To derive the master equation under a continuous measurement we first define the two measurement operators

$$M_0=\mathbb{I}-\left(R~/~2+iH\right)dt \tag 1$$ $$M_1=c\sqrt{dt}, \tag 2 $$

where $M_0$ is known as the no-measurement measurement operator and $M_1$ the measurement operator, i.e. the detection of an event and $R$ and $H$ two hermitian operators, and where $H$ would normally correspond to the Hamiltonian of the isolated quantum system without any measurement. If the state matrix after the measurement of time dt is $$\rho\left(t+dt\right)=\sum_rM_r\rho(t)M_r^\dagger, \tag 3$$

where the sum is over the possible measurements. Then when substituting Eq $(1)$ & Eq $(2)$ into Eq $(3)$, we arrive at the master equation for a continuous measurement :

$$\dot{\rho}(t)=-[H,\rho(t)]+c\rho(t) c^\dagger -\frac{1}{2}\left(c^\dagger c\rho(t)+\rho(t)c^\dagger c\right), \tag 4$$

where the dot denotes the derivative with respect to time. My question concerns the form of Eq $(1)$, in many textbooks (Quantum Measurements and Control by Wiseman and others) it is simply stated without or with minimal explanation.

  1. Why does the no-measurement measurement operator have this form?
  2. What does a no-measurement measurement mean?
  3. What does the imaginary unit in front of $H$ physically mean?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.