Abstract

Interaction with a thermal environment decoheres the quantum state of a mechanical oscillator. When the interaction is sufficiently strong, such that more than one thermal phonon is introduced within a period of oscillation, quantum coherent oscillations are prevented. This is generally thought to preclude a wide range of quantum protocols. Here we show that the combination of pulsed optomechanics techniques with coherent control can overcome this limitation, allowing ground-state cooling, general linear quantum nondemolition measurements, optomechanical state swaps, and quantum-state preparation and tomography without requiring quantum coherent oscillations. Finally, we show how the protocol can break the usual thermal limit for classical sensing of impulse forces.

© 2017 Optical Society of America

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References

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    [Crossref]

2017 (2)

Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, “Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution,” Nat. Nano 12, 776–783 (2017).
[Crossref]

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
[Crossref]

2016 (2)

J. S. Bennett, K. Khosla, L. S. Madsen, M. R. Vanner, H. Rubinsztein-Dunlop, and W. P. Bowen, “A quantum optomechanical interface beyond the resolved sideband limit,” New J. Phys. 18, 053030 (2016).
[Crossref]

A. R. Kermany, J. S. Bennett, G. A. Brawley, W. P. Bowen, and F. Iacopi, “Factors affecting the f x Q product of 3c-SiC microstrings: what is the upper limit for sensitivity?” J. Appl. Phys. 119, 055304 (2016).
[Crossref]

2015 (2)

M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” Ann. Phys. 527, 15–26 (2015).
[Crossref]

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement-based control of a mechanical oscillator at its thermal decoherence rate,” Nature 524, 325–329 (2015).
[Crossref]

2014 (3)

C. Graf, B. W. Barr, A. S. Bell, F. Campbell, A. V. Cumming, S. L. Danilishin, N. A. Gordon, G. D. Hammond, J. Hennig, E. A. Houston, S. H. Huttner, R. A. Jones, S. S. Leavey, H. Luck, J. Macarthur, M. Marwick, S. Rigby, R. Schilling, B. Sorazu, A. Spencer, S. Steinlechner, K. A. Strain, and S. Hild, “Design of a speed meter interferometer proof-of-principle experiment,” Class. Quantum Grav. 31, 215009 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
[Crossref]

2013 (8)

S. A. McGee, D. Meiser, C. A. Regal, K. W. Lehnert, and M. J. Holland, “Mechanical resonators for storage and transfer of electrical and optical quantum states,” Phys. Rev. A 87, 053818 (2013).
[Crossref]

T. P. Purdy, R. W. Peterson, and C. A. Regal, “Observation of radiation pressure shot noise on a macroscopic object,” Science 339, 801–804 (2013).
[Crossref]

A. Kronwald, F. Marquardt, and A. A. Clerk, “Arbitrarily large steady-state bosonic squeezing via dissipation,” Phys. Rev. A 88, 063833 (2013).

M. J. Woolley and A. A. Clerk, “Two-mode back-action-evading measurements in cavity optomechanics,” Phys. Rev. A 87, 063846 (2013).

T. P. Purdy, P.-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3, 031012 (2013).

A. H. Safavi-Naeini, S. Groblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref]

K. E. Khosla, M. R. Vanner, W. P. Bowen, and G. J. Milburn, “Quantum state preparation of a mechanical resonator using an optomechanical geometric phase,” New J. Phys. 15, 043025 (2013).
[Crossref]

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 3295 (2013).
[Crossref]

2012 (5)

E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
[Crossref]

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, 013603 (2012).
[Crossref]

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys. 8, 393–397 (2012).
[Crossref]

S. Forstner, S. Prams, J. Knittel, E. D. van Ooijen, J. D. Swaim, G. I. Harris, A. Szorkovszky, W. P. Bowen, and H. Rubinsztein-Dunlop, “Cavity optomechanical magnetometer,” Phys. Rev. Lett. 108, 120801 (2012).
[Crossref]

E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nano 7, 509–514 (2012).
[Crossref]

2011 (9)

K. Stannigel, P. Rabl, A. S. Sorensen, M. D. Lukin, and P. Zoller, “Optomechanical transducers for quantum-information processing,” Phys. Rev. A 84, 042341 (2011).
[Crossref]

G. Milburn and M. Woolley, “An introduction to quantum optomechanics,” Acta Phys. Slovaca 61, 483–601 (2011).
[Crossref]

M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, C. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, “Pulsed quantum optomechanics,” Proc. Natl. Acad. Sci. USA 108, 16182–16187 (2011).
[Crossref]

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref]

R. Riviere, S. Deleglise, S. Weis, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanical sideband cooling of a micromechanical oscillator close to the quantum ground state,” Phys. Rev. A 83, 063835 (2011).
[Crossref]

G. Anetsberger, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Cavity optomechanics and cooling nanomechanical oscillators using microresonator enhanced evanescent near-field coupling,” C. R. Phys. 12, 800–816 (2011).
[Crossref]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[Crossref]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

2010 (2)

J. B. Hertzberg, T. Rocheleau, T. Ndukum, M. Savva, A. A. Clerk, and K. C. Schwab, “Back-action-evading measurements of nanomechanical motion,” Nat. Phys. 6, 213–217 (2010).
[Crossref]

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, “Strong and tunable nonlinear optomechanical coupling in a low-loss system,” Nat. Phys. 6, 707–712 (2010).
[Crossref]

2009 (3)

G. Anetsberger, O. Arcizet, Q. P. Unterreithmeier, R. Riviere, A. Schliesser, E. M. Weig, J. P. Kotthaus, and T. J. Kippenberg, “Near-field cavity optomechanics with nanomechanical oscillators,” Nat. Phys. 5, 909–914 (2009).
[Crossref]

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009).
[Crossref]

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

2008 (2)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref]

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

2006 (1)

A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C. Schwab, “Cooling a nanomechanical resonator with quantum back-action,” Nature 443, 193–196 (2006).
[Crossref]

2004 (1)

S. L. Danilishin, “Sensitivity limitations in optical speed meter topology of gravitational-wave antennas,” Phys. Rev. D 69, 102003 (2004).
[Crossref]

2003 (2)

Y. Chen, “Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector,” Phys. Rev. D 67, 122004 (2003).
[Crossref]

W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositions of a mirror,” Phys. Rev. Lett. 91, 130401 (2003).
[Crossref]

2002 (2)

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
[Crossref]

P. Purdue, “Analysis of a quantum nondemolition speed-meter interferometer,” Phys. Rev. D 66, 022001 (2002).
[Crossref]

2001 (1)

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

1994 (1)

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref]

1990 (1)

V. B. Braginsky and F. J. Khalili, “Gravitational wave antenna with QND speed meter,” Phys. Lett. A 147, 251–256 (1990).
[Crossref]

1983 (1)

A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
[Crossref]

1981 (1)

R. Benguria and M. Kac, “Quantum Langevin equation,” Phys. Rev. Lett. 46, 1–4 (1981).
[Crossref]

1980 (1)

C. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75–79 (1980).
[Crossref]

1978 (1)

V. Braginskii, Y. I. Vorontsov, and F. Y. Khalili, “Optimal quantum measurements in detectors of gravitation radiation,” JETP Lett. 27, 296 (1978).

Aichele, T.

A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum state reconstruction of the single-photon Fock state,” Phys. Rev. Lett. 87, 050402 (2001).
[Crossref]

Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
[Crossref]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
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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, 013603 (2012).
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G. Milburn and M. Woolley, “An introduction to quantum optomechanics,” Acta Phys. Slovaca 61, 483–601 (2011).
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M. R. Vanner, I. Pikovski, and M. S. Kim, “Towards optomechanical quantum state reconstruction of mechanical motion,” Ann. Phys. 527, 15–26 (2015).
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C. R. Phys. (1)

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Class. Quantum Grav. (1)

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V. Braginskii, Y. I. Vorontsov, and F. Y. Khalili, “Optimal quantum measurements in detectors of gravitation radiation,” JETP Lett. 27, 296 (1978).

Nano Lett. (1)

A. H. Ghadimi, D. J. Wilson, and T. J. Kippenberg, “Radiation and internal loss engineering of high-stress silicon nitride nanobeams,” Nano Lett. 17, 3501–3505 (2017).
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Nat. Commun. (1)

M. R. Vanner, J. Hofer, G. D. Cole, and M. Aspelmeyer, “Cooling-by-measurement and mechanical state tomography via pulsed optomechanics,” Nat. Commun. 4, 3295 (2013).
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Nat. Nano (2)

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Nat. Phys. (4)

I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and C. Brukner, “Probing Planck-scale physics with quantum optics,” Nat. Phys. 8, 393–397 (2012).
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Nature (8)

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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Groblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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The optical annihilation operator after the beam splitter is af=0.99ai+0.01(di+β)≈ai+0.1β, where ai is the annihilation operator of the light in the top port, and di is the annihilation operator of the displacement point with coherent amplitude β. We have neglected the input vacuum noise (di) since it contributes only 0.01 vacuum noise to the final operator af.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of the protocol. Half- and quarter-wave plates (HWP and QWP, respectively), a polarizing beam splitter (PBS), and a switchable beam splitter (SBS) are used to initially direct the interaction pulse along paths 1–4, along which the first optomechanical interaction takes place. When the interaction pulse reaches the top port of the highly reflective beam splitter (RBS), its coherent amplitude is changed by the displacement pulse. Since almost all of the interaction pulse reflects from the RBS, any quantum correlations between it and the oscillator remain, as the pulse interacts with the mechanical oscillator a second time. After the second optomechanical interaction, the SBS switches out the pulse to a homodyne measurement device instead of directing it along path four a second time.

Fig. 2.
Fig. 2.

(a) Conditional variance for the measurement of different quadratures (ϕ=π/2,π/4, and π/8), using the double (solid lines) or single (dashed lines) interaction protocols. (b)–(d) Conditional variance as a function of mechanical quadrature angle, ϕ, for the double (solid lines) and single (dashed lines) interaction protocols for λ=10, λ=1, and λ=0.1 respectively. (a)–(d) Assume a bath temperature of 1 K, ωM/2π=100  kHz, and γ/2π=1  Hz. Gray dashed lines indicate the ground-state variance.

Fig. 3.
Fig. 3.

Conditional variance of the final oscillator momentum using the double (solid lines) and single (dashed lines) interaction protocols at different temperatures. A temperature of 100 K corresponds to 1.3×103 phonons exchanged per cycle for ωM/2π=100  kHz and γ/2π=1  Hz.

Equations (12)

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HI/=g0α(a+a)(b+b)+g0α2(b+b).
U=exp[iλ2XMXL]eiθbbexp[iλ1XMXL]=R(θ)χ(λ1λ2)U(XLXMϕ),
XMXMcosθ+PMsinθXLλ1sinθ,
PMPMcosθXMsinθXL[λ2λ1cosθ],
XLXL,
PLPLGXMϕ+XLλ2λ1sinθ.
X˙M=ωMPM,
P˙M=ωMXMγPM+2γξ,
XM,out=XMcosθ+PMsinθ+ξXλ1XLsinθ,
PM,out=PMcosθXMsinθ+ξPXL(ηλ2λ1cosθ)λ21ηδX1,
XL,out=ηXL+ηη2δX1+1ηδX2,
PL,out=ηPL+ηη2δP1+1ηδP2+ηλ1λ2sinθXLGXMϕηλ2ξX,