Intro - Quantum optics

sub-discipline in atoms, molecular, and optics physics

Quantum optics focuses on the simplest quantum objects, usually light and few-level atoms, where quantum mechanics appears in its purest form without the complications of more complex systems.

Quantum electrodynamics in meida has been the backbone of quantum optics for the last 50 years.
the theory of irreversible 不可逆转 quantum processes that has been highly developed in the theory of the laser or in the models of the quantum measurement process
{cosmology} quantum fluctuations of light→quantum nonlocality→quantum cryptography; {astrophysics} laboratory analogues of the event horizon.

Axioms

Superposition Principle
eigenstates $|\alpha \rangle$, measurement value $a\in \mathbb R$ with probabilities $p_a$ [Born’s probability interpreation]
- $|\Psi\rangle=\sum_a\langle a|\Psi\rangle|a\rangle.$ We can expand the state vector before the measurement in terms of eigenstates.
- $\langle a|\Psi\rangle\in \mathbb C\to p_a\in \mathbb R$. The simplest possible expression for the transition probability is the ratio $p_a=\frac{\left|\langle a|\psi\rangle\right|^2}{\langle\psi|\psi\rangle}.$

Reproduce the basic formalism of quantum mechanics

consider only normalized states: $p_a=\left|\langle a|\psi\rangle\right|^2=\langle\psi|a\rangle\langle a|\psi\rangle,\langle\psi|\psi\rangle=1.$
the eigenstate produce the measurement result $a$ with certainty[determined state]→they must be orthonormal $\langle a|a'\rangle=\delta_{a'a}$
the system of eigenvectors must be complete.$\sum_a|a\rangle\langle a|=I=\hat 1$
- the average of the measurement values $\langle A\rangle=\sum_a ap_a=a\langle\psi|a\rangle\langle a|\psi\rangle=\langle\psi|(a|a\rangle\langle a|)|\psi\rangle$, where we have introduced the operator $\hat A=a|a\rangle\langle a|$, with eigenvalues $a\in \mathbb R$, eigenvectors$|a\rangle$
- The operator $\hat A$ is Hermitian: $\hat A^\dagger=\hat A$

Suppose that after the measurement of the observable quantity $\hat A$, the object is in the eigenstate $|a\rangle=\hat A|\psi\rangle.$When we subsequently perform another measurement of the observable $\hat B$

$$ b=\langle b|a\rangle,|b\rangle=\hat B|\psi_1\rangle=\hat B\hat A|\psi_0\rangle.\\ $$

The measurement result $b$ is statistically uncertain if the overlap $p(b|a)=|\langle b|a\rangle|^2=|\langle a|b\rangle|^2<1$
- In this case, the two operator do not commute $[\hat A,\hat B]\equiv\hat A\hat B-\hat B\hat A\ne 0$
  
  Otherwise, they would share the same system of eigenstates $|\psi_0\rangle\to |\psi_1\rangle\to p_b=1.$
  
  $\hat C=[\hat A,\hat B]$ is an anti-hermitian. $\hat C^\dagger=-\hat C${the operator of the observable is hermitian.}
- Incompatible observables correspond to non-commuting operators, cause mutual statistical uncertainty which can be quantified in uncertainty relations.

Concerning the composition of physical objects.$\text{first: }(a_1,a_2)\text{,repeated: }(a_1,a_2) \text{ or }(a_1),(a_2)$

It is natural to assume that independent measurements correspond to factorized eigenstates $|a_1,a_2\rangle=|a_1\rangle\otimes |a_2\rangle$
However, the superposition of two different states will not factorize in general.

Quantum statistic

We have an ensemble of pure states woth probabilities $\rho_n$

$$ \begin{aligned}\left\langle A\right\rangle & =\sum_{n}\rho_{n}\langle\psi_{n}|\hat{A}|\psi_{n}\rangle \\&=\sum_{a}\sum_{n}\rho_{n}\langle\psi_{n}|\hat{A}|a\rangle\langle a|\psi_{n}\rangle \\&=\sum_{a}\langle a|\sum_{n}\rho_{n}|\psi_{n}\rangle\langle\psi_{n}|\hat{A}|a\rangle.\end{aligned} $$

Introducing the density operator(state operator, density matrix)