11.2 Wigner function and phase-space representations

Wigner functions and phase-space representations are like quantum GPS, helping us navigate the weird world of quantum states. They show us where particles might be and how fast they're moving, all in one neat picture.

These tools are crucial for quantum state tomography, letting us peek inside quantum systems. By mapping out the Wigner function, we can reconstruct the full quantum state, revealing its quirks and non-classical features.

Wigner Function as Quasi-Probability

Definition and Properties

Top images from around the web for Definition and Properties

• 

  Wigner function = Fourier transform + Coordinate rotation – foreXiv View original

  Is this image relevant?

• 

  Wigner function for SU(1,1) – Quantum View original

  Is this image relevant?

• 

  Phase Space Path Integral Representation for Wigner Function View original

  Is this image relevant?

• 

  Wigner function = Fourier transform + Coordinate rotation – foreXiv View original

  Is this image relevant?

• 

  Wigner function for SU(1,1) – Quantum View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Properties

• 

  Wigner function = Fourier transform + Coordinate rotation – foreXiv View original

  Is this image relevant?

• 

  Wigner function for SU(1,1) – Quantum View original

  Is this image relevant?

• 

  Phase Space Path Integral Representation for Wigner Function View original

  Is this image relevant?

• 

  Wigner function = Fourier transform + Coordinate rotation – foreXiv View original

  Is this image relevant?

• 

  Wigner function for SU(1,1) – Quantum View original

  Is this image relevant?

1 of 3

• The Wigner function, W(x,p), is a quasi-probability distribution that represents the quantum state of a system in phase space, where x is the position and p is the momentum
• Defined as the Fourier transform of the density matrix in the position representation, providing a connection between the wave function and the phase-space representation
• The Wigner function is a real-valued function, but it can take on negative values, which distinguishes it from classical probability distributions and highlights the non-classical nature of quantum states
• Satisfies certain properties, such as normalization and the correspondence principle, which states that the Wigner function reduces to the classical probability distribution in the limit of large quantum numbers

Marginal Distributions and Observables

• The marginal distributions of the Wigner function, obtained by integrating over either position or momentum, yield the probability distributions for position and momentum, respectively
• Can be used to calculate expectation values of observables, such as position, momentum, and their higher moments, by integrating the product of the Wigner function and the corresponding phase-space function
• Provides a powerful tool for visualizing the evolution of quantum states under the influence of Hamiltonians or quantum operations
• The time evolution of the Wigner function can be described by the quantum Liouville equation, which is analogous to the classical Liouville equation for the evolution of phase-space distributions

Properties of the Wigner Function

Calculation and Characteristics for Various States

• The Wigner function for a pure state, described by a wave function ψ(x), can be calculated using the definition $W(x,p) = (1/πℏ) ∫ ψ*(x-y) ψ(x+y) exp(2ipy/ℏ) dy$
• For a coherent state, which is a minimum uncertainty state that closely resembles a classical state, the Wigner function is a Gaussian distribution centered at the expectation values of position and momentum
• Squeezed states, which have reduced uncertainty in one quadrature at the expense of increased uncertainty in the other, exhibit Wigner functions that are elongated or compressed along the squeezed quadrature
• Fock states, which are eigenstates of the number operator with a fixed number of photons, have Wigner functions that exhibit oscillations and negative regions, indicating their non-classical nature

Non-Classicality and Entanglement

• The Wigner function for a thermal state, which describes a system in thermal equilibrium, is a Gaussian distribution with a width determined by the temperature
• Entangled states, such as the two-mode squeezed state, have Wigner functions that exhibit correlations between the phase-space variables of the entangled modes
• The negativity of the Wigner function can be used as a measure of non-classicality, quantifying the degree to which a quantum state deviates from classical behavior
• Phase-space methods can be applied to investigate the properties of quantum states in various physical systems, such as quantum optics, quantum information processing, and quantum many-body systems

Phase-Space Representations: Wigner vs Others

Husimi Q-Function

• The Husimi Q-function, Q(α), is a positive-definite phase-space distribution obtained by convolving the Wigner function with a Gaussian function in phase space
  • Defined as $Q(α) = (1/π) ⟨α|ρ|α⟩$, where $|α⟩$ is a coherent state and $ρ$ is the density matrix
  • Provides a smoothed version of the Wigner function and is always non-negative, making it a true probability distribution
• Useful for studying the classical limit of quantum states and identifying the most probable regions in phase space

Glauber-Sudarshan P-Function

• The Glauber-Sudarshan P-function, P(α), is a phase-space distribution that expresses the density matrix as a weighted sum of coherent state projectors
  • Defined as $ρ = ∫ P(α) |α⟩⟨α| d²α$, where $ρ$ is the density matrix and $|α⟩$ is a coherent state
  • Can be highly singular and may not always exist as a well-behaved function, especially for non-classical states
• Provides a way to represent quantum states in terms of classical-like coherent states, which is particularly useful for studying the quantum-classical correspondence

Relationships and Reconstruction

• The Wigner function, Q-function, and P-function are related through convolution and Fourier transform relationships, allowing the reconstruction of one representation from another
• The Q-function can be obtained by convolving the Wigner function with a Gaussian function in phase space, while the P-function is the Fourier transform of the characteristic function of the Wigner function
• These relationships enable the study of quantum states from different perspectives and the extraction of complementary information

Visualizing Quantum States in Phase Space

Evolution and Dynamics

• Phase-space representations provide a powerful tool for visualizing the evolution of quantum states under the influence of Hamiltonians or quantum operations
• The time evolution of the Wigner function can be described by the quantum Liouville equation, which is analogous to the classical Liouville equation for the evolution of phase-space distributions
• Visualizing the dynamics of quantum states in phase space can provide insights into the interplay between quantum and classical behavior, as well as the emergence of non-classical features (interference patterns, negative regions)

Applications and Physical Systems

• Phase-space methods can be applied to investigate the properties of quantum states in various physical systems, such as quantum optics, quantum information processing, and quantum many-body systems
• In quantum optics, phase-space representations are used to study the properties of light fields, such as coherence, squeezing, and entanglement (continuous-variable systems)
• In quantum information processing, phase-space methods can be employed to characterize the performance of quantum gates, study the effects of decoherence, and develop error-correction schemes (qubits, quantum circuits)
• In quantum many-body systems, phase-space techniques can be used to investigate the quantum-classical transition, the emergence of collective phenomena, and the properties of quantum phase transitions (Bose-Einstein condensates, spin systems)