Coherent expansion of the motional state of a massive nanoparticle beyond its linear dimensions

Abstract

Quantum mechanics predicts that massive particles exhibit wave-like behavior. Matterwave interferometry has been able to validate such predictions through ground-breaking experiments involving microscopic systems like atoms and molecules. The wavefunction of such systems coherently extends over a distance much larger than their size, an achievement that is incredibly challenging for massive and more complex objects. Yet, reaching similar level of coherent diffusion will enable tests of fundamental physics at the genuinely macroscopic scale, as well as the development of quantum sensing apparata of great sensitivity. We report on experimentally achieving an unprecedented degree of position diffusion in a massive levitated optomechanical system through frequency modulation of the trapping potential. By starting with a pre-cooled state of motion and employing a train of sudden pulses yet of mild modulation depth, we surpass previously attained values of position diffusion in this class of systems to reach diffusion lengths that exceed the physical dimensions of the trapped nanoparticle.

Figures (2)

• Figure 1: Experimental setup for pulse-driven coherent wakepacket expansionPanel A: Schematic of experimental setup. A silica nanoparticle is trapped by an optical tweezers in vacuum. An acusto-optical modulator (AOM) controlled by a signal generator (SG) and lock-in amplifier (LI) modulates the power of the laser, which serves as the mechanism to modulate the intensity of the driving laser and thus control the trapping potential of the particle. The oscilloscope (O'scope), triggered by a synchronous signal from the SG, records the signal from the particle. The polarising beam splitter (PBS) and $\lambda/4$ waveplate routes the particle's movement signal towards a photo-detector. The particle displacement is detected to implement parametric and electric cooling mechanisms to lower the energy of the translational degrees of freedom of the system in three directions (only one direction shown in the figure). Panel B: Time sequence of laser modulation $S(t)$ for expanding the mechanical state with timings for lowering the power $\tau_{low}$ and rising it back up $\tau_{high}$. Panel C: Phase-space distribution of position and momentum for the nanoparticle. From top-left subpanel we show the evolution of an isotropic initial state at 4.2mK within a 0.8ms timeframe.
• Figure 2: Panel A: Time-evolution of the position standard-deviation $\sigma_{z}$, experiment vs simulations. The length of the semi-major axis $\sigma_{max}$ increases exponentially in a timescale of about 1ms all the way up to 124nm, which is of the order of the thermal spread (green dashed line) and radius of the nanoparticle used in the experiment (blue dashed line). In the same time-window, the semi-minor axis undergoes an initial contraction, as one would expect from the dynamics. However, such contraction is then followed by an expansion occurring at roughly the same rate as the growth of the semi-major axis, suggesting additional heating mechanisms. The brown dots are the results of our Langevin simulations where we added a Gaussian noise to the harmonic frequency $\omega_z$. Panel B: State Coherence $C(\hat{\rho})$ (blue trace) and purity $P(\hat{\rho})$ (red line) evolutions being computed as in Eq. \ref{['eq.coherence.gauss']} and Eq. \ref{['eq.purity.gauss']} respectively. The state's coherence initially increases, up to $154.7 \, \mu s$ when it peaks at 0.939pm. This point corresponds to the semi minor axis at its smallest value $0.155 \,\text{nm}$. Afterwards the state becomes broader, and coherence starts to decrease. After an initial plateau, the purity decreases exponentially. After around 1 ms, it follows a similar oscillatory behavior as coherence. Panel C: Comparison of coherence (blue line) and position standard-deviation (orange line) at the beginning of the protocol. We highlighted the maximal value of coherence, which is larger than the initial one, and the minimal value of the position standard-deviation. Panel D: Purity evolution at the beginning of the protocol, where the initial plateau is clearly visible.