Quantum Optics Group

• As classical counterparts utilizing both digit and analog information processing, quantum qubits in continuous variables (CV) provide a complementary family to the fragile discrete variables only with single photons and photon pairs.
• Continuous variables (CV) states, including coherent and squeezed states, are intrinsically multi-particle quantum superpositions and may be used to probe the quantum-to-classical transition, as well as the generation of Einstein-Podolsky-Rosen states.
• To have a fully understanding about quantum wave function, we have implemented the quantum state tomography, including the Wigner function and density matrix for quantum wave-packets.

• We report our implementation of a squeezer, by generating squeezed vacuum state at 1064 nm through a bow-tie optical parametric oscillator cavity operated below the threshold. With the help of our home-made balanced homodyne detector (BHD), characterized with a Common Mode Rejection Ratio (CMRR) more than 80dB. Detection of 10 dB vacuum noise squeezing at 1064 nm [1].

 

Targets:

The astrophysical reach of current and future ground-based gravitational-wave detectors is mostly limited by quantum noise, induced by vacuum fluctuations entering the detector output port. The replacement of this ordinary vacuum field with a squeezed vacuum field has proven to be an effective strategy to mitigate such quantum noise and it is currently used in advanced detectors.

However, current squeezing cannot improve the noise across the whole spectrum because of the Heisenberg uncertainty principle: when shot noise at high frequencies is reduced, radiation pressure at low frequencies is increased. A broadband quantum noise reduction is possible by using a more complex squeezing source, obtained by reflecting the squeezed vacuum off a Fabry-Perot cavity, known as filter cavity.

 

Achievements:

• First demonstration of a frequency-dependent squeezed vacuum source able to reduce quantum noise of advanced gravitational-wave detectors in their whole observation bandwidth. The experiment uses a suspended 300-m-long filter cavity, similar to the one planned for KAGRA, Advanced Virgo, and Advanced LIGO, and capable of inducing a rotation of the squeezing ellipse below 100 Hz [1].

Next:


The plan is to have filter cavities like these installed for the next round of observing that should start in 2022. Eventually, this and other improvements are expected to give an eightfold increase in detection rate, for both black hole and neutron star mergers, which are observable, respectively, at low and high frequencies.
Squeezing into the Tunnel !!

• Based on the niche of silicon photonics technologies and semiconductor industries in Taiwan, we plan to integrate these enabling technologies, based our expertise on photonic qubits (single photon source, entangled photon pair, squeezed light), integrated optical components based on silicon photonics, and photon detector arrays (single photon avalanche diode, homodyne detector), for the realization of "Scalable Quantum Photonic Chips on Silicon".
• The target is to incorporate our current fabrication technologies into a large-scale quantum system, and to realize multiplexing silicon photonic quantum chips with the ability to perform fault-tolerant quantum computation at room temperature.

• Exploiting the correspondence between the Wigner distribution function and the frequency-resolved optical gating (FROG) measurement, we experimentally demonstrate the existence of chessboard-like interference patterns with a time-bandwidth product smaller than that of a transform-limited pulse in the phase-space representation of compass states. Using superpositions of four electric pulses as the realization of compass states, we have shown via direct measurements that displacements leading to orthogonal states can be smaller than limits set by uncertainty relations. In the experiment we observe an exactly chronocyclic correspondence to the sub-Planck structure in the interference pattern appearing for the superposition of two Schrodinger-cat-like states in a position-momentum phase space [1].
• We present a phase-space study of a non-Hermitian Hamiltonian with PT symmetry based on the Wigner distribution function. For an arbitrary complex potential, we derive a generalized continuity equation for the Wigner function flow and calculate the related circulation values. Studying the vicinity of an exceptional point, we show that a PT-symmetric phase transition from an unbroken PT -symmetry phase to a broken one is a second-order phase transition [2].
• We derive a continuity equation for the evolution of the SU(2) Wigner function under nonlinear Kerr evolution. We give explicit expressions for the resulting quantum Wigner current, and discuss the appearance of the classical limit. We show that the global structure of the quantum current significantly differs from the classical one, which is clearly reflected in the form of the corresponding stagnation lines [3].

Achievements:

• We rule out PT-symmetry as a fundamental theory, but it still could be useful as an effective theory and an interesting model for open systems in classical optics [1].
• Due to noisy environment, quantum signals are in general fragile and hard to be detected. Through weak measurement to enlarge the quantum signal, we disclose the way to simulate pseudo-Hermitian system quantum theory [6].
• For the quantification of steerability, we prove that a strengthened version of the steering fraction is a convex steering monotone and demonstrate how it is related to two other steering monotones, namely, steerable weight and steering robustness [3].
• For a bipartite state with equal local dimension d, we prove that one can obtain work gain under Landauer’s erasure process on one party in the identically and independently distributed limit when the corresponding fully entangled fraction is larger than d. As a step to link quantum thermodynamics and quantum nonlocality, we also provide a simple picture to approximate the optimal work extraction and suggest a potential thermodynamic interpretation of the fully entangled fraction for isotropic states [4].

Next:

Achievements:

• How to improve our understanding of noisy quantum walks in networks with computer vision? we came up with an automated approach to study quantum transport properties and predict a possibility of a quantum advantage in particle transfer, with a specific neural network called classical-quantum convolutional neural network (CQCNN) [1].

Next:

Implementation of Quantum Machine Learning in IBM Q Experience!

• Quantum-noise squeezing and correlations are two key quantum properties that can exhibit completely different characteristics when compared to the predictions of the classical theory. Almost all the proposed applications to quantum measurements and quantum information treatment utilize either one or both of these properties.
• The quantum nonlinear Schrodinger equation (QNLSE) has been widely used as a model equation for studying the quantum effects of bosonic solitons, i.e., the evolution equation of a one-dimensional bosonic system with δ-function interaction under the second quantization framework. In particular, solitons in optical fibers have been known to serve as a platform for demonstrating macroscopic quantum properties in optical fields, such as quadrature squeezing, amplitude squeezing, and both intra-pulse and inter-pulse correlations.
• Meanwhile, low-absorption, slow-light and quantum memory make electromagnetically induced transparency (EIT) to be a promising ingredient in the development of quantum technologies. In the scenario, quantum optical pulse propagation in EIT, quantum squeezing generation in coherent population trapping media, large cross-phase modulation at few-photon level, and quantum correlated light generation as well as multiple fields correlation have been actively studied.

• Solitons after collision are rigorously proved to become quantum entangled in the sense that their quadrature components of suitably selected internal modes satisfy the inseparability criterion [10].
  
• The quantum-noise squeezing and quantum correlations are revealed for fiber solitons [2], Bragg solitons [1, 3], gap solitons in Bose-Einstein condensate [7], breathers [5, 6], bounded solitons in mode-locked fiber lasers [4, 8].
• A general quantum theory of self-induced transparency 􏰋(SIT)􏰃 solitons is developed with nonlinear quantum effects of atoms taken into account [9].
• The quantum fluctuation effects in the time-of-flight (TOF) experiment for a condensate released from an optical-lattice potential is studied within the truncated Wigner approximation [12].
  
• Beyond the adiabatic approximation, we develop a quantum theory for optical probe pulses propagating in electromagnetically-induced-transparency (EIT) media by including Langevin noise operators and asking the field operator to satisfy bosonic commutation relation [15, 16, 19].
• Considering matter wave bright solitons from weakly coupled Bose-Einstein condensates trapped in a double-well potential, we study the formation of macroscopic non- classical states, including Schrodinger-cat superposition state and maximally path entangled N00N-state. We have shown that solitons, as quantum nonlinear structured field objects, allow super-Heisenberg (SH) sensitivity, i.e. beyond Heisenberg limit, is one of the principal problems for current quantum metrology [17, 18].

Next:

Quantum Metrology with Quantum Solitons !

• Ite A. Yu (余怡德); Department of Physics, National Tsing Hua University, Hsinchu
  [4] You-Lin Chuang, RKL, and Ite A. Yu, "Generation of quantum entanglement based on electromagnetically induced transparency media," submitted for publication (April 2020); [arXiv: 1906.12025].
  
  [3] You-Lin Chuang, RKL, and Ite A. Yu, "Generation of quantum entanglement based on electromagnetically induced transparency media," submitted for publication (April 2020); [arXiv: 1906.12025].
  
  [2] You-Lin Chuang, RKL, and Ite A. Yu, "Optical-density-enhanced squeezed-light generation without optical cavities," Phys. Rev. A 96, 053818 (2017) [download].
  
  [1] You-Lin Chuang, Ite A. Yu, and RKL, "Quantum theory for pulse propagation in electromagnetically-induced-transparency media beyond the adiabatic approximation," Phys. Rev. A 91, 063818 (2015) [download].
  
  
  
• Daw-Wei Wang (王道維); Department of Physics, National Tsing Hua University, Hsinchu
  [2] Ching-Hao Wang, Tzay-Ming Hong, RKL, and Daw-Wei Wang, "Particle-wave duality in quantum tunneling of a bright soliton," Optics Express 20, 22675 (2012) [download].
  

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  [1] Shiang Fang, RKL, Daw-Wei Wang, "Quantum fluctuations and condensate fraction during time-of-flight expansion," Phys. Rev. A 82, 031601(R) (2010). [download]
  
  
  
• Chih-Sung Chuu (褚志崧); Department of Physics, National Tsing Hua University, Hsinchu
  
  
• Mark Ming-Chang Lee (李明昌); Institute of Photonics Technologies, National Tsing Hua University, Hsinchu



• Shang-Da Yang (楊尚達); Institute of Photonics Technologies, National Tsing Hua University, Hsinchu
  [1] Ludmila Praxmeyer, Chih-Cheng Chen, Popo Yang, Shang-Da Yang, and RKL, "Direct measurement of time-frequency analogs of sub-Planck structures," Phys. Rev. A 93, 053835 (2016) [download].
  
  
  
• Chi-Yang Lee (李紫原); Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu
  [3] Jeng-Yi Lee, Min-Chiao Tsai, Po-Chin Chen, Tin-Tin Chen, Kuei-Lin Chan, Chi-Young Lee , and RKL, "Thickness effect on light absorption and scattering for nanoparticles in shape of hollow-spheres," J. Phys. Chem: C 119, 25754 (2015) [download].
  
  [2] Min-Chiao Tsai, Jeng-Yi Lee, Ya-Chen Chang, Min-Han Yang, Tin-Tin Chen, I-Chun Chang, Pei-Chi Lee, Hsin-Tien Chiu, RKL, and Chi-Young Lee, "Scattering resonance enhanced dye absorption of dye sensitized solar cells at optimized hollow structure size," J. Power Sources 268, 1 (2014) [download].
  
  [1] Min-Chiao Tsai, Jeng-Yi Lee, Po-Chin Chen, Yuan-Wei Chang, Ya-Chen Chang, Min-Han Yang, Hsin-Tien Chiu, I-Nan Lin, RKL, and Chi-Young Lee, "Effects of size and shell thickness of TiO2 hierarchical hollow spheres on photocatalytic behavior: An experimental and theoretical study," Applied Catalysis B: Environmental 147, 499 (2014) [download].
  
  
• Yinchieh Lai (賴暎杰); Department of Photonics, National Chiao-Tung University, Hsinchu
  [10] Yinchieh Lai and RKL, "Entangled Quantum Nonlinear Schrodinger Solitons," Phys. Rev. Lett. 103, 013902 (2009); also selected in Virtual Journal of Nanoscale Science & Technology 20, Issue 2 (2009); and selected in Virtual Journal of Quantum Information 9, Issue 7 (2009) [download].
  
  [9] RKL and Yinchieh Lai, "Quantum squeezing and correlation of self-induced transparency solitons," Phys. Rev. A 80, 033839 (2009) [download].
  
  [8] RKL, Yinchieh Lai, and Boris A. Malomed, "Photon-number fluctuation and correlation of bound soliton pairs in mode-locked fiber lasers," Opt. Lett. 34, 3084 (2005) [download].
  
  [7] RKL, Elena A. Ostrovskaya, Yuri S. Kivshar, and Yinchieh Lai, "Quantum-noise properties of matter-wave gap solitons," Phys. Rev. A 72, 033607 (2005) [download].
  
  [6] RKL, Yinchieh Lai, and Yuri S. Kivshar, "Quantum correlations in soliton collisions," Phys. Rev. A 71, 035801 (2005); also selected in Virtual Journal of Quantum Information 5, Issue 4 (2005) [download].
  
  [5] RKL, Yinchieh Lai, and Boris Malomed, "Generation of photon-number entangled soliton pairs through interactions," Phys. Rev. A 71, 013816 (2005); also selected in Virtual Journal of Quantum Information 5, Issue 2 (2005) [download].
  
  [4] RKL, Yinchieh Lai, and Boris Malomed, "Quantum correlations in bound-soliton pairs and trains in fiber lasers," Phys. Rev. A 70, 063817 (2004); also selected in Virtual Journal of Quantum Information 5, Issue 1 (2005) [download].
  
  [3] RKL and Yinchieh Lai, "Quantum theory of fiber Bragg grating solitons," J. Opt. B: Quantum and Semiclassical Optics 6, S638 (special issue 2004) in memoriam of Hermann Anton Haus, 1925-2003 [download].
  
  [2] RKL and Yinchieh Lai, and Boris A. Malomed, "Quantum Fluctuations around Bistable Solitons in the cubic-quintic nonlinear Schrodinger equation," J. Opt. B: Quantum and Semiclassical Optics 6, 367 (2004) [download].
  

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  [1] RKL and Yinchieh Lai, "Amplitude-squeezed fiber-Bragg-grating solitons," Phys. Rev. A 69, 021801(R) (2004); also selected in Virtual Journal of Nanoscale Science & Technology 9, Issue 8 (2004) [download].
  
  
  
• Tsin-Fu Jiang (江進福); Institute of Physics, National Chiao-Tung University, Hsinchu
  [1] YuanYao Lin, RKL, Yee-Mou Kao, and Tsin-Fu Jiang, "Band structures of a dipolar Bose-Einstein condensate in one-dimensional lattices," Phys. Rev. A 78, 023629 (2008) [download].
  
  
  
• Tien-chang Lu (盧廷昌); Department of Photonics, National Chiao-Tung University, Hsinchu
  [3] Yu-Chi Wang, Heng Li, Yu-Heng Hong, Kuo-Bin Hong, Fang-Chung Chen, Chia-Hung Hsu, RKL, Claudio Conti, Tsung Sheng Kao, and Tien-Chang Lu, "Flexible Organometal-Halide Perovskite Lasers for Speckle Reduction in Imaging Projection," ACS Nano 13, 5421 (2019) [download].
  
  [2] Kou-Bin Hong, Chun-Yan Lin, Tsu-Chi Chang, Wei-Hsuan Liang, Ying-Yu Lai, Chien-Ming Wu, You-Lin Chuang, Tien-Chang Lu, Claudio Conti, and RKL, "Lasing on nonlinear localized waves in curved geometry," Optics Express 25, 29068 (2017) [download].
  
  [1] Yonan Su, Chun-Yan Lin, Ray-Ching Hong, Wen-Xing Yang, Chien-Chung Jeng, Tien-Chang Lu, and RKL "Lasing on surface states in vertical-cavity surface-emission lasers," Opt. Lett. 39, 5582 (2014) [download].
  
  
  
• Po-Tsung Lee (李柏璁); Department of Photonics, National Chiao-Tung University, Hsinchu
  
  
• Ching-Yi Lai (賴青沂); Department of Electrical Engineering, National Chiao-Tung University
  [1] Li-Yi Hsu, Ching-Yi Lai, You-Chia Chang, Chien-Ming Wu, and RKL, "Carrying an arbitrarily large amount of information using a single quantum particle," Phys. Rev. A 102, 022620 (2020); [download].
  
  
  
• Yen-Hung Chen (陳彥宏); Department of Photonics, National Central University, Taoyuan
  
  
• Queena Yi-Shan Lee (李依珊); Department of Electrical Engineering, National Central University, Taoyuan
  
  
• Ming-Feng Shih (石明豐): Department of Physics, National Taiwan University, Taipei
  [2] Ray-Ching Hong, Chun-Yan Lin, You-Lin Chuang, Chien-Ming Wu, Yonan Su, Jeng Yi Lee, Chien-Chung Jeng, Ming-Feng Shih, and RKL, "Resonance in modulation instability from non-instantaneous nonlinearities," Opt. Lett. 43, 3329 (2018) [download].
  
  [1] Ming Shen, Yonan Su, Ray-Ching Hong, YuanYao Lin, Chien-Chung Jeng, Min-Feng Shih, and RKL, "Observation of phase boundaries in spontaneous optical pattern formation," Phys. Rev. A 91, 023810 (2015) [download].
  
  
  
• Chien-Chung Jeng (鄭建宗), Department of Physics, National Chung Hsing University, Taichung
  [10] Ray-Ching Hong, Chun-Yan Lin, You-Lin Chuang, Chien-Ming Wu, Yonan Su, Jeng Yi Lee, Chien-Chung Jeng, Ming-Feng Shih, and RKL, "Resonance in modulation instability from non-instantaneous nonlinearities," Opt. Lett. 43, 3329 (2018) [download].
  
  [9] Chien-Chung Jeng, Yonan Su, Ray-Ching Hong, and RKL, "Control modulation instability in photorefractive crystals by the intensity ratio of background to signal fields," Optics Express 23, 10266 (2015) [download].
  
  [8] Ming Shen, Yonan Su, Ray-Ching Hong, YuanYao Lin, Chien-Chung Jeng ,, Min-Feng Shih, and RKL, "Observation of phase boundaries in spontaneous optical pattern formation," Phys. Rev. A 91, 023810 (2015) [download].
  
  [7] Yonan Su, Chun-Yan Lin, Ray-Ching Hong, Wen-Xing Yang, Chien-Chung Jeng , Tien-Chang Lu, and RKL "Lasing on surface states in vertical-cavity surface-emission lasers," Opt. Lett. 39, 5582 (2014) [download].
  
  [6] L. Chen, Q. Wang, Ming Shen, H. Zhao, Y.Y. Lin, Chien-Chung Jeng , RKL, and W. Krolikowski, "Nonlocal dark solitons under competing cubic-quintic nonlinearities," Opt. Lett. 38, 13 (2013) [download].
  
  [5] Ming Shen, J.-J. Zheng, Q. Kong, YuanYao Lin, Chien-Chung Jeng, RKL, and W. Krolikowski, "Stabilization of counter-rotating vortex pairs in nonlocal media," Phys. Rev. A 86, 013827 (2012) [download].
  
  [4] Chandroth P. Jisha, Kuei-Chu Hsu, YuanYao Lin, Ja-Hon Lin, Chien-Chung Jeng, and RKL, "Tunable pattern transitions in a liquid-crystal-monomer mixture using two-photon polymerization," Opt. Lett. 37, 4931 (2012) [download].
  
  [3] Ming Shen, YuanYao Lin, Chien-Chung Jeng, and RKL, "Vortex pairs in nonlocal nonlinear meida," J. Opt. 14, 065204 (2012) [download].
  
  [2] Ming Shen, Qian Kong, Chien-Chung Jeng, Li-Juan Ge, RKL, and Wieslaw Krolikowski, "Instability suppression of vector-necklace-ring solitons in nonlocal media," Phys. Rev. A 83, 023825 (2011) [download].
  
  [1] Chien-Chung Jeng, YuanYao Lin, Ray-Ching Hong, and RKL, "Optical Pattern Transitions from Modulation to Transverse Instabilities in Photorefractive Crystals," Phys. Rev. Lett. 102, 153905 (2009) [download].
  
  
  
• Watson Kuo (郭華丞), Department of Physics, National Chung Hsing University, Taichung
  [1] Yu-Han Chang, Raul A. Robles Robles, Vanna Chrismas Silalahi, Cen-Shawn Wu, Gang Wang, Giulia Marcucci, Laura Pilozzi, Claudio Conti, RKL, and Watson Kuo, "Direct Observation of Topological Protected Edge States in Slow-Light," submitted for publication (Apr. 2020); [arXiv: 2004.09282].
  
  
  
• Jen-Hsu Chang (張仁煦), Graduate School of National Defense, National Defense University, Taoyuan
  [1] Chun-Yan Lin, Jen-Hsu Chang, Gershon Kurizki, and RKL, "Solitons supported by intensity-dependent dispersion," Opt. Lett. 45, 1471 (2020) [download].
  
  [2] Jen-Hsu Chang, Chun-Yan Lin, and RKL, "Nonlinear Effective Masses Schrodinger Equation with Harmonic Oscillator," submitted for publication (Aug. 2019); [arXiv: 1911.12477].
  
  
  
• Shin-Tza Wu (吳欣澤), Department of Physics, National Chung Cheng University, Chiayi
  [1] Popo Yang, Ivan F. Valtierra, Andrei B. Klimov, Shin-Tza Wu, RKL, and Luis L. Sanchez-Soto, and Gerd Leuchs, "The Wigner flow on the sphere," Physica Scripta 94, 044001 (2019); for the New Focus issue: Quantum Optics and Beyond- in honour of Wolfgang Schleich [download].
  
  
  
• Yeong-Cherng Liang (梁永成), Department of Physics, National Cheng Kung University, Tainan
  [1] Chung-Yun Hsieh, Yeong-Cherng Liang, and RKL, "Quantum steerability: Characterization, quantification, superactivation, and unbounded amplification," Phys. Rev. A 94, 062120 (2016) [download].
  
  
  
• Li-Yi Hsu(徐立義); Department of Physics, Chung Yuan Christian University, Taoyuan
  [1] Li-Yi Hsu, Ching-Yi Lai, You-Chia Chang, Chien-Ming Wu, and RKL, "Carrying an arbitrarily large amount of information using a single quantum particle," Phys. Rev. A 102, 022620 (2020); [download].
  
  
  
• Cheng-Ling Lee (李澄琳); Department of Physics, National United University, Miaoli
  [3] Cheng-Ling Lee, RKL, and Yee-Mou Kao, "Lagrange-multiplier-constrained optimization for designing narrowband dispersionless fiber Bragg gratings," Optical Engineering 47, 015005 (2008) [download].
  
  [2] Cheng-Ling Lee, RKL, and Yee-Mou Kao, "Synthesis of long-period fiber gratings with a Lagrange multiplier optimization method," Optics Comm. 281, 61 (2008) [download].
  
  [1] Cheng-Ling Lee, RKL, and Yee-Mou Kao, "Design of multichannel DWDM fiber Bragg grating filters by Lagrange multiplier constrained optimization," Optics Express 14, 11002 (2006) [download].
  
  
  
• YuanYao Lin (林元堯); Department of Photonics, National Sun Yat-sen University, Kaohsiung
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• Jeng Yi Lee (李政誼); Department of Photonics, National Dong Hwa University, Hualien
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Textbook

Text Book: S. Friedberg, A. Insel, and L. Spence, "Linear Algebra," 4th Edition (Pearson).

Reference Book

Gilbert Strang, "Introduction to Linear Algebra," International 5th Edition (Wellesley-Cambridge Press).

RK's goals

You should go through the whole Textbook.

Instead of repeating what you can find in the Textbook, I will illustrate the content from Scratch, by raising questions to you first.

You need to have Quiz (40\% to the semester score) on every Wednesday.

Syllabus

• Course Introduction: [link for my scratch notes]
• Part I: Vector Spaces
• [Vector Space], [Subspaces], [Bases], Dimensions, Linear Independence, Dual Spaces,
• Linear Transformation and Matrices
• [Quiz 1], Quiz 2, Quiz 3, Quiz 4
• Part II: Elementary Matrix Operation
• Solving Linear Equations, Elimination, Inverse Matrices
• Composition, Invertibility and Isomorphisms
• Inner Product, Norm, Gram-Schmidt Orthogonalization Process
• Matrix Operations, Rank, Inverse, Determinants,
• Part III: Operators
• Normal, Self-Adjoint, Unitary, and Orthogonal
• Singular Value Decomposition
• Spectral Theorem, Bilinear and Quadratic Forms

Evaluation

Quiz > 12, 40%
Exams x 3, 60%

Office hours

RK: 1:00-5:00 PM, Wednesday, at R 911, Delta Hall, or by appointment.
TA time: 6:30-8:30 PM, Wednesday, at R 217, Delta Hall

Expected Outputs:

Quantum properties of Electromagnetic Fields;

Non-classical light and its generation, measurement, and applications;

Interaction between photon-atoms;

Test of Quantum Mechanics by Optics;

Syllabus

Introduction to Quantum Optics

Quantum Theory,

Quantum Field theory of Light,

Number states and Coherent States,

Squeezed States and Phase Space,

Simple Optical Instruments:

Photon-atom interaction:

Rabi oscillation,

Jaynes-Cummings Hamiltonian,

Dicke model,

Cavity-Quantum Electro-Dynamics (Cavity-QED),

Electromagnetically Induced Transparency (EIT),

Optical Parametric Oscillator (OPO),

Dissipative Systems

Applications of Quantum Optics:

Entanglement,

Horizons,

Gravitational Wave Detectors,

Test of Quantum Mechanics,

Quantum Information Processing

References:

U. Leonhardt, "Essential Quantum Optics," Cambridge (2010).

G. S. Agarwal, "Quantum Optics", Cambridge University (2013).

D. F. Walls and G. J. Milburn, "Quantum Optics," 2nd Ed. Springer (2008).

M. Fox, "Quantum Optics, an introduction," Oxford (2006).

C. C. Gerry and P. L. Knight, "Introductory Quantum Optics,"" Cambridge (2005).

Y. Yamamoto and A. Imamoglu, "Mesoscopic Quantum Optics," Wiley (1999).

M. O. Scully, and M. S Zubairy, "Quantum Optics," Cambridge (1997).