ENGLISH ABSTRACT: We present here a new generalized Time-Frequency-Represenation (TFR) for use in describing ultra-fast laser pulses. The TFR is developed in terms of Fourier- Hermite-Gauss (FHG) polynomials on a von Neumann TFR lattice, by expanding the lattice in a third dimension, thus forming a lattice cube. The temporal and spectral coefficients of the Hermite-Gauss (HG) clearly are inherently functionally Fourier transform invariant. The enhanced numerical complexity of the FHG TFR is greatly reduced by exploiting the translation property of Hermite polynomials by means of Pascal matrices. Although the new FHG TFR represents an over complete basis set, it can be reduced by a subset selection to a complete basis. This method and the accompanying overlap integral is then developed and the complex orthogonality and similarity of the temporal and spectral overlap integral matrices is then analytically proved. Numerically the Pascal matrices are unstable. The new two correlation FHG TFR lattice cubes present an improvement on the traditional TFRs in that they have the advantage that it is unnecessary to compute the undesirable inverse overlap matrix to reconstruct the signal, temporally and spectrally, i.e. they contain all the information necessary to reconstruct the signal. It is then proven that it is a digital requirement to double the original proposed bandwidths of the signal inputs, here and also for the von Neumann TFR. The Hermite-Gauss polynomials correspond to the number states {nk} of the Glauber-Sudarshan coherent states in Fock space. A classical correspondence analogy between Glauber-Sudarshan coherent states in Fock space and the temporal FHG TFR is then considered under certain conditions and thus allows for a comparison of the amplitudes between the two decompositions for each kkk = !k/c, culminating in an optimization procedure to determine a “classical” coherent state correlation TFR. Application simulation results of quantum coherent control of IR ultra-short laser pulse interaction with octahedral molecules utilizing an optimal genetic algorithm are presented. A representative shaped laser pulse is used throughout to compare various TFRs.
ENGLISH ABSTRACT: We present here a new generalized Time-Frequency-Represenation (TFR) for use in describing ultra-fast laser pulses. The TFR is developed in terms of Fourier- Hermite-Gauss (FHG) polynomials on a von Neumann TFR lattice, by expanding the lattice in a third dimension, thus forming a lattice cube. The temporal and spectral coefficients of the Hermite-Gauss (HG) clearly are inherently functionally Fourier transform invariant. The enhanced numerical complexity of the FHG TFR is greatly reduced by exploiting the translation property of Hermite polynomials by means of Pascal matrices. Although the new FHG TFR represents an over complete basis set, it can be reduced by a subset selection to a complete basis. This method and the accompanying overlap integral is then developed and the complex orthogonality and similarity of the temporal and spectral overlap integral matrices is then analytically proved. Numerically the Pascal matrices are unstable. The new two correlation FHG TFR lattice cubes present an improvement on the traditional TFRs in that they have the advantage that it is unnecessary to compute the undesirable inverse overlap matrix to reconstruct the signal, temporally and spectrally, i.e. they contain all the information necessary to reconstruct the signal. It is then proven that it is a digital requirement to double the original proposed bandwidths of the signal inputs, here and also for the von Neumann TFR. The Hermite-Gauss polynomials correspond to the number states {nk} of the Glauber-Sudarshan coherent states in Fock space. A classical correspondence analogy between Glauber-Sudarshan coherent states in Fock space and the temporal FHG TFR is then considered under certain conditions and thus allows for a comparison of the amplitudes between the two decompositions for each kkk = !k/c, culminating in an optimization procedure to determine a “classical” coherent state correlation TFR. Application simulation results of quantum coherent control of IR ultra-short laser pulse interaction with octahedral molecules utilizing an optimal genetic algorithm are presented. A representative shaped laser pulse is used throughout to compare various TFRs.
Recently we introduced the von Neumann representation as a joint time-frequency description for femtosecond laser pulses. Here we show that the von Neumann basis can be implemented into an evolutionary algorithm for adaptive optimization in coherent control. We perform simulations that demonstrate the efficiency compared to other parametrizations in the frequency domain. We also illustrate pulse-shape simplification by basis-function reduction. Essential structures using the von Neumann basis are retained without losing control performance significantly. In an optical demonstration experiment we show the practicality by producing double pulses with a given time separation. Adaptive control in time-frequency space will be especially valuable for quantum systems requiring specific transition frequencies at definite times.
The theoretical description of broadband, multimode quantum pulses undergoing a second-order χ(2)-nonlinear interaction can be quite intricate, due to the large dimensionality of the underlying phase space. However, in many cases only a few broadband (temporal) modes are relevant before and after the nonlinear interaction. Here we present an efficient framework to calculate the relation between the quantum states at the input and output of a nonlinear element in their respective relevant modes. Since the number of relevant input and output modes may differ, resulting in an open quantum system, we introduce the generalized Bloch-Messiah decomposition (GBMD), reducing the description to an equal number of input and output modes. The GBMD enables us to calculate the multimode Wigner function of the output state by convolving the rescaled Wigner function of the reduced input quantum pulse with a multivariate Gaussian phase-space function. We expand on this result by considering two examples input states: A Fock state in a single broadband mode and a two-mode squeezed vacuum, both in the THz-frequency regime, up-converted to a single output broadband mode of optical frequencies. We investigate the effect, the convolution and thermalization due to entanglement breakage have on the output Wigner function by calculating the von Neumann entropy of the output Wigner function. The methods presented here can be used to optimize the amplification or frequency conversion of broadband quantum states, opening an avenue to the generation and characterization of optical quantum states on ultrafast time scales.
We present a new primitive for quantum algorithms that implements a discrete Hermite transform efficiently, in time that is polylogarithmic in the dimension and the inverse of the allowable error. This transform, which maps basis states to states whose amplitudes are proportional to the Hermite functions, can be interpreted as the Gaussian analogue of the Fourier transform. Our algorithm is based on a method to exponentially fast-forward the evolution of the quantum harmonic oscillator, giving a simulation algorithm with nearly optimal circuit complexity for a fundamental Hamiltonian more than four decades after Feynman posed the simulation of quantum physics as an application of quantum computers.
Creating and controlling well-defined single-photon states is important for many quantum enhanced technologies. Information can be encoded in any degree of freedom associated with a single-photon field excitation, for example in the polarization or transverse spatial mode structure. The ability to encode multiple qubits in these states is desirable for high data transmission rates and increased information processing capacity. The spectral-temporal domain offers a large Hilbert space for encoding information, well suited to integrated optical architectures owing to the low cross-talk between channels. Indeed, time-frequency encoding is an integral component of existing information technologies infrastructure. Thus, complete, coherent control of the spectral-temporal mode structure of light is essential to advancing optical quantum technologies. Arbitrary control of single-photon states has been demonstrated in the polarization and spatial-momentum degrees of freedom however, is yet to be established in the spectral-temporal domain. Recently it was shown that spectral-temporal pulse shaping is experimentally feasible using nonlinear optical methods. However, such techniques can introduce noise photons, are challenging to implement deterministically and require specially prepared auxiliary pump pulses. Furthermore, they are typically only possible at relatively low repetition rates to avoid damage of nonlinear optical materials. Here, we demonstrate deterministic pulse shaping of single-photon wave packets by introducing a time-varying phase. Applying linear or quadratic phase to the wave packet introduces a spectral shift or spectral broadening, respectively. Achieving significant spectral manipulation requires temporal-phase modulation on the order of π. The phase must also vary on the same time scale as the optical pulses, approximately 1 ps in duration here. This relatively rapid phase variation is achieved using a fast electro-optic phase modulator. The modulator is driven by a time-varying voltage that is synchronized with a pulse train of single-photon wave packets for accurate temporal phase control. In this thesis, we experimentally demonstrate control of the spectral-temporal state of single-photon wave packets using this method. Heralded single-photon wave packets are generated by spontaneous parametric down conversion pumped by a frequency-doubled Titanium-Sapphire laser oscillator with 80 MHz repetition rate. To achieve significant modulation of single-photon pulses a 10 V peak-to-peak radio-frequency signal of 40 GHz drives the phase modulator, which requires approximately 3.6 V to achieve a π phase shift. The signal is synchronized with the oscillator pulse train using custom designed electronics. Details of these electronics are presented, resulting in a robust phase and amplitude controllable 40 GHz voltage source, phase locked to the single-photon pulse train. To demonstrate the utility of this pulse-shaping method we experimentally show spectral shearing of almost 1 nm for single-photon wave packets with 830 nm central wavelength and 1 nm bandwidth. Measurement of the second-order intensity correlation before and after the wave-packet manipulation remains constant within uncertainty of the measurement, showing that the quantum nature of the single-photon source is not deteriorated by the modulator. Preservation of the wave-packet coherence is verified by two-photon interference between spectrally sheared and non-modulated single-photons.
The intent of this study is to determine the feasibility of diatomics as molecular quantum computing candidates and shed insight into the use of such experimental laser pulse shaping methods to represent quantum logic gates. Four appropriate rovibrational states of model diatomic molecules are encoded as the qubit states. A set of 2-qubit quantum logic gates (ACNOT, CNOT, NOT, Hadamard) are represented by amplitude and phase shaped laser pulses. The combinations of amplitudes and phases that produce the optimal laser pulse representation, for each quantum logic gate, are determined by a Genetic Algorithm optimization routine. The theoretical laser pulse shaping is analogous to current experimental frequency-domain pulse shaping apparatus with amplitude and phase control at individual frequencies. A model set of diatomics is sampled in order to determine a relationship between optimal laser pulse shaping and the choice of diatomic molecule. We show that the choice of diatomic molecule greatly influences the ability to produce optimal laser pulse shapes to represent quantum logic gates. Tuneable parameters specific to laser pulse shaping instruments are varied to determine their effect on optimal pulse production. They include varying the number of amplitude and phase components, adjusting the number of frequency components, and altering the frequency resolution which is synonymous with altering the laser pulse duration. A time domain analytic form of the original frequency domain laser pulse function is derived, providing a useful means to infer the laser pulse dependencies on these parameters. Initially, we show that the appropriate choice of rovibrational state qubits of carbon monoxide (12C16O) and the use of simple shaped binary pulses, 2 amplitude and 2 phase components, can provide significant control for specific quantum gates. Further amplitude variation at each frequency component is shown to be a crucial requirement for optimal laser pulse shaping, whereas phase variation provides minimal contribution. We show that the generation of optimal laser pulse shapes is highly dependent upon the frequency resolution and increasing the number of frequency components provides incremental improvements to optimal laser pulses.
Quantum physics holds promise to revolutionise a number of technologies by enabling performance beyond what is feasible with classical resources, such as secure communications, precision measurement and high-speed simulation and computation. To realise this, quantum information must be encoded into one of the physical degrees of freedom of light. The time-frequency (TF) domain provides a infinite-dimensional space which is naturally compatible with integrated optical platforms. Additionally, future quantum networks will require coherent coupling of quantum systems that do not necessarily exhibit TF mode overlap. The ability to generate, coherently control and measure a variety of TF states of quantum light is therefore a necessary step towards implementing optical quantum technologies. Developing the building blocks for future optical quantum applications calls for an assortment of methods for measuring the properties of TF states of quantum light. Whilst much progress has been made towards this goal recently, deficiencies with existing methods remain. Efficient spectrally-resolved detection away from the telecoms wavelength range and a robust approach to single-photon state characterisation that is sensitive to the spectral phase have so far proven challenging. Additionally, accessing quantum advantages requires measurements that are sensitive to the nonclassical correlations that can exist between separate subsystems. However, until now a flexible approach to determining the full quantum state of systems of multiple time/frequency entangled photons has been lacking. Here we develop a toolbox of techniques for the measurement of TF quantum states of light for few-photon systems, and apply these to the task of precision measurement. Firstly, we propose, construct and demonstrate efficient single-photon spectrometers based on dispersive time-to-frequency mapping, using chirped fibre Bragg gratings to apply the necessary group delay dispersion. We use two such devices operating in coincidence to realise a full phase-sensitive reconstruction of the TF state of a single photon, by spectrally resolving the output of a nonclassical interference process with an external reference. We also demonstrate a complementary, optical reference-free method. This characterisation tool, an implementation of electro-optic spectral shearing interferometry with coincident spectrally-resolved single-photon detection, melds techniques from ultrafast metrology and quantum optics. We proceed to demonstrate the flexibility of this device by reconstructing a wide range of test pulses generated with a pulse shaper. By implementing this technique in conjunction with spectrally-resolved detection of a second photon, we realise complete characterisation of the TF wave function of a spectrally-entangled broadband photon pair. This method is robust to a wide range of inputs and is generalisable to spectrally mixed states and across a wide range of wavelengths, including to spectrally non-degenerate photon pairs. To demonstrate, we generate an entangled photon pair with controllable TF entanglement and perform a full reconstruction. Lastly, we present a new technique for using TF entangled photon pairs to realise quantum-enhanced measurement of a time delay in a quantum channel in the presence of environmental noise. This scheme is based on entanglement-enhanced rejection of background light. A simple experiment demonstrating the feasibility of such a measurement is presented.