupation number N c,i . Neglecting anon-RWA pre-factor to N c,i , it is defined as the spectrumof the cavity input evaluated at the qubit frequency, i.e., N c,i = (cid:82) ∞−∞ d τ (cid:104) f † in ,i ( τ ) f in ,i (0) (cid:105) free e − iω q τ and we have N c, = 0 for the first node.More explicitly, for the case of N = 2 nodes, the cas-caded noise is simply determined by the output of thefirst cavity and it can be expressed as N c, = 2 κ f C ( ω q ),with the steady-state correlation matrix C ( ω ) definedin App. B. Assuming that the first qubit is on resonancewith one of its associated OM normal modes, we obtainfor κ, | G | (cid:28) ω r = ∆ c : N c, ≈ κ f κ (cid:18) γ m N m γ op + κγ op | G | ω r (cid:19) χ , (42)where χ = 1 for weak OM coupling ( | G | ≤ κ/
2) and χ = | G | / ( | G | − κ ) ≤ | G | > κ/ N c, ∼ N ex (Γ − ) Γ /γ op ∼ N , .This means that the OMT at the second node providesexactly the filtering needed to suppress the noise origi-nating from the first node (cf. Sec. II E). Therefore, weconclude that exchanging a single photon between twonodes of the quantum network is in principle possible.For completeness, we briefly consider the general caseof many nodes. To get a quantitative picture, it is againinstructive to consider the simple case of ζ = 0 and κ = 0, such that for γ m → γ m N m . In this case, Eq. (41)holds exactly and we obtain N c,i = (cid:88) n
Up to now, the dynamics of the cascaded qubit networkhas been discussed on general grounds. We now turn toa specific application, namely the transfer of a quantumstate from one qubit to the next. This has been thetopic of our recent work [42] and we provide here thedetails of the protocol. The main problem in transferringa quantum state | ψ (cid:105) = α | (cid:105) + β | (cid:105) between two nodesof a cascaded network according to | ψ (cid:105) | (cid:105) → | (cid:105) | ψ (cid:105) , (44)lies in the possibility that the photon emitted by thefirst node could pass the second node instead of beingreabsorbed (see Fig. 4(a)). It was first realized by Cirac et al. [11] in the context of atomic cavity QED that suchphoton loss can be avoided by choosing appropriate time-dependent control pulses in the nodes, which leads toa deterministic state transfer protocol. In our setting,which is closely related, we can adiabatically tune theeffective qubit-fiber couplings Γ i ( t ) as well as the barequbit frequencies ω iq ( t ). We proceed by first deriving theideal pulse-shapes needed for the state transfer and thencomment on their realization.
1. Pulse Shapes for state transfer
We ignore all imperfections for the moment and baseour derivation on the ideal cascaded ME as given inEq. (31) for N = 2 qubits. Along the lines of Ref. [11]we require that the system remains in a pure state µ ( t ) = | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | during the entire evolution, which isequivalent to requiring the output of the photo-detectorindicated in Fig. 4(a) to be exactly zero. It is clearfrom Eqs. (32),(33) that this is the case if we can en-force the so-called dark state condition S ( t ) | ψ ( t ) (cid:105) = 0for all times, where the time-dependence of the jumpoperator is attributed to the time-dependence of Γ i ( t ).Under this constraint the evolution of the system iscompletely characterized by the Schroedinger equation ∂ t | ψ (cid:105) = − i ( H eff + H ) | ψ (cid:105) , with H eff given in Eq. (33)and H = (cid:80) i ˜ ω iq ( t ) σ iz / θ i that had been absorbed in Sec. III A and expand thestate vector in terms of three time-dependent amplitudes u , v , v as follows: | ψ ( t ) (cid:105) = αu ( t ) e i Φ + ( t ) | (cid:105) (45)+ β (cid:104) v ( t ) e i Φ − ( t ) | (cid:105) + v ( t ) e i Φ − ( t )+ iφ ( t ) | (cid:105) (cid:105) . (46)Here, the phase factors have been introduced for laterconvenience according to Φ ± ( t ) = (cid:82) tt d s (˜ ω q ( t ) ± ˜ ω q ( t ))and φ ( t ) = arg { J } = θ − θ . The dark-state condi-tion S| ψ (cid:105) = 0 now reads √ Γ v + √ Γ v = 0 and the2Schr¨odinger equation amounts to ˙ u = 0 and˙ v = − Γ v , ˙ v = (cid:18) − Γ − iδ ( t ) (cid:19) v − (cid:112) Γ Γ v , (47)where δ ( t ) = ˜ ω q ( t ) − ˜ ω q ( t ) + ˙ φ ( t ) is an effective detuning.We will first derive the pulses for δ ( t ) = 0, which is ageneralized resonance condition taking the varying phaseof the cascaded coupling into account.Achieving perfect state transfer means that we findtime-dependent Γ i ( t ) such that the solution of Eq. (47)satisfies the dark state condition as well as the boundaryconditions v ( t i ) = | v ( t f ) | = 1 , v ( t f ) = v ( t i ) = 0 , (48)where t i and t f are the initial and final times, respec-tively. In reality, however, one will have to tolerate slightviolations of these boundary conditions due to finite pulselengths, etc. To find suitable pulses we make use of thefollowing time-symmetry argument [11, 55]: If a pho-ton is emitted by the first qubit, then, upon reversingthe direction of time, we would see a perfect reabsorp-tion. We can exploit this by ensuring that the emittedphoton shape is invariant under time-reversal and use atime-reversed control pulse for the second qubit. As aresult, the absorption process in the second node is atime-reversed copy of the emission in the first and maythus be – at least in principle – perfect. In the presentformulation the role of the photon shape is played by a ( t ) ≡ √ Γ v and we can generate a class of pulses byrequiring its time-derivative to be of the form ˙ a = f ( t ) a with some f ( t ) satisfying f ( t ) = − f ( − t ), provided weassume t f = − t i . Upon using Eq. (47) this yields a dif-ferential equation for the pulse shape:˙Γ = Γ + 2 f ( t )Γ . (49)After choosing a suitable (i.e. bounded and positive) so-lution one may use it to calculate v ( t ) and the quantitiesin the second node are then given by v ( t ) = − v ( − t )and Γ ( t ) = Γ ( − t ), which solve the Schr¨odinger equa-tion and satisfy the dark-state-condition. Note also, thatrelated time-reversal arguments can be employed for re-absorbing almost arbitrary photon wave-packets, as hasbeen discussed for formally similar atomic systems [56].We assume that there is a maximal achievable decayrate Γ max and give two example pulse-shapes. First, for f ( t ) = − Γ max sign( t ) / a ( t ) ∝ e − Γ max | t | / and a solution of Eq. (49) is given byΓ ( t <
0) = Γ max e Γ max t − e Γ max t , Γ ( t ≥
0) = Γ max . (50)The resulting dynamics is illustrated in Fig. 4(b), wherethe pulse-length has been chosen to be T p ≡ t f − t i = 8 / Γ max such that | v ( t f ) | < − . As a second example,we consider f ( t ) = − ct for c > ( t ) = exp( − ct ) (0) − √ π √ c Erf ( √ ct ) (51)where the parameters c and Γ (0) have to be adjustedsuch that the boundary conditions (48) are met. Again,we display an example with | v ( t f ) | < − in Fig. 4(c).Note that the pulse-length T p = 12 / Γ max is slightlylonger than for the previous pulse, since the maximaldecay is reached only shortly.
2. State transfer between two nodes
To discuss the implementation of the state transfer pro-tocol in more detail, we focus on the pulse shape givenin Eq. (51). We expect the other pulse (cf. Eq. (50)) toperform worse in terms of non-RWA noise due to the factthat the maximum decay is required over a long period.The following discussion is, however, valid in general.The task is to tune the OM parameters in such a waythat the resulting effective decays Γ i ( t ) in the two nodesfollow the prescribed time-evolution ˜Γ i ( t ), and we firstconsider varying the OM couplings G i ( t ) as depicted inFig. 2(c). The key step is then to solve the equationΓ i ( G i ( t )) = ˜Γ i ( t ) with κ = γ m = 0 for the desiredtemporal variation of the OM coupling G i ( t ), where weuse the full relation Eq. (19) to also capture non-RWAcorrections. Note that the renormalization of ∆ c has tobe taken into account and we have further ignored the ef-fective qubit shifts ∆ i as well as the varying phase of thecascaded coupling. This means that the resonance con-dition is in general not satisfied, i.e., we have δ ( t ) (cid:54) = 0.In order to correct this, we can adjust the bare qubit fre-quencies and in a first iteration set ω iq → ω iq + δω iq ( t ),with δω q ( t ) = − δω q ( t ) = δ ( t ) /
2. This shift is of order λ /γ op and the new ˜ ω iq approximately satisfy the reso-nance condition with corrections being of higher order in λ . In principle, shifting the bare qubit frequencies alsomodifies the Γ i ( t ) which therefore deviate from the pre-scribed ˜Γ i ( t ). However, this deviation is negligible sincethe shifts δω iq are small compared to the width γ op ofthe resonances in Γ( ω q , G ) . We have thus obtained thepulse ( δω q ( t ) , G ( t ) , δω q ( t ) , G ( t )) that realizes the de-sired control pulse ˜Γ i ( t ) and satisfies the resonance con-dition.An example for a resulting pulse is displayed inFig. 5(a) and we note that the adiabaticity conditionsare well satisfied: Since G ( t ) is tuned from 0 up to its To satisfy all of these constraints exactly, one may perform aminimization of the functional C [ δω iq ( t ) , G i ( t )] = (cid:82) t f t i d t [( δ ( t )) + (cid:80) i (Γ i ( t ) − ˜Γ i ( t )) ]. T p , we have on average ˙ G/G max ∼ /T p ∝ Γ max (cid:28) γ op ,as is required for the adiabatic elimination to hold andis therefore valid by assumption. In addition, for trans-lating the desired pulse G ( t ) into applied laser power viaEq. (11), we need ˙ G/G max (cid:28) κ , which is implied by theprevious relation. To assess the impact of imperfectionswe can now evaluate all quantities in the effective ME(37) for potentially finite γ m , κ , N m , as will be donebelow. Finally, as an alternative to tuning the OM cou-plings G i , we can also vary the cavity frequencies in orderto regulate the effective decays, as depicted in Fig. 2(d)and a corresponding pulse ∆ c,i ( t ) is shown in Fig. 5(b).
3. Discussion
The quality of the state transfer can be estimated byaveraging the fidelity F ( ψ ) = (cid:104) ˜ ψ | Tr { µ ( t f ) }| ˜ ψ (cid:105) (52)over all transmitted single-qubit states | ψ (cid:105) = α | (cid:105) + β | (cid:105) .Here, µ ( t ) is the solution of the ME (37) with initial con-dition µ (0) = | ψ (cid:105)(cid:104) ψ | ⊗ | (cid:105)(cid:104) | and | ˜ ψ (cid:105) = αe i Φ + ( t ) | (cid:105) − βe i Φ − ( t )+ iφ ( t ) | (cid:105) is the target state including the variousphases that are accumulated due to the varying qubitfrequencies and phase of the cascaded coupling. We alsotake into account a decay of the qubit coherences ac-cording to e − t/T by adding a term T (cid:80) i D [ σ iz ] µ to theME, where T is the qubit’s intrinsic coherence time. Theleading contributions to the fidelity are expected to scaleas F ≈ − κ κ − C γ m N m κ − C κ ω r − C κλ T , (53)where the C i are numerical constants of order one andthe four terms correspond to intrinsic cavity loss, ther-mal noise, Stokes scattering (we chose G max ∼ κ for thepulses), and intrinsic qubit dephasing. The constants C i depend on the pulse-shape and can be optimized depend-ing on the experimental parameters. Figs. 5(c),(e) dis-play the fidelities obtained in Ref. [42] for varying G i ( t ),while Figs. 5(d),(f) show results for varying ∆ c,i ( t ). Itcan be seen that the two control schemes do not differsignificantly, although we will use the former in our esti-mates below.To conclude this section, we briefly discuss the resultsfor an implementation with spin- and charge qubits, thedetails of which are described Sec. V. For spin qubits,we have λ/ π on the order of 50 kHz if we choose ω r / π ≈ κ/ π ≈ κ , Γ m ) / π = (50 ,
10) kHz, aswell as T ≈
10 ms, we obtain state transfer fidelitiesof
F ≈ .
85, where all imperfections contribute approx-imately equally. However, with recent improvements of T -times for spin qubits [57], fidelities beyond F ≈ . (e)(a) (b)(c) (d)(f) FIG. 5. (Color online) Performance of state transfer proto-col using the pulse of Fig. 4(c). The left and right columnsshow realizations via tuning of OM couplings and cavity fre-quencies, respectively. (a),(b) Resulting control pulses for κ = 0 . ω r (see also Fig. 2). (c),(d) Fidelities as functionsof κ and Γ m , where κ = 1 /T = 0. (e),(f) Fidelities as func-tions of κ and 1 /T , with κ = Γ m = 0. For the left columnwe deduce the parameters of Eq. (53) to be C ≈ C ≈ . C ≈ .
5, and for the right column C remains the samewhile the others are slightly worse, i.e., C ≈ C ≈ seem feasible for our parameters. In contrast, the fidelityfor charge qubits is clearly dominated by intrinsic qubitdephasing. In this case, one can choose a larger resonatorfrequency ω r / π ≈
50 MHz and κ/ π ∼ λ/ π ∼ T ≈ µ s. Again, we obtain F ≈ . ∼ .
1. In this case, strategies to overcomethis problem would have a significant impact on the per-formance of the state transfer. In particular, one couldnumerically optimize the pulse-shapes for gate-speed ordepart from the adiabatic elimination and treat the fulldynamics of the OMT, yielding faster gate-sequences. Weconclude that state transfer operations with both typesof solid-state qubits are feasible with present-day tech-nology.
IV. OPTICAL ON-CHIP NETWORKS
So far we have considered networks where quantum in-formation is transmitted through a continuum of modessupported by an optical fiber using pulse-shaping tech-4niques. While this is a natural setting for long-distancequantum communication, one may also imagine situa-tions where the different nodes of the quantum networkare closely spaced and could instead be linked by opticalresonators which exhibit a discrete set of modes. Thisscenario could be relevant for quantum communicationand entanglement distribution within quantum process-ing architectures built on a chip, with optical channelsproviding a fast and robust alternative to other communi-cation schemes, as discussed in the introduction. There-fore, we consider a setup as shown in Fig. 6(a), whereeach qubit is linked via an OMT to a “bus”-cavity oflength L , which may, e.g., be implemented by a shortoptical fiber. Similar to the long-distance network con-sidered above, the system is described by a Hamiltonianof the form H = (cid:88) i (cid:0) H i node + H i cav-link (cid:1) + H link + H env , (54)where the structure of H i node is the same as in Eq. (2),with the exception that we do not apply laser drives tothe nodes (see below). The bus is modeled by an ad-ditional cavity with discrete modes separated by a freespectral range of δω/ π (cid:38) L (cid:46)
10 cm. We take this spacing to be much larger thanthe other couplings and decays and therefore, we can re-strict our discussion to a single fiber mode with destruc-tion operator d and frequency ω close to the frequencies ω ic of the nodal cavities. In this case, we obtain H link = ω d † d + i ( E e − iω L t d † − E ∗ e iω L t d ) , (55) H i cav-link = h i d † c i + h ∗ i d c † i , (56)where we have also included a laser of frequency ω L whichcoherently drives the fiber mode. Here, we have neglectedadditional degenerate modes in the nodes which do notcouple to the fiber. They will not be classically occupiedand will thus not couple significantly to the mechanicalresonance (see Sec. II C).To proceed, we can follow the basic steps described inSec. III: First, we introduce the QLEs for the OM degreesof freedom to describe dissipative effects [49]:˙ d = − i [ d , H (cid:48) ] − κ f d − (cid:112) κ f f ( t ) , (57a)˙ c i = − i [ c i , H (cid:48) ] − κ c i − √ κ f ,i ( t ) , (57b)˙ b i = − i [ b i , H (cid:48) ] − γ m b − √ γ m ξ i ( t ) . (57c)Here, H (cid:48) = H − H env and κ , γ m / f ,i ( t ), ξ i ( t ) as introduced after Eqs. (7),(9).In addition, we have introduced a decay rate κ f andvacuum noise operator f ( t ) for the fiber mode. Sec-ond, we determine the classical steady state by trans-forming all photonic operators to a frame rotating atthe laser frequency and subsequently making the replace-ments d → d + α , c i → c i + α i and b i → b i + β i . De-manding the c-number contributions to the transformed (a)(b) o p t o - m e c h a n i c a l s y s t e m FIG. 6. (Color online) Phonon-photon bus realized via a fibercavity. (a) Schematic illustration of the setup. Each circlerepresents an OM mode and the long box indicates the fibercavity which is driven by a laser of frequency ω L . Wavy arrowsindicate cavity decay channels (red, single-headed) and me-chanical diffusion (green, double-headed). (b) Level schemefor N = 2 identical nodes (black levels) corresponding to theresults presented in Tab. I. The two outer levels signify thetwo qubits to be coupled. For N > a, ± )become N − w a ± = (˜ b a ± ˜ c a ) / √
2, as well as w s = Kδ ˜ b s − Gδ c and w s ± = √ ( Kδ ˜ b s ± ˜ c s + Gδ c ), where δ = √ G + K . QLEs to vanish then yields ( i > α = − ( i ∆ c + κ f ) α − i (cid:88) i h i α i + E , (58a)˙ α i = − ( i ∆ ci + κ ) α i − ih ∗ i α − ig α i ( β i + β ∗ i ) , (58b)˙ β i = − ( iω r + γ m / β i − ig | α i | , (58c)where we have introduced the detunings of the cavitiesfrom the laser drive according to ∆ c = ω − ω L and∆ ci = ω ic − ω L . Below, we are interested in the caseof identical nodes, where the above system admits for auniform steady-state solution with identical α i and β i .For | α i | (cid:29) H (cid:48) → H osc + (cid:88) i (cid:20) ω q σ iz + λ (cid:16) σ − i b † i + σ + i b i (cid:17)(cid:21) , (59)where H osc is given by H osc = ∆ c d † d + (cid:88) i (cid:16) d † h i c i + h.c. (cid:17) (60)+ (cid:88) i (cid:104) ω r b † i b i + ∆ ci c † i c i + ( G i c † i + G ∗ i c i )( b i + b † i ) (cid:105) . Here, G i = g α i is the enhanced OM coupling and wehave redefined ∆ ci − | G i | /ω r → ∆ ci . In summary, wehave thus obtained a system of qubits which is connectedby a finite linear network of harmonic oscillators.The linear oscillator network H osc can in principle beused to resonantly propagate excitations from one qubitto another. However, this would require a certain degreeof control over the various couplings to route an excita-tion to its target node and also has the drawback thatthe oscillators’ decay channels have a strong effect. Forthese reasons we will focus on an off-resonant scheme:The qubits chosen to take part in the gate operation aretuned close to one of the normal modes of the intermedi-ate OM system described by H osc . If ¯∆ is the detuningfrom that mode they experience an effective interactionat rate J ∼ λ / ¯∆, provided λ (cid:28) | ¯∆ | . On the otherhand, a decay γ of the chosen normal mode will lead toan induced qubit-decay Γ ∼ λ γ/ ¯∆ , which can be sup-pressed for larger detunings according to Γ /J ∼ γ/ ¯∆, aslong as the intrinsic noise-processes of the qubit do notplay a role (see below). To exploit this scaling, we needof course well-resolved OM normal modes. A. Two node network
We illustrate the resulting interactions for N = 2nodes and postpone matters regarding the scalability toSec. IV D. To begin with, we analyze the mode-structureof the OM system described by H osc , which will providethe basis for understanding the induced qubit dynam-ics. Assuming identical nodes, we introduce symmet-ric and anti-symmetric combinations of all system andnoise operators (i.e. ˜ c s/a = ( c ± c ) / √
2, etc.). Thesecond term in Eq. (60) then simply reads Kd † ˜ c s + h.c.with K = √ h and the QLEs for the oscillator network(Eq. (57) with H (cid:48) → H osc ) decouple into a symmetricand an anti-symmetric set:˙˜ v ν ( t ) = − ˜ M ν ˜ v ν ( t ) − ˜ R ν ( t ) ν = s, a . (61)Here, the degrees of freedom have been grouped into˜ v s = (˜ b s , ˜ c s , d , ˜ b † s , ˜ c † s , d † ) T and ˜ v a = (˜ b a , ˜ c a , ˜ b † a , ˜ c † a ) T , andthe noise vectors ˜ R ν ( t ) are mutually uncorrelated. Ex-plicit expressions for the ˜ M ν and ˜ R ν ( t ) can be found inApp. C and we take G, K > G (cid:29) κ , γ m , mechanical and optical modeshybridize to produce normal modes at ω a ± ≈ ω r ± G .Here and in what follows we assume ∆ c = ∆ c = ω r andthe results in the present subsection are obtained for OMRWA ( ζ = 0). In contrast, the symmetric subsystem isan augmented OM system including also the fiber cavitywith normal modes at ω s = ω r and ω s ± ≈ ω r ± δ , where δ = G + K . Therefore, symmetric and anti-symmetricnormal modes are staggered as shown in Fig. 6(b) and for G, K (cid:29) κ , κ f , γ m the modes are well resolved, as canbe seen from the expressions for their widths γ listed inTab. I.We proceed to eliminate the OM degrees of freedom onthe basis of the assumption that they are detuned fromthe qubits by much more than λ . In terms of the new OMmodes, the qubit-resonator interaction in Eq. (59) reads H int = λ (cid:88) ν = s,a (cid:16) ˜ σ + ν ˜ b ν + h.c. (cid:17) , (62)and since the ˜ b ν are mutually uncorrelated they may beeliminated independently, as is done in App. C. However,for well-resolved normal modes it is reasonable to furtherassume that also correlations between modes of the sameset are small. To be specific, we introduce normal modes w ν via v ν = U ν w ν , such that the transformations U ν diagonalize ˜ M ν . This yields˙ w νj = − ( iω νj + γ νj ) w νj − (cid:88) k ( U ν ) − jk ˜ R νk , (63)where iω νj + γ νj are the eigenvalues of ˜ M ν . In addition,the qubit-resonator interaction becomes H int = 12 (cid:88) (cid:48) ν,j (cid:16) ˜ λ νj ˜ σ + ν w νj + h.c. (cid:17) , ˜ λ νj = λ ( U ν ) j , (64)where we have introduced the effective couplings ˜ λ νj andthe primed sum is restricted to normal modes with fre-quencies ω νj >
0. In the present RWA this is exact, but itpresents an approximation otherwise. Now, correlationsbetween the normal modes w νj of a given set ν arise due tothe noise terms in Eq. (63), which scale as √ γ m , (cid:112) κ f , √ κ . Since these quantities have to be small anyway, weneglect these correlations and the normal modes w νj thenevolve independently. As a result, the general eliminationprocedure as outlined in App. A becomes very simple andcalculating the necessary correlation functions (A5)-(A8)for each w νj yields a ME that is just a sum over the fivenormal modes listed in Tab. I:˙ µ = − i (cid:2) J ( σ σ − + σ − σ ) , µ (cid:3) (65)+ 12 (cid:88) (cid:48) ν,j (cid:110) Γ νj ( N νj + 1) D [˜ σ − ν ] µ + Γ νj N νj D [˜ σ + ν ] µ (cid:111) , mode ( ν, j ) ω γ ˜ λ N th N non − RWA (a, ± ) ω r ± G γ m κ √ λ γ m N m γ a ± G G ± ω r ) (s,0) ω r K δ γ m + G δ κ f Kδ λ K δ γ m N m γ s κ K κ f ω r + O (cid:18) Kω r (cid:19) (s, ± ) ω r ± δ G δ γ m + κ K δ κ f G √ δ λ G δ γ m N m γ s ± g κ κ f + κ + g κ ) K ω r + O (cid:18) Kω r (cid:19) TABLE I. Characterization of OM normal modes ( δ = √ G + K ): The different columns display eigenfrequencies ω anddecay rates γ as obtained by diagonalizing ˜ M ν , as well as quantities entering the effective ME (65) in the independent-modeapproximation, i.e., effective couplings ˜ λ , thermal occupation numbers N th and non-RWA occupation numbers N non − RWA .The first four columns were obtained in OM RWA and we have generally made use of κ , κ f , γ m (cid:28) G, K and assumed∆ c = ∆ c = ω r to simplify the expressions. The last column displays results to first order in the non-RWA terms (see text)and we show only leading terms in G /ω r , K /ω r for γ m → g ≡ G/K (cid:46) Here, J = (cid:80) (cid:48) ν,j J νj is the desired effective coupling and wehave dropped terms renormalizing the qubit frequenciesfor simplicity, since they are the same for both qubits.The couplings and decay rates are directly related to thenormal modes as described in Tab. I and owing to ourindependent-mode approximation they have the form ofsimple Lorentzians ( ¯∆ νj = ω q − ω νj ): J νj = (˜ λ νj ) jν ( − δ νa ( γ νj ) + ( ¯∆ νj ) , (66a)Γ νj = (˜ λ νj ) γ νj ( γ νj ) + ( ¯∆ νj ) . (66b)Hence, we have obtained the expected scaling Γ νj /J νj ∼ γ νj / ¯∆ νj when close to a resonance. The alternating signof the J νj is rooted in the fact that eliminating a modeof symmetry ν yields a shift of the operator ˜ σ + ν ˜ σ − ν ∝ ( − δ νa ( σ +1 σ − + σ − σ +2 ). Note that when tuning thequbits somewhere in between two modes, these have op-posite symmetry (cf. Fig. 6(b)) and their contributionsto J add up constructively, since the relevant detunings¯∆ νj also have opposite signs.Finally, the influence of the thermal noise on thequbits is characterized by the effective occupation num-bers N νj ∼ γ m N m /γ νj (see Tab. I for details). This meansthat the thermal noise is suppressed by an OM coolingeffect, provided that we have γ m N m (cid:46) κ , κ f . B. Non-RWA effects
The effect of the non-RWA terms is to admix a counter-rotating portion to each OM normal mode, giving rise tonew noise terms as well as quantitative corrections. Inthe case of the full OM coupling the eigenmode trans-formation U ν is not block diagonal anymore and theeigenmodes in Eq. (63) are driven by counter-rotatingnoise operators. To estimate their effect we can intro-duce new noise operators with effective non-zero occupa-tion numbers by means of a Bogoliubov transformationand then proceed as above and treat the normal modes independently. For illustration, the eigenvectors of theanti-symmetric system ˜ M a are, to lowest order in thenon-RWA parameter ζ and for vanishing dissipation, w a ± ≈ √ (cid:16) ˜ b a ± ˜ c a (cid:17) − ζG √ G ± ω r ) (cid:16) ˜ b † a ± ˜ c † a (cid:17) . (67)Calculating, also to first order in ζ , the matrix ( U a ) − from these eigenvectors then allows to estimate the effec-tive occupation number due to Stokes scattering: N a ± , non − RWA ≈ G G ± ω r ) , (68)where we have neglected non-RWA corrections to thethermal noise. For the symmetric modes the expressionsbecome quite lengthy, so we display in Tab. I only theleading order terms in G /ω r , K /ω r . C. Discussion: √ SWAP gate
The effective two-qubit interaction obtained in Eq. (65)can be used to swap the states of the qubits or to performthe entangling √ SWAP operation, which amounts to ex-posing the qubits to this interaction for a time t g = π/ J .To asses the optimal operation point for such operationswe consider the creation of a maximally entangled state | ψ (cid:105) = ( | (cid:105) − i | (cid:105) ) / √ | (cid:105) by applying the √ SWAP gate. For this purpose,we also account for a finite qubit coherence time T byadding the term T (cid:80) i D [ σ iz ] to the ME (65), which isthen solved to first order in the Lindblad terms. The re-sulting approximation of the fidelity F = (cid:104) ψ | µ ( t g ) | ψ (cid:105) for this operation can be written as F ≈ − π (cid:88) (cid:48) ν,j Γ νj | J | (1 + 2 N νj ) − π | J | T , (69)which is a function of the qubit frequency (cf. Eq. (66)).Generally speaking, for K (cid:38) G we expect higher fidelitiesif the qubits are tuned in between the central resonances(s,0), (a, ± ), as can be argued from Tab. I: Compared7to the outer resonances (s, ± ), they have larger weights˜ λ and thus larger J . Further, their width γ is slightlysmaller for the relevant case γ m (cid:28) κ ≈ κ f .In the limit where the induced decay Γ dominates overintrinsic dephasing, i.e., Γ (cid:29) /T , we see from the sec-ond term in Eq. (69) that there is a point between eachpair of neighboring resonances where J/ Γ is optimal. For G = K such points can, e.g., be found half-way be-tween resonances ( s,
0) and ( a, ± ) at ω q ≈ ω r ± G/ κ = κ f (cid:28) G F opt ≈ − π κ G (cid:18) γ m N m κ + G ω r (cid:19) − π Gλ T . (70)In this limit, the infidelity is thus dominated by cavity-induced qubit decay (first term in parentheses), providedthe thermal noise of the resonator is suppressed due to γ m N m (cid:46) κ . The last term in parentheses representsnon-RWA corrections and is small for moderate cou-plings, as required by stability constraints (cf. Sec. II).In the opposite limit, where intrinsic qubit dephasingdominates (1 /T (cid:29) Γ), we have to tune very close to asingle resonance ( ν, j ) to obtain large J and hence a fastgate. There is, however, a trade-off between minimizingthe dephasing (small ¯∆) and the optomechanically in-duced decay Γ (large ¯∆). The optimal detuning in thiscase is given by ¯∆ opt ≈ [(˜ λ νj ) γ νj (1 + 2 N νj ) T / / whichyields a fidelity of F opt ≈ − π (cid:115) γ νj (1 + 2 N νj )(˜ λ νj ) T . (71)There is, however, also the limitation (cid:12)(cid:12) ¯∆ opt (cid:12)(cid:12) (cid:29) λ im-posed by the elimination procedure.Figure 7 shows the fidelities for an implementation withspin and charge qubits as discussed in more detail inSec. V. Apart from the expression (69) based on theindependent-mode approximation (dashed) we also dis-play the result of the full elimination, Eq. (C8), show-ing reasonable agreement. For an ideal superconductingqubit, the fidelities are rather good and with T ≈ µ swe still obtain F (cid:38) .
85 for the parameters of Fig. 7(a),which shows that the performance is limited by intrin-sic dephasing. On the other hand, for the spin qubita resonator of lower frequency is better suited (yieldinghigher zero-point motion), which also puts limits on theOM coupling G . As a result, the leading term in Eq. (70)proportional to κ /G is larger than for the charge qubit,meaning that the intrinsic cavity decays become a moresevere limitation in this case. For an ideal spin qubit weobtain fidelities of ∼ . T ≈
10 ms we still find
F ≈ . − .
9. Strictly speaking, these estimates areonly valid for small infidelities, but they are neverthe-less useful for identifying interesting regimes of operation (a) (b)
FIG. 7. (Color online) Fidelities for generating a maximallyentangled state between N = 2 nodes (see text). Solid curves:results of the full elimination presented in App. C, dashedcurves: approximate expressions according to Eq. (69). (a)Parameters suitable for charge qubit: ω r / π = 50 MHz, λ/ π = 3 . G = 0 . ω r , K = 0 . ω r , and decoherencerates ( κ , κ f , Γ m ) / π = (50 , ,
10) kHz. The blue (darkgray) curves correspond to vanishing dephasing 1 /T = 0and the green (light gray) curves to T = 2 µ s. (b) Pa-rameters for spin qubit: ω r / π = 7 . λ/ π = 40 kHz, G = K = 0 . ω r and decoherence rates as in (a). Blue (darkgray) curves: 1 /T = 0, green (light gray) curves: T = 10 ms. and we recognize the two limiting cases discussed above.As in the case of the long-distance state transfer, onemight – especially in the case of the charge qubit – profitfrom leaving the adiabatic regime in order to shorten thegate-sequence and to reduce the impact of intrinsic qubitdephasing. D. Outlook: Scalability
When considering an N -node network the moststraight-forward way to perform a gate operation be-tween two given qubits is to detune the nodal cavitiesof all other nodes sufficiently far, such that the networkreduces to the two-node problem, which leads us to ex-pect similar fidelities. Since this requires only moderatetemporal control of the cavity frequencies ω ic , this couldbe achieved by applying, e.g., strain or heat to thosecavities not meant to participate (see, e.g., Ref. [58] fortuning of the toroidal cavities discussed in Sec. V). Notethat if the fiber has to be extended to accommodate thenew nodes, the cavity-fiber couplings scale as h ∝ / √ L ,which has to be taken into account.We now consider N -qubit interactions and the basisfor the discussion is again the normal-mode structure ofthe underlying N -node OM network. For identical nodes,only the “center-of-mass” cavity mode ˜ c = (cid:80) i c i / √ N couples to the fiber while all other modes decouple. Asa result, the level scheme in Fig. 6(b) remains valid withthe modification that the modes ( a, ± ) become ( N − h ∝ / √ N and hence K = √ N h = const. As couldbe expected, the calculation presented in App. C showsthat the induced coherent qubit dynamics is given by8 (a) (b)(c) magnetictipelectrodestaperedfiber
FIG. 8. (Color online) Realization with toroidal cavities anddoubly clamped beams. (a) Illustration of a single toroidalcavity coupled to an optical fiber as well as to a doublyclamped beam. (b) Coupling of the doubly clamped beamto a spin qubit via magnetic fields. (c) Coupling of the dou-bly clamped beam to a charge qubit via electrostatic forces. J N S + S − , where S − = (cid:80) i σ − i and in general J N ∝ /N .For illustration, we note that if the qubits are tuned closeto the symmetric mode ( s,
0) at ω r , we can neglect theanti-symmetric ones provided G (cid:38) N κ and write theeffective N -qubit ME (C2) as˙ µ ≈ − i (cid:2) J N S + S − , µ (cid:3) (72)+ Γ coll N coll + 1) D [ S − ] µ + Γ coll N coll D [ S + ] µ , where the coefficients are related to the two-node re-sult via J N ∼ J/N and Γ coll ∼ s /N , and N coll is an effective occupation number. We hence obtainΓ coll /J N ∼ const and conclude that decays induced bythe OMTs do not limit the number of nodes, providedwe are close enough to the mode (s,0). In contrast, whentuning to the anti-symmetric modes, there are always N − /J N ∝ N . Fi-nally, note that even for the symmetric modes there arealways the intrinsic decoherence processes of the qubitsthat set an absolute time-scale and limit the number ofnodes via the requirement J N T ∝ JT /N (cid:29) V. IMPLEMENTATIONS
In the preceeding sections, we have mainly discussedthe OMT on general grounds and conditions for low-noiseoperation have been given. Here, we describe the realiza-tion of the OMT within specific physical systems and theresulting numbers have already been used in Sec. III andIV to illustrate the performance of the proposed schemes.Generally, implementing the OMT poses the challenge ofcombining high- Q optical cavities with high- Q mechan-ical resonators exhibiting an appreciable zero-point mo-tion, i.e., a low effective mass. Further, the mechanicalresonators must interact with the qubits, which requiressuitable coupling schemes. Linking the different nodes of a quantum network can be achieved with standardoptical fibers, while for short distances, integrated nano-photonic circuits [43, 59] might be a promising alterna-tive, which we, however, do not explore here. A. Toroidal cavities and doubly clamped beams
As a possible candidate system for the OMT we pro-pose to use a setup similar to the one recently demon-strated by Anetsberger et al. [60]. There, a doublyclamped beam of stressed SiN is positioned in the evanes-cent field of a whispering-gallery mode (WGM) sup-ported by a microtoroidal cavity made of silica. Thisarrangement gives rise to the standard OM coupling andthe toroid is further coupled to a tapered fiber for driv-ing and interrogation as shown in Fig. 8(a). To obtaina resonator-qubit coupling we propose to attach a cou-pling element to the resonator (e.g. a magnetic tip or anelectrode as discussed below), which resides sufficientlyfar outside the evanescent field of the WGM such thatthe optical quality factor is not degraded. To be specific,we envision to use a toroid of diameter 2 R ≈ µ m andthe second harmonic mode of a doubly clamped beam oflength l ≈ µ m positioned as in Fig. 8(a). The couplingto qubit and cavity can then occur at different antin-odes and the coupling element is located at a distance of d (cid:38) . µ m from the rim of the toroid, which is sufficientfor our purposes (see below).The elastic modes of a doubly-clamped SiN beam oflength l , width w and thickness t are well described bystandard elastic theory [60, 61]. For the case that thebeam is exposed to high tensile stress, the modes re-semble those of a vibrating string with frequencies givenby ω n = (cid:112) σ /ρ m πn/l , where n = 1 , , . . . labels themode, ρ m is the mass density and σ the internal ten-sile stress. In contrast, for low internal tensile stress and w = t the frequencies scale as ω n = c n (cid:112) E/ ρ m t/l with c ≈ . c ≈ . E is Young’smodulus. We are interested in the second harmonic ofthe beam and hence set ω r = ω . When quantizing theresonator, the effective mass of each mode is given byan integral over the mode’s displacement profile u n ( z )according to m eff = ml − (cid:82) l d z u n ( z ), where m = t lρ m is the real mass. For the relevant mode we find that m eff ≈ m/ ρ m = 2800 kg/m , E = 160 GPa,and σ up to ∼ ω r / π ≈ . . .
50 MHz for abeam with dimensions ( l, w, t ) = (15 , . , . µ m isrealistic and the respective zero-point motions are in therange a = (cid:112) (cid:126) / m eff ω r ≈ . . . . . × − m. Demon-strated Q -factors for these devices range from 10 to1 . × [60, 61] and for Q = 2 × we obtain, e.g., amechanical decoherence rate of γ m N m / π ≈
10 kHz at asupport temperature of T ≈
100 mK.The microtoroidal cavity is modeled by a pair of9(by symmetry) degenerate counter-propagating WGMmodes, whose field distributions we denote by E L,R ( r ).The theoretical quality factors (including radiation lossand absorption in silica) are beyond 10 , whereasdemonstrated values reach up to Q = 4 × [63].When the SiN beam is brought into the near-field ofthese modes, they generally experience a frequency shiftas well as a cross-coupling. Simple perturbation theoryon the level of the wave equation shows that the rele-vant overlap integrals are I µν ∝ (cid:82) resonator d r ∆ (cid:15) E ∗ µ E ν with ∆ (cid:15) = n − n res the refractive index ofthe resonator. However, a resonator oriented as shownin Fig. 8(a) is quite different from a point-scatterer andhence does not scatter between the modes. To be specific,we obtain the scaling I LR /I RR ∼ exp[ − k R/χ ] (cid:28) k = 2 π/λ c and χ − = 250 nm is the field decayconstant outside the cavity [60]. Therefore, we can takethe OM coupling to be of the type assumed in Eq. (3)with g ∝ ∂I RR /∂z .We now turn to linking the different nodes: Whenthe toroidal cavities are side-coupled to the fiber, themodes of a given propagation direction experience a uni-directional coupling, as needed for the cascaded quantumnetwork of Sec. III. However, for standing wave modes, asthey occur when the degeneracy of the WGM-modes islifted by an amount h (cid:38) κ, G or alternatively in Fabry-P´erot or photonic band-gap cavities, the cascaded cou-pling has to be achieved by additional non-reciprocal op-tical elements, such as optical circulators or Faraday ro-tators. While this may complicate the structure of thenetwork, the idea of the OMT remains valid.In the on-chip setting, the nodes are linked by a cavityof length L and refractive index n fib , such that its mode-spacing is given by δω = πn fib c/L . The single-mode de-scription of Sec. IV is valid as long as δω is much largerthan all other frequency scales, which poses the restric-tion L (cid:46)
10 cm for mechanical frequencies of ω r / π (cid:46)
100 MHz. Such a cavity could, e.g., be realized by ter-minating an optical fiber with coated end mirrors. Forthe mirrors demonstrated in Ref. [64] (amplitude loss of85 ppm) we obtain the naive estimate κ f / π ∼
10 kHzfor the decay rate of a cavity with L = 10 cm and assum-ing a power-attenuation of 1 dB/km, the intrinsic lossesare of the same order. B. Spin and charge qubits
In principle, the proposed OMT works for all basictypes of qubits that can be coherently coupled to a me-chanical resonator. In the following, we discuss twoprominent examples.
1. Spin qubits
Prime examples for spin qubits [12] are phosphordonors in silicon [13] or the ground state triplet of NV centers in diamond [14]. Both systems exhibit excellentcoherence properties, i.e., T = 1 . T = 10 ms for phosphor donors [13], with morerecent experiments even achieving T on the order of sec-onds [57]. Also, coherent control of both systems viaexternal fields has been demonstrated.The coupling to the mechanical resonator is accom-plished by attaching a magnetic tip to the beam at theposition of an antinode as shown in Fig. 8(b). This tipproduces a magnetic field at the location of the spinwhich depends on the resonator displacement, and to firstorder in the latter we obtain H q = 12 (cid:126)B · (cid:126)σ + λ b + b † ) σ z , (73)where (cid:126)σ is the vector of Pauli matrices, while (cid:126)B is thefree qubit splitting, into which the constant componentof the tip’s field has been absorbed. The qubit-resonatorcoupling is given by λ = g s µ B a |∇ B | / (cid:126) , with g s ≈ µ b the Bohr magneton,and |∇ B | the field gradient produced by the tip at thelocation of the qubit. Provided that the qubit splittingcan be arranged to be (cid:126)B = ( ω q , ,
0) with ω q ≈ ω r ,we may store the qubit in the σ x -eigenstates and after arelabeling of axes ( x ↔ z ) and subsequent RWA directlyobtain the coupling of Eq. (2), which was the startingpoint for our analysis.For qubits whose natural energy splitting ω does notmatch the resonator frequency (i.e. ω (cid:29) ω r ), we canapply a classical drive of frequency ω d to bridge thegap between the two energy scales. In this case wetake (cid:126)B ( t ) = (2Ω cos( ω d t ) , , ω ), where Ω is the Rabi-frequency. After moving to a frame rotating at ω d wedrop rapidly oscillating terms and transform to a neweigenbasis basis rotated by an angle θ . The latter is givenby tan θ = Ω / ∆, where ∆ = ω − ω d is the detuning ofthe qubit from the drive. The qubit Hamiltonian thenbecomes H q = ω q σ z + λ b + b † ) (cos( θ ) σ z − sin( θ ) σ x ) , (74)with the effective energy splitting ω q = √ ∆ + Ω (cid:28) ω .Provided that we have λ (cid:28) ω q ∼ ω r , we may drop allrapidly oscillating terms in a RWA and retain only H q − res = λ θ ) (cid:0) σ − b † + σ + b (cid:1) , (75)which is of the form given in Eq. (2) if we redefinesin( θ ) λ → λ .We take the magnetic tip coupling the beam to thequbit to be fabricated of highly magnetic Co Fe whichexhibits a magnetization of M = 2 . µ [35]. Fora cone-shaped tip of height and radius 100 nm we es-timate the field gradient 25 nm below the tip to be |∇ B | = 10 T/m. Using a resonator of low frequency ω r / π ≈ a ≈ . × − m, we obtain a coupling of λ/ π ≈
50 kHz.0Finally, we estimate the degradation of the toroid’s qual-ity factor due to the presence of the magnetic tip, whichis placed a distance d (cid:38) . µ m away from the rimof the toroid as discussed above. We make the con-servative assumptions that all light which hits the tipgets lost and obtain Q (cid:38) πe χd V mode /λA tip . Here, V mode = (cid:82) d r (cid:15) r ( r ) | E ( r ) | / | E max | is the mode volumeof the WGM, A tip the cross-sectional area of the tip and χ the decay constant of the evanescent cavity field. Us-ing the values A tip ≈ − µ m , V mode = 90 µ m [63], χ − = 250 nm [60] we obtain Q (cid:38) which is wellabove the intrinsic Q quoted above.Based on these numbers, the results presented inSec. III C and Sec. IV C show that quantum networkingoperations for spin qubits are feasible, both in the long-and short-distance setups.
2. Charge qubits
A second promising qubit candidate for the setup de-scribed in this work is the superconducting charge qubit.It has evolved from the original Cooper pair box qubit[66] to more recent designs such as the transmon [67, 68].For the latter, coherence times of T = 2 µ s have beenobserved [69] and recently, the relaxation times have beenpushed up to T = 200 µ s [70]. Control or measurementtasks can, for example, be performed in a circuit QEDarchitecture [20].Coupling a charge-based qubit to a mechanical res-onator is most easily achieved by pushing excess chargesonto an electrode located on the resonator as depicted inFig. 8(c). The resulting coupling amounts to a position-dependent capacitance and has been widely discussedand also implemented in experiment [38–41]. However,since the intrinsic level splitting ω of superconductingqubits is typically in the GHz range, we need to bridgethe gap to the MHz resonator frequency ω r in order toobtain a Jaynes-Cummings coupling. Therefore, we mod-ulate the gate voltages as described in Ref. [40, 71] withfrequency ω d = ω − ω r and amplitude V r , which yieldsthe coupling of Eq. (2) with λ = 2 (cid:126) E C a d C r V r e . (76)Here, E C is the charging energy of the qubit, C r theequilibrium resonator capacitance, and d the equilib-rium electrode separation of the coupling capacitance.For E C /h ≈
20 GHz, d ≈
100 nm, and V r a couple ofvolts one easily obtains values in the MHz regime (seealso Refs. [39, 41]). If it is necessary to place the qubitfurther away from the resonator, the coupling could alsobe mediated by an intermediate normal-conducting wire,or additional superconducting circuitry [72].We estimate the optical Q -factor due to the presenceof the electrode, which we assume to cover one third ofthe resonator and to be thinner than the typical skin-depth of ∼ Q -factor is given by Q (cid:38) e χd V mode / ( V electrode (cid:15) (cid:48) r ) [73], where (cid:15) (cid:48) r is the imaginarypart of the relative permittivity of the tip and V electrode =2 . × − µ m its volume. Using (cid:15) (cid:48) r <
10 for noble metalsat optical frequencies [74] we find Q (cid:38) , which againexceeds the intrinsic Q .As has been discussed in Sec. III and IV, the fideli-ties for quantum networking operations resulting fromthe above numbers are mainly limited by intrinsic qubit-dephasing. Improvements of T would thus have a bigpositive impact on the performance and also, especiallysuperconducting qubits would profit from leaving theregime where the evolution of the OM degrees of free-dom is adiabatic. VI. CONCLUSIONS & OUTLOOK
In conclusion, we have discussed the potential of micro-mechanical resonators to mediate interactions betweendark qubits and light. We have described in detail thedynamics of a single OMT and found that the condi-tions for low-noise operation overlap with those for OMground-state cooling. Further, we have derived an effec-tive description of a multiqubit cascaded network that issuitable for long-distance quantum communication, dis-cussed the relevant noise-sources in such systems, andpresented an efficient protocol for state transfer betweentwo nodes. In addition, we have shown how OMTs can beutilized in an on-chip setting to perform entangling gatesbetween qubits that are spaced by less than ∼
10 cm.Here, OMTs might provide an alternative to other cou-pling schemes [6–9, 37] for scalable quantum processors.We point out that the ingredients needed for the OMTare quite generic and may be implemented with a vari-ety of solid-state qubits and different OM systems in theoptical, as well as the microwave domain [72, 75]. In thiswork, we have specifically discussed a potential realiza-tion for spin and charge qubits, showing that long- andshort-distance communication is feasible for present-daytechnology. The gate-schemes presented in this work onlyprovide a starting point for optimized and more complexprotocols. In particular, the long-distance state trans-fer protocol could be extended by schemes to correct forphoton loss (see, e.g., Ref. [76]), while the on-chip gatescould profit from spin-echo techniques to reduce the im-pact of qubit dephasing. In both settings, the limitationsposed by imperfect qubits could be overcome by depart-ing from the adiabtic desciption of the OMT and devis-ing fully dynamical gate sequences. This seems especiallyworthwhile for charge qubits, where the qubit-resonatorcoupling λ can be made large rather easily. The benefitwould be shorter gate-sequences at the price of possiblystronger requirements on controllability.More generally, the OMT can be used to realize com-plex qubit-light interactions, which may have applica-tions in the implementation of nonlinear optical deviceson a few-photon level. These are particularly interesting1in the context of integrated nanophotonic systems, wherethe proposed device naturally fits in. ACKNOWLEDGMENTS
We gratefully acknowledge discussions with KlemensHammerer. This work is supported by ITAMP, NSF,CUA, DARPA, the Packard Foundation, and the Dan-ish National Research Foundation. Work in Innsbruck issupported by the Austrian Science Fund (FWF) throughSFB FOQUS and the EU network AQUTE.
Appendix A: Elimination of the optomechanicalsystem
We briefly describe the elimination of the OM degreesof freedom to obtain an effective description of the qubitdynamics. We denote by ρ the density matrix of the com-plete N -node system composed of qubit and OM degreesof freedom and our goal is to derive an effective ME forthe reduced qubit density operator µ = Tr om { ρ } to sec-ond order in the qubit-resonator coupling λ by means ofprojection operator techniques [54]. The way the variousOM systems interact with each other is left unspecifiedin this Appendix and we lump the whole OM dynamicsinto a Liouvillian L om equivalent to the linearized OMQLEs of the setup under consideration. In an interac-tion picture with respect to the free qubit Hamiltonian ω q (cid:80) i σ iz / ρ = L ( t ) ρ ≡ ( L om + L int ( t )) ρ , (A1) L int ( t ) ρ = − i λ (cid:88) i (cid:104) b i σ + i e iω q t + b † i σ − i e − iω q t , ρ (cid:105) , (A2)where L int describes the qubit-resonator interactions.Note that σ − i , b i can be nodal operators (Sec. III andApp. B) or normal mode operators (Sec. IV and App. C)and that the commutators of the σ − i , σ + i will not enterfrom now on. We define the projector on the relevantpart of the density matrix as P ρ = Tr om { ρ } ⊗ ρ omss , where ρ omss is the steady state of the OM system in the absenceof the qubits, defined by L om ρ omss = 0. We also intro-duce Q = 1 − P and note that P = P , Q = Q and PQ = QP = 0. To derive an effective equation of motionfor µ we project Eq. (A1) on the P - and Q -subspaces andformally integrate the equation for the Q -part, where weassume for simplicity that at the initial time Q ρ ( t ) = 0.The formal expression for the P -part is then P ˙ ρ = PL ( t ) P ρ + PL ( t ) Q (cid:90) tt d s T exp (cid:20)(cid:90) ts d τ QL ( τ ) Q (cid:21) QL ( s ) P ρ ( s ) , where T exp[ . . . ] is the time-ordered exponential. Consid-erable simplification is brought about by exploiting the relations L om P = 0 (steady state) , PL om = 0 ( L om preserves trace) , PL int P ρ = 0 (vanishing of Tr om { ρ omss b i } = (cid:104) b i (cid:105) free ) , where the last conditions follow from the fact that wehave removed any classical forces from the descriptionof the OM systems. These relations allow us to drop L om everywhere except for the exponent and since weare interested in the dynamics to second order in λ wemay subsequently drop L int in the exponent leaving uswith P ˙ ρ ( t ) = PL int ( t ) (cid:90) t − t d τ e L om τ L int ( t − τ ) P ρ ( t − τ ) . (A3)Upon expanding the interaction Liouvillian and takingthe trace implicit in the left-most P -operation we ob-tain terms of the following form (neglecting transientsby sending t → −∞ )˙ µ ∝ λ (cid:90) ∞ d τ e ± iω q τ f ( τ ) L A L B µ ( t − τ ) , where f ( τ ) = (cid:104) A ( τ ) B (0) (cid:105) free = Tr om { Ae L om τ Bρ omss } is aresonator-resonator correlation function evaluated in thesteady state of the OM system and L A,B are associatedqubit Liouvillians. For illustration, we assume that f ( τ )has a dominant contribution f ( τ ) ∝ e ( i Ω − γ ) τ and with (cid:15) = i (Ω ± ω q ) − γ integration by parts then yields˙ µ ∝ λ (cid:15) L A L B µ ( t ) − λ (cid:15) (cid:90) ∞ d τ e (cid:15)τ L A L B ˙ µ ( t − τ ) . By iterating this equation we obtain an expansion in λ/ | (cid:15) | and for λ (cid:28) | (cid:15) | the first, Markovian term domi-nates. If this is true for all terms in Eq. (A3) and forall frequency components of the correlation functions,we may neglect the non-Markovian corrections, whichis equivalent to replacing ρ ( t − τ ) → ρ ( t ) in Eq. (A3).In general, this procedure is thus valid if the OM nor-mal modes which couple to the qubits decay much fasterthan λ − or if they are detuned from the qubits by muchmore than λ . The final result can be compactly writ-ten as a Markovian ME with coefficients given by one-sided Fourier transforms of resonator correlation func-tions evaluated at the qubit frequency:˙ µ = − λ (cid:88) i,j (cid:104) S ij ( ω q )( σ + i σ − j µ − σ − j µσ + i ) (A4)+ T ij ( ω q )( σ − i σ + j µ − σ + j µσ − i ) + h.c. (cid:105) , where we have dropped terms rotating at exp[ ± i ω q t ]that contain operators such as σ + i σ + j based on the as-2sumption λ (cid:28) ω q . The coefficients are given by S ij ( ω ) = (cid:90) ∞ d τ (cid:104) b i ( τ ) b † j (0) (cid:105) free e iωτ , (A5) T ij ( ω ) = (cid:90) ∞ d τ (cid:104) b † i ( τ ) b j (0) (cid:105) free e − iωτ , (A6)and for later convenience we also introduce X ij ( ω ) = S ij ( ω ) − T ∗ ij ( ω ) and Y ij ( ω ) = T ij ( ω ) + T ∗ ji ( ω ) such that X ij ( ω ) = (cid:90) ∞ d τ (cid:104) [ b i ( τ ) , b † j (0)] (cid:105) free e iωτ , (A7) Y ij ( ω ) = (cid:90) ∞−∞ d τ (cid:104) b † i ( τ ) b j (0) (cid:105) free e − iωτ . (A8)It will turn out that interaction- and decay-rates aregiven by X ij ( ω ), while Y ij ( ω ) occurs in diffusion terms. Appendix B: Master equation for cascaded setup
Here, we derive the effective ME for N qubits cou-pled to a chain of cascaded OM systems as discussedin Sec. III. Also, for N = 1 we reproduce the results ofSec. II in a different language. To begin with, we rewritethe general result (A4) to separate single- and multiqubitterms (this rewrite is exact):˙ µ = (cid:88) i (cid:26) − i ∆ i (cid:2) σ iz , µ (cid:3) + Γ i N i + 1) D [ σ − i ] µ + Γ i N i D [ σ + i ] µ (cid:27) − (cid:88) i (cid:54) = j (cid:0) J ij (cid:2) σ + i , σ − j µ (cid:3) + J ∗ ij (cid:2) µσ + j , σ − i (cid:3)(cid:1) + (cid:88) i (cid:54) = j D ij (cid:2)(cid:2) σ + j , µ (cid:3) , σ − i (cid:3) , (B1)where D [ a ] µ = 2 aµa † − a † a µ − µ a † a denotes a Lindbladterm with jump operator a . Here, we have defined theeffective frequency shifts ∆ i = ∆ ,i + ∆ th ,i with contri-butions∆ ,i = λ { X ii ( ω q ) } , ∆ th ,i = − λ { T ii ( ω q ) } , (B2)as well as the effective decay rates Γ i and occupationnumbers N i ,Γ i = λ { X ii ( ω q ) } , N i = λ i Y ii ( ω q ) . (B3)Further, the multiqubit coupling and diffusion rates aregiven by J ij = λ X ij ( ω q ) , D ij = λ Y ij ( ω q ) , (B4) respectively, and we stress that the cascaded nature ofthe coupling will only become evident below, where weevaluate J ij from the physics of the underlying, elimi-nated system.The OM steady state correlation functions (A5)-(A8)determining the effective dynamics are conveniently cal-culated from the QLEs (30) and the input-output relation(28) by Fourier or Laplace transformation. To give theresulting expressions in a compact form we use the OMresponse matrices A i ( ω ) = ( M i − iω ) − and in addition,we define the single-node transfer matrices C i ( ω ) thatdescribe how the fiber-field is modified when bypassingnode i . They are given by C i ( ω ) = ( P − κ if P A i ( ω ) P ),where P = diag[0 , , , T ij ( ω ) defined by ( i > j > T jj ( ω ) = A j ( ω ) , T ij ( ω ) = − (cid:112) κ if κ jf A i ( ω ) C i − ( ω ) · · · C j +1 ( ω ) P A j ( ω ) . In addition, to aid the description of the common fiberinput, we also introduce ( i > T i ( ω ) = − (cid:112) κ if A i ( ω ) C i − ( ω ) · · · C ( ω ) . (B5)To determine the correlation functions X ij ( ω ) we makeuse of the quantum regression theorem [49] with the ini-tial condition (cid:104) [ b i (0) , b † j (0)] (cid:105) free = δ ij and solve the re-sulting equations in the Laplace domain. Since node i isonly driven by nodes j < i , we obtain X ij ( ω ) = (cid:40) i < j T ij ( ω ) for i ≥ j (B6)which, via Eq. (B4), turns Eq. (B1) into a cascaded ME.In order to specify the noise terms in the ME wesolve Eq. (30) in the frequency domain, with the Fouriertransformation of some quantity f ( t ) defined as f ( ω ) = √ π (cid:82) d t e iωt f ( t ) (which induces [ f ( ω )] † = f † ( − ω )). Asa result, we get v i ( ω ) = T i ( ω ) I ( ω ) − (cid:80) j ≤ i T ij ( ω ) R j ( ω ),where I ( ω ) ≡ I ( ω ) = (0 , f in ( ω ) , , f in ( ω )) T is the fiberinput of the first node. Due to the δ -correlated nature ofthe noise sources we can express the second moments ofthe OM operators as follows ( i, j = 1 . . . N label nodesand k, l = 1 . . . (cid:104) v i † k ( ω (cid:48) ) v jl ( ω ) (cid:105) free = C ijkl ( ω ) δ ( ω (cid:48) + ω ) , (B7)Here, we have introduced the matrices C ij ( ω ), which di-rectly give the desired correlators for the ME via Y ij ( ω ) = C ij ( ω ). They can be expressed in terms of the transfermatrices and noise statistics according to C ij ( ω ) = ( T i ( ω )) ∗ I ( T j ( ω )) T + min( i,j ) (cid:88) n =1 ( T in ( ω )) ∗ r n ( T jn ( ω )) T , (B8)3where the matrices r i characterize the noise correlationsof node i by means of (cid:104) R i † k ( t ) R il ( t (cid:48) ) (cid:105) = r ikl δ ( t − t (cid:48) ), while I describes the fiber input, (cid:104) I † k ( t ) I l ( t (cid:48) ) (cid:105) = I kl δ ( t − t (cid:48) ).From the statistics quoted in the main text it followsthat r i = diag[ γ im N im , , γ im ( N im + 1) , κ ] and I =diag[0 , , , th ,i , which we do exemplarily for the first node. Us-ing the quantum regression theorem we obtain the rele-vant correlation function T ( ω ) = (cid:80) k A ∗ k ( ω ) (cid:104) v † k v (cid:105) free ,where the steady state moments are given by the fre-quency integral (cid:104) v i † k v il (cid:105) free = (cid:82) d ω π C iikl ( ω ), which can beevaluated exactly [52]. If the qubit is on resonance witha normal mode of the OMT we obtain for ∆ c = ω r ∆ th ∆ ≈ γ m N m G /κ + κ ω r , (B9)where we have neglected higher orders in κ/ω r , G/ω r aswell as non-RWA corrections to the thermal part. Fromthe low noise conditions identified in Sec. II E it followsthat | ∆ th | (cid:28) | ∆ | . If, however, the qubit is off-resonantwith the OM system as for G → | ∆ th | (cid:46) | ∆ | for most cases of interest and finally notethat ∆ th can be compensated to the same amount as theunavoidable shift ∆ by tuning of the bare qubit frequen-cies. Appendix C: Master equation for setup with fibercavity
Here, we derive the effective qubit dynamics for thecase of the on-chip setup discussed in Sec. IV. Note thatthe results given here apply to the case of identical nodes.
1. Network with N identical nodes According to the second term in Eq. (60), only the”center-of-mass” mode couples to the fiber cavity andwe therefore introduce normal modes for the cavities ac-cording to ˜ c i = V ij c j with V j = 1 / √ N , and similarly formechanical resonators and qubits. The remaining rowsof V are constructed from the constraint V T V = , suchthat the remaining terms in Eqs. (59),(60) are left invari-ant. As a result, we can write the QLEs describing theOM variables as N decoupled sets˙˜ v i ( t ) = − ˜ M i ˜ v i ( t ) − ˜ R i ( t ) , i = 1 . . . N , (C1)where ˜ v = (˜ b , ˜ c , d , ˜ b † , ˜ c † , d † ) T and ˜ v i ≥ =(˜ b i , ˜ c i , ˜ b † i , ˜ c † i ) T . Note that the systems i ≥ R i ( t ) can be constructedeasily from the original QLEs (57) and the or-thogonality of the transformation V ensures that they are mutually uncorrelated. We have ˜ R ( t ) =( √ γ m ˜ ξ , √ κ ˜ f , , (cid:112) κ f f , √ γ m ˜ ξ † , √ κ ˜ f † , , (cid:112) κ f f † ) T and ˜ R i ≥ ( t ) = ( √ γ m ˜ ξ i , √ κ ˜ f ,i , √ γ m ˜ ξ † i , √ κ ˜ f † ,i ) T .The statistics of these operators are given by (cid:104) ( ˜ R ik ( t )) † ˜ R il ( t (cid:48) ) (cid:105) = ˜ r ikl δ ( t − t (cid:48) ) with diagonal correlationmatrices ˜ r = diag[ γ m N m , , , γ m ( N m + 1) , κ , κ f ]and ˜ r i ≥ = diag[ γ m N m , , γ m ( N m + 1) , κ ]. Finally, thedrift matrices of the systems (C1) are ˜ M i ≥ = M , where M is defined in Eq. (15) with κ → κ and˜ M = i ω r G ∗ ζG G ∆ c K ∗ ζG K ∆ c − ζG ∗ − ω r − G − ζG ∗ − G ∗ − ∆ c − K − K ∗ − ∆ c + ˜ D , where ˜ D = diag[ γ m / , κ , κ f , γ m / , κ , κ f ]. Here, K = √ N h is the coupling of the center-of-mass mode to thebus cavity and the full OM coupling is again obtained for ζ = 1, while in RWA the off-diagonal blocks are dropped( ζ = 0).We proceed to eliminate the OM degrees of freedomby applying the results of App. A based on the inter-action H int = λ (cid:80) ν (˜ σ + ν ˜ b ν + h.c.) / N sets of normal modes described byEq. (C1) are mutually uncorrelated, i.e., (cid:104) ˜ b i ( t )˜ b † j (0) (cid:105) free ∝ δ ij , and we may hence eliminate them one by one. There-fore, we drop the superfluous second index on the cor-relation functions (A5)-(A8) and finally note that thecorrelators for sets i = 2 . . . N are all identical, that is,˜ X i ≥ ( ω ) = ˜ X ( ω ), etc. The effective ME (A4) may thenbe written in terms of collective and local contributions:˙ µ = − i (cid:2) J N S + S − , µ (cid:3) − i ∆2 [ S z , µ ] (C2)+ 12 (cid:110) Γ coll ( N coll + 1) D [ S − ] µ + Γ coll N coll D [ S + ] µ (cid:111) + 12 N (cid:88) i =1 (cid:110) Γ loc ( N loc + 1) D [ σ − i ] µ + Γ loc N loc D [ σ + i ] µ (cid:111) . Here, we have introduced the collective operators S ± = (cid:80) i σ ± i and S z = (cid:80) i σ iz , and the interaction and decayrates are given by J N = λ N Im { ˜ X − ˜ X } , ∆ = λ { ˜ X } , (C3)Γ coll = λ N Re { ˜ X − ˜ X } , Γ loc = λ { ˜ X } , (C4)where we neglected a thermal contribution to ∆. Thecollective and local occupation numbers are determinedby Γ coll N coll = λ N ( ˜ Y − ˜ Y ) , Γ loc N loc = λ Y , (C5)4and all correlation functions are understood to be eval-uated at the qubit frequency ω q . They can be eval-uated as for the cascaded setting and with ˜ A i ( ω ) =( ˜ M i − iω ) − we obtain ˜ X i ( ω ) = ˜ A i ( ω ) and ˜ Y i ( ω ) =[ ˜ A i ∗ ( ω )˜ r i ˜ A iT ( ω )] .
2. Two node network
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