Frequency conversion is an enabling process in many quantum information protocols. In this letter we study fundamental limits to high efficiency frequency conversion imposed by time ordering corrections. Using the Magnus expansion, we argue that these corrections, which are usually considered detrimental, can be used to increase the efficiency of conversion under certain circumstances. The corrections induce a nonlinear behaviour in the probability of upconversion as a function of the pump intensity, significantly modifying the sinusoidal Rabi oscillations that are otherwise expected. Finally, by using a simple scaling argument, we explain why cascaded frequency conversion devices attenuate time ordering corrections, allowing the construction of near ideal quantum pulse gates.
Frequency conversion is an enabling process in many quantum information protocols. In this letter we study fundamental limits to high efficiency frequency conversion imposed by time ordering corrections. Using the Magnus expansion, we argue that these corrections, which are usually considered detrimental, can be used to increase the efficiency of conversion under certain circumstances. The corrections induce a nonlinear behaviour in the probability of upconversion as a function of the pump intensity, significantly modifying the sinusoidal Rabi oscillations that are otherwise expected. Finally, by using a simple scaling argument, we explain why cascaded frequency conversion devices attenuate time ordering corrections, allowing the construction of near ideal quantum pulse gates.
Frequency conversion (FC) is an enabling process in many quantum information protocols. Recently, it has been observed that upconversion efficiencies in single-photon, mode-selective FC are limited to around 80%. In this Letter, we argue that these limits can be understood as time-ordering corrections (TOCs) that modify the joint conversion amplitude of the process. Furthermore, using a simple scaling argument, we show that recently proposed cascaded FC protocols that overcome the aforementioned limitations act as "attenuators" of the TOCs. This observation allows us to argue that very similar cascaded architectures can be used to attenuate TOCs in photon generation via spontaneous parametric downconversion. Finally, by using the Magnus expansion, we argue that the TOCs, which are usually considered detrimental for FC efficiency, can also be used to increase the efficiency of conversion in partially mode-selective FC.
This thesis presents a study of photon generation and conversion processes in nonlinear optics. Our results extend beyond the first order perturbative regime, in which only pairs of photons are generated in spontaneous parametric down-conversion or four wave mixing. They also allow us to identify the limiting factors for achieving unit efficiency frequency conversion, and to correctly account for the propagation of bright nonclassical photon states generated in the nonlinear material. This is done using the Magnus expansion, which allows us to incorporate corrections due to time-ordering effects in the time evolution of the state, while also respecting the quantum statistics associated with squeezed states (in photon generation) and single photons or coherent states (in photon conversion). We show more generally that this expansion should be the preferred strategy when dealing with any type of time-dependent Hamiltonian that is a quadratic form in the creation and annihilation operators of the fields involved. Using the Magnus expansion, simple figures of merit to estimate the relevance of these time-ordering corrections are obtained. These quantities depend on the group velocities of the modes involved in the nonlinear process and provide a very simple physical picture of the interactions between the photons at different times. These time-ordering effects are shown to be important only when the photons and pump beam overlap within the nonlinear material for a significant amount of time. We also discuss the possibility of preparing Fock states with many photons using parametric down-conversion sources and realistic photon-number-resolving detectors. After discussing photon generation processes, we study photon conversion processes in which a nonlinear material and a classical strong pump field are used modify the frequency of a second incoming low-intensity field. We show that time-ordering effects limit the efficiency of this conversion process, re-examine strategies to get around this problem, and show that it should be possible to have a clear observation of modified upconversion probabilities due to these effects, and thus have an experimental demonstration of time-ordering effects in nonlinear optics.
We demonstrate the utility of frequency upconversion in a non-linear crystal for quantum information applications. By combining a bright “escort” beam with a signal photon inside of a periodically poled lithium niobate (PPLN) crystal, we are able to achieve near unit conversion efficiency. We use this conversion process to create high-efficiency infrared single-photon detectors by upconverting infrared photons to visible ones and using efficient silicon avalanche photo diodes to detect the upconverted photon. We have achieved a net system detection efficiency of 56%, far superior to previous methods for infrared photon detection. We also demonstrate the phase coherence of the upconversion process, which enables it to convert photonic qubits from one wavelength to another, potentially allowing quantum networks of disparate operating wavelengths to communicate efficiently. This quantum “transduction” feature is observed by upconverting photons passed between two unbalanced Michelson interferometers that create and measure time-bin qubits, similar to those used in many quantum cryptography systems, and verifying that the phase of the qubits is preserved. The Rabi oscillation-like behavior of upconversion is also demonstrated, allowing for the creation and manipulation of highly non-degenerate frequency qubits. These states may be used in conjunction with other degrees of freedom such as polarization, time-bin, or orbital angular momentum states to enlarge the usable single-photon Hilbert space and create a rich vocabulary for transmission of quantum information.
Frequency conversion (FC) and type-II parametric down-conversion (PDC) processes serve as basic building blocks for the implementation of quantum optical experiments: type-II PDC enables the efficient creation of quantum states such as photon-number states and Einstein-Podolsky-Rosen (EPR)-states. FC gives rise to technologies enabling efficient atom-photon coupling, ultrafast pulse gates and enhanced detection schemes. However, despite their widespread deployment, their theoretical treatment remains challenging. Especially the multi-photon components in the high-gain regime as well as the explicit time-dependence of the involved Hamiltonians hamper an efficient theoretical description of these nonlinear optical processes. In this paper, we investigate these effects and put forward two models that enable a full description of FC and type-II PDC in the high-gain regime. We present a rigorous numerical model relying on the solution of coupled integro-differential equations that covers the complete dynamics of the process. As an alternative, we develop a simplified model that, at the expense of neglecting time-ordering effects, enables an analytical solution. While the simplified model approximates the correct solution with high fidelity in a broad parameter range, sufficient for many experimental situations, such as FC with low efficiency, entangled photon-pair generation and the heralding of single photons from type-II PDC, our investigations reveal that the rigorous model predicts a decreased performance for FC processes in quantum pulse gate applications and an enhanced EPR-state generation rate during type-II PDC, when EPR squeezing values above 12 dB are considered.
Frequency conversion (FC) and type-II parametric down-conversion (PDC) processes serve as basic building blocks for the implementation of quantum optical experiments: type-II PDC enables the efficient creation of quantum states such as photon-number states and Einstein–Podolsky–Rosen (EPR)-states. FC gives rise to technologies enabling efficient atom–photon coupling, ultrafast pulse gates and enhanced detection schemes. However, despite their widespread deployment, their theoretical treatment remains challenging. Especially the multi-photon components in the high-gain regime as well as the explicit time-dependence of the involved Hamiltonians hamper an efficient theoretical description of these nonlinear optical processes. In this paper, we investigate these effects and put forward two models that enable a full description of FC and type-II PDC in the high-gain regime. We present a rigorous numerical model relying on the solution of coupled integro-differential equations that covers the complete dynamics of the process. As an alternative, we develop a simplified model that, at the expense of neglecting time-ordering effects, enables an analytical solution. While the simplified model approximates the correct solution with high fidelity in a broad parameter range, sufficient for many experimental situations, such as FC with low efficiency, entangled photon-pair generation and the heralding of single photons from type-II PDC, our investigations reveal that the rigorous model predicts a decreased performance for FC processes in quantum pulse gate applications and an enhanced EPR-state generation rate during type-II PDC, when EPR squeezing values above 12 dB are considered.
We study time-ordering corrections to the description of spontaneous parametric down-conversion (SPDC), four-wave mixing (SFWM), and frequency conversion using the Magnus expansion. Analytic approximations to the evolution operator that are unitary are obtained. They are Gaussian preserving, and allow us to understand order-by-order the effects of time ordering. We show that the corrections due to time ordering vanish exactly if the phase-matching function is sufficiently broad. The calculation of the effects of time ordering on the joint spectral amplitude of the photons generated in SPDC and SFWM are reduced to quadrature.