CContinuous Variable Multipartite Vibrational Entanglement
Mehdi Abdi ∗ Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran
A compact scheme for the preparation of macroscopic multipartite entanglement is proposed and analyzed.In this scheme the vibrational modes of a mechanical resonator constitute continuous variable (CV) subsystemsthat entangle to each other as a result of their interaction with a two-level system (TLS). By properly driving theTLS, we show that a selected set of modes can be activated and prepared in a multipartite entangled state. Wefirst study entanglement properties of a three-mode system by evaluating the genuine multipartite entanglement.And investigate its usefulness as a quantum resource by computing the quantum Fisher information. Moreover,the robustness of the state against the qubit and thermal noises is studied, proving a long-lived entanglement. Toexamine the scalability and structural properties of the scheme, we derive an effective model for the multimodesystem through elimination of the TLS dynamics. This work provides a step towards a compact and versatiledevice for creating multipartite noise-resilient entangled state in vibrational modes as a resource for CV quantummetrology, quantum communication, and quantum computation.
I. INTRODUCTION
Quantum entanglement is a pivotal resource in many quan-tum technologies, from quantum communications [1–3] andquantum computations [4–6], to quantum metrology wherethe Heisenberg limit is approached only by employing amultipartite entangled state [7–9]. Meanwhile, as mechani-cal systems have proven to be very efficient in weak forcesensing both at macroscopic level [10, 11] and mesoscopicscales [12, 13], the quantum sensing thus would require in-vestigations on the entangled mechanical oscillators [14–20].From a different perspective, the entangled mechanical res-onators can serve as nodes of a quantum network [21–23].Interaction of the mechanical resonators with other quan-tum systems such as optical cavities [24–26], electronic [27,28], and spin degrees of freedom [29] opens an avenue foremploying their vibrational modes for quantum informationprocessing [30–33]. An opposite route is to use the me-chanical resonators for controlling [34–37] or intermediatinginteractions among other physical systems [38, 39]. Suchschemes have already been sought and used for developingscalable quantum networks [40, 41]. In either case, the sizeof the system can be considerably miniaturized—compared tothe electromagnetic counterparts—due to the smaller wave-length and stationary nature of the vibrational modes. Mul-timode optomechanical systems have been investigated fromvarious points of view both theoretically [42–44] and exper-imentally [45–48]. Recently the multimode circuit quantumacousto-dynamical (cQAD) systems based on bulk [49–51]and surface [52–56] acoustic resonators have been subject toan intense research, propelling towards the scalable quantuminformation with vibrational modes [57–61].In this work, we put forth a scheme that allows for creat-ing multipartite vibrational entangled states in a controllableway via their coupling to a two-level system (TLS). Insteadof several mechanical systems, we propose to employ thenormal modes of a single resonator. Thus, significantly re-ducing spatial extent, complexity, and noise of the system. ∗ [email protected] The TLS serves in mediating a controllable interaction be-tween the modes. The scheme is in principle implementablein surface acoustic wave (SAW) cavities and high overtonebulk acoustic resonators (HBAR) coupled to superconductingqubits in a cQAD system [62, 63] or can be envisaged in othersetups such as flexural modes of a membrane combined withan embedded optically active lattice defect [64, 65]. Here, westudy a regime where the TLS decay rate dominates its cou-pling to the vibrational modes and show that the TLS can beadiabatically eliminated from the system dynamics to obtain anetwork of effectively interacting modes with an adjustableinteraction structure. The latter is achieved by modulatingdrive frequency of the TLS at the eigenfrequencies of a set ofvibrational modes, see Fig. 1. We show that a multipartite en-tangled state of CV systems is attainable under experimentallyrealizable conditions and study its robustness against the sys-tem noises. Furthermore, by computing the quantum Fisherinformation (QFI) of the state we prove that these states areuseful for enhanced quantum sensing at the Heisenberg limit,e.g. for detection of gravitational waves and dark matter evi-dences [66–68].
II. MODEL
The frequency spectrum { ω k } ( k = , , · · · , N ) of the vi-brational eigenmodes of a mechanical resonator are deter-mined by its geometrical properties [65, 69]. These vibra-tional degrees-of-freedom can couple to a two-level system invarious schemes, e.g. a superconducting qubit through piezo-electric effects in a cQAD device [62] or to the electronic spintwo-level system of an embedded optically active lattice de-fect [70–74]. The dynamics of such a system is governed bythe Hamiltonianˆ H = ( ∆ ˆ σ z + Ω ˆ σ x ) + N ∑ k = [ ω k ˆ b † k ˆ b k + g k ˆ σ z ( ˆ b k + ˆ b † k )] , (1)where ˆ σ x and ˆ σ z are the Pauli matrices of a TLS driven witha Rabi frequency Ω and at detuning ∆ . The vibrational modesare expressed by the phonon annihilation (creation) opera-tors ˆ b k (ˆ b † k ) with the nontrivial canonical commutation re- a r X i v : . [ qu a n t - ph ] F e b FIG. 1. (a) A TLS coupled to multiple vibrational modes of a me-chanical resonator can intermediate their mutual interactions, whichallows for creating a multipartite entangled state among those modes.This can be implemented e.g. in: (b) A HBAR whose longitudinalmodes couple to a superconducting qubit via a piezoelectric device.Or (c) the flexural modes of a membrane interacting with an embed-ded atom-like defect. (d) In a multimode scheme the set of modesare activated (here 1 to 6) by employing a properly modulated drivesthat form two-mode squeezing interactions. The interaction is shownas a graph for two different modulation schemes: At the mode fre-quencies (e) and the half sum frequencies (f). The edge thicknessrepresents the interaction strength, while the active modes are high-lighted in yellow. lation [ ˆ b k , ˆ b † l ] = δ kl . The dimensionless canonical operatorsof each mode are related to these bosonic operators throughˆ b k = ( ˆ x k + i ˆ p k ) / √ [ ˆ x k , ˆ p l ] = i δ kl .Interaction of the TLS with the spectrum of the mechani-cal modes leads to their mutual interactions [Fig. 1(a)]. Thiscan be readily seen by adiabatic elimination of the TLS fromthe equations of motion. The process is valid when the TLSdecay rate Γ is greater than its coupling strength to any me-chanical mode of interest, for details see Appendix A. We alsoassume that the TLS is driven at far off-resonance ( ∆ (cid:29) Γ , Ω )and apply a proper polaron transformation [75]. The resultinginteraction Hamiltonian is not in resonance with the desiredinteractions, and thus, does not provide appreciable entangle-ments. To ‘activate’ the interactions, we propose to modulatethe TLS drive Ω ( t ) = Ω ∑ i cos ( w i t ) at the proper frequencies w j [33, 76]. This brings us at the following interaction Hamil-tonian after applying a rotating wave approximation (RWA)ˆ H RWA = ∑ k , l G k , l (cid:0) B tms k , l ˆ b k ˆ b l − B qst k , l ˆ b k ˆ b † l (cid:1) + H . c ., (2)where G k , l is the effective coupling rate and B tms and B qst arethe weighted adjacency matrices that determine the strengthof two-mode squeezing and quantum state transfer interac-tions, respectively, see Appendix B for the details. When themechanical spectrum has non-commensurate frequencies, anydesired subset of modes are activated by setting { w i } ⊆ { ω k } with the cardinality card { w i } = M , where M is the num-ber of active modes. In the language of graphs, the sys-tem forms a complete-graph. Such interactions with enoughstrength lead to a state which is equivalent to a multipar-tite CV GHZ-state [77–80]. In a commensurate spectrum, the other modes get involved in the interactions by the samechoice of modulation frequencies. However, their couplingstrength is smaller than the active modes and only negligiblycontribute in the entanglement dynamics [Fig. 1(e)]. Alterna-tively, one modulates the drive at half of the mode sum fre-quencies w i = ( ω k + ω l ) for getting a better connected graph[Fig. 1(f)]. Nonetheless, the numerical results suggest that theeffect is incremental at the cost of a more complicated modu-lation.In a HBAR cQAD device the longitudinal modes of a bulkacoustic wave cavity with high quality factors couple to asuperconducting qubit through a piezoelectric interface, seee.g. [51]. The HBAR modes form a spectrum of equallyspaced frequencies ω k = k δ FSR ( k = , , · · · ) with free spec-tral range δ FSR which is determined by the cavity length andmedium, hence, forming a commensurate spectrum. A fewtechniques can be envisaged for introducing anharmonicity tothe mode spacing [61], and thus, enhancing efficiency of theactivating protocol. However, here we set our focus on thesimple commensurate setup for experimental feasibility andconcreteness.
III. TRIANGLE SYSTEM
The full system is described by Hamiltonian (1) where thenonlinear essence of the qubit gives rise to non-Gaussianityof the mechanical state. As the simplest network, first we in-vestigate a triangle; a system made of three lowest vibrationalmodes [Fig. 1]. The dynamics is described by the quantumoptical master equation˙ ρ = i ¯ h (cid:2) ρ , ˆ H (cid:3) + Γ (cid:8) ( n ω q + ) D ˆ σ − [ ρ ] + n ω q D ˆ σ + [ ρ ] (cid:9) + (cid:101) Γ D ˆ σ z [ ρ ]+
12 3 ∑ k = γ k (cid:8) ( n ω k + ) D ˆ b k [ ρ ] + n ω k D ˆ b † k [ ρ ] (cid:9) , (3)where the Lindblad superoperators are D ˆ o [ ρ ] ≡ o ρ ˆ o † − ˆ o † ˆ o ρ − ρ ˆ o † ˆ o . Here, ω q is the TLS level splitting, while Γ and (cid:101) Γ are the qubit relaxation and decoherence rate, respectively.The mechanical damping rates are γ k ≡ ω k / Q with the qualityfactor Q . The bosonic thermal occupation number at temper-ature T is n ω = ( exp { ¯ h ω / k B T } − ) − with the Boltzmannconstant k B . In our analysis, we assume that the mechanicalmodes as well as the TLS are initialized in their ground-stateby some cooling mechanism [73, 74] and a modulated elec-tromagnetic drive excites the TLS.In numerical evaluation of Eq. (3), we consider a vibra-tional spectrum with δ FSR / π =
20 MHz and Q = inter-acting with the TLS ( ω q / π =
10 GHz and Γ / π =
20 MHz)with coupling rates g k = g (cid:46) Γ , δ FSR [48, 51, 55, 81]. Un-less specified otherwise, we take dilation refrigerator temper-ature of T =
10 mK for the reservoir, set ∆ = Ω for getting (cid:104) ˆ σ z (cid:105) ≈ − (cid:101) Γ =
0. By choosing anoptimal value for the drive power Ω and coupling strengths,a long-lived tripartite mechanical entangled state is attained t r i p a r ti t e s e p a r a b l e (d)(a) (b) . . . . . . . . . (c) FIG. 2. Time evolution of the TPE (a) for different coupling rates(the curve labels show g / Γ ) at the Rabi frequency Ω = Γ , and(b) for different Rabi frequencies at g = . Γ . (c) Robustness oflongtime E | | : Density plot showing the tripartite entangled andseparable parameter regions versus reservoir temperature and TLSpure dephasing rate (cid:101) Γ evaluated at t = τ FSR for Ω = Γ and g = . Γ . (d) The maximum normalized QFI as a function of cou-pling strength and drive amplitude. that sustains the qubit and thermal noise for thousands of thelongest mechanical period, τ FSR ≡ π / δ FSR , which is hun-dreds of nanoseconds.The results are summarized in Fig. 2, where we present thegenuine multipartite entanglement E | | and the normalizedquantum Fisher information F Q as defined in Refs. [82, 83]and given in Appendix E. In Fig. 2(a), time evolution of theentanglement in the three-mode system is plotted for differ-ent coupling rates, while in Fig. 2(b) its evolution is shownfor various drive amplitudes. The entanglement curves ex-hibit a fast rise followed by a rapid decay for large couplingrates and/or low Rabi frequencies, which survives for severalhundreds of oscillations. Remarkably, in the opposite regime(weaker g and/or higher Ω ) the system is dragged towardsa quasi-stationary tripartite entanglement (TPE). To study ro-bustness of the entanglement against TLS pure dephasing aswell as the thermal noise at higher temperatures, we take the E | | for g = . Γ and Ω = Γ at t = τ FSR as a rep-resentative long-living TPE and show the effect of these twomajor sources of noise in Fig. 2(c). Interestingly, the systemremains tripartite entangled, though fragile, for a wide rangeof noise parameters. In higher temperatures TLS thermaliza-tion is the main source of decoherence. In a setup with a highenough ω q TPE survives even up to T ∼
10 K, where the me-chanical noises destroys the entanglement (not shown). Fi-nally, we compute the maximum QFI in measuring collectivemechanical position ˆ X = ∑ i = ˆ x i in our scheme and study itsdependence on the coupling strength g and Rabi frequency Ω [Fig. 2(d)]. The results suggest that the QFI increases byoperating at high coupling strengths and drive amplitudes. In-terestingly, the maximum QFI even reaches the Heisenberg limit F Q =
3, which means at least 1 .
73 times enhancementin the sensitivity compared to a separable state, is obtainedfor moderate g values. It is worth mentioning here that themechanical entangled state is non-Gaussian thanks to the non-linear nature of the TLS. This is confirmed by computing theentropy distance of the density matrix from a reference Gaus-sian state as discussed in the Appendix E. IV. MULTIPARTITE ENTANGLED STATE
It is computationally expensive to study systems with largernumber of modes through (3) since it becomes memory inten-sive. Therefore, to expand our investigations to larger systemsand testing our strategy for generating a multipartite entan-gled state, we employ an effective Gaussian model. The effec-tive model is derived for both the Hamiltonian and the TLS-induced noise and damping on the mechanical modes throughadiabatic elimination of the TLS. Thanks to its Gaussian na-ture, the system is thus fully characterized by the covariancematrix (CM) V when displaced to the mean values of thecanonical operators. To find the dynamics of CM, one formsa vector of quadrature operators u = [ ˆ x , ˆ p , · · · , ˆ x M , ˆ p M ] (cid:124) andderives the quantum Langevin equation of its elements, seeAppendix C for the details. In the compact form one gets˙ u = − A u + n , where n is the vector of noise operators and thetime-dependent drift matrix is given by A ( t ) = κ − ω · · · ω + G , ( t ) κ · · · G , M ( t ) · · · κ M − ω M G M , ( t ) · · · ω M + G M , M ( t ) κ M , (4)where G k , l ( t ) ≡ ∆ ( n ω q + ) g k g l ω k ω l Ω ( t ) is the coupling strengthof the modes to each other in the effective model. Here, κ k = γ k + (cid:101) γ k is the total damping rate composed of the intrinsic γ k (a) (b)(c) (d) FIG. 3. Genuine N -partite entanglement for a set of six modes as afunction of the coupling strength g at Ω = Γ and t = τ FSR :(a) N =
3, (b) N =
4, (c) N =
5, and (d) N = FIG. 4. The bipartite entanglement between the fundamental modeand the k th mode E | k as a function of coupling rate g : (a) Logarith-mic negativity at T =
10 mK. (b) The entangled regions at five dif-ferent temperatures. In both plots the values correspond to Ω = Γ at t = τ FSR . and the TLS-induced damping rates (cid:101) γ k ≡ g k [ S ( ω k ) −S ( − ω k )] with S ( ω ) the TLS steady-state fluctuation spectrum [75].The CM of the system is readily computed via the followingequation ˙ V = AV + VA (cid:124) + D , (5)where D is the diffusion matrix, i.e. the matrix of noise cor-relators [84, 85]. In this work we assume that the noises areGaussian and Markovian. This brings us to D = (cid:76) Mk = D k with D k ≡ [ γ k ( n ω k + ) + (cid:101) γ k ( ˜ n ω k + )] I , where I is a 2 × n ω ≡ S ( − ω ) / [ S ( ω ) − S ( − ω )] .The chosen set of parameters ensures validity of the TLSadiabatic elimination. The convergence and reliability of thenumerical results are verified as outlined in Appendix D. Werestrict our analysis to M =
6, as this is the maximum num-ber of modes that we can afford given the computational re-sources. The genuine N -partite entanglement values (3 ≤ N ≤
6) are evaluated at t = τ FSR as a function of g and pre-sented in Fig. 3. Two sets for N = , , G k , l is proportional to the inverse mode fre-quencies, the fundamental mode has the strongest couplingto the active modes. From this perspective, the stronger en-tanglement exhibited in the first set with an almost monotonicgrowth with the coupling rate is intuitive. Therefore, the fun-damental mode plays a central role in structure of the entan-gled cluster. We should emphasize that the system is origi-nally nonlinear, and thus, one expects non-Gaussian essencein the above discussed multimode mechanical states that is notperceptible in our Gaussian effective model. V. SCALABILITY
The fundamental mode is central in the entanglement struc-ture of our scheme and at the same time it is the most vulnera-ble mode to the thermal noise. Therefore, it is crucial to learnabout the depth of its influence in the vibrational spectrum. In other words, how strongly does it entangle to the higher ordermodes of the resonator and how robust it is. Hence, we single-out the fundamental and k th modes to study their bipartite en-tanglement E | k [Fig. 4(a)]. The logarithmic negativity [86] ofthe bipartite system of ρ H ⊗ H k is evaluated for 100 modes at t = τ FSR as a function of g when the qubit is excited bya drive modulated at ω and ω k . In Fig. 4(b) the residual en-tanglement, the parameter region where E | k >
0, is presentedat different temperatures. The modes with lower frequenciesassume stronger entanglement to the fundamental mode as ex-pected. Remarkably, by increasing the temperature they re-tain their entanglement quite firmly, with small change in theamount (not shown). Meanwhile, the high-frequency modespreserve their entanglement with the fundamental mode, pro-vided the coupling strength to the intermediating TLS g islarge enough. Summary and conclusion.—
To summarize, we have intro-duced and studied a compact and versatile device for creatinga network of continuous variable multipartite entangled states.The vibrational modes of a single resonator constitute nodesof the network. These modes are coupled and entangled as aresult of their interaction with a driven-dissipative two-levelsystem. We have proposed to activate a desired set of modesby modulating the TLS drive tone at their respective frequen-cies. This results-in a long-living noise-resilient genuine mul-tipartite entangled state that is useful for quantum metrologyas the QFI of their state shows. We have also studied the struc-ture of entanglement for a hexagon network, proving the cen-tral role of the fundamental mode in the network. Finally, in-fluence of the fundamental mode on the mechanical spectrumshows that at the cryogenic temperatures hundreds of modescan join the network.
ACKNOWLEDGMENTS
The author acknowledges Martin B. Plenio for careful read-ing and comments on the manuscript. This work was sup-ported by Iran National Science Foundation (INSF) via grantNo. 98005028. The support by STDPO and IUT throughSBNHPCC is acknowledged.
Appendix A: Adiabatic elimination
In this appendix we present the details of our procedurein deriving the effective Hamiltonian (2) starting from theoriginal Hamiltonian (1). We start by removing the qubit-mechanical interaction by applying the polaron transforma-tion ˆ U = exp { i ˆ σ z ˆ P } with the collective dimensionless mo-mentum [87] ˆ P = − i ∑ k g k ω k ( ˆ b k − ˆ b † k ) . (A1)We are then brought toˆ U ˆ H ˆ U † = ∆ σ z + Ω (cid:0) ˆ σ + e − i ˆ P + ˆ σ − e + i ˆ P (cid:1) + ∑ k ω k ˆ b † k ˆ b k , (A2)where we have discarded a constant term.The adiabatic elimination is performed by assuming thatthe qubit decay rate is larger than its coupling strength to anyof the mechanical modes Γ (cid:29) { g k } . In this regime the to-tal density matrix can be approximated by ρ ( t ) (cid:39) ρ TLS ( t ) ⊗ ρ Mec ( t ) . Therefore, a mean-field approximation becomes ap-plicable. Then the following equations describe the qubit dy-namics [88] (cid:104) ˙ˆ σ + (cid:105) = − (cid:16) Γ ( n ω q + ) − i ∆ (cid:17) (cid:104) ˆ σ + (cid:105) − i Ω e + i (cid:104) ˆ P (cid:105) (cid:104) ˆ σ z (cid:105) , (A3a) (cid:104) ˙ˆ σ − (cid:105) = − (cid:16) Γ ( n ω q + ) + i ∆ (cid:17) (cid:104) ˆ σ − (cid:105) + i Ω e − i (cid:104) ˆ P (cid:105) (cid:104) ˆ σ z (cid:105) , (A3b) (cid:104) ˙ˆ σ z (cid:105) = − Γ (cid:16) + ( n ω q + ) (cid:104) ˆ σ z (cid:105) (cid:17) − i Ω (cid:0) (cid:104) ˆ σ + (cid:105) e − i (cid:104) ˆ P (cid:105) − (cid:104) ˆ σ − (cid:105) e + i (cid:104) ˆ P (cid:105) (cid:1) , (A3c)where for the sake of simplicity an ideal qubit with no pure de-phasing (cid:101) Γ = (cid:104) ˆ σ ± (cid:105) ss = (cid:104) ˆ σ z (cid:105) ss = (cid:104) ˆ σ ± (cid:105) ss = ± i Ω (cid:16) Γ ( n ω q + ) ∓ i ∆ (cid:17) e ± i (cid:104) ˆ P (cid:105) ( n ω q + ) (cid:16) Γ ( n ω q + ) + ∆ + Ω (cid:17) , (A4a) (cid:104) ˆ σ z (cid:105) ss = − Γ ( n ω q + ) + ∆ ( n ω q + ) (cid:16) Γ ( n ω q + ) + ∆ + Ω (cid:17) . (A4b)A simple set of equations is found by operating the system atthe far detuned regime where ∆ (cid:29) Γ , Ω (cid:104) ˆ σ ± (cid:105) ss ≈ Ω ∆ ( n ω q + ) e ± i (cid:104) ˆ P (cid:105) , (A5a) (cid:104) ˆ σ z (cid:105) ss ≈ − . (A5b)By plugging these in (A2) for the qubit operators we arrive atˆ H ad = ∑ k ω k ˆ b † k ˆ b k + Ω ∆ ( n ω q + ) cos ( ˆ P − (cid:104) ˆ P (cid:105) ) , (A6)after discarding a constant shift in the energy. This Hamilto-nian is valid for the above discussed regime and adequatelyaddresses the system dynamics after the TLS passes its tran-sient dynamics t > Γ − . The collective momentum then isexpected to exhibit small fluctuations around its mean valueTr (cid:8) ( ˆ P − (cid:104) ˆ P (cid:105) ) ρ Mec ( t ) (cid:9) (cid:28)
1. This allows us to expand the co-sine and keep up to the quadratic termscos ( ˆ P − (cid:104) ˆ P (cid:105) ) ≈ − ( ˆ P − (cid:104) ˆ P (cid:105) ) = − (cid:0) ˆ P + (cid:104) ˆ P (cid:105) − (cid:104) ˆ P (cid:105) ˆ P (cid:1) . The second term in the parentheses gives a constant energy,while the third causes a shift in the phase space. We skip both and writeˆ H ad ≈ ∑ k ω k ˆ b † k ˆ b k + ∑ k , l G k , l (cid:16) ˆ b k ˆ b l − ˆ b k ˆ b † l + H.c. (cid:17) , (A7)where the effective coupling rate is G k , l ≡ Ω ∆ ( n ω q + ) g k g l ω k ω l . (A8)To check the validity of mean-field approximation in thediscussed regime, we numerically solve the qubit equations in(A3) replacing ˆ P with a real number and compare the resultswith the solution of quantum optical master equation (3). InFig. A1 the two solutions are compared against each other.One clearly verifies that the solutions are very close in theadiabatic elimination regime.On the dissipation side of the dynamics, the TLS-mechanical interaction causes extra dissipation in the mechan-ical modes. This is formulated effectively through second or-der perturbation theory [89]. The additional damping rate foreach mechanical mode is then (cid:101) γ k ≡ g k [ S ( ω k ) − S ( − ω k )] . Ac-cording to the quantum dissipation-fluctuation theorem thereis an accompanying thermal noise with occupation number˜ n ω k ≡ S ( − ω k ) / [ S ( ω k ) − S ( − ω k )] . Here, the TLS steady-state fluctuation spectrum is given by S ( ω ) = Re (cid:90) ∞ ds (cid:2) (cid:104) ˆ σ z ( s ) ˆ σ z ( ) (cid:105) ss − (cid:104) ˆ σ z (cid:105) (cid:3) e i ω s . (A9)The spectrum is evaluated using quantum regression theo-rem [88]. The resulting expression is cumbersome. There-fore, we only summarize the equations that lead to it in thesame lines of Ref. [75]. We express the spectrum of TLS as S ( ω ) = Re { C ( s = − i ω ) } , where C is the third element ofthe following vector C ( s ) = ( sI − M ) − v , (A10)where I is a 3 × M and vector v are defined through Eqs. (A3) and their steady-state solutionsas MF MF
FIG. A1. Comparison of the expectations values for the TLS opera-tors: the exact (solid lines) versus mean-field approximation (dashedlines), which are solutions to Eqs. (A3). See the text for details. M = − Γ ( n ω q + ) + (cid:101) Γ + i ∆ − i Ω − Γ ( n ω q + ) + (cid:101) Γ − i ∆ + i Ω − i Ω i Ω − Γ ( n ω q + ) , v = −(cid:104) ˆ σ + (cid:105) ss ( + (cid:104) ˆ σ z (cid:105) ss )+ (cid:104) ˆ σ − (cid:105) ss ( − (cid:104) ˆ σ z (cid:105) ss ) − (cid:104) ˆ σ z (cid:105) . (A11)These TLS-induced damping and noises are taken into ac-count in studying the dynamics of effective fully mechanicalsystem, see Appendix C. Appendix B: Rotating Wave Approximation
The Hamiltonian (A7) provides us with interactions of two-mode squeezing and state-transfer between all modes, andthus, is potentially rich for quantum information processingpurposes. However, none of the above interactions are onresonance. This becomes clear by moving to the interactionpicture of ˆ H = ∑ k ω k ˆ b † k ˆ b k . In this appendix we show that aselected set of modes can resonantly brought into desired in-teraction by applying a modulated drive to the qubit. In otherwords, by replacing the Ω with Ω ( t ) = Ω ∑ i cos w i t , (B1)with properly chosen modulation frequencies w i that will be-come clear, shortly. To show this, we move to the interactionpicture of ˆ H and arrive at (cid:101) H ad ≈ ∑ k , l G k , l ( t ) (cid:2) ˆ b k ˆ b l e − i ( ω k + ω l ) t − ˆ b k ˆ b † l e − i ( ω k − ω l ) t + H.c. (cid:3) , where now G k , l are time dependent through the Rabi fre-quency, see Eq. (A8). The drive in (B1) then gives us Ω ( t ) = Ω ∑ i , j cos ( w i t ) cos ( w j t )= Ω ∑ i , j (cid:110) cos (cid:2) ( w i + w j ) t (cid:3) + cos (cid:2) ( w i − w j ) t (cid:3)(cid:111) = Ω ∑ i , j (cid:104) e i ( w i + w j ) t + e i ( w i − w j ) t + c.c. (cid:105) . In the numerical analysis that is performed in the manuscriptfor multimode systems larger than three we use Hamiltonian(A7) and derive the Langevin equations, see Appendix C.Nonetheless, it is instructive to see that how a time dependentdrive can link the mechanical modes to each other. There-fore, by setting the above equation back in (cid:101) H ad and perform-ing a rotating wave approximation (RWA), only few of themode interactions can survive. When the mode frequenciesare non-commensurate the desired set of modes with frequen-cies { ω k } can be activated by simply setting w k = ω k for anyset of modes. This givesˆ H RWA = M ∑ k , l G k , l ˆ p k ˆ p l , (B2)where G k , l is the same as (A8) but with Ω replaced by Ω .Note that the interaction with and among the rest of spectrum become counter-rotating and thus are ignored. An alternativemodulation strategy is to drive at half of the sum frequenciessuch that w i = ω k , l : = ( ω k + ω l ) . In this case the effectiveHamiltonian in RWA is in the following formˆ H (cid:48) RWA = M ∑ k , l G k , l ( ˆ x k ˆ x l + ˆ p k ˆ p l ) , (B3)Concerning the two-mode squeezing interaction, bothschemes result the same interaction graphs as it is conceivedfrom Fig. B1.Nevertheless, in our proposed setup the set of mode fre-quencies { ω k } are commensurate. This indeed involves a fewundesired modes in the system dynamics. Another cause ofthe commensurate spectrum is that the coupling rate coef-ficients become weighted as some interactions receive drivefrom other drive tones. Therefore, the RWA Hamiltonian inour setup becomesˆ H RWA = ∑ k , l G k , l (cid:0) B tms k , l ˆ b k ˆ b l − B qst k , l ˆ b k ˆ b † l + H.c. (cid:1) , (B4)where B tms and B qst are the adjacency matrices of the two-mode squeezing and quantum state-transfer interactions, re-spectively. Fig. B2 shows the weight of each interaction viaan illustrative presentation of the matrix elements of the ad-jacency matrix F k , l ≡ G k , l B tms k , l for two different modulationschemes described above. We notice that even though themodulated drive is tuned for a set of target modes (the greensquare), the other modes are also connected (the blue square).By taking into account the inverse frequencies that appear inthe interaction strengths, the effect of these higher modes be-comes negligible. This indeed is also shown as the graphs inFig. 1 in the main text. (c)(b)(a) FIG. B1. (a) When the spectrum is non-Commensurate any desiredset of modes can be activated by simply modulating the TLS driveon their frequencies (red arrows) or at half of the sum frequencies(green arrows). The activated modes form a complete graph, thoughweighted, for the two-mode squeezing interaction in both modulationstrategies as illustrated for in: (b) mode frequencies modulation and(c) half of the sum frequencies modulation. (a) (b) (c) (d)
FIG. B2. Normalized matrix elements of the effective two-mode squeezing coupling matrices F k , l : (a) and (b) for commensurate spectrumwith modulations at half sum frequencies and the mode frequencies, respectively. In (c) and (d) the same are shown for a noncommensuratespectrum. The green square highlights the target set of modes, whereas the blue square illustrates the set of activated modes that are directlycoupled to the target set. Appendix C: Mechanical Langevin equations
The Hamiltonian (A7) is Gaussian and allows us to studythe system dynamics through the covariance matrix of the me-chanical canonical operators. The quantum Langevin equa-tions (QLEs) include the effect of environment on dynamicsof the system operators are used to obtain time evolution ofthe covariance matrix. We find the following QLEs for themotion of the mechanical modes:˙ b k = − ( κ k + i ω k ) ˆ b k − i ∑ l G k , l ( t )( ˆ b l + ˆ b † l ) + ˆ ξ k , (C1)where κ k ≡ γ k + (cid:101) γ k is the total damping rate stemming from thesupport, γ k , and the TLS decoherences, (cid:101) γ k , as discussed in Ap-pendix A. The total noise operator is also divided into the sup-port noise and the TLS induced noise ˆ ξ k ≡ √ γ k ˆ b in k + (cid:112)(cid:101) γ k ˜ b in k .Assuming that the intrinsic motional noise is Markovian, thenoise operator has the following non-vanishing correlationfunctions (cid:104) ˆ ξ k ( t ) ˆ ξ † l ( t (cid:48) ) (cid:105) = (cid:2) γ k ( n ω k + ) + (cid:101) γ k ( (cid:101) n ω k + ) (cid:3) δ ( t − t (cid:48) ) δ kl , (C2a) (cid:104) ˆ ξ † k ( t ) ˆ ξ l ( t (cid:48) ) (cid:105) = (cid:2) γ k n ω k + (cid:101) γ k (cid:101) n ω k (cid:3) δ ( t − t (cid:48) ) δ kl , (C2b)By forming a vector of Hermitian operators u ≡ [ ˆ x , ˆ p , · · · , ˆ x M , ˆ p M ] (cid:124) , one arrives at the following com-pact form of Eqs. (C1) ˙ u = A u + n , (C3)where A is given by (4) and n = [ ˆ ξ x , ˆ ξ p , · · · , ˆ ξ xM , ˆ ξ pM ] is thevector of noise operators with ˆ ξ k ≡ ( ˆ ξ xk + i ˆ ξ pk ) / √
2. The co-variance matrix dynamics is readily calculated from (C3) asEq. (5) in the main text [85].
Appendix D: Numerical method
For performing the numerical integration over the quantumoptical master equation (3) we use the QuTiP package [90].The Hilbert space H truncation was done by minimizing theerror. Our estimation for the error by tracking the convergencecurve shows that for a truncation at dim {H} = × × × ≈ .
73% is committed, which is theminimum possible error given the computational resources athand.The study of the original Hamiltonian with more than threemechanical modes is not feasible because of the huge de-mands for the computational memories. We, therefore, haveturned to use the Gaussian effective Hamiltonian which isvalid in the regime discussed in Appendices A and C. To nu-merically prove its validity, we compare the results obtainedfrom (3) and (5). In Fig. D1 the bipartite entanglement E | obtained from the original and effective models are plotted forthree different coupling strength values. The results confirmsthat the behavior of both models coincides for weak couplingstrengths. This is indeed as one would expect as the adiabaticelimination is valid when the TLS arrives at its quasi-steadystate. The deviation emerges as the coupling rate assumeshigher values. Such that the effective model overestimates theTLS-induced decoherence. Appendix E: Entanglement, quantum Fisher information, andnon-Gaussianity
In this appendix we deal with the details of the entangle-ment properties, represent the measures used in the main text,and study the non-Gaussianity of the entangled state.In order to analyze the entanglement structure of the sys-tem, we first show that our scheme indeed leads to a GHZ-state. For the sake of clarity and simplicity the case of non-commensurate modes for which the interaction is pure ˆ p k ˆ p l isdiscussed, see Eq. (B2). The results are easily generalized to asystem with commensurate mode spectrum. In the interactionpicture, after a time interval t the Hamiltonian (B2) transformsthe canonical operators asˆ x k → ˆ x k − t ∑ l G k , l ˆ p l , (E1a)ˆ p k → ˆ p k . (E1b)The effective coupling constant G k , l is inversely proportionalto the mode frequencies G k , l ∝ ( ω k ω l ) − . Therefore, the modewith the lowest frequency couples more strongly to the othernodes in the network [Fig. B1]. This suggests a star-shaped FIG. D1. Comparison of the entanglement E | as an illustration of the consistency: the original Hamiltonian (orange) versus the effectiveHamiltonian (blue) for three different coupling strengths. Here, Ω = Γ and the other parameters are the same as in the main text. graph for the network with the lowest mode at the center.Therefore, the position transformations can be approximatelysimplified to ˆ x → ˆ x − ∑ k χ k ˆ p k for the ‘first’ mode andˆ x k → ˆ x k − χ k ˆ p for the rest with χ k ≡ tG , k . The entanglement of a state isnot affected by local unitary transformations. Hence, after alocal π / − rotation on the first-mode one gets ˆ x → ˆ p andˆ p → − ˆ x . Then it is easy to verify that state of the system ata given time t becomes such that M ∑ k = ˆ p k ( t ) = M ∑ k = ˆ p k ( ) e − χ k , (E2a)ˆ x k ( t ) − ˆ x l ( t ) = ˆ x k ( ) e − χ k − ˆ x l ( ) e − χ l , (E2b)with ( k , l = , , · · · M ) . After a long enough drive one has χ k → ∞ , where the system reaches a Greenberger-Horne-Zeilinger entangled state [77, 79].We use the genuine multipartite entanglement measure in-troduced in Ref. [82] to quantify the entanglement in our sys-tem. The measure exploits monogamy property of the entan-glement such that E | , , ··· , M = M ∑ j = E | j + M ∑ k > j M ∑ j = E | j | k + · · · + E | |···| M , where the underline denotes the focus party and E is a propermeasure of entanglement [91]. The genuine residual N -partiteentanglement is then calculated as the minimum over all per-mutations of the subsystem indices E | |···| M ≡ min { E i | i |···| i M } . (E3)The quantum Fisher information determines the Cramer-Rao bound in parameter estimation and saturates to the Heisenberg limit for a fully entangled system [9]. Hence, itreflects the degree of the multipartite entanglement [92–94].For a mixed state ρ and observable ˆ O the QFI is defined as F Q [ ρ , ˆ O ] = ∑ k , l |(cid:104) k | ˆ O | l (cid:105)| ( λ k − λ l ) λ k + λ l , (E4)where λ k and | k (cid:105) are the eigenvalues and eigenvectors of ρ , re-spectively. The sum is over indices that λ k + λ l > X M ≡ ∑ Mk = ˆ x k asthe observable and introduce F Q ≡ M F Q , the normalized QFI.This quantity is then upperbounded by M for a fully entangledsystem.Finally, in order to examine the non-Gaussian nature ofthe entangled state in the triangle system we compute thenon-Gaussianity measure of the state using the one intro-duced in Ref. [95]: δ NG of a given state ρ is defined as thedistance of its entropy from a reference Gaussian state ρ G : δ NG ≡ S ( ρ G ) − S ( ρ ) where S ( ρ ) = − Tr { ρ log ρ } is the vonNeumann entropy. 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