.. _sphx_glr_api_gallery_optics_quantum_optics:

Quantum optics
--------------

Light quantized in the truncated Fock basis: Dirac's creation/annihilation
ladder operators and vacuum fluctuations, Wigner's phase-space
quasi-probability distribution and the negativity of a single-photon Fock
state, Glauber's coherent states and their Poissonian photon statistics,
the Jaynes-Cummings model's vacuum Rabi oscillation and collapse-and-
revival, Kimble/Dagenais/Mandel's photon antibunching via the Fano factor,
Hanbury Brown and Twiss's intensity-correlation bunching, and Slusher's
observation of squeezed, sub-vacuum quadrature noise.


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    <div class="sphx-glr-thumbcontainer" tooltip="Roy Glauber identified the coherent states |\alpha\rangle -- eigenstates of the annihilation operator, \hat a|\alpha\rangle = \alpha|\alpha\rangle -- as the quantum states that most closely reproduce a classical, stable-amplitude light wave: a positive Gaussian in phase space, displaced away from the origin, whose photon number follows a Poissonian distribution -- the quantum-mechanical signature of what an ideal, shot-noise-limited laser beam actually is. coherent_state constructs this state directly in a truncated Fock basis (dimension cutoff),">

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  .. image:: /api/gallery/optics/quantum_optics/images/thumb/sphx_glr_plot_coherent_state_statistics_thumb.png
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  :doc:`/api/gallery/optics/quantum_optics/plot_coherent_state_statistics`

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      <div class="sphx-glr-thumbnail-title">Glauber's coherent states: displaced vacuum, Poissonian statistics</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Hanbury Brown and Twiss measured intensity correlations between two separated detectors and found bunching in thermal starlight -- the second-order correlation function">

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  .. image:: /api/gallery/optics/quantum_optics/images/thumb/sphx_glr_plot_hanbury_brown_twiss_thumb.png
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  :doc:`/api/gallery/optics/quantum_optics/plot_hanbury_brown_twiss`

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      <div class="sphx-glr-thumbnail-title">The Hanbury Brown-Twiss effect: bunching, coherent, and antibunched light</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Edwin Jaynes and Fred Cummings introduced the minimal fully quantum model of light-matter interaction: a single two-level atom coupled to a single quantized cavity mode. In the rotating-wave approximation, the Hamiltonian JaynesCummingsModel builds is">

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  .. image:: /api/gallery/optics/quantum_optics/images/thumb/sphx_glr_plot_jaynes_cummings_rabi_thumb.png
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  :doc:`/api/gallery/optics/quantum_optics/plot_jaynes_cummings_rabi`

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      <div class="sphx-glr-thumbnail-title">The Jaynes-Cummings model: vacuum Rabi oscillation and collapse-and-revival</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Dirac gave the first fully quantum treatment of radiation by promoting each field mode to a quantum harmonic oscillator, with non-commuting annihilation and creation operators \hat a,\hat a^\dagger satisfying [\hat a,\hat a^\dagger]=1. Photon number becomes the oscillator&#x27;s excitation number, \hat n=\hat a^\dagger\hat a, with \hat a and \hat a^\dagger respectively removing and adding one field quantum -- exactly the ladder structure Einstein&#x27;s 1905 light quanta had implied but never derived. This example builds \hat a and \hat a^\dagger directly with annihilation_operator and creation_operator, verifies the commutation relation and the ladder action on Fock states, and shows the vacuum-fluctuation fact behind Dirac&#x27;s other major result: even the vacuum \lvert0\rangle is not &quot;nothing&quot; to \hat a^\dagger, which is exactly what makes spontaneous emission unavoidable.">

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  .. image:: /api/gallery/optics/quantum_optics/images/thumb/sphx_glr_plot_ladder_operators_thumb.png
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  :doc:`/api/gallery/optics/quantum_optics/plot_ladder_operators`

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      <div class="sphx-glr-thumbnail-title">Dirac's quantization of the electromagnetic field: ladder operators</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Kimble, Dagenais, and Mandel measured the first direct evidence that light can arrive one photon at a time, with a vanishing probability of detecting two photons simultaneously -- no classical wave, however dim, can produce sub-Poissonian photon statistics. The Fano factor F = \mathrm{Var}(\hat n)/\langle\hat n\rangle equals 1 for a Poissonian (coherent, classical-limit) source and 0 for a number (Fock) state -- the fully antibunched, sub-Poissonian extreme their photon-counting statistics approached. fock_state and coherent_state give the two limiting photon-number distributions directly, from which the Fano factor is computed as an ordinary first/second moment.">

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  .. image:: /api/gallery/optics/quantum_optics/images/thumb/sphx_glr_plot_photon_antibunching_thumb.png
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  :doc:`/api/gallery/optics/quantum_optics/plot_photon_antibunching`

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      <div class="sphx-glr-thumbnail-title">Kimble, Dagenais, and Mandel: photon antibunching and the Fano factor</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Richard Slusher and coworkers produced the first experimentally observed squeezed state of light. In terms of the dimensionless field quadratures \hat x = (\hat a+\hat a^\dagger)/\sqrt2 and \hat p = (\hat a-\hat a^\dagger)/(i\sqrt2), a coherent state divides the quantum uncertainty of the electromagnetic field equally between the two conjugate quadratures; a squeezed state redistributes that uncertainty unequally, reducing the noise in one quadrature below the vacuum (shot-noise) variance of 1/2 at the unavoidable cost of increasing it in the other, while the uncertainty product itself remains bounded below. squeezed_state constructs the squeeze-then-displace state">

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  .. image:: /api/gallery/optics/quantum_optics/images/thumb/sphx_glr_plot_squeezed_light_thumb.png
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  :doc:`/api/gallery/optics/quantum_optics/plot_squeezed_light`

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      <div class="sphx-glr-thumbnail-title">Slusher et al.: squeezed light and sub-vacuum quadrature noise</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Eugene Wigner introduced a quasi-probability distribution W(x,p) that represents a quantum state jointly over the dimensionless quadratures x = (a+a^\dagger)/\sqrt2 and p = (a-a^\dagger)/(i\sqrt2), reproducing the correct marginal probability densities upon integration along either axis, while permitting negative values elsewhere -- a signature with no classical phase-space analogue. compute_wigner_function evaluates W(x,p) = \sum_{n,m}\rho_{nm}K_{nm}(x,p) on a phase-space grid from the density matrix \rho=\lvert\psi\rangle\langle\psi\rvert of an arbitrary Fock-basis state vector; for a number (Fock) state \lvert n\rangle (diagonal \rho) this reduces to the closed form">

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  .. image:: /api/gallery/optics/quantum_optics/images/thumb/sphx_glr_plot_wigner_function_fock_state_thumb.png
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  :doc:`/api/gallery/optics/quantum_optics/plot_wigner_function_fock_state`

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      <div class="sphx-glr-thumbnail-title">Wigner's phase-space distribution: negativity of a Fock state</div>
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.. toctree::
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   /api/gallery/optics/quantum_optics/plot_coherent_state_statistics
   /api/gallery/optics/quantum_optics/plot_hanbury_brown_twiss
   /api/gallery/optics/quantum_optics/plot_jaynes_cummings_rabi
   /api/gallery/optics/quantum_optics/plot_ladder_operators
   /api/gallery/optics/quantum_optics/plot_photon_antibunching
   /api/gallery/optics/quantum_optics/plot_squeezed_light
   /api/gallery/optics/quantum_optics/plot_wigner_function_fock_state