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In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a squeezed coherent state is a quantum state that is usually described by two
non-commuting observables
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
having continuous spectra of
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
. Examples are position
and momentum
of a particle, and the (dimension-less) electric field in the amplitude
(phase 0) and in the mode
(phase 90°) of a light wave (the wave's
quadratures). The product of the standard deviations of two such
operators obeys the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
:
:
and
, respectively.
Trivial examples, which are in fact not squeezed, are the ground state
of the
and the family of
coherent state
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harm ...
s
. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with
in "natural oscillator units" and
. (In literature different normalizations for the quadrature amplitudes are used. Here we use the normalization for which the sum of the ground state variances of the quadrature amplitudes directly provide the zero point quantum number
).
The term squeezed state is actually used for states with a standard deviation below that of the ground state for one of the operators or for a linear combination of the two. The idea behind this is that the circle denoting the uncertainty of a coherent state in the
quadrature phase
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
space (see right) has been "squeezed" to an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
of the same area. Note that a squeezed state does not need to saturate the uncertainty principle.
Squeezed states of light were first produced in the mid 1980s.
[R. E. Slusher et al., ''Observation of squeezed states generated by four wave mixing in an optical cavity'', Phys. Rev. Lett. 55 (22), 2409 (1985)
] At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e.
. As of 2017, squeeze factors larger than 10 (10 dB) have been directly observed.
Mathematical definition
The most general
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
that satisfies the identity above is the squeezed coherent state (we work in units with
)
:
where
are constants (a normalization constant, the center of the
wavepacket
In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diffe ...
, its width, and the expectation value of its
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
). The new feature relative to a
coherent state
In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harm ...
is the free value of the width
, which is the reason why the state is called "squeezed".
The squeezed state above is an
eigenstate of a linear operator
:
and the corresponding
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
equals
. In this sense, it is a generalization of the ground state as well as the coherent state.
Operator representation
The general form of a squeezed coherent state for a quantum harmonic oscillator is given by
:
where
is the
vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
,
is the
displacement operator In the quantum mechanics study of optical phase space, the displacement operator for one mode is the shift operator in quantum optics,
:\hat(\alpha)=\exp \left ( \alpha \hat^\dagger - \alpha^\ast \hat \right ) ,
where \alpha is the amount of dis ...
and
is the
squeeze operator In quantum physics, the squeeze operator for a single mode of the electromagnetic field is
:\hat(z) = \exp \left ( (z^* \hat^2 - z \hat^) \right ) , \qquad z = r \, e^
where the operators inside the exponential are the ladder operators. It is a ...
, given by
:
_operated_below_threshold_produces_squeezed_vacuum,_whereas_the_same_OPO_operated_above_threshold_produces_bright_squeezed_light._Bright_squeezed_light_can_be_advantageous_for_certain_quantum_information_processing_applications_as_it_obviates_the_need_of_sending_
local_oscillator
In electronics, a local oscillator (LO) is an electronic oscillator used with a mixer to change the frequency of a signal. This frequency conversion process, also called heterodyning, produces the sum and difference frequencies from the frequenc ...
_to_provide_a_phase_reference,_whereas_squeezed_vacuum_is_considered_more_suitable_for_quantum_enhanced_sensing_applications._The_
AdLIGO_and_
GEO600
GEO600 is a gravitational wave detector located near Sarstedt, a town 20 km to the south of Hanover, Germany. It is designed and operated by scientists from the Max Planck Institute for Gravitational Physics, Max Planck Institute of Quantum Opti ...
_gravitational_wave_detectors_use_squeezed_vacuum_to_achieve_enhanced_sensitivity_beyond_the_standard_quantum_limit.
_Atomic_spin_squeezing
For_squeezing_of_two-level_neutral_atom_ensembles_it_is_useful_to_consider_the_atoms_as_spin-1/2_particles_with_corresponding_
angular_momentum_operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
s_defined_as
:
J_v=\sum_^N_j_v^
where_
v=_and_
j_v^_is_the_single-spin_operator_in_the_
v-direction._Here_
J_z_will_correspond_to_the_population_difference_in_the_two_level_system,_i.e._for_an_equal_superposition_of_the_up_and_down_state_
J_z=0._The_
J_x−
J_y_plane_represents_the_phase_difference_between_the_two_states._This_is_also_known_as_the_
Bloch_sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
_picture._We_can_then_define_uncertainty_relations_such_as_
\Delta_J_z_\cdot_\Delta_J_y_\geq_\left, \Delta_J_x\/2._For_a_coherent_(unentangled)_state,_
\Delta_J_z=\Delta_J_y=\sqrt/2._Squeezing_is_here_considered_the_redistribution_of_uncertainty_from_one_variable_(typically_
J_z)_to_another_(typically_
J_y)._If_we_consider_a_state_pointing_in_the_
J_x_direction,_we_can_define_the_Wineland_criterion_for_squeezing,_or_the_metrological_enhancement_of_the_squeezed_state_as
:
\chi^2=\left(\frac\frac\right)^2.
This_criterion_has_two_factors,_the_first_factor_is_the_spin_noise_reduction,_i.e._how_much_the_quantum_noise_in_
J_z_is_reduced_relative_to_the_coherent_(unentangled)_state._The_second_factor_is_how_much_the_coherence_(the_length_of_the_Bloch_vector,_
\left, J_x\)_is_reduced_due_to_the_squeezing_procedure._Together_these_quantities_tell_you_how_much_metrological_enhancement_the_squeezing_procedure_gives._Here,_metrological_enhancement_is_the_reduction_in_averaging_time_or_atom_number_needed_to_make_a_measurement_of_a_specific_uncertainty._20 dB_of_metrological_enhancement_means_the_same_precision_measurement_can_be_made_with_100_times_fewer_atoms_or_100_times_shorter_averaging_time.
_Experimental_realizations
There_has_been_a_whole_variety_of_successful_demonstrations_of_squeezed_states._The_first_demonstrations_were_experiments_with_light_fields_using_
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s_and_
non-linear_optics
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typica ...
_(see_
optical_parametric_oscillator
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave (called "pump") with frequency \omega_p into two output waves of lower frequency (\omega_s, \omega_i) by mean ...
)._This_is_achieved_by_a_simple_process_of_four-wave_mixing_with_a_
\chi^_crystal;_similarly_travelling_wave_phase-sensitive_amplifiers_generate_spatially_multimode_quadrature-squeezed_states_of_light_when_the_
\chi^_crystal_is_pumped_in_absence_of_any_signal._
Sub-Poissonian In mathematics, a super-Poissonian distribution is a probability distribution that has a larger variance than a Poisson distribution with the same mean. Conversely, a sub-Poissonian distribution has a smaller variance.
An example of super-Poissonia ...
_current_sources_driving_semiconductor_laser_diodes_have_led_to_amplitude_squeezed_light.
Squeezed_states_have_also_been_realized_via_motional_states_of_an_
ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
_in_a_trap,_
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
_states_in_
crystal_lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
s,_and_spin_states_in_neutral_
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
_ensembles._Much_progress_has_been_made_on_the_creation_and_observation_of_spin_squeezed_states_in_ensembles_of_neutral_atoms_and_ions,_which_can_be_used_to_enhancement_measurements_of_time,_accelerations,_fields,_and_the_current_state_of_the_art_for_measurement_enhancement_is_20 dB.
_Generation_of_spin_squeezed_states_have_been_demonstrated_using_both_coherent_evolution_of_a_coherent_spin_state_and_projective,_coherence-preserving_measurements._Even_macroscopic_oscillators_were_driven_into_classical_motional_states_that_were_very_similar_to_squeezed_coherent_states._Current_state_of_the_art_in_noise_suppression,_for_laser_radiation_using_squeezed_light,_amounts_to_15 dB_(as_of_2016),_which_broke_the_previous_record_of_12.7 dB_(2010).
_Applications
Squeezed_states_of_the_light_field_can_be_used_to_enhance_precision_measurements._For_example,_phase-squeezed_light_can_improve_the_phase_read_out_of_ interferometric_measurements_(see_for_example_gravitational_wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s)._Amplitude-squeezed_light_can_improve_the_readout_of_very_weak_ spectroscopic_signals.
Spin_squeezed_states_of_atoms_can_be_used_to_improve_the_precision_of_atomic_clock
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
s._This_is_an_important_problem_in_atomic_clocks_and_other_sensors_that_use_small_ensembles_of_cold_atoms_where_the_ quantum_projection_noise_represents_a_fundamental_limitation_to_the_precision_of_the_sensor.
Various_squeezed_coherent_states,_generalized_to_the_case_of_many_ degrees_of_freedom,_are_used_in_various_calculations_in_quantum_field_theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
,_for_example_Unruh_effect
The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...
_and_Hawking_radiation
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
,_and_generally,_particle_production_in_curved_backgrounds_and_Bogoliubov_transformation
In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous ...
s.
Recently,_the_use_of_squeezed_states_for_quantum_information_processing
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
_in_the_continuous_variables_(CV)_regime_has_been_increasing_rapidly._Continuous_variable_quantum_optics_uses_squeezing_of_light_as_an_essential_resource_to_realize_CV_protocols_for_quantum_communication,_unconditional_quantum_teleportation_and_one-way_quantum_computing._This_is_in_contrast_to_quantum_information_processing_with_single_photons_or_photon_pairs_as_qubits._CV_quantum_information_processing_relies_heavily_on_the_fact_that_squeezing_is_intimately_related_to_quantum_entanglement,_as_the_quadratures_of_a_squeezed_state_exhibit_sub-shot-noise_quantum_correlations.
_See_also
*Negative_energy
Negative energy is a concept used in physics to explain the nature of certain fields, including the gravitational field and various quantum field effects.
Gravitational potential energy
Gravitational potential energy can be defined as being n ...
*Nonclassical_light
Nonclassical light is light that cannot be described using classical electromagnetism; its characteristics are described by the quantized electromagnetic field and quantum mechanics.
The most common described forms of nonclassical light are the fo ...
*Optical_phase_space
In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an ''optical system''. For any such system, a plot o ...
*Quantum_optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...
*Squeeze_operator In quantum physics, the squeeze operator for a single mode of the electromagnetic field is
:\hat(z) = \exp \left ( (z^* \hat^2 - z \hat^) \right ) , \qquad z = r \, e^
where the operators inside the exponential are the ladder operators. It is a ...
_References
_External_links
Tutorial_about_quantum_optics_of_the_light_field
{{DEFAULTSORT:Squeezed_Coherent_State
Quantum_opticshtml" ;"title=" \mathrm \rangle \langle \mathrm , ]\\ &= \frac \sum_^\infty \tanh^(r)
, n \rangle \langle n, , \end
with an effective average number of photons \widetilde = \sinh^2(r).
Based on the presence of a mean field
Squeezed states of light can be divided into squeezed vacuum and bright squeezed light, depending on the absence or presence of a non-zero mean field (also called a carrier), respectively. An optical parametric oscillator
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave (called "pump") with frequency \omega_p into two output waves of lower frequency (\omega_s, \omega_i) by mean ...
operated below threshold produces squeezed vacuum, whereas the same OPO operated above threshold produces bright squeezed light. Bright squeezed light can be advantageous for certain quantum information processing applications as it obviates the need of sending local oscillator
In electronics, a local oscillator (LO) is an electronic oscillator used with a mixer to change the frequency of a signal. This frequency conversion process, also called heterodyning, produces the sum and difference frequencies from the frequenc ...
to provide a phase reference, whereas squeezed vacuum is considered more suitable for quantum enhanced sensing applications. The AdLIGO and GEO600
GEO600 is a gravitational wave detector located near Sarstedt, a town 20 km to the south of Hanover, Germany. It is designed and operated by scientists from the Max Planck Institute for Gravitational Physics, Max Planck Institute of Quantum Opti ...
gravitational wave detectors use squeezed vacuum to achieve enhanced sensitivity beyond the standard quantum limit.
Atomic spin squeezing
For squeezing of two-level neutral atom ensembles it is useful to consider the atoms as spin-1/2 particles with corresponding angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
s defined as
:J_v=\sum_^N j_v^
where v= and j_v^ is the single-spin operator in the v-direction. Here J_z will correspond to the population difference in the two level system, i.e. for an equal superposition of the up and down state J_z=0. The J_x−J_y plane represents the phase difference between the two states. This is also known as the Bloch sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
picture. We can then define uncertainty relations such as \Delta J_z \cdot \Delta J_y \geq \left, \Delta J_x\/2. For a coherent (unentangled) state, \Delta J_z=\Delta J_y=\sqrt/2. Squeezing is here considered the redistribution of uncertainty from one variable (typically J_z) to another (typically J_y). If we consider a state pointing in the J_x direction, we can define the Wineland criterion for squeezing, or the metrological enhancement of the squeezed state as
:\chi^2=\left(\frac\frac\right)^2.
This criterion has two factors, the first factor is the spin noise reduction, i.e. how much the quantum noise in J_z is reduced relative to the coherent (unentangled) state. The second factor is how much the coherence (the length of the Bloch vector, \left, J_x\) is reduced due to the squeezing procedure. Together these quantities tell you how much metrological enhancement the squeezing procedure gives. Here, metrological enhancement is the reduction in averaging time or atom number needed to make a measurement of a specific uncertainty. 20 dB of metrological enhancement means the same precision measurement can be made with 100 times fewer atoms or 100 times shorter averaging time.
Experimental realizations
There has been a whole variety of successful demonstrations of squeezed states. The first demonstrations were experiments with light fields using laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s and non-linear optics
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typica ...
(see optical parametric oscillator
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave (called "pump") with frequency \omega_p into two output waves of lower frequency (\omega_s, \omega_i) by mean ...
). This is achieved by a simple process of four-wave mixing with a \chi^ crystal; similarly travelling wave phase-sensitive amplifiers generate spatially multimode quadrature-squeezed states of light when the \chi^ crystal is pumped in absence of any signal. Sub-Poissonian In mathematics, a super-Poissonian distribution is a probability distribution that has a larger variance than a Poisson distribution with the same mean. Conversely, a sub-Poissonian distribution has a smaller variance.
An example of super-Poissonia ...
current sources driving semiconductor laser diodes have led to amplitude squeezed light.
Squeezed states have also been realized via motional states of an ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
in a trap, phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
states in crystal lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
s, and spin states in neutral atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
ensembles. Much progress has been made on the creation and observation of spin squeezed states in ensembles of neutral atoms and ions, which can be used to enhancement measurements of time, accelerations, fields, and the current state of the art for measurement enhancement is 20 dB. Generation of spin squeezed states have been demonstrated using both coherent evolution of a coherent spin state and projective, coherence-preserving measurements. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states. Current state of the art in noise suppression, for laser radiation using squeezed light, amounts to 15 dB (as of 2016), which broke the previous record of 12.7 dB (2010).
Applications
Squeezed states of the light field can be used to enhance precision measurements. For example, phase-squeezed light can improve the phase read out of interferometric measurements (see for example gravitational wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s). Amplitude-squeezed light can improve the readout of very weak spectroscopic signals.
Spin squeezed states of atoms can be used to improve the precision of atomic clock
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
s. This is an important problem in atomic clocks and other sensors that use small ensembles of cold atoms where the quantum projection noise represents a fundamental limitation to the precision of the sensor.
Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, for example Unruh effect
The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...
and Hawking radiation
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
, and generally, particle production in curved backgrounds and Bogoliubov transformation
In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous ...
s.
Recently, the use of squeezed states for quantum information processing
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
in the continuous variables (CV) regime has been increasing rapidly. Continuous variable quantum optics uses squeezing of light as an essential resource to realize CV protocols for quantum communication, unconditional quantum teleportation and one-way quantum computing. This is in contrast to quantum information processing with single photons or photon pairs as qubits. CV quantum information processing relies heavily on the fact that squeezing is intimately related to quantum entanglement, as the quadratures of a squeezed state exhibit sub-shot-noise quantum correlations.
See also
*Negative energy
Negative energy is a concept used in physics to explain the nature of certain fields, including the gravitational field and various quantum field effects.
Gravitational potential energy
Gravitational potential energy can be defined as being n ...
*Nonclassical light
Nonclassical light is light that cannot be described using classical electromagnetism; its characteristics are described by the quantized electromagnetic field and quantum mechanics.
The most common described forms of nonclassical light are the fo ...
*Optical phase space
In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an ''optical system''. For any such system, a plot o ...
*Quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...
*Squeeze operator In quantum physics, the squeeze operator for a single mode of the electromagnetic field is
:\hat(z) = \exp \left ( (z^* \hat^2 - z \hat^) \right ) , \qquad z = r \, e^
where the operators inside the exponential are the ladder operators. It is a ...
References
External links
Tutorial about quantum optics of the light field
{{DEFAULTSORT:Squeezed Coherent State
Quantum optics> \mathrm \rangle \langle \mathrm , \ &= \frac \sum_^\infty \tanh^(r)
, n \rangle \langle n, , \end
with an effective average number of photons \widetilde = \sinh^2(r).
Based on the presence of a mean field
Squeezed states of light can be divided into squeezed vacuum and bright squeezed light, depending on the absence or presence of a non-zero mean field (also called a carrier), respectively. An optical parametric oscillator
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave (called "pump") with frequency \omega_p into two output waves of lower frequency (\omega_s, \omega_i) by mean ...
operated below threshold produces squeezed vacuum, whereas the same OPO operated above threshold produces bright squeezed light. Bright squeezed light can be advantageous for certain quantum information processing applications as it obviates the need of sending local oscillator
In electronics, a local oscillator (LO) is an electronic oscillator used with a mixer to change the frequency of a signal. This frequency conversion process, also called heterodyning, produces the sum and difference frequencies from the frequenc ...
to provide a phase reference, whereas squeezed vacuum is considered more suitable for quantum enhanced sensing applications. The AdLIGO and GEO600
GEO600 is a gravitational wave detector located near Sarstedt, a town 20 km to the south of Hanover, Germany. It is designed and operated by scientists from the Max Planck Institute for Gravitational Physics, Max Planck Institute of Quantum Opti ...
gravitational wave detectors use squeezed vacuum to achieve enhanced sensitivity beyond the standard quantum limit.
Atomic spin squeezing
For squeezing of two-level neutral atom ensembles it is useful to consider the atoms as spin-1/2 particles with corresponding angular momentum operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
s defined as
:J_v=\sum_^N j_v^
where v= and j_v^ is the single-spin operator in the v-direction. Here J_z will correspond to the population difference in the two level system, i.e. for an equal superposition of the up and down state J_z=0. The J_x−J_y plane represents the phase difference between the two states. This is also known as the Bloch sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
picture. We can then define uncertainty relations such as \Delta J_z \cdot \Delta J_y \geq \left, \Delta J_x\/2. For a coherent (unentangled) state, \Delta J_z=\Delta J_y=\sqrt/2. Squeezing is here considered the redistribution of uncertainty from one variable (typically J_z) to another (typically J_y). If we consider a state pointing in the J_x direction, we can define the Wineland criterion for squeezing, or the metrological enhancement of the squeezed state as
:\chi^2=\left(\frac\frac\right)^2.
This criterion has two factors, the first factor is the spin noise reduction, i.e. how much the quantum noise in J_z is reduced relative to the coherent (unentangled) state. The second factor is how much the coherence (the length of the Bloch vector, \left, J_x\) is reduced due to the squeezing procedure. Together these quantities tell you how much metrological enhancement the squeezing procedure gives. Here, metrological enhancement is the reduction in averaging time or atom number needed to make a measurement of a specific uncertainty. 20 dB of metrological enhancement means the same precision measurement can be made with 100 times fewer atoms or 100 times shorter averaging time.
Experimental realizations
There has been a whole variety of successful demonstrations of squeezed states. The first demonstrations were experiments with light fields using laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
s and non-linear optics
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typica ...
(see optical parametric oscillator
An optical parametric oscillator (OPO) is a parametric oscillator that oscillates at optical frequencies. It converts an input laser wave (called "pump") with frequency \omega_p into two output waves of lower frequency (\omega_s, \omega_i) by mean ...
). This is achieved by a simple process of four-wave mixing with a \chi^ crystal; similarly travelling wave phase-sensitive amplifiers generate spatially multimode quadrature-squeezed states of light when the \chi^ crystal is pumped in absence of any signal. Sub-Poissonian In mathematics, a super-Poissonian distribution is a probability distribution that has a larger variance than a Poisson distribution with the same mean. Conversely, a sub-Poissonian distribution has a smaller variance.
An example of super-Poissonia ...
current sources driving semiconductor laser diodes have led to amplitude squeezed light.
Squeezed states have also been realized via motional states of an ion
An ion () is an atom or molecule with a net electrical charge.
The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
in a trap, phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
states in crystal lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
s, and spin states in neutral atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
ensembles. Much progress has been made on the creation and observation of spin squeezed states in ensembles of neutral atoms and ions, which can be used to enhancement measurements of time, accelerations, fields, and the current state of the art for measurement enhancement is 20 dB. Generation of spin squeezed states have been demonstrated using both coherent evolution of a coherent spin state and projective, coherence-preserving measurements. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states. Current state of the art in noise suppression, for laser radiation using squeezed light, amounts to 15 dB (as of 2016), which broke the previous record of 12.7 dB (2010).
Applications
Squeezed states of the light field can be used to enhance precision measurements. For example, phase-squeezed light can improve the phase read out of interferometric measurements (see for example gravitational wave
Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s). Amplitude-squeezed light can improve the readout of very weak spectroscopic signals.
Spin squeezed states of atoms can be used to improve the precision of atomic clock
An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwee ...
s. This is an important problem in atomic clocks and other sensors that use small ensembles of cold atoms where the quantum projection noise represents a fundamental limitation to the precision of the sensor.
Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, for example Unruh effect
The Unruh effect (also known as the Fulling–Davies–Unruh effect) is a kinematic prediction of quantum field theory that an accelerating observer will observe a thermal bath, like blackbody radiation, whereas an inertial observer would observe ...
and Hawking radiation
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...
, and generally, particle production in curved backgrounds and Bogoliubov transformation
In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous ...
s.
Recently, the use of squeezed states for quantum information processing
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
in the continuous variables (CV) regime has been increasing rapidly. Continuous variable quantum optics uses squeezing of light as an essential resource to realize CV protocols for quantum communication, unconditional quantum teleportation and one-way quantum computing. This is in contrast to quantum information processing with single photons or photon pairs as qubits. CV quantum information processing relies heavily on the fact that squeezing is intimately related to quantum entanglement, as the quadratures of a squeezed state exhibit sub-shot-noise quantum correlations.
See also
*Negative energy
Negative energy is a concept used in physics to explain the nature of certain fields, including the gravitational field and various quantum field effects.
Gravitational potential energy
Gravitational potential energy can be defined as being n ...
*Nonclassical light
Nonclassical light is light that cannot be described using classical electromagnetism; its characteristics are described by the quantized electromagnetic field and quantum mechanics.
The most common described forms of nonclassical light are the fo ...
*Optical phase space
In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an ''optical system''. For any such system, a plot o ...
*Quantum optics
Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have b ...
*Squeeze operator In quantum physics, the squeeze operator for a single mode of the electromagnetic field is
:\hat(z) = \exp \left ( (z^* \hat^2 - z \hat^) \right ) , \qquad z = r \, e^
where the operators inside the exponential are the ladder operators. It is a ...
References
External links
Tutorial about quantum optics of the light field
{{DEFAULTSORT:Squeezed Coherent State
Quantum optics