👀Quantum Optics Unit 2 – Quantum States of Light

Foundations of Quantum Optics

• Quantum optics studies the quantum mechanical properties of light and its interaction with matter at the microscopic level
• Combines principles from quantum mechanics and classical optics to describe the behavior of photons and their interactions with atoms and molecules
• Explores phenomena such as quantum entanglement, superposition, and quantum coherence in the context of light
• Lays the groundwork for understanding the quantum nature of light and its potential applications in quantum technologies (quantum computing, quantum cryptography, quantum sensing)
• Provides a theoretical framework for describing the generation, manipulation, and detection of quantum states of light
• Incorporates concepts from quantum field theory to describe the quantization of the electromagnetic field and the creation and annihilation of photons
• Investigates the role of quantum fluctuations and quantum noise in optical systems and their impact on the precision of measurements

Classical vs. Quantum Light

• Classical light described by Maxwell's equations treats light as continuous electromagnetic waves with well-defined amplitude and phase
• Quantum light described by quantum mechanics treats light as discrete photons with particle-like properties and probabilistic behavior
• Classical light exhibits wave-like phenomena such as interference and diffraction, while quantum light exhibits particle-like phenomena such as the photoelectric effect and photon antibunching
• Quantum light displays non-classical properties that cannot be explained by classical optics, such as sub-Poissonian photon statistics and violation of Bell's inequality
• Classical light sources (incandescent bulbs, LEDs) emit light with a Poissonian distribution of photon numbers, while quantum light sources (single-photon emitters, squeezed light) can produce non-classical photon statistics
• Quantum light enables secure communication through quantum key distribution, while classical light is vulnerable to eavesdropping
• Quantum light allows for the creation of entangled photon states, which have no classical counterpart and form the basis for quantum information processing

Quantum States of Light: An Overview

• Quantum states of light describe the quantum mechanical properties of photons and their statistical behavior
• Can be represented mathematically using state vectors in a Hilbert space or density matrices for mixed states
• Include Fock states (number states), coherent states, squeezed states, and entangled photon states, each with distinct properties and applications
• Fock states have a well-defined number of photons and exhibit sub-Poissonian photon statistics, making them suitable for quantum cryptography and quantum metrology
• Coherent states resemble classical light with a Poissonian distribution of photon numbers and are generated by ideal lasers
• Squeezed states have reduced quantum noise in one quadrature at the expense of increased noise in the orthogonal quadrature, enabling precision measurements beyond the standard quantum limit
• Entangled photon states exhibit strong correlations between multiple photons that cannot be explained by classical theories and are essential for quantum information processing tasks (quantum teleportation, quantum computing)

Fock States and Number States

• Fock states, also known as number states, are quantum states of light with a well-defined number of photons
• Denoted as $|n\rangle$, where $n$ is the number of photons in the state
• Eigenstates of the photon number operator $\hat{n} = \hat{a}^\dagger \hat{a}$, where $\hat{a}^\dagger$ and $\hat{a}$ are the creation and annihilation operators, respectively
  • The creation operator $\hat{a}^\dagger$ adds a photon to the state: $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$
  • The annihilation operator $\hat{a}$ removes a photon from the state: $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$
• Exhibit sub-Poissonian photon statistics, meaning the variance in the photon number is less than the mean photon number
• Can be generated using single-photon sources (quantum dots, nitrogen-vacancy centers in diamond) or through the process of spontaneous parametric down-conversion
• Form the basis for quantum communication protocols (quantum key distribution) and quantum metrology applications (photon-number-resolving detectors)
• Fock states with different photon numbers are orthogonal to each other, meaning they can be perfectly distinguished by a suitable measurement

Coherent States

• Coherent states are quantum states of light that most closely resemble classical light waves with a well-defined amplitude and phase
• Denoted as $|\alpha\rangle$, where $\alpha$ is a complex number representing the amplitude and phase of the state
• Eigenstates of the annihilation operator $\hat{a}$, satisfying $\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$
• Can be generated by ideal lasers operating above the lasing threshold, where the gain medium is continuously pumped to maintain a steady-state population inversion
• Exhibit Poissonian photon statistics, meaning the variance in the photon number is equal to the mean photon number
  • The probability of measuring $n$ photons in a coherent state follows a Poisson distribution: $P(n) = \frac{|\alpha|^{2n}}{n!}e^{-|\alpha|^2}$
• Have a minimum uncertainty product between the amplitude and phase quadratures, making them useful for precision measurements and quantum metrology
• Coherent states with different amplitudes and phases are not orthogonal to each other, meaning they cannot be perfectly distinguished by a single measurement
• Play a crucial role in continuous-variable quantum information processing (quantum teleportation, quantum key distribution with continuous variables)

Squeezed States

• Squeezed states are quantum states of light with reduced quantum noise in one quadrature at the expense of increased noise in the orthogonal quadrature
• Can be generated by nonlinear optical processes (parametric down-conversion, four-wave mixing) or by using a squeezed light source (optical parametric oscillator)
• Enable precision measurements beyond the standard quantum limit set by the Heisenberg uncertainty principle for coherent states
• Squeezed vacuum states have no coherent amplitude but exhibit reduced quantum noise in one quadrature
  • Squeezed vacuum states are used in gravitational wave detectors (LIGO) to enhance the sensitivity of the interferometer
• Squeezed coherent states combine the properties of coherent states and squeezed states, having a non-zero coherent amplitude and reduced quantum noise in one quadrature
• Squeezed states can be characterized by the squeezing parameter $r$ and the squeezing angle $\theta$, which determine the amount and direction of the squeezing
• Quantum entanglement can be generated by combining two squeezed states on a beam splitter, creating Einstein-Podolsky-Rosen (EPR) entangled states
• Squeezed states find applications in quantum metrology, quantum imaging, and continuous-variable quantum information processing

Entangled Photon States

• Entangled photon states are quantum states in which two or more photons exhibit strong correlations that cannot be explained by classical theories
• Entanglement can occur in various degrees of freedom (polarization, frequency, spatial mode, time-bin) and between multiple photons
• Bell states are maximally entangled two-photon states that form the basis for many quantum information processing tasks (quantum teleportation, superdense coding)
  • The four Bell states are $|\Phi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|HH\rangle \pm |VV\rangle)$ and $|\Psi^{\pm}\rangle = \frac{1}{\sqrt{2}}(|HV\rangle \pm |VH\rangle)$, where $H$ and $V$ denote horizontal and vertical polarization, respectively
• Entangled photon states can be generated through the process of spontaneous parametric down-conversion (SPDC) in nonlinear crystals or by using quantum dots and atomic systems
• Violate Bell's inequality, demonstrating the incompatibility of quantum mechanics with local hidden-variable theories
• Enable secure quantum communication (quantum key distribution), quantum teleportation, and quantum computing with photonic qubits
• Greenberger-Horne-Zeilinger (GHZ) states and cluster states are examples of multi-photon entangled states that are essential for quantum error correction and measurement-based quantum computing

Measuring and Manipulating Quantum Light

• Quantum states of light can be measured and manipulated using various techniques and devices
• Photon counting can be performed using single-photon detectors (avalanche photodiodes, superconducting nanowire detectors) to measure the number of photons in a state
• Homodyne detection measures the amplitude and phase quadratures of a quantum state by interfering it with a strong local oscillator on a beam splitter
  • Enables the reconstruction of the quantum state through quantum tomography techniques (maximum likelihood estimation, Bayesian inference)
• Quantum state tomography allows for the complete characterization of a quantum state by performing a series of measurements in different bases
• Quantum logic gates (single-qubit gates, two-qubit gates) can be implemented using linear optical elements (beam splitters, phase shifters) and measurement-induced nonlinearities
• Quantum memories can be used to store and retrieve quantum states of light using atomic ensembles or solid-state systems (rare-earth-doped crystals, nitrogen-vacancy centers in diamond)
• Quantum frequency conversion enables the transfer of quantum states between different wavelengths, facilitating the integration of quantum systems with telecommunications infrastructure
• Quantum state engineering techniques (photon addition, photon subtraction, photon catalysis) allow for the creation of non-classical states of light with tailored properties

Applications in Quantum Technologies

• Quantum states of light find numerous applications in various quantum technologies
• Quantum cryptography uses entangled photon states or squeezed states to enable secure communication through protocols like BB84 and continuous-variable quantum key distribution
• Quantum computing with photonic qubits relies on entangled photon states and linear optical quantum gates to perform quantum information processing tasks
  • Boson sampling, a problem that is hard for classical computers, can be solved using photonic quantum devices
• Quantum metrology exploits squeezed states and entangled photon states to achieve precision measurements beyond the standard quantum limit (gravitational wave detection, optical magnetometry)
• Quantum imaging techniques (ghost imaging, sub-shot-noise imaging) utilize the quantum correlations between photons to enhance image resolution and sensitivity
• Quantum sensing uses entangled photon states to detect weak signals and improve the sensitivity of sensors (quantum illumination, quantum radar)
• Quantum simulation with photonic systems can be used to study complex quantum systems and solve problems in condensed matter physics and chemistry
• Quantum networks rely on entangled photon states to establish long-distance quantum communication and distribute quantum resources between nodes
  • Quantum repeaters, which use quantum memories and entanglement swapping, are essential for extending the range of quantum communication