A density matrix ρ(t) yields probabilistic information about the outcome of measurements on a quantum system. We introduce here the past quantum state, which, at time T, accounts for the state of a quantum system at earlier times t

We review the introduction of likelihood functions and Fisher information in classical estimation theory, and we show how they can be defined in a very similar manner within quantum measurement theory. We show that the stochastic master equations describing the dynamics of a quantum system subject to a definite set of measurements provides likelihood functions for unknown parameters in the system dynamics, and we show that the estimation error, given by the Fisher information, can be identified by stochastic master equation simulations. For large parameter spaces we describe and illustrate the efficient use of Markov Chain Monte Carlo sampling of the likelihood function.

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 ...

We investigate the thermodynamics of a combined Dicke-and Isingmodel which exhibits a rich phenomenology arising from the second order and quantum phase transitions from the respective models. The partition function is calculated using mean field theory, and the free energy is analyzed in detail to determine the complete phase diagram for the system. The analysis reveals both first-and second-order Dicke phase transitions into a super-radiant state, and the cavity mean-field in this regime acts as an effective magnetic field, which restricts the Ising chain dynamics to parameter ranges away from the Ising phase transition. Physical systems with a first order phase transitions are natural candidates for metrology and calibration purposes, and we apply filter theory to show that the sensitivity of the physical system to temperature and external fields reaches the 1/N Heisenberg limit.

We analyze the quantum jumps of an atom interacting with a cavity field. The strong atomfield interaction makes the cavity transmission depend on the time dependent atomic state, and we present a Hidden Markov Model description of the atomic state dynamics which is conditioned in a Bayesian manner on the detected signal. We suggest that small variations in the observed signal may be due to spatial motion of the atom within the cavity, and we represent the atomic system by a number of hidden states to account for both the small variations and the internal state jump dynamics. In our theory, the atomic state is determined in a Bayesian manner from the measurement data, and we present an iterative protocol, which determines both the atomic state and the model parameters. As a new element in the treatment of observed quantum systems, we employ a Bayesian approach that conditions the atomic state at time t on the data acquired both before and after t and we show that the state assignment by this approach is more decisive than the usual conditional quantum states, based on only earlier measurement data. arXiv:1312.5827v2 [quant-ph] 2 May 20142