Cooling a Micro-mechanical Beam by Coupling it to a Transmission Line
We study a method to cool down the vibration mode of a micro-mechanical beam using a capacitively-coupled superconducting transmission line. The Coulomb force between the transmission line and the beam is determined by the driving microwave on the transmission line and the displacement of the beam. When the frequency of the driving microwave is smaller than that of the transmission line resonator, the Coulomb force can oppose the velocity of the beam. Thus, the beam can be cooled. This mechanism, which may enable to prepare the beam in its quantum ground state of vibration, is feasible under current experimental conditions.
pacs:85.85.+j, 45.80.+r, 84.40.Az
Mechanical resonators Cleland (2002); Blencowe (2004) have important applications in high precision displacement detection Caves1980 (); Bocko1996 (); LaHaye et al. (2004), mass detection Buks and Yurke (2006), quantum measurements Braginskybook1992 (), and studies of quantum behavior of either mechanical motion Mancini et al. (2002); Marshall et al. (2003); Eisert et al. (2004); Wei et al. (2006); Xue et al. (2007a, b) or phonons Hu and Nori (1996); Hu1996b (); Hu and Nori (1997); Hu1999 (). Recently, proposals Savel’ev and Nori (2004); Savel’ev et al. (2006a); Sergey et al. (2007) have been made for implementing qubits by using buckling nanoscale bars with quantized motion. Casimir effects on nanoscale mechanical device were also studied Chan2001 (); Munday et al. (2005, 2006); Capasso et al. (2007). However, in previous studies (see, e.g., Refs. Savel’ev and Nori, 2004; Savel’ev et al., 2006a; Sergey et al., 2007) of quantized mechanical resonators (and macroscopic quantum phenomena Leggett (2002) in mechanical resonators, see, e.g., Refs. Mancini et al., 2002; Marshall et al., 2003; Eisert et al., 2004), it is necessary to prepare the mechanical resonators into their vibrational ground states. Therefore, one needs to cool the mechanical resonators down to ultra-low temperatures to put them into their ground states. For example, a temperature below one milli-Kevin is necessary for cooling a 20 MHz mechanical resonator to its vibrational ground state.
To reach temperatures below one milli-Kelvin, which is beyond the capability of present dilution refrigerators, alternative cooling mechanisms are now being explored. Using optomechanical couplings, the cooling of mechanical resonators was recently demonstrated experimentally Metzger and Karrai (2004); Arcizet et al. (2006); Gigan et al. (2006); Kleckner and Bouwmeester (2006); Schliesser et al. (2006); Poggio et al. (2007). To observe the quantized motion of a mechanical resonator, one should be able to cool the mechanical resonator down to its ground state of vibration and to detect the phonon number state. Besides optomechanical cooling, electronic cooling Martin et al. (2004); Zhang et al. (2005); Wineland et al. (2006); Naik et al. (2006); Wang2007 (); Zhao2007 () was also studied. For instance, theoretical proposals for cooling a mechanical resonator were considered by coupling it either to a two-level system Martin et al. (2004); Zhang et al. (2005); Wang2007 (), to an ion Hensinger2005 () or to an LC circuit Wineland et al. (2006). An experimental demonstration of cooling a mechanical resonator by the quantum back-action of a superconducting single-electron transistor was recently reported Naik et al. (2006). Most of these cooling experiments (e.g., Refs. Metzger and Karrai, 2004; Arcizet et al., 2006; Gigan et al., 2006; Kleckner and Bouwmeester, 2006) focus on cooling mechanical resonators with a frequency lower than 1 MHz, with a mechanical quality factor higher than . It is difficult to experimentally cool mechanical resonators to their quantum ground state of vibrations because of the weak coupling between the mechanical resonators and the cooling media for optomechanical systems (see, e.g., Ref. Bernad et al., 2006).
Recently, the strong coupling between a one dimensional (1D) transmission line resonator (TLR) and a solid state qubit You and Nori (2003, 2005) was achieved Wallraff et al. (2004), and the detection of photon number states was also demonstrated Schuster et al. (2007). Based on these experimental developments, here we consider replacing the Fabry-Pérot cavity used in previous cooling proposals Metzger and Karrai (2004) by a 1D TLR, in order to cool a micron-scale bar.
The working mechanism of our proposal here is similar to the cooling of a tiny mirror in a Fabry-Pérot cavity Metzger and Karrai (2004). This cooling mechanism can be summarized as follows. A force on the mirror is coupled to the light intensity inside the cavity. This intensity does not change instantaneously with each mirror displacement. The delayed response of the intensity to a change in the mirror displacement leads to a force that can either agree or oppose the motion of the mirror, depending on whether the laser frequency is bigger or smaller than the cavity resonant frequency Milonni2004 (). By including this intensity-dependent force, in addition to a thermal force on the mirror, the mirror can be cooled.
In our proposal here, the TLR, whose frequency is determined by its overall capacitance and inductance, acts as a cavity. The beam is placed near the middle of the TLR and capacitively coupled to the TLR. When the mechanical beam has a displacement, the overall capacitance of the TLR changes, thereby the resonant frequency of the TLR also changes. Now let us consider the case where the TLR is driven by a microwave with fixed frequency. Any displacement of the beam will change, after a delay, the voltage between the TLR and the beam (and also the force between them). Recall that here we are considering two coupled oscillators: the TLR and the mechanical beam. The rf microwave drive acts directly on the TLR, and indirectly on the mechanical beam. After the transients are gone, the driven damped oscillator (here, the TLR) exhibits a steady-state response which is delayed with respect to the drive. In other words, the beam displacement changes the TLR’s oscillation frequency . Since the frequency of the drive is fixed, this change in will affect the steady-state amplitude of the TLR oscillator, which will be reached after some delay. The displacement of the beam (i.e., the action on the TLR), causes a delayed reaction (i.e., a delayed back-action) force from the TLR to the beam. The delay is determined by the damping rate of the TLR. When the frequency of the microwave is smaller than the resonant frequency of the TLR, this back-action force opposes the motion the beam, thereby damping the Brownian motion of the beam.
This cooling mechanism studied here is also related to the mechanism recently employed in Refs. Brown2007, ; Wineland et al., 2006. There, cooling is produced by a capacitive force which is phase-shifted relative to the cantilever motion. In their set-up, when the cantilever oscillates, its motion modulates the capacitance of an LC circuit, therefore modulating its resonant frequency. This resonant frequency, and the potential across the capacitance, is modulated relative to the fixed frequency of the applied rf drive. The modulated force linked to this potential shifts the resonant frequency of the cantilever Brown2007 (); Wineland et al. (2006). Because of the finite response time of the LC circuit, there is a phase lag in the force, relative to the motion. When the rf frequency is smaller than the resonant frequency, the phase lag produces a force that opposes the cantilever velocity, producing damping. When this damping is realized without introducing too much noise in the force, then the cantilever is cooled.
Our analysis, presented below, shows that it is possible to cool a 2 MHz beam, initially at mK, down to its quantum vibrational ground state at around mK. This is a cooling factor of about . Our proposed device, which is a combination of the devices in Refs. Wallraff et al., 2004; Naik et al., 2006, should be realizable in experiments. Moreover, because of its on-chip structure, our device has some practical advantages to be integrated in dilution refrigerators and be operated on; while optomechanical systems need an additional optical system.
Our proposed device is illustrated in Fig. 1(a). A doubly-clamped micro-beam is placed in the middle of a 1D superconducting TLR formed by thin coplanar striplines. The central stripline has a length , with a capacitance and an inductance , per unit length. For not-very-high frequencies, the equivalent circuit of the stripline is an infinite series of inductors with each node capacitively connected to the ground, as shown in Fig. 1(b). It can be described as a series of resonators that accommodate different resonant modes Wallraff et al. (2004). Since the length of the micro-beam is much smaller than that of the 1D TLR, we consider the voltage in the middle of the 1D TLR to be the voltage on the beam. Here, we only consider the mode with the largest coupling, i.e., the lowest mode Wallraff et al. (2004) coupled to the beam. The 1D TLR is coupled to both, two semi-infinite TLRs to the left and right, via the capacitors , and the beam via the capacitor . Thus the boundary conditions and the voltage of the 1D TLR are modified by these additional capacitors. When , the circuit can be approximated by a 1D TLR with a modified frequency
with . Actually, due to its coupling to the environment, the 1D superconducting TLR acts as a cavity with finite quality factor , where is the damping rate of the 1D TLR.
The fundamental vibration mode of the doubly-clamped beam can be approximated by a mechanical resonator with frequency and effective mass . The beam is coupled to a conductor (the 1D TLR) via a capacitor, and its equivalent circuit is illustrated in Fig. 2. The beam is exposed to a Coulomb force from the 1D TLR. Please note that for the case we studied in this paper, the amplitudes of the oscillates are small and thus the beam is essentially in the linear regime. For a review of nonlinear oscillators, see, e.g., Ref. Dykman1984, .
Iii Coulomb force on the beam
This force gives rise to a cooling mechanism which is similar to the cavity-cooling of the vibrating mirror in Ref. Metzger and Karrai, 2004. As shown in Fig. 2(a), it is assumed that the beam vibrates around its equilibrium position with an amplitude , which is much smaller than the distance between its equilibrium position and the TLR, i.e., . The averaged Coulomb force on the beam can be written as Wineland et al. (2006)
when . Here, is the capacitance between the beam and TLR for . Assuming that an external driving source acts on the central TLR via the capacitor , will reach a steady amplitude after a time delay . To first order in , Eq. (2) can be rewritten as
where the effective elastic constant of the Coulomb force on the beam by the TLR is
The term , which is independent of the displacement of the beam, will change the equilibrium position of the beam. does not contribute to the cooling of the beam, and can be canceled by applying an appropriate dc voltage between the TLR and the beam. Therefore, hereafter it will be omitted. The term
describes the coupling strength between the beam and the TLR. is the total capacitance of the TLR for . And
is a dimensionless parameter determined by the ratio
takes its maximum value on resonance . The typical behavior of , versus the detuning
is plotted in Fig. 3. Here, is the frequency of the TLR for . There is an optimal detuning point for the driving microwave where takes its maximum value. As shown in Fig. 3, the sign of the effective elastic constant of the Coulomb force is determined by the detuning between the frequency of the driving microwave and that of the TLR . When , additional damping is induced by the Coulomb force, cooling the beam because of its delayed response to the displacement of the beam.
Iv Cooling mechanism
We define the effective temperature of the fundamental vibration mode of the beam according to the equipartition law
where is the Boltzmann constant, a modified elastic constant of the beam after considering the existence of the TLR. In some papers (e.g. in Refs. Metzger and Karrai, 2004; Poggio et al., 2007) the effective temperature is defined/estimated from the original elastic constant of the beam, instead of the effective elastic constant of the beam. For the case when the effective elastic constant of the force is much smaller than the original elastic constant of the beam, the definition in Eq. (9) and the one in Refs. Metzger and Karrai, 2004; Poggio et al., 2007 give almost the same result of the effective temperature. However, please note that for a large effective elastic constant of the force , one should not neglect the modification of the elastic constant of the beam.
The Coulomb force from the TLR has two effects on the mean kinetic energy of the beam. First, because of its delayed response to the displacement of the beam, the Coulomb force introduces additional damping in the beam motion, thereby increases or reduces the mean kinetic energy of the beam. Below, it is shown that the damping rate of the beam increases when . Second, fluctuations in the Coulomb force from the TLR also introduce additional noise in the motion of the beam, thereby increasing the mean kinetic energy of the beam. The balance of these two competing effects gives the theoretical lower limit of the attainable effective temperature by this cooling mechanism.
To evaluate the cooling effect of the Coulomb force from TLR, we use the following equation of motion for the beam Metzger and Karrai (2004):
where describes the coupling strength between the beam and its thermal environment. Here, is the quality factor of the beam; the elastic constant of the beam; the thermal noise force on the beam, with a spectral density Sidles et al. (1995)
is the temperature of the environment. is the Coulomb force on the beam, acting on the beam via a delay-function:
for . Using the Laplace transform, we obtain the mean-squared motion of the beam:
There are three measurable effects on the vibration mode of the beam from : a modified effective elastic constant :
a modified damping rate :
with ; and additional noise in the motion of the beam generated by the fluctuation of .
For the case , neglecting fluctuations of in Eq. (10), the steady value of the mean-squared motion of the beam is given by
which defines an effective temperature of the beam. For the case when or , the frequency of the beam will be greatly shifted away from the original one, resulting in weak cooling of the beam Wang2007 ().
V Effects of fluctuations of
In the above discussions, we did not consider the effect of fluctuations of on the effective temperature. Equation (15) shows that the damping rate of the beam is modified because of the existence of the TLR. Thus, the effective temperature is changed, (see Eqs. (9) and (16)), however, according to the fluctuation and dissipation theorem, dampings are always accompanied with noises. We now study noises on the beam introduced by the force of the TLR. Actually, There are several noise sources affecting , such as fluctuations in the driving microwave, back-action due to measurements on the TLR, and thermal noise in the TLR. Among these noise sources, the thermal noise provides an intrinsic limit of the fluctuations of . Therefore, a lower limit of the spectral density of can be obtained by considering the voltage fluctuation from the thermal noise in the TLR, which is given by , with the effective resistor in the TLR. The voltage fluctuation gives rise to a fluctuation of the charge on the capacitor , giving rise to fluctuations of on the beam through the capacitor . Since for small vibration amplitudes of the beam, we find that the thermal noise in the TLR gives a fluctuation of on the beam
Assuming an Ohmic friction for the beam, the temperature and the damping rate of the beam and the spectral density of the noise on the beam have the following relationWeissbook2001 ()
For not very low temperature near the beam’s resonant frequency we find that an effective temperature of the beam could be related to the spectral density and the damping rate of the beam: . Therefore, after considering the fluctuation and dissipation theorem we further modify the effective temperature of the beam to
where we only take into account the thermal noise in the TLR. Indeed, the attainable lowest effective temperature of the beam would be higher than the above limit, since there are also other fluctuations acting on the beam, from both the driving microwave and the back-action of the measurement on the beam. These fluctuations would add more noises, which depend on the special parameters of the circuit for the driving microwave and the circuit for the measurement, e.g., the noise from amplifiers Brown2007 (), in the numerator of Eq. (19). We do not address them here.
Vi Cooling ability
Now let us estimate the cooling effect . Using experimentally feasible parameters Wallraff et al. (2004); Naik et al. (2006), we take GHz, KHz, m, aF, , and N/m. When the driving power of the microwave is set at mV, the effective spring constant of the Coulomb force can be as large as 0.55 N/m for optimal detuning of the driving microwave. It is possible to obtain stronger coupling between the beam and the TLR by increasing the driving power of the microwave, as long as the voltage between the beam and the TLR is kept below the breakdown voltage. Using the parameters list above and assuming , the cooling effect depends on the oscillating frequency of the beam and the detuning , as shown in Fig. 4. In Fig. 4, we assume , and then take the effective spring constant in Eq. (14). Otherwise, the optimal value of to reach the lowest will be slightly drifting away from unity Wang2007 (). The best cooling effect on a 200 KHz beam is estimated to be for the parameters given above. Therefore, if this beam is precooled by the dilution refrigerator to a temperature of 1 K, it can be further cooled down to 0.36 mK using the TLR.
For a 2 MHz beam, we use a stronger microwave mV. The best cooling effect is about with MHz. If the beam is precooled by the dilution refrigerator to a temperature of 50 mK, it could be further cooled down to 0.07 mK by the TLR. This implies that the thermal phonons in the 2 MHz beam will be less than 0.24, where a quantum description is expected Marquardt et al. (2007); Wilson-Rae et al. (2007). It should be noticed that, when the beam reaches a quantum regime a quantum theory is expected to give the cooling efficiency in the quantum regime.
Above we do not consider matching the impedance of the TLR to that of the conventional microwave components Wallraff et al. (2004). To obtain an optimal impedance, e.g. 50 , of TLR, one needs to carefully design the geometry of the TLR and the beam. If one simply considers the TLR as a straight coplanar transmission line, then might be 1 pF for a 10 GHz TLR. Thus, a larger is necessary for this larger to maintain the cooling effects described above.
The fabrication of superconducting TLR now is good enough to provide a 1D TLR with an electrical quality factor as high as . The reduced effective temperature of the beam can be inferred from the power spectrum of the 1D TLR around the oscillating frequency , whose integral is proportional to the effective temperature. Detection of microwave photon was achieved in recent experiments, capable of resolving a single microwave photon number Schuster et al. (2007). Therefore, in principle the information of the beam could be inferred by detecting the field in the TLR.
The working principle of our proposal is similar to that in the optomechanical cooling by a Fabry-Pérot cavity Metzger and Karrai (2004). The cooling or heating is determined by the detuning between the driving laser and cavity, which is determined by the detuning between the driving microwave and the TLR in our case. In both cases, a high mechanical quality factor is needed, since it measures heating effects on the beam by its thermal environments.
Our proposal is also similar to the one in Ref. Wineland et al., 2006, which deals with a cantilever and a coupled LC circuit. Here we consider a doubly-clamped beam coupled to a co-planar transmission line resonator (TLR). Besides considering different physical systems, in Ref. Wineland et al., 2006, they analyze only two special detunings between the frequency of the driven microwave and the resonant frequency of the LC circuit. Here we present a more general result for the effective elastic constant versus the detuning, valid for all values of the detuning between the frequency of the driven microwave and the resonant frequency of the TLR. We also explain how the damping rate of the beam and the noise on the beam is changed by the TLR. Our studies enable us to optimize the setup of experimental parameters for achieving a lower effective temperature of the beam, as shown in Fig. 4.
Since the best cooling is obtained when , the cooling efficiency of optical-cavity cooling would be efficient for beams with tens of MHz, or even higher frequency, considering current experimental parameters. For a typical optical cavity, with a resonant frequency of Hz, the damping rate is about Hz, for an optical quality factor , making it favorable for cooling a 100 MHz beam. However, to cool a beam with a MHz vibration frequency, the optimal damping rate of the optical cavity would also be MHz. This requires an optical quality factor for a tiny mirror. A high mechanical quality factor is also required at the same time, which is a great challenge for the fabrication of the tiny mirror. However, in our case, the damping rate of the TLR can be as small as 200 KHz. The damping rate of the TLR can be easily increased to match the frequency of the beam by attaching an additional circuit to the TLR, while it is very difficult to decrease the damping rate of an optical cavity to match the lower-frequency beam. Thus a MHz-beam could be cooled down to its quantum ground state and also reach the regime , where the cavity line width is much smaller than the mechanical frequency and the corresponding cavity detuning. Then the photon sidebands could be resolved when the beam is cooled down to the quantum regime Marquardt et al. (2007). A recent interesting study of the lower limit for resonator-based side-band cooling can be found in Ref. Grajcar2007 ().
We thank M. Grajcar, S. Ashhab and K. Maruyama for helpful discussions. FN was supported in part by the US National Security Agency (NSA), Army Research Office (ARO), Laboratory of Physical Sciences (LPS), the National Science Foundation grant No. EIA-0130383, and the JSPS CTC program. YDW was partially supported by the JSPS KAKENHI (No.18201018).
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