Continuous mode cooling and phonon routers for phononic quantum networks

The mechanical quantum channel which is used to communicate between the nodes can, in general, be represented by a chain of N_ch coupled mechanical resonators,

$$\begin{equation} H_{\rm channel}= \sum_{\ell=1}^{N_{\rm ch}} \frac{p_\ell^2}{2m} +\frac{1}{2} m \omega_0^2 x_\ell^2\, +\, \frac{k}{2}\sum_{\ell=1}^{N_{\rm ch}-1} (x_\ell-x_{\ell+1})^2. \end{equation} \tag{ 3 }$$

Here x_ℓ and p_ℓ are the position and momentum operators, m is the effective mass, ω₀ is the bare oscillation frequency and the spring constant k accounts for the nearest-neighbor coupling. For small N_ch the coupled resonators form a discrete set of collective eigenmodes. In this case quantum state transfer and quantum information processing protocols between two qubits can be implemented by addressing only a single collective mode, as has been discussed in the context of trapped ion systems [26] or coupled nanomechanical [23] and OM [28] resonators.

In this paper, we are primarily interested in the opposite regime of extended arrays, N_ch ≫ 1. In this limit H_channel can be represented by a dense set of plane wave modes, $H_{\rm channel} = \sum _q \omega _q b_q^\dagger b_q$, where [b_q,b^†_q'] = δ_q,q', ω_q is the phonon dispersion relation, and for a lattice spacing a, the momentum label q is restricted to the first Brillouin zone $q\in (-\frac {\pi }{a},\frac {\pi }{a}]$. This scenario is realized, for example, in large arrays of coupled nanomechanical beams [37] or in phononic band gap structures [32], where each resonator ℓ corresponds to a vibrational mode of a unit cell, typically of size a ∼ 1 μm. In the continuum limit we can identify a frequency range Δω away from the band edges in which the coupled resonator array exhibits an approximately linear dispersion $\omega _q\approx \tilde \omega _0 + c |q|$, where c is the effective speed of sound and $\tilde \omega _0$ a frequency offset. In this case it is convenient to introduce the normalized displacement field Φ(z) = Φ_R(z) + Φ_L(z), where

$$\begin{equation} \Phi_{\rm R} (z) =\sqrt{\frac{2c}{L}} \sum_{q>0} {\rm e}^{ {\rm i}q z} b_q, \quad \Phi_{\mathrm{L}} (z)=\sqrt{\frac{2c}{L}} \sum_{q>0} {\rm e}^{ -{\rm i}q z} b_{-q}, \end{equation} \tag{ 4 }$$

describe right- and left-moving mechanical excitations of the phononic channel. Under the assumption of a linear dispersion these fields obey [Φ_R(z,t),Φ^†_L(z',t')] = 0 and

$$\begin{equation} [ \Phi_{{\rm R/L}}(z,t), \Phi^\dagger_{{\rm R/L}}(z',t')] \simeq {\rm e}^{-{\rm i}\tilde \omega_0 (t-t')} \delta( t-t' \mp (z-z')/c). \end{equation} \tag{ 5 }$$

By assuming that the frequency of the local mechanical modes ω_m also lies within this frequency range, the coupling between the local resonators and the waveguide modes can be approximated by

$$\begin{equation} H_{\rm int}\simeq \sqrt{\frac{\gamma}{2}} \int_0^{L} \mathrm{d}z\, \sum_{i=1}^N \delta(z-z_i) \left( b_i^\dagger \Phi(z) + b_i \Phi^\dagger(z)\right)\kern-2pt, \end{equation} \tag{ 6 }$$

where L is the length of the waveguide and z_i are the positions of the nodes along the waveguide (see figure 1(b)). Below we identify γ as the total decay rate of the local resonator modes into the waveguide. Note that equations (5) and (6) are valid for times |t − t'| > Δω⁻¹ and distances |z − z'| > a. An explicit and more detailed derivation of these results is given in appendix A for the case of a simple coupled resonator chain.

Under the validity of equations (5) and (6) and in the limit where ω_m and the bandwidth Δω are large compared to the other characteristic frequency scales, we can eliminate the waveguide modes and use an input–output formalism [38] to describe the effective dynamics of the coupled nodes. Using (5) and (6) and making a standard Born–Markov approximation, we can derive a set of coupled quantum Langevin equations (QLEs) [38]. For each node we obtain

$$\begin{equation} \dot b_i= \mathrm{i} [H_{\rm node},b_i] - \frac{\gamma}{2} b_i - \sqrt{\gamma_{\rm R} }b^{\rm R}_{i,{\rm in}}(t)- \sqrt{\gamma_{\rm L} }b^{\rm L}_{i,{\rm in}}(t), \end{equation} \tag{ 7 }$$

where γ = γ_R + γ_L is the total decay rate of phonons into the waveguide. For side-coupled phonon cavities we would usually have γ_R = γ_L, but below we describe scenarios where the emission either to the left or to the right is effectively switched off. In equation (7) we have defined incoming fields (see figure 1(c))

$$\begin{equation} b^{\rm L}_{i,{\rm in}}= \lim_{\epsilon \rightarrow 0} \Phi_{\rm L}(z_i+\epsilon),\quad b^{\rm R}_{i,{\rm in}}= \lim_{\epsilon \rightarrow 0} \Phi_{\rm R}(z_i-\epsilon), \end{equation} \noindent \tag{ 8 }$$

which specify the left- and right-moving waveguide modes before they interact with b_i. Similarly, we introduce the scattered outgoing fields

$$\begin{equation} b^{\rm L}_{i,{\rm out}}= \lim_{\epsilon \rightarrow 0} \Phi_{\rm L}(z_i-\epsilon),\quad b^{\rm R}_{i,{\rm out}}= \lim_{\epsilon \rightarrow 0} \Phi_{\rm R}(z_i+\epsilon). \end{equation} \noindent \tag{ 9 }$$

The dynamics of the whole network is then described by the QLEs (7) together with the input–output relations at each node,

$$\begin{equation} b^{{\rm R,L}}_{i, {\rm out}}(t)= b^{{\rm R,L}}_{i,{\rm in}}(t) +\sqrt{\gamma_{{\rm R,L}}} b_i(t), \end{equation} \noindent \tag{ 10 }$$

and the free propagation

$$\begin{equation} b^{\rm R}_{i, {\rm in}}(t)= b^{\rm R}_{i-1,{\rm out}}(t-\tau_{i,i+1}),\quad b^{\rm L}_{i, {\rm in}}(t)= b^{\rm L}_{i+1,{\rm out}}(t-\tau_{i,i+1}), \end{equation} \noindent \tag{ 11 }$$

where τ_i,i+1 = |z_i+1 − z_i|/c. Under realistic conditions we must also account for intrinsic phonon losses, backscattering and rethermalization, which we address below.

The phonon network described above is formally equivalent to optical quantum networks and can, in principle, be used in a similar way to transfer quantum states from one node to another [29, 31]. However, in contrast to high-frequency optical fields, the thermal population $N_{\rm th}= (\exp (\hbar \omega _{\rm m}/k_{\rm B} T) -1)^{-1}$ of the phonon modes is usually not negligible. For phonon frequencies in the MHz to GHz regime and temperatures T ∼ 0.1–1 K, the number of thermal noise phonons in the waveguide will exceed the quantum signal encoded in a single phonon. In addition, as shown in figure 1(c), phonons emitted from a single side-coupled node will naturally propagate along two directions, i.e. γ_L = γ_R. In larger networks or 2D arrangements, the inability to route propagating phonons severely limits an efficient implementation of quantum communication protocols.

In the remainder of the paper, we consider an extended setup as shown in figure 2 and describe how the integration of additional OM control elements can be used to overcome these fundamental limitations of phononic networks. The two key ingredients in this setup are the following:

• (i)  
  an OM noise filter, which is used to suppress thermal noise within the relevant transmission bandwidth;
• (ii)  
  coherent OM phonon circulators or phonon routers, which allow for directed propagation and switchable routing of phonons through the network.

Zoom In Zoom Out Reset image size

Before addressing the suppression of thermal noise in extended phonon waveguides, let us first briefly review the standard scenario for OM cooling [39, 40], where the frequency ω_c of an optical cavity mode is modulated by the displacement of a single mechanical mode. The optical cavity is driven by a coherent laser field of frequency ω_d = ω_c + δ and in a frame rotating with ω_d the Hamiltonian for the OMS is given by

$$\begin{equation} H_{\rm om}= - \delta a^{\dagger}a+\omega_{{\rm m}}b^{\dagger}b+ga^{\dagger}a(b+b^{\dagger})+\mathrm{i}\,\mathcal{E}(a^{\dagger}-a). \end{equation} \tag{ 12 }$$

Here the bosonic operators a and b represent the optical and mechanical modes, respectively, ω_m is the mechanical oscillation frequency, g is the OM coupling constant and $\mathcal {E}$ is the strength of the external driving field. Terms rotating with the optical frequency have been neglected by a rotating wave approximation (RWA). The OM interaction, as described by the third term in equation (12), derives from a linear dependence of the optical resonance frequency on the position quadrature of the mechanical mode [41, 42]. Including dissipation through cavity decay and intrinsic mechanical losses, the dynamics of the OMS is described by the QLEs

$$\begin{equation} \dot{a}=(\mathrm{i}\delta-\kappa)a-\mathrm{i}g a (b+b^{\dagger})+\mathcal{E} -\sqrt{2\kappa}a_{\rm in}, \end{equation} \tag{ 13 }$$

$$\begin{equation} \dot{b}=(-\mathrm{i}\omega_{{\rm m}}-\gamma_0/2)b-\mathrm{i}g a^{\dagger}a-\sqrt{\gamma_0} b_{0,{\rm in}}, \end{equation} \tag{ 14 }$$

where κ is the optical field decay rate and γ₀ = ω_m/Q₀ is the mechanical damping rate for an intrinsic mechanical quality factor Q₀. The δ-correlated operators a_in and b_0,in represent the vacuum input noise of the optical field and the thermal mechanical noise, respectively. Within the relevant frequency range the latter is characterized by a non-vanishing equilibrium occupation number 〈b^†_0,in(t)b_0,in(t')〉 = N_thδ(t − t').

In the limit of strong driving, the external field displaces both the optical and the mechanical modes by a large classical amplitude $\alpha =\langle a\rangle =\mathcal {E}/(\kappa -{\rm i}\delta (\alpha ))$ and β = 〈b〉 = −g|α|²/ω_m, where δ(α) = δ + 2g²|α|²/ω_m. For |α| ≫ 1, we can make a unitary transformation a → α + a and b → β + b and linearize the OM coupling around the classical mean values,

$$\begin{equation} H_{\rm om}\simeq -\delta a^{\dagger}a+\omega_{{\rm m}}b^{\dagger}b+g(\alpha a^{\dagger}+\alpha^{\ast}a)(b+b^{\dagger})\kern-2pt, \end{equation} \tag{ 15 }$$

where we have redefined δ(α) → δ. If the external driving field is (near) resonant with the red mechanical sideband of the optical cavity, i.e. if δ ≃ − ω_m, and if ω_m ≫ κ,|gα| we can make an RWA with respect to ω_m and obtain a beam-splitter Hamiltonian

$$\begin{equation} H_{\rm om}\simeq H_{\rm bs}= (\omega_{{\rm m}}+\delta)b^{\dagger}b+g(\alpha b a^{\dagger}+\alpha^{\ast}b^{\dagger}a), \end{equation} \tag{ 16 }$$

which describes a (near) resonant conversion of phonons into photons (and vice versa). Combined with the decay from the optical cavity, it allows for cooling: incident (noise) phonons are up-converted to photons and decay from the optical cavity.

The QLEs for the linearized OMS can be conveniently solved in the Fourier domain as outlined in more detail in appendix B. In the relevant weak coupling and sideband resolved regime (|gα| < κ ≪ ω_m = −δ), we then obtain the standard result for the mechanical fluctuation spectrum,

$$\begin{equation} \langle b^\dagger(\omega) b(\omega') \rangle\simeq \left[ \frac{(\gamma_0+\gamma_{\rm op}) \bar N }{(\omega-\omega_{\rm m})^2+(\gamma_0+\gamma_{\rm op})^2/4}\right]\delta(\omega-\omega'), \end{equation} \tag{ 17 }$$

where γ_op = 2|gα|²/κ is the optically induced mechanical damping rate and $\bar N=N_{\rm th} \gamma _0/(\gamma _0+\gamma _{\rm op})+ \kappa ^2/(4\omega _{\rm m}^2)$ the reduced steady-state resonator occupation number. In the following, we are interested in OMS where both γ_op ≫ γ₀ and single-mode ground-state cooling $\bar N < 1$ can be achieved.

The single-mode cooling scheme can be generalized to cool mechanical networks consisting of a few coupled resonators only one of which is OM cooled, as shown in figure 4(b). In this case the linearized OM Hamiltonian is given by

$$\begin{equation} H_{\rm om}\simeq -\delta a^{\dagger}a + \sum_{j=1}^N \omega_{{\rm m}}b_j^{\dagger}b_j- \sum_{\langle i,j\rangle } K_{i,j} \left(b_i b_{j}^\dagger + b_i^\dagger b_{j}\right) +g\alpha\left(a^{\dagger} b_1+ ab_1^\dagger\right)\kern-2pt, \end{equation} \tag{ 18 }$$

where the K_i,j denote the nearest-neighbor couplings and we have already performed an RWA assuming that K_i,j,g|α|,κ ≪ ω_m. The resulting QLEs can be written as

$$\begin{equation} \dot{a}=({\rm i}\,\delta-\kappa)a-{\rm i}\,\alpha g b_1-\sqrt{2\kappa}a_{\rm in}, \end{equation} \tag{ 19 }$$

$$\begin{equation} \dot{b}_1=-({\rm i}\,\omega_{{\rm m}}+\gamma_0 / 2)b_1+{\rm i}\,K_{1,2} b_2 -{\rm i}\,g\alpha^{\ast}a -\sqrt{\gamma_0}b_{0,{\rm in}}^{(1)}, \end{equation} \tag{ 20 }$$

$$\begin{equation} \dot{b}_{1<j<N}=-({\rm i}\,\omega_{{\rm m}}+\gamma_0 /2)b_j+\mathrm{i}(K_{j-1,j} b_{j-1} + K_{j,j+1}b_{j+1}) -\sqrt{\gamma_0}b_{0,{\rm in}}^{(j)}, \end{equation} \tag{ 21 }$$

$$\begin{equation} \dot{b}_N=-({\rm i}\,\omega_{{\rm m}}+\gamma_0 / 2)b_N+\mathrm{i}\, K_{N-1,N} b_{N-1} -\sqrt{\gamma_0}b_{0,{\rm in}}^{(N)} , \end{equation} \tag{ 22 }$$

where the b^(j)_0,in are mutually uncorrelated noise operators associated with intrinsic mechanical damping of each mode.

Zoom In Zoom Out Reset image size

In figure 4(c) we plot the correlation spectrum 〈b^†_N(ω)b_N(ω')〉 = S(ω)δ(ω − ω') for a chain of N = 10 resonators with K_i,i+1 = K assuming that the b₁ mode is optically cooled. For reference the dashed line shows the undamped case gα = 0. In this case the heights of the peaks at frequencies ω_n = ω_m − 2Kcos(nπ/(N + 1)) are given by S(ω_n) = (4N_th/γ₀)|c_n(N)|², where $c_n(j)=\sqrt {2/(N+1)}\sin ( n j\pi /(N+1))$ are the normalized mode distributions. When the OM coupling is turned on, the peaks are broadened and their height is reduced, which corresponds to an overall cooling of the individual eigenmodes. However, we see that cooling occurs in a highly non-uniform way and only the few modes close to the cavity resonance are cooled efficiently. While the details depend on the ratios between K, g|α| and κ, we find that this behavior is quite generic and a precursor to the features we identify in the following for the continuous waveguide limit.

Starting from the multi-mode setting shown in figure 4(b), let us now address the limit N → ∞ of a continuous mode waveguide as depicted in figure 3 by retaining the OM coupling to the first mode b ≡ b₁, but describing the other phonon modes b_j>1 in terms of continuous left- and right-propagating fields Φ_L,R(z). If K_1,2 ≪ K_i,i+1 ≡ K, we can adapt the input–output formalism introduced in section 2 to model the coupling between the localized mode b and the rest of the waveguide in terms of the scattering relation

$$\begin{equation} b_{\rm out}(t)= b_{\rm in}(t) + \sqrt{\gamma} b(t). \end{equation} \tag{ 23 }$$

Here b_in(t) = Φ_L(z → 0,t) and b_out(t) = Φ_R(z → 0,t) are the incident and reflected waveguide fields and γ ≈ K²_1,2/K is the characteristic phonon decay rate into the waveguide. Ignoring other, intrinsic loss channels for the moment, we obtain

$$\begin{equation} \dot{a}=({\rm i}\delta-\kappa)a-{\rm i}\alpha g (b+b^{\dagger})-\sqrt{2\kappa}a_{\rm in}, \end{equation} \tag{ 24 }$$

$$\begin{equation} \dot{b}=-({\rm i}\omega_{{\rm m}}+\gamma/2)b-\mathrm{i}g(\alpha a^{\dagger}+\alpha^{\ast}a)-\sqrt{\gamma}b_{{\rm in}} , \end{equation} \tag{ 25 }$$

for the linearized OM dynamics of the local mode and we see that the problem of cooling a continuous waveguide is formally identical to the single-mode cooling considered above. However, instead of looking at the occupation of the local mode b we are now interested in the spectrum of the reflected waveguide field b_out(t), given a thermal incident field 〈b^†_in(t)b_in(t')〉 = N_thδ(t − t'). To do so, we solve the QLEs in Fourier space and write the result as

$$\begin{equation} \vec{A}_{\rm out}(\omega)=\mathcal{S}(\omega)\vec{A}_{\rm in}(\omega), \end{equation} \tag{ 26 }$$

where $\vec {A}_{j}=(a_{j}(\omega ),a^{\dagger }_{j}(-\omega ),b_{j}(\omega ),b^{\dagger }_{j}(-\omega ))^{\rm T}$ with j = 'in', 'out'. The matrix $\mathcal {S}(\omega )$ is a 4 × 4 scattering matrix and a more detailed derivation of equation (26) is given in appendix B. For given input noise correlations $C_{\rm in}(\omega ,\omega ^\prime )=\langle \vec{A}_{\rm in}(\omega )\otimes \vec{A}_{\rm in}^{\rm T}(\omega ')\rangle$, we obtain the output correlation matrix $C_{\rm out}(\omega ,\omega ^\prime )=\langle \vec{A}_{\rm out}(\omega )\otimes \vec{A}_{\rm out}^{\rm T}(\omega ')\rangle =\mathcal {S}(\omega )C_{\rm in}(\omega ,\omega ')\mathcal {S}^{\rm T}(\omega ')$.

3.3.1. Optomechanical noise filters

The quantity that we are interested in is the filtered noise spectrum N_F(ω) of the reflected field, which is defined by

$$\begin{equation} \langle b_{\rm out}^{\dagger}(\omega) b_{\rm out}(\omega')\rangle=N_{\rm F}(\omega) \delta(\omega-\omega'). \end{equation} \tag{ 27 }$$

Figure 5(a) shows the typical frequency dependence of N_F(ω). As expected, we see a strong suppression of thermal noise around the mechanical frequency ω_m. For δ = −ω_m and under the condition |gα| < κ ≪ ω_m, the dynamics of the OMS is well approximated by the beam-splitter Hamiltonian (16) and we obtain

$$\begin{eqnarray} &&N_{{\rm F}} (\omega)\simeq N_{\rm th}\left(1- \frac{4 \kappa^2 \gamma_{\rm op} \gamma}{\kappa^2(\gamma_{\rm op} + \gamma)^2 +(\gamma-2\kappa)^{2}(\omega-\omega_{{\rm m}})^2+4(\omega-\omega_{{\rm m}})^{4}}\right). \end{eqnarray} \tag{ 28 }$$

We see that, under these approximations, the 'impedance matching' condition γ_op = γ [25] results in a complete cancellation N_F(ω = ω_m) = 0 of the reflected noise on resonance. In this case the optical decay rate matches the mechanical waveguide coupling and the OMS acts effectively as a double-sided phonon cavity with the thermal mechanical bath on the one side and the optical T = 0 bath on the other side.

Zoom In Zoom Out Reset image size

Under realistic conditions the noise cancellation described by equation (28) will not be perfect, and to account for various imperfections in the system, we assume in our discussion below a spectrum of the form

$$\begin{equation} N_{{\rm F}} (\omega)= N_{\rm th}- (N_{\rm th}-N_0) \frac{ \tilde \gamma^2}{(\omega-\tilde \omega_{\rm m})^2\,+\,\tilde \gamma^2}, \end{equation} \tag{ 29 }$$

where $\tilde \gamma$ and $\tilde \omega _{\rm m}$ now include small corrections of the width and the center of the dip and N₀ is a non-vanishing noise floor. Intrinsic limitations that lead to a finite N₀ are already contained in the full OM interaction itself, where for any finite ω_m off-resonant Stokes-scattering terms in the linearized Hamiltonian (15) induce corrections to the ideal beam-splitter interaction. However, as shown in figure 5(b), by accounting for these effects up to second order in 1/ω_m we only obtain a small shift of the spectral dip $\tilde \omega _{\rm m}\simeq \omega _{\rm m}-g^2 |\alpha |^2/(2\omega _{{\rm m}})^2$ and an offset N₀ ≃ |gα|²/(4ω²_m), which is negligible under the weak coupling conditions mentioned above. A more crucial limitation for the OM noise filter arises from intrinsic mechanical losses γ₀ of the localized mechanical mode, which can be accounted for by introducing an additional decay rate γ₀ and the associated noise operator b_0,in(t) in the QLE (25) (see appendix B). This leads to a finite N₀ ≃ 4N_thγγ₀/(γ_op + γ + γ₀)². This means that, while for the optimized case with γ = γ₀ + γ_op the local mode b can be highly excited, also the condition γ₀N_th/γ_op ≪ 1 must be fulfilled to achieve good noise filtering.

3.3.2. Propagation losses and scattering

The spectrum N_F(ω) determines the mechanical noise of the outgoing displacement field b_out(t) ≡ Φ_R(z → 0,t) immediately after the OM device and for an ideal phonon waveguide this noise spectrum will be the same at all positions z > 0. However, in a realistic setting, scattering losses in the waveguide and the associated noise lead to a rethermalization of the noise spectrum and N_F(ω) → N_F(ω,z). For an approximately linear dispersion relation, the propagation of the outgoing field can be modeled by

$$\begin{equation} \left(\frac{\partial}{\partial t}+ c \frac{\partial}{\partial z}\right) \Phi_{\rm R}(z,t) = -\left({\rm i}\tilde \omega_0 +\frac{c}{2l_\gamma} \right) \Phi_{\rm R}(z,t) -\frac{c}{ \sqrt{l_\gamma}} \Phi_{\rm th}(z,t), \end{equation} \tag{ 30 }$$

where l_γ is the phonon mean free path in the waveguide and Φ_th(z,t) is a thermal noise field with 〈Φ^†_th(z,t)Φ_th(z',t')〉 = N_thδ(z − z')δ(t − t'). In appendix A.1 we outline a microscopic derivation of this result for a coupled mechanical resonator array, where l_γ = c/γ₀ can be connected to the intrinsic damping rate γ₀ of the individual resonators in the array. For z > 0 and in a frame rotating with $\tilde \omega _0$, we obtain

$$\begin{eqnarray} &&\fl \Phi_{\rm R}(z,t) = \mathrm{e}^{-\frac{z}{2l_\gamma} } \Phi_{\rm R}\left(0,t-\frac{z}{c}\right) - \frac{1}{\sqrt{ l_{\gamma}}} \int_0^z \mathrm{d}z' \, \mathrm{e}^{-\frac{(z-z')}{2l_\gamma} }\Phi_{\rm th}\left(z',t-\frac{(z-z')}{c}\right)\kern-2pt, \end{eqnarray} \tag{ 31 }$$

and from this result we can derive the full noise spectrum along the waveguide

$$\begin{equation} N_{\rm F}(\omega,z)= \mathrm{e}^{-\frac{z}{l_\gamma} } N_{\rm F}(\omega,0) + N_{\rm th}\left(1- \mathrm{e}^{-\frac{z}{l_\gamma}}\right)\kern-2pt. \end{equation} \tag{ 32 }$$

This result describes the rethermalization of the noise dip on a length scale given by the phonon mean free path. For the relevant distances z ≪ l_γ, the noise spectrum can still be described by the Lorentzian shape given in equation (29), but the noise floor N₀(z) ≃ N₀ + (z/l_γ)N_th increases linearly with the distance to the OM noise filter. Note that similar quantitative conclusions follow from the standard approach of treating propagation losses in terms of a series of beam splitters [25, 38], but the present analysis also provides a direct connection to the underlying microscopic phonon scattering mechanism.

To the extent that the scatterers are linear, phonon backscattering gives rise to small overall losses, which are on an equal footing with the thermalization discussed above.

As a first step we describe the implementation of an effective qubit network with tunable qubit–waveguide couplings Γ_1,2(t) by eliminating the dynamics of the local phonon modes. We start from the full set of QLEs, which, for each node j = 1,2, derive from the Hamiltonian in (2) and are given by

$$\begin{equation} \dot{\sigma}_-^j=-{\rm i}\Delta_{\rm q}(t) \sigma_-^j + {\rm i}\,\lambda_j(t)\sigma_z^j b_j, \end{equation} \tag{ 33 }$$

$$\begin{equation} \dot{\sigma}_z^j=-2{\rm i}\,\lambda_j(t)( \sigma_+^j b_j - b_j^\dagger \sigma_-^j) , \end{equation} \tag{ 34 }$$

$$\begin{equation} \dot{b}_j=-{\rm i}\,\omega_{\rm m} b_j-\frac{\gamma}{2} b_j -{\rm i}\,\lambda_j(t)\sigma^j_- -\sqrt{\gamma}b_{j,{\rm in}}. \end{equation} \noindent \tag{ 35 }$$

The input–output relations are

$$\begin{equation} b_{j, {\rm out}}(t)= b_{j,{\rm in}}(t) + \sqrt{\gamma} b_j(t)\quad\mathrm{and}\quad b_{2,{\rm in}}(t)= b_{1,{\rm out}}(t-\tau_{12}), \end{equation} \noindent \tag{ 36 }$$

where b_1,in(t) ≡ b_in(t) describes the incident waveguide field before interacting with the first node. Hereafter, we absorb the retardation time τ₁₂ by redefining time-shifted operators and control fields for the second node [38] and for simplicity we focus on the resonant case in which Δ^j_q(t) = ω_m.

We are interested in the regime in which the decay γ of the the local phonon modes into the waveguide is fast compared to the coupling λ(t). This allows us to adiabatically eliminate the phonon modes and to derive an effective description in terms of the qubit degrees of freedom only. In the frame rotating with ω_m and to lowest order in λ, (35) can be solved to give

$$\begin{equation} b_j(t)\simeq -\frac{2\mathrm{i}\,\lambda_j(t)}{\gamma} \sigma_-^j(t)- \sqrt{\gamma} \int_{-\infty}^t \mathrm{d}s \, \mathrm{e}^{-\frac{\gamma}{2}(t-s)} \, \mathrm{e}^{{\rm i}\,\omega_{\rm m} s} b_{j,{\rm in}}(s). \end{equation} \noindent \tag{ 37 }$$

After reinserting this expression into equations (33) and (34), we obtain

$$\begin{equation} \dot{\sigma}_-^j=- \frac{\Gamma_j(t)}{2} \sigma_-^j -\sqrt{\Gamma_j(t)} \sigma_z^j B_{j,{\rm in}}, \end{equation} \tag{ 38 }$$

$$\begin{equation} \dot{\sigma}_z^j=-\Gamma_j(t)\left(\mathbbm{1} +\sigma_z^j\right) + 2\sqrt{\Gamma_j(t)}\left( \sigma_+^jB_{j,{\rm in}} +B_{j,{\rm in}}^\dagger \sigma_-^j\right). \end{equation} \noindent \tag{ 39 }$$

Here Γ_j(t) = 4λ²_j(t)/γ are tunable qubit decay rates and

$$\begin{equation} B_{j,{\rm in}}(t)=\frac{\mathrm{i}\,\gamma}{2} \int_{-\infty}^t \mathrm{d}s \, \mathrm{e}^{-\frac{\gamma}{2}(t-s)} \, \mathrm{e}^{{\rm i}\,\omega_{\rm m} s}b_{j,{\rm in}}(s), \end{equation} \noindent \tag{ 40 }$$

the associated effective noise operators which on the timescale γ⁻¹ obey [B_j,in(t),B^†_j,in(t')] ≈ δ(t − t'). Provided that λ(t) and σ¹₋(t) also vary slowly on this time scale, we find that

$$\begin{equation} B_{2,{\rm in}}(t)\simeq - B_{1,{\rm in}}(t) + \sqrt{\Gamma_1(t)} \sigma_-^1(t). \end{equation} \noindent \tag{ 41 }$$

Thus, we obtain a new set of effective quantum network equations for the qubit operators with tunable decay rates Γ_j(t). We emphasize that while in the following this effective description allows a simplified discussion of the state-transfer protocol, it is not necessary to consider the regime λ ≪ γ and a perfect state transfer can also be achieved using the full model [29].

Our goal is to implement a quantum state transfer between the two qubits, i.e.

$$\begin{equation} |\psi_0\rangle=(\alpha|0\rangle_1+\beta |1\rangle_1)|0\rangle_2 \rightarrow |0\rangle_1(\alpha |0\rangle_2+ \beta|1\rangle_2), \end{equation} \noindent \tag{ 42 }$$

which is achieved via coherent emission and reabsorption of a single phonon in the waveguide. A solution to this problem was first described for optical networks [29], where it has been shown that by identifying appropriate pulses Γ_j(t), a perfect quantum state transfer can be implemented. Let us briefly summarize the main idea behind this scheme, for the moment assuming zero temperature. In this case the waveguide is initially in the vacuum state |vac〉 and the dynamics of the whole system can be restricted to the zero- and one-excitation subspace. Then, for an initial two-qubit state |ψ₀〉, we can define the amplitudes v_j(t) = 〈vac|〈0₁,0₂|σ^j₋(t)|ψ₀〉|vac〉. From the QLEs (38) and (39) and the input–output relation (41), it follows that these amplitudes evolve according to

$$\begin{equation} \dot v_1(t)=-\frac{\Gamma_{1}(t)}{2}v_1(t), \end{equation} \tag{ 43 }$$

$$\begin{equation} \dot v_2(t)=-\frac{\Gamma_{2}(t)}{2}v_1(t)-\sqrt{\Gamma_{1}(t)\Gamma_2(t)}v_1(t), \end{equation} \noindent \tag{ 44 }$$

and for initial amplitudes v_j(t₀), the general solutions is given by

$$\begin{equation} v_1(t)=\mathcal{G}_{1}(t,t_{0})v_1(t_0), \end{equation} \tag{ 45 }$$

$$\begin{equation} v_2(t)=\mathcal{G}_{2}(t,t_{0})v_2(t_0)+\mathcal{T}(t,t_{0})v_1(t_0). \end{equation} \noindent \tag{ 46 }$$

Here $\mathcal {G}_{j}(t,t_{0})=\mathrm {e}^{-\int _{t_{0}}^{t}\mathrm {d}s\;\Gamma _{j}(s)/2}$ and

$$\begin{equation} \mathcal{T}(t,t_{0})=-\int_{t_{0}}^{t}\mathrm{d}t'\; \mathcal{G}_{2}(t,t')\sqrt{\Gamma_{1}(t')\Gamma_{2}(t')}\mathcal{G}_{1}(t',t_{0}) \end{equation} \tag{ 47 }$$

is the state-transfer amplitude. For the initial state given in equation (42), v₂(t₀) = 0, and therefore a perfect transfer is achieved if at some final time t_f the amplitudes approach

$$\begin{equation} \mathcal{G}_1(t_{\rm f},t_0)\rightarrow 0\quad\mathrm{and}\quad |\mathcal{T}(t_{\rm f},t_0)|\rightarrow 1. \end{equation} \tag{ 48 }$$

While the first condition is easily fulfilled for sufficiently long but otherwise arbitrary pulses, satisfying the second one requires control over their shape. Perfect state transfer is only possible if the total number of excitations is conserved, i.e. if no population gets lost from the one-excitation subspace. This means that the norm $\mathcal {G}_1^2(t,t_0)+\mathcal {T}^2(t,t_0)=1$ for all times, which after taking the time derivative of this condition yields

$$\begin{equation} \sqrt{\Gamma_{1}(t)}v_1(t)+\sqrt{\Gamma_{2}(t)}v_2(t)=0. \end{equation} \tag{ 49 }$$

This result also follows from the requirement that the total scattered field after the second node vanishes at all times, i.e. B_2,out(t)|ψ₀〉|vac〉 = 0. Therefore equation (49) is often referred to as the dark-state condition.

A set of pulses Γ₁(t) and Γ₂(t) that realize a perfect state transfer can always be found numerically by choosing Γ₁(t) such that $\mathcal {G}_{1}(t_{{\rm f}},t_{0})\rightarrow 0$ and then solve equations (43) and (44) iteratively, choosing Γ₂(t) at each time such that the dark-state condition (49) is fulfilled. Further, simple analytical expressions for pulses that realize a perfect state transfer may be obtained from a time-inversion argument [29]. Without loss of generality, we can assume that [t_f,t₀] = [ − τ_p/2,τ_p/2], where τ_p is the pulse length. A control pulse Γ₁(t) for the first qubit gives rise to a wave packet $a(t)=\sqrt {\Gamma _{1}(t)}\mathcal {G}_{1}(t,t_{0})v_{1}(t_{0})$ in the waveguide. For reasons of time-inversion symmetry, the inverted wave packet a(−t) is fully absorbed if the reverse pulse Γ₁(−t) is applied. This implies that, in the special case of a time-symmetric wave packet a(t) = a(−t), the wave packet generated at the first node will be fully absorbed if we choose Γ₂(t) = Γ₁(−t). Describing the symmetric wave packet by the differential equation $\dot {a}(t)=g(t)a(t)$, where g(t) is anti-symmetric g(t) = −g(−t), one can derive the following differential equation for the pulse-shape Γ₁(t) from (43) [43]:

$$\begin{equation} \dot{\Gamma}_{1}=\Gamma_{1}^{2}+2g(t)\Gamma_{1}. \end{equation} \tag{ 50 }$$

For the simplest choice of g(t), i.e. g(t) = sign(t)(Γ_max/2), where Γ_max is the maximal decay rate, the solution reads

$$\begin{equation} \Gamma_{1}(t)=\left\{\begin{array}{@{}c@{\quad}c@{\quad}c@{}} \left(\frac{\mathrm{e}^{-\Gamma_{\rm max}t}}{2-\mathrm{e}^{-\Gamma_{\rm max}t}}\right)\!\Gamma_{\rm max} & {\rm if} & t<0\\ \Gamma_{\rm max} & {\rm if} & t\geqslant 0\end{array}\right.\quad\mathrm{and}\quad \Gamma_{2}(t)=\Gamma_{1}(-t). \end{equation} \tag{ 51 }$$

We will use this specific pulse shape for our numerical examples discussed below.

Now we consider the same state-transfer problem but in the case when the in-field of the quantum channel is characterized by a non-vanishing noise spectrum 〈B^†_in(ω)B_in(ω')〉 = N(ω)δ(ω − ω'), which can be either a flat thermal background N(ω) = N_th or the filtered noise dip N(ω) ≃ N_F(ω) as discussed in section 3.3.

4.3.1. Effective thermal occupation number

In the presence of thermal excitations in the quantum channel we must necessarily go beyond the single excitation subspace and a closed analytic solution to the state-transfer problem is no longer available. To gain more intuition on the impact of noise on the state-transfer fidelity, let us first consider a single qubit. We assume that at time t = t₀ the qubit is prepared in its ground state and we then switch on the coupling to the waveguide Γ₁(t), for example, using the pulse shape defined in equation (51). In the regime where the noise amplitude is low, N(ω) ≪ 1, we can linearize the QLEs (38) and (39) and obtain for the final qubit excitation 〈σ¹₊(t_f)σ¹₋(t_f)〉 ≃ N_eff ≪ 1, where

$$\begin{eqnarray} &&N_{\rm eff}=\int_{t_0}^{t_{\rm f}} \mathrm{d}t_1 \mathrm{d}t_2 \,\sqrt{\Gamma_1(t_1)\Gamma_1(t_2)} \mathcal{G}_1(t_{\rm f},t_1) \mathcal{G}_1(t,t_2) \langle B^\dagger_{{\rm in}}(t_1)B_{{\rm in}}(t_2)\rangle. \end{eqnarray} \tag{ 52 }$$

In the case of incident δ-correlated thermal noise and assuming that the pulse duration is sufficiently long, i.e. t_f − t₀ ≫ Γ⁻¹_max, we find that N_eff = N_th. This means that during the state transfer an erroneous excitation or de-excitation probability for the qubits of ∼N_th can be expected.⁵ In turn, for the filtered noise spectrum N_F(ω) defined in equation (29) we obtain

$$\begin{eqnarray} &&N_{\rm eff}=\int_{-\infty}^\infty {\rm d}\omega \, |F(\omega)|^2 N_{\rm F}(\omega) \simeq\frac{2\gamma N_{0}+\Gamma_{\rm max}N_{\rm th}}{2\gamma+\Gamma_{\rm max}}, \end{eqnarray} \tag{ 53 }$$

where in general $F(\omega )=\frac {1}{\sqrt {2\pi }}\int _{t_0}^{t_{\rm f}} \mathrm {d}t \mathrm{e}^{{\rm i}\omega t}\sqrt{\Gamma _1(t)}\mathcal {G}_1(t_{\rm f},t)$ and the integral in equation (53) has been evaluated for the pulse shape given in equation (51). This shows that using the OM noise filter to clean the waveguide, the effective occupation number can be considerably reduced compared to N_th, assuming that N₀ ≪ 1 and that the bandwidth of the transfer pulse fits within the noise dip. This is illustrated in figure 6, where we plot the spectral overlap between |F(ω)|² and N_F(ω) and the resulting N_eff for different ratios γ/Γ_max.

Zoom In Zoom Out Reset image size

4.3.2. Discussion

To verify our approximate analytic predictions we convert the cascaded QLEs into an equivalent cascaded master equation [38] and simulate the full state transfer numerically. Since a master equation for the two qubits can only be derived for δ-correlated noise, we include the OM noise filter as a third subsystem to emulate the spectral variations of the incident noise, as outlined in appendix C. In figure 6(c) and (d) we simulate the transfer of a single-qubit excitation from node 1 to node 2 and compare the case of a thermal quantum channel where N(ω) = N_th to the case when the OM noise filter is switched on and N(ω) = N_F(ω). We see that even for N_th = 0.5 the thermal noise in a phonon quantum network already almost completely washes out the transferred signal, while in combination with the noise filter a high-fidelity state transfer is still possible. In figure 6(d) we also compare the full numerical results with a master equation for the qubits only, assuming a δ-correlated noise with an effective occupation number N_eff. We find that apart from small corrections when pulses are switched on and off, this reduced model describes the state transfer very accurately, which shows that, indeed, N_eff is the relevant parameter for a quantum channel with a filtered noise spectrum.

As a specific example, we consider a potential realization of phonon networks using the on-chip OM structures discussed in [25]. We assume local phonon modes of frequency ω_m = 2π × 4 GHz and a decay into the waveguide of γ = 2π × 25 MHz. For a quality factor Q₀ = 10⁶ the intrinsic decoherence rate is γ₀ = 2π × 4 kHz. For these parameters we plot in figure 7 the fidelity $\mathcal {F}$ for transferring the superposition state $|\psi \rangle =(|0\rangle +|1\rangle )/\sqrt {2}$ between two nodes of a phonon network as a function of N_th and for different effective qubit decay rates γ₀ ≪ Γ_max < γ. For the numerical simulations we have used the effective model with N_eff as obtained in equation (53) and with N₀ = γ₀N_th/γ. The fidelity is defined as the overlap between the target state and the actual state after the transfer, i.e. $\mathcal {F}={\rm Tr}\{ \rho _{\rm tar}\rho (t_{{\rm f}})\}$. We see that compared to the case where no cooling is applied and the fidelity already drops significantly for N_th ∼ 0.1, the presence of the OM noise filter can push this bound to much larger occupation numbers. In particular, in the example considered here, this means that instead of requiring temperatures of T ⩽ 100 mK, the OM cooling scheme allows the implementation of high-fidelity state-transfer protocols at much more convenient liquid helium temperatures T = 4 K where N_th ∼ 20. Note that for the assumed value of Γ_max = 10⁻²γ = 2π × 250 kHz, the total transfer time of ∼1 μs is compatible with decoherence times that are achievable with various different solid-state qubits. A few specific examples will be discussed in section 6.

Zoom In Zoom Out Reset image size

To illustrate the basic idea of a non-reciprocal phonon device let us consider the minimal setting of a three-port circulator as shown in figure 8(a). The localized phonon modes b_j=1,2,3 are mutually tunnel coupled with a strength |t| and coupled to phonon waveguides with a decay rate γ. The Hamiltonian is given by

$$\begin{equation} H_{\rm circ} =\sum_i \omega_{\rm m} b^\dagger_i b_i + t\!\left(b_{1}b_{2}^{\dagger} {\rm e}^{{\rm i}\varphi} +b_{2}b_{3}^{\dagger} +b_{3}b_{1}^{\dagger} +{\rm H.c.} \right), \end{equation} \tag{ 54 }$$

assuming that the resonator frequencies ω_m are all equal. The crucial term in this setup is a complex tunneling amplitude te^iφ between two of the resonators, which cannot be absorbed into local redefinitions of the b_i. Therefore, this phase can be associated with an effective magnetic field for the phonon modes. Previously, such a setting was described for superconducting microwave cavities, where the effective magnetic field can be engineered using superconducting qubits and external fluxes [44, 45]. Below we describe a non-magnetic approach to achieve this complex hopping amplitude in OMS.

Zoom In Zoom Out Reset image size

Including the coupling to the waveguides and in a frame rotating with the resonator frequencies ω_m, the QLEs read

$$\begin{equation} \left(\begin{array}{@{}c@{}}\dot{b_{1}}\\ \dot{b_{2}}\\ \dot{b_{3}} \end{array}\right)=-\left(\begin{array}{@{}ccc@{}} \gamma/2 & {\rm i}\, t\,{\rm e}^{-{\rm i}\varphi} & \mathrm{i}\,t\\ {\rm i}\, t\,{\rm e}^{{\rm i}\varphi} & \gamma/2 & \mathrm{i}\, t \\ \mathrm{i} \,t & \mathrm{i}\,t & \gamma/2 \end{array}\right)\left(\begin{array}{@{}c@{}}b_{1}\\ b_{2}\\ b_{3}\end{array}\right) - \sqrt{\gamma} \left(\begin{array}{@{}c@{}}b_{1,{\rm in}}\\ b_{2,{\rm in}}\\ b_{3,{\rm in}}\end{array}\right), \end{equation} \tag{ 55 }$$

with the input–output relations $b_{j,\rm out}=b_{j,\rm in}+\sqrt {\gamma }b_{j}$. The above set of QLEs can be solved in Fourier space and the input relation can be applied to obtain the full scattering matrix $\mathcal {S}(\omega )$ of this system. By choosing φ = π/2 and t = γ/2, such that the decays into the waveguides match the tunneling amplitudes, we find that for frequencies around ω ≃ ω_m (ω ≃ 0 in the rotating frame), the scattering matrix is given by

$$\begin{equation} \left(\begin{array}{@{}c@{}}b_{1,\rm out}\\ b_{2,\rm out}\\ b_{3,\rm out}\end{array}\right)= \left(\begin{array}{@{}ccc@{}}0 & 1 & 0\\ 0 & 0 & \mathrm{i}\\ \mathrm{i} & 0 & 0 \end{array}\right)\left(\begin{array}{@{}c@{}}b_{1,\rm in}\\ b_{2,\rm in}\\ b_{3,\rm in}\end{array}\right). \end{equation} \tag{ 56 }$$

Up to factors i, which can be absorbed in the definitions of the operators, equation (56) describes a perfect circulator. For φ = −π/2 we obtain a similar result, but with the scattering direction reversed, i.e. b_1,out = i b_3,in, b_2,out = b_1,in and b_3,out = i b_2,in. In figure 8(c) we plot the scattering probabilities $|\mathcal {S}_{1\rightarrow j}(\omega )|^2$, defined as $b_{j,{\rm out}}(\omega )= \mathcal {S}_{1\rightarrow j}(\omega ) b_{\rm 1,in}(\omega )$, as a function of the frequency ω (relative to ω_m) and for t = γ/2. We see the emergence of the ideal circulator relations close to resonance. We also find that these features are robust when we add an additional intrinsic loss rate γ₀ = γ/20.

To implement a complex tunnel amplitude te^iφ between two localized mechanical modes, we propose a setup as shown in figure 8(b). Here, the localized phonon mode b₃ is coupled to modes b₁ and b₂ mechanically with a (real) tunneling amplitude t, while the tunneling between b₁ and b₂ is mediated by two driven optical cavities. The Hamiltonian for this system is

$$\begin{eqnarray} H_{\rm circ} &=&\sum_{i=1}^3 \omega_{\rm m} b_i^\dagger b_i +t (b_1 b_3^\dagger +b_2 b_3^\dagger + {\rm H.c.}) + H_{\rm c} + H_{\rm om}. \end{eqnarray} \tag{ 57 }$$

In the frame rotating with the frequency of an external driving field, the Hamiltonian for the two coupled optical cavities is

$$\begin{equation} H_{\rm c}= -\sum_{i=1,2} \delta_i a_i^\dagger a_i - J (a_1 a_2^\dagger + a_1^\dagger a_2) +\mathrm{i} \sum_{i=1,2} \mathcal{E}_i \left(a_i^\dagger {\rm e}^{{\rm i}\phi_i} - {\rm H.c.}\right)\kern-2pt, \end{equation} \tag{ 58 }$$

where $\mathcal {E}_i$ are the strengths and ϕ_i the phases of the optical driving fields. Finally,

$$\begin{equation} H_{\rm om} = \sum_{i=1,2} g a_i^\dagger a_i (b_i+b_i^\dagger) \end{equation} \tag{ 59 }$$

describes the local OM interactions.

As discussed in section 3, in the limit of strong driving we can change to a displaced representation and linearize the OM coupling around the classical expectation values α_i = 〈a_i〉. In the present case, the classical field amplitudes are given by

$$\begin{equation} \alpha_1= \frac{(\kappa -{\rm i} \delta_2 )\mathcal{E}_1 {\rm e}^{{\rm i}\phi_1}+ \mathrm{i} J \mathcal{E}_2 {\rm e}^{{\rm i}\phi_2}}{(\kappa- {\rm i}\delta_1)(\kappa- {\rm i}\delta_2) +J^2}, \end{equation} \tag{ 60 }$$

$$\begin{equation} \alpha_2=\frac{(\kappa -{\rm i} \delta_1 )\mathcal{E}_2 {\rm e}^{{\rm i}\phi_2}+ \mathrm{i} J \mathcal{E}_1 {\rm e}^{{\rm i}\phi_1}}{(\kappa- {\rm i}\delta_1)(\kappa- {\rm i}\delta_2) +J^2}, \end{equation} \tag{ 61 }$$

where we have assumed that both cavities decay with the same rate κ and we have absorbed nonlinear shifts of the detunings by a redefinition of δ_i. We write α_i = |α_i|e^iφ_i and obtain the linearized OM coupling

$$\begin{equation} H_{\rm om} = \sum_{i=1,2} g |\alpha_i|( {\rm e}^{-{\rm i}\varphi_i}a_i + {\rm e}^{{\rm i}\varphi_i}a_i^\dagger) (b_i+b_i^\dagger). \end{equation} \tag{ 62 }$$

Note that for given δ_i, J and κ the external control parameters $\mathcal {E}_i$ and ϕ_i provide enough flexibility to independently adjust the phases φ_i and keep |α₁| ≈ |α₂|.

We focus on the specific case δ_i = δ, |α_i| = α and introduce symmetric and anti-symmetric optical modes $a_{\pm }=(a_{1}\pm a_{2})/\sqrt {2}$ with detunings δ_± = δ ± J. Further, we assume that −δ_± ≈ ω_m ≫ g|α_i|, which allows us to make a RWA with respect to ω_m. Then, after changing to a frame rotating with ω_m, the total Hamiltonian is given by

$$\begin{eqnarray} &&\fl H_{\rm circ}= \!-\! \sum_{\nu=\pm} \Delta_\nu a_\nu^\dagger a_\nu + t\!(b_1 b_3^\dagger +b_2 b_3^\dagger + {\rm H.c.}) +\frac{g \alpha}{\sqrt{2}}[ a^\dagger_+({\rm e}^{{\rm i} \varphi_1} b_1+ {\rm e}^{{\rm i}\varphi_2} b_2) + a^ \dagger_-({\rm e}^{{\rm i} \varphi_1} b_1\!-\!{\rm e}^{{\rm i}\varphi_2} b_2) + {\rm H.c.}],\nonumber\\ \end{eqnarray} \tag{ 63 }$$

where Δ_ν = δ_ν + ω_m. In a final step we assume that |Δ_ν| ≫ gα and treat the OM coupling using second-order perturbation theory. Apart from small frequency shifts for the mechanical modes this results in an effective tunneling Hamiltonian

$$\begin{equation} H_{\rm circ}\simeq t b_1 b_3^\dagger + t b_2 b_3^\dagger + t_{\rm eff} {\rm e}^{{\rm i}\varphi} b_{1}b_{2}^{\dagger} + {\rm H.c.}, \end{equation} \tag{ 64 }$$

where φ = φ₁ − φ₂ and

$$\begin{equation} t_{\rm eff}=\frac{g^2\alpha^2}{2}\left(\frac{1}{\Delta_+}-\frac{1}{\Delta_-}\right). \end{equation} \tag{ 65 }$$

Thus, by choosing the external control parameters such that φ = ± π/2 and that t_eff matches the bare mechanical tunneling amplitude t, the setup shown in figure 8(b) realizes a switchable three-port phonon circulator, as discussed in the previous subsection. Note that the interaction with the optical modes also introduces an additional decay channel with rate $\gamma _{\rm op}\approx g^2\alpha ^2\kappa /\bar \Delta ^2$, where $\bar \Delta ^{-2}=\Delta _+^{-2}+ \Delta _-^{-2}$. Compared to the noise filtering scheme described above, this requires slightly lower cavity decays satisfying $\kappa \ll |\bar \Delta | \ll \omega _{\rm m}$.

As a specific example let us consider the phonon waveguide scenario discussed above with typical phonon frequencies ω_m ≈ 2π × 4 GHz and a phonon–waveguide coupling of γ = 2π × 25 MHz. If we choose δ_i = −ω_m we obtain t_eff = g²α²/J and the conditions t_eff = γ/2 can be achieved for gα ≈ 2π × 110 MHz and J = 2π × 1 GHz. For the same parameters a value of κ ⩽ 2π × 50 MHz is sufficient to suppress the optically induced decay rate γ_op = 2 g²α²κ/J² below the value γ_op/γ ⩽ 0.05 shown in figure 8(c). This is only slightly below the demonstrated values for κ in state-of-the-art OM crystal cavities [22, 25].

As described in section 2, a 1D phonon channel can be realized by fabricating a large array of coupled nanomechanical resonators with a bare frequency ω_m and nearest-neighbor coupling K < ω_m. This scenario has been experimentally studied, for example in [37], where the resonators were charged up and coupled via electrostatic interactions. Alternatively, the resonators can also be coupled mechanically via bridges or the support [46]. Both approaches are suitable for realizing phonon channels with frequencies ω_m ranging from MHz to a few hundreds of MHz, where also very high Q-values around Q ∼ 10⁵–10⁶ are now routinely achieved. At T ≈ 100 mK this frequency range corresponds to a few tens to a few hundreds of thermal phonons, which can still be efficiently suppressed using the proposed OM noise filtering scheme.

A more practical and very versatile approach for implementing phonon waveguides based on phononic bandgap materials has in recent years attracted increasing attention [32, 33]. Here, a 2D structure with periodically varying mechanical properties is designed such that a complete band gap in the mechanical dispersion relation appears. Then, phonon wave guides can be realized by introducing line defects into these structures, to which the phonons are confined as long as their frequency lies within the bandgap of the bulk phononic crystal. With this approach phonon waveguides with frequencies ω_m ≈ 1–10 GHz can be realized, where even at T = 4 K the thermal occupation of the waveguide is only a few tens of quanta. Further, as indicated in figure 1(b), such waveguides can be easily combined with localized phonon cavities and integrated OM devices [25].

The implementation of phonon quantum networks also requires the realization of coherent and controllable interfaces between mechanics and stationary qubits. Here the ability of mechanical systems to respond to weak optical, electrical and magnetic forces enables the coupling of mechanical resonators to a wide variety of spin- or charge-based qubits and a few prototype examples are outlined in the following.

6.2.1. Spin qubits

Qubits encoded in localized spin degrees of freedom, for example in spins associated with nitrogen vacancy (NV) impurities in diamond [47] or phosphor donors in silicon [48], can be coupled to mechanical motion using magnetized tips [13, 14, 49]. A strong magnetic field gradient ∇B from the tip induces a position-dependent Zeeman shift of the spin, resulting in an interaction of the form

$$\begin{equation} H_{\rm node}= \frac{\omega_0}{2} \sigma_z + \omega_{\rm m} b^\dagger b + \lambda (b+b^\dagger)\sigma_z + \frac{\Omega}{2}\cos(\omega_{\mathrm{mw}} t) \sigma_x. \end{equation} \tag{ 66 }$$

Here ω₀ ∼ GHz is the bare spin splitting and λ = g_sμ_Ba₀∇B/(2ℏ) is the Zeeman shift per zero point motion a₀, where g_s ≃ 2 and μ_B is the Bohr magneton. For nano-scale mechanical resonators with frequencies in the few MHz regime, this coupling can be substantial and reach values λ/(2π) ≈ 10–100 kHz [50]. Based on this coupling the use of mechanical transducers to mediate spin–spin interactions in small resonator arrays has been proposed previously [23], and the present techniques can be used to extend these ideas to larger networks. To achieve an effective Jaynes–Cummings-type interaction as given in equation (2), the spin is driven by a near-resonant microwave field with frequency ω_mw = ω₀ + δ and Rabi frequency Ω as described by the last term in equation (66). Then, by changing into a dressed spin basis and making an RWA with respect to ω₀, the effective interaction is given as [43, 50]

$$\begin{equation} H_{\rm node}\simeq \frac{\Delta_{\rm q}}{2} \tilde \sigma_z + \omega_{\rm m} b^\dagger b + \lambda \sin(\theta) (\tilde \sigma_+ b + \tilde \sigma_- b^\dagger), \end{equation} \tag{ 67 }$$

where $\Delta _{\rm q}=\sqrt {\delta ^2+\Omega ^2}\sim \omega _{\rm m}$ and tan(θ) = Ω/δ. The qubit–resonator coupling can be controlled by adiabatically varying the effective detuning Δ_q or the mixing angle θ.

Instead of using external magnetic field gradients, alternative schemes to strongly couple spins to mechanical motion have recently been suggested for carbon nanotubes [51, 52]. Here a single electron is localized on the nanotube and couples to its vibration via spin–orbit interactions. In this case even larger couplings λ/(2π) ≈ 0.5 MHz and larger mechanical frequencies >100 MHz are achievable. Although a controlled fabrication and positioning of carbon nanotubes is still challenging, a phonon quantum bus could be realized by electrically coupling the nanotube to other mechanical resonators, which can be fabricated in a more controlled manner.

6.2.2. Superconducting qubits

6.2.3. Quantum dots and defect centers

The bending of a mechanical resonator locally deforms the lattice configuration of the host material and by that shifts the electronic states associated with artificial quantum dots or naturally occurring defect centers in solids. In the bulk this deformation potential interaction is usually responsible for the line broadening of optical transitions and other decoherence effects, but in the case of confined modes it can also lead to a significant coherent coupling to single phonons. In [59] the coupling of a quantum dot to the fundamental bending mode of a doubly clamped beam has been considered, leading to a deformation potential coupling of the form

$$\begin{equation} H_{\rm def} = \lambda (b+b^\dagger)|e\rangle\langle e|, \quad \hbar \lambda \approx (\Xi_{\rm e}-\Xi_{\rm g}) \frac{a_0 t}{2l^2}, \end{equation} \tag{ 68 }$$

where |e〉 denotes the electronically excited state, l is the beam length, t is its thickness and a₀ is the zero point motion. The Ξ_e and Ξ_g are deformation potential constants for the ground and excited electronic states, and for typical values Ξ_e,g ∼ 1–10 eV and micron-sized beams a coupling strength of λ/2π ≈ 10 MHz can be achieved. This is already comparable to the radiative lifetime Γ_e of the electronically excited state, but can, in principle, be pushed to values λ ∼ 1 GHz using much smaller, phononic Bragg cavities as suggested in [60].

As a potential scenario where the deformation potential interactions could be used to implement a controllable qubit–phonon interface, we consider an NV center embedded in a diamond nanoresonator. The electronic ground state of the NV defect is a spin triplet, where two states |0〉 = |m_s = −1〉 and |1〉 = |m_s = 1〉 can be used to encode quantum information. The qubit states in the electronic ground state, which are highly immune against external perturbations, can be coupled via an optical Raman process involving an electronically excited state, which, in contrast, is strongly coupled to phonons [61]. In appendix D we adiabatically eliminate the dynamics of the excited state and derive an effective spin–phonon interaction of the form

$$\begin{equation} H_{\rm eff}\simeq \lambda_{\rm eff}(t) (\sigma_+ b + \sigma_- b^\dagger). \end{equation} \tag{ 69 }$$

Here the σ_± are Pauli operators for the qubit subspace and for optimized laser detunings λ_eff(t) = 4λΩ₀(t)Ω₁(t)/ω²_m is a tunable coupling where Ω_0,1(t) are the optical Rabi frequencies. Note that this coherent interaction is accompanied by dissipation at an average rate $\bar \Gamma _{\rm eff}(t) = 4\lambda \Omega _0(t)\Omega _1(t)/\omega _{\rm m}^2$ and the ratio $\lambda _{\rm eff} /\bar \Gamma _{\rm eff} =\lambda /\Gamma$ is not affected by the off-resonant Raman coupling. This is in contrast to cavity QED where the optical field couples to the atomic coherence and not to the population of the excited state as described by H_def.

To model propagation losses in the phonon waveguide we add an intrinsic loss channel with rate γ₀ for each of the waveguide modes. Making the RWA right from the beginning, the dynamics of the whole resonator array is modeled by the coupled QLEs,

$$\begin{equation} \dot b_\ell= -\left(\mathrm{i}(\omega_0+K)+\frac{\gamma_0}{2}\right) b_\ell + \frac{\mathrm{i}K}{2} \left(b_{\ell-1} + b_{\ell +1}\right) -\sqrt{\gamma_0} b_{\ell,{\rm in}}(t), \end{equation} \noindent \tag{ A.10 }$$

where b_ℓ is the bosonic operator for the mechanical resonator at site ℓ and the b_ℓ,in(t) are uncorrelated thermal noise operators 〈b^†_ℓ,in(t)b_ℓ',in(t')〉 = N_thδ_ℓ,ℓ'δ(t − t'). In the corresponding momentum representation $b_q=1/\sqrt {N_{\rm ch}} \sum _\ell {\rm e}^{{\rm i}qa\ell } b_\ell$ and $b_{q,{\rm in}}=1/\sqrt {N_{\rm ch}} \sum _\ell {\rm e}^{{\rm i}qa\ell } b_{\ell ,{\rm in}}$, the coupling is diagonal

$$\begin{equation} \dot b_q= -\left({\rm i}\,\omega_q+\frac{\gamma_0}{2}\right) b_q -\sqrt{\gamma_0} b_{q,{\rm in}}(t), \end{equation} \noindent \tag{ A.11 }$$

where as above $\omega _q=\omega _0+K(1-\cos (qa))\approx \tilde \omega _0 + c |q|$. From this equation we obtain the propagation equation for the right-moving field $\Phi _{\rm R}(z,t)=\sqrt {2c/L}\sum _{q>0} {\rm e}^{{\rm i}qz} b_q(t)$,

$$\begin{equation} \left(\frac{\partial}{\partial t}+ c \frac{\partial}{\partial z}\right)\! \Phi_{\rm R}(z,t) = -\left({\rm i}\tilde \omega_0 + \frac{\gamma_{0}}{2}\right)\!\Phi_{\rm R}(z,t) - \sqrt{\gamma_{0} c}\, \Phi_{\rm th}(z,t). \end{equation} \noindent \tag{ A.12 }$$

Here the thermal noise field is defined as

$$\begin{equation} \Phi_{\rm th}(z,t)= \sqrt{\frac{2}{L}}\sum_{q>0} {\rm e}^{{\rm i}qz} b_{q,{\rm in}}(t), \end{equation} \noindent \tag{ A.13 }$$

such that it is normalized as 〈Φ^†_th(z,t)Φ_th(z',t')〉 = N_thδ(z − z')δ(t − t').

We can use the same approach to solve for the stationary state of the multi-mode system described in section 3.2. In the following we assume for simplicity the validity of the RWA, such that the QLEs for the equations for the annihilation operators $\vec{A}=(a,b_1,\ldots b_N)^{\mathrm {T}}$ form a closed set

$$\begin{equation} \partial_t \vec{A}(t)= -\mathcal{M} \vec{A}(t)- \sqrt{2\kappa} \vec{A}_{\rm in}(t) -\sqrt{\gamma_0} \vec{B}_{0,{\rm in}}(t), \end{equation} \tag{ B.7 }$$

where now $\vec{A}_{\rm in}(t)= (a_{\rm in} (t) ,0,\ldots , 0)^{\mathrm {T}}$ and $\vec{B}_{0,{\rm in}}(t) = (0, b^{(1)}_{0,{\rm in}} (t) ,\ldots , b^{(N)}_{0,{\rm in}} (t))^{\mathrm {T}}$ and the matrix $\mathcal {M}$ can be derived from equations (19)–(22). Following the same steps as above, we obtain

$$\begin{equation} \langle b_i^\dagger (\omega) b_i(\omega')\rangle= \gamma_0 N_{\rm th} \left( \sum_{j=1}^{N} \left| \mathcal{X}(\omega)_{i+1,j+1 }\right|^2\right)\delta(\omega-\omega'), \end{equation} \tag{ B.8 }$$

which we have used for evaluating the multi-mode cooling results presented in figure 4(c).