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Preparation and reconstruction of arbitrary path and polarization states in photons

Alfredo Ricci Vásquez

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

The purpose of this monograph is to show how different discrete variables of light can be used to generate qubits. In particular, the tomographic preparation and reconstruction process for the polarization and path variables in photons is shown. In addition, the preparation of two-qubit states in a single photon using these variables is done, and these states are used to generate statistical mixtures states on the path. Finally, an experimental setup is proposed for the reconstruction of two-qubit states in a single photon. During the monograph, two different tomography techniques are used: Standard quantum state tomography and quantum state tomography through the use of mutually unbiased bases

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The purpose of this monograph is to show how different discrete variables of light can be used to generate qubits. In particular, the tomographic preparation and reconstruction process for the polarization and path variables in photons is shown. In addition, the preparation of two-qubit states in a single photon using these variables is done, and these states are used to generate statistical mixtures states on the path. Finally, an experimental setup is proposed for the reconstruction of two-qubit states in a single photon. During the monograph, two different tomography techniques are used: Standard quantum state tomography and quantum state tomography through the use of mutually unbiased bases
Quantum state tomography (QST), the process through which the density matrix of a quantum system is characterized from measurements of specific observables, is a fundamental pillar in the fields of quantum information and computation. In this work, we propose a simple QST method to reconstruct the density matrix of two qubits encoded in the polarization and path degrees of freedom of a single photon, which can be realized with a single linear-optical setup. We demonstrate that the density matrix can be fully described in terms of the Stokes parameters related to the two possible paths of the photon, together with a quantum version of the two-point Stokes parameters introduced here. Our findings put forward photonic circuits for the investigation of the dynamics of open quantum systems.
Quantum state tomography (QST), the process through which the density matrix of a quantum system is characterized from measurements of specific observables, is a fundamental pillar in the fields of quantum information and computation. In this work, we propose a simple QST method to reconstruct the density matrix of two qubits encoded in the polarization and path degrees of freedom of a single photon, which can be realized with a single linear-optical setup. We demonstrate that the density matrix can be fully described in terms of the Stokes parameters related to the two possibles paths of the photon, together with a quantum version of the two-point Stokes parameters introduced here. Our findings put forward photonic circuits for the investigation of the dynamics of open quantum systems.
Tomography allows one to estimate the density matrix describing the state an ensemble of quantum systems are prepared in (for example, polarization tomography determines the polarization state of a beam of identically prepared photons). In general, it is not possible to uniquely decompose the density matrix into its pure state components. Agarwal et al. proposed a protocol which, for a mixture composed of any two pure states of a qubit (with arbitrary probabilities), allows an observer to infer not only the density matrix but the identity of those specific pure states and their weights - the additional requirement being that the qubits arrive in pairs, where both qubits in each pair are in the same state. We experimentally demonstrate this learning-from-pairs concept using photons in the polarization degree of freedom. We use tomography to measure a sequence of single photons and make use of their time-of-arrival information to 'pair up' the photons after the measurement. From here we are able to infer the photons' polarization states and their respective probabilities, and we demonstrate this for various different choices of polarization states and ratios. Finally, we investigate our ability to discriminate between two equal mixtures of distinct pairs of orthogonal polarization states. We find that on the order of approx. 10e4 photons is typically enough to achieve tomography fidelities of approximately 0.9999. This is sufficient to discriminate between two different preparations of the same mixed state, differing by angles of less than 5 degrees between the pure states used in the two preparations.
The measurement of quantum states is one of the most important problems in quantum mechanics. We introduce a quantum state tomography technique in which the state of a qubit is reconstructed, while the qubit remains undetected. The key ingredients are (i) employing an additional qubit, (ii) aligning the undetected qubit with a known reference state by using path identity, and (iii) measuring the additional qubit to reconstruct the undetected qubit state. We establish theoretically and demonstrate experimentally the method with photonic polarization states. The principle underlying our method could also be applied to quantum entities other than photons.
We have performed quantum-state tomography on several different single-photon entangled states, that is, states in which a single photon is shared between two possible paths. We begin by doing a tomographic reconstruction of density matrices in the subspace where a single photon is shared by two spatial modes. In this subspace we are able to create high-fidelity, path-entangled states. Also within this subspace we use the Akaike information criterion and Monte Carlo simulations to help us estimate the amount and source of state drift in our system. We find that the primary state drift is due to phase drifts and fluctuations on the order of $\ensuremath{\pi}/400$ in our experimental apparatus. We then use the single-photon subspace density matrices and further measurements to estimate density matrices in the larger space consisting of two modes containing up to one photon each. In this larger space we find that the concurrence of the density matrices is $C\ensuremath{\cong}0.08$ and is greater than 0 by at least 45 standard deviations, indicating that our states are indeed entangled.
Quantum state tomography is used in quantum communication, computing, and cryptography to characterize quantum states enabling assessment and control of quantum systems. Rapidly scaling quantum technologies require fast, accurate, and simple techniques, but current tomography methods are bulky, hard to scale, and slow. To overcome these limitations, we experimentally demonstrate Fourier tomography [1], of single-photon and entangled photon pairs generated by a quantum dot. As shown in Fig. 1(a) for two-photon tomography, the setup consists of a continuously rotating quarter waveplate, a fixed linear polarizer and a single photon detector per qubit. As the waveplates rotates with different speed, we record the coincidences rate between the two detectors. This signal can be expressed as a Fourier series whose coefficients can be related to the polarization state of light. Fig. 1(b) shows the simulated coincidence rate as the waveplate rotates for input Bell states∣Φ+⟩\vert \Phi^{+}\rangle∣Φ+⟩and∣Φ−⟩\left\vert\Phi^{-}\right\rangle∣Φ−⟩. By capturing the coincidences variations as a Fourier series through a single rotating element and fixed polarizer, we simplify the tomography process, providing an accurate, and simple approach suitable for scalable quantum systems.