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The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting

Self-calibrating tomography for non-unitary processes

Nicolás Quesada, Agata M. Brańczyk, Daniel F. V. James

Published Jan 1, 2013• 1 citation
Publisher

Abstract

We extend the idea of using a incompletely characterized process to do quantum state tomography by including the possibility that the process itself is not unitary. We study a to level atom whose dipole moment is unknown and that undergoes spontaneous emission and is irradiated by a laser whose phase and intensity can be controlled at will. We show that by using five different parameter settings of the electric field of the laser it is possible two reconstruct the state, and obtain the unknown spontaneous emission rate and dipole moment of the atom.

DOI
10.1364/qim.2013.w6.38

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We extend the idea of using a incompletely characterized process to do quantum state tomography by including the possibility that the process itself is not unitary. We study a to level atom whose dipole moment is unknown and that undergoes spontaneous emission and is irradiated by a laser whose phase and intensity can be controlled at will. We show that by using five different parameter settings of the electric field of the laser it is possible two reconstruct the state, and obtain the unknown spontaneous emission rate and dipole moment of the atom.
Quantum state tomography relies on the ability to perform well-characterized unitary operations, which are not always available. This places restrictions on which physical systems can be characterized using such techniques. We develop a formalism that circumvents the requirement for well-characterized unitary operations by treating unknown parameters in the state and unitary process on the same footing, thereby simultaneously performing state and partial process tomography. Our formalism is generalized to ddd-level systems, and we provide a specific example for a V\mathsf{V}V-type three-level atomic system whose transition dipole moments are not known. We show that it is always possible to retrieve not only the unknown state but also the process parameters, except for a set of zero measure in the state-parameter space.
We introduce and experimentally demonstrate a technique for performing quantum state tomography (QST) on multiple-qubit states despite incomplete knowledge about the unitary operations used to change the measurement basis. Given unitary operations with unknown rotation angles, our method can be used to reconstruct the density matrix of the state up to local rotations as well as recover the magnitude of the unknown rotation angle. We demonstrate high-fidelity self-calibrating tomography on polarization-encoded one- and two-photon states. The unknown unitary operations are realized in two ways: using a birefringent polymer sheet—an inexpensive smartphone screen protector—or alternatively a liquid crystal wave plate with a tuneable retardance. We explore how our technique may be adapted for QST of systems such as biological molecules where the magnitude and orientation of the transition dipole moment is not known with high accuracy.
The usual form of electromagnetic forces is given by but for neutral atoms q = 0. The next order of force is the dipole term, but this also vanishes because neutral atoms have no inherent dipole moment. However, a dipole moment can be induced by a field, and this is most efficient if the field is alternating near the atomic resonance frequency. Since these frequencies are typically in the optical range, dipole moments are efficiently induced by shining nearly resonant light on the atoms. Laser cooling and trapping rely on the interaction between laser light and atoms to exert a controllable force on the atoms, and many sophisticated schemes have been developed using the special properties of the interaction. This chapter begins with the simplest schemes for exerting optical forces on atoms, namely a single-frequency light field interacting with a two-level atom. Although this is the simplest possible scheme, it is pedagogically valuable because it shows many of the features that will be encountered further on. The ultimate temperature using such a scheme is limited, but lower temperatures can be achieved with multilevel atoms. This topic is addressed later in the chapter. Two kinds of optical forces When an atom absorbs nearly resonant light, it makes a transition to the excited state, and its subsequent return to the ground state can be either by spontaneous or by stimulated emission. The nature of the optical force that arises from these two different processes is quite different, and will be described separately [1]. The spontaneous emission case is different from the familiar quantum mechanical calculations using state vectors to describe the state of the system, because spontaneous emission causes the state of the system to evolve from a pure state into a mixed state (see Exercise 6.9). Spontaneous emission is an essential ingredient for the dissipative nature of the optical forces in monochromatic light, and the density matrix is needed to describe it (see Chap. 6). When an atom absorbs light, its energy excites the atom and its angular momentum changes the electron's orbit. Both of these are internal atomic properties. However, the linear momentum of the light, can change only the external, translational motion of the atom.
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.