Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity

Nature Communications, Jan 2018

Mechanical resonators are promising systems for storing and manipulating information. To transfer information between mechanical modes, either direct coupling or an interface between these modes is needed. In previous works, strong coupling between different modes in a single mechanical resonator and direct interaction between neighboring mechanical resonators have been demonstrated. However, coupling between distant mechanical resonators, which is a crucial request for long-distance classical and quantum information processing using mechanical devices, remains an experimental challenge. Here, we report the experimental observation of strong indirect coupling between separated mechanical resonators in a graphene-based electromechanical system. The coupling is mediated by a far-off-resonant phonon cavity through virtual excitations via a Raman-like process. By controlling the resonant frequency of the phonon cavity, the indirect coupling can be tuned in a wide range. Our results may lead to the development of gate-controlled all-mechanical devices and open up the possibility of long-distance quantum mechanical experiments.

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Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity

NATURE COMMUNICATIONS | Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity Gang Luo 0 1 Zhuo-Zhi Zhang Guang-Can Guo 0 1 Lin Tian Guo-Ping Guo 2 Hai-Ou Li 0 1 Gang Cao 0 1 Ming Xiao 0 1 0 CAS Key Laboratory of Quantum Information, University of Science and Technology of China , Hefei, 230026 Anhui , China 1 Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China , Hefei, 230026 Anhui , China 2 School of Nature Sciences Mechanical resonators are promising systems for storing and manipulating information. To transfer information between mechanical modes, either direct coupling or an interface between these modes is needed. In previous works, strong coupling between different modes in a single mechanical resonator and direct interaction between neighboring mechanical resonators have been demonstrated. However, coupling between distant mechanical resonators, which is a crucial request for long-distance classical and quantum information processing using mechanical devices, remains an experimental challenge. Here, we report the experimental observation of strong indirect coupling between separated mechanical resonators in a graphene-based electromechanical system. The coupling is mediated by a far-offresonant phonon cavity through virtual excitations via a Raman-like process. By controlling the resonant frequency of the phonon cavity, the indirect coupling can be tuned in a wide range. Our results may lead to the development of gate-controlled all-mechanical devices and open up the possibility of long-distance quantum mechanical experiments. - ;,: () 0 9 8 7 6 5 4 3 2 1 T enables the storage and manipulation of phonon states in he rapid development of nanofabrication technology micro- and nano-mechanical resonators1?5. Mechanical resonators with quality factors6 exceeding 5 million and frequencies7,8 in the sub-gigahertz range have been reported. These advances have paved the route to controllable mechanical devices with ultralong memory time9. To transfer information between different mechanical modes, tunable interactions between these modes are required10. While different modes in a single mechanical resonator can be coupled by parametric pump3,4,11?16 and neighboring mechanical resonators can be coupled via phonon processes through the substrate2 or direct contact interaction17, it is challenging to directly couple distant mechanical resonators. Here, we observe strong effective coupling between mechanical resonators separated at a distance via a phonon cavity that is significantly detuned from these two resonator modes. The coupling is generated via a Raman-like process through virtual excitations in the phonon cavity and is tunable by varying the frequency of the phonon cavity. Typically, a Raman process can be realized in an atom with three energy levels in the ? form18,19. The two lower energy levels are each coupled to the third energy level via an optical field with detunings. When these two detunings are tuned to be equal to each other, an effective coupling is formed between the lower two levels. To our knowledge, tunable indirect coupling in electro-mechanical systems has not been demonstrated before. The physical mechanism of this coupling is analogous to the coupling between distant qubits in circuit quantum electrodynamics20,21, where the interaction between qubits is induced by virtual photon exchange via a superconducting microwave resonator. Results Sample characterization. The sample structure is shown in Fig. 1a, where a graphene ribbon22,23 with a width of ~1 ?m and ~5 layers is suspended over three trenches (2 ?m in width, 150 nm in depth) between four metal (Ti/Au) electrodes. This configuration defines three distinct electromechanical resonators: R1, R2 and R3. The metallic contacts S and D3 are each 2 ?m wide and D1 and D2 are each 1.5 ?m wide, which leads to a 7-?m separation between the centers of R1 and R3 (see Supplementary Methods and Supplementary Fig. 1). All measurements are performed in a dilution refrigerator at a base temperature of approximately 10 mK and at pressures below 10?7 torr. The suspended resonators are biased by a dc gate voltage (VgDi C for the ith resonator) and actuated by an ac voltage (VgAiC for the ith resonator with driving frequency fgi = ?d/2?) through electrodes (gi for the ith resonator) underneath the respective resonators. To characterize the spectroscopic properties of the resonators, a driving tone is applied to one or more of the bottom gates with frequency ?d, and another microwave tone with frequency ?d + ?? is applied to the contact S. A mixing current (Imix = Ix + jIy) can then be obtained at D3 (D1 and D2 are floated during all measurements) by detecting the ?? signal with a lock-in amplifier fixed at zero phase during all measurements (see Supplementary Methods and Supplementary Fig. 2). Figure 1b shows the measured mixing current as a function of the dc gate voltage and the ac driving frequency on R3, where the oblique lines represent the resonant frequencies of the resonator modes. We denote the resonant frequency of the ith resonator as fmi = ?mi/2?. This plot shows that dfm3=dVgD3C 7:7 MHz=V when jVgD3Cj>5 V. The frequencies of the resonators can hence be tuned in a wide range (see Supplementary Note 1 and Supplementary Fig. 3 for results of R1 and R2), which allows us to adjust the mechanical modes to be on or off resonance with each other. The quality factors (Q) of the resonant modes are determined by fitting the measured spectral widths (see Supplementary Fig. 8) at low driving powers (typically ?50 dBm). Figure 1c shows the spectral dependence of R3, which gives a linewidth of ?3/2? ~ 28 kHz at a resonant frequency of fm3 ~ 98.05 MHz. The resulting quality factor is Q ~ 3500. The quality factors of the other two resonators are similar, at ~3000. Strong coupling between neighboring resonators. Neighboring resonators in this system couple strongly with each other, similar to previous studies on gallium arsenide2 and carbon nanotube17. Figure 1d, e shows the spectra of the coupled modes (R1, R2) and (R2, R3), respectively, by plotting the mixed current Ix as a function of gate voltages and driving frequencies. In Fig. 1e, the voltage VgD3C is fixed at 10.5 V, with a corresponding resonant frequency fm3 = 101.15 MHz, and VgD2C is scanned over a range with fm2 being near-resonant to fm3. A distinct avoided level crossing appears when fm2 approaches fm3, which is a central feature of two resonators with direct coupling. From the measured data, we extract the coupling rate between these two modes as ?23/2? ~ 200 kHz, which is the energy splitting when fm2 = fm3. In Supplementary Note 2 and Supplementary Fig. 4, we fit the measured spectrum with a single two-mode model using this coupling rate. Similarly in Fig. 1d, by fixing VgD1C at 10.5 V and scanning the voltage VgD2C, we obtain the coupling rate between R1 and R2 as ?12/2? ~ 240 kHz. There are several possible origins for the coupling between two adjacent resonators in this system. One coupling medium is the substrate and the other medium is the graphene ribbon itself. Mechanical energy can be transferred in a solid-state material by phonon propagation, as demonstrated in several experiments2,17,24. Second, because adjacent resonators share lattice bonds, the phonon energy can transfer in the graphene ribbon. The dependence of the coupling strength on the width of the drain contacts is still unknown (see Supplementary Fig. 5 for another sample). The measured coupling strength satisfies the strong coupling condition with ?23 ?2; ?3. Defining the cooperativity for this phonon?phonon coupling system as C ? ?223=?2?3, we find that C = 44. A similar strong coupling condition can be found between modes R1 and R2. By adjusting the gate voltages of these three resonators, R2 can be successively coupled to both R1 and R3 (see Fig. 1f). For comparison, we study the coupling strength between modes R1 and R3. The frequency fm2 of resonator R2 is set to be detuned from fm1 and fm3 by 700 kHz in Fig. 2a. In the dashed circle, we observe a near-perfect level crossing when fm1 approaches fm3, which indicates a negligible coupling between these two modes, with ?13 ?1; ?3 (also see Supplementary Fig. 6). Raman-like coupling between well-separated resonators. The three resonator modes in our system are in the classical regime. The Hamiltonian of these three classical resonators can be written as: 3 1 Hc ? X i 2 ?pi2 ? ?2mixi2? ? ?12x1x2 ? ?23x2x3; ?1? where ?ij ? ?ijpffi?ffiffiffimffiffiiffiffi?ffiffiffimffiffiffij is a coupling parameter between i- and jth resonators, ppi is the effective momentum and xi is the effective coordinate of the oscillation for the ith resonator, respectively. Let xi ? qffi2ffi?ffi1ffimffiffiffii??i ? ?i? and pi ? ipffi?ffi2ffimffiffii??i ?i?, with ?i and ?i being complex numbers. The Hamiltonian in Eq. g1 D1 g2 D2 g3 D3 R2 Vg2DC R3 Vg3DC Bias-T Bias-T Bias-T V Ag1C( d) V Ag2C( d) V Ag3C( d) dlx /d d (pA/Hz) Resonator 3 (R3) Imix (? ) d Coupling between R1 and R2 12 23 12 23 Here, we have applied the rotating-wave approximation and neglected the ?i?j and ?i ?j terms. This approximation is valid when ?mi ?12; ?23. This Hamiltonian describes the direct couplings between neighboring resonators (R1, R2) and (R2, R3). Through these couplings, the mechanical modes hybridize into three normal modes, and an effective coupling between modes R1 and R3 can be achieved. If the resonators work in the quantum regime, ?i and ?i can be quantized into the annihilation and creation operators of a quantum harmonic oscillator, respectively. We study the hybridization of this three-mode system by fixing the gate voltages (mode frequencies) of modes R1 and R2, and sweeping the gate voltage of R3 over a wide range. The spectrum of this system depends strongly on the detuning between modes R1 and R2, which is defined as ?12 = 2?(fm2?fm1). In Fig. 2a, ?12/ 2? ~ 70 kHz. Similar to Fig. 1f, modes R3 and R1 show a level crossing. Moreover, we observe a large avoided level crossing between modes R2 and R3 when the frequency fm3 approaches fm2, indicating strong coupling between these two modes. Hence, even with strong couplings between all neighboring resonators, the effective coupling between the distant modes R1 and R3 is still negligible when the frequency of mode R2 is significantly far off 87.5 ) z H M ( 87.1 2 /d 86.7 86.3 Experiment R2 Vg2DC = 7.64 V 12 86.3 86.7 87.1 87.5 87.9 d /2 (MHz) 86.65 86.95 d /2 (MHz) resonance from the other two modes. On the contrary, when the detuning ?12 is lowered to ~180 kHz, a distinct avoided level crossing between modes R1 and R3 is observed, as shown inside the dashed circle in Fig. 2b. With coupling strengths ?12/2? = 240 kHz and ?23/2? = 170 kHz extracted from the measured data, we plot the theoretical spectra of the normal modes in this three-mode system given by Eq. (2), for ?12/2? = 700 and 180 kHz in Fig. 2c, d, respectively. Our result shows good agreement between theoretical and experimental results. With direct couplings between neighboring resonators, an effective coupling between the two distant resonators R1 and R3 can be obtained via their couplings to mode R2. The effective coupling can be viewed as a Raman process, as illustrated in Fig. 3a. Here mode R2 functions as a phonon cavity that connects the mechanical resonators R1 and R3 via virtual phonon excitations. The physical mechanism of this effective coupling is similar to that of the coupling between distant superconducting qubits via a superconducting microwave cavity20. The detuning between the phonon cavity and the other two modes ?12 can be used as a control parameter to adjust this effective coupling. To derive the effective coupling, we consider the case of ?12 = ?32 = ?, where ?32/2? = fm2?fm3 and j?j ?12; ?23. The avoided level crossing between modes R1 and R3 can be extracted at this point. Using a perturbation theory approach, we obtain the effective Hamiltonian between modes R1 and R3 as (see Methods for details) 2 2 Heff ? ?? ? ?4 ?12??1?1 ? ?? ? ?4 ?23??3?3 ? ?13 ? ? 2 ? 1 3 ? ?3?1?: Here, an effective coupling is generated between R1 and R3 with magnitude ?13 = ?12?23/2?, and the resonant frequencies of each mode are shifted by a small term. The effective coupling ?13 in the Hamiltonian depends strongly on the detuning ?. Thus, the effective coupling between R1 and R3 can be controlled over a wide range by varying the frequency (gate voltage) of resonator R2. The effective coupling strength ?13 between R1 and R3 as a function of ?12 is shown in Fig. 3b. Each data point is obtained by changing the gate voltage of R2 and repeating the measurements in Fig. 2a, b (see Supplementary Fig. 7). Over a large range of detuning, the effective coupling is larger than the linewidths of the resonators ?1,2,3/2?, with ?13 > 30 kHz. The red line shows the results using perturbation theory. The experimental data indicate good agreement with the theoretical results. Discussion In summary, we have demonstrated indirect coupling between separated mechanical resonators in a three-mode electromechanical ?3? a 13 12 12 13 23 R1 R2 R3 Virtual phonon bath b 100 ) z HM 80 ( system constructed from a graphene ribbon. Our study suggests that coupling between well-separated mechanical modes can be created and manipulated via a phonon cavity. These observations hold promise for a wide range of applications in phonon state storage, transmission, and transformation. In the current experiment, the sample works in an environment subjected to noise and microwave heating with typical temperatures as high as 100 mK and phonon numbers reaching ~24. By cooling the mechanical resonators to lower temperatures25?28, quantum states could be manipulated via this indirect coupling29,30. Furthermore, in the quantum limit, by coupling the mechanical modes to solid-state qubits, such as quantum-dots and superconducting qubits17,31,32, this system can be utilized as a quantum data bus to transfer information between qubits33,34. Future work may lead to the development of graphene-based mechanical resonator arrays as phononic waveguides24 and quantum memories35 with high tunabilities. Methods Theory of three-mode coupling. We describe this three-mode system with the Hamiltonian (? = 4) Ht ? X?mi?i ?i ? ?212 ??1?2 ? ?1?2? ? ?223 ??2?3 ? ?2?3?; ?4? where ?ij is the coupling between mechanical resonators i and j. The couplings between the resonators induce hybridization of the three modes. The hybridized normal modes under this Hamiltonian can be obtained by solving the eigenvalues of the matrix 0 ?12 2 0 ?23 2 where ?ij/2? = fmi?fmj is the frequency difference between Ri and Rj. The eigenvalues of this matrix correspond to the frequencies of the normal modes, i.e., the peaks in the spectroscopic measurement. We consider the special case of ?12 = ?23 = ?, with j?j ?12; ?23, in the three-mode system. Here, the eigenvalues of the normal modes can be derived analytically. One eigenvalue is ?? = ?, which corresponds to the eigenmode with ??0 ? qffi?ffiffiffi2ffiffiffi?ffiffiffiffiffi?ffiffiffi21ffiffi2ffiffiffi?ffiffiffiffiffi?ffiffiffi22ffiffi3ffi. The corresponding normal modes are With j?j ?12; ?23, for ? > 0, ??? is nearly degenerate with ??, and a? ? ? ??12ap1?ffi2ffiffi??ffi?ffiffiffiffi?ffiffi0ffiffi?ffiffiffiffiffiffi?ffiffiffi?ffiffiffi?ffiffi?ffiffiffi??ffi23a3? : ?8? ?0? ?0 ? ? ??12 ? ?223?=4?. The mode ??+ 2 ??? ?12?1 ? ?23?3 : qffi?ffiffiffi21ffiffi2ffiffiffi?ffiffiffiffiffiffiffiffi2ffiffiffi ?23 The mode ??? has frequency ?? ??122 ? ?223?=4?, with ??? ? ?2. The normal modes now become separated into two nearly degenerate modes {??,??+}, which are superpositions of modes ?1 and ?3, and a third mode ??? that is significantly off resonance from the other two modes. The nearly degenerate modes can be viewed as a hybridization of ?1 and ?3 by an effective coupling ?13??y1?3 ? ?y ?1?=2. The magnitude of this effective coupling is equal to the 3 frequency splitting between the nearly degenerate modes, with 2 2 ?13 ? ??12 ? ?23?=2?. A similar result can be derived for ? < 0, where ?? ? ? ??122 ? ?223?=4?, with ??? given by the expression in Eq. (8), and ??? ??122 ? ?223?=4? with ??+ ? ?2. The effective coupling rate can be derived with a perturbative approach on the matrix M. When |?| ?12, ?23, the dynamics of ?1 and ?3 is governed by matrix M ? 0@ ???12??243?122 ?142???23223 1A: ?10? 4? ? ? 4? This matrix tells us that because of their interaction with the middle mode ?2, 2 2 the frequency of mode ?1 (?3) is shifted by ?4?12 (?4?23), which is much smaller than |?|. Meanwhile, an effective coupling is generated between these two modes with magnitude ?13 ? ?122??23. The effective Hamiltonian for ?1 and ?3 can be written as 2 2 Heff ? ?? ? 4? ?23??3?3 ? ?213 ??1?3 ? ?3?1?: ?12??1?1 ? ?? ? 4? ?11? The effective coupling can be controlled over a wide range by varying the frequency of the second mode ?2. ?9? ?5? Data availability. The remaining data contained within the paper and Supplementary files are available from the author upon request. ?6? ?7? Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article?s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article?s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/. This work was supported by the National Key R&D Program of China (Grant No. 2016YFA0301700), the NSFC (Grants Nos . 11625419 , 61704164 , 61674132 , 11674300 , 11575172, and 91421303), the SPRP of CAS (Grant No. XDB01030000), and the Fundamental Research Fund for the Central Universities . L.T. is supported by the National Science Foundation under Award No. DMR-0956064 and PHY-1720501 and the UC Multicampus-National Lab Collaborative Research and Training under Award No . LFR17 - 477237 . This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication. Additional information Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- 018-02854-4.


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Gang Luo, Zhuo-Zhi Zhang, Guang-Wei Deng, Hai-Ou Li, Gang Cao, Ming Xiao, Guang-Can Guo, Lin Tian, Guo-Ping Guo. Strong indirect coupling between graphene-based mechanical resonators via a phonon cavity, Nature Communications, 2018, DOI: 10.1038/s41467-018-02854-4