Quantum Optics V Density Operator and Optical Bloch equations
fengxiaot Lv4

Density Operator is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, without describing their combined state.

Density Operator

Pure states and Mixed states

Pure state A pure state is a quantum state that could be represented by a ket in state space.

Mixed state A mixed state is a physical state not coherently but statistically composed of many pure states.

If a group of pure states are denoted by ψi\ket{\psi_i}, with probability pip_i , the mixed state it represents is then written as {pi,ψi}\left\{p_i,\ket{\psi_i}\right\}.

Density Operator

Density operator The density operator of a system in the mixed state {pi,ψi}\left\{p_i,\ket{\psi_i}\right\} is defined as

ρ=ipiψiψi\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i |

If we define the density operator of a pure state system ρi=ψiψi\rho_i = |\psi_i\rangle\langle\psi_i |, then ρ=piρi\rho = \sum p_i \rho_i.

Density Matrix If the state space has a pre-selected basis ui\ket{u_i}, the density matrix element is

ρmn=uiρuj=ipiumψiψiun\rho_{mn} = \langle u_i |\rho | u_j \rangle = \sum_i p_i \braket{u_m | \psi_i}\braket{\psi_i | u_n}

Corollary: The trace of density operator is 1. Trρ=1\operatorname{Tr} \rho = 1 .

Expectation Value The expectation value of an observation operator OO is Tr{ρO}\operatorname{Tr} \{\rho \,O\}.

O(t)=Tr{ρ(t)O(t)}\braket{O} (t) = \operatorname{Tr} \{\rho (t) O(t)\}

Hermite The density matrix is a Hermite, positive semidefinite matrix.

ρ=PΛP    (ρ)ρ=ρ=PΛP\sqrt{\rho} = P^{\dagger} \sqrt{\Lambda} P \implies ( \sqrt{\rho} )^\dagger\sqrt{\rho}=\rho=P^\dagger \Lambda P


Schrodinger Picture

Since pure states ψi\ket{\psi_i} are independent from each other and evolve respectively

{iddtψi(t)=H(t)ψi(t)ψi(t0)=ψi\begin{cases} \mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t} \ket{\psi_i (t)} = H(t) \ket{\psi_i (t)} \\ \ket{\psi_i (t_0)} = \ket{\psi_i} \end{cases}

and at time tt the density operator is ρ(t)=ipiρi(t)=ipiψi(t)ψi(t)\rho(t) = \sum_i p_i \rho_i (t) = \sum_i p_i |\psi_i(t) \rangle\langle\psi_i (t)|, thus we can derive

iddtρ(t)=[H(t),ρ(t)]\mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t} \rho(t) = [H(t),\rho(t)]

and dρSdt=ρSt\frac{\mathrm{d} \rho^S}{\mathrm{d} t} = \frac{\partial \rho^S}{\partial t}.


If, in addition, the system undergoes at random instants a succession of brief, weak collisions with other systems and decays due to the environment, their average effect can be represented by the addition to iddtρ(t)=[H(t),ρ(t)]\mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t} \rho(t) = [H(t),\rho(t)] of a relaxation operator.

(ddtρii)rel=jiΓijρii+jiΓjiρjj\left(\frac{\mathrm{d}}{\mathrm{d} t} \rho_{i i}\right)_\text{rel}=-\sum_{j \neq i} \Gamma_{i \to j} \,\rho_{i i}+\sum_{j \neq i} \Gamma_{j \to i} \,\rho_{jj}

(ddtρij)rel=γijρij\left(\frac{\mathrm{d}}{\mathrm{d} t} \rho_{i j}\right)_\text{rel}=-\gamma_{i j} \rho_{i j}

in which the off-diagonal elements of the density matrix ρij\rho_{ij} termed coherences since they depend on the relative phases of the i\ket{i} and j\ket{j} components of the system wavefunction.

If we only consider the spontaneous emission, the relaxation rates are

Γeg=Γge=Γγij=Γ2+γc\Gamma_{e\to g} = -\Gamma_{g \to e} =\Gamma\qquad\gamma_{ij}=\frac{\Gamma}{2}+\gamma_c

which could be proved in quantum theory of damping.