Figure 1. (left) Colorized SEM image of a qubit-coupled NEMS, which was fabricated in collaboration with Pierre Echternach at JPL. (blue=SiN, grey=Al). The nanoresonator is the fundamental in-plane flexural mode of the doubly-clamped SiN nanostructure, with resonant frequency /2 ~ 60MHz. We use a Cooper-pair box (CPB) as the qubit, with transition frequency /2 ~ 10 GHz. The NEMS and CPB are biased at different DC electric potentials. The flexural motion of the nanoresonator thus modulates the electrostatic energy of the qubit. Actuation and read-out of the nanoresonator's motion is done capacitively through the NEMS gate by RF reflectometry. (right) As verified by preliminary experiments in the Roukes Group, the electrostatically-coupled NEMS-CPB system is formally equivalent to that of a harmonic oscillator linearly coupled to a two-level quantum system with interaction strength /2 ~ MHz.

Introduction

Researchers in a diverse range of fields are rapidly developing technologies that should soon enable quantum-limited control and measurement of mechanical objects ranging in size from nano- and micro-fabricated structures to massive optical mirrors. The motivations for these studies span from the development of force detectors for ultra-sensitive nuclear magnetic imaging and the observation of gravitational waves to quantum information processing to the study of foundational issues in quantum mechanics such as decoherence and the quantum-classical divide^{1, 2}.

Figure 2. Schematic of CPB-coupled NEMS.

For over a decade, it has been appreciated that the frontier of NEMS falls within the quantum regime^{3, 4}. Nanomechanical devices have been utilized for measurements of the quantum of thermal conductance⁵ and for displacement detection with sensitivity approaching limits set by the Heisenberg Uncertainty Principle^{6, 7}. NEMS resonators with frequencies in excess of 1 GHz have been achieved⁸, and 10-100's MHz are routine; this makes viable the use of standard cryogenic refrigeration to cool individual nanomechanical modes to low thermal occupation numbers, where the intrinsic fluctuations in the nanoresonator's energy become dominated by quantum noise, the so-called zero-point fluctuations (the world record to-date is held by Keith Schwab's group, who have measured a thermal occupation of n_th ~ 25 for a 21 MHz NEMS mode⁷). Additionally, at milli-Kelvin temperatures, the quality factors of NEMS modes have been demonstrated in excess of 10⁵,⁷ indicating weak coupling between the NEMS mode and its environment, and promising measurably-long lifetimes for highly non-classical states of motion, such as number states (a.k.a. Fock states) and entangled states, two hallmarks of quantum nature.

Figure 3. Expected NEMS resonance frequency shift vs. CPB gate charge. It shows that "dispersive" shift depends on the CPB gate charge and is maximized at the degeneracy point (gate charge = 1 electron). The shift changes polarity depending on the CPB state. This effect is used to probe the qubit dynamics in our experiment.

The manipulation and measurement of quantum NEMS has received extensive theoretical treatment in the literature in the last decade. One of the most promising approaches (and certainly the most versatile) that has been put forth is the integration of the superconducting qubits in NEMS.⁹ First proposed by Keith Schwab, Miles Blencowe and Andrew Armour in 2002, qubit-coupled NEMS (Figure 1) are formally analogous to cavity quantum electrodynamics (CQED) systems¹⁰. In fact, in certain limits, the Hamiltonian for the system is similar to the Jaynes-Cummings (JC) Hamiltonian⁹: The qubit plays the role of a two-level atom; the nanomechanical resonator serves as the harmonic oscillator; and the two systems couple linearly through the nanoresonator's displacement (Figure 1b). In CQED, the physics captured by the JC model has enabled researchers to utilize single atoms to control and measure quantum states of macroscopic electromagnetic resonators. Crucially, with the spectacular development of superconducting qubit technology and the field of circuit QED¹¹, techniques for the measurement and control of superconducting qubits are now readily available to be applied for the exploration of quantum NEMS.

Figure 4. (a) Nanomechanical frequency shift versus voltage V_cpb and flux Φ applied to the CPB qubit. We see a clear periodic modulation of the frequency shift due to the dispersive coupling to the CPB. (b-d) The data agrees well with our theoretical model.

Research

Current efforts in the Roukes milli-Kelvin lab are focused on performing the first-ever measurements of the coupling between a NEMS resonator and a superconducting qubit, the Cooper-pair box¹². In this work, we have demonstrated that the qubit-coupled NEMS system is indeed well described by the simple model of a two-level quantum system coupled to a harmonic oscillator, and have laid the groundwork for implementing many of the advanced proposals to realize quantum states of NEMS.

The interaction between the NEMS and CPB is achieved via electrostatic coupling (Figure 1a and Figure 2). The capacitance between the CPB island and a nearby, metallized nanoresonator is modulated through the displacement of the resonator about its equilibrium position. In the presence of a DC voltage applied between the nanoresonator and CPB island, the nanoresonator's displacement thus modulates the electrostatic energy of the CPB. For typical parameters, the two systems and their interaction are well-described by a dispersive JC-type model⁹, resulting in a CPB-state-dependent shift in the frequency of the NEMS mode (Figure 3).

Figure 5. (a-d) Shift in nanomechanical frequency as a function of CPB voltage V_cpb and flux Φ with microwaves applied to the CPB. The measured hyperbolic pattern in the frequency shift reflects the increased probability of the CPB occupying the excited state when the microwaves are resonant with the CPB transition. (e) The CPB transition frequency and expected resonance contours for the applied microwave frequencies.

For the sake of low-power dissipation (the measurements are done at milli-Kelvin temperatures) and ease of implementation in this first round of experiments, we have used a combination of capacitive detection and RF reflectometry¹³ for measuring the CPB-state-dependent shift in the NEMS' resonant frequency. Figure 4a demonstrates the frequency shift of the NEMS mode measured while tuning the CPB in parameter space along its ground state energy band. The periodic modulation of the NEMS frequency as a function of voltage, Vcpb, and flux, Φ, applied to the CPB is well-described by the simple model of dispersive coupling between a two-level system and harmonic oscillator (Figure 4 b-d).

Figure 6. At a fixed flux bias point, the qubit is driven with microwaves that are much lower than the CPB transition frequency (6.5GHz in this plot). An interference pattern is observed to develop as a function of microwave amplitude Vµ and CPB gate voltageV_cpb. This is the outcome of the interference between successive coherent Landau-Zener transitions in the CPB.

Applying microwaves to the CPB qubit, we have also been able to perform spectroscopy of the CPB (Figure 5). When the qubit is driven with microwaves we observe that the NEMS' frequency shift decreases at points in CPB parameter space where the CPB transition frequency is near resonance with the microwaves. At these points, the qubit undergoes Rabi oscillations between the ground and excited states, thus resulting, on average, in a much smaller shift in the NEMS frequency.

When the microwave frequency is relatively slow with respect to the CPB transition frequency, we observe an interference pattern that is the result of successive LZ tunneling events between the CPB ground and excited states. (Figure 6) The observation of Landau-Zener interference is indicative of coherent dynamics occurring in the qubit and is the first example of nanomechanical read-out of quantum interference.

Personnel
Michael Roukes, PI
Matt LaHaye, Ph.D., Postdoc
Junho Suh, Graduate Student
Collaborators: (JPL) and Keith Schwab (Caltech)

References

1. K.C. Schwab & M.L. Roukes, Physics Today, 58, 36 (2005).
2. 2008 Gordon Conference: Mechanical Systems in the Quantum Regime
3. A.N. Cleland & M.L. Roukes, Proceedings of the 24th Int'l Conference on the Physics of Semiconductors (World Scientific, 1999).
4. M.L. Roukes, Physics World 14, 25 (2001).
5. K. Schwab, E.A. Henriksen, J.M. Worlock & M.L. Roukes, Nature 404, 974 (2000).
6. M.D. LaHaye, O. Buu, B. Camarota, K.C. Schwab, Science 304, 74 (2004).
7. A. Naik et al., Nature 443, 193 (2006).
8. X.M.H. Huang, C.A. Zorman, M. Mehregany, M.L. Roukes, Nature 421, 496 (2003).
9. For example see:
   • A.D. Armour, M.P. Blencowe & K.C. Schwab, K.C, Phys. Rev. Lett. 88, 148301 (2002)
   • E.K. Irish & K.C. Schwab, Phys. Rev. B 68, 155311 (2003)
   • I. Martin, A. Shnirman, L. Tian & P. Zoller, Phys. Rev. B 69, 125339 (2004)
   • A.N. Cleland & M.R. Geller, Phys. Rev. Lett. 93, 070501 (2004)
   • L.F. Wei, Yu-xi Liu, C.P. Sun & Franco Nori, Phys. Rev. Lett. 97 237201 (2006)
   • A.A. Clerk & D. Wahyu Utami, Phys. Rev. A 75 042302 (2007)
   • K. Jacobs, A.N. Jordan & E.K. Irish, Euro. Phys. Lett. 82, 18003 (2008)
   • A.D. Armour & M.P. Blencowe, New J. Phys. 10, 095004 (2008).
10. S. Haroche & J.M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford Univ. Press, 2006).
11. R.J. Schoelkopf & S.M Girvin, Nature 451, 664 (2008).
12. M.D. LaHaye, J. Suh, P.M. Echternach, K. Schwab & M.L. Roukes, In press at Nature (2009).
13. P. A. Truitt, J. B. Hertzberg, C. C. Huang, K. L. Ekinci, K. C. Schwab, Nano Letters 7, 120 (2007).