15.2 Nonlinear optics for quantum state generation

Nonlinear optics plays a crucial role in quantum state generation. By exploiting nonlinear processes like SPDC and FWM, scientists can create entangled photon pairs, squeezed states, and single photons essential for quantum applications.

These techniques form the backbone of experimental quantum optics. From OPOs generating squeezed light to FWM in photonic fibers producing single photons, nonlinear processes enable the creation and manipulation of quantum states of light.

Nonlinear Optical Processes in Quantum Optics

Fundamentals of Nonlinear Optics

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• Nonlinear optical processes involve the interaction of light with matter where the material system's response is nonlinearly dependent on the optical field's strength
• In nonlinear optics, the medium's polarization is not linearly proportional to the applied electric field, leading to phenomena such as:
  • Frequency mixing
  • Harmonic generation
  • Parametric processes
• The nonlinear susceptibility tensor characterizes the strength of a material's nonlinear optical response
  • Second-order (χ^(2)) and third-order (χ^(3)) susceptibilities are the most relevant for quantum optics applications

Importance of Nonlinear Optics in Quantum Optics

• Nonlinear optical processes are crucial for generating non-classical states of light essential for quantum optics and quantum information processing, such as:
  • Entangled photon pairs
  • Squeezed states
  • Single photons
• Phase matching is a critical condition in nonlinear optical processes
  • Ensures efficient energy transfer between the interacting waves
  • Enables the generation of quantum states of light with high efficiency and purity

Entangled Photon Pairs via SPDC

Spontaneous Parametric Down-Conversion (SPDC) Process

• SPDC is a second-order nonlinear optical process where a high-energy pump photon is converted into two lower-energy photons (signal and idler) in a nonlinear crystal
• SPDC occurs probabilistically when phase-matching conditions are satisfied, ensuring energy and momentum conservation between the pump, signal, and idler photons
• The efficiency of SPDC and the quality of the generated entangled photon pairs depend on factors such as:
  • Nonlinear coefficient of the crystal
  • Phase-matching bandwidth
  • Spatial mode overlap

Entanglement in SPDC

• The generated signal and idler photons are entangled in various degrees of freedom, depending on the phase-matching configuration and crystal properties, such as:
  • Polarization
  • Frequency
  • Spatial mode
• Type-I and Type-II phase matching are two common SPDC configurations, resulting in different types of entanglement:
  • Type-I SPDC produces signal and idler photons with the same polarization, entangled in other degrees of freedom (frequency or spatial mode)
  • Type-II SPDC generates signal and idler photons with orthogonal polarizations, entangled in polarization and other degrees of freedom
• SPDC is widely used as a source of entangled photon pairs for applications in:
  • Quantum communication
  • Quantum cryptography
  • Quantum metrology

Squeezed States of Light with OPOs

Optical Parametric Oscillators (OPOs)

• OPOs are devices that exploit second-order nonlinear optical processes to generate squeezed states of light
  • Squeezed states have reduced quantum noise in one quadrature at the expense of increased noise in the orthogonal quadrature
• An OPO consists of a nonlinear crystal placed inside an optical cavity
  • Parametric down-conversion occurs in the presence of a strong pump field, leading to the amplification of the signal and idler fields
• When operated below the oscillation threshold, an OPO generates squeezed vacuum states, which exhibit quantum noise reduction in one quadrature of the electromagnetic field

Characterizing Squeezed States

• The squeezing parameter and the squeezing angle characterize the degree of noise reduction and the orientation of the squeezed quadrature, respectively
  • Can be controlled by adjusting the pump power and the phase of the pump field relative to the cavity
• OPOs can be designed to generate squeezed states in various frequency bands (visible to near-infrared) by choosing appropriate nonlinear crystals and cavity configurations
• Squeezed states of light generated by OPOs have applications in quantum-enhanced metrology
  • Gravitational wave detection, where they can improve the sensitivity of interferometric measurements beyond the standard quantum limit

Four-Wave Mixing for Photon Generation

Four-Wave Mixing (FWM) Process

• FWM is a third-order nonlinear optical process involving the interaction of four waves in a nonlinear medium
  • Results in the generation of new frequency components and quantum states of light
• In the spontaneous FWM process, two pump photons are annihilated, and a signal and an idler photon are created, satisfying energy and momentum conservation
• The efficiency and spectral properties of photons generated by FWM can be engineered by controlling:
  • Dispersion and nonlinearity of the medium
  • Phase-matching conditions

Single Photon Generation with FWM

• FWM can generate single photons using a single pump field and a nonlinear medium with a large χ^(3) nonlinearity, such as:
  • Photonic crystal fiber
  • Silicon waveguide
• The spontaneous FWM process produces photon pairs
  • By detecting one photon (idler) of the pair, the presence of the other photon (signal) is heralded, effectively creating a single-photon source
• The quality of single photons generated by FWM depends on factors such as:
  • Purity of the quantum state
  • Heralding efficiency
  • Photon indistinguishability

Correlated Photon Pairs with FWM

• FWM can also generate correlated photon pairs, where the signal and idler photons exhibit strong temporal and spectral correlations
• The correlated photon pairs generated by FWM can be used for applications in quantum communication and quantum information processing, such as:
  • Quantum key distribution
  • Quantum state teleportation