Fisher Information and the Quantum Cramér-Rao Sensitivity Limit of Continuous Measurements

Abstract

Precision measurements with quantum systems rely on our ability to trace the differences between experimental signals to variations in unknown physical parameters. In this Letter we derive the Fisher information and the ensuing Cramér-Rao sensitivity limit for parameter estimation by continuous measurements on an open quantum system. We illustrate our theory by application to resonance fluorescence from a laser driven two-state atom and we show that photon counting and homodyne detection records yield different sensitivity to the atomic parameters, while none of them exceed our general result.PACS numbers: 03.65. Yz, 02.50.Tt, 42.50:Dv Quantum systems find wide applications in high precision measurements, e.g., as clocks and as probes of the strength of perturbations and of inertial effects. For the situation where an initially prepared quantum system is measured after being subject to an unknown interaction, much theoretical effort has been devoted to identify which are the ideal initial states for such experiments and by which kind of measurement does one optimally distinguish among close candidate values of the quantity probed [1]. In this Letter, we derive the quantum sensitivity limit for a different situation where continuous measurements are performed on the radiation emitted over time by an open quantum system. Quantum trajectory analyses [2-4] have been applied to simulate how the interplay of random measurement outcomes and quantum measurement back action gradually filters the candidate values for system parameters [5][6][7][8][9]. While the achievement of such filters depend on the detection scheme applied, we present in this Letter a new, general theory which fundamentally limits how properties of open quantum systems, are revealed by the continuous interaction with their environment.Our analysis applies for example to conventional fluorescence detection where the radiation emitted spontaneously on a laser driven atomic two-level transition is a function of the detuning, driving field strength and the atomic decay rate which can therefore be determined, e.g., by fitting the mean fluorescence intensity at different driving frequencies to a Lorenzian frequency profile. It is easy to understand that this fit improves with the total number of detected photons, and thus with accumulation time. The fluctuations in the time dependent fluorescence signal, however, also contribute important information, since following each detection event, the atom jumps to the ground state and is subsequently excited by the laser field. Unlike the mean fluorescence intensity which is power broadened and which saturates at high laser driving power, the distribution of time intervals between photo detection events is an oscillatory function Figure 1: (Color online) A quantum system interacts with the quantized radiation field, the environment, through emission of photons. Under continuous probing until time T the emitter quantum state evolves in a stochastic manner, while in the absence of observation, the full system and ...

Dynamical phase transitions as a resource for quantum enhanced metrology

“…The same result has already been discussed in [21,22] and can be derived similarly to the discretized version (A1) by using the CMPS which describes the state of the system and the output in continuous time [14],…”

“…Our approach connects to recent work on parameter estimation with single stationary states of open quantum systems [20][21][22]. We overcome the problem of mixed states by considering the combined state of the system and output.…”

Cramér-Rao bound for time-continuous measurements in linear Gaussian quantum systems

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“…While several strategies based on time-continuous measurements and feedback have been proposed for quantum state engineering, in particular with the main goal of generating steady-state squeezing and entanglement [5,[7][8][9][10][11][12][13][14][15] or to study and exploit trajectories of superconducting qubits [16,17], less attention has been devoted to parameter estimation. Notable exceptions are the estimation of a magnetic field via a continuously monitored atomic ensemble [18], the tracking of a varying phase [19][20][21], the estimation of Hamiltonian and environmental parameters [22][23][24][25][26][27][28][29], and optimal state estimation for a cavity optomechanical system [30].…”

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