Continuous Measurement and Quantum Trajectories

A continuously monitored system produces a record: photon clicks, a noisy field quadrature, or a calibrated voltage trace. A quantum trajectory is the state conditioned on that record up to the present. Averaging trajectories with their correct probabilities recovers the corresponding unconditioned evolution. Different detectors can produce different trajectories for the same unconditioned master equation.

Required background. Generalized Measurements and Instruments and Quantum Instruments supply the outcome-resolved update. Markovian Master Equations supplies the ensemble dynamics, and Weak Measurements motivates repeated small updates.

A trajectory uses both events and their absence

For a short-time instrument $\mathcal I_r^{dt}$, the probability of record increment $r$ is the trace of $\mathcal I_r^{dt}(\rho_c)$. Divide by that probability to update the conditional state. Taking an appropriate continuous-time limit produces a stochastic master equation.

For ideal monitoring of a single radiative decay channel $c=\sqrt\gamma\,|g\rangle\langle e|$, a click has probability

$$\Pr(dN=1\mid\rho_c)=\gamma(\rho_c)_{ee}\,dt +O(dt^2).$$

A detected decay leaves the atom in $|g\rangle$. Between clicks, the absence of a count also changes predictions. With perfect detection, the unnormalized no-click evolution uses $H_{\rm eff}=H-i\hbar c^\dagger c/2$, and its squared norm is the no-click probability.

No-click evolution does not always move a state toward the ground state. An atom initially known to be excited stays excited conditional on no emission in this ideal undriven model; only the probability of that record decreases. A superposition instead changes its relative ground and excited amplitudes. The initial state and detector efficiency matter.

Start with Continuous Monitoring, Measurement Records, and Stochastic Master Equations, then work through Quantum Jump Trajectories.

Counting and quadrature records reveal different information

Readout	What the conditional model uses
Photon Counting	Event times and a state-dependent intensity; inefficiency and dark counts alter the likelihood.
Homodyne Detection	One noisy output-field quadrature selected by a local-oscillator phase.
Heterodyne Detection	Two noisy quadrature records, with the added vacuum noise required by their joint measurement.

Diffusive Trajectories develops the continuous-current description. In a normalized ideal homodyne convention,

$$dY_t=m_c(t)\,dt+dW_t, \qquad dW_t^2=dt,$$

where $m_c$ is the signal predicted from the conditional state. Noise in a single short interval scales as $\sqrt{dt}$, while its mean signal scales as $dt$. A raw infinitesimal current is therefore not a precise reading of an instantaneous observable. Information accumulates through a record and an appropriate filter.

Unravelings shows how different monitoring choices generate different ensembles. A mathematical stochastic decomposition need not correspond to an implemented detector, so a physical interpretation requires specifying the readout.

Filtering is causal; smoothing uses later information

Quantum Filtering recursively estimates the current state using records available so far. The innovation is the observed increment minus its predicted mean. With a correct model, it has zero conditional mean and the stated noise variance.

Bayesian Quantum Measurement relates likelihood updating to the physical instrument. A likelihood alone does not determine backaction. Retrodiction using later data generally needs additional objects, such as a forward state and a backward effect; it is not the same causal state used for real-time control.

Feedback from Measurement Records uses past and present information to choose later controls. Detector bandwidth, delay, and unobserved channels can change performance. Feedback changes the physical dynamics, so averaging its trajectories should be compared with the corresponding feedback model, not automatically with the original uncontrolled master equation.

Test a trajectory calculation in two ways

For a fixed monitoring setup without feedback, the ensemble average satisfies $\rho(t)=\mathbb E[\rho_c(t)]$. Compare it with a direct master-equation solution. Also test the records: waiting-time distributions for counting, or innovation statistics for a diffusive filter. Correct mean populations alone do not ensure that the simulated detector record is correct.

Time-step error and sampling error are separate. Refining the step cannot remove finite-trajectory fluctuations, and adding trajectories cannot fix an inconsistent integration scheme. The Computational Notebooks develops both checks.

Exercises

Missed photons. An undriven atom starts excited, its survival probability at time $t$ is $s=e^{-\gamma t}$, and each emitted photon is detected independently with efficiency $\eta$. Find the probability of no observed count and the conditional excited population.

Solution

No count means either no emission, with probability $s$, or an emission that was missed, with probability $(1-\eta)(1-s)$. Thus $P_0=1-\eta+\eta s$, and the conditional excited population is $s/P_0$. At $\eta=1$ it is one; at $\eta=0$ it is $s$, the unconditional population. Inefficient no-click conditioning generally leaves a mixture.

Signal averaging. For a constant known signal $m$ in $dY=m\,dt+dW$, find the variance of the estimator $Y_T/T$.

Solution

$Y_T=mT+W_T$, with $\operatorname{Var}W_T=T$. Therefore $\operatorname{Var}(Y_T/T)=1/T$ in this normalized record convention. Longer integration improves the estimate of a constant signal. For a changing quantum state, the filter must also account for the state dynamics and measurement backaction.

References

• Carmichael, H. J. An Open Systems Approach to Quantum Optics. Springer, 1993. Counting trajectories and optical master equations.
• Jacobs, K. Quantum Measurement Theory and its Applications. Cambridge University Press, 2014. Continuous measurement and stochastic state estimation.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Detector models, filtering, unravelings, and feedback.