2025 WiSe in Universität Heidelberg · Lecture Notes by Offensive77

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Key Tricks Performed in This Topic

Statistical Operator

The statistical operator (density operator) reflects the statistical
attributes of a quantum system. Analogous to a probability density
function, it allows the definition of expectation values and
correlations.

where is the statistical operator.

For the energy eigenbasis :

• Population
  

• Coherence
  

Time evolution:

Reduced Spaces and Operators

The total Hilbert space is
.

The Hamiltonian is written as

More explicitly,

where

connects the system and the bath.

Reduced density operators are obtained via partial trace:

In general,

Schrödinger Interaction Picture

Define

The interaction picture is made up of

Usually, results obtained in the interaction picture are transformed
back to the Schrödinger picture.

Orthogonal Projectors

Introduce projectors and :

They separate the statistical operator into relevant and correlation
parts.

How We Develop the Theory

1. Density operators instead of wavefunctions.
2. von Neumann equation instead of Schrödinger equation.
3. Schrödinger picture as starting point.
4. Repeated insertion of .

Mean-Field Approximation

The von Neumann equation reads

Tracing over the bath gives

This evolution is unitary and contains no dissipation.

Nakajima–Zwanzig Equation

In the interaction picture:

The second term defines the memory kernel.

Bath as Harmonic Oscillators

Assume the bath consists of harmonic oscillators:

Linearized system–bath coupling:

Spectral Density

Define

It characterizes the absorption strength of the bath.

Redfield Equation

Population decay:

Population lifetime:

Coherence lifetime:

Usually, . That’s why the qubits in quantum computers are so subject to the perturbation from environment.

Detailed Balance

This guarantees thermal equilibrium of system in a (maybe not) long run.