Quantum Jump Trajectories

A quantum jump trajectory is a conditioned evolution in which the observer monitors discrete events such as photon detections, tunneling events, or emitted quanta. Between detections the state evolves smoothly under a no-jump rule; when a detector clicks, the state is updated by a jump operator.

Quantum jumps are not an additional postulate beyond measurement and open-system dynamics. They are one unraveling of a Lindblad master equation: a way to represent the same unconditional evolution as an ensemble of conditioned records.

For the direct-detection record model, see Photon Counting. For the monitoring viewpoint that explains records, conditional states, and no-click information before the jump equations, see Continuous Monitoring. For the general stochastic-equation framework, see Stochastic Master Equations.

Lindblad Starting Point

First assume that all displayed channels are counted separately with unit efficiency, and that the initial state is pure when a state-vector trajectory is used. Rates are absorbed into collapse operators $L_m$, which have units of inverse square-root time. The unconditional master equation is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \sum_m \mathcal D[L_m]\rho,$$

where

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

In a jump unraveling, each monitored $L_m$ labels a possible detected event. For spontaneous emission of a two-level atom,

$$L=\sqrt{\Gamma}\,\sigma_-, \qquad \sigma_-=\lvert g\rangle\langle e\rvert.$$

The event is a detected emitted photon, and the state jumps toward the ground state.

Effective Non-Hermitian Hamiltonian

Define

$$H_{\mathrm{eff}} = H - \frac{i\hbar}{2} \sum_mL_m^\dagger L_m.$$

Between jumps, a pure state evolves according to the unnormalized rule

$$\lvert\tilde\psi(t+dt)\rangle = \left( I-\frac{i}{\hbar}H_{\mathrm{eff}}dt \right) \lvert\psi(t)\rangle.$$

The norm decreases:

$$\langle\tilde\psi(t+dt)\vert\tilde\psi(t+dt)\rangle = 1 - dt \sum_m \langle L_m^\dagger L_m\rangle_{\psi} + O(dt^2).$$

The lost norm is the probability that a jump occurs during the interval $dt$.

Jump Probabilities

The probability for a jump of type $m$ during $dt$ is

$$p_m(t) = dt\, \langle\psi(t)\vert L_m^\dagger L_m\vert\psi(t)\rangle.$$

If jump $m$ occurs, the normalized state update is

$$\lvert\psi\rangle \longrightarrow \frac{L_m\lvert\psi\rangle} {\sqrt{\langle L_m^\dagger L_m\rangle_\psi}}.$$

If no jump occurs, the state is normalized after the no-jump evolution:

$$\lvert\psi\rangle \longrightarrow \frac{ \left( I-\frac{i}{\hbar}H_{\mathrm{eff}}dt \right) \lvert\psi\rangle } { \left\| \left( I-\frac{i}{\hbar}H_{\mathrm{eff}}dt \right) \lvert\psi\rangle \right\| }.$$

This no-jump update is also measurement backaction: not seeing a detector click is information.

Density-Matrix Form

For a mixed conditioned state, the jump rate is

$$r_m(t) = \operatorname{Tr}(L_m^\dagger L_m\rho_c(t)).$$

The jump update is

$$\rho_c \longrightarrow \frac{L_m\rho_c L_m^\dagger}{r_m}.$$

The unnormalized no-jump update is

$$\tilde\rho_{\mathrm{no}} = \rho_c - \frac{i}{\hbar} \left( H_{\mathrm{eff}}\rho_c - \rho_c H_{\mathrm{eff}}^\dagger \right)dt.$$

Then normalize:

$$\rho_{\mathrm{no}} = \frac{\tilde\rho_{\mathrm{no}}} {\operatorname{Tr}\tilde\rho_{\mathrm{no}}}.$$

This displayed no-jump formula assumes every channel in $H_{\mathrm{eff}}$ is observed. If some channels $K_j$ are unobserved, add $\sum_j\mathcal D[K_j]\rho_c\,dt$ to the unnormalized update and include only observed channels in the non-Hermitian loss term. Missed events must be averaged over, not treated as events known not to have occurred.

Recovery of the Master Equation

For one small time step, average over no-jump and jump possibilities. To first order in $dt$,

$$\rho(t+dt) = \rho - \frac{i}{\hbar}[H,\rho]dt + \sum_m \left( L_m\rho L_m^\dagger - \frac12 \{L_m^\dagger L_m,\rho\} \right)dt.$$

Therefore

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \sum_m\mathcal D[L_m]\rho.$$

The jump trajectory is a selective description. The Lindblad equation is recovered by ignoring the detection record and averaging over all records.

Waiting Times

The jump rate is also called the hazard rate. For a pure state,

$$\lambda(t) = \sum_m \langle L_m^\dagger L_m\rangle_{\psi(t)}.$$

The hazard is evaluated on the normalized state conditioned on no earlier jump in the interval. If $S(t)$ is that no-jump probability, then

$$\dot S(t)=-\lambda(t)S(t),\qquad S(t)=\exp\!\left[-\int_0^t\lambda(s)\,ds\right],\qquad w(t)=\lambda(t)S(t).$$

A dark component can give $S(\infty)>0$, so $w$ need not integrate to one: some runs never jump. For ideal complete monitoring, $S$ is also the squared norm of the unnormalized no-jump state. If the hazard is constant,

$$S(t)=e^{-\lambda t},$$

and the waiting-time density is

$$w(t)=\lambda e^{-\lambda t}.$$

For driven atoms, cavities, and multilevel systems, $\lambda(t)$ depends on the conditioned state. Waiting-time distributions can reveal antibunching, shelving, coherent driving, and detector inefficiency.

Example: Spontaneous Emission

For a two-level atom with

$$L=\sqrt{\Gamma}\,\sigma_-,$$

and no driving, the excited state has jump rate

$$\lambda=\Gamma.$$

Starting from $\lvert e\rangle$, the no-jump state remains proportional to $\lvert e\rangle$ while its norm decays:

$$\|\tilde\psi(t)\|^2 = e^{-\Gamma t}.$$

The waiting-time density for the emitted photon is

$$w(t)=\Gamma e^{-\Gamma t}.$$

After the jump,

$$\lvert e\rangle \longrightarrow \lvert g\rangle.$$

With no repumping or driving, no further emissions occur.

Multiple Channels

If several detectors are monitored, each has its own collapse operator $L_m$ and counting increment $dN_m$. The probability for more than one jump in the same infinitesimal interval is $O(dt^2)$ and is neglected in the Itô limit.

The jump record identifies which channel clicked. If the detector does not resolve channels, the effective measurement operators are coarser, and the conditioned update changes.

This is why a Lindblad operator is not automatically a unique physical detector. The same unconditional master equation can have different unravelings depending on what is measured in the environment.

Detection Efficiency

Real detectors miss events. If a fraction $\eta$ of emissions is detected, one can split a collapse channel $L$ into an observed and an unobserved channel:

$$L_{\mathrm{obs}} = \sqrt{\eta}\,L$$

and

$$L_{\mathrm{miss}} = \sqrt{1-\eta}\,L$$

at the level of collapse channels. The observed channel produces jumps in the record. The unobserved channel contributes Lindblad decoherence or damping without a known event time.

When $\eta\lt1$, the conditional state generally becomes mixed because some environmental information is lost. For the initially excited, undriven atom, the exact no-detected-click probability is

$$S_{\mathrm{det}}(t)=1-\eta+\eta e^{-\Gamma t},\qquad p_e(t\mid\text{no detected click})=\frac{e^{-\Gamma t}}{S_{\mathrm{det}}(t)}.$$

The first expression includes runs whose photon was missed. It is not $e^{-\eta\Gamma t}$: after a missed emission there is no excitation left to emit again. This is a simple check on an inefficient counting filter.

Simulation Algorithm

For ideal complete monitoring, a basic Monte Carlo wave-function step for pure states is:

1. Compute all rates $p_m=dt\,\langle L_m^\dagger L_m\rangle$.
2. Draw a random number.
3. If a jump occurs, choose channel $m$ with probability proportional to $p_m$ and apply $L_m$.
4. If no jump occurs, evolve with $H_{\mathrm{eff}}$.
5. Normalize the state.
6. Repeat and average many trajectories.

The time step should satisfy

$$\sum_m p_m\ll1$$

so that the chance of multiple events in one step is negligible. The coherent propagation must also resolve the Hamiltonian time scales or use an accurate propagator. A small click probability does not guarantee a small coherent integration error. Adaptive waiting-time methods can be more efficient when rates vary slowly or jumps are rare.

Relation to Diffusive Trajectories

Jump trajectories arise when the environment is monitored by counting events. Diffusive Trajectories arise when the emitted field is mixed with a local oscillator or measured as a continuous noisy quadrature.

The unconditional master equation can be the same in both cases. What changes is the measurement performed on the environment and therefore the conditioned state update.

Exercises

Jump probability

For $L=\sqrt{\Gamma}\sigma_-$ and a state

$$\lvert\psi\rangle = \alpha\lvert g\rangle+\beta\lvert e\rangle,$$

compute the probability of a detected emission in time $dt$.

Solution

The jump probability is

$$p = dt\, \langle\psi\vert L^\dagger L\vert\psi\rangle.$$

Since

$$L^\dagger L = \Gamma\sigma_+\sigma_- = \Gamma\lvert e\rangle\langle e\rvert,$$

one gets

$$p=\Gamma|\beta|^2dt.$$

No-jump norm

Show that the no-jump norm loss equals the total jump probability to first order in $dt$.

Solution

The unnormalized no-jump state is

$$\lvert\tilde\psi'\rangle = \left( I-\frac{i}{\hbar}H_{\mathrm{eff}}dt \right) \lvert\psi\rangle.$$

Using

$$H_{\mathrm{eff}} = H-\frac{i\hbar}{2}\sum_mL_m^\dagger L_m,$$

and keeping terms through first order,

$$\langle\tilde\psi'\vert\tilde\psi'\rangle = 1 - dt \sum_m \langle L_m^\dagger L_m\rangle_\psi.$$

Thus the norm loss is

$$dt \sum_m \langle L_m^\dagger L_m\rangle_\psi,$$

which is the total jump probability.

Waiting time

A two-level atom starts in $\lvert e\rangle$ and decays with rate $\Gamma$ without driving. For unit-efficiency detection, what is the probability that no photon has been detected by time $t$?

Solution

The jump hazard is constant:

$$\lambda=\Gamma.$$

The survival probability is

$$S(t)=e^{-\Gamma t}.$$

This is the probability that no photon has been detected by time $t$.

Ensemble recovery

Why does averaging jump trajectories recover the Lindblad master equation?

Solution

In each small interval, one averages over mutually exclusive possibilities: no jump or one jump of type $m$. The jump contribution gives $L_m\rho L_m^\dagger dt$. The no-jump contribution gives the Hamiltonian commutator and the anticommutator loss terms

$$- \frac12 \{L_m^\dagger L_m,\rho\}dt.$$

Adding these terms over all channels gives

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \sum_m\mathcal D[L_m]\rho.$$

References

• H. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• J. Dalibard, Y. Castin, and K. Mølmer, “Wave-function approach to dissipative processes in quantum optics,” Physical Review Letters 68, 580–583 (1992).
• R. Dum, P. Zoller, and H. Ritsch, “Monte Carlo simulation of the atomic master equation for spontaneous emission,” Physical Review A 45, 4879–4887 (1992).
• K. Jacobs, Quantum Measurement Theory and its Applications, Cambridge University Press (2014).
• K. Mølmer, Y. Castin, and J. Dalibard, “Monte Carlo wave-function method in quantum optics,” Journal of the Optical Society of America B 10, 524–538 (1993).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).