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Quantum Theory of Transmission Line Resonator-Assisted Cooling of a Micromechanical Resonator

We propose a quantum description of the cooling of a micromechanical flexural oscillator by a one-dimensional transmission line resonator via a force that resembles cavity radiation pressure. The mechanical oscillator is capacitively coupled to the central conductor of the transmission line resonator. At the optimal point, the micromechanical oscillator can be cooled close to the ground state, and the cooling can be measured by homodyne detection of the output microwave signal.

Recently, other optomechanical-like cooling schemes have been proposed to replace the optical cavity by a radio-frequency (RF) circuit [40,41] or a one-dimensional transmission line resonator (TLR) [42]. However, the theoretical understanding of the cooling schemes via a RF circuit in Refs. [40,41] or via a TLR in Ref. [42] is based on a classical description of the motion of the MR. A quantum-mechanical description of the motion of the MR, in a similar system consisting of a mechanical resonator capacitively coupled to a superconducting coplanar waveguide (which is an example of a TLR), was discussed recently in Ref. [43], which focused on studying the entanglement between the MR and the TLR without considering the cooling of MR. Most recently, Teufel et al. [44] considered the cooling of a MR by applying directly the theoretical analysis of the cavity-assisted back-action cooling scheme [34] to a superconducting microwave resonator. They also presented experimental data about the cooling effect on the MR due to the microwave radiation field. The quantum-mechanical description of TLRassisted cooling of a MR has also been investigated in Ref. [45] via embedding a SQUID [46], which allows to control the coupling strength between MR and TLR by controlling the flux through the SQUID.
There are some practical advantages [47,48] in the microwave TLR schemes. The TLR is realized in a thin on-chip superconducting film and is easily pre-cooled by standard dilution refrigeration techniques. It is ready to be integrated with quantum circuits containing Josephson junctions which may offer a sensitive measurement and a connection with quantum information processing. In addition, the size of the mechanical resonator could be much smaller than the wavelength of the radiation in the TLR, unlike in optical cavity experiments that work with reflection.
In this paper, we present a quantum-mechanical description and use it to investigate the motion of the MR when it is coupled capacitively to a driven TLR as in Ref. [42] where the calculation was carried out in a semi-classical framework. The Hamiltonian of our TLR-assisted model is also studied in Refs. [43,44], and is very similar to that of a MR coupled to a driven optical cavity via radiation pressure coupling [32,33,34,35,36,37,38,39]. We study the TLRassisted passive back-action cooling of a MR in detail by using a quantum Langevin description (without taking into account quantum entanglement [43] between the MR and the TLR). One of the main results of our work is to show that the MR can be cooled close to its ground state using realistic parameters: final effective mean phonon numbers below 1 can be reached assuming an initial temperature of 10 mK which can be achieved using a dilution refrigerator. We discuss in detail how such a ground state cooling of the MR can be obtained for all kind of parameter choices in practice.

II. MODEL AND HAMILTONIAN
The system that we consider is shown schematically in Fig. 1: a MR is fixed on both ends (or a cantilever fixed on one end) located at the center of the TLR and is coupled capacitively to the central conductor of the TLR [49]. We restrict the description to the fundamental flexural mode of oscillation of the MR which is modeled as a harmonic oscillator of frequency ω b and effective mass m. The TLR is driven by an external microwave at a frequency ω d and can be modeled as a single mode LC resonator with frequency ω ′ a = 1/ √ L a C a (the second mode of the TLR [50]), where L a is the inductance and C a the capacitance of the TLR. . The external microwave drive field enters from the left and drives the TLR. The signal at the output on the right end can be used to measure the motion of the MR via homodyne detection [32,44,47,61].
The Hamiltonian of the system reads (1) The first line describes the free Hamiltonian of the TLR and the MR, respectively, with lowering (rising) operator of the TLR mode a (a † ), and the position (momentum) operator of the MR x (p) which satisfy [a, a † ] = 1 and [x, p] = i . The first term in the second line is the capacitive coupling between the TLR and the MR. Actually, it describes the capacitive energy between them. The MR and the TLR are assumed to form a capacitor with the capacitance C g (x) ≈ C 0 g (1 − x/d) (for small displacement) depending on the position of the MR along the x-direction (d is the initial equilibrium distance without the coupling and C 0 g the corresponding initial capacitance; typically d ∼ 1 µm [47]). The capacitor is assumed to be placed in the center of the structure, i.e., its voltage is given by the antinode voltage of the second mode: V = V rms (a † + a) (where V rms = ω ′ a /C a is the rms voltage [49]), since the length of the MR is usually much shorter than that of the TLR: L ∼ cm ≫ l ∼ 10 − 100 µm. The last term in Eq. (1) describes the input driving of the TLR by an external microwave field with the coupling strength [22,36,49] where κ is the decay rate of the TLR, P is the input external microwave drive power. Here, the non-rotating wave terms like ae −iω d t and a † e iω d t have been ignored since we keep |ε| ≪ ω ′ a ∼ ω d . Usually, the fundamental oscillation frequency is of the order of 2π× (10 3 -10 6 ) Hz for micromechanical resonators and 2π× (10 7 -10 9 ) Hz for nanomechanical resonators; the TLR frequency can be made to be of the order of 2π × 10 GHz. Here we will focus on the case of a micro-MR for which the condition ω b ≪ ω ′ a is satisfied. In the interaction picture with respect to ω d a † a and neglecting the rapidly-oscillating terms, the Hamiltonian reads where g 0 := C 0 g V 2 rms /( d) is a real coupling constant; ∆ 0 = ω a − ω d is the detuning, and ω a = ω ′ a + C 0 g V 2 rms / is the modified frequency of the TLR shifted by the coupling between TLR and MR.
This Hamiltonian resembles that used in cavityassisted cooling schemes [32,33,34,35,36,37]. This suggests that the capacitive-coupling scheme in a microwave TLR can be used to cool the MR like in the case of radiation-pressure cooling in an optical cavity.

III. QUANTUM LANGEVIN EQUATIONS AND FINAL MEAN PHONON NUMBER
The dynamics is also determined by fluctuationdissipation processes that affect both the TLR and the mechanical mode. They are taken into account in a fully consistent way by the quantum Langevin equations [51]: Here a in (a † in ) is the noise operator due to the external microwave drive, and ξ(t) denotes the quantum Brownian force that the resonator is subject to. They satisfy [51] where is the mean number of thermal microwave photons of the TLR, k B is the Boltzmann constant, T the temperature of the environment, and γ b the damping rate of the MR. For simplicity, we have assumed that both the bath correlated to the external microwave drive field and the one connected to the MR have the same temperature [43]. We now perform a similar calculation as that given in Refs. [32,36,37,52]. The steady-state solution of the quantum Langevin equations (3) can be obtained by first replacing the operators by their average and then setting d ... /dt = 0. Hence we can get the steady-state values as where ∆ = ∆ 0 − g 0 x is the effective detuning. In Eq. (6), we can also take | a | 2 + 1 2 ≃ | a | 2 , since we will focus on the case | a | ≫ 1 which can be achieved by controlling the input power of the external microwave drive.
Rewriting each operator as a c-number steady-state value plus an additional fluctuation operator, and neglecting the nonlinear terms (since we have chosen | a | ≫ 1), we obtain a set of linear quantum Langevin equations (see Eq. (A1)) and then solve for the spectrum of the position and momentum of the MR as in Refs. [32,36,37,52], see Appendix A.
Using the fluctuation spectra of the MR as given in Eqs. (A10,A11), we can define the final mean phonon number in the steady state [36] as where the variances of position and momentum are This allows us to define the effective temperature T eff as In the next section, we will consider the cooling of the MR by discussing its final effective mean phonon number (or equivalently, its effective temperature) in detail.

IV. COOLING OF THE MR
The final effective mean phonon number of the MR can be calculated directly by evaluating the integral in Eq. (8) numerically and using Eq. (7). Alternatively, instead of being evaluated directly, Eq. (8) can also be evaluated analytically using the approximation scheme described in the following.
The effective mechanical damping rate γ eff b (ω) = γ b + γ ca (ω) can be significantly increased, γ eff b (ω) ≫ γ b , when |g 0 a | is very large, see Eq. (A16). Let us consider the most interesting regime when the significantly increased effective mechanical damping rate is less than the mechanical frequency: , and also less than the decay rate of TLR: |γ eff b (ω)| < κ [33,34,37,53]. In this regime, the effective frequency is [36,54] according to Eq. (A15), and the effective susceptibility is peaked around the points ω = ±ω eff b (ω) ≃ ±ω b . Then one can get an approximate expression for the variance where the effective thermal noise spectrum S ′ th (ω) and the induced noise spectrum S ′ ca (ω) are the symmetrized parts of S th (ω) and S ca (ω), respectively: Similarly, one can obtain Figure 2 shows the variance of position δx 2 as a function of the effective detuning ∆. The dashed lines correspond to a numerical evaluation of the integral in Eq. (8). The solid lines describe the approximate results obtained through Eq. (10) which can be seen to agree perfectly with the exact numerical evaluation. We checked that this is also the case for the variance of the momentum δp 2 .
In Eq. (10), the induced noise spectrum S ′ ca (ω b ) increases (heats) the motion of the MR. On the other hand, when the effective damping rate is enhanced: |γ eff b (ω b )| > γ b , the mechanical motion will reduce, that means cooling. Mathematically, the cooling effect would dominate the heating effect when the effective damping rate is sufficiently increased. Actually, this is right when the significantly increased effective damping rate is positive for positive detuning. However, it is not the case when γ eff b (ω b ) is negative and γ eff b (ω b ) ≫ γ b (for negative detuning ∆ < 0). That is because the stability conditions, derived using Ref. [55], are satisfied only for positive detuning [ 32,52,60]. In fact, a negative effective damping means the amplitude motion of the MR will be amplified which will lead to an instability [56,57,58].
In what follows, we will focus on the case of positive detuning ∆ > 0. According to Eqs. (7,10,13), one has where is the initial mean thermal excitation phonon number of the MR; is the induced mean phonon number due to the capacitive coupling between the MR and the TLR. As discussed above, the significant reduced value of n f b in Eq. (14) can only be obtained when the additional damping rate γ ca (ω b ) (or effective damping rate γ eff b (ω b )) is positive and much larger than the original one (but still less than the decay rate of TLR and less than the frequency of the MR as discussed before): This can be satisfied by enhancing the value of |g 0 a |, that is, by controlling the capacitive coupling strength g 0 and increasing the external microwave drive power P to make |ε| large (equivalently, | a | will be large). In practice, the capacitive coupling strength g 0 would be limited by the realistic system, and the external microwave drive strength ε would also be limited according to the validity of the rotating wave approach as we mentioned before. Here we put the length of the MR l as large as 10 − 100 µm and the distance between the MR and TLR d as small as 1 µm (see Fig. 1) in order to get a large capacitance C 0 g which will lead to large g 0 , and fix |ε| = ω a /8 for all the numerical calculations. Then for a significantlyincreased effective damping rate, the final mean phonon number reduces to In order to get the ground state cooling, that is, n f b ≪ 1, both n ca and n b γ b /γ ca (ω b ) should be much less than 1. Especially, if the γ ca (ω b ) is significantly increased enough to make then n f b approaches the limit n ca : Now we discuss the possible minimal value of n ca by discussing all kinds of parameters, e.g., κ, ∆, and N . From Eq. (16), it is obvious that the optimal value of κ satisfies the high-quality cavity limit and the optimal detuning satisfies ∆ = ω 2 b + κ 2 ≈ ω b . Then the corresponding induced mean phonon number n ca is The optimal N needs a sufficiently low initial temperature of the bath which is limited in practice to the experimental dilute refrigerator temperatures. For the superconducting TLR scheme, its microwave frequency is of the order of 2π × 10 10 Hz. For the initial temperature T 1 K, k B T ω a and N 1, the groundstate cooling of the MR is not possible. Therefore, initial temperature less than 100 mK are required to achieve ground-state cooling.
Our result on the limiting value in Eq. (22) is consistent with that in other optical schemes except the limit of initial temperatures. In the optical cavity case ω a ≫ k B T (even at room temperature) and therefore N ≃ 0, the optimal value of n ca becomes which is just the case of resolved sideband cooling as discussed in the optomechanical cooling schemes [22,23,25,26,28]. We would like to point out that these references also mention another cooling limit: the Doppler cooling limit, which is realized in our system when N ≃ 0, ∆ = ω 2 b + κ 2 , and κ 2 ≫ ω 2 b in Eq. (16): On the other hand, if N ≫ κ 2 /4ω 2 b , the induced mean phonon number n ca in Eq. (22) becomes n ca → N . In the classical limit when the initial temperature is so high that N ≈ k B T /( ω a ) ≫ 1, one has which is also given in Ref. [31]. The Doppler cooling limit in Eq. (24) and the classical cooling limit in Eq. (25) preclude ground state cooling. We will focus on the resolved sideband cooling in this paper. The final mean phonon number n f b is plotted as a function of the effective detuning ∆ and the decay rate κ of the TLR in Fig. 3. It is clear that one can obtain a significant suppression of the mechanical motion of the MR in the positive detuning range ∆ ≃ ω b . The optimal cooling is obtained for κ 2 ≪ ω 2 b , which agrees with both the above analysis and that in other treatments of radiation-pressure cooling [22,23,25,26,42].
Physically, as discussed in the back-action optomechanical cooling schemes in optical cavities (Refs. [33,34,36,37,60]), the external driving microwave is scattered by the "TLR + MR" system mostly to the first Stokes sideband (ω d − ω b ) and the first anti-Stokes sideband (ω d + ω b ). The generation of an anti-Stokes photon will cool the MR by taking away a phonon of the MR. On the contrary, the generation of a Stokes photon will heat the MR by creating a phonon. When the effective detuning ∆ > 0, the microwave field of the TLR (with the frequency ω a ≡ ω d +∆ 0 ≈ ω d +∆) interacts with the first anti-Stokes sideband (ω d +ω b ) more than it interacts with the first Stokes sideband (ω d −ω b ), and cooling will occur. This is the physical reason why the positive effective detuning (∆ > 0) will lead to cooling. In the high-quality cavity limit κ < ω b , the anti-Stokes (Stokes) sideband is resolved, and the corresponding cooling (heating) process is prominent. Especially, for the optimal effective detuning ∆ ≈ ω b , the frequency of the TLR is resonant with that of the anti-Stokes sideband, which will apparently lead to optimal cooling. This physical discussion is consistent with the calculation presented above. Figure 3 suggests there is a finite optimal value of κ for a fixed effective detuning. That is because one should have both a small value of n ca and a large effective damping rate γ ca (ω b ) in order to get strong cooling: κ should not be too large, since n ca depends somewhat on the value of κ 2 /ω 2 b ; κ should not be too small, since γ ca (ω b ) → 0 when κ → 0 [59]. In the cooling process, the thermal energy of the MR is mainly first transferred to the TLR, and then leaks out of the TLR through the bath coupled to the TLR. When the decay rate of the TLR is too small: κ → 0, the energy leakage out of the TLR is too weak, and one could not obtain a strong cooling. We would like to emphasize that the results shown in Fig. 3 are based on the approximate expressions Eqs. (10,13), where the condition of the so-called weakcoupling limit [34,44] has been assumed, that is, the effective damping rate of the MR should be always less than the decay rate of the cavity and less than the frequency of the MR |γ eff b (ω b )| < κ, ω b . Normally the weakcoupling is satisfied but not in some special cases. In Fig. 4(b), the weak-coupling condition is violated when κ/ω b < 0.1 at the optimal effective detuning ∆ = ω b . Beyond the weak coupling limit, Fig. 4(a) shows that the approximate treatment through Eqs. (10,13) ceases to be valid. Then one should discuss the cooling, e.g., effective mean phonon number in Eq. (7), by using the numerical evaluation of the integral in Eq. (8). But going beyond the weak-coupling limit, the contribution from the posi-tion variance is not equivalent to that from the momentum variance any more [36]. In other words, the energy equipartition is not satisfied. That means it is hard to define an effective temperature since it is not in a strict thermal state.
According to the above analysis, both the high-quality cavity and weak-coupling limit should be satisfied, so the optimal decay rate of the TLR is better taken to be κ ≈ 0.1ω b for the typical parameters in Fig. 4. The weak-coupling condition depends only weakly on the initial temperature T and the original damping rate of the MR γ b in the cooling process. In what follows, we will consider the optimal decay rate at κ ≈ 0.1ω b for different parameters T and γ b , for which the weak coupling limit is always satisfied.
In Fig. 5, the ratio of final effective temperature T eff to bath temperature T is plotted as a function of the effective detuning ∆ for the optimal κ ≈ 0.1ω b . Apparently, here the weak coupling limit is satisfied (according to the above analysis in Fig. 4). For initial temperatures T = 100, 30, and 10 mK, the corresponding initial mean phonon numbers are , 980, and 330, with the final mean phonon number n f b ≈ 1.6, 0.5, and 0.16, respectively. It is obvious that a significant cooling of the MR is obtained and lower initial temperatures will generally lead to better cooling. For an initial temperature T = 10 mK, which can be realized experimentally by using a dilution refrigerator, the MR (with the frequency ω b ∼ 4 MHz) can be cooled close to the ground state since the final mean phonon number n f b ≈ 0.16 For the other parameters see Fig. 2 .
For an initial temperature of T = 10 mK as in Fig. 5, the final mean phonon number would in principle be n c a (in Eq. (16)), which is much less than that obtained in Fig. 5: 16. This is because n f b → n ca only when the condition in Eq. (19) is satisfied. Unfortunately, it is not the case for the parameters in Fig. 5. A possible way to approach this condition is to increase the quality factor of the MR. In Fig. 6, the final effective mean phonon number is plotted as a function of the effective detuning ∆ for different quality factors of MR Q b (≡ ω b /n b ): Q b = 4 × 10 4 (typically, see Ref. [44]); Q b = 10 5 , 4 × 10 5 , 10 6 (expected in the near future). The corresponding minimal n f b ≈ 0.16, 0.06, 0.02, 0.01. One can find the cooling is better for a higher quality factor of the MR. The cooling discussed above can be measured by a homodyne detection method like that given in the scheme of cavity-assisted radiation-pressure cooling of a MR [32,44,47,61]. The motion of the MR can be detected by monitoring the output microwave signal (e.g., the field phase quadrature) of the TLR (as seen in Fig. 1) since the measurement of the output spectrum corresponds to a faithful measurement of the MR motion [47].

V. CONCLUSION
We have found that a MR with frequency ω b ∼ 2π×10 6 Hz can be cooled close to its ground state when it is coupled to a typical TLR (ω a ∼ 2π × 10 10 Hz). Actually, by considering the optimal parameters in this scheme, that is, assuming the high-quality cavity limit (κ 2 ≪ ω 2 b ), a positive optimal effective detuning (∆ ≈ ω b ), a low initial temperature (e.g., T = 10 mK in order that N ≈ 10 −27 ≃ 0), a high quality factor of the MR (Q b ≡ ω b /γ b 10 4 ), and both strong external input microwave drive power P and strong capacitive coupling strength g 0 to get the significantly increased positive effective damping rate (γ b ≪ γ eff b (ω b ) ≈ γ ca (ω b )), we find that resolved sideband cooling of the MR occurs. The possible minimal value of the final effective phonon number could approach the induced mean phonon number: n f b → n ca ≈ κ 2 /4ω 2 b ≪ 1. Moreover, one should also consider the condition of weak-coupling limit, that is, the significantly increased effective damping rate γ eff b (ω b ) should be less than both ω b and κ (in the high-quality cavity limit, one only needs γ eff b (ω b ) < κ). This condition requires that κ must not be too small though lower κ will lead to lower n ca . As shown in Fig. 2 and its discussion, there will be an optimal range of κ. For the typical parameters in this cooling scheme, we take κ ∼ 0.1ω b (though this is not the optimal result in general). We find that a MR with ω b = 4 MHz can be cooled close to its ground state with the final effective mean phonon number in the steady state: n f b ≈ 0.16 (for a typical quality factor Q b = 4 × 10 4 ) or n f b ≈ 0.01 (for a high quality factor Q b = 10 6 ) by using the resolved sideband cooling scheme when it is coupled to a driven TLR (with the frequency ω a = 8 × 10 10 Hz).
We would like to stress the condition in Eq. (19), which can lead to n f b → n ca . As pointed out in the discussion of Figs. (5,6), this is not always satisfied. A possible way to approach this condition is to increase the quality factor of the MR. For example, if the quality factor of the MR is high enough (e.g., Q b > 10 7 ), one would have the optimal n f b → n ca ≈ κ 2 /4ω 2 b = 0.0025, for which the MR is cooled much closer to the ground state.
The back-action self-cooling scheme presented here is similar to the optical-cavity-assisted cooling scheme [33,34]. In both cases, the MR can be cooled close to its ground state using resolved sideband cooling which is possible in the limit of a high-quality cavity. But it seems that this limit (e.g., κ = 0.1ω b , for ω b = 4 × 10 6 Hz) is easier to reach in the microwave TLR than that in the optical cavity. In the case of a TLR (typically ω a = 8×10 10 Hz), quality factors of Q a = ω a /κ = 2 × 10 5 have been seen in experiments [47,62]. However, in the case of an optical cavity (ω a ∼ 10 15 Hz), the corresponding quality factor should be Q a = ω a /κ ∼ 2.5 × 10 9 , which is hard to achieve since the typical cavity quality factor is Q a ∼ 10 7−8 [49].
To conclude, we have studied the self-cooling of a mechanical resonator that is capacitively coupled to a transmission-line resonator. The discussion was based on a linearized quantum Langevin equation. The cooling method presented here is similar to the self-cooling of a MR coupled to an optical cavity by radiation pressure. By using the optimal parameters discussed above, the MR can be cooled close to its ground state in the high-quality cavity and weak-coupling limit.