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

• 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 Heisenberg uncertainty principle
• The squeezing parameter (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 nonlinear optical processes
  • Parametric down-conversion
  • 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 quadrature squeezing
• 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 thermal state 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 $g^{(2)}(0)$ 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: $ΔX^2 ΔP^2 ≥ (1/4)^2$

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 $ΔX^2 = (1/4)e^{-2r}$, where r is the squeezing parameter
  • The variance of the anti-squeezed quadrature is given by $ΔP^2 = (1/4)e^{2r}$
• 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
• Quantum metrology utilizes squeezed states to enable sub-shot-noise measurements and enhance the precision of parameter estimation in optical systems

Quantum communication and information processing

• Quantum communication 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