Glossary

2.2 Squeezed states and thermal states

4 min read•july 30, 2024

Squeezed states and thermal states are key players in quantum optics. They showcase the weird and wonderful world of quantum light, where uncertainty can be manipulated and photons behave in unexpected ways.

These states have practical applications too. Squeezed states improve precision measurements and enable secure communication, while thermal states help us understand how light behaves at different temperatures. Understanding both is crucial for advancing quantum technologies.

Squeezed states: properties and generation

Properties of squeezed states

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  • Squeezed states are quantum states of light with reduced uncertainty in one quadrature (amplitude or phase) at the expense of increased uncertainty in the conjugate quadrature, while still satisfying the
  • The (denoted as r) quantifies the degree of squeezing in a squeezed state
    • Larger values of r result in greater squeezing effect
  • Squeezed states can be classified as amplitude-squeezed (reduced uncertainty in the amplitude quadrature) or phase-squeezed (reduced uncertainty in the phase quadrature) depending on the quadrature with reduced uncertainty
  • The electric field of a squeezed state can be represented as a combination of the coherent state and the squeezing operator acting on the vacuum state
  • The photon number distribution of a squeezed state is different from that of a coherent state, exhibiting sub-Poissonian or super-Poissonian statistics depending on the type of squeezing

Generation of squeezed states

  • Squeezed states can be generated through
    • Four-wave mixing
    • Interaction of light with nonlinear media (nonlinear crystals)
  • The generation process involves the interaction of a strong pump field with a nonlinear medium, resulting in the production of correlated photon pairs or
  • The properties of the generated squeezed state depend on the characteristics of the nonlinear interaction (phase matching, pump power, crystal length)
  • Experimental techniques for generating squeezed states include optical parametric oscillators (OPOs) and optical parametric amplifiers (OPAs)

Thermal states: characteristics and photon statistics

Characteristics of thermal states

  • Thermal states are mixed quantum states that describe the equilibrium state of a quantum system in contact with a thermal reservoir at a given temperature
  • The density matrix of a is given by the Gibbs distribution, which depends on the temperature and the Hamiltonian of the system
  • Thermal states have higher entropy compared to pure states, reflecting the statistical mixture of different photon number states
  • The quantum state purity of a thermal state decreases with increasing temperature, approaching a maximally mixed state at high temperatures
  • The electric field of a thermal state has random phase fluctuations, resulting in a lack of phase coherence

Photon statistics of thermal states

  • Thermal states exhibit a Bose-Einstein photon number distribution, characterized by an average photon number that depends on the temperature and the frequency of the mode
  • The photon statistics of thermal states follow a super-Poissonian distribution, with a variance larger than the mean photon number
    • This contrasts with the Poissonian statistics of coherent states, where the variance equals the mean
  • The probability of observing n photons in a thermal state decreases exponentially with increasing n
  • The second-order correlation function g(2)(0)g^{(2)}(0) for thermal states is equal to 2, indicating photon bunching and strong intensity fluctuations

Quadrature variances for squeezed states

Quadrature operators and variances

  • The quadrature operators, denoted as X and P, represent the amplitude and phase quadratures of the electromagnetic field, respectively
  • For a squeezed state, the variance of one quadrature (e.g., X) is reduced below the standard quantum limit (SQL) of 1/4, while the variance of the conjugate quadrature (e.g., P) is increased above the SQL
  • The product of the quadrature variances for a squeezed state always satisfies the Heisenberg uncertainty principle: ΔX2ΔP2≥(1/4)2ΔX^2 ΔP^2 ≥ (1/4)^2

Calculating quadrature variances

  • The quadrature variances of a squeezed state can be calculated using the squeezing parameter (r) and the squeezing angle (φ)
    • The variance of the squeezed quadrature is given by ΔX2=(1/4)e−2rΔX^2 = (1/4)e^{-2r}, where r is the squeezing parameter
    • The variance of the anti-squeezed quadrature is given by ΔP2=(1/4)e2rΔP^2 = (1/4)e^{2r}
  • The squeezing parameter (r) determines the degree of squeezing, with larger values of r resulting in greater squeezing and increased quadrature variance in the anti-squeezed quadrature
  • The squeezing angle (φ) determines the orientation of the squeezing ellipse in phase space, specifying the quadrature with reduced variance

Applications of squeezed states in quantum optics

Precision measurements

  • Squeezed states find applications in precision measurements, as they can enhance the sensitivity of optical measurements beyond the standard quantum limit
    • Gravitational wave detection using interferometers (LIGO) can benefit from squeezed states to improve the signal-to-noise ratio and increase the detection sensitivity
    • Squeezed states can be used in optical magnetometry to enhance the sensitivity of magnetic field measurements
  • utilizes squeezed states to enable sub-shot-noise measurements and enhance the precision of parameter estimation in optical systems

Quantum communication and information processing

  • protocols, such as continuous-variable quantum key distribution (CV-QKD), can employ squeezed states to achieve secure communication with increased key rates and improved security against eavesdropping
  • Squeezed states are used in quantum information processing, serving as a resource for continuous-variable quantum computation and quantum simulation
  • The generation and manipulation of squeezed states are essential for studying fundamental aspects of quantum optics
    • Nonclassical light
    • Quantum entanglement
    • Quantum-to-classical transition
  • Quantum imaging techniques, such as ghost imaging and quantum illumination, can utilize squeezed states to improve image quality and enhance the detection of weak signals in the presence of background noise

Key Terms to Review (18)

Annihilation operator: The annihilation operator is a mathematical operator used in quantum mechanics and quantum optics that reduces the number of particles in a given quantum state by one. It plays a crucial role in describing the quantization of fields, allowing for the manipulation and understanding of quantum states, such as squeezed states and thermal states.
Bose-Einstein Distribution: The Bose-Einstein Distribution is a statistical distribution that describes the occupancy of quantum states by indistinguishable bosons, such as photons or atoms, at thermal equilibrium. This distribution is crucial in understanding phenomena in quantum optics, particularly regarding squeezed and thermal states, where bosonic particles can occupy the same quantum state without restriction, leading to unique statistical behaviors distinct from classical particles.
Cavendish Experiment: The Cavendish Experiment, conducted by Henry Cavendish in 1797-1798, was a groundbreaking scientific experiment designed to measure the force of gravitational attraction between masses and, ultimately, to determine the density of the Earth. This experiment was pivotal in providing a quantitative understanding of gravitational forces, leading to the calculation of the gravitational constant and establishing foundational concepts in gravitational theory.
Classical noise: Classical noise refers to random fluctuations in signals that can interfere with measurements or communications in quantum systems. It is a crucial concept as it limits the precision and accuracy of quantum measurements, influencing the behavior of both squeezed and thermal states. Understanding classical noise helps in designing systems that mitigate its effects and improve quantum state preparation and manipulation.
Creation Operator: A creation operator is a mathematical tool used in quantum mechanics that adds a quantum of energy or excitation to a particular state of a quantum system, such as a mode of the electromagnetic field. This operator plays a crucial role in defining and manipulating the quantum states of particles, particularly in contexts where the quantization of fields is essential, such as in squeezed and thermal states. By working in tandem with annihilation operators, creation operators enable the exploration of various physical phenomena within quantum optics.
Heisenberg Uncertainty Principle: The Heisenberg Uncertainty Principle states that it is fundamentally impossible to simultaneously know both the exact position and exact momentum of a particle. This principle highlights a fundamental limit of measurement in quantum mechanics, revealing that the more accurately one property is measured, the less accurately the other can be controlled or determined.
LIGO observations: LIGO observations refer to the detection of gravitational waves by the Laser Interferometer Gravitational-Wave Observatory, which captures tiny ripples in spacetime caused by massive astronomical events like merging black holes or neutron stars. These observations have revolutionized our understanding of the universe and provide insights into phenomena that cannot be studied through traditional electromagnetic radiation. The data obtained from LIGO has been crucial for studying quantum optics, particularly in understanding squeezed states and their role in enhancing measurement sensitivity.
Nonlinear optical processes: Nonlinear optical processes refer to interactions of light with matter that result in changes to the properties of light, such as frequency, phase, and intensity, in a way that is not directly proportional to the light's intensity. These processes enable various advanced technologies in quantum optics, including frequency conversion and the generation of squeezed states. They play a crucial role in manipulating quantum states of light, leading to applications in fields like quantum communication and metrology.
Parametric down-conversion: Parametric down-conversion is a quantum optical process where a single photon from a higher energy state is converted into two lower energy photons, called signal and idler photons. This process is significant for generating entangled photon pairs, which are crucial for various applications in quantum optics, such as quantum communication and quantum computing, and plays a vital role in the generation of squeezed states and understanding thermal states.
Phase Angle: The phase angle is a measure of the relative position of a waveform within a cycle, often expressed in degrees or radians. It plays a critical role in understanding quantum states, particularly in squeezed and thermal states, where it impacts how quantum systems evolve and interact. The phase angle helps characterize the distribution of uncertainties in quantum measurements, influencing the behavior of light and matter at the quantum level.
Quadrature squeezing: Quadrature squeezing refers to a phenomenon in quantum optics where the uncertainty in one quadrature of a light field is reduced below the standard quantum limit, while the uncertainty in the conjugate quadrature is increased. This effect enhances the precision of measurements and is crucial in applications like quantum information and gravitational wave detection. It shows how quantum mechanics allows for manipulation of light beyond classical limits, providing insight into squeezed states and their interaction with thermal states.
Quantum communication: Quantum communication refers to the use of quantum mechanics principles to transmit information securely and efficiently, often leveraging phenomena like entanglement and superposition. This form of communication ensures that any eavesdropping attempts can be detected, making it an essential technology for secure information transfer.
Quantum metrology: Quantum metrology is a field that leverages quantum mechanics to enhance the precision and accuracy of measurements. By utilizing quantum states of light and matter, this discipline enables the development of new techniques that surpass classical limitations in measuring physical quantities. This capability is particularly relevant in various applications, where improved measurement precision can lead to significant advancements in technology and fundamental science.
Quantum noise reduction: Quantum noise reduction refers to the techniques used to minimize uncertainty and fluctuations in quantum systems, particularly in measurements, enhancing the precision and sensitivity of these measurements. This concept is closely tied to the manipulation of quantum states, like squeezed states, which can exhibit reduced uncertainty in one quadrature at the expense of increased uncertainty in another, leading to better performance in applications such as quantum optics and information processing.
Squeezed coherent state: A squeezed coherent state is a quantum state of light that combines the properties of both coherent states and squeezed states, leading to reduced uncertainty in one quadrature variable while increasing uncertainty in the conjugate variable. This manipulation of uncertainty makes it possible to achieve lower noise levels than classical light sources, which is important for applications in quantum optics and information processing. The squeezed coherent state thus plays a critical role in enhancing measurement precision and improving the performance of quantum technologies.
Squeezed vacuum state: A squeezed vacuum state is a specific type of quantum state where the uncertainty in one quadrature of the electromagnetic field is reduced below the standard quantum limit, while the uncertainty in the conjugate quadrature increases. This results in a state that exhibits less noise in one parameter, which is crucial for applications like quantum optics and quantum information processing.
Squeezing parameter: The squeezing parameter quantifies the degree of squeezing in quantum states, specifically relating to the uncertainty in measurements of non-commuting observables. It is a critical feature in the study of quantum optics, indicating how much the quantum uncertainty in one variable can be reduced at the expense of increased uncertainty in another, thereby demonstrating enhanced precision for specific measurements. This concept plays a vital role in generating squeezed states and understanding their applications in quantum state generation and manipulation.
Thermal state: A thermal state is a quantum state that describes a system in thermal equilibrium, characterized by a specific temperature and represented by a density operator that follows the Boltzmann distribution. In this context, it highlights the statistical properties of many quantum systems, where particles are distributed among energy levels according to their thermal energy. Understanding thermal states is essential for analyzing phenomena such as photon statistics and the interplay between squeezed states and thermal noise.
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