Measurement of Motion beyond the Quantum Limit by Transient Amplification

Abstract

Through simultaneous but unequal electromechanical amplification and cooling processes, we create a method for nearly noiseless pulsed measurement of mechanical motion. We use transient electromechanical amplification (TEA) to monitor a single motional quadrature with a total added noise −8.5 ± 2.0 dB relative to the zero-point motion of the oscillator, or equivalently the quantum limit for simultaneous measurement of both mechanical quadratures. We demonstrate that TEA can be used to resolve fine structure in the phase-space of a mechanical oscillator by tomographically reconstructing the density matrix of a squeezed state of motion. Without any inference or subtraction of noise, we directly observe a squeezed variance 2.8 ± 0.3 dB below the oscillator's zero-point motion.The past ten years has seen a dramatic improvement in the ability to measure and control the quantum state of macroscopic mechanical oscillators. Much of this progress results from advances in the parametric coupling of these oscillators to optical cavities or resonant electrical circuits. These related fields of optomechanics and electromechanics have demonstrated the ability to cool mechanical oscillators to near their motional ground state [1], entangle mechanical oscillators with each other [2, 3] or with other degrees of freedom [4], and create squeezed states of motion [5][6][7]. To verify the successful creation of these non-classical states, electromechanical and optomechanical methods have also enabled measurements of mechanical motion with near 50% quantum efficiency [8, 9], or equivalently an added noise equal to the zero-point motion of the oscillator, the quantum limit for simultaneous measurement of both mechanical quadratures [10].These advances have encouraged notions of using nonclassical states of motion to test quantum mechanics at larger scales, sensing forces with quantum enhanced precision, and enabling quantum transduction between disparate physical systems [11]. But as mechanical oscillators are prepared in more profoundly quantum states [12, 13], with finer features in oscillator phase-space, the measurement efficiency must further improve to resolve these fine features and to use them to realize a quantum advantage.Reaching higher levels of efficiency with existing methods is hindered by fundamental and technical limitations, which seem difficult to overcome. In electromechanical and optomechanical devices, the state of motion can be converted without gain or added noise into a propagating electric field, and one quadrature component of the field can be measured nearly noiselessly [4, 8]. However, the loss experienced by the field traveling between the device and the amplifier has prevented quantum efficiency much greater than 50%. To improve measurement ef-ficiency, the device can be used as its own parametric amplifier, emitting an electric field that encodes an amplified copy of the mechanical oscillator's state, thereby overcoming any subsequent loss and inefficiency of the following measurement chain. Using this ...

Ground State Cooling of an Ultracoherent Electromechanical System

“…( 1), we extract t coh ≈ 140 ms. This is three orders of magnitude larger than for state-of-the-art electromechanical systems [14,10]. However, further work will be needed to fully confirm the coherence of the mechanical system, ruling out e. g. excess decoherence by dephasing [21].…”

“…Analogous to cavity optomechanics [3], this coupling is at the heart of a broad set of phenomena and techniques of interest in quantum science and technology. They range from ground-state cooling of the mechanics [4,5,6], via entanglement and squeezing [7,8,9,10], to coherent microwave-optical [11,12] (see also [13] and references therein) and superconducting qubit-mechanical interfaces [14,15,16,17].…”

“…Here, Q = Ω m /Γ m is the mechanical quality factor defined as the ratio of the mechanical (angular) frequency Ω m and its energy decay rate Γ m ; T the resonator temperature; and k B the reduced Planck and the Boltzmann constant, respectively; and nth ≈ k B T / Ω m is the equivalent occupation of the thermal bath. For state-of-the-art electromechanical systems with Q ∼ 10 5 operated at milliKelvin temperatures, typical coherence times are in the range of 100 microseconds [14,10].…”

Combining Floquet and Lyapunov techniques for time-dependent problems in optomechanics and electromechanics

“…Under the RWA, where the time-dependent parts in Eq. (27) are ignored, we are left with just the Brillouin zone n = 0, i.e, the matrix A (0) . To evaluate the mechanical squeezing we need the submatrix describing the mechanical covariances.…”

“…A crucial obstacle for a more widespread application of these techniques is the explicit time dependence of the driving electromagnetic fields. Dissipative preparation of mechanical states [4,[9][10][11][12][13][18][19][20][21][22][23][24][25] and tomographic backactionevading measurements of mechanical motion [26][27][28][29][30][31][32][33][34] rely on driving the system with multiple fields at different frequencies while parametric squeezing requires modulation of the optical spring [5,[35][36][37]; both of these approaches result in time-dependent optomechanical Hamiltonians. The steady-state Lyapunov equation can then only be applied under the rotating wave approximation (RWA) which neglects fast oscillating terms in the interaction and only keeps those that are resonant.…”