Stochastic Path Integral Analysis of the Continuously Monitored Quantum Harmonic Oscillator
arXiv:2103.06111 · doi:10.1103/PRXQuantum.3.010327
Abstract
We consider the evolution of a quantum simple harmonic oscillator in a general Gaussian state under simultaneous time-continuous weak position and momentum measurements. We deduce the stochastic evolution equations for position and momentum expectation values and the covariance matrix elements from the system's characteristic function. By generalizing the Chantasri-Dressel-Jordan (CDJ) formalism (Chantasri et al.~2013 and 2015) to this continuous variable system, we construct its stochastic Hamiltonian and action. Action extremization gives us the equations for the most-likely readout paths and quantum trajectories. For steady states of the covariance matrix elements, the analytical solutions for these most-likely paths are obtained. Using the CDJ formalism we calculate final state probability densities exactly starting from any initial state. We also demonstrate the agreement between the optimal path solutions and the averages of simulated clustered stochastic trajectories. Our results provide insights into the time dependence of the mechanical energy of the system during the measurement process, motivating their importance for quantum measurement engine/refrigerator experiments.
arXiv comment15 pages, 7 figures
References in corpus (69)
- Cavity Optomechanics
- Introduction to Quantum Noise, Measurement and Amplification
- Strong dispersive coupling of a high finesse cavity to a micromechanical membrane
- Quantum squeezing of motion in a mechanical resonator
- Remote quantum entanglement between two micromechanical oscillators
- Entangled massive mechanical oscillators
- A Straightforward Introduction to Continuous Quantum Measurement
- Feedback-control of quantum systems using continuous state-estimation
- Non-classical correlations between single photons and phonons from a mechanical oscillator
- Second Law of Thermodynamics with Discrete Quantum Feedback Control
- Measurement-based quantum control of mechanical motion
- Squeezing of quantum noise of motion in a micromechanical resonator
- Observation of Radiation Pressure Shot Noise on a Macroscopic Object
- Measurement of the quantum zero-point motion of a nanomechanical resonator
- Quantum information with Gaussian states
- Feedback cooling of a cantilever's fundamental mode below 5 mK
- Measurement and control of a mechanical oscillator at its thermal decoherence rate
- IBM Q Experience as a versatile experimental testbed for simulating open quantum systems
- Quantum feedback: theory, experiments, and applications
- Quantum nondemolition measurement of a nonclassical state of a massive object
- Observing a quantum Maxwell demon at work
- Hanbury Brown and Twiss interferometry of single phonons from an optomechanical resonator
- Optical dilution and feedback cooling of a gram-scale oscillator to 6.9 mK
- Quantum back action evading measurement of motion in a negative mass reference frame
- Mapping the optimal route between two quantum states
- Phonon counting and intensity interferometry of a nanomechanical resonator
- Dynamics of simultaneously measured non-commuting observables
- Information-to-work conversion by Maxwell's demon in a superconducting circuit-QED system
- Information gain and loss for a quantum Maxwell's demon
- Efficient Quantum Measurement Engine
- Quantum nondemolition measurement of mechanical squeezed state beyond the 3 dB limit
- Quantum back-action evading measurement of collective mechanical modes
- Quantum control of bosonic modes with superconducting circuits
- Measurement quantum mechanics and experiments on quantum Zeno effect
- Quantum limits on phase-preserving linear amplifiers
- How to project qubits faster using quantum feedback
- Charging a quantum battery with linear feedback control
- Observing and Verifying the Quantum Trajectory of a Mechanical Resonator
- Quantum measurement and the first law of thermodynamics: the energy cost of measurement is the work value of the acquired information
- Action principle for continuous quantum measurement
- Conditional and unconditional Gaussian quantum dynamics
- Continuous quantum measurement with independent detector cross-correlations
- Squeezing a thermal mechanical oscillator by stabilized parametric effect on the optical spring
- Quantum State Smoothing
- Improving broadband displacement detection with quantum correlations
- Decoherence and the theory of continuous quantum measurements
- Towards Optomechanical Quantum State Reconstruction of Mechanical Motion
- Stochastic path integral formalism for continuous quantum measurement
- Appearance and disappearance of quantum correlations in measurement-based feedback control of a mechanical oscillator
- Quantum measurement engines and their relevance for quantum interpretations
- Prediction and retrodiction with continuously monitored Gaussian states
- Qubit state monitoring by measurement of three complementary observables
- Quantum trajectories and their statistics for remotely entangled quantum bits
- Anatomy of Fluorescence: Quantum trajectory statistics from continuously measuring spontaneous emission
- Exact solution for a non-Markovian dissipative quantum dynamics
- Generation of mechanical interference fringes by multi-photon counting
- Quantum caustics in resonance fluorescence trajectories
- Continuous Quantum Monitoring of Position of Nonlinear Oscillators
- Prediction and Characterization of Multiple Extremal Paths in Continuously Monitored Qubits
- Quantum State Smoothing for Linear Gaussian Systems
- Measuring Fluorescence to Track a Quantum Emitter's State: A Theory Review
- Simultaneous weak measurement of non-commuting observables
- Chaos in Continuously Monitored Quantum Systems: An Optimal Path Approach
- Multi-time correlators in continuous measurement of qubit observables
- Smoothing of Gaussian quantum dynamics for force detection
- Optical read out and feedback cooling of a nanostring optomechanical cavity
- General criteria for quantum state smoothing with necessary and sufficient criteria for linear Gaussian quantum systems
- Correlators exceeding one in continuous measurements of superconducting qubits
- Linear Gaussian Quantum State Smoothing: Understanding the optimal unravelings for Alice to estimate Bob's state
Cited by in corpus (12)
- Detecting single gravitons with quantum sensing
- Efficiently Fuelling a Quantum Engine with Incompatible Measurements
- Pontryagin-Optimal Control of a non-Hermitian Qubit
- Thermodynamics of Quantum Measurement and the Demon's Arrow of Time
- Optimal quantum parametric feedback cooling
- Optimal Zeno Dragging for Quantum Control: A Shortcut to Zeno with Action-based Scheduling Optimization
- Continuous feedback protocols for cooling and trapping a quantum harmonic oscillator
- Simultaneous Measurements of Noncommuting Observables. Positive Transformations and Instrumental Lie Groups
- Quantum state-preparation control in noisy environment via most-likely paths
- Homodyne Measurement of a Non-Hermitian Qubit Undergoing Fluorescence
- Sequential Quantum Measurements and the Instrumental Group Algebra
- CDJ-Pontryagin Optimal Control for General Continuously Monitored Quantum Systems