Design of strongly nonlinear graphene nanoelectromechanical systems in quantum regime

Quantum entities, such as photons, spins, ions, and artificial atoms (e.g., quantum dots, superconducting circuits, etc.), lie at the heart of quantum information processing. They serve as basic physical building blocks^1–5 to encode fundamental quantum bits (qubits) and states, exploiting their quantum phenomena (e.g., superposition, entanglement) in the processing, storage, and communication of quantum information. Among many candidates and variants, superconducting qubits,^2,6,7 trapped ions,³ and photonic integrated circuits⁴ have emerged as leading hardware building blocks to construct integrated quantum processors for implementation of algorithms in the noisy intermediate scale quantum (NISQ) technology era.⁸ These advances have spurred the development of the first quantum processing platforms and catalyzed fledgling demonstrations of “quantum supremacy” (or “quantum advantage”), a milestone that refers to the demonstration of computational capability (in solving certain problems) that is not achievable (in feasible time) by even the most powerful classical supercomputers.^9–11 

Amidst these remarkable advances, there are concerns about challenges in device integration and scaling beyond NISQ devices toward distributed architectures and fully fault-tolerant processors. For example, a superconducting qubit essentially consists of two components: a capacitor and a Josephson junction (JJ), together operating as a nonlinear LC resonator (artificial atom) in the microwave frequency band.^2,6,7 In order to minimize deleterious perturbations from low-frequency charge noise, while preserving useful nonlinearity or anharmonicity, the energy in the Josephson junction should be much larger than the energy stored in the capacitor (i.e., operating the qubit in the transmon regime^6,7). To achieve this, the size of the capacitor in the superconducting qubit needs to be increased, leading to a fairly large footprint: the typical planar dimension of the single transmon qubit is around ∼300–500 μm.¹² Although 3D integration is being developed,¹³ the size of the quantum processors would still occupy considerably large volume when one accounts for the integrated components (e.g., linear microwave cavity or planar resonators) used to control, program, and read out the transmon qubit states.

To address these challenges and to facilitate miniaturization, the employment of resonant nano/microelectromechanical systems (N/MEMS) in quantum devices and circuits, and engineering N/MEMS in the quantum regime (with electrical or optical transduction), have been envisioned and actively explored.^14,15 N/MEMS have very small footprints and hold great promise for further miniaturization thanks to advanced nanofabrication processes. A number of important milestones and breakthroughs have been achieved in coupling N/MEMS flexural, thin film bulk acoustic, or surface acoustic wave resonators with superconducting qubits and pushing them into their quantum regime.^16–23 Encouragingly, over the recent decade, there have been several successful demonstrations of the preparation of these resonators with high probability in their quantum ground states.^17–28 These achievements have been done either by cryogenic cooling of devices to milli-Kelvin (mK) temperatures^{17–23,26–28} or cooling mechanical modes below environmental temperatures using radiation pressure back-action (e.g., sideband cooling^24,25). Critically, the quantum behavior of phonons in mechanical resonators can be coupled to microwave cavities or superconducting qubits, enabling coherent control and detection of phonon states. Thanks to excellent coherence of mechanical modes enabled by high quality (Q) factors up to Q ≈10⁹–10¹⁰ (Refs. 26 and 27), phonons could possibly act as long-lived quantum memories or transducers that can store, transmit, and receive quantum states.

Despite these remarkable achievements, N/MEMS resonators have not been employed as qubits in quantum information processing yet. The potential energy of mechanical resonators depends quadratically on their displacement when device motion is small (or strictly, in its linear regime where amplitude ∝ force). In the quantum regime, this can be described very accurately by using a quantum harmonic resonator model^14,29 with strong linearity, which is valid for all the demonstrated quantum MEMS and acoustic resonators, especially those with high-stiffness acoustic modes and high signal ceilings in their linear dynamic ranges. Accordingly, these quantum MEMS resonators exhibit a harmonic spectrum with equally separated energy levels of ΔE = ℏω, making the transition energy between ground and first excited level (E₀→E₁) indistinguishable from those of other transitions (E_n+1→E_n+2, where n is an integer with n ≥ 0, representing a sequence of energy levels).^14,29 Toward the desired anharmonicity, NEMS resonators with much lower linear stiffness and much earlier onset of nonlinearity^30,31 than their MEMS counterparts deserve careful investigations. Some schemes with certain complexity, for example, utilizing buckling instability³² or coupling mechanical motion to two quantum dots in a carbon nanotube,³³ have been proposed to introduce anharmonicity, but concepts of NEMS qubits based on simpler and more generic resonator designs have not been investigated yet.

The MEMS and acoustic resonators that have been demonstrated to enter the quantum regime hitherto have been mainly carved out of 3D crystalline materials or films including Si,^26,27 aluminum nitride (AlN),^17–19 quartz,²³ aluminum,^24,25 and lithium niobate (LiNbO₃).^20–22 Meanwhile, in a further miniaturized regime and toward molecular/atomic-scale systems, van der Waals (vdW) crystals, such as graphene and atomic layer heterostructures, have recently emerged as an attractive platform for making two-dimensional (2D) NEMS resonators with orders of magnitude smaller footprint and volume, technical merits, and promising perspectives for being incorporated in quantum systems.^31,34–40 In addition to superior elastic properties (Young's modulus of 1 TPa and strain limit of 25%),⁴¹ which are suitable for achieving ultrahigh-frequency mechanical modes necessary for accessing the quantum regime, 2D vdW crystals offer atomically thin layers with attractive and tailorable electronic, photonic, and phononic properties; thus engineering parameters of 2D NEMS could enable new characteristics, high performance, and diverse functionalities for quantum transduction and interconnect applications. Moreover, in contrast to conventional N/MEMS, resonators based on 2D vdW materials exhibit ultrabroad electronic frequency tunability up to Δf/f ∼ 430% (Ref. 40), providing an efficient means of controlling frequency. Finally, the Q factors of 2D NEMS can be greatly improved at cryogenic temperatures with up to Q ≈1.05 × 10⁶ demonstrated recently (Ref. 36), promising long relaxation times, which are critical for high fidelity quantum operations. To date, despite the encouraging progress of 2D NEMS resonators in the classical regime and possessing many advantageous attributes for quantum applications, the operation of 2D NEMS devices in the quantum regime has not been demonstrated.

In this Letter, we present the analysis, modeling, and design of graphene 2D NEMS resonators [Fig. 1(a)] to investigate the feasibility of making NEMS qubits that can retain strong Duffing nonlinearity (quartic in terms of potential energy) even for mean square flexural displacement in the ground state. We find that graphene NEMS can have adequate anharmonicity in their potential energy thanks to ultrasmall mass and strong Duffing nonlinearity (a universal nonlinear mechanism for mechanical systems^30,31), and thus may enable highly miniaturized qubits.

To start our analysis, we first deduce conditions that must be achieved for a flexural mode of a doubly clamped graphene NEMS resonator [Fig. 1(a)] to operate as a qubit. First, generally speaking, the temperature of the 2D graphene NEMS resonator, and hence its thermal energy, should be negligible compared to the energy level splitting of the qubit states. This condition will enable high fidelity ground state preparation and limit thermally induced transitions between states. As a rule of thumb, it is gauged by the thermal occupation factor N_th, which is given by^14,29

where ℏ is the reduced Planck constant, k_B is the Boltzmann constant, T is the temperature, and ω_m = 2πf_m is the angular frequency of the mechanical resonance mode in the absence of nonlinearity. As discussed below, the nonlinearity of the NEMS resonator derives from a tensioning effect that produces a positive anharmonicity. This leads to an increase in energy level spacing that depends on the specific energy level n, which causes deviations from the Bose–Einstein occupation factor [Eq. (1)]. Nonetheless, because of the positive anharmonicity, Eq. (1) provides an upper-bound estimate on N_th for a given ω_m and T. From Eq. (1), the larger the ratio ℏω_m/(k_BT) of the resonance mode, the more likely the mode is to be found in the ground state when at thermal equilibrium. Thus, for proper comparison with today's mainstream or leading qubit modalities, we should limit attention to the modes of NEMS resonators that have sufficiently high ℏω_m/(k_BT) to access the deep quantum regime, e.g., N_th ≪10⁻².

Figure 2(a) shows the calculated N_th using Eq. (1) for the NEMS temperature at 10 mK, which is readily achievable using commercial dilution refrigerators. It is clear that for NEMS mode resonance frequencies above a critical frequency of f_c,10mK ≈0.96 GHz, the deep quantum regime of N_th ≤10⁻² is achieved. For accessing the much lower N_th ≤10⁻³ and N_th ≤10⁻⁴, the device frequency should be increased above f_c,10mK ≈ 1.44 GHz and f_c,10mK ≈1.92 GHz, respectively. If we relax the temperature to modest 30 mK, the critical frequency increases to f_c,30mK ≈2.88 GHz for achieving N_th = 10⁻².

Assuming the effect of the nonlinearity of the 2D NEMS resonators only serves to increase the spacing of excited state energy levels compared to the quantum harmonic resonator model, an upper (lower) bound on the occupation probability of the excited (ground) energy states can be estimated from²⁹ 

With N_th = 10⁻², the estimated occupation probability is P₀ ≈0.99 (ground energy state) and P₁ ≈0.0098 (first excited energy state), which is on par with state-of-the-art steady-state superconducting ground and first excited state thermal populations. Achieving the more aggressive N_th = 10⁻³ and N_th = 10⁻⁴ (yielding P₀ ≈0.999 and P₀ ≈0.9999, respectively) would require careful engineering of phonon thermal conduction channels;^42,43 but if it could be achieved, it would yield an important advantage over superconducting qubits, which can be driven out of equilibrium with the milli-Kelvin environment due to black-body radiation propagating through microwave circuitry from higher temperatures stages.^44,45

To determine the proper resonator dimensions to engineer the required frequencies for quantum operations in the milli-Kelvin regime, we consider the graphene NEMS resonator illustrated in Fig. 1(a). This device consists of a single atomic layer with a thickness of b = 0.335 nm. With nonzero tension, the device behaves as a doubly clamped membrane (simple lateral extrusion of a 1D string in the width direction) due to its negligible flexural rigidity. The linear resonance frequency of this graphene membrane NEMS resonator scales with its length L and built-in strain ε:

where ρ is the mass density and E_Y is Young's modulus. Figure 2(b) shows the calculated device length and strain level to achieve the linear resonance frequency f_m = 0.96, 1.44, and 1.92 GHz. These device dimensions can be achievable using advanced lithography techniques such as extreme ultraviolet lithography⁴⁶ or electron beam lithography.^47,Figure 2(b) also displays scaling of the resonance frequency. As the device length becomes shorter and built-in strain goes higher, the resonance frequency increases, and the frequency to put the graphene device in the quantum regime f_m >0.96 GHz can be attainable by adjusting these two parameters. The strain level required to operate the graphene NEMS in the quantum regime can be achieved by precisely tuning the strain using electrostatic force^31,34–40 or stretching the graphene membrane employing MEMS actuators,^48,49 which have already been demonstrated in engineering 2D NEMS in the classical regime.

We now turn our focus to the nonlinearity of the graphene 2D NEMS resonator in the quantum regime by considering intrinsic stiffening Duffing nonlinearity, which originates from additional tension induced by dynamic vibration of the resonator.^30,31,50 The device motion with Duffing nonlinearity can be described by

where z(t) = z₀cos(ωt) is the harmonic oscillation displacement with amplitude z₀, m_eff is the effective mass, k₁ is the first order (linear) spring constant (stiffness), and k₃ = m_effα is the third-order spring constant with α being the Duffing nonlinear coefficient, and F = F(t) is the driving force. For the doubly clamped membrane resonator, m_eff = 0.5ρLwb, where w is the width of the device. The coefficient α can be obtained from the resonator mode shape, $α=EY[∫0L[∂u(x)∂x]2dx]2/[2ρL∫0L[u(x)]2dx]$⁠, where u(x) is the mode shape of the graphene NEMS resonator $u(x)=2 sin (πxL)$ (Ref. 50). For the first resonance mode, we obtain $α=EYπ4/(2ρL4)$ for doubly clamped membrane (or string) resonators. With α given in terms of material and geometric parameters, we can write the full potential energy for the graphene NEMS resonator by including both linear and nonlinear (Duffing) spring constants:

The effects of the nonlinearity on the quantum dynamics of the NEMS can now be modeled through a simple analysis following the standard canonical quantization of z and p (position and momentum). Utilizing the quantized version of Eq. (5) in the Schrödinger equation, one can then solve for the energy eigenstates of the nonlinear NEMS and compare to the harmonic case (Fig. 3). Furthermore, one can determine the degree of anharmonicity in the energy spectrum by calculating absolute Δf₂₁–Δf₁₀ or relative anharmonicity (Δf₂₁–Δf₁₀)/Δf₁₀, where Δf₁₀ = f₁–f₀ is the transition frequency between the ground state f₀ and the first level f₁, and Δf₂₁ = f₂–f₁ is the transition frequency between the first (f₁) and second (f₂) levels. This quantification of anharmonicity enables direct comparison with existing qubit modalities.

We investigate the degree of anharmonicity of graphene NEMS resonators by evaluating discrete energy eigenstates E with different device geometric parameters. We set the device's linear frequency at f_m = 1.44 GHz and calculate the linear stiffness k₁ = m_eff(2πf_m)² and nonlinear stiffness k₃ = m_effα = 0.25wbE_Yπ⁴/L³. Note the set f_m can be achieved at different lengths L by adjusting strain in the device according to the strain-length trade-off in Fig. 2(b). The potential energy U(z) is then obtained via Eq. (5), and both the calculated U(z) and m_eff are incorporated into the Schrödinger equation

where Ψ(z) is the wave function. We then numerically solve Eq. (6) by using an open-source MATLAB package, Chebfun,⁵¹ to determine the discrete eigenstates E in graphene NEMS.

Figures 3(a)–3(c) show the calculated discrete frequency (energy) states in the graphene NEMS resonators with different lengths L = 10, 24.9, and 30 nm, and fixed width w = 10 nm and thickness b = 0.335 nm (single-layer). Since α is positive here and Duffing nonlinearity is stiffening (increasing the frequency), this ensures that such graphene NEMS Duffing resonators in the quantum regime will have lower N_th than the calculated values shown in Fig. 2(a). Given the device dimensions, the Duffing nonlinear term (m_effαz⁴/4) can be comparable to the harmonic energy term (k₁z²/2), which modifies the shape of the potential energy [blue curves in Figs. 3(a)–3(c)]. For the device with L = 10 nm, its ground, first, and second states are at f₀ ≈1.40 GHz, f₁ ≈4.84 GHz, and f₂ ≈9.28 GHz, which is a substantial increase compared to the harmonic model where the energy levels are at f₀ ≈0.72 GHz, f₁ ≈2.16 GHz, and f₂ ≈3.60 GHz, respectively, thus exhibiting strong anharmonicity of the graphene NEMS qubits. Importantly, this strong anharmonicity modifies the thermal occupation factor and ground-state occupation probability to N_th ≈7.16 × 10⁻⁸ and P₀ ≈0.999 999 93, putting the device in much deeper quantum regime, compared with a purely linear resonator. Moreover, the transition frequency from the ground state to the first excited level is Δf₁₀ ≈3.44 GHz, while the frequency required to excite a phonon from the first level to the second level is Δf₂₁ ≈4.44 GHz, leading to Δf₂₁–Δf₁₀ ≈1.00 GHz with (Δf₂₁–Δf₁₀)/Δf₁₀ ≈28.8% and suggesting a highly nonlinear qubit based on graphene 2D NEMS. For the device with L = 24.9 nm [Fig. 3(b)], we obtain Δf₁₀ ≈1.53 GHz and Δf₂₁ ≈1.60 GHz, leading to Δf₂₁–Δf₁₀ ≈77 MHz and (Δf₂₁–Δf₁₀)/Δf₁₀ ≈5.0%, which is the relative anharmonicity level often seen in transmon qubits, and yielding N_th ≈6.61 × 10⁻⁴ and P₀ ≈0.9993. For the even longer device with L = 30 nm [Fig. 3(c)], Δf₁₀ ≈1.48 GHz, and Δf₂₁ ≈1.52 GHz, it still offers anharmonicity of Δf₂₁–Δf₁₀ ≈34 MHz and (Δf₂₁–Δf₁₀)/Δf₁₀ ≈2.3%, with N_th ≈8.40 × 10⁻⁴ and P₀ ≈0.9992. Figure 3(d) depicts predicted Δf₁₀, Δf₂₁, and Δf₂₁–Δf₁₀ with varying L. Assuming anharmonicity of (Δf₂₁–Δf₁₀)/Δf₁₀ >5%, which is comparable to typical fractional frequency differences in superconducting qubits,⁵² is required to make usable high-performance qubits, we find that the device length should be L <24.9 nm [the pink region in Fig. 3(d)]. In the region of L >24.9 nm, as L increases and goes to L >40 nm, the Duffing nonlinearity becomes less effective, showing Δf₁₀ ≈ Δf₂₁ ≈1.46 GHz, which approaches the uniform frequency gap between the levels of the quantum harmonic resonator, f_m = 1.44 GHz.

To further understand anharmonicity in the graphene NEMS resonators, we calculate Δf₂₁–Δf₁₀ with varying L and w to provide guidelines and scaling laws for designing graphene NEMS qubit candidates. Figure 4 displays Δf₂₁–Δf₁₀ as a function L and w with three different f_m values. We find that the device can attain the same amount of anharmonicity by trading L vs w. Aside from that, Fig. 4 reveals that for higher f_m, device dimensions should be reduced to introduce strong enough anharmonicity required by the NEMS qubit. We plot contour lines for both absolute anharmonicity Δf₂₁–Δf₁₀ = 200 MHz and relative anharmonicity (Δf₂₁–Δf₁₀)/Δf₁₀ = 5%, as shown in Figs. 4(a)–4(c). The device dimensions should be on the left side of the contour lines to enable adequate anharmonicity, according to the absolute or relative measures listed above.

We envision that quantum state manipulation in the graphene NEMS can be implemented by coupling the graphene NEMS qubits to optical,^15,26,27 microwave,^35–37 and acoustic cavities. For example, the qubit state could be initialized to the ground state or excited state by performing an anti-Stokes sideband process^24,25 (i.e., transferring the thermal phonon population from the graphene NEMS to a cavity for preparing the ground state or transferring an excitation from the cavity into the NEMS to prepare the excited state). Next, the qubit state could be manipulated and controlled capacitively by applying RF microwave signals to a nearby electrode [Fig. 1(a)] or via mechanical vibrations by utilizing a piezoelectric film. For example, single qubit Z gates on the Bloch sphere could be performed via the capacitive frequency pulling of the NEMS by applying a DC voltage offset to the gate electrode. Similarly, X and Y rotations could also be performed capacitively by applying sinusoidal signals at the qubit frequency on top of a DC offset—in both cases, the gate speed could be tuned in situ by adjusting the magnitude of the DC voltage. Subsequently, the resulting quantum states could be measured by utilizing another set of sideband anti-Stokes processes, thus transferring the phonon from the NEMS mode to the cavity. Finally, one could implement a variety of two-qubit operations between graphene NEMS qubits through different techniques, such as phonon-mediated swap gates via an intervening acoustic cavity^39,53,54 or remote entanglement utilizing Stokes and anti-Stokes operations via optomechanical interactions.⁵⁵ 

A decoherence source in the proposed graphene NEMS qubit could derive from phonon decay channels, with T₁ of the graphene NEMS qubit estimated by the Q factor of the mode.⁵³ At cryogenic temperature (∼15 mK), a high Q factor of Q ≈1.05 × 10⁶ (Ref. 36) has been demonstrated in graphene NEMS. Assuming similar Qs in the proposed graphene NEMS qubits at ∼10 mK, the estimated T₁ coherence time for the device with Δf₁₀ = 1.70 GHz [L = 19.2 nm and w = 10 nm in Fig. 4(b)] is T₁ ≈ Q/(2πΔf₁₀) ≈0.1 ms, which is comparable to T₁ values in state-of-the-art superconducting qubit technology.² Moreover, bilayer graphene Moiré superlattices with carefully engineered twist (the magic angle at ∼1.1°) offer gate tunable superconductivity below T_C ∼0.5 to 1.7 K,⁵⁶ thus promising superconducting graphene NEMS with quantum mechanical anharmonicity (Fig. 3), when the Moiré structures are integrated into the device depicted in Fig. 1(a).

In conclusion, we have designed and analytically evaluated graphene NEMS toward qubits, which are expected to offer anharmonicity in the quantum regime with ultrasmall device footprints. Computational results show that, thanks to the atomically thin structure, Duffing nonlinearity significantly alters the potential energy of graphene 2D NEMS and provides strong anharmonicity in single-layer graphene membranes with length ∼30 nm or below. It thus appears achievable to engineer a new type of highly miniaturized qubits that could be densely integrated in scalable platforms, enabling functions for quantum memory⁵⁷ and quantum electromechanical simulations of complex, many-body systems.⁵⁸ 

J.L. and P.X.-L.F. are thankful for the support from the National Science Foundation (NSF) via the EFRI ACQUIRE program (Grant No. EFMA-1641099) and the NSF CAREER Award (Grant No. ECCS-2015708). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of AFRL. Approved for Public Release; Distribution Unlimited. PA #: AFRL-2021-4212.