Abstract
Nanomechanical resonators (NRs) with very high frequency have a great potential for mass sensing with unprecedented sensitivity. In this study, we propose a scheme for mass sensing based on the NR capacitively coupled to a Cooperpair box (CPB) driven by two microwave currents. The accreted mass landing on the resonator can be measured conveniently by tracking the resonance frequency shifts because of mass changes in the signal absorption spectrum. We demonstrate that frequency shifts induced by adsorption of ten 1587 bp DNA molecules can be well resolved in the absorption spectrum. Integration with the CPB enables capacitive readout of the mechanical resonance directly on the chip.
1 Introduction
Nanoelectromechanical systems (NEMS) offer new prospects for a variety of important applications ranging from semiconductorbased technology to fundamental science [1]. In particular, the minuscule masses of NEMS resonators, combined with their high frequencies and high resonance quality factors, are very appealing for mass sensing [27]. These NEMSbased mass sensing employs tracking the resonance frequency shifts of the resonators due to mass changes. The most frequently used techniques for measuring the resonance frequency are based on optical detection [8]. Though inherently simple and highly sensitive, this technique is susceptible to temperature fluctuation noise because it usually generates heat and heat conduction. On the other hand, it has experimentally been demonstrated that capacitive detection is less affected to noise than optical detection in ambient atmosphere [9]. Capacitive detection is realized by connecting NEMS resonator with standard microelectronics, such as complementary metaloxidesemiconductor (CMOS) circuitry [10]. Here, we propose a scheme for mass sensing based on a coupled nanomechanical resonator (NR)Cooperpair box (CPB) system.
The basic superconducting CPB consists of a lowcapacitance superconducting electrode weakly linked to a superconducting reservoir by a Josephson tunnel junction. Owing to its controllability [1114], a CPB has been proposed to couple to the NR to drive an NR into a superposition of spatially separated states and probe the decay of the NR [15], to prepare the NR in a Fock state and perform a quantum nondemolition measurement of the Fock state [16], and to cool the NR to its ground state [17]. Recently, this coupled CPBNR system has been realized in experiments [18,19] and the resonance frequency shifts of the NR could be monitored by performing microwave (MW) spectroscopy measurement. Based on the abovementioned achievements, in this article, we investigate the signal absorption spectrum of the CPB qubit capacitively coupled to an NR in the simultaneous presence of a strong control MW current and a weak signal MW current. Theoretical analysis shows that two sideband peaks appear at the signal absorption spectrum, which exactly correspond to the resonance frequency of the NR. Therefore, the accreted mass landing on the NR can be weighed precisely by measuring the frequency shifts because of mass changes of the NR in the signal absorption spectrum. Similar mass sensing scheme has been proposed recently in a hybrid nanocrystal coupled to an NR by our group [20], which is based on a theoretical model. However, recent experimental achievements in the coupled CPBNR system [18,19] make it possible for our proposed mass sensing scheme here to be realized in future.
2 Model and theory
In our CPBNR composite system shown schematically in Figure 1, the NR is capacitively coupled to a CPB qubit consisting of two Josephson junctions which form a SQUID loop. A strong control MW current and a weak signal MW current are simultaneously applied in a MW line through the CPB to induce the oscillating magnetic fields in the Josephson junction SQUID loop of the CPB qubit. Besides, a direct current I_{b }is also applied to the MW line to control the magnetic flux through the SQUID loop and thus the effective Josephson coupling of the CPB qubit. The Hamiltonian of our coupled CPBNR system reads:
Figure 1. Schematic diagram of an NR capacitively coupled to a CPB. Two MW currents with frequency ω_{c }and ω_{s }and a direct current I_{b }are applied in the MW line to control the magnetic flux Φ_{x }through the CPB loop.
where H_{CPB }is the Hamiltonian of the CPB qubit described by the
pseudospin 1/2 operators σ_{z }and σ_{x }=
σ_{+ }+ σ_{}. ω_{q
}= 4E_{c}(2n_{g } 1)/ħ is the
electrostatic energy difference and E_{J0 }is the maximum Josephson
energy. 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 total
capacitance and n_{g }= (C_{b}V_{b }+
C_{g}V_{g})/(2e) is the dimensionless
polarization charge (in units of Cooper pairs), where C_{b }and
V_{b }are, respectively, the capacitance and voltage between the NR
and the CPB island, C_{g }and V_{g }are, respectively,
the gate capacitance and voltage of the CPB qubit, and C_{J }is the
capacitance of each Josephson junction. Displacement (by x) of the NR leads to
linear modulation of the capacitance between NR and CPB,
C_{b}(x) ≈ C_{b}(0) +
(∂C_{b}/∂x)x, which modulates the
electrostatic energy of the CPB qubit, resulting in the capacitive coupling constant
where Δ = ω_{q } ω_{c }is the
detuning of the qubit resonance frequency and the control current frequency, δ
= ω_{s } ω_{c }is the detuning of the
signal current and the control current, μ =
μ_{0}SE_{J0}/(8rΦ_{0})
is the effective 'electric dipole moment' of the qubit, and
The dynamics of the coupled CPBNR system in the presence of dissipation and dephasing is described by the following master equation [21]
where ρ is the density matrix of the coupled system, T_{1
}is the qubit relaxation time, τ_{ϕ }is the qubit pure
dephasing time, and γ is the decay rate of the NR which is given by
γ = ω_{n}/Q.
Using the identity
where
Note that if the pure dephasing rate is neglected, i.e.,
where
where
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 auxiliary function
The population inversion k_{0 }of the CPB is determined by
p_{1 }is a parameter corresponding to the linear susceptibility
The real and imaginary parts of χ(ω_{s}) characterize, respectively, the dispersive and absorptive properties.
The coupled CPBNR system has been proposed to measure the vibration frequency of
the NR
by calculating the absorption spectrum [23]. On the other hand, NRs have widely been used as mass sensors by measuring
the resonant frequency shift because of the added mass of the bound particles. The
mass
sensing principle is simple. NRs can be described by harmonic oscillators with an
effective mass m_{eff}, a spring constant k, and a mechanical
resonance frequency
where
3 Numerical results and discussion
In what follows, we choose the realistically reasonable parameters to demonstrate
the
validity of mass sensing based on the coupled CPBNR system. Typical parameters of
the
CPB (charge qubit) are E_{C}/ħ = 40 GHz and
E_{J0}/ħ = 4 GHz such that E_{C
}≫ E_{J }[27]. Experiments by many researchers have demonstrated CPB eigenstates with
excited state lifetime of up to 2 μs and coherence times of a
superpositions states as long as 0.5 μs, i.e., T_{1 }=
2μs, and T_{2 }= 0.5 μs [13,28,29]. NR with resonance frequency ω_{n }= 2π
× 133 MHz, quality factor Q = 5000, and effective mass
m_{eff }= 73 fg has been used for zeptogramscale mass sensing [5]. Besides, coupling constant λ between the CPB and NR can be
chosen as λ = 0.1ω_{n }= 2π ×
13.3 MHz [16]. We assume S = 1 μm^{2}, r = 10
μm, and
Firstly, we would show the principle of measuring the resonance frequency of the NR
in
the coupled CPBNR system. Figure 2a illustrates the absorption of
the signal current as a function of the detuning Δ_{s
}(Δ_{s }= ω_{s }
ω_{q}). The absorption (Im(χ)) has been
normalized with its maximum when the control current is resonant with the CPB qubit
(Δ_{c }= 0). Mollow triplet, commonly known in atomic and
some artificial twolevel 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
mode
of the NR. Our proposed mass sensing scheme is just based on these new features in
the
absorption spectrum. An intuitive physical picture explaining these peaks can be given
in the energy level diagram shown in Figure 2b. The Hamiltonian of
the coupled system without the externally applied current can be diagonalized [33,34] in the eigenbasis of
Figure 2. Scaled absorption spectrum of the signal current as a function of the detuning Δ_{s }and energy level diagram of the coupled system. (a) Scaled absorption spectrum of the signal current as a function of the detuning Δ_{s }without landing any masses 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 be well 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 μs, T_{2 }= 0.05 μs, and Ω_{c }= 3.
Next, we illustrate how to measure the mass of the particles landing on the NR based
on
the above discussions. Unlike traditional mass spectrometers, nanomechanical mass
sensors do not require the potentially destructive ionization of the test sample,
are
more sensitive to large biomolecules, such as proteins and DNA, and could eventually
be
incorporated on a chip [6]. Here, we use the functionalized 1587 bp long dsDNA molecules with mass
m_{DNA }≈ 1659 zg (1 zg = 10^{21 }g) [35], and assume for simplicity that the mass adds uniformly to the mass of the
overall NR and changes the resonance frequency of the NR by an amount given by Equation
19. Figure 2c demonstrates the signal absorption as a function of
Δ_{s }before and after a binding event of ~ 10
functionalized 1587 bp DNA molecules in the vicinity of the resonance frequency of
the
NR. 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
the
NR. According to Equation 19, we can obtain the mass of the accreted DNA molecule:
Figure 3. Plot of frequency shifts versus the number of DNA molecules landing on two different masses of NRs. Other parameters used are ω_{n }= 835 MHz, λ_{0 }= 0.01, Δ_{c }= 0. Q = 5000, T_{1 }= 0.25 μs, T_{2 }= 0.05 μs, and Ω_{c }= 3.
In order to demonstrate the novelty of our proposed mass sensing scheme, we plot Figure 4 to illustrate how the vibration mode of NR and the control current affect the spectral features. Figure 4a shows the absorption spectrum of the signal field through the CPB system without the influence of the NR (coupling off) in the absence of the control field (control off), which shows the standard resonance absorption profile. However, when the coupling turns on, the center of the curve shifts from the resonance ω_{s }= ω_{q }a bit, as shown in Figure 4b. This is because of the coupling λ_{0 }between the CPB and the NR [16,36]. Figure 4c demonstrates the absorption spectrum of the signal 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 artificial twolevel system [31,32]. None of the above situations can be used to measure the resonance frequency of the NR. However, when the coupled CPBNR system is driven by a strong control field and a weak signal field simultaneously, the resonance frequency of the NR be measured from the absorption spectrum of the signal field, as shown in Figure 4d. The spectral linewidth of the two sideband peaks that corresponds to the resonance frequency of the NR is much narrower than the peak in the center, since the damping rate of the NR is much smaller than the decay rate of the CPB qubit. Therefore, such a coupled CPBNR system is proposed here to measure the resonance frequency of the NR when the control field is resonant with the CPB qubit (ω_{c }= ω_{q}). By measuring the frequency shift of the NR before and after the adsorption of particles landing on it, we can obtain the accreted mass according to Equation 19.
Figure 4. Signal current absorption spectrum as a function of the detuning Δ_{s }considering the effects of NR and the control field. Other parameters are Δ_{c }= 0, Q = 5000, T_{1 }= 0.25 μs, T_{2 }= 0.05 μs, and ω_{n }= 835 MHz.
4 Conclusion
To conclude, we have demonstrated that the coupled NRCPB system driven by two MW currents can be employed as a mass sensor. In this coupled system, the CPB serves as an auxiliary system to read out the resonance frequency of the NR. Therefore, the accreted mass landing on the NR can be weighed conveniently by measuring the frequency shifts in the signal absorption spectrum. In addition, the use of onchip capacitive readout will prove especially advantageous for detection in liquid environments of low or arbitrarily varying optical transparency, as well as for operation at cryogenic temperatures, where maintenance of precise optical component alignment becomes difficult.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
CJ finished the main work of this article, including deducing the formulas, plotting the figures, and drafting the manuscript. BC and JJL participated in the discussion and provided some useful suggestion. KDZ conceived of the idea and participated in the coordination.
Acknowledgements
The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Nos. 10774101 and 10974133) and the National Ministry of Education Program for Training Ph.D.
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