A Equivalent Circuit Quantisation

# Quartz-superconductor quantum electromechanical system

## Abstract

We propose and analyse a quantum electromechanical system composed of a monolithic quartz bulk acoustic wave (BAW) oscillator coupled to a superconducting transmon qubit via an intermediate LC electrical circuit. Monolithic quartz oscillators offer unprecedentedly high effective masses and quality factors for the investigation of mechanical oscillators in the quantum regime. Ground-state cooling of such mechanical modes via resonant piezoelectric coupling to an LC circuit, which is itself sideband cooled via coupling to a transmon qubit, is shown to be feasible. The fluorescence spectrum of the qubit, containing motional sideband contributions due to the couplings to the oscillator modes, is obtained and the imprint of the electromechanical steady-state on the spectrum is determined. This allows the qubit to function both as a cooling resource for, and transducer of, the mechanical oscillator. The results described are relevant to any hybrid quantum system composed of a qubit coupled to two (coupled or uncoupled) thermal oscillator modes.

###### pacs:
42.50.-p, 85.25.Cp, 85.50.-n

## I Introduction

Recent experiments have demonstrated the cooling of macroscopic mechanical oscillators to their quantum ground state (1); (2); (3), as well as the generation of quantum squeezed states (4); (5); (6). This work provides a foundation for the demonstration of entangled quantum states of mechanical modes (7) and enhanced sensing capabilities (8). In most of the experimental demonstrations (2); (3); (4); (5); (6), the mechanical motion of membranes and beams modulates parameters of a high-frequency electromagnetic cavity mode, forming a cavity optomechanical system (9); (10). Driving of the cavity mode enables cooling of the mechanical oscillator, analogous to the laser cooling of trapped ions to their motional ground state (11). However, parametric coupling of this type is ineffective for quartz bulk acoustic wave (BAW) oscillators.

Fortunately, quartz oscillators can be directly coupled to electrical circuits due to the piezoelectric effect (12). Resonant coupling of a film BAW oscillator to a superconducting qubit has been realised (1), and indeed, provided the first observation of a macroscopic mechanical degree of freedom in its quantum ground state. However, unlike film BAW oscillators, monolithic BAW oscillators offer exceptional mechanical properties including large effective masses and extremely high quality factors (13); (14). This makes them an attractive platform not only for the pursuit of quantum optics experiments with phonons, but also for tests of the limits of quantum mechanics itself (15); (16), high-frequency gravitational wave detection (17), and tests of Lorentz symmetry (18).

On the other hand, their large geometric size makes coupling to them challenging. Most critically, large unavoidable stray capacitance between the BAW oscillator electrodes reduces the amplitude of the oscillating voltage, and the corresponding coupling strength to electrical circuits becomes impractically small. To maximise the coupling between an electrical circuit and the mechanics we propose a scheme where the stray capacitance itself forms an electrical circuit with an additional external shunting inductor. Tuning the circuit into resonance with a particular mechanical mode of the BAW oscillator allows for the direct coupling of phonons to photons with greater coupling strength than is possible via the more conventional detuned capacitive coupling schemes (19). Now the electrical circuit will not be at a sufficiently high frequency to be in its quantum ground state, even in a cryogenic environment. Thus, the circuit itself must be cooled: this may be achieved via sideband coupling (20); (21); (22) to a superconducting transmon qubit (23); (24). The latter forms a circuit QED system (25); (26), albeit one in which the circuit is at a much lower frequency than the transmon (27); (28). This infrastructure also provides the hardware for quantum state control beyond the ground state.

Note that the cooling and measurement of a macroscopic mechanical oscillator via direct coupling to a quantum two-level system has been studied both theoretically (29); (31); (30); (32); (33); (34) and experimentally (35); (36); (37); (38); (40); (39); (41); (42). However, in our case the direct coupling between a quartz oscillator and a transmon is too weak (35), and hence effective coupling is not feasible without an intermediate tank circuit.

The study of hybrid quantum systems composed of a solid-state quantum two-level system, an electrical circuit mode, and a mechanical oscillator mode, has attracted considerable interest recently. In theoretical work, Restrepo et al. have solved the corresponding Hamiltonian problem (with the full radiation pressure interaction) in terms of qubit-cavity-mechanical polaritons (43). Accounting for dissipative dynamics, they have also described the possibility of cooling and unconventional phonon statistics. Note that this solution is inapplicable here since in our case the bare electromechanical coupling Hamiltonian is quadratic, and therefore the qubit-circuit polariton number does not commute with the total Hamiltonian. Others have discussed state engineering possibilities enabled by a cavity-mediated interaction between a qubit and a mechanical oscillator (44) and by three-body interactions (45).

In terms of experimental work, Pirkkalainen et al. have coupled a microwave cavity to a mechanical oscillator via a qubit (46). Differently from our proposal, the qubit is used as a mechanism for coupling to the mechanics rather than as an auxiliary cooling system. They did, however, observe motional sidebands in this work. They subsequently used the intermediate qubit to greatly enhance the effective optomechanical coupling (47). Lecocq et al. have used a phase qubit to control a mechanical oscillator, with the interaction mediated via a microwave electrical circuit (48). Here, time-dependent control was used for the measurement of the mechanical oscillator. The key difference from our proposal, aside from the absence of a quartz oscillator, is that in this work the microwave electrical circuit is itself at a relatively high frequency, being near-resonant with the qubit and at a far higher frequency than the mechanical oscillator. The optomechanical interaction in their case is the driven, linearised optomechanical interaction. As noted, such coupling is difficult for quartz BAW oscillators.

Here we propose and thoroughly analyse a quantum electromechanical system composed of a quartz BAW oscillator coupled to a transmon via an intermediate electrical circuit. In Sec. II we give an overview of BAW oscillator technology, provide an equivalent electrical circuit, and use it to obtain a Hamiltonian description of the system. In Sec. III we determine the steady-state of the system in both an adiabatic limit and a sideband picture, demonstrating the feasibility of ground-state cooling. In Sec. IV, we calculate the qubit fluorescence spectrum analytically in the adiabatic limit and numerically in a sideband picture. We demonstrate the existence of motional sidebands, potentially enabling transduction of the mechanical motion.

## Ii System

### ii.1 BAW Oscillators

The mechanical part of our system is provided by a BAW oscillator. BAW oscillators, originally developed in the frequency control community, can be divided into three main groups: High-Overtone Bulk Acoustic Resonators (HBAR) (49), Thin-Film Bulk Acoustic Resonators (FBAR) (50) and single-crystal (monolithic) BAW oscillators (51). The latter group is mainly composed of bulk quartz devices, which can be used to achieve the highest frequency stability in the RF band. For these devices, an acoustic wave is excited in the thickness of a single crystal plate clamped from the sides via electrodes.

As noted above, a film BAW oscillator (FBAR) has already been measured in its quantum ground state (1). Their very high resonance frequencies mean that they can be prepared in their quantum ground state with high fidelity in a cryogenic environment. Further, they can be integrated on a chip due to their small size and low mass. On the other hand, FBARs have relatively low quality factors. In contrast, single-crystal BAW oscillators (52); (53); (13) have losses limited only by material properties with logarithmic temperature dependencies leading to extraordinary quality factors at cryogenic temperatures (54).

In particular, we consider BVA-type phonon trapping single-crystal BAW oscillators (55). Such an oscillator is a thin plate with one curved surface allowing effective trapping of acoustic phonons in the plate centre (14). This trapping results in the spatial separation of the vibrating parts of the plate from points of suspension, and consequently to the material acoustic loss limit. Several modes of vibration are possible, and each mode of vibration gives a series of overtones corresponding to a different number of acoustic half-waves in the device thickness. We could model many mechanical modes via a parallel connection of branches as per the well-known Butterworth-Van Dyke model (56). However, since the quality factors and resonance frequencies are high, the modes are well-resolved in frequency space and we are justified in considering the coupling to one mechanical mode alone.

### ii.2 Hamiltonian

Now we formulate a minimal model of the proposed system for the purpose of analysis. The system consists of a mechanical oscillator, in the form of a quartz oscillator, coupled to a superconducting tank circuit, which is itself sideband coupled to a superconducting circuit in the form of a transmon. The coupling between the mechanical oscillator and the circuit is due to the piezoelectric nature of quartz. The inductor of the circuit may be realised using a series of DC SQUIDs (57); (58), which can be tuned via an external magnetic field to match the resonance of the desired overtone of the quartz oscillator. The system is represented schematically in Fig. 1(a). There is also a small direct coupling between the mechanical oscillator and the transmon. We may write down an equivalent electrical circuit for this electromechanical system (23), including an equivalent electrical representation of the mechanical oscillator (12); (56).

The equivalent (dissipation-free) electrical circuit is shown in Fig. 1(b). It may be quantised in the standard manner (60), as described in App. A, and the resulting Hamiltonian is

 ^H = ℏ∑pωp^a†p^ap+4EC^n2+EJ(1−cos^ϕ) (1) +ℏgmc(^am+^a†m)(^ac+^a†c) +2e∑pVp0βpt(^ap+^a†p)^n,

where the index is summed over the set , denoting the mechanical mode and the electrical circuit mode (i.e., the tank circuit), respectively. Throughout this Article, the index shall be used in this way. The transmon mode is described by the observables (the number of Cooper pairs transferred between the superconducting islands of the transmon) and (the phase difference between the islands). The resonance frequencies of the mechanical oscillator and electrical circuit oscillator are given (in terms of equivalent electrical circuit parameters) by and , respectively. The transmon charging energy is and the Josephson energy is where is the applied magnetic flux and is the magnetic flux quantum. In the transmon coupling terms in Eq. (1), denotes the rms ground-state voltage fluctuations of the equivalent electrical circuit modes. The effective capacitances, electromechanical coupling () and oscillator-transmon couplings () are complicated functions of the equivalent circuit capacitances and inductances. They are fully specified in App. A.

We truncate the transmon mode to its two lowest-lying energy levels and use Pauli operators defined in the uncoupled eigenbasis of the resulting qubit. Assuming qubit driving of amplitude and frequency , implemented via modulation of the flux bias (22), the Hamiltonian (1) takes the form

 ^HS = ^HmcS+^HqS+∑p^HpqS, (2a) ^HmcS = ℏ∑pωp^a†p^ap+ℏgmc(^am+^a†m)(^ac+^a†c), (2b) ^HqS = ℏ(Ω/2)^σz−ℏ(Ed/2)cosωdt^σz, (2c) ^HpqS = ℏgpq(^ap+^a†p)^σx, (2d)

where the level splitting of the transmon qubit is

 ℏΩ=√8ECEJ−EC, (3)

and the couplings of the qubit to the oscillators are given by

 ℏgpq=eVp0βpt(EJ/2EC)1/4. (4)

The subscript in Eq. (2a) indicates that the Hamiltonian is specified in a Schrödinger picture. The subscript in Eq. (4) indicates that the transmon is being approximated as a qubit.

For the purpose of determining the steady-state of the system, it is useful to remove the trivial time-dependences in the Hamiltonian (2a) and retain only near-resonant couplings between the oscillators and the qubit. To do so, we apply the unitary transformation

 ^U = exp[i∑pωp^a†p^apt+i(ωd/2)^σzt (5) −iEd/(2ωd)sinωdt^σz],

to (2a) under the assumption that the qubit frequency is much higher than the oscillator frequencies (i.e., ). This leaves the interaction picture Hamiltonian (I subscript),

 ^HI = ^HqI+^HmcI+∑p^HpqI, (6a) ^HqI = ℏ(δd/2)^σz, (6b) ^HmcI = ℏgmc(^ame−iωmt+^a†me+iωmt) (6c) ×(^ace−iωct+^a†ce+iωct) ≈ ℏgmc(^a†m^ac+^am^a†c), ^HpqI = ℏgpq(^ape−iωpt+^a†pe+iωpt) (6d) ×+∞∑n=−∞[J−n(Ed/ωd)ei(n+1)ωdt^σ+ +Jn(Ed/ωd)ei(n−1)ωdt^σ−] ≈ ℏ¯gpq(^ape−iωpt+^a†pe+iωpt)^σx,

where we have assumed that the mechanical oscillator mode is resonant with the electrical circuit mode and made a rotating-wave approximation on the electromechanical coupling in (6c). Further, we have introduced as the detuning between the qubit level splitting and the qubit drive frequency, and

 ¯gpq=gpqJ1(Ed/ωd), (7)

as the sideband-reduced couplings in Eq. (6d), where denotes a Bessel function of the first kind.

### ii.3 Dissipation

The mechanical oscillator and electrical circuit modes are assumed to be linearly damped at rates into independent Markovian environments with thermal occupations . The qubit is assumed to be damped into a Markovian environment with relaxation, excitation and dephasing rates being denoted by , , and , respectively. Therefore, the master equation describing the evolution of the system density matrix is

 ˙ρ = −iℏ[^H,ρ]+Ldmcρ+Ldqρ=Lρ, (8a) Ldmcρ = ∑p[γp(¯np+1)D[^ap]ρ+γp¯npD[^a†p]ρ], (8b) Ldqρ = γ↓D[^σ−]ρ+γ↑D[^σ+]ρ−(γϕ/4)[^σz,[^σz,ρ]],

where is given by (6a), we have introduced the notation for the Liouvillian of the entire system, and and for the Lindbladians (i.e., dissipation) of the oscillator modes and qubit, respectively. As is usual, denotes the dissipative superoperator whose action is given by

 D[^c]ρ=^cρ^c†−12^c†^cρ−12ρ^c†^c. (9)

### ii.4 Parameters

In order to proceed with the analysis, let us first consider the parameters expected for the proposed system. The parameters appearing in the Hamiltonian (2a)-(2d) and in the master equation (8a) are quoted here; the equivalent electrical circuit parameters from which they are derived are quoted in App. A. The mechanical and electrical circuit resonance frequencies, and qubit level splitting, are , , and , respectively. The direct mechanics-circuit, circuit-qubit, and mechanics-qubit couplings are , , and , respectively.

The qubit driving conditions are set by and , such that . The effective (sideband-reduced) coupling rates are then and .

The mechanical and electrical circuit mode damping rates are and , respectively, and the corresponding environmental thermal occupations are (assuming a cryogenic environment, with ). The qubit relaxation, excitation and pure dephasing are given by , , and , respectively.

We note that the parameters specified place us well into the resolved-sideband regime (9); (10); (11), here defined by the condition:

 ωc,ωm≫γt, (10)

where is the total qubit decoherence rate, given by

 γt=γ↓+γ↑+2γϕ. (11)

The parameters specified are also such that we are in an adiabatic regime (61) in which the qubit is damped rapidly compared with other relevant time-scales in the system. More precisely, this is here defined by the condition:

 γ↓≫gmc,¯gpq,γp. (12)

The system shall subsequently be analysed in both of these regimes.

The qubit may be used to cool both the mechanical and electrical circuit modes. This may be efficiently achieved in the resolved-sideband regime and the adiabatic limit. Analysis of the system is then facilitated by the adiabatic elimination of the qubit (62), which can (after some further approximations) result in a linear, time-invariant, Markovian description of the dynamics of the reduced system (i.e., the mechanical oscillator and the electrical circuit mode). The analytical adiabatic limit results shall be validated using numerical results obtained in a sideband picture.

Adiabatic elimination in the presence of a time-dependent coupling, as in the Hamiltonian (6d), may be treated using a projection operator approach (61); (62). The calculation is detailed in App. B.1. We obtain a master equation in Lindblad form for the reduced density matrix of the two oscillator modes, ,

 ˙ρs = −iℏ[^HmcI,ρs]+Ldmcρs+∑p(−iδp[^a†p^ap,ρs] (13) +γ−peD[^ap]ρs+γ+peD[^a†p]ρs +¯gpq¯g¯pq{G(+ωp)[^a¯pρs,^a†p] +G(−ωp)[^a†¯pρs,^ap]+H.c.}),

where denotes “not ” (i.e., if and if ), is given by Eq. (6c), is given by Eq. (8b), and the oscillator cooling/heating rates and frequency shifts due to the qubit coupling are

 γ∓pe = 2¯g2pqR[G(±ωp)], (14a) δp = ¯g2pqI[G(+ωp)−G(−ωp)]. (14b)

Here is the fluctuation spectrum of the uncoupled qubit,

 G(ω)=∫+∞0dτeiωτTrq[^σxeLqτ^σxρq], (15)

where the action of the qubit Liouvillian appearing in Eq. (15) is given by

 Lqρ=−iℏ[^HqI,ρ]+Ldqρ, (16)

with and given by Eqs. (6b) and (LABEL:eq:qubitdissipation), respectively. The appearing in Eq. (15) denotes the steady-state density matrix of the uncoupled qubit (i.e., ).

Evaluating Eq. (15) and substituting the result into Eqs. (14a) and (14b) yields

 γ−pe = 4¯g2pqγt(γ↓γ↓+γ↑+γ↑γ↓+γ↑γ2tγ2t+16ω2p), (17a) γ+pe = 4¯g2pqγt(γ↑γ↓+γ↑+γ↓γ↓+γ↑γ2tγ2t+16ω2p), (17b) δp = 2¯g2pq4ωpγ2t+16ω2p. (17c)

Now the frequency shifts have negligible impact on the steady-state provided that , which is equivalent to the requirement that and . These inequalities are comfortably satisfied for our anticipated parameters, and so we henceforth neglect the frequency shifts for the analytical evaluation of the steady-state. Further, for our anticipated parameters, the cross-terms in Eq. (13) are small compared with the larger oscillator relaxation and excitation rates, and we henceforth neglect those terms. Thus, we consider the master equation

 ˙ρs = −iℏ[^HmcI,ρs]+Ldmcρs (18) +∑p(γ−peD[^ap]ρs+γ+peD[^a†p]ρs).

Given Eq. (18), the steady-state of the reduced system is readily obtained. Setting the direct electromechanical coupling to zero (), we find that . That is, each oscillator mode is independently cooled due to its coupling to the driven qubit, as expected (29). Now for the quartz-superconductor quantum electromechanical system proposed, the direct qubit-mechanics coupling is weak, such that the electrical circuit mode is directly cooled while the mechanical mode is cooled sympathetically due to its coupling to the circuit. The general result is slightly complicated, but if we take the optimal qubit driving condition (which is ), and make the additional assumptions , then we find

 ⟨^a†c^ac⟩ = ¯ncγcγc+γ−ce, (19a) ⟨^a†m^am⟩ = ¯ncγcγc+γ−ce4g2mc4g2mc+(γm+γ−me)(γc+γ−ce) (19b) +¯nm(γm+γ−me)(γ−ce+γc)(γm+γ−me)(γ−ce+γc)+4g2mc.

Consequently, subject to the stated assumptions, ground-state cooling of the mechanical mode requires

 γ−ce > γc, (20a) 4g2mc > (γm+γ−me)(γ−ce+γc). (20b)

The former is simply a necessary but not sufficient requirement for ground-state cooling of the electrical circuit mode. The latter is a requirement for ground-state cooling of the mechanical mode; the direct electromechanical coupling must be made sufficiently large compared with the product of the effective damping rates of the two oscillator modes.

The effective parameters that emerge in the adiabatic limit in our case can be evaluated, giving (for the circuit-qubit coupling effective parameters) , , and . The corresponding effective parameters for the mechanical-qubit coupling are very small: , , and , and have a negligible impact on the dynamics of the system. However, for the sake of generality, we retain both couplings to the qubit in our subsequent analysis as the same considerations could apply to a variety of hybrid quantum systems. Now, these parameters satisfy the requirements for ground-state cooling, Eqs. (20a) and (20b), though not by so great a margin that the results obtained in the adiabatic limit should be accepted without validation via numerical calculations.

### iii.2 Sideband Picture

The Hamiltonian (6a)-(6d) is explicitly time-dependent. For convenient numerical analysis, we seek a time-independent description of the system that is valid outside of the adiabatic limit. We can obtain such a description by assuming that the qubit is driven on its red sideband corresponding to the oscillator resonance frequencies (i.e., ), making an additional unitary transformation on (6a), and making another rotating-wave approximation on the oscillator-qubit couplings. This leads to the simple sideband Hamiltonian (21),

 ^HSB = ^HmcI+∑p^HpqI,RWA, (21a) ^HpqI,RWA = ℏ¯gpq(^ap¯σ++^a†p¯σ−), (21b)

where we stress that the Pauli operators are defined in a different frame from that in Eq. (6a), indicated by the overbar notation. Now the steady-state of the entire system can be easily obtained by the direct numerical integration of the master equation (8a) with the Hamiltonian (21a). The sideband picture retains the description of the qubit excitation and near-resonant motional sideband, while neglecting the off-resonant motional sideband. It provides a reliable approximation provided that the resolved-sideband condition (10) is well-satisfied, as is the case for our proposed system.

The steady-state mechanical occupation determined numerically in the sideband picture is shown in Fig. 2, and compared with analytical results obtained in the adiabatic limit. The mechanical occupation is shown as a function of the qubit relaxation rate, for a range of electrical circuit mode intrinsic damping rates. These results demonstrate that ground-state cooling is feasible with the proposed quartz-superconductor quantum electromechanical system. We see that the analytical adiabatic limit results provide a good approximation provided that the condition (12) is well-satisfied. With our assumed couplings, is satisfied on the right-hand-side of the plot, but not on the left-hand-side of the plot. To exploit the analytical results in an actual experiment, we would want to place the system into the adiabatic regime.

For completeness, an adiabatic elimination in this sideband picture is also given in App. B.2. Also note that the master equation (8a) with the Hamiltonian (21a) is closely related to the dissipative Jaynes-Cummings model, which can actually be solved analytically using continued fractions (63). However, the addition of another oscillator renders this approach inapplicable here.

## Iv Qubit Spectrum

From Sec. III it is clear that by appropriately driving the qubit, we can cool the electromechanical system. Here we show that by measuring the fluorescence of the driven qubit, we can transduce the electromechanical system. In particular, the coupling of the qubit to the oscillators results in sidebands on the qubit spectrum which are related to the steady-state number expectations of the coupled oscillators, similarly to the motional sidebands observed in the fluorescence spectrum of a trapped ion (64) or a cavity optomechanical system (65).

This motional sideband contribution to the qubit fluorescence spectrum may be determined analytically in the limit of weak oscillator-qubit couplings using a perturbative expansion of the Liouvillian, an approach pioneered by Cirac and co-workers in the case of a trapped ion (66). Going beyond the trapped ion case, here the qubit is coupled to two oscillator modes, which are themselves also mutually coupled to each other. Additionally in our case, and again in contrast to the trapped ion case, each oscillator mode is damped into a reservoir, and that reservoir is assumed to be at some finite temperature. The calculation that we describe here is also relevant to other experimental systems composed of one (59) or two (46) thermal oscillators coupled to a qubit.

Now the quantity that we wish to evaluate is the qubit fluorescence spectrum, given by

 S[ω]=Re∫+∞0dte−iωt⟨^σ+(t)^σ−(0)⟩ss. (22)

This is to be evaluated under the Hamiltonian (6a) and the master equation (8a), such that the frequency in (22) is defined relative to the qubit drive frequency.

To proceed further, we express the Hamiltonian (6a) in a Schrödinger picture for the mechanical and electrical circuit modes, leaving

 ^HSI = ^HqI+^HmcS,RWA+∑p^HpqSI, (23a) ^HmcS,RWA = ℏωc∑p^a†p^ap+ℏgmc(^a†m^ac+^am^a†c), (23b) ^HpqSI = ℏ¯gpq(^ap+^a†p)^σx, (23c)

where is given by Eq. (6b).

Now we must evaluate the spectrum (22) under the Hamiltonian (23a) and the master equation (8a). To do so, we make a polaron transformation (67) on the Pauli operators given by the unitary transformation

 ^U = exp[i∑p~gpq(^ap+^a†p)^σz/2], (24a) ~gpq = ¯gpq/(γ↓+γ↑), (24b)

with being the sideband-reduced oscillator-qubit couplings scaled by the sum of the relaxation and excitation rates of the qubit. This transformation makes the spectrum (22) explicitly dependent on the mechanical and electrical circuit mode operators as

 S[ω] = Re∫+∞0dte−iωt ×⟨σ+(t)e+i∑p~gpqqp(t)σ−(0)e−i∑p~gpqqp(0)⟩ss,

where are oscillator quadrature operators, which are invariant under the transformation (24a). Note that the post-polaron-transformation operators are distinguished from the pre-polaron-transformation operators by the lack of a hat; this notation shall be used throughout the remainder of the Article. The polaron-transformed Hamiltonian and master equation then take the same form as (23a) and (8a), respectively, to first-order in , albeit in terms of the post-polaron-transformation (i.e., hatless) operators.

Using the quantum regression theorem (61); (68), the spectrum (IV.1) may be expressed as

 S[ω] = Re∫+∞0dte−iωtTr[σ+ei∑p~gpqqpμ(t)], (26a) μ(t) = eLtσ−e−i∑p~gpqqpρss, (26b)

where the Liouvillian corresponds to the master equation (8a) with the Hamiltonian (23a) expressed in terms of post-polaron-transformation operators, and is the steady-state density matrix of this system (i.e., ).

In the adiabatic limit in which the qubit is damped rapidly compared with other system rates, defined by the condition (12), the are small parameters. Therefore, the spectrum (26a) may be calculated perturbatively in ; this calculation is described in detail in App. C.

Ultimately, we find that the spectrum consists of contributions from the excitation of the uncoupled qubit and the motional sideband contributions due to coupling to the oscillator modes. Thus, the qubit fluorescence spectrum may be written as

 S[ω]=Sq[ω]+∑pSp[ω], (27)

where is the fluorescence spectrum of the uncoupled qubit,

 Sq[ω] = Re∫+∞0dte−iωt⟨σ+(t)σ−(0)⟩ss (28) = γ↑γ↓+γ↑γt/2(γt/2)2+(ω−δd)2.

In evaluating the qubit fluctuation spectrum in Eq. (28) we have neglected a correction first-order in , which is assumed to be small.

The in Eq. (27) are the motional sideband contributions, given by

 Sp[ω] = Missing or unrecognized delimiter for \left (29)

where is summed over the set . Here, the components of the spectrum are expressed as a sum over the eigenvalues of the Liouvillian of the master equation (8a) with the Hamiltonian (23a). More precisely, and are the eigenvalues of the oscillator part of the Liouvillian, with and without (respectively) the renormalisation due to the qubit coupling, given by Eqs. (17a)-(17c). The prime notation on the summation in Eq. (29) indicates that eigenvalues with real components of the order of the qubit relaxation rate are explicitly excluded from the summation; they will not make a significant contribution since this rate is assumed to be large. The functions and appearing in Eq. (29) are the uncoupled qubit correlation functions evaluated at ,

 r(λmc) = ∫+∞0dte+λmct⟨[σ−(t),σx(0)]⟩ss, (30a) t(λmc) = ∫+∞0dte−λmct⟨σx(t)σ−(0)⟩ss +∫+∞0dte+λmct⟨σx(0)σ−(t)⟩ss.

Eqs. (30a) and (LABEL:eq:tapp) are evaluated explicitly in App. C.4. The operators appearing in the moments of oscillator-space quadrature operators in Eq. (29) are projection operators on the oscillator space corresponding to the oscillator-space eigenvalues .

Given Eqs. (27)-(29), our tasks are now to evaluate the moments of oscillator quadrature operators and projection operators, and then evaluate the summation in Eq. (29).

This is first performed under the assumption that

 16g2mc<(γc,eff−γm,eff)2, (31)

where we have introduced the new effective oscillator decay rates,

 γp,eff=γp+γ−pe−γ+pe. (32)

The assumption (31), which is expected to be comfortably satisfied in our system, means that the electromechanical coupling does not affect the imaginary part of the oscillator Liouvillian eigenvalues. This assumption allows us to approximate the projection operators on the oscillator space corresponding to these eigenvalues as simply the product of the projection operators of the mechanical mode and the electrical circuit mode treated independently. Crucially, we can then assume that all the oscillator cross-correlations (i.e., moments with ) in (29) are zero. The calculation is detailed in App. C.5.

Setting (i.e., qubit driving on the red oscillator sideband of the qubit) and assuming , the motional sideband contributions to the spectrum may be decomposed into upper and lower sideband components (relative to the qubit drive frequency). This leads to

 Sp[ω] = Sup[ω]+Slp[ω], (33a) Sup[ω] = 8¯g2pq(γt+~γp)2~γp,eff4(ω−ωp−δp)2+~γ2p,eff⟨^a†p^ap⟩ss, (33b) Slp[ω] = 8¯g2pq(γt+~γp)2+16ω2p ×~γp,eff4(ω+ωp+δp)2+~γ2p,eff(⟨^a†p^ap⟩ss+1),

where the damping rates with tildes are introduced (assuming and ) via

 2~γp = γm+γc±√(γc−γm)2−16g2mc, (34a) 2~γp,eff = γm,eff+γc,eff±√(γc,eff−γm,eff)2−16g2mc,

with the signs corresponding to . The motional sideband contributions for arbitrary are quoted in App. C.5.

We see that the upper motional sideband (33b) is a Lorentzian located at , resonant with the qubit and so is enhanced, while the lower motional sideband (LABEL:eq:lower) is a Lorentzian located at , detuned by twice the oscillator resonance frequencies from the qubit resonance, and so is suppressed. Crucially, the upper motional sideband contributions are proportional to the number expectation of the oscillator modes. Assuming knowledge of the relevant system parameters, one can then measure the steady-state oscillator occupations. The contributions arising from the different oscillator modes can be distinguished by their difference in linewidths. Note that the mechanical motional sideband contribution is broadened by virtue of its direct coupling to the electrical circuit mode (and the circuit motional sideband is correspondingly narrowed), as expected from Eqs. (33b) and (LABEL:eq:decaytilderateseff). This provides an experimental signature of the direct electromechanical coupling.

Note that Eqs. (33b) and (LABEL:eq:lower) also incorporate the usual motional sideband asymmetry between up-conversion and down-conversion processes which is attributable to the possibility (impossibility) of absorption (emission) processes from the quantum ground state (66). Further, we note the similarity of each motional sideband contribution here, obtained after a number of simplifying approximations, to those obtained when the auxiliary cooling system is itself an oscillator (70). This similarity arises due to the fact that in the adiabatic regime () the qubit decays before a scattering process is likely to attempt to excite it again, such that the distinction between a qubit and an oscillator is not very important.

Now Eq. (LABEL:eq:decaytilderateseff) describes the hybridisation of the effective oscillator damping rates of the oscillator modes for ; they become equal at . For , the damping rates of the two oscillator modes are the same, and we see the emergence of two non-degenerate electromechanical normal modes. The evaluation of Eq. (29) in the case that the condition (31) does not hold is outlined in App. C.6.

The uncoupled qubit and upper motional sideband contributions (electrical circuit and mechanical) to the total qubit fluorescence spectrum, as given by Eqs. (28) and (33b) respectively, are plotted in Fig. 3. The results are plotted for the anticipated experimental parameters listed in Sec. II.4. Clearly, the total qubit fluorescence spectrum is dominated by the motional sideband contribution from the coupling to the electrical circuit mode. Note that under the assumed (red sideband) driving conditions, the lower motional sidebands are suppressed by four orders of magnitude compared with the upper motional sidebands.

The upper motional sideband should allow the direct measurement of the steady-state electrical circuit occupation, which in the parameter regime specified by Eq. (20b), is essentially equal to the steady-state mechanical occupation. As noted above, the oscillator contributions to the spectrum are distinguishable via their linewidths, such that independent transduction of the oscillator modes in this way is possible, in principle. Unfortunately, the difference in power spectral densities would make this very challenging for the system considered here. However, the circuit motional sideband itself arises, in part, from coupling to the phononic excitations of the quartz oscillator. The interpretation of these results in terms of the circuit and transmon as a composite transducer (71) for the quartz mechanical oscillator system shall be described elsewhere.

### iv.2 Sideband Picture

As for the calculation of the steady-state, an alternative to the adiabatic limit is provided by the sideband picture. The calculations in this picture are unconstrained by the adiabatic condition (12), but more strongly constrained by the sideband resolution condition (10). It is expected that the system under consideration shall be well-described by the sideband picture. Hence, numerical calculations in this description enable us to check and assess the validity of the useful analytical results obtained in the adiabatic limit.

The spectrum may be conveniently calculated numerically using the time-independent Hamiltonian (21a). It is given by

 ¯S[ω] = Re∫+∞0dte−iωt⟨¯σ+(t)¯σ−(0)⟩ss (35) = S[ω+δd],

where is as defined in Eq. (22). The correlation function in (35) is obtained using the quantum regression theorem (68); (61), as

 ⟨¯σ+(t)¯σ−(0)⟩ss = limt′→∞Tr[¯σ+(t′+t)¯σ−(t′)ρ(t′)] = limt′→∞Tr[¯σ+e−Lt¯σ−ρ(t′)eLt],

where denotes the Liouvillian of Eq. (8a) with the Hamiltonian (21a). That is, the required correlation function follows from the solution of the master equation (8a) subject to the initial condition . The analytical results provide a good approximation to the numerical results in the same parameter regime that the steady-state was well-approximated by the numerical results, as indicated in Fig. 2.

## V Conclusions

Quartz BAW oscillators provide an attractive platform for the pursuit of quantum optics experiments with phonons. The difficulty of directly coupling such an oscillator to higher-frequency modes of superconducting electrical circuits led to the proposal of coupling the quartz oscillator to a superconducting transmon qubit via an intermediate tank circuit, resonant with the quartz oscillator. Ground-state cooling of a quartz BAW oscillator mode via sideband driving of the qubit coupled to the tank circuit is shown to be feasible. The qubit fluorescence spectrum is evaluated, with the contributions from the coupled oscillator modes determined. The mechanical and electrical circuit modes may be transduced through the observation of motional sidebands in the qubit spectrum. Cooling and measurement of high- modes of a quartz oscillator should provide a platform for future experiments in quantum phononics.

## Vi Acknowledgments

We wish to acknowledge support from a UNSW Canberra Early Career Researcher grant and from the Australian Research Council grant CE110001013.

## Appendix A Equivalent Circuit Quantisation

We can write down the Lagrangian corresponding to the equivalent electrical circuit depicted in Fig. 1(b). The Lagrangian, , consists of the kinetic and potential energy contributions,

 T = 12CmV2m+12CcV2c+12CctV2ct+12CtV2t, (37a) V = Φ2m2Lm+Φ2c2Lc+EJ(1−cosϕt), (37b)

respectively, where the m, c, and t subscripts denote the mechanical, electrical circuit, and transmon modes, respectively, and is the phase difference across the superconducting islands of the transmon. Applying Kirchhoff’s voltage law around the loops of the equivalent circuit yields , , , and . Given the Lagrangian in terms of the generalised coordinates and the corresponding velocities, we can obtain the generalised momenta and then the corresponding Hamiltonian via a Legendre transformation (69); . We find

 H = Q2c2~Cc+Φ2c2Lc+Q2m2~Cm+Φ2