Diffusive Trajectories

A diffusive trajectory is a conditioned quantum evolution driven by a continuous noisy measurement record. Instead of isolated detector clicks, the observer sees a fluctuating current, voltage, or quadrature record. The conditional state changes by small stochastic increments, and averaging over all records recovers the same Lindblad master equation that describes the unobserved ensemble.

The standard examples are homodyne detection, heterodyne detection, dispersive qubit readout, weak continuous position measurement, and quantum state diffusion. In each case the record is noisy because the measurement extracts only infinitesimal information in each time interval.

For the conceptual entry point to continuous records and conditional states, see Continuous Monitoring. For the general stochastic-equation framework, see Stochastic Master Equations. For event-conditioned monitoring, see Quantum Jump Trajectories.

What Makes the Trajectory Diffusive

The word diffusive means that the record is modeled by Wiener increments rather than Poisson counting increments. For a Wiener process,

$$\mathbb E[dW_t]=0, \qquad dW_t^2=dt.$$

The stochastic increment is of order $\sqrt{dt}$, not of order $dt$. This is why the Itô rule $dW_t^2=dt$ cannot be discarded.

A diffusive measurement record has the schematic form

$$dY_t = \text{conditional signal}\,dt + dW_t.$$

The noisy part $dW_t$ is not an optional imperfection. It is the shot noise, vacuum noise, or measurement imprecision that remains when the observer extracts information continuously rather than projectively.

Innovations

The state update is driven by the innovation: the part of the observed record that was not already predicted by the conditional state. If

$$dY_t = \mu_t\,dt + dW_t,$$

then

$$dW_t = dY_t-\mu_t\,dt.$$

Here $\mu_t$ is the conditional expectation of the measurement signal. The innovation has zero conditional mean:

$$\mathbb E[dW_t\mid\rho_c(t)]=0.$$

This is the continuous-time version of Bayesian updating. The innovation term describes the correction due to the newly observed fluctuation. The state also changes through its deterministic Hamiltonian and dissipative drift, even when that innovation is zero.

Homodyne SME

Consider one monitored output channel with collapse operator $c$ and detection efficiency $0\le\eta\le1$. A common normalized homodyne stochastic master equation is

$$d\rho_c = \mathcal L\rho_c\,dt + \sqrt{\eta}\, \mathcal H[c]\rho_c\,dW_t,$$

where

$$\mathcal H[c]\rho = c\rho+\rho c^\dagger - \operatorname{Tr}\!\left[(c+c^\dagger)\rho\right]\rho.$$

The corresponding measurement record is

$$dY_t = \sqrt{\eta}\, \langle c+c^\dagger\rangle_c\,dt + dW_t,$$

with

$$\langle X\rangle_c = \operatorname{Tr}(X\rho_c).$$

The unconditional generator $\mathcal L$ includes the Hamiltonian and all monitored and unmonitored Lindblad channels. If the monitored channel is also the only dissipative channel,

$$\mathcal L\rho = - \frac{i}{\hbar}[H,\rho] + \mathcal D[c]\rho,$$

where

$$\mathcal D[c]\rho = c\rho c^\dagger - \frac12\{c^\dagger c,\rho\}.$$

The stochastic term changes the conditioned state from one record to another. The deterministic term gives the ensemble drift.

Homodyne Phase

Homodyne detection measures a field quadrature selected by the phase of a local oscillator. With phase $\phi$, replace $c$ by $e^{-i\phi}c$ in the innovation term:

$$d\rho_c = \mathcal L\rho_c\,dt + \sqrt{\eta}\, \mathcal H[e^{-i\phi}c]\rho_c\,dW_t.$$

The record is

$$dY_t = \sqrt{\eta}\, \langle e^{-i\phi}c+e^{i\phi}c^\dagger\rangle_c\,dt + dW_t.$$

Changing $\phi$ changes which output quadrature is monitored. The unconditional master equation is unchanged, but the conditioned trajectory is not.

The field relation between the monitored output and the system operator is supplied by Input–Output Theory. The local-oscillator detection model is treated in Homodyne Detection.

Heterodyne Trajectories

Splitting the output and measuring orthogonal quadratures gives two independent innovations, $dW_x^2=dW_y^2=dt$ and $dW_xdW_y=0$. In the same collapse-operator convention,

$$d\rho_c={}\mathcal L\rho_c\,dt +\sqrt{\eta/2}\,\mathcal H[c]\rho_c\,dW_x +\sqrt{\eta/2}\,\mathcal H[-ic]\rho_c\,dW_y.$$

The factor $1/\sqrt2$ comes from the signal split; the unconditional generator still contains one $\mathcal D[c]$. With $d\zeta=(dW_x+i\,dW_y)/\sqrt2$, the complex record is $dZ=\sqrt\eta\langle c\rangle_cdt+d\zeta$, where $d\zeta\,d\zeta^*=dt$ and $d\zeta^2=0$. See Heterodyne Detection for the derivation and added-noise interpretation.

Continuous Qubit Readout

For a continuous measurement of $\sigma_z$, take

$$c=\sqrt{\gamma}\,\sigma_z.$$

The unconditional dissipator is pure dephasing:

$$\mathcal D[c]\rho = \gamma(\sigma_z\rho\sigma_z-\rho).$$

The diffusive SME is

$$d\rho_c = - \frac{i}{\hbar}[H,\rho_c]dt + \gamma\mathcal D[\sigma_z]\rho_c\,dt + \sqrt{\eta\gamma}\, \mathcal H[\sigma_z]\rho_c\,dW_t.$$

The record is

$$dY_t = 2\sqrt{\eta\gamma}\, \langle\sigma_z\rangle_c\,dt + dW_t.$$

The ensemble state loses coherence in the $\sigma_z$ basis. Without a competing Hamiltonian, the accumulated record gradually resolves one of the two $\sigma_z$ eigenstates.

For $H=0$, write

$$\rho_c = \frac12 \left( I+x\sigma_x+y\sigma_y+z\sigma_z \right).$$

Direct evaluation of the Itô equation gives

$$\begin{aligned} dx &= -2\gamma x\,dt - 2\sqrt{\eta\gamma}\,xz\,dW_t, \\ dy &= -2\gamma y\,dt - 2\sqrt{\eta\gamma}\,yz\,dW_t, \\ dz &= 2\sqrt{\eta\gamma}\,(1-z^2)\,dW_t. \end{aligned}$$

The $z$ coordinate has no deterministic drift in this convention, but its noise amplitude vanishes at $z=\pm1$. These endpoints are stable under the measurement update. The transverse components decay in the ensemble, while individual efficient trajectories can remain pure when all relevant information is monitored. To check this rather than infer it from the drift, put $R^2=x^2+y^2+z^2$. On the pure-state surface $R^2=1$, Itô’s rule gives

$$dR^2=-4\gamma(1-\eta)(1-z^2)\,dt.$$

The quadratic noise terms cancel the deterministic purity loss exactly when $\eta=1$. Omitting them would incorrectly predict mixing even for ideal monitoring. For $\eta<1$, a state away from a measurement eigenstate initially loses purity.

Purification and Ensemble Decoherence

Diffusive trajectories make a useful distinction visible:

ensemble state:
  average over all possible noisy records

conditional state:
  state assigned after one actual record is known

The ensemble state can decohere while the conditional state becomes purer. There is no contradiction because averaging over records erases the information that selected a particular trajectory.

Detection efficiency controls how much information is available. When $\eta=1$ and no other unmonitored channels are present, a pure initial state can remain pure along a trajectory. When $\eta\lt1$, some information escapes unobserved, and the conditional state generally becomes mixed.

Same Master Equation, Different Unraveling

A Lindblad equation does not determine a unique trajectory picture. The same dissipator $\mathcal D[c]\rho$ can be unraveled in different ways:

• counting the emitted quanta gives quantum jumps;
• homodyne monitoring gives a real diffusive trajectory;
• heterodyne monitoring gives a two-quadrature diffusive trajectory;
• ignoring the environment gives the unconditional master equation.

Thus a Lindblad operator is not automatically a literal detector click. The monitoring scheme determines the record, and the record determines the conditioned update.

This is especially important in quantum optics. Spontaneous emission can be represented by jumps when photons are counted, but by diffusion when the emitted field is mixed with a strong local oscillator and a quadrature current is recorded. The general comparison is Unravelings.

Relation to Instruments

Diffusive SMEs can be obtained as limits of weak quantum instruments. Over a short time interval $dt$, a weak measurement produces a small outcome $dy$ with a broad Gaussian distribution. The conditioned update has the form

$$\rho \longrightarrow \frac{\mathcal I_{dy}(\rho)} {\operatorname{Tr}\mathcal I_{dy}(\rho)}.$$

Expanding the outcome-dependent completely positive map to the correct order in $dt$ and $dW_t$ yields the stochastic master equation. The trace of the unnormalized update gives the likelihood density for the observed infinitesimal record.

This perspective connects diffusive trajectories to Quantum Instruments and Measurement Backaction rather than adding a separate measurement rule.

Numerical Use

Diffusive equations are usually simulated by Euler–Maruyama or by higher-order stochastic integrators. A basic time step uses

$$dW_t = \sqrt{dt}\,\xi, \qquad \xi\sim\mathcal N(0,1).$$

Then update $\rho_c$ using the Itô SME and check:

• the trace remains one for normalized equations;
• eigenvalues stay nonnegative within numerical tolerance;
• the ensemble average converges to the Lindblad master equation;
• the generated record has the correct conditional mean and variance;
• the same random increment is used consistently in the record and the state update.

Naive large time steps can produce nonphysical negative eigenvalues even when the continuum equation is valid. A normalized Kraus/instrument step preserves positivity by construction and is often preferable. An unnormalized linear equation by itself is not a cure: an inaccurate integration of that equation can also lose positivity. Check time-step convergence and compare the ensemble average with a deterministic master-equation calculation.

Exercises

Innovation Mean

For the homodyne record

$$dY_t = \sqrt{\eta}\, \langle c+c^\dagger\rangle_c\,dt + dW_t,$$

define the innovation

$$dW_t = dY_t - \sqrt{\eta}\, \langle c+c^\dagger\rangle_c\,dt.$$

Show that the innovation has zero conditional mean.

Solution

By definition of the record model,

$$\mathbb E[dY_t\mid\rho_c] = \sqrt{\eta}\, \langle c+c^\dagger\rangle_c\,dt.$$

Therefore

$$\mathbb E[dW_t\mid\rho_c] = \mathbb E[dY_t\mid\rho_c] - \sqrt{\eta}\, \langle c+c^\dagger\rangle_c\,dt =0.$$

The innovation is precisely the unpredictable part of the observed increment.

Ensemble Average

For

$$d\rho_c = \mathcal L\rho_c\,dt + \sqrt{\eta}\mathcal H[c]\rho_c\,dW_t,$$

explain why averaging over records recovers $d\rho/dt=\mathcal L\rho$.

Solution

Take the conditional expectation at time $t$:

$$\mathbb E[dW_t\mid\rho_c(t)]=0.$$

Thus the stochastic term has no direct contribution to the mean increment at order $dt$:

$$\mathbb E[ \sqrt{\eta}\mathcal H[c]\rho_c\,dW_t ]=0.$$

Linearity of $\mathcal L$ gives

$$\frac{d}{dt}\mathbb E[\rho_c] = \mathcal L\,\mathbb E[\rho_c].$$

Identifying $\rho=\mathbb E[\rho_c]$ gives the unconditional master equation.

Why the squared noise matters

For the displayed qubit equations, evaluate $d(x^2+y^2+z^2)$ on $x^2+y^2+z^2=1$ with $\eta=1$.

Solution

The ordinary drift contributes $-4\gamma(x^2+y^2)dt$. The Itô correction contributes

$$(dx)^2+(dy)^2+(dz)^2 =4\gamma[z^2(x^2+y^2)+(1-z^2)^2]dt =4\gamma(1-z^2)dt.$$

Since $x^2+y^2=1-z^2$, the two cancel. The stochastic term is proportional to $z(1-R^2)dW$ and also vanishes on this surface. Thus an initially pure state stays pure in the ideal continuous equation.

Bloch-Coordinate Update

For $H=0$ and $c=\sqrt{\gamma}\sigma_z$, use

$$\mathcal H[\sigma_z]\rho = (1-z^2)\sigma_z-zx\sigma_x-zy\sigma_y$$

to derive the stochastic equation for $dz$.

Solution

Write

$$\rho = \frac12 \left( I+x\sigma_x+y\sigma_y+z\sigma_z \right).$$

The stochastic part of the SME is

$$d\rho_{\mathrm{stoch}} = \sqrt{\eta\gamma}\, \mathcal H[\sigma_z]\rho\,dW_t.$$

The coefficient of $\sigma_z$ in $d\rho$ is $dz/2$. The coefficient of $\sigma_z$ in the right-hand side is

$$\sqrt{\eta\gamma}\,(1-z^2)dW_t.$$

Therefore

$$dz = 2\sqrt{\eta\gamma}\,(1-z^2)dW_t.$$

Why Not Jumps?

A damped cavity has collapse operator $c=\sqrt{\kappa}a$. Explain why this does not by itself tell you whether the trajectory should jump or diffuse.

Solution

The collapse operator specifies the unconditional dissipator $\mathcal D[c]\rho$. It does not specify how the environment is measured. If photons leaking from the cavity are counted, the conditioned dynamics has quantum jumps. If the output field is mixed with a local oscillator and a quadrature current is measured, the conditioned dynamics is diffusive. If no record is kept, only the unconditional master equation remains.

References

• A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time, Springer (2009).
• V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear filtering,” Journal of Multivariate Analysis 42, 171–201 (1992).
• L. Bouten, R. van Handel, and M. R. James, “An introduction to quantum filtering,” SIAM Journal on Control and Optimization 46, 2199–2241 (2007).
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• N. Gisin and I. C. Percival, “The quantum-state diffusion model applied to open systems,” Journal of Physics A: Mathematical and General 25, 5677–5691 (1992).
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).