The Density Matrix Formalism

Author

Daniel Fischer

Introduction

Up to this point, we have assumed that the two–level system is always in a pure quantum state that can be written as a coherent superposition of the bare states. This is expressed as

\[ \left\lvert \psi \right\rangle= c_e \left\lvert e \right\rangle+ c_g \left\lvert g \right\rangle= \lvert c_e\rvert e^{-i\varphi_e}\left\lvert e \right\rangle+ \lvert c_g\rvert e^{-i\varphi_g}\left\lvert g \right\rangle. \tag{1}\]

In such a description the complex amplitudes \(c_e\) and \(c_g\) are assumed to have well-defined magnitudes and phases at all times. The phase difference between the two components determines, for example, the ability of the system to undergo coherent population transfer via Rabi oscillations.

However, real physical systems are rarely perfectly isolated. Effects such as spontaneous decay, collisions, fluctuations of external fields, or more generally any coupling to unobserved degrees of freedom (the “environment”) lead to a loss of phase information, commonly referred to as decoherence. After sufficiently long times, the phases \(\varphi_e\) and \(\varphi_g\) become unknown or fluctuate so strongly that the system can no longer be represented by a single coherent wavefunction. Instead, it must be regarded as being in a statistical mixture of ground and excited states:

The wavefunction formalism alone is not capable of simultaneously describing (i) coherent dynamics (Rabi oscillations) and (ii) incoherent processes (population decay, decoherence).

To handle both aspects within one unified framework, we turn to the density matrix formalism.

By the end of this chapter, you should be able to:

Understand why the wavefunction description is insufficient for open quantum systems.
Define and interpret the density operator and the density matrix.
Distinguish clearly between pure and mixed states and understand how each appears in the density matrix.
Compute expectation values using the trace rule,
\[\left\langle \hat A \right\rangle= \operatorname{Tr}(\hat\rho \hat A).\]
Derive the equation of motion for the density matrix,
\[\dot{\hat\rho} = \frac{1}{i\hbar}[\hat H,\hat \rho],\]
known as the Liouville–von Neumann equation.
Prepare for the introduction of dissipative terms and the optical Bloch equations in later sections.

1. The density matrix formalism

A powerful and conceptually elegant solution to the limitations of the wavefunction description is the density matrix (or density operator) formalism. Instead of describing a state by a vector in Hilbert space, we represent the state by an operator:

\[ \hat{\rho} = \left\lvert \psi \right\rangle\!\!\left\langle \psi \right\rvert. \]

This object contains full information about:

Populations of the basis states (diagonal elements).
Coherences, i.e., the phase relations between basis states (off-diagonal elements).

For the present two–level system we can write, using the operator matrix elements defined earlier (see Equation 1 in Two–Level Systems — The Theoretical Framework):

\[ \hat{\rho} = \begin{pmatrix} \rho_{ee} & \rho_{eg} \\ \rho_{ge} & \rho_{gg} \end{pmatrix} = \begin{pmatrix} \lvert c_e\rvert^2 & c_e c_g^\ast \\ c_e^\ast c_g & \lvert c_g\rvert^2 \end{pmatrix}. \tag{2}\]

At this stage the density matrix is simply a rewriting of our familiar wavefunction. Its power becomes clear when we consider states that cannot be represented by a single pure state \(\left\lvert \psi \right\rangle\).

1.1 Pure states

A pure state is one that can be written as a coherent superposition such as Equation 1. In this case, the density matrix is constructed directly from the state vector, so its elements have a strict and unambiguous interpretation:

The diagonal elements are the populations of the excited and ground states in that specific state vector:

\[ \rho_{ee} = \lvert c_e\rvert^2,\qquad \rho_{gg} = \lvert c_g\rvert^2,\qquad \rho_{ee} + \rho_{gg} = 1. \]

These numbers come from the amplitudes \(c_e\) and \(c_g\) and therefore describe a single quantum state, not an ensemble.
The off-diagonal elements encode the phase relation between the amplitudes and therefore quantify coherence:

\[ \rho_{eg} = c_e c_g^\ast = \lvert c_e\rvert \lvert c_g\rvert \, e^{-i(\varphi_e - \varphi_g)}. \]

These off-diagonal terms vanish only when the system is in an eigenstate (\(\left\lvert e \right\rangle\) or \(\left\lvert g \right\rangle\)) and are otherwise nonzero. As long as \(\rho_{eg}\neq 0\), the system retains well-defined phase information and can undergo coherent dynamics, such as Rabi oscillations.

1.2 Mixed states

A mixed state represents an ensemble of systems or repeated preparations in which the state is not given by a single wavefunction. The density matrix still has the same mathematical form, but its entries have a different physical meaning:

The diagonal elements now represent ensemble probabilities:

\[ \rho_{ee} = p_e,\qquad \rho_{gg} = p_g,\qquad p_e + p_g = 1, \]

where \(p_e\) and \(p_g\) are the classical probabilities of finding the system in \(\ket e\) or \(\ket g\). These are not \(\lvert c_e\rvert^2\) and \(\lvert c_g\rvert^2\) because there is no single underlying state vector in a mixed state.
The off-diagonal elements, which in a pure state carry coherent phase information, now vanish:

\[ \langle \rho_{eg} \rangle = \big\langle \lvert c_e\rvert \lvert c_g\rvert e^{-i(\varphi_e - \varphi_g)} \big\rangle = 0. \]

The reason is that the relative phase \(\varphi_{eg}\) varies randomly across realizations of the ensemble and therefore averages out. Thus, a mixed state can still have a definite population imbalance (\(p_e\neq p_g\)), but it carries no coherence.

1.3 Experimental distinction between pure and mixed states

How can one tell experimentally whether a state is pure or mixed?

Pure states can undergo coherent evolution.
If the system is pure (\(\rho_{eg}\neq 0\)), applying a resonant coherent field produces Rabi oscillations. By choosing the pulse area appropriately one can rotate the state vector to any point on the Bloch sphere, including a full transfer to \(\ket g\) or \(\ket e\).
Mixed states cannot support coherent dynamics.
When driven with the same coherent field, a mixed state (with \(\rho_{eg}=0\)) shows no oscillations. The populations evolve only incoherently (e.g., due to optical pumping or spontaneous decay), and the system remains a probabilistic mixture rather than a coherent superposition.

This operational difference—coherent vs. incoherent response to a resonant field—provides a direct experimental signature distinguishing pure from mixed quantum states.

1.4 Partially coherent states

In real systems the loss of coherence is usually gradual rather than sudden. A state with

\[ 0 < \lvert\rho_{eg}\rvert < \sqrt{\rho_{ee}\rho_{gg}} \]

is neither fully coherent nor fully incoherent; it retains some phase information but not enough to behave like a pure state. Such states appear naturally in open quantum systems and their dynamics exhibit damped Rabi oscillations whose decay rate reflects the loss of coherence.

2. Expectation values from the density matrix

One of the most useful features of the density matrix is that it provides a basis-independent formula for expectation values.

Starting from

\[ \hat\rho = \left\lvert \psi \right\rangle\!\!\left\langle \psi \right\rvert, \]

consider the expectation value of a linear operator \(\hat{A}\):

\[ \begin{aligned} \langle \hat{A} \rangle &= \langle \psi | \hat{A} \mathbb{1} | \psi \rangle \\ &= \langle \psi | \hat{A}(\left\lvert e \right\rangle\!\!\left\langle e \right\rvert+ \left\lvert g \right\rangle\!\!\left\langle g \right\rvert) | \psi \rangle \\ &= \langle \psi | \hat{A} | e \rangle \langle e | \psi \rangle + \langle \psi | \hat{A} | g \rangle \langle g | \psi \rangle \\ &= \langle e | \left( | \psi \rangle \langle \psi | \hat{A} \right) | e \rangle + \langle g | \left( | \psi \rangle \langle \psi | \hat{A} \right) | g \rangle \\ &= \operatorname{Tr}(\hat{\rho}\hat{A}), \end{aligned} \]

where Tr denotes the trace (sum of diagonal elements). This trace rule remains valid whether the state is pure or mixed.

3. Time evolution: the Liouville–von Neumann equation

To derive the equation of motion for the density matrix, we start from the Schrödinger equation,

\[ i\hbar \frac{\partial}{\partial t}\left\lvert \psi \right\rangle= \hat{H}\left\lvert \psi \right\rangle, \qquad -i\hbar \frac{\partial}{\partial t}\left\langle \psi \right\rvert= \left\langle \psi \right\rvert\hat{H}. \]

Taking the time derivative of \(\hat\rho = \left\lvert \psi \right\rangle\!\!\left\langle \psi \right\rvert\) gives:

\[ \frac{\partial}{\partial t}\hat{\rho} = \frac{\partial}{\partial t} \left( \left\lvert \psi \right\rangle\!\!\left\langle \psi \right\rvert\right) = \frac{1}{i\hbar} \bigl( \hat{H}\left\lvert \psi \right\rangle\!\!\left\langle \psi \right\rvert- \left\lvert \psi \right\rangle\!\!\left\langle \psi \right\rvert\hat{H} \bigr) = \frac{1}{i\hbar} [\hat{H}, \hat{\rho}]. \tag{3}\]

This is the Liouville–von Neumann equation, the analogue of the Schrödinger equation in the density-matrix formalism. It describes unitary, fully coherent time evolution. In later chapters we will extend it by including additional terms that model spontaneous decay and dephasing, leading to the optical Bloch equations.

Key Takeaways

The wavefunction formalism cannot describe decoherence or statistical mixtures.
The density matrix captures both populations and coherences in a unified framework.
Pure states have nonzero off-diagonal elements; mixed states do not.
Expectation values of observables are computed via
\[\left\langle \hat A \right\rangle= \operatorname{Tr}(\hat\rho\,\hat A).\]
The density matrix obeys the Liouville–von Neumann equation
\[\dot{\hat\rho} = \frac{1}{i\hbar}[\hat H,\hat\rho].\]
This framework is essential for describing open-system dynamics and for understanding real optical systems with decay, dephasing, and noise.