Quantum Optics IV Optical Bloch Equation

The density operator provides a unified framework for describing quantum systems in both pure and mixed states, making it indispensable for analyzing open quantum systems. For a two-level system with dissipation, this description leads to the Optical Bloch Equations.

Density Operator

Pure states and Mixed states

Pure state A pure state is a quantum state that can 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 $\ket{\psi_i}$, with probability $p_i$ , the mixed state it represents is then written as $\left\{p_i,\ket{\psi_i}\right\}$.

Density Operator

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

$$\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i |$$

If we define the density operator of each pure state as $\rho_i = |\psi_i\rangle\langle\psi_i |$, then $\rho = \sum p_i \rho_i$.

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

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

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

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

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

Hermite The density matrix is a Hermite, positive semi-definite matrix.

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

Evolution

Schrodinger Picture

In the Schrödinger picture, each pure state $|\psi_i(t)\rangle$ evolves independently under the Schrödinger equation,

$$\mathrm{i}\hbar \frac{\mathrm{d}}{\mathrm{d}t} |\psi_i(t)\rangle = H(t) |\psi_i(t)\rangle$$

with initial condition $|\psi_i(t_0)\rangle = |\psi_i\rangle$. The density operator at time $t$ is $\rho(t) = \sum_i p_i |\psi_i(t)\rangle\langle\psi_i(t)|$. From this, one can derive the von Neumann equation for the density operator:

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

For a closed system in the Schrödinger picture, the total time derivative is $\frac{\mathrm{d}\rho}{\mathrm{d}t} = \frac{\partial \rho}{\partial t}$.

Relaxation

Real physical systems are often coupled to an environment, leading to decoherence and relaxation. The average effect of such weak, random interactions (e.g., spontaneous emission or collisions) can be modeled by adding phenomenological relaxation terms to the von Neumann equation.

For the populations (diagonal elements $\rho_{ii}$), these terms describe transitions between levels:

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

For the coherences (off-diagonal elements $\rho_{ij}$), they describe phase randomization and decay:

$$\left(\frac{\mathrm{d}\rho_{ij}}{\mathrm{d}t}\right)_{\text{rel}} = -\gamma_{ij} \rho_{ij}$$

For a two-level atom with states $|g\rangle$ (ground) and $|e\rangle$ (excited), considering only spontaneous emission from $|e\rangle$ to $|g\rangle$ at rate $\Gamma$ and additional collisional dephasing at rate $\gamma_c$, the rates are:

$$\Gamma_{e \to g} = \Gamma, \quad \Gamma_{g \to e} = 0, \quad \gamma_{eg} = \gamma_{ge} = \frac{\Gamma}{2} + \gamma_c$$

The factor of $\Gamma/2$ for the coherence decay is required for consistency with the full quantum mechanical treatment (e.g., the Lindblad form).

Optical Bloch Equation

Two-level System

We now consider a two-level atom with ground state $|g\rangle$ and excited state $|e\rangle$, interacting with a classical monochromatic electromagnetic field. The system Hamiltonian, within the dipole and rotating-wave approximations (RWA), is

$$H = \frac{\hbar\omega_0}{2} \sigma_z - \frac{\hbar\Omega}{2} \left( e^{\mathrm{i}\omega t} \sigma^\dagger + e^{-\mathrm{i}\omega t} \sigma \right)$$

where

• $\omega_0$ is the atomic transition frequency,
• $\Omega = (\mathbf{d} \cdot \mathbf{E})/\hbar$ is the Rabi frequency, proportional to the dipole moment $\mathbf{d}$ and the field amplitude $\mathbf{E}$,
• $\sigma = |g\rangle\langle e|$ and $\sigma^\dagger = |e\rangle\langle g|$ are the lowering and raising operators,
• $\sigma_z = |e\rangle\langle e| - |g\rangle\langle g|$ is the population difference operator.

To remove the explicit time dependence of the Hamiltonian, we transform to a frame rotating at the laser frequency $\omega$. This is done via the unitary transformation $U = \exp[-\mathrm{i}(\omega t/2) \sigma_z]$. The density operator in the interaction picture is $\rho_I = U^\dagger \rho U$, and it evolves under the effective Hamiltonian:

$$H_I = - \frac{\hbar\Omega}{2} (\sigma^\dagger + \sigma) + \frac{\hbar\Delta}{2} \sigma_z$$

where the detuning is $\Delta = \omega - \omega_0$.

In the basis $\{|e\rangle, |g\rangle\}$, the density matrix is

$$\rho = \begin{pmatrix} \rho_{ee} & \rho_{eg} \\ \rho_{ge} & \rho_{gg} \end{pmatrix}$$

with $\rho_{gg} = 1 - \rho_{ee}$.

Optical Bloch Equation

Including spontaneous emission and pure dephasing, the system’s dynamics are governed by a Lindblad master equation. The full equation is:

$$\frac{\mathrm{d}\rho}{\mathrm{d}t} = -\frac{\mathrm{i}}{\hbar} [H_I, \rho] + \mathcal{D}[\sqrt{\Gamma} \sigma] \rho + \mathcal{D}[\sqrt{2\gamma_c} (\sigma_z/2)] \rho$$

where the dissipator is $\mathcal{D}[L]\rho = L\rho L^\dagger - \frac{1}{2} L^\dagger L \rho - \frac{1}{2} \rho L^\dagger L$.

From this master equation, we derive the equations of motion for the density matrix elements, known as the Optical Bloch Equation

$$\frac{\mathrm{d}\rho_{ee}}{\mathrm{d}t} = -\Gamma \rho_{ee} - \frac{\mathrm{i}\Omega}{2} (\rho_{ge} - \rho_{eg})$$

$$\frac{\mathrm{d}\rho_{gg}}{\mathrm{d}t} = +\Gamma \rho_{ee} + \frac{\mathrm{i}\Omega}{2} (\rho_{ge} - \rho_{eg})$$

$$\frac{\mathrm{d}\rho_{eg}}{\mathrm{d}t} = -\left( \frac{\Gamma}{2} + \gamma_c + \mathrm{i}\Delta \right) \rho_{eg} - \frac{\mathrm{i}\Omega}{2} (\rho_{ee} - \rho_{gg})$$

with $\rho_{ge} = \rho_{eg}^*$.

The total decoherence rate for the optical coherence $\rho_{eg}$ is thus

$$\gamma_{eg} = \gamma_{ge} = \frac{\Gamma}{2} + \gamma_c$$