The suppression of the feedback noise from the detection amplifier is also observed, up to 4dB. In this context, I set my experimental goal as obtaining measurements of the dynamics of a nanomechanical resonator coupled to a superconducting quantum circuit.
Theory
The x displacement of the nanoresonator results in linear capacitance modulation between the nanoresonator and the CPB, NR( ) NR(0) CNR. The correction of the energy levels due to Hˆint and hence the dispersive shift of the nanoresonator ΔωNR / (2)π are then calculated using second-order time-independent perturbation theory.
Sample fabrication
K =α ω , where Mgeom ≈9×10−16kg is the estimated geometric mass and α=0.48 is found by assuming that the CPB couples to the average displacement of the nanoresonator over the length of the CPB island Q Typical nanoresonator quality factor for the range of.
Experimental set-up
Measurement circuit
The output port of the directional coupler feeds VRF to the drive gate electrode of the nanoresonator for excitation. And the input port of the directional coupler feeds the reflected component Vr through a Miteq AFS series cryogenic amplifier to the room temperature electronics that compose the phase-locked loop (Fig. 2).
Capacitive detection of mechanical motion
A small change in the frequency of the nanoresonator ΔωNR / (2 )π leads to a correction given by ΔΓ≈ jQΔωNR/ωNR. The VCO output is split into two signals: one to excite the nanoresonator (VRF) and the other provides the lock-in reference input (REF).
Results
Dispersive shift of mechanical resonance
3c shows the frequency response of the nanoresonator's amplitude (Main) and phase (Inset) at two values of Vcpb for fixed Φand. The CPB is formed from Al and placed on one side of the nanoresonator at a distance of ~ 300 nm.
Periodicity in V cpb and Φ
The background increase in ωNR during the measurement was not typical of most magnetic field motions, which were taken over a much smaller range. For the data in Figure 7, a different current source and a larger background field were used.
Correction for charge drift
This allowed us to use the cards without microwaves applied to correct for the charge dissipation of the cards with microwaves applied. Black line in (B) is a fit of cross section “A” to two Gaussian peaks to determine charge offset and background change from trace to trace.
Microwave spectroscopy of CPB
We can estimate the charge energy Ec and Josephson energy EJ of the qubit from spectroscopy by plotting ωμ /(2π) against the dashed green lines indicate the resonance bands we expect for the applied microwave frequency ωμ /(2π), based on a calculation of ΔE from the full. At the lowest microwave frequencies (Fig. 9a–b), the observed ). nCPB Φ -dependence of the microwave resonance appears to be consistent with the coupling of the CPB island to an incoherent charge fluctuation.49 However, we have no apparent dependence of the additional resonant features on thermal cycles or the application of background electric and magnetic fields not observed. .
Additionally, we find that by increasing )ωμ /(2π (Fig. 9c–d), the qualitative agreement between the observed and expected )(nCPB,Φ -dependence improves.
Landau-Zener tunneling of CPB with strong microwave excitation
Vμ taken at the expense of degeneracy, demonstrating the expected .. periodic dependence and Lorentzian form of the interference maxima. 2 = π ωμ . in the CPB gate line, which is in reasonable agreement with measurements of the damping at room temperature. The contour lines overlapping the Landau-Zener interferogram in Figure 11a indicate locations in )(Vcpb,Vμ space where the phase of the CPB wave function is a multiple of 2π.
First, we observe increased damping of the NEMS when tuning the CPB to the charge degeneracy point.
Discussion
These parameters have already been demonstrated with single-electron transistors.15 Such a significant but realistic improvement of the coupling, together with the implementation of circuit-QED (cQED) architecture52 to reduce qubit damping and provide independent readout of the qubit , should push these experiments into the highly dispersive limit. For example, assuming the parameters realized in a recent experiment66, pushing the mechanical noise to 10% of the vacuum level would require modulating the gate electrode of the nanoresonator with an amplitude of 300 mV. When the nanoresonator is capacitively coupled to the CPB, this charge-voltage ratio affects the motion of the nanoresonator, and its resonance exhibits a CPB state-dependent shift13.
The use of qubit nonlinearity to parametrically pump the nanoresonator also significantly reduces the direct electrostatic driving of the resonator, which occurs simultaneously with the parametric modulation when pumping through the geometric capacitance67.
Parametric excitation of harmonic oscillator: amplification and noise squeezing
Parametric amplification and oscillation
To obtain a parametric gain, we set VG and Φ, then turn on the resonator excitation )Vωcos(ω0t+ϕ) and apply a pump V2ωcos(2ω0t) to the CPB gate electrode.
Nonlinear dissipation
The PLL circuit used in the resonance shift measurement also monitors the magnitude simultaneously, giving us information about the nanoresonator's quality factor with respect to the gate charge. It is clear that additional dissipation arises due to coupling to the CPB, which becomes maximal at the degeneracy point. One possible explanation could be the resonance frequency fluctuations due to the stochastic process of quasiparticle poisoning.
From top to bottom, the circles correspond to the measured gains with Vω=3.6nV, 6.3nV, 11nV. The black dashed line is a fit to Eq.
Noise squeezing
The first term, on the other hand, increases when Rm decreases; i.e. that the coupling to the amplifier is increased. And since only Sx,BA depends on the mechanical Q, the noise floor is below the motion peak of the shear noise spectrum Sx,ADD. Here we do not include the thermomechanical noise from the nanoresonator, since the noise temperature of the amplifier (~30 K) is much higher than the sample temperature.
The minimum power noise is then, .. where kB is the Boltzmann constant and TN is the minimum noise temperature of the amplifier.
Discusssion
Exciting such states would require reducing the thermal occupation number N of the nanomechanical mode to a value close to its quantum ground state (i.e. Achieving the quantum ground state of a 6 GHz micromechanical resonator was recently demonstrated using conventional dilution cooling 11. Changes to the geometry of the sample should produce a factor of 10, or more, increase in the electrostatic coupling λ.
A recent demonstration of the cooling of a micro-mechanical resonator to the ground state was done in this way11.
Theory
Superconducting coplanar waveguide resonator
This effect highlights the need for further studies on the characterization and reduction of fluctuations from the ubiquitous two-level system bath to enable reaching the quantum ground state in nanomechanical resonators. Connection to a CPW open resonator with input and output capacitors as shown in Figure 17c, resonant response. The coupling adds dissipation due to loss due to external loads, and the input/output quality factor due to the load is given by Qin/out =C/(ω0Cin2/outZ0)87.
From the circuit model, it can be seen that the electromagnetic energy stored in the resonator when the resonator is driven by an input power is Pun at ωR.
Back-action cooling of mechanical motion
Because the force has a delay corresponding to the turn-off time of the CPW resonator, net work is done by the mechanical resonator, creating additional opto-mechanical damping γopt. The effective mode temperature of the mechanical resonator is classically given by Teff =T0γm/(γm+γopt), where TO is the initial temperature without microwave power.88. Here xzp is the zero-point motion of the mechanical resonator, κ is the CPW resonance linewidth, np is the average number of microwave pump photons, and the sideband resolved limit is assumed to be κ/4ωm <<1.
The total width of the mechanical line. where γm0 is the intrinsic linewidth of the mechanical resonance without broadening of the microwave back action.
Experimental Setup
Device parameters
Cryogenic circuit
The end of the input line is connected to the sample housing, and the output of the sample housing goes to two series-connected cryogenic isolators, providing more than 40 dB of total isolation. A UT-85 niobium coaxial cable is used to connect the insulators to the input of the HEMT cryogenic amplifier, and the output of the amplifier is connected to a UT-141 CuNi cable that goes to the top of the heatsink.
Room temperature circuit
A half-wavelength section of waveguide is coupled through the input and output capacitances from the tapered ports. At the voltage antinode of the fundamental resonance at 11.8 GHz, (dotted circle), a nanoscale suspended beam is coupled to the ground plane, and capacitively coupled to the center conductor of the microwave resonator. It is etched and suspended from the substrate with an aluminum layer as a mask, which also acts as a conductive layer to connect the.
After cold attenuators that attenuate the room temperature blackbody radiation, the signal propagates through the microwave resonator, and is amplified by the cryogenic high electron mobility transistor (HEMT) amplifier.
Results
Thermomechanical noise
This spread becomes much more prominent when using a cryogenic bias tee (Anritsu K250) (Fig. 18b). A possible reason for this increased dispersion could be the ferromagnetic material inside the bias tee; these kinds of materials exhibit abnormally high heat capacity at low temperatures. Data fall on a line that crosses zero (green line), indicating that the mechanical state follows the temperature of the bath and obeys the equipartition theorem. b) noise effect with bias tee (red circles).
Below 100 mK, the bias tee creates power noise and drives the nanoresonator, resulting in large variations in the thermomechanical noise power.
Back-action cooling of mechanical motion
This excess noise is attributed to two-level systems interacting resonantly with the microwave field. One possible explanation for this excessive heating is the coupling of the two-level oscillations to the bath. As in the electromagnetic case (equation 13), the resonant interaction of two-level systems with a phonon field shows saturation effects in dissipation94,.
Here Na is the density of states of the two-level systems per unit volume (J−1m−3), M is the coupling between the two-level system and the strain (∂(energy)/∂(strain)), ρ is the mass density , v: speed of sound, and Ic is the critical acoustic intensity (=h2ρv3/(2M2T1T2)).
Discussion
The microwave-driven response of the qubit was measured using the dispersive shift as a probe. The process also compresses the counteracting noise from the sense amplifier, down to 4 dB. D1, the fractional change of the resonant frequency shows a logarithmic dependence on temperature, and the damping is roughly ~T.
As discussed in Chapter 4, the back-cooling limit is determined by the occupancy of the CPW resonator, provided there are no other heating sources.
