Quantum control with Lyapunov function and bang–bang solution in the optomechanics system

Abstract

We propose a quantum control scheme with the help of Lyapunov control function in the optomechanics system. The principle of the idea is to design suitable control fields to steer the Lyapunov control function to zero as t → ∞ while the quantum system is driven to the target state. Such an evolution makes no limit on the initial state and one needs not manipulate the laser pulses during the evolution. To prove the effectiveness of the scheme, we show two useful applications in the optomechanics system: one is the cooling of nanomechanical resonator and the other is the quantum fluctuation transfer between membranes. Numerical simulation demonstrates that the perfect and fast cooling of nanomechanical resonator and quantum fluctuation transfer between membranes can be rapidly achieved. Besides, some optimizations are made on the traditional Lyapunov control waveform and the optimized bang–bang control fields makes Lyapunov function V decrease faster. The optimized quantum control scheme can achieve the same goal with greater efficiency. Hence, we hope that this work may open a new avenue of the experimental realization of cooling mechanical oscillator, quantum fluctuations transfer between membranes and other quantum optomechanics tasks and become an alternative candidate for quantum manipulation of macroscopic mechanical devices in the near future.

Graphical abstract

Keywords

bang–bang solution / quantum control / Lyapunov control / optomechanics system

References

[1] T. J. Kippenberg and K. J. Vahala , Cavity optomechanics: Back-action at the mesoscale, Science 321(5893), 1172(2008)

[2] M. Aspelmeyer , T. J. Kippenberg , and F. Marquardt , Cavity optomechanics, Rev. Mod. Phys. 86(4), 1391(2014)

[3] I. Wilson-Rae , N. Nooshi , W. Zwerger , and T. J. Kippenberg , Theory of ground state cooling of a mechanical oscillator using dynamical backaction, Phys. Rev. Lett. 99(9), 093901(2007)

[4] Y. C. Liu , Y. F. Xiao , X. S. Luan , and C. W. Wong , Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics, Phys. Rev. Lett. 110(15), 153606(2013)

[5] X. Y. Lü , Y. Wu , J. R. Johansson , H. Jing , J. Zhang , and F. Nori , Squeezed optomechanics with phase-matched amplification and dissipation, Phys. Rev. Lett. 114(9), 093602(2015)

[6] A. Szorkovszky , A. C. Doherty , G. I. Harris , and W. P. Bowen , Mechanical squeezing via parametric amplification and weak measurement, Phys. Rev. Lett. 107(21), 213603(2011)

[7] K. Jähne , C. Genes , K. Hammerer , M. Wallquist , E. S. Polzik , and P. Zoller , Cavity-assisted squeezing of a mechanical oscillator, Phys. Rev. A 79(6), 063819(2009)

[8] M. Asjad , G. S. Agarwal , M. S. Kim , P. Tombesi , G. D. Giuseppe , and D. Vitali , Robust stationary mechanical squeezing in a kicked quadratic optomechanical system, Phys. Rev. A 89(2), 023849(2014)

[9] X. L. Huang , T. Wang , and X. X. Yi , Effects of reservoir squeezing on quantum systems and work extraction, Phys. Rev. E 86(5), 051105(2012)

[10] L. Tian and H. L. Wang , Optical wavelength conversion of quantum states with optomechanics, Phys. Rev. A 82(5), 053806(2010)

[11] C. Genes , D. Vitali , P. Tombesi , S. Gigan , and M. Aspelmeyer , Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes, Phys. Rev. A 77(3), 033804(2008)

[12] A. Nunnenkamp , K. Børkje , J. G. E. Harris , and S. M. Girvin , Cooling and squeezing via quadratic optomechanical coupling, Phys. Rev. A 82(2), 021806(2010)

[13] S. Mancini , D. Vitali , and P. Tombesi , Optomechanical cooling of a macroscopic oscillator by homodyne feedback, Phys. Rev. Lett. 80(4), 688(1998)

[14] R. W. Peterson , T. P. Purdy , N. S. Kampel , R. W. Andrews , P. L. Yu , K. W. Lehnert , and C. A. Regal , Laser cooling of a micromechanical membrane to the quantum backaction limit, Phys. Rev. Lett. 116(6), 063601(2016)

[15] J. Y. Yang , D. Y. Wang , C. H. Bai , S. Y. Guan , X. Y. Gao , A. D. Zhu , and H. F. Wang , Ground-state cooling of mechanical oscillator via quadratic optomechanical coupling with two coupled optical cavities, Opt. Express 27(16), 22855(2019)

[16] F. Marquardt , J. P. Chen , A. A. Clerk , and S. M. Girvin , Quantum theory of cavity-assisted sideband cooling of mechanical motion, Phys. Rev. Lett. 99(9), 093902(2007)

[17] A. H. Safavi-Naeini and O. Painter , Proposal for an optomechanical traveling wave phonon–photon translator, New J. Phys. 13(1), 013017(2011)

[18] K. Stannigel , P. Komar , S. J. M. Habraken , S. D. Bennett , M. D. Lukin , P. Zoller , and P. Rabl , Optomechanical quantum information processing with photons and phonons, Phys. Rev. Lett. 109(1), 013603(2012)

[19] M. Tsang , Cavity quantum electro-optics (II): Inputoutput relations between traveling optical and microwave fields, Phys. Rev. A 84(4), 043845(2011)

[20] X. W. Xu , Y. J. Zhao , and Y. X. Liu , Entangled-state engineering of vibrational modes in a multimembrane optomechanical system, Phys. Rev. A 88(2), 022325(2013)

[21] X. W. Xu , Y. X. Liu , C. P. Sun , and Y. Li , Mechanical PT symmetry in coupled optomechanical systems, Phys. Rev. A 92(1), 013852(2015)

[22] Y. D. Wang and A. A. Clerk , Reservoir-engineered entanglement in optomechanical systems, Phys. Rev. Lett. 110(25), 253601(2013)

[23] V. Fiore , Y. Yang , M. C. Kuzyk , R. Barbour , L. Tian , and H. Wang , Storing optical information as a mechanical excitation in a silica optomechanical resonator, Phys. Rev. Lett. 107(13), 133601(2011)

[24] Y. D. Wang and A. A. Clerk , Using interference for high fidelity quantum state transfer in optomechanics, Phys. Rev. Lett. 108(15), 153603(2012)

[25] S. S. U and A. Narayanan , Mechanical switch for state transfer in dual-cavity optomechanical systems., Phys. Rev. A 88(3), 033802(2013)

[26] K. C. Schwab and M. L. Roukes , Putting mechanics into quantum mechanics, Phys. Today 58(7), 36(2005)

[27] T. Kippenberg and K. Vahala , Cavity opto-mechanics, Opt. Express 15(25), 17172(2007)

[28] V. Fiore , Y. Yang , M. C. Kuzyk , R. Barbour , L. Tian , and H. L. Wang , Storing optical information as a mechanical excitation in a silica optomechanical resonator, Phys. Rev. Lett. 107(13), 133601(2011)

[29] Y. D. Wang and A. A. Clerk , Using interference for high fidelity quantum state transfer in optomechanics, Phys. Rev. Lett. 108(15), 153603(2012)

[30] S. Barzanjeh , M. Abdi , G. J. Milburn , P. Tombesi , and D. Vitali , Reversible optical-to-microwave quantum interface, Phys. Rev. Lett. 109(13), 130503(2012)

[31] H. K. Li , X. X. Ren , Y. C. Liu , and Y. F. Xiao , Photonphoton interactions in a largely detuned optomechanical cavity, Phys. Rev. A 88(5), 053850(2013)

[32] M. D. La Haye , O. Buu , B. Camarota , and K. C. Schwab , Approaching the quantum limit of a nanomechanical resonator, Science 304(5667), 74(2004)

[33] J. J. Li and K. D. Zhu , All-optical mass sensing with coupled mechanical resonator systems, Phys. Rep. 525(3), 223(2013)

[34] B. Pepper , R. Ghobadi , E. Jeffrey , C. Simon , and D. Bouwmeester , Optomechanical superpositions via nested interferometry, Phys. Rev. Lett. 109(2), 023601(2012)

[35] P. Sekatski , M. Aspelmeyer , and N. Sangouard , Macroscopic optomechanics from displaced single-photon entanglement, Phys. Rev. Lett. 112(8), 080502(2014)

[36] A. Carlini , A. Hosoya , T. Koike , and Y. Okudaira , Timeoptimal quantum evolution, Phys. Rev. Lett. 96(6), 060503(2006)

[37] N. Khaneja , R. Brockett , and S. J. Glaser , Time optimal control in spin systems, Phys. Rev. A 63(3), 032308(2001)

[38] P. D’ Alessandro and E. De Santis , Controlled invariance and feedback laws, IE EE Trans. Automat. Contr. 46(7), 1141(2001)

[39] J. P. Palao and R. Kosloff , Quantum computing by an optimal control algorithm for unitary transformations, Phys. Rev. Lett. 89(18), 188301(2002)

[40] M. Mirrahimi , P. Rouchon , and G. Turinici , Lyapunov control of bilinear Schrödinger equations, Automatica 41, 1987(2005)

[41] X. T. Wang and S. G. Schirmer , Entanglement generation between distant atoms by Lyapunov control, Phys. Rev. A 80(4), 042305(2009)

[42] W. Wang , L. C. Wang , and X. X. Yi , Lyapunov control on quantum open systems in decoherence-free subspaces, Phys. Rev. A 82(3), 034308(2010)

[43] Y. H. Chen , W. Qin , and F. Nori , Fast and high-fidelity generation of steady-state entanglement using pulse modulation and parametric amplification, Phys. Rev. A 100(1), 012339(2019)

[44] C. H. Dong , V. Fiore , M. C. Kuzyk , and H. Wang , Optomechanical dark mode, Science 338(6114), 1609(2012)

[45] D. Garg , A. K. Chauhan , and A. Biswas , Adiabatic transfer of energy fluctuations between membranes inside an optical cavity, Phys. Rev. A 96(2), 023837(2017)

[46] Y. H. Chen , Z. C. Shi , J. Song , and Y. Xia , Invariant-based inverse engineering for fluctuation transfer between membranes in an optomechanical cavity system, Phys. Rev. A 97(2), 023841(2018)

[47] Y. H. Kang , Z. C. Shi , B. H. Huang , J. Song , and Y. Xia , Deterministic conversions between Greenberger–Horne–Zeilinger states and W states of spin qubits via Lietransform-based inverse Hamiltonian engineering, Phys. Rev. A 100(1), 012332(2019)

[48] Y. H. Kang and Y. Xia , Unconventional geometric phase gate of transmon qubits with inverse Hamiltonian engineering, IE EE J. Sel. Top. Quantum Electron. 26(3), 6700107(2020)

[49] Y. H. Kang , Z. C. Shi , B. H. Huang , J. Song , and Y. Xia , Flexible scheme for the implementation of nonadiabatic geometric quantum computation, Phys. Rev. A 101(3), 032322(2020)

[50] Y. H. Kang , Y. H. Chen , Z. C. Shi , B. H. Huang , J. Song , and Y. Xia , One-step implementation of N‐qubit nonadiabatic holonomic quantum gates with superconducting qubits via inverse hamiltonian engineering, Ann. Phys. 531(7), 1800427(2019)

[51] Y. Wang , C. S. Hu , Z. C. Shi , B. H. Huang , J. Song , and Y. Xia , Accelerated and noise‐resistant protocol of dissipation‐based Knill–Laflamme–Milburn state generation with Lyapunov control, Ann. Phys. 531(7), 1900006(2019)

[52] Y. H. Zhou , H. Z. Shen , and X. X. Yi , Unconventional photon blockade with second-order nonlinearity, Phys. Rev. A 92(2), 023838(2015)

[53] Z. C. Zhang , J. C. Pei , Y. P. Wang , and X. G. Wang , Measuring orbital angular momentum of vortex beams in optomechanics, Front. Phys. 16(3), 32503(2021)

[54] X. B. Yan , H. L. Lu , F. Gao , F. Gao , and L. Yang , Perfect optical nonreciprocity in a double-cavity optomechanical system, Front. Phys. 14(5), 52601(2019)

[55] M. M. Zhao , Z. Qian , B. P. Hou , Y. Liu , and Y. H. Zhao , Optomechanical properties of a degenerate nonperiodic cavity chain, Front. Phys. 14(2), 22601(2019)

[56] J. H. Liu , Y. B. Zhang , Y. F. Yu , and Z. M. Zhang , Photonphonon squeezing and entanglement in a cavity optomechanical system with a flying atom, Front. Phys. 14(1), 12601(2019)

[57] S. Liu , J. H. Shen , R. H. Zheng , Y. H. Kang , Z. C. Shi , J. Song , and Y. Xia , Optimized nonadiabatic holonomic quantum computation based on Förster resonance in Rydberg atoms, Front. Phys. 17(2), 21502(2022)

[58] Y. H. Kang , Z. C. Shi , J. Song , and Y. Xia , Heralded atomic nonadiabatic holonomic quantum computation with Rydberg blockade, Phys. Rev. A 102(2), 022617(2020)

[59] Y. H. Kang , Z. C. Shi , J. Song , and Y. Xia , Effective discrimination of chiral molecules in a cavity, Opt. Lett. 45(17), 4952(2020)

[60] D. D’ Alessandro , Introduction to Quantum Control and Dynamics, Taylor and Francis Group, Boca Raton, 2007

[61] Z. C. Shi , X. L. Zhao , and X. X. Yi , Robust state transfer with high fidelity in spin-1/2 chains by Lyapunov control, Phys. Rev. A 91(3), 032301(2015)

[62] X. X. Yi , X. L. Huang , C. F. Wu , and C. H. Oh , Driving quantum systems into decoherence-free subspaces by Lyapunov control, Phys. Rev. A 80(5), 052316(2009)

[63] S. C. Hou , M. A. Khan , X. X. Yi , D. Dong , and I. R. Petersen , Optimal Lyapunov-based quantum control for quantum systems, Phys. Rev. A 86(2), 022321(2012)

[64] J. T. Sheng , X. R. Wei , C. Yang , and H. B. Wu , Selforganized synchronization of phonon lasers, Phys. Rev. Lett. 124(5), 053604(2020)

[65] W. W. Zhou , S. G. Schirmer , M. Zhang , and H. Y. Dai , Bang–bang control design for quantum state transfer based on hyperspherical coordinates and optimal time–energy control, J. Phys. A Math. Theor. 44(10), 105303(2011)

[66] X. T. Wang , S. Vinjanampathy , F. W. Strauch , and K. Jacobs , Ultraefficient cooling of resonators: Beating sideband cooling with quantum control, Phys. Rev. Lett. 107(17), 177204(2011)

[67] L. Tian , Ground state cooling of a nanomechanical resonator via parametric linear coupling, Phys. Rev. B 79(19), 193407(2009)

[68] K. Nishio , K. Kashima , and J. Imura , Effects of time delay in feedback control of linear quantum systems, Phys. Rev. A 79(6), 062105(2009)

[69] X. X. Yi , S. L. Wu , C. Wu , X. L. Feng , and C. H. Oh , Time-delay effects and simplified control fields in quantum Lyapunov control, J. Phys. At. Mol. Opt. Phys. 44(19), 195503(2011)

[70] D. Stefanatos , J. Ruths , and J. S. Li , Frictionless atom cooling in harmonic traps: A time-optimal approach, Phys. Rev. A 82(6), 063422(2010)

[71] X. J. Lu , X. Chen , J. Alonso , and J. G. Muga , Fast transitionless expansions of Gaussian anharmonic traps for cold atoms: Bang–singular–bang control, Phys. Rev. A 89(2), 023627(2014)

[72] M. Palmero , E. Torrontegui , D. Guéry-Odelin , and J. G. Muga , Fast transport of two ions in an anharmonic trap, Phys. Rev. A 88(5), 053423(2013)

[73] M. Palmero , R. Bowler , J. P. Gaebler , D. Leibfried , and J. G. Muga , Fast transport of mixed-species ion chains within a Paul trap, Phys. Rev. A 90(5), 053408(2014)

[74] Y. C. Ding , T. Y. Huang , K. Paul , M. J. Hao , and X. Chen , Smooth bang–bang shortcuts to adiabaticity for atomic transport in a moving harmonic trap, Phys. Rev. A 101(6), 063410(2020)

[75] S. Balasubramanian , S. Y. Han , B. T. Yoshimura , and J. K. Freericks , Bang–bang shortcut to adiabaticity in trapped-ion quantum simulators, Phys. Rev. A 97(2), 022313(2018)

[76] D. Vitali , S. Gigan , A. Ferreira , H. R. Böhm , P. Tombesi , A. Guerreiro , V. Vedral , A. Zeilinger , and M. Aspelmeyer , Optomechanical entanglement between a movable mirror and a cavity field, Phys. Rev. Lett. 98(3), 030405(2007)

[77] C. S. Hu , Z. Q. Liu , Y. Liu , L. T. Shen , H. Z. Wu , and S. B. Zheng , Entanglement beating in a cavity optomechanical system under two-field driving, Phys. Rev. A 101(3), 033810(2020)

[78] G. De Chiara , M. Paternostro , and G. M. Palma , Entanglement detection in hybrid optomechanical systems, Phys. Rev. A 83(5), 052324(2011)

[79] L. Tian and H. L. Wang , Optical wavelength conversion of quantum states with optomechanics, Phys. Rev. A 82(5), 053806(2010)

RIGHTS & PERMISSIONS

2022 Higher Education Press