1Institut für Physik, Universität Rostock, Albert-Einstein-Straße 23, 18059 Rostock, Germany
2ZARM, University of Bremen, Am Fallturm 2, 28359 Bremen, Germany
3Institut für Physik, Humboldt Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
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Abstract
The role of entanglement in determining the non-classicality of a given interaction has gained significant traction over the last few years. In particular, as the basis for new experimental proposals to test the quantum nature of the gravitational field. Here we show that the rate of gravity mediated entanglement between two otherwise isolated optomechanical systems can be significantly increased by modulating the optomechanical coupling. This is most pronounced for low mass, high frequency systems – convenient for reaching the quantum regime – and can lead to improvements of several orders of magnitude, as well as a broadening of the measurement window. Nevertheless, significant obstacles still remain. In particular, we find that modulations increase decoherence effects at the same rate as the entanglement improvements. This adds to the growing evidence that the constraint on noise (acting on the position d.o.f) depends only on the particle mass, separation, and temperature of the environment and cannot be improved by novel quantum control. Finally, we highlight the close connection between the observation of quantum correlations and the limits of measurement precision derived via the Cramér-Rao Bound. An immediate consequence is that probing superpositions of the gravitational field places similar demands on detector sensitivity as entanglement verification.
Featured image: Two Fabry-Pérot cavities brought in close proximity. The moving elements interact via their gravitational fields. To second order this leads to an $hat x_1 hat x_2$ coupling and allows the generation entanglement between the mechanical modes. This is exchanged back and forth with the cavity fields, which themselves become entangled around multiples of the mechanical period $1/omega_m$.
Popular summary
Such an experiment, however, is expected to be exceedingly difficult – with the entanglement rate depending on the mass, separation and superposition size (or more generally, variance) that can be achieved during coherent evolution. The latter in particular poses a significant obstacle to optomechanical systems, which are often regarded as one of the most attractive platforms for tests of macroscopic quantum physics. These rely on the radiation pressure of a light field to drive the dynamics of a mechanical oscillator, the variance of which depends on that of the field as well as the strength of the optomechanical coupling. However, achieving a high photon number variance is typically difficult, and so the conventional approach is to simply increase the number of photons in a (possibly squeezed) coherent light field. This cannot be done without limit, as if the mechanical elements are driven too hard they risk colliding. As a result, predicted entanglement times are typically much longer than even the most optimistic noise timescales.
To address this problem, we show that if the optomechanical coupling strength, $k$, can be modulated close to the mechanical resonance, then the entanglement rate is significant enhanced – potentially by several orders of magnitude. This is because radiation pressure leads to a force on the mechanical elements proportional to the coupling, and so modulating $k$ is equivalent to resonantly driving the oscillator. As the force is also proportional to the photon number, fluctuations in the field are passed on to mechanics, but now enhanced by the increased displacement. This means that in ideal, zero noise conditions, state of the art systems could potentially generate appreciable entanglement on the order of seconds.
Unfortunately, however, we find that decoherence is also enhanced by a commensurate amount. This adds to a growing number of results – across multiple platforms – suggesting the existence of an unavoidable noise limit that depends only on the interaction term and the environment, i.e. it cannot be mitigated by local control or preparation of the mechanical states. In practice, this has severe implications for any attempts to probe gravity through entanglement tests. We highlight further difficulties by quantifying the extreme levels of control needed over the dynamics, which in the nonlinear optomechanical setting is reflected in both the degree that the mechanical frequencies of each oscillator must match as well as the measurement timing precision. Similar constraints should be expected in any experiment where entanglement is transferred from mechanical to ancilla degrees of freedom.
Finally, we compare our analysis against a variety of approximation methods, showing that these are often sufficient to obtain an accurate entanglement rate. In particular, by characterising the measurement sensitivity when two optomechanical systems are considered as a sensor-source pair, we find that the time needed to detect the quantum fluctuations in the source roughly coincides with that required to establish witnessable entanglement. This highlights the close connection to quantum metrology, and underlines the importance of improving sensor performance when attempting to access the quantum gravity regime.
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[1] K. Eppley and E. Hannah. “The necessity of quantizing the gravitational field”. Foundations of Physics 7, 51–68 (1977).
https://doi.org/10.1007/BF00715241
[2] A. Peres and D. R. Terno. “Hybrid classical-quantum dynamics”. Phys. Rev. A 63, 022101 (2001).
https://doi.org/10.1103/PhysRevA.63.022101
[3] D. R. Terno. “Inconsistency of quantum—classical dynamics, and what it implies”. Foundations of Physics 36, 102–111 (2006).
https://doi.org/10.1007/s10701-005-9007-y
[4] G. Amelino-Camelia, C. Lämmerzahl, F. Mercati, and G. M. Tino. “Constraining the energy-momentum dispersion relation with planck-scale sensitivity using cold atoms”. Phys. Rev. Lett. 103, 171302 (2009).
https://doi.org/10.1103/PhysRevLett.103.171302
[5] J. D. Bekenstein. “Is a tabletop search for planck scale signals feasible?”. Phys. Rev. D 86, 124040 (2012).
https://doi.org/10.1103/PhysRevD.86.124040
[6] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner. “Probing planck-scale physics with quantum optics”. Nature Physics 8, 393–397 (2012).
https://doi.org/10.1038/nphys2262
[7] C. Anastopoulos and B. L. Hu. “Probing a gravitational cat state”. Classical and Quantum Gravity 32, 165022 (2015).
https://doi.org/10.1088/0264-9381/32/16/165022
[8] M. Carlesso, A. Bassi, M. Paternostro, and H. Ulbricht. “Testing the gravitational field generated by a quantum superposition”. New Journal of Physics 21, 093052 (2019).
https://doi.org/10.1088/1367-2630/ab41c1
[9] A. Albrecht, A. Retzker, and M. B. Plenio. “Testing quantum gravity by nanodiamond interferometry with nitrogen-vacancy centers”. Phys. Rev. A 90, 033834 (2014).
https://doi.org/10.1103/PhysRevA.90.033834
[10] A. D. K. Plato, C. N. Hughes, and M. S. Kim. “Gravitational effects in quantum mechanics”. Contemporary Physics 57, 477–495 (2016).
https://doi.org/10.1080/00107514.2016.1153290
[11] S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn. “Spin entanglement witness for quantum gravity”. Phys. Rev. Lett. 119, 240401 (2017).
https://doi.org/10.1103/PhysRevLett.119.240401
[12] C. Marletto and V. Vedral. “Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity”. Phys. Rev. Lett. 119, 240402 (2017).
https://doi.org/10.1103/PhysRevLett.119.240402
[13] C. Wan. “Quantum superposition on nano-mechanical oscillator”. PhD thesis. Imperial College, London. (2017). url: https://spiral.imperial.ac.uk/handle/10044/1/74060.
https://spiral.imperial.ac.uk/handle/10044/1/74060
[14] E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter. “Everything you always wanted to know about locc (but were afraid to ask)”. Communications in Mathematical Physics 328, 303–326 (2014).
https://doi.org/10.1007/s00220-014-1953-9
[15] N. Matsumoto, S. B. Cataño Lopez, M. Sugawara, S. Suzuki, N. Abe, K. Komori, Y. Michimura, Y. Aso, and K. Edamatsu. “Demonstration of displacement sensing of a mg-scale pendulum for mm- and mg-scale gravity measurements”. Phys. Rev. Lett. 122, 071101 (2019).
https://doi.org/10.1103/PhysRevLett.122.071101
[16] S. B. Cataño Lopez, J. G. Santiago-Condori, K. Edamatsu, and N. Matsumoto. “High-$q$ milligram-scale monolithic pendulum for quantum-limited gravity measurements”. Phys. Rev. Lett. 124, 221102 (2020).
https://doi.org/10.1103/PhysRevLett.124.221102
[17] M. Rademacher, J. Millen, and Y. L. Li. “Quantum sensing with nanoparticles for gravimetry: when bigger is better”. Advanced Optical Technologies 9, 227–239 (2020).
https://doi.org/10.1515/aot-2020-0019
[18] C. Montoya, E. Alejandro, W. Eom, D. Grass, N. Clarisse, A. Witherspoon, and A. A. Geraci. “Scanning force sensing at micrometer distances from a conductive surface with nanospheres in an optical lattice”. Applied optics 61, 3486–3493 (2022).
https://doi.org/10.1364/AO.457148
[19] F. Armata, L. Latmiral, A. D. K. Plato, and M. S. Kim. “Quantum limits to gravity estimation with optomechanics”. Phys. Rev. A 96, 043824 (2017).
https://doi.org/10.1103/PhysRevA.96.043824
[20] S. Qvarfort, A. Serafini, P. F. Barker, and S. Bose. “Gravimetry through non-linear optomechanics”. Nature Communications 9, 3690 (2018).
https://doi.org/10.1038/s41467-018-06037-z
[21] S. Qvarfort, A. D. K. Plato, D. E. Bruschi, F. Schneiter, D. Braun, A. Serafini, and D. Rätzel. “Optimal estimation of time-dependent gravitational fields with quantum optomechanical systems”. Phys. Rev. Research 3, 013159 (2021).
https://doi.org/10.1103/PhysRevResearch.3.013159
[22] A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen. “Mechanical Squeezing via Parametric Amplification and Weak Measurement”. Physical Review Letters 107, 213603 (2011).
https://doi.org/10.1103/PhysRevLett.107.213603
[23] J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, and P. F. Barker. “Cavity cooling a single charged levitated nanosphere”. Physical Review Letters 114, 123602 (2015).
https://doi.org/10.1103/PhysRevLett.114.123602
[24] P. Z. G. Fonseca, E. B. Aranas, J. Millen, T. S. Monteiro, and P. F. Barker. “Nonlinear dynamics and strong cavity cooling of levitated nanoparticles”. Physical Review Letters 117, 173602 (2016).
https://doi.org/10.1103/PhysRevLett.117.173602
[25] E. B. Aranas, P. Z. G. Fonseca, P. F. Barker, and T. S. Monteiro. “Split-sideband spectroscopy in slowly modulated optomechanics”. New Journal of Physics 18, 113021 (2016).
https://doi.org/10.1088/1367-2630/18/11/113021
[26] W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester. “Towards quantum superpositions of a mirror”. Phys. Rev. Lett. 91, 130401 (2003).
https://doi.org/10.1103/PhysRevLett.91.130401
[27] D. Kafri, J. M. Taylor, and G. J. Milburn. “A classical channel model for gravitational decoherence”. New Journal of Physics 16, 065020 (2014).
https://doi.org/10.1088/1367-2630/16/6/065020
[28] H. Miao, D. Martynov, H. Yang, and A. Datta. “Quantum correlations of light mediated by gravity”. Physical Review A 101, 063804 (2020).
https://doi.org/10.1103/PhysRevA.101.063804
[29] C. Anastopoulos and B. L. Hu. “Problems with the newton-schrödinger equations”. New Journal of Physics 16, 085007 (2014).
https://doi.org/10.1088/1367-2630/16/8/085007
[30] C. Anastopoulos and B. L. Hu. “Quantum superposition of two gravitational cat states”. Classical and Quantum Gravity 37, 235012 (2020).
https://doi.org/10.1088/1361-6382/abbe6f
[31] V. Sudhir, M. G. Genoni, J. Lee, and M. S. Kim. “Critical behavior in ultrastrong-coupled oscillators”. Phys. Rev. A 86, 012316 (2012).
https://doi.org/10.1103/PhysRevA.86.012316
[32] T. Krisnanda, G. Y. Tham, M. Paternostro, and T. Paterek. “Observable quantum entanglement due to gravity”. npj Quantum Information 6, 12 (2020).
https://doi.org/10.1038/s41534-020-0243-y
[33] S. Qvarfort, A. Serafini, A. Xuereb, D. Braun, D. Rätzel, and D. E. Bruschi. “Time-evolution of nonlinear optomechanical systems: Interplay of mechanical squeezing and non-gaussianity”. Journal of Physics A: Mathematical and Theoretical 53, 075304 (2020).
https://doi.org/10.1088/1751-8121/ab64d5
[34] S. Mancini, V. I. Man’ko, and P. Tombesi. “Ponderomotive control of quantum macroscopic coherence”. Phys. Rev. A 55, 3042–3050 (1997).
https://doi.org/10.1103/PhysRevA.55.3042
[35] S. Bose, K. Jacobs, and P. L. Knight. “Preparation of nonclassical states in cavities with a moving mirror”. Phys. Rev. A 56, 4175–4186 (1997).
https://doi.org/10.1103/PhysRevA.56.4175
[36] O. Gühne and G. Tóth. “Entanglement detection”. Physics Reports 474, 1–75 (2009).
https://doi.org/10.1016/j.physrep.2009.02.004
[37] E. Shchukin and W. Vogel. “Inseparability criteria for continuous bipartite quantum states”. Phys. Rev. Lett. 95, 230502 (2005).
https://doi.org/10.1103/PhysRevLett.95.230502
[38] E. V. Shchukin and W. Vogel. “Nonclassical moments and their measurement”. Phys. Rev. A 72, 043808 (2005).
https://doi.org/10.1103/PhysRevA.72.043808
[39] M. T. Naseem, A. Xuereb, and Ö. E. Müstecaplıoğlu. “Thermodynamic consistency of the optomechanical master equation”. Physical Review A 98, 052123 (2018).
https://doi.org/10.1103/PhysRevA.98.052123
[40] A. Matsumura and K. Yamamoto. “Gravity-induced entanglement in optomechanical systems”. Phys. Rev. D 102, 106021 (2020).
https://doi.org/10.1103/PhysRevD.102.106021
[41] W. H. Zurek. “Decoherence, einselection, and the quantum origins of the classical”. Rev. Mod. Phys. 75, 715–775 (2003).
https://doi.org/10.1103/RevModPhys.75.715
[42] Á. Rivas, A. D. K. Plato, S. F. Huelga, and M. B. Plenio. “Markovian master equations: a critical study”. New Journal of Physics 12, 113032 (2010).
https://doi.org/10.1088/1367-2630/12/11/113032
[43] S. L. Adler, A. Bassi, and E. Ippoliti. “Towards quantum superpositions of a mirror: an exact open systems analysis — calculational details”. Journal of Physics A: Mathematical and General 38, 2715–2727 (2005).
https://doi.org/10.1088/0305-4470/38/12/013
[44] H. B. G. Casimir and D. Polder. “The influence of retardation on the london-van der waals forces”. Physical Review 73, 360 (1948).
https://doi.org/10.1103/PhysRev.73.360
[45] P. Rodriguez-Lopez. “Casimir energy and entropy in the sphere-sphere geometry”. Phys. Rev. B 84, 075431 (2011).
https://doi.org/10.1103/PhysRevB.84.075431
[46] J. Chiaverini, S. J. Smullin, A. A. Geraci, D. M. Weld, and A. Kapitulnik. “New experimental constraints on non-newtonian forces below 100 $mu$ m”. Physical Review Letters 90, 151101 (2003).
https://doi.org/10.1103/PhysRevLett.90.151101
[47] T. W. van de Kamp, R. J. Marshman, S. Bose, and A. Mazumdar. “Quantum gravity witness via entanglement of masses: Casimir screening”. Physical Review A 102, 062807 (2020).
https://doi.org/10.1103/PhysRevA.102.062807
[48] C. K. Law. “Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation”. Physical Review A 51, 2537–2541 (1995).
https://doi.org/10.1103/PhysRevA.51.2537
[49] O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, and J. I. Cirac. “Optically levitating dielectrics in the quantum regime: Theory and protocols”. Physical Review A 83, 013803 (2011).
https://doi.org/10.1103/PhysRevA.83.013803
[50] A. Serafini. “Quantum continuous variables: A primer of theoretical methods”. CRC Press. (2017).
https://doi.org/10.1201/9781315118727
[51] J. Millen, T. S. Monteiro, R. Pettit, and A. N. Vamivakas. “Optomechanics with levitated particles”. Reports on Progress in Physics 83, 026401 (2020).
https://doi.org/10.1088/1361-6633/ab6100
[52] D. E. Bruschi. “Time evolution of two harmonic oscillators with cross-kerr interactions”. Journal of Mathematical Physics 61, 032102 (2020).
https://doi.org/10.1063/1.5121397
[53] C. M. DeWitt and D. Rickles. “The role of gravitation in physics: report from the 1957 chapel hill conference”. Volume 5. epubli. (2011).
[54] F. Schneiter, S. Qvarfort, A. Serafini, A. Xuereb, D. Braun, D. Rätzel, and D. E. Bruschi. “Optimal estimation with quantum optomechanical systems in the nonlinear regime”. Phys. Rev. A 101, 033834 (2020).
https://doi.org/10.1103/PhysRevA.101.033834
[55] D. F. Walls. “Squeezed states of light”. Nature 306, 141–146 (1983).
https://doi.org/10.1038/306141a0
[56] S. Ast, M. Mehmet, and R. Schnabel. “High-bandwidth squeezed light at 1550 nm from a compact monolithic ppktp cavity”. Opt. Express 21, 13572–13579 (2013).
https://doi.org/10.1364/OE.21.013572
[57] J. Z. Bernád, L. Diósi, and T. Geszti. “Quest for quantum superpositions of a mirror: high and moderately low temperatures”. Physical review letters 97, 250404 (2006).
https://doi.org/10.1103/PhysRevLett.97.250404
[58] S. Rijavec, M. Carlesso, A. Bassi, V. Vedral, and C. Marletto. “Decoherence effects in non-classicality tests of gravity”. New Journal of Physics 23, 043040 (2021).
https://doi.org/10.1088/1367-2630/abf3eb
[59] S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert. “Observation of non-markovian micromechanical brownian motion”. Nature Communications 6, 7606 (2015).
https://doi.org/10.1038/ncomms8606
[60] M. Ludwig, K. Hammerer, and F. Marquardt. “Entanglement of mechanical oscillators coupled to a nonequilibrium environment”. Phys. Rev. A 82, 012333 (2010).
https://doi.org/10.1103/PhysRevA.82.012333
[61] A. Datta and H. Miao. “Signatures of the quantum nature of gravity in the differential motion of two masses”. Quantum Science and Technology 6, 045014 (2021).
https://doi.org/10.1088/2058-9565/ac1adf
[62] B. Dakić, V. Vedral, and Č. Brukner. “Necessary and sufficient condition for nonzero quantum discord”. Physical review letters 105, 190502 (2010).
https://doi.org/10.1103/PhysRevLett.105.190502
[63] A.O. Caldeira and A.J. Leggett. “Quantum tunnelling in a dissipative system”. Annals of Physics 149, 374–456 (1983).
https://doi.org/10.1016/0003-4916(83)90202-6
[64] B. L. Hu, J. P. Paz, and Y. Zhang. “Quantum brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise”. Phys. Rev. D 45, 2843–2861 (1992).
https://doi.org/10.1103/PhysRevD.45.2843
[65] M. B. Plenio. “Logarithmic negativity: A full entanglement monotone that is not convex”. Phys. Rev. Lett. 95, 090503 (2005).
https://doi.org/10.1103/PhysRevLett.95.090503
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