Spectroscopy of classical environmental noise with a qubit subjected to projective measurements

Abstract

We show theoretically how a correlation of multiple measurements on a qubit undergoing pure dephasing can be expressed as environmental noise filtering. Measurement of such correlations can be used for environmental noise spectroscopy, and the family of noise filters achievable in such a setting is broader than the one achievable with a standard approach, in which dynamical decoupling sequences are used. We illustrate the advantages of this approach by considering a case of noise spectrum with sharp features at very low frequencies. We also show how appropriately chosen correlations of a few measurements can detect the non-Gaussian character of certain environmental noises, particularly the noise affecting the qubit at the so-called optimal working point.

Relationship between subjecting the qubit to dynamical decoupling and to a sequence of projective measurements

“…This equation relates the expectation value of the last measurement ofσ x in a sequence of n measurements to a combination of decoherence signals (also expectation values ofσ x ) measured after subjecting the qubit to all the possible sequences of 1 1 Q andσ x operations. This result is a generalization of an analogous relation obtained in [29] within a much more restricted setting: for a qubit experiencing pure dephasing due to external classical noise. Here we do not assume anything about qubitenvironment coupling, and the environment is treated quantum mechanically.…”

“…showing that, for any given initial state, the state after two measurements can be written as a convex combination of states obtained by making a single measurement at time t 2 with no control pulse, and with π pulse at time t 1 . This relation is the basis of the classical environmental noise characterization scheme by two single-shot measurements described in [27], which has been recently generalized to n > 2 measurements and explicitly connected to dynamical-decoupling based noise spectroscopy in [29]. For the converse relation, it suffices to consider only the case of ((x) , (x) , .…”

“…An example of an earlier appearance of such a structure can be found in [27], where pure dephasing due to classical noise was considered. Such a case of environment being a source of external classical noise is considered in a more general setting with multiple measurements in [29].…”

“…It is possible to exert some degree of control over the system by subjecting it to an appropriate sequence of measurements, typically also involving post-selection [20,21], but the system control protocols typically used to perform some form of characterization of environmen- * sakuldee@ifpan.edu.pl † lcyw@ifpan.edu.pl tal dynamics affecting the system (often called "environmental noise spectroscopy") are based on sequences on unitary operations. While multiple measurements on a qubit are known to allow for characterization and changing the state of an environment that is static during its interaction with the qubit [22][23][24][25], only quite recently the possibility of performing such characterization of dynamics of environment by performing only measurements on the system has gathered more attention [26][27][28][29][30].…”

“…Examples include observation that positive operator valued measures (POVMs) can be explained in terms of projective measurements [31,32]; derivation of noisy quantum channel decoding efficiency bound via the expansion over projective sequential measurements [33]; projective measurement(-preparations) reconstruction of non-Markovian dynamics over limited controls [34], and explanation of non-Markovian control in terms of alternative formalism of quantum stochastic process constructed from a set of measurement-preparation pairs [17]. For this work, the most interesting is the observation that a widely used [12,13,[35][36][37][38][39][40][41][42][43][44] dynamical decoupling based environmental noise spectroscopy scheme [12,13], can be emulated by a protocol in which only measurements are performed on the qubit [27][28][29], when the environment can be treated as a source of classical external noise acting on the qubit. In the more general case of an environment treated fully quantum mechanically, it has been shown recently in Ref.…”