**Project OPUS X of the National Science Centre entitled**

*“Theoretical foundations of qubit-based environmental noise spectroscopy – quantum vs classical and Gaussian vs non-Gaussian noise” (2016-2019)
*

**Objectives**

Nontrivially quantum properties of physical systems are extremely fragile to interactions with the „rest of the world”, i.e. with the environment of a given system. Due to ubiquity of this decoherence process, realistic quantum systems are always open, and this feature of them stands in the way of possible applications of their truly quantum properties. One can, however, treat decoherence not as an obstacle, but employ the large sensitivity of qubits to external stimuli. A system which is coherently controlled, small, and spatially localized, e.g. a spin qubits (an NV center in diamond or a quantum dot), can be used to sense environmental noise with nanoscale resolution.

Recently proposed qubit manipulation and measurement protocols, in development of which the environment was assumed to be a source of classical noise having Gaussian statistics (i.e. being fully characterized by its spectral density), have been used in experiments aimed at characterization of environmental fluctuations occurring in the immediate vicinity of solid-state based qubits. However, the conditions, under which such a classical approximation for decoherence proces is valid, are not well understood.

Our main goal is theoretical understanding of the signal obtained in experiments on environmental noise spectroscopy without assuming that the noise is classical and Gaussian. Furthermore, we want to develop such methods of qubit manipulation and readout, with which it will be possible to detect to what degree the environmental noise is non-Gaussian or quantum.

**Main results obtained so far:**

1) Development of theoretical model of nuclear noise coupling transversally to spin qubits interacting with nuclear bath in quantum dots [1].

2) A comprehensive review of state-of-the-art theoretical understanding of dynamical-decoupling based noise spectroscopy with qubits [2]. This review contains also a few previously unpublished results concerning quantum Gaussian noise.

3) Analysis of accuracy of popularly employed protocols of dynamical-decoupling based noise spectroscopy [4]. In this paper we have shown that when using them to reconstruct Gaussian-shaped spectrum of noise one obtains artificial power-law tails in the spectrum of noise. We have proposed a modified noise spectroscopy protocol that avoids this pitfall.

4) Analysis of generation of entanglement between a system and its environment [5,12,16] and proposal of protocol in which measurements on qubit only give information on this entanglement [9].

5) Extension of protocols of dynamical-decoupling based noise spectroscopy to a case of an environment characterized by a spectral density consisting of isolated peaks [11].

6) Calculation of dephasing of NV qubits interacting with nuclear environment: using two qubits to gain insight into noise correlations and sense the non-Gaussian nature of environmental noise [6], using a single qubit to witness truly quantum features of environmental dynamics causing decoherence [15].

7) Comprehensive discussion of noise spectroscopy without dynamical decoupling, only with multiple subsequent measurements performed on the system [13,14]

8) Last but not least: a careful discussion of features of environment and qubit-environment coupling that enable the use of “classical noise approximation” for modeling of environmental influence on the quantum system [17].

**Publications**

[19] D. Kwiatkowski, Ł. Cywiński, and J. K. Korbicz

*Appearance of objectivity for NV centers interacting with dynamically polarized nuclear environment*

New J. Phys. **23**, 043036 (2021)

arXiv:2012.02855

[18] K. Roszak and Ł. Cywiński,

*Qubit-environment entanglement generation and the spin echo*

Phys. Rev. A **103**, 032208 (2021)

arXiv:2007.02656

[17] P. Szańkowski and Ł. Cywiński,

*Noise representations of open system dynamics*

Sci. Rep. **10**, 22189 (2021)

arXiv:2003.09688

[16] M. Strzałka, D. Kwiatkowski, Ł. Cywiński, and K. Roszak

*Qubit-environment negativity versus fidelity of conditional environmental states for a nitrogen-vacancy-center spin qubit interacting with a nuclear
environment*

Phys. Rev. A

**102**, 042602 (2020)

arXiv:1911.08867

[15] D. Kwiatkowski, P. Szańkowski, and Ł. Cywiński,

*Influence of nuclear spin polarization on spin-echo signal of an NV-center qubit*

Phys. Rev. B **101**, 155412 (2020)

arXiv:1909.06438

[14] F. Sakuldee and Ł. Cywiński

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

Phys. Rev. A **101**, 042329 (2020)

arXiv:1907.05165

[13] F. Sakuldee and Ł. Cywiński

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

Phys. Rev. A **101**, 012314 (2020)

arXiv:1907.01784

[12] K. Roszak and J. K. Korbicz

*Entanglement and objectivity in pure dephasing models*

Phys. Rev. A **100**, 062127 (2020)

arXiv:1904.08261

[11] P. Szańkowski

*The dynamical decoupling spectroscopy of environment characterized by discrete spectrum*

Phys. Rev. A **100**, 052115 (2019)

arXiv:1904.02001

[10] F. Sakuldee and Ł. Cywiński

*Characterization of a quasistatic environment with a qubit*

Phys. Rev. A **99**, 062113 (2019)

arXiv:1903.06463

[9] K. Roszak, D. Kwiatkowski, and Ł. Cywiński

*How to detect qubit-environment entanglement generated during qubit dephasing*

Phys. Rev. A **100**, 022318 (2019)

arXiv:1810.09217

[8] J. Krzywda, P. Szańkowski, and Ł. Cywiński

*The dynamical-decoupling-based spatiotemporal noise spectroscopy*

New J. Phys. **21**, 043034 (2019)

arXiv:1809.02972

[7] J. Krzywda, P. Szańkowski, J. Chwedeńczuk, and Ł. Cywiński

*Decoherence-assisted detection of entanglement of bipartite states*

Phys. Rev. A **98**, 022329 (2018)

arXiv:1806.08728

[6] D. Kwiatkowski and Ł. Cywiński

*Decoherence of two entangled spin qubits coupled to an interacting sparse nuclear spin bath: application to nitrogen vacancy centers*

Phys. Rev. B **98**, 155202 (2018)

arXiv:1806.06845

[5] K. Roszak

*Criteria for system-environment entanglement for systems of any size in pure-dephasing evolutions*

Phys. Rev. A **98**, 052344 (2018)

arXiv:1708.05535

[4] P. Szańkowski and Ł. Cywiński

*Accuracy of dynamical decoupling based spectroscopy of Gaussian noise*

Phys. Rev. A **97**, 032101 (2018)

arXiv:1708.05535

[3] J. Krzywda, Ł. Cywiński, and P. Szańkowski

*Localization of a magnetic moment using a two-qubit probe*

Phys. Rev. A **96**, 042108 (2017)

arXiv:1706.02948

[2] P. Szańkowski, G. Ramon, J. Krzywda, D. Kwiatkowski, and Ł. Cywiński

*Environmental noise spectroscopy with qubits subjected to dynamical decoupling*

J. Phys.: Condens. Matter. **29**, 333001 (2017)

arXiv:1705.02262

[1] F. K. Malinowski, F. Martins, Ł. Cywiński, M. S. Rudner, P. D. Nissen, S. Fallahi, G. C. Gardner, M. J. Manfra, C. M. Marcus, and F. Kuemmeth

*Spectrum of the Nuclear Environment for GaAs Spin Qubits*

Phys. Rev. Lett. **118**, 177702 (2017)

arXiv:1701.01855

For all our papers related to this topic, see Noise spectroscopy with qubits