1 Introduction

Nanoelectromechanical systems (NEMS) offer new prospects for a variety of importantapplications ranging from semiconductor-based technology to fundamental science [1]. In particular, the minuscule masses of NEMS resonators, combined with theirhigh frequencies and high resonance quality factors, are very appealing for mass sensing [27]. 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 [8]. 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 [9]. Capacitive detection is realized by connecting NEMS resonator with standardmicroelectronics, such as complementary metal-oxide-semiconductor (CMOS) circuitry [10]. 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 [1114], 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 [15], to prepare the NR in a Fock state and perform a quantum non-demolitionmeasurement of the Fock state [16], and to cool the NR to its ground state [17]. 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 [20], 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

In our CPB-NR composite system shown schematically in Figure 1,the NR is capacitively coupled to a CPB qubit consisting of two Josephson junctionswhich form a SQUID loop. A strong control MW current and a weak signal MW current aresimultaneously applied in a MW line through the CPB to induce the oscillating magneticfields in the Josephson junction SQUID loop of the CPB qubit. Besides, a direct currentI b is also applied to the MW line to control the magnetic fluxthrough the SQUID loop and thus the effective Josephson coupling of the CPB qubit. TheHamiltonian of our coupled CPB-NR system reads:

Figure 1
figure 1

Schematic diagram of an NR capacitively coupled to a CPB. Two MW currentswith frequency ω c and ω s and adirect current I b are applied in the MW line to control themagnetic flux Φ x through the CPB loop.

H = H CPB + H NR + H int ,
H CPB = 1 2 ω q σ z - 1 2 E J0 cos π Φ x ( t ) Φ 0 σ x ,
H NR = ω n a a ,
H int = λ ( a + a ) σ z .

where H CPB is the Hamiltonian of the CPB qubit described by thepseudospin -1/2 operators σ z and σ x =σ + + σ -. ω q = 4E c(2n g - 1)/ħ is theelectrostatic energy difference and E J0 is the maximum Josephsonenergy. Here, E C = e 2/2C Σ is the charging energy with C Σ = C b + C g + 2C J being the CPB island's totalcapacitance and n g = (C b V b +C g V g)/(2e) is the dimensionlesspolarization charge (in units of Cooper pairs), where C b andV b are, respectively, the capacitance and voltage between the NRand the CPB island, C g and V g are, respectively,the gate capacitance and voltage of the CPB qubit, and C J is thecapacitance of each Josephson junction. Displacement (by x) of the NR leads tolinear modulation of the capacitance between NR and CPB,C b(x) ≈ C b(0) +(∂C b/∂x)x, which modulates theelectrostatic energy of the CPB qubit, resulting in the capacitive coupling constantλ= 4 n g NR E C 1 C b C b x Δ x z p , where n g NR = C b V b 2e and Δ x z p = 2 m ω n is the zero-point uncertainty of the NR with effectivemass m and resonance frequency ω n . The couplingbetween the MW line and the CPB qubit in the second term of Equation 2 results from thetotally externally applied magnetic flux Φ x (t) =Φ q(t) + Φ b through the CPBqubit loop of an effective area S with Φ 0 =h/(2e) being the flux quantum. Here,Φ q(t) =μ 0 SI(t)/(2πr), with r being the distance between the MW line and the qubit and μ 0 being the vacuum permeability. Φ q(t) andΦ b are controlled, respectively, by the MW currentI ( t ) = c cos ( ω c t ) + s cos ( ω s t + δ ) and the direct current I b in the MWline. For convenience, we assume the phase factor δ' = 0 because it is notdifficult to demonstrate that the results of this article are not dependent on the valueof δ'. By adjusting the direct current I b and the MWcurrent I(t) such that Φ bΦ q (t) andπΦ b/Φ 0 = π /2,we can obtain E J cos π Φ x ( t ) Φ 0 - E J π Φ q ( t ) Φ 0 . In a rotating frame at the control frequencyω c, the total Hamiltonian can now be written as

H = 1 2 Δ σ z + ω n a a + λ ( a + a ) σ z + Ω ( σ + + σ - ) + μ s ( σ + e - i δ t + σ - e i δ t ) ,

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/(8 0)is the effective 'electric dipole moment' of the qubit, and Ω=μ c is the effective 'Rabi frequency' of the controlcurrent.

The dynamics of the coupled CPB-NR system in the presence of dissipation and dephasingis described by the following master equation [21]

d ρ d t = - i [ H , ρ ] + 1 2 T 1 [ σ - ] + γ 2 [ a ] + 1 4 τ ϕ [ σ z ] ,

where ρ is the density matrix of the coupled system, T 1 is the qubit relaxation time, τ ϕ is the qubit puredephasing time, and γ is the decay rate of the NR which is given byγ = ω n /Q. [ D ] , describing the incoherent decays, is the Lindbladoperator for an operator and is given by:

[ D ] = 2 D ρ D - D D ρ - ρ D D .

Using the identity O ˙ = T r ( O ρ ˙ ) for an operator O and a density matrix ρ in Equation 6, we obtain the following Bloch equations for the coupled CPB-NRsystem:

d σ - d t = - 1 T 2 + i Δ σ - - i Q σ - + i Ω σ z + i μ s σ z e - i δ t ,
d σ z d t = - 1 T 1 ( σ z + 1 ) - 2 i Ω ( σ + - σ - ) - 2 i μ ( s σ + e - i δ t - s * σ - e i δ t ) ,
d 2 Q d t 2 + γ d Q d t + ω r 2 Q = - 4 ω r 3 λ 0 σ z ,

where λ 0 = λ 2 ω n 2 and T 2 is the qubit dephasing timesatisfying

1 T 2 = 1 2 T 1 + 1 τ ϕ .

Note that if the pure dephasing rate is neglected, i.e., 1 τ ϕ =0, then T 2 = 2T 1. Inorder to solve the above equations, we first take the semiclassical approach byfactorizing the NR and CPB qubit degrees of freedom, i.e.,〈 -〉 = 〈Q〉〈σ -〉, which ignores any entanglement betweenthese systems. For simplicity, we define p = μσ -,k = σ z and then we have

d p d t = - 1 T 2 - i ( Δ + Q ) p + i μ 2 k ,
d k d t = - 1 T 1 ( k + 1 ) - 4 Im ( p * ) ,
d 2 Q d t 2 + γ d Q d t + ω r 2 Q = - 4 λ 0 ω r 3 k

where = c + s e - i δ t . In order to solve the above equations, we make the ansatz〈p(t)〉 = p 0 +p 1 e -iδt+p -1 e iδt,〈k(t)〉 = k 0 +k 1 e -iδt+k -1 e iδt, and〈Q(t)〉 = Q 0 +Q 1 e -iδt+Q -1 e iδt [22]. Upon substituting these equations into Equations 12-14 and upon working tothe lowest order in s but to all orders in c , we obtain in the steady state:

p 1 = μ 2 s T 2 k 0 2 T 1 T 2 B ( δ 0 + 2 i ) ( C + Ω c 2 ) + E ( B - δ 0 ) A E ( B - δ 0 ) .


A = Δ c - 4 λ 0 ω 0 k 0 - δ 0 - i , B = Δ c - 4 λ 0 ω 0 k 0 + δ 0 + i , C = 4 λ 0 ω 0 k 0 η Ω c 2 ( Δ c - 4 λ 0 ω 0 k 0 - i ) , D = 4 λ 0 ω 0 k 0 η Ω c 2 ( Δ c - 4 λ 0 ω 0 k 0 + i ) , E = 2 T 1 T 2 A ( D + Ω c 2 ) - 2 T 1 T 2 B ( C + Ω c 2 ) - A B ( T 1 T 2 δ 0 + i ) .

Here, dimensionless variables ω 0 =ω r T 2, γ 0 =γT 2, Ω c =ΔT 2, and Δ c =ΔT 2 are introduced for convenience and the auxiliaryfunction

η = ω 0 2 ω 0 2 - i γ 0 δ 0 - δ 0 2 .

The population inversion k 0 of the CPB is determined by

( k 0 + 1 ) [ ( Δ c - 4 λ 0 ω 0 k 0 ) 2 + 1 ] + 4 Ω c 2 k 0 T 1 T 2 = 0 .

p 1 is a parameter corresponding to the linear susceptibility χ ( 1 ) ( ω s ) = p 1 s = ( μ 2 T 2 ) χ ( ω s ) , where the dimensionless linear susceptibilityχ(ω s ) is given by

χ ( ω s ) = 2 T 1 T 2 B ( δ 0 + 2 i ) ( C + Ω c 2 ) + E ( B - δ 0 ) A E ( B - δ 0 ) k 0 .

The real and imaginary parts of χ(ω s )characterize, respectively, the dispersive and absorptive properties.

The coupled CPB-NR system has been proposed to measure the vibration frequency of the NRby calculating the absorption spectrum [23]. On the other hand, NRs have widely been used as mass sensors by measuringthe resonant frequency shift because of the added mass of the bound particles. The masssensing principle is simple. NRs can be described by harmonic oscillators with aneffective mass m eff, a spring constant k, and a mechanicalresonance frequency ω n = k m eff . When a particle adsorbs to the resonator andsignificantly increases the resonator's effective mass, therefore, the mechanicalresonance frequency reduces. Mass sensing is based on monitoring the frequency shiftΔω of ω n induced by the adsorption to theresonator. The relationship between Δω with the deposited massΔm is given by

Δ m = - 2 m eff ω n Δ ω = - 1 Δ ω ,

where = ( - 2 m eff ω n ) - 1 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 [24]. For optical detection, as device dimensions are scaled far below thedetection wavelength, diffraction effects become pronounced and will limit thesensitivity of this approach [25]. 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 CE J [27]. 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 [5]. Besides, coupling constant λ between the CPB and NR can bechosen as λ = 0.1ω n = 2π ×13.3 MHz [16]. We assume S = 1 μ m2, r = 10μ m, and c =200μ A [30], therefore, we can obtain μ/ħ =μ 0 SE J0/(8ħrϕ 0)≈ 30 GHzA-1 and Ω c =Ω T 2 = ( μ ) c T 2 =3. 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.

Firstly, we would show the principle of measuring the resonance frequency of the NR inthe coupled CPB-NR system. Figure 2a illustrates the absorption ofthe signal current as a function of the detuning Δ s (Δ s = ω s -ω q ). The absorption (Im(χ)) has beennormalized with its maximum when the control current is resonant with the CPB qubit(Δ c = 0). Mollow triplet, commonly known in atomic andsome artificial two-level system [31, 32], appears in the middle part of Figure 2a. However,there are also two sharp peaks located exactly at Δ s ω n in the sidebands of the absorption spectrum,which corresponds to the resonant absorption and amplification of the vibrational modeof the NR. Our proposed mass sensing scheme is just based on these new features in theabsorption spectrum. An intuitive physical picture explaining these peaks can be givenin the energy level diagram shown in Figure 2b. The Hamiltonian ofthe coupled system without the externally applied current can be diagonalized [33, 34] in the eigenbasis of | ± , N ± = | ± z e ( λ / ω n ) ( a a ) | N , with the eigenenergies E ±ħ/2ω q +ħω n (N - λ 0), wherethe CPB qubit states |±〉 z are eigenstates ofσ z with the excited state |+〉 z = |e〉 and the ground state |-〉 z =|g〉, the resonator states |N ±〉 areposition-displaced Fock states. Transitions between |-, N -〉and |+, (N + 1)+〉 represent signal absorption centered atω c + ω n (the rightmost solidline in Figure 2b). Besides, transitions between |+,N +〉 and |-, (N + 1)-〉 indicateprobe amplification (the leftmost solid line in Figure 2b) becauseof a three-photon process, involving simultaneous absorption of two control photons andemission of a photon at frequency ω c -ω n . The middle dashed lines in Figure 2a corresponds to the transition where the signal frequency is equal to thecontrol frequency. Therefore, Figure 2a provides a method tomeasure the resonance frequency of the NR. If we first tune the frequency of the controlMW current to be resonant with the CPB qubit (ω c =ω q ) and scan the signal frequency across the CPB qubitfrequency, then we can easily obtain the resonance frequency of the NR from the signalabsorption spectrum.

Figure 2
figure 2

Scaled absorption spectrum of the signal current as a function of the detuning Δ s and energy level diagram of thecoupled system. (a) Scaled absorption spectrum of the signal currentas a function of the detuning Δ s without landing anymasses on the NR. (b) The energy level diagram of the CPB coupled to an NR.(c) Signal absorption spectrum as a function of Δ s before (black solid line) and after (red dashed line) a binding event of ~10 functionalized 1587 bp long dsDNA molecules. Frequency shift of 95 kHz can bewell resolved in the spectrum. Other parameters used are ω n = 835 MHz, λ 0 = 0.01, Δ c = 0, Q = 5000, T 1 = 0.25 μ s,T 2 = 0.05 μ s, and Ω c = 3.

Next, we illustrate how to measure the mass of the particles landing on the NR based onthe above discussions. Unlike traditional mass spectrometers, nanomechanical masssensors do not require the potentially destructive ionization of the test sample, aremore sensitive to large biomolecules, such as proteins and DNA, and could eventually beincorporated on a chip [6]. Here, we use the functionalized 1587 bp long dsDNA molecules with massm DNA ≈ 1659 zg (1 zg = 10-21 g) [35], and assume for simplicity that the mass adds uniformly to the mass of theoverall NR and changes the resonance frequency of the NR by an amount given by Equation19. Figure 2c demonstrates the signal absorption as a function ofΔ s before and after a binding event of ~ 10functionalized 1587 bp DNA molecules in the vicinity of the resonance frequency of theNR. We can see clearly that there is a resonance frequency shift Δω =-95 kHz after the adsorption of the DNA molecules because of the increased mass of theNR. According to Equation 19, we can obtain the mass of the accreted DNA molecule:Δm=- 2 m eff ω n Δω=16590 zg , about the mass of 10 functionalized 1587 bp long dsDNAmolecules. Therefore, such a coupled CPB-NR system can be used to weigh the externalaccreted mass landing on the NR by measuring the frequency shift in the signalabsorption spectrum when the control current is resonant with the CPB qubit. Plot offrequency shifts versus the number of DNA molecules landing on two different masses ofNRs. Other parameters used are ω n = 835 MHz,λ 0 = 0.01, Δ c = 0, Q =5000, T 1 = 0.25 μ s, T 2 = 0.05μ s, and Ω c = 3. Mass responsivity is an important parameter to evaluate the performance of aresonator for mass sensing. Figure 3 plots the frequency shifts asa function of the number of DNA molecules landing on the NR for two different kinds ofNRs. One is ω n = 2π × 133 MHz(m eff = 73 fg), the other is ω n =2π × 190 MHz (m eff = 96 fg) [2, 3]. The mass responsivities, which can be obtained from the slope of the line,are, respectively, ||5.72 Hz/zg and ||6.21 Hz/zg . Smaller mass of the nanoresonator enables higher massresponsivity. Here, we have assumed that the DNA molecules land evenly on the NR andthey remain on it. In fact, the position on the surface of the resonator where thebinding takes place is one factor that strongly affects the resonance frequency shift.The maximization in mass responsivity is obtained if the landing takes places at theposition where the resonator's vibrational amplitude is maximum. For the doubly clampedNR used in our model, maximum shift is achieved at the center for the fundamental modeof vibration, while the minimum shift exists at the clamping points. This statisticaldistribution of frequency shifts has been investigated by building the histogram ofevent probability versus frequency shift for small ensembles of sequential singlemolecule or single nanoparticle adsorption events [6, 7].

Figure 3
figure 3

Plot of frequency shifts versus the number of DNA molecules landing on twodifferent masses of NRs. Other parameters used are ω n = 835 MHz, λ 0 = 0.01, Δ c = 0. Q = 5000, T 1 = 0.25 μ s,T 2 = 0.05 μ s, and Ω c = 3.

In order to demonstrate the novelty of our proposed mass sensing scheme, we plot Figure4 to illustrate how the vibration mode of NR and the controlcurrent affect the spectral features. Figure 4a shows theabsorption spectrum of the signal field through the CPB system without the influence ofthe NR (coupling off) in the absence of the control field (control off), which shows thestandard resonance absorption profile. However, when the coupling turns on, the centerof the curve shifts from the resonance ω s =ω q a bit, as shown in Figure 4b. Thisis because of the coupling λ 0 between the CPB and the NR [16, 36]. Figure 4c demonstrates the absorption spectrum of thesignal field when the control field turns on in the absence of the NR (coupling off).This is the commonly known Mollow triplet, which appears in atomic and some artificialtwo-level system [31, 32]. None of the above situations can be used to measure the resonance frequencyof the NR. However, when the coupled CPB-NR system is driven by a strong control fieldand a weak signal field simultaneously, the resonance frequency of the NR be measuredfrom the absorption spectrum of the signal field, as shown in Figure 4d. The spectral linewidth of the two sideband peaks that corresponds to theresonance frequency of the NR is much narrower than the peak in the center, since thedamping rate of the NR is much smaller than the decay rate of the CPB qubit. Therefore,such a coupled CPB-NR system is proposed here to measure the resonance frequency of theNR when the control field is resonant with the CPB qubit (ω c =ω q ). By measuring the frequency shift of the NR beforeand after the adsorption of particles landing on it, we can obtain the accreted massaccording to Equation 19.

Figure 4
figure 4

Signal current absorption spectrum as a function of the detuning Δ s considering the effects of NR and thecontrol field. Other parameters are Δ c = 0, Q = 5000, T 1 = 0.25 μ s, T 2 = 0.05 μ s, and ω n = 835 MHz.

4 Conclusion

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.