Introduction to quantum optics

These are the lecture notes for a course that I am teaching at Zhiyuan College of Shanghai Jiao Tong University (available at www.youtube.com/derekkorg), though the first draft was created for a previous course I taught at the University of Erlangen-Nuremberg in Germany. It has been designed for students who have only had basic training on quantum mechanics, and hence, the course is suited for people at all levels (say, from the end of the bachelor all the way into the PhD). The notes are a work in progress, meaning that some proofs and many figures are still missing. However, I’ve tried my best to write everything in such a way that a reader can follow naturally all arguments and derivations even with these missing bits. Also a few chapters are left to add, including one on mathematical methods to analyze the dynamics of open systems, and another introducing the plethora of current experimental platforms where the tools and ideas developed in these notes are being currently implemented

Floquet theory for temporal correlations and spectra in time-periodic open quantum systems: Application to squeezed parametric oscillation beyond the rotating-wave approximation

Open quantum systems can display periodic dynamics at the classical level either due to external periodic modulations or to self-pulsing phenomena typically following a Hopf bifurcation. In both cases, the quantum fluctuations around classical solutions do not reach a quantum-statistical stationary state, which prevents adopting the simple and reliable methods used for stationary quantum systems. Here we put forward a general and efficient method to compute two-time correlations and corresponding spectral densities of time-periodic open quantum systems within the usual linearized (Gaussian) approximation for their dynamics. Using Floquet theory we show how the quantum Langevin equations for the fluctuations can be efficiently integrated by partitioning the time domain into one-period duration intervals, and relating the properties of each period to the first one. Spectral densities, like squeezing spectra, are computed similarly, now in a two-dimensional temporal domain that is treated as a chessboard with one-period x one-period cells. This technique avoids cumulative numerical errors as well as efficiently saves computational time. As an illustration of the method, we analyze the quantum fluctuations of a damped parametrically-driven oscillator (degenerate parametric oscillator) below threshold and far away from rotating-wave approximation conditions, which is a relevant scenario for modern low-frequency quantum oscillators. Our method reveals that the squeezing properties of such devices are quite robust against the amplitude of the modulation or the low quality of the oscillator, although optimal squeezing can appear for parameters that are far from the ones predicted within the rotating-wave approximation.Comment: Comments and constructive criticism are welcom

Deep recurrent networks predicting the gap evolution in adiabatic quantum computing

One of the main challenges in quantum physics is predicting efficiently the dynamics of observables in many-body problems out of equilibrium. A particular example occurs in adiabatic quantum computing, where finding the structure of the instantaneous gap of the Hamiltonian is crucial in order to optimize the speed of the computation. Inspired by this challenge, in this work we explore the potential of deep learning for discovering a mapping from the parameters that fully identify a problem Hamiltonian to the full evolution of the gap during an adiabatic sweep applying different network architectures. Through this example, we find that a limiting factor for the learnability of the dynamics is the size of the input, that is, how the number of parameters needed to identify the Hamiltonian scales with the system size. We demonstrate that a long short-term memory network succeeds in predicting the gap when the parameter space scales linearly with system size. Remarkably, we show that once this architecture is combined with a convolutional neural network to deal with the spatial structure of the model, the gap evolution can even be predicted for system sizes larger than the ones seen by the neural network during training. This provides a significant speedup in comparison with the existing exact and approximate algorithms in calculating the gap

Noncritical generation of nonclassical frequency combs via spontaneous rotational symmetry breaking

Synchronously pumped optical parametric oscillators (SPOPOs) are optical cavities driven by mode-locked lasers, and containing a nonlinear crystal capable of down-converting a frequency comb to lower frequencies. SPOPOs have received a lot of attention lately because their intrinsic multimode nature makes them compact sources of quantum correlated light with promising applications in modern quantum information technologies. In this work we show that SPOPOs are also capable of accessing the challenging and interesting regime where spontaneous symmetry breaking confers strong nonclassical properties to the emitted light, which has eluded experimental observation so far. Apart from opening the possibility of studying experimentally this elusive regime of dissipative phase transitions, our predictions will have a practical impact, since we show that spontaneous symmetry breaking provides a specific spatiotemporal mode with large quadrature squeezing for any value of the system parameters, turning SPOPOs into robust sources of highly nonclassical light above threshold

Nonlinear optical Galton board

We generalize the concept of optical Galton board (OGB), first proposed by Bouwmeester et al. {[}Phys. Rev. A \textbf{61}, 013410 (2000)], by introducing the possibility of nonlinear self--phase modulation on the wavefunction during the walker evolution. If the original Galton board illustrates classical diffusion, the OGB, which can be understood as a grid of Landau--Zener crossings, illustrates the influence of interference on diffusion, and is closely connected with the quantum walk. Our nonlinear generalization of the OGB shows new phenomena, the most striking of which is the formation of non-dispersive pulses in the field distribution (soliton--like structures). These exhibit a variety of dynamical behaviors, including ballistic motion, dynamical localization, non--elastic collisions and chaotic behavior, in the sense that the dynamics is very sensitive to the nonlinearity strength.Comment: 8 pages, 8 figure

Deep Learning of Quantum Many-Body Dynamics via Random Driving

Neural networks have emerged as a powerful way to approach many practical problems in quantumphysics. In this work, we illustrate the power of deep learning to predict the dynamics of a quantummany-body system, where the training is based purely on monitoring expectation values of observables under random driving. The trained recurrent network is able to produce accurate predictions for driving trajectories entirely different than those observed during training. As a proof of principle, here we train the network on numerical data generated from spin models, showing that it can learn the dynamics of observables of interest without needing information about the full quantum state.This allows our approach to be applied eventually to actual experimental data generated from aquantum many-body system that might be open, noisy, or disordered, without any need for a detailedunderstanding of the system. This scheme provides considerable speedup for rapid explorations andpulse optimization. Remarkably, we show the network is able to extrapolate the dynamics to times longer than those it has been trained on, as well as to the infinite-system-size limit

Spontaneous symmetry breaking as a resource for noncritically squeezed light

In the last years we have proposed the use of the mechanism of spontaneous symmetry breaking with the purpose of generating perfect quadrature squeezing. Here we review previous work dealing with spatial (translational and rotational) symmetries, both on optical parametric oscillators and four-wave mixing cavities, as well as present new results. We then extend the phenomenon to the polarization state of the signal field, hence introducing spontaneous polarization symmetry breaking. Finally we propose a Jaynes-Cummings model in which the phenomenon can be investigated at the single-photon-pair level in a non-dissipative case, with the purpose of understanding it from a most fundamental point of view.Comment: Review for the proceedings of SPIE Photonics Europe. 11 pages, 5 figures

Quantum coherent control of highly multipartite continuous-variable entangled states by tailoring parametric interactions

The generation of continuous-variable multipartite entangled states is important for several protocols of quantum information processing and communication, such as one-way quantum computation or controlled dense coding. In this article we theoretically show that multimode optical parametric oscillators can produce a great variety of such states by an appropriate control of the parametric interaction, what we accomplish by tailoring either the spatio-temporal shape of the pump, or the geometry of the nonlinear medium. Specific examples involving currently available optical parametric oscillators are given, hence showing that our ideas are within reach of present technology.Comment: 14 pages, 5 figure

Generating highly squeezed Hybrid Laguerre-Gauss modes in large-Fresnel-number Degenerate Optical Parametric Oscillators

We theoretically describe the quantum properties of a large Fresnel number degenerate optical parametric oscillator with spherical mirrors that is pumped by a Gaussian beam. The resonator is tuned so that the resonance frequency of a given transverse mode family coincides with the down-converted frequency. After demonstrating that only the lower orbital angular momentum (OAM) Laguerre-Gauss modes are amplified above threshold, we focus on the quantum properties of the rest of (classically empty) modes. We find that combinations of opposite OAM (Hybrid Laguerre-Gauss modes) can exhibit arbitrary large quadrature squeezing for the lower OAM non amplified modes.Comment: 10 pages, 3 figures and 2 table