Mass spectrometry based on a coupled Cooper-pair box and nanomechanical resonator system
© Jiang et al; licensee Springer. 2011
Received: 20 August 2011
Accepted: 31 October 2011
Published: 31 October 2011
Nanomechanical resonators (NRs) with very high frequency have a great potential formass sensing with unprecedented sensitivity. In this study, we propose a scheme formass sensing based on the NR capacitively coupled to a Cooper-pair box (CPB) drivenby two microwave currents. The accreted mass landing on the resonator can be measuredconveniently by tracking the resonance frequency shifts because of mass changes inthe signal absorption spectrum. We demonstrate that frequency shifts induced byadsorption of ten 1587 bp DNA molecules can be well resolved in the absorptionspectrum. Integration with the CPB enables capacitive readout of the mechanicalresonance directly on the chip.
Nanoelectromechanical systems (NEMS) offer new prospects for a variety of importantapplications ranging from semiconductor-based technology to fundamental science . In particular, the minuscule masses of NEMS resonators, combined with theirhigh frequencies and high resonance quality factors, are very appealing for mass sensing [2–7]. These NEMS-based mass sensing employs tracking the resonance frequencyshifts of the resonators due to mass changes. The most frequently used techniques formeasuring the resonance frequency are based on optical detection . Though inherently simple and highly sensitive, this technique is susceptibleto temperature fluctuation noise because it usually generates heat and heat conduction.On the other hand, it has experimentally been demonstrated that capacitive detection isless affected to noise than optical detection in ambient atmosphere . Capacitive detection is realized by connecting NEMS resonator with standardmicroelectronics, such as complementary metal-oxide-semiconductor (CMOS) circuitry . Here, we propose a scheme for mass sensing based on a coupled nanomechanicalresonator (NR)-Cooper-pair box (CPB) system.
The basic superconducting CPB consists of a low-capacitance superconducting electrodeweakly linked to a superconducting reservoir by a Josephson tunnel junction. Owing toits controllability [11–14], a CPB has been proposed to couple to the NR to drive an NR into asuperposition of spatially separated states and probe the decay of the NR , to prepare the NR in a Fock state and perform a quantum non-demolitionmeasurement of the Fock state , and to cool the NR to its ground state . Recently, this coupled CPB-NR system has been realized in experiments [18, 19] and the resonance frequency shifts of the NR could be monitored by performingmicrowave (MW) spectroscopy measurement. Based on the above-mentioned achievements, inthis article, we investigate the signal absorption spectrum of the CPB qubitcapacitively coupled to an NR in the simultaneous presence of a strong control MWcurrent and a weak signal MW current. Theoretical analysis shows that two sideband peaksappear at the signal absorption spectrum, which exactly correspond to the resonancefrequency of the NR. Therefore, the accreted mass landing on the NR can be weighedprecisely by measuring the frequency shifts because of mass changes of the NR in thesignal absorption spectrum. Similar mass sensing scheme has been proposed recently in ahybrid nanocrystal coupled to an NR by our group , which is based on a theoretical model. However, recent experimentalachievements in the coupled CPB-NR system [18, 19] make it possible for our proposed mass sensing scheme here to be realized infuture.
2 Model and theory
where Δ = ω q - ω c is thedetuning of the qubit resonance frequency and the control current frequency, δ = ω s - ω c is the detuning of thesignal current and the control current, μ =μ 0 SE J0/(8rΦ 0)is the effective 'electric dipole moment' of the qubit, and is the effective 'Rabi frequency' of the controlcurrent.
The real and imaginary parts of χ(ω s )characterize, respectively, the dispersive and absorptive properties.
where is defined as the mass responsivity. However, themeasurement techniques are rather challenging. For example, electrical measurement isunsuitable for mass detections based on very high frequency NRs because of the generatedheat effect . For optical detection, as device dimensions are scaled far below thedetection wavelength, diffraction effects become pronounced and will limit thesensitivity of this approach . Moreover, in any actual implementation, frequency stability of the measuringsystem as well as various noise sources, including thermomechanical noise generated bythe internal loss mechanisms in the resonator and Nyquist-Johnson noise from the readoutcircuitry [3, 26] will also impose limits to the sensitivity of measurement. Here, we candetermine the frequency shifts with high precision by the MW spectroscopy measurementbased on our coupled CPB-NR system.
3 Numerical results and discussion
In what follows, we choose the realistically reasonable parameters to demonstrate thevalidity of mass sensing based on the coupled CPB-NR system. Typical parameters of theCPB (charge qubit) are E C/ħ = 40 GHz andE J0/ħ = 4 GHz such that E C ≫ E J . Experiments by many researchers have demonstrated CPB eigenstates withexcited state lifetime of up to 2 μ s and coherence times of asuperpositions states as long as 0.5 μ s, i.e., T 1 =2μ s, and T 2 = 0.5 μ s [13, 28, 29]. NR with resonance frequency ω n = 2π × 133 MHz, quality factor Q = 5000, and effective massm eff = 73 fg has been used for zeptogram-scale mass sensing . Besides, coupling constant λ between the CPB and NR can bechosen as λ = 0.1ω n = 2π ×13.3 MHz . We assume S = 1 μ m2, r = 10μ m, and , therefore, we can obtain μ/ħ =μ 0 SE J0/(8ħrϕ 0)≈ 30 GHzA-1 and . The experiments of our proposed mass sensing schemeshould be done in situ within a cryogenically cooled, ultrahigh vacuumapparatus with base pressure below 10-10 Torr.
To conclude, we have demonstrated that the coupled NR-CPB system driven by two MWcurrents can be employed as a mass sensor. In this coupled system, the CPB serves as anauxiliary system to read out the resonance frequency of the NR. Therefore, the accretedmass landing on the NR can be weighed conveniently by measuring the frequency shifts inthe signal absorption spectrum. In addition, the use of on-chip capacitive readout willprove especially advantageous for detection in liquid environments of low or arbitrarilyvarying optical transparency, as well as for operation at cryogenic temperatures, wheremaintenance of precise optical component alignment becomes difficult.
The authors gratefully acknowledge the support from the National Natural ScienceFoundation of China (Nos. 10774101 and 10974133) and the National Ministry ofEducation Program for Training Ph.D.
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