Among the main goals of quantum science is the control of nano- and micromechanical oscillators at the quantum level. One powerful and well-developed tool for this purpose is quantum optomechanics: the study and engineering of the interaction of light with mechanical systems via radiation pressure.1This interaction can, for instance, be used to cool a nano- or micromechanical oscillator to the ground state, thus counteracting thermalization and decoherence.2Given that many optomechanical effects can be interpreted in terms of nonlinear quantum optics, quantum control protocols are already at hand and only need an appropriate translation.
Figure 1. Schematic of an optomechanical system consisting of a Fabry-Perot cavity driven by an external laser. Incoming radiation exchanges momentum with the movable mirror on the right. γM: Spring friction coefficient. k: Optical mode decay rate. λ: Optical mode wavelength. m: Mirror mass. ωC: Optical mode frequency. ωL: Input laser quasi-resonant frequency. ωM(t): Spring characteristic frequency. P(t): Input laser power.
Figure 2. Asymptotic quantum features in the presence of mechanical modulation, as a function of Ω/ωMand ∊. EN, MAX: Maximum of entanglement, measured via logarithmic negativity. ∊: Mechanical modulation strength. Ω: Modulating frequency (mechanical). ωM: Modulated mechanical frequency. Δq2MIN: Minimum of generalized quadratures of the mirror (below 1/2, the oscillator is in a squeezed state). Parameters used in the simulation: γM/(2π) = 1Hz. k = 1.34MHz. λ = 1064nm. m = 150ng.ωC- ωL = ωM0. ωM0/(2π) = 1MHz. P0 = 10mW. T = 0.1K.
Figure 3. Asymptotic quantum features in the presence of both mechanical frequency modulation and input laser power modulation, as a function of relative phase φ(solid black line). For comparison, the same quantities are plotted when only the mechanical frequency (solid blue line) or the input laser power (dashed red line) is modulated. η: Laser power modulation strength. nMAX: Maximum number of phonons in the mirror. φ: Relative phase. Additional parameters used in the simulation: Ω1 = Ω2 = 2ωM0.
In the last two decades, a variety of interesting applications in quantum optomechanics have been proposed. An example is the generation of entanglement—a key resource in quantum information processing—between radiation and a moving mirror in a Fabry-Perot (FP) cavity.3Another example is the generation of position-squeezed mechanical states, helpful in increasing the precision of interferometry and detection experiments.4But successful implementation of many of these applications has proven elusive, mainly because the level of light/matter interaction attained to date has frequently been insufficient to enter the so-called strong coupling regime.5What is needed is a way to enhance the visibility of the desired quantum properties using today's technology.
A solution first proposed by Mari and Eisert4 would apply periodic modulations to certain system parameters. The resulting modulation induced on the system response could show a periodic strengthening of quantum effects. That being so, are there optimal modulations for which this increase is maximal? And would modulating more than one parameter at once produce a substantial improvement?
To tackle these issues in a realistic way, we consider the simple case shown in Figure 1. It consists of an FP cavity with a movable mirror acting as a mechanical oscillator. The mirror's motion evolves under the action of both radiation pressure—exerted by the photons of an externally driven optical mode—and thermal noise. We first apply a modulation to the mechanical frequency, of the form ωM(t) = ωM0[1+∊ cos(Ω1t)]. We then add a second modulation on the input laser power P(t) = P0[1+η cos(Ω2t+φ)] and study the interplay between the two. Fixing all parameters to state-of-the-art values, we simulate the system dynamics and characterize various quantum properties in the asymptotic stationary regime.6
Starting with the single-modulation picture, we vary ∊ and Ω1 across a wide range of values and search for an optimal value for each. As the plots in Figure 2 show, a frequency of Ω1 ∼ 2ωM0gives the best performance at all strengths ∊. Also obvious from the plots is the fact that quantum effects increase monotonically with respect to ∊ up to a threshold value, at which point the system becomes unstable. This behavior can be explained as a resonance between the modulation frequency Ω1 and the natural frequency of evolution of the system correlations, which for our case is close to 2ωM0. When the modulation is too strong, the energy absorbed by the system is not compensated by dissipation, and the evolution has no asymptotic steady state. Comparison with the case ∊ = 0 shows that modulation can increase entanglement by a factor of up to three while generating considerable levels of squeezing, a feature that completely disappears when the modulation is turned off.
Next, we modulate the input laser power and look for possible interference patterns between the two modulations. The modulation strengths ∊ and η chosen produce comparable levels of squeezing when applied individually. We choose frequencies optimized at Ω1 = Ω2 = 2ωM0. Figure 3 shows certain quantum properties as a function of relative phase φ. While entanglement is affected very little, we find rather drastic changes in the mirror squeezing and its energy, with evident constructive/destructive interference effects showing up. From this, we conclude that careful choice of the phase φ can enhance squeezing, producing values unattainable in the presence of only one modulation.
In summary, we have explored the capabilities of periodically modulated optomechanical systems and have shown how to exploit suitable modulation regimes to produce a strong increase in the visibility of quantum effects such as entanglement and squeezing. Next steps could include perfecting our analysis by revisiting our mathematical methods and contextualizing our work to a specific experimental setup: namely, levitated optomechanics,7 a good candidate for future implementations.
We thank Rosario Fazio for useful discussions and comments. This work was supported by MIUR through FIRB-IDEAS project RBID08B3FM.
Alessandro Farace, Vittorio Giovannetti
Scuola Normale Superiore di Pisa Sezione Fisica
Alessandro Farace is a PhD candidate in the Quantum Transport and Information (QTI) group at Scuola Normale Superiore (SNS).
Vittorio Giovannetti is an associate professor of physics in the QTI group at SNS. He specializes in quantum information and quantum optics.
1. G. J. Milburn, M. J. Woolley, An introduction to quantum optomechanics, Acta Phys. Slov. 61(5), p. 483-601, 2012.
2. C. Genes, A. Mari, D. Vitali, P. Tombesi, Quantum effects in optomechanical systems, Adv. At. Mol. Opt. Phys. 57, p. 33-86, 2009.
3. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, M. Aspelmeyer, Optomechanical entanglement between a movable mirror and a cavity field, Phys. Rev. Lett. 98, p. 030405, 2007.
4. A. Mari, J. Eisert, Gently modulating optomechanical systems, Phys. Rev. Lett. 103, p. 213603, 2009.
5. E. Verhagen, S. Deleglise, S. Weis, A. Schliesser, T. J. Kippenberg, Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode, Nature 482, p. 63-67, 2012.
6. A. Farace, V. Giovannetti, Enhancing quantum effects via periodic modulations in optomechanical systems, Phys. Rev. A
7. D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, P. Zoller, Cavity opto-mechanics using an optically levitated nanosphere, Proc. Nat'l Acad. Sci. U.S.A. 107, p. 1005-1010, 2010.