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{{Short description|Quantum noise channel describing loss of phase coherence}}
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== Overview ==
'''Phase damping''', also called '''pure dephasing''', is a model of [[Physics:Quantum decoherence|quantum decoherence]] in which the relative phase between components of a quantum state becomes uncertain. In a qubit, this means that a superposition such as <math>\alpha |0\rangle+\beta |1\rangle</math> gradually loses the off-diagonal terms of its [[Density matrix|density matrix]], while the probabilities <math>|\alpha|^2</math> and <math>|\beta|^2</math> are left unchanged.

The channel is useful because it separates loss of coherence from loss of energy. In [[Physics:Quantum Amplitude damping|amplitude damping]], an excited state can relax into a lower-energy state. In phase damping, by contrast, the energy populations are preserved, but the phase relation needed for interference is degraded. This makes it a basic example in [[Physics:Quantum Open systems|open quantum systems]], [[Physics:Quantum channel|quantum channels]], and [[Quantum information|quantum information]].

== Key ideas ==
Phase damping describes a situation where an environment gains information about the phase, path, frequency, or energy splitting of a quantum system without necessarily causing a transition between energy levels. The system can therefore look unchanged if only energy is measured, while interference experiments reveal that coherence has been lost.

For a single qubit written in the basis <math>\{|0\rangle,|1\rangle\}</math>, a general density matrix has the form

<math display="block">
\rho =
\begin{pmatrix}
\rho_{00} & \rho_{01}\\
\rho_{10} & \rho_{11}
\end{pmatrix}.
</math>

A simple phase-damping process leaves the diagonal entries fixed and suppresses the off-diagonal entries:

<math display="block">
\rho \longmapsto
\begin{pmatrix}
\rho_{00} & \lambda \rho_{01}\\
\lambda \rho_{10} & \rho_{11}
\end{pmatrix},
\qquad 0\leq \lambda \leq 1.
</math>

Here <math>\lambda</math> is a coherence factor. When <math>\lambda=1</math>, no dephasing has occurred. When <math>\lambda=0</math>, all phase coherence in the chosen basis has been removed.

== Channel description ==
As a [[Physics:Quantum Kraus operators|Kraus-operator]] channel, phase damping can be represented in several equivalent ways. One common qubit form is

<math display="block">
\mathcal{E}(\rho) = (1-p)\rho + p Z\rho Z,
</math>

where <math>Z</math> is the Pauli phase operator and <math>p</math> is a phase-flip probability. In matrix form this channel preserves the diagonal entries and multiplies the off-diagonal entries by <math>1-2p</math>. A related parametrization uses a non-negative decay rate <math>\Gamma_\phi</math>:

<math display="block">
\rho_{01}(t)=e^{-\Gamma_\phi t}\rho_{01}(0),
\qquad
\rho_{00}(t)=\rho_{00}(0),\quad
\rho_{11}(t)=\rho_{11}(0).
</math>

This exponential form appears in many Markovian models of dephasing. More general environments can produce non-exponential decay, revivals, or correlated noise, especially when the environment has memory or when several qubits are affected by the same fluctuating field.

== Bloch-sphere picture ==
On the [[Bloch sphere]], a qubit state is represented by a vector with components <math>(x,y,z)</math>. Phase damping in the computational basis leaves the <math>z</math>-component unchanged and shrinks the transverse components:

<math display="block">
(x,y,z) \longmapsto (\lambda x,\lambda y,z).
</math>

Thus states near the north or south pole are relatively unaffected, while superpositions around the equator move inward toward the vertical axis. This picture makes the physical meaning clear: phase information is carried by the transverse components, and phase damping erases that information without driving the system upward or downward in energy.

== Physical examples ==
Pure dephasing can arise when a qubit's transition frequency fluctuates because of uncontrolled external fields, material defects, charge noise, magnetic noise, or thermal motion in the surrounding apparatus. In atomic and optical systems it may appear as random phase diffusion. In solid-state systems it is often associated with slow fluctuations in the local electromagnetic environment.

In quantum-computing hardware, phase damping is one of the processes measured by a coherence time commonly denoted <math>T_2</math>. Energy relaxation is associated with <math>T_1</math>, while pure dephasing contributes additional loss of phase coherence. A common relation is

<math display="block">
\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\phi},
</math>

where <math>T_\phi</math> is the pure-dephasing time. The exact interpretation depends on the physical platform and on the noise model used in the experiment.

== Relation to decoherence ==
Phase damping is one of the simplest mathematical models of decoherence. It explains how a quantum system can become effectively classical in a particular basis: the probabilities remain, but the interference terms that distinguish a coherent superposition from a classical mixture disappear.

This process is also central to the distinction between population dynamics and coherence dynamics. Master-equation descriptions, including the [[Physics:Quantum Lindblad equation|Lindblad equation]], often contain separate terms for relaxation and dephasing. Keeping these effects distinct is important when designing error models, pulse sequences, and [[Physics:Quantum error correction|quantum error correction]] strategies.

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

== References ==
{{reflist|3}}

* {{Cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010 |edition=10th anniversary |isbn=978-1-107-00217-3}}
* {{Cite book |last1=Breuer |first1=Heinz-Peter |last2=Petruccione |first2=Francesco |title=The Theory of Open Quantum Systems |publisher=Oxford University Press |year=2002 |isbn=978-0-19-852063-4}}
* {{Cite book |last1=Carmichael |first1=Howard |title=An Open Systems Approach to Quantum Optics |publisher=Springer |year=1993 |isbn=978-3-540-56634-2}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum Phase damping|1}}

Physics:Quantum Phase damping - Revision history

Maintenance script: Add Quantum Phase damping page