Simultaneous cooling and entanglement of mechanical modes of a micromirror in an optical cavity

Laser cooling of a mechanical mode of a resonator by the radiation pressure of a detuned optical cavity mode has been recently demonstrated by various groups in different experimental configurations. Here, we consider the effect of a second mechanical mode with a close but different resonance frequency. We show that the nearby mechanical resonance is simultaneously cooled by the cavity field, provided that the difference between the two mechanical frequencies is not too small. When this frequency difference becomes smaller than the effective mechanical damping of the secondary mode, the two cooling processes interfere destructively similarly to what happens in electromagnetically induced transparency, and cavity cooling is suppressed in the limit of identical mechanical frequencies. We show that also the entanglement properties of the steady state of the tripartite system crucially depend upon the difference between the two mechanical frequencies. If the latter is larger than the effective damping of the second mechanical mode, the state shows fully tripartite entanglement and each mechanical mode is entangled with the cavity mode. If instead, the frequency difference is smaller, the steady state is a two-mode biseparable state, inseparable only when one splits the cavity mode from the two mechanical modes. In this latter case, the entanglement of each mechanical mode with the cavity mode is extremely fragile with respect to temperature.

[1]  M. Pinard,et al.  Cooling of a Mirror by Radiation Pressure , 1999 .

[2]  B. Chui,et al.  Single spin detection by magnetic resonance force microscopy , 2004, Nature.

[3]  O. Arcizet,et al.  Resolved Sideband Cooling of a Micromechanical Oscillator , 2007, 0709.4036.

[4]  P. Meystre,et al.  Optical bistability and mirror confinement induced by radiation pressure , 1983 .

[5]  H. Gaub,et al.  Dynamic single-molecule force spectroscopy: bond rupture analysis with variable spacer length , 2003 .

[6]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[7]  Stefano Mancini,et al.  Entangling macroscopic oscillators exploiting radiation pressure. , 2002, Physical review letters.

[8]  Stefano Mancini,et al.  Scheme for teleportation of quantum states onto a mechanical resonator. , 2003, Physical review letters.

[9]  Simón.  Peres-horodecki separability criterion for continuous variable systems , 1999, Physical review letters.

[10]  S. Gigan,et al.  Self-cooling of a micromirror by radiation pressure , 2006, Nature.

[11]  V. Giovannetti,et al.  Phase-noise measurement in a cavity with a movable mirror undergoing quantum Brownian motion , 2000, quant-ph/0006084.

[12]  P. Nussenzveig,et al.  Classical analog of electromagnetically induced transparency , 2001, quant-ph/0107061.

[13]  M. Roukes,et al.  Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems , 2003, physics/0309075.

[14]  J. Cirac,et al.  Separability and Distillability of Multiparticle Quantum Systems , 1999, quant-ph/9903018.

[15]  G. Vidal,et al.  Computable measure of entanglement , 2001, quant-ph/0102117.

[16]  Sylvain Gigan,et al.  Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes , 2007, 0705.1728.

[17]  Michael L. Roukes,et al.  Putting mechanics into quantum mechanics , 2005 .

[18]  Dirk Bouwmeester,et al.  Sub-kelvin optical cooling of a micromechanical resonator , 2006, Nature.

[19]  M. Pinard,et al.  Self-cooling of a movable mirror to the ground state using radiation pressure , 2007, 0707.2038.

[20]  Law.  Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[21]  Samuel L. Braunstein,et al.  Quantum-state transfer from light to macroscopic oscillators , 2003 .

[22]  J. Ignacio Cirac,et al.  Separability Properties of Three-mode Gaussian States , 2001 .

[23]  Stefano Mancini,et al.  Stationary entanglement between two movable mirrors in a classically driven Fabry–Perot cavity , 2006, quant-ph/0611038.

[24]  Edith Innerhofer,et al.  An all-optical trap for a gram-scale mirror. , 2006, Physical review letters.

[25]  Christoph Simon,et al.  Towards quantum superpositions of a mirror , 2004 .

[26]  P. Meystre,et al.  Entangling the ro-vibrational modes of a macroscopic mirror using radiation pressure , 2007, 0712.1657.

[27]  Vladimir B. Braginsky,et al.  Quantum Measurement , 1992 .

[28]  A. Sopczak.  Neutral Higgs boson mass constraints in the minimal supersymmetric standard model from searches in $\rm e^+e^-$ collisions , 1999 .

[29]  T. Briant,et al.  Radiation-pressure cooling and optomechanical instability of a micromirror , 2006, Nature.

[30]  F. Illuminati,et al.  Extremal entanglement and mixedness in continuous variable systems , 2004, quant-ph/0402124.

[31]  Daniel Sigg,et al.  Optical dilution and feedback cooling of a gram-scale oscillator to 6.9 mK. , 2007, Physical review letters.

[32]  Quantum Langevin Equation , 1981 .

[33]  Seth Lloyd,et al.  Macroscopic entanglement by entanglement swapping. , 2006, Physical review letters.

[34]  M Pinard,et al.  High-sensitivity optical monitoring of a micromechanical resonator with a quantum-limited optomechanical sensor. , 2006, Physical review letters.

[35]  B. Muzykantskii,et al.  ON QUANTUM NOISE , 1995 .

[36]  S. Gigan,et al.  Erratum: Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes [Phys. Rev. A 77, 033804 (2008)] , 2009 .

[37]  M. Wolf,et al.  Bound entangled Gaussian states. , 2000, Physical review letters.

[38]  Florian Marquardt,et al.  Quantum theory of cavity-assisted sideband cooling of mechanical motion. , 2007, Physical review letters.

[39]  A. Heidmann,et al.  Quantum limits of cold damping with optomechanical coupling , 2001, quant-ph/0107138.

[40]  J Eisert,et al.  Creating and probing multipartite macroscopic entanglement with light. , 2007, Physical review letters.

[41]  Stefano Mancini,et al.  Macroscopic mechanical oscillators at the quantum limit through optomechanical cooling , 2003 .

[42]  T J Kippenberg,et al.  Theory of ground state cooling of a mechanical oscillator using dynamical backaction. , 2007, Physical review letters.

[43]  H J Mamin,et al.  Feedback cooling of a cantilever's fundamental mode below 5 mK. , 2007, Physical review letters.

[44]  A. Gozzini,et al.  Light-pressure bistability at microwave frequencies , 1985 .

[45]  Stefano Mancini,et al.  Optomechanical Cooling of a Macroscopic Oscillator by Homodyne Feedback , 1998 .

[46]  Aires Ferreira,et al.  Optomechanical entanglement between a movable mirror and a cavity field , 2007, 2007 European Conference on Lasers and Electro-Optics and the International Quantum Electronics Conference.

[47]  Cirac,et al.  Inseparability criterion for continuous variable systems , 1999, Physical review letters.

[48]  Antoine Heidmann,et al.  Entangling movable mirrors in a double-cavity system , 2005 .

[49]  V. Hahn,et al.  Stability theory , 1993 .

[50]  K. Vahala,et al.  Radiation Pressure Cooling of a Micromechanical Oscillator Using Dynamical Backaction , 2006 .

[51]  B. Camarota,et al.  Approaching the Quantum Limit of a Nanomechanical Resonator , 2004, Science.

[52]  Stefano Mancini,et al.  Mirror quiescence and high-sensitivity position measurements with feedback , 2001, quant-ph/0111067.

[53]  J. Cirac,et al.  Classification of multiqubit mixed states: Separability and distillability properties , 1999, quant-ph/9911044.

[54]  S. Girvin,et al.  Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane , 2007, Nature.