Pulsed Quantum-State Reconstruction of Dark Systems

Abstract

We propose a novel strategy to reconstruct the quantum state of dark systems, i.e., degrees of freedom that are not directly accessible for measurement or control. Our scheme relies on the quantum control of a two-level probe that exerts a state-dependent potential on the dark system. Using a sequence of control pulses applied to the probe makes it possible to tailor the information one can obtain and, for example, allows us to reconstruct the density operator of a dark spin as well as the Wigner characteristic function of a harmonic oscillator. Because of the symmetry of the applied pulse sequence, this scheme is robust against slow noise on the probe. The proof-of-principle experiments are readily feasible in solid-state spins and trapped ions.Introduction.-The measurement of the quantum state of a system is a prerequisite ingredient in most modern quantum experiments, ranging from fundamental tests of quantum mechanics [1, 2] to various quantuminformation-processing tasks [3][4][5]. However, even with the rapid progress in the coherent manipulation and quantum-state tomography of several quantum systems, such as photons [6,7], electron spins [8-10], atomic qubits [11], superconducting circuits [12,13], and mechanical resonators [14,15], many quantum systems still remain difficult to access for a direct observation of their state, systems we will refer to as dark. In order to circumvent the requirement of such a direct access, a promising technique is to employ an auxiliary quantum system as a measurement probe, on which measurements as well as coherent manipulations can be performed [16][17][18][19][20][21][22][23]. Interferometry [24] based on such a measurement probe allows us to extract information on a target system [25][26][27][28][29][30]. Nevertheless, it still remains a key challenge to achieve a full quantum-state tomography of dark systems without requiring any direct control.

Pulsed characteristic-function measurement of a thermalizing harmonic oscillator

“…i.e., a probe-state dependent constant force acting on the harmonic oscillator [14,33], with the same strength but opposite signs depending on the probe state. Such a coupling can be realized in a variety of electromechanical systems, for example, when an electronic spin is coupled to the motional degree of freedom of a mechanical element emanating a magnetic field or is embedded in a mechanical oscillator subject to a spatially inhomogeneous magnetic field [34][35][36].…”

“…More details on the feasibility of our method in such experimental realizations have been discussed in Ref. [14] for the undamped case of the harmonic oscillator. Cooling and heating of trapped ions can be well described by a master equation of the form (1) [40][41][42].…”

“…The situation changes appreciably in systems where such a direct access is not attainable. In this kind of scenario, an established method is to employ a readily accessible measurement probe [9][10][11][12][13][14][15], often given by two-level quantum systems. For example, such a procedure has been employed in a well-known probe-based direct measurement of the Wigner function [9].…”

“…These offdiagonal elements can then be directly connected to values of the Wigner characteristic function at desired points in reciprocal phase space. We generalize a recently-proposed pulsed reconstruction [14] to a sequence of temporally non-equidistant pulses applied to the probe, resulting in the possibility for a better reconstruction of the quantum state of the target harmonic oscillator. We additionally account for decoherence of the harmonic oscillator, described using a Lindblad master equation for a Markovian damping by a finite-temperature environment [16,17].…”

Probing quantum chaos in multipartite systems

“…Experimentally, the characteristic function (3) can be measured by introducing an auxiliary qubit coupled to the system. This technique known as single-qubit interferometry has been widely used in measuring LE [44,45], work statistics [46][47][48], Lee-Yang zeros [49][50][51], OTOC [52], quantum-state reconstruction of dark systems [53], full distribution of many-body observables [43], and SFF [54]. The key procedure is to perform a controlled X gate conditioning on the auxiliary qubit, i.e., U (t) = |1 1| ⊗ exp(itX) + |0 0| ⊗ 1.…”