# 4 Quantum trajectories ## 4.1 Introduction - 4.2: Simplest quantum trajectory, which involves jumps - 4.3: Photon counting measurements on the bath, correlation functions for measurement records - 4.4: Adds a coherent field (local oscillator) before detection: homodyne detection - 4.5: Heterodyne detection, more general diffusive trajectories - 4.6: Conditioned evolution of a damped driven two-level atom - 4.7: Continuous measurement in the Heisenberg picture - 4.8: Imperfect detection, inefficiency, thermal and squeezed bath noise, dark noise and finite detector bandwidth - 4.9: Mesoscopic electronics ## 4.2 Quantum jumps ### 4.2.1 Master equations and continuous measurements Unconditioned evolution with *monitoring* \\[ \rho(t + dt) = \sum_r\mathcal{J}[\hat{M}\_r(dt)]\rho(t). \tag{4.4} \\] Measurement operators \begin{align} \hat{M} _0(dt) &= \hat{1} - (\hat{c}^\dagger\hat{c}/2 + i\hat{H})dt, \tag{4.5}\\\\ \hat{M} _1(dt) &= \sqrt{dt}\hat{c}. \tag{4.6} \end{align} ### 4.2.2 Stochastic evolution The stochastic increment \\(dN(t)\\) of photodetections up to time \\(t\\) \begin{align} dN(t)^2 &= dN(t), \tag{4.13}\\\\ \operatorname{E}[dN(t)] &= \langle \hat{M} _1^\dagger(dt)\hat{M} _1(dt) \rangle = dt \langle \psi(t)|\hat{c}^\dagger\hat{c}|\psi(t) \rangle. \tag{4.14} \end{align} The nonlinear *stocahstic Schrödinger equation* (SSE) \\[ d|\psi(t)\rangle = \left[ dN(t) \left( \frac{\hat{c}}{\sqrt{\langle \hat{c}^\dagger\hat{c}\rangle(t)}} - 1 \right) + dt \left( \frac{\langle \hat{c}^\dagger\hat{c}\rangle(t)}{2} - \frac{\hat{c}^\dagger\hat{c}}{2} - i\hat{H} \right) \right] |\psi(t)\rangle. \tag{4.19} \\] Nonlinear superoperators \begin{align} \mathcal{G}[\hat{r}]\rho &= \frac{\hat{r}\rho\hat{r}^\dagger}{\operatorname{Tr}[\hat{r}\rho\hat{r}^\dagger]} - \rho, \tag{4.23}\\\\ \mathcal{H}[\hat{r}]\rho &= \hat{r}\rho + \rho\hat{r}^\dagger - \operatorname{Tr}[\hat{r}\rho + \rho\hat{r}^\dagger]\rho. \tag{4.24} \end{align} Stocahstic evolution of a projector \\(\hat{\pi}(t) = |\psi(t)\rangle\langle\psi(t)|\\) \\[ d\hat{\pi}(t) = \left\\{ dN(t)\mathcal{G}[\hat{c}] - dt\mathcal{H} \left[i\hat{H} + \frac{1}{2}\hat{c}^\dagger\hat{c}\right] \right\\}\hat{\pi}(t). \tag{4.22} \\] ### 4.2.3 Quantum trajectories for simulations Unnormalized evolution \\[ \frac{d}{dt}|\tilde{\psi}(t)\rangle = -\left(\sum_\mu \hat{c}^\dagger _\mu\hat{c} _\mu/2 + i\hat{H}\right) |\tilde{\psi}(t)\rangle. \tag{4.34} \\] ## 4.3 Photodetection ### 4.3.2 Output correlation functions Photocurrent \\[ I(t) = dN(t)/dt. \tag{4.39} \\] *Stochastic master equation* (SME) \\[ d\rho _I(t) = \left\\{ dN(t)\mathcal{G}[\hat{c}] - dt\mathcal{H} \left[i\hat{H} + \frac{1}{2}\hat{c}^\dagger\hat{c}\right] \right\\}\rho _I(t). \tag{4.40} \\] Jump probability \\[ \operatorname{E}[dN(t)|I(s), s < t] = dt\operatorname{Tr}[\hat{c}^\dagger\hat{c}\rho _I(t)]. \tag{4.41} \\] Mean photocurrent \\[ \operatorname{E}[I(t)] = \operatorname{Tr}[\hat{c}^\dagger\hat{c}\rho(t)] = \operatorname{E}[\operatorname{Tr}[\hat{c}^\dagger\hat{c}\rho _I(t)]]. \tag{4.42} \\] Autocorrelation function \\[ F^{(2)}(t, t+\tau) = \operatorname{E}[I(t+\tau)I(t)]. \tag{4.43} \\] Assume \\(\rho(t)\\) is given. Then \\[ F^{(2)}(t, t+\tau) = \operatorname{Tr}[\hat{c}^\dagger\hat{c}e^{\mathcal{L}\tau}\hat{c}\rho(t)\hat{c}^\dagger] + \operatorname{Tr}[\hat{c}^\dagger\hat{c}\rho(t)]\delta(\tau). \tag{4.50} \\] ### 4.3.3 Coherent field input Initial state of the bath \\[ [\hat{1} + \beta d\hat{B}^\dagger]|0\rangle,\quad \langle d\hat{B}\rangle = \beta dt. \\] Jump measurement operator \\[ \hat{M} _1(dt) = (dt)^{1/2}(\hat{c} + \beta). \tag{4.53} \\] No-jump measurement operator \\[ \hat{M} _0(dt) = \hat{1} - \left[ i\hat{H} + \frac{1}{2}\hat{c}^\dagger\hat{c} + \hat{c}^\dagger\beta \right]dt. \tag{4.55} \\] SSE \\[ d|\psi _I(t)\rangle = \left[ dN(t)\left( \frac{\hat{c}+\beta}{\sqrt{\langle(\hat{c}^\dagger+\beta^*)(\hat{c}+\beta)\rangle _I(t)}} - 1 \right) + dt\left( \frac{\langle\hat{c}^\dagger\hat{c}\rangle _I(t)}{2} - \frac{\hat{c}^\dagger\hat{c}}{2} + \frac{\langle\hat{c}^\dagger\beta+\beta^ *\hat{c}\rangle _I(t)}{2} - \hat{c}^\dagger\beta - i\hat{H} \right) \right]|\psi _I(t)\rangle. \tag{4.56} \\] Unconditioned master equation \\[ \dot{\rho} = \mathcal{D}[\hat{c}]\rho - i[\hat{H}+i\beta^ *\hat{c}-i\hat{c}^\dagger\beta, \rho]. \tag{4.57} \\] In this situation, **photons in the bath can excite the system**, which corresponds to the \\(-\hat{c}^\dagger\beta dt\\) term in Eq. (4.55). The system is **driven** by the coherent field. ## 4.4 Homodyne detection ### 4.4.1 Adding a local oscillator Consider a different unraveling of the usual master equation with the vacuum bath \\[ d\rho = -idt[\hat{H}, \rho] + dt\mathcal{D}[\hat{c}]\rho. \tag{4.58} \\] Measurement operators \begin{align} \hat{M} _1(dt) &= (dt)^{1/2}(\hat{c} + \gamma), \tag{4.60}\\\\ \hat{M} _0(dt) &= \hat{1} - dt\left[i\hat{H} + \frac{1}{2}\hat{c}^\dagger\hat{c} + \hat{c}\gamma^ * + \frac{1}{2}|\gamma|^2 \right]. \tag{4.61} \end{align} This can be achieved by mixing the bath with a strong coherent field (equivalent to displacement of the bath annihilation operator) before detection. In this situation, **photons in the bath cannot excite the system**, which can be seen from the absence of the \\(\hat{c}^\dagger\\) term in Eq. (4.61). When \\(\gamma\\) is real, \\[ \operatorname{E}[dN(t)/dt|I(s), s < t] = \operatorname{Tr}[(\gamma^2 + \gamma\hat{x} + \hat{c}^\dagger\hat{c})\rho _I(t)], \tag{4.64} \\] where the two system quadratures are defined by \\[ \hat{x} = \hat{c} + \hat{c}^\dagger,\qquad \hat{y} = -i(\hat{c} - \hat{c}^\dagger). \tag{4.65} \\] SME \\[ d\rho _I(t) = \left\\{ dN(t)\mathcal{G}[\hat{c} + \gamma] + dt\mathcal{H} \left[-i\hat{H} - \frac{1}{2}\hat{c}^\dagger\hat{c} - \gamma\hat{c}\right] \right\\}\rho _I(t). \tag{4.66} \\] SSE \\[ d|\psi _I(t)\rangle = \left[ dN(t)\left( \frac{\hat{c}+\gamma}{\sqrt{\langle(\hat{c}^\dagger+\gamma)(\hat{c}+\gamma)\rangle _I(t)}} - 1 \right) + dt\left( \frac{\langle\hat{c}^\dagger\hat{c}\rangle _I(t)}{2} - \frac{\hat{c}^\dagger\hat{c}}{2} + \frac{\langle\hat{c}^\dagger\gamma+\gamma\hat{c}\rangle _I(t)}{2} - \gamma\hat{c} - i\hat{H} \right) \right]|\psi _I(t)\rangle. \tag{4.67} \\] ### 4.4.2 The continuum limit When \\(\gamma \to \infty\\), it is possible to approximate the photocurrent by a continuous function of time. The SME yields a continuous evolution: \\[ d\rho _J(t) = -i[\hat{H}, \rho _J(t)]dt + dt\mathcal{D}[\hat{c}]\rho _J(t) + dW(t)\mathcal{H}[\hat{c}]\rho _J(t), \tag{4.72} \\] where \\(dW(t)\\) is the *Wiener increment* satisfying \\[ dW(t)^2 = dt,\qquad \operatorname{E}[dW(t)] = 0. \tag{4.73, 4.74} \\] The homodyne signal after removing the constant local oscillator contribution \\[ J _\text{hom}(t) \equiv \lim _{\gamma\to\infty} \frac{\delta N(t) - \gamma^2\delta t}{\gamma\delta t} = \langle\hat{x}\rangle _J(t) + \xi(t), \tag{4.75} \\] where \\(\xi(t)=dW(t)/dt\\).
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