Download An Introduction to: Coherent and Squeezed states

An Introduction to:
Coherent and Squeezed states
Michael R. Vanner
Bertlmann Seminar: 16th March 2011
They are states obeying the Schrödinger equation for a harmonic potential.
A coherent state is a pure state
with a circularly symmetric
Gaussian phase space distribution
which has equal position and
momentum noise.
A squeezed state has an
asymmetric Gaussian phase space
distribution. The semi-minor axis of
the distribution must be narrower
than the width of a coherent state.
It is not necessarily pure.
Referred to as “quantum noise” , origin is commutator / Heisenberg uncertainty.
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They are basic components of modern day quantum optics.
Much insight into the term “vacuum” can be gained.
Useful tools in experimental physics.
(Interferometry in particular.)
Fundamental quantum limits to experimental sensitivity can
be understood and even overcome.
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Part I:
Coherent States
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Minimum uncertainty Gaussian wavepackets
First example found that obeys the Correspondence Principle
Schrödinger, Naturwissenschaften 14, 664 (1926)
Dirac also
Born, Wiener
1920s
Operator method for photon creation and annihilation.
(QFT development)
Glauber, Phys. Rev. 131, 2766 (1963)
“Coherent and Incoherent States of the Radiation Field”
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Coherent State Eigenvalue Equation
Displacement Operators
Rotation
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It is useful to work in units of “quantum noise”
and introduce the dimensionless quadratures.
i.e. Phase-space distances are expressed in
terms of coherent state widths.
Coherent state position wavefunction:
Translated vacuum noise.
From double displacement
This is a displacement.
The momentum wavefunction is related to the above via the Fourier transform. (For pure states only.)
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Freely Propagating Light
Breitenbach et al., Nature 387, 471 (1997)
Cavity Confined Microwave Radiation
Deleglise et al., Nature 455, 510 (2008)
Atomic Ensemble Spin
(Bloch sphere shown)
Motion of a trapped ion
Leibfried et al., PRL 77, 4281 (1996)
Teper et al., Phys. Rev. A 78, 051803 (2008)
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Hang on! Isn’t this amazing!
Input State
Output State
This is a unique property of coherent states.
Zavatta et al., New Journal of Physics 10, 123006 (2008)
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Poissonion statistics are common in random processes where
only integer values are allowed e.g. particle counting.
Poisson Distribution
Fully characterized by the mean
This an example of “shot noise”.
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Super-Poissonian statistics
Poissonian statistics
Sub-Poissonian statistics
Thermal state
Coherent state
Non-classical e.g.
number state.
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Part II:
Squeezed States
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There are many methods to generate squeezed optical fields.
One common technique is to use parametric amplification.
Input
Output
The crystal responds to the electric
field of the pump which modulates
the potential as seen by the input.
Pump
The ground state of the potential is modulated in time: squeezed vacuum.
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As pump is at 2ω creation and
annihilation are pairwise processes.
Scully and Zubairy, Quantum Optics (1997)
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“Quantum noise” places a fundamental limit on the sensitivities
of many experiments e.g. Gravitational Wave Detectors.
Less noise in P quad.
Inteferometers detect small
phase shifts.
Signal
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The annihilation operator can be
considered to fully represent the field.
However, it is clearly not Hermitian.
One can then fully characterize a field by
knowing all properties of all its quadratures
These can be accessed via homodyne detection.
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In quantum optics a common definition of “quantumness” is:
A state is considered quantum if it cannot be written as an
incoherent mixture of coherent states.
See e.g. Leonhardt, Measuring the Quantum State of Light (1997)
This implies squeezed states are quantum states and
Coherent states are classical states.
There is only a quantum world.
Danny Greenberger, 2005
The origin of the noise of a coherent state is then not an issue.
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You may have heard about Wigner or Q functions. Intimately
linked with coherent states is the P representation.
Coherent states form an overcomplete basis set
We are then motivated to write:
See e.g. Walls and Milburn,
Quantum Optics (1994)
P(α) is analogous to but not stricktly a probability function.
This is because different coherent states are not orthogonal.
e.g. the P function is a delta function for a coherent state.
Also, we see that whether the P function exists is linked to our
previous defintion of quantumness.
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Breitenbach et al., Nature 387, 471 (1997)
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• Coherent and squeezed states are basic elements of modern
day quantum optics.
• Consequently, this seminar also provided an introduction to
continuous variable quantum optics.
• An understading of these states is crucial in order to
understand other more striking states such as a quantum
optical Schrödinger-cat states.
Deleglise et al., Nature 455, 510 (2008)
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