Sub-Planck structures revealed by Wigner function of a state evolving in a χ^{(3)} medium [2]

Dr. habil. Magdalena Stobińska specializes in theoretical quantum optics since her studies at the Faculty of Physics UW. She has gained a broad expertise in this field. Its range covers e.g. single- and multi-particle quantum states, quantum interference, interactions of light with matter, nonlinear effects, entanglement and quantum state engineering. Below, we present a list of topics Dr. habil. Stobińska has worked over years. They constitute a knowledge base our Group capitalizes on when tackling new problems.

In [1] a new method of **witnessing squeezing and entanglement** of one- and two-mode mixed Gaussian states of light was developed. This approach is based on Hanbury-Brown and Twiss (HBT) interference. Nonclassical character of the states is exhibited by violation of derived inequalities. They can be employed to test entanglement in balanced homodyne experiments, which allows to perform measurements of continuous variables in the regime of high fluctuations. These results were noticed by the scientific community and adopted for developing entanglement witnesses for other systems, e.g. spin chains and ring cavities.

**The Fokker-Planck equation** which describes the time evolution of the Wigner function in a χ^{(3)} nonlinear medium, e.g. the Kerr medium, was derived in [2] and solved numerically. The results revealed the quantum nature of the evolving state. This is one of the first papers presenting how sub-Planck structures emerge in the Wigner function. The approach is universal and allows to investigate arbitrary states evolving in any medium characterized by a χ^{(3)} nonlinearity, e.g. optical fibers, ion traps, optical microcavities, nanomechanical oscillators, Rydberg atoms. Furthermore, an experimental protocol for generation of coherent state superpositions was proposed and shown to be suitable for ions in the Paul and stylus traps.

An experimental technique for **measuring the second-order coherence function of light using the HBT interferometer with homodyne detection** was described in [3]. The experiment was carried out in the regime of continuous variables. A phenomena of photon antibunching was demonstrated, which directly confirms the quantum nature of light. These results proved to be useful for describing the nonclassicality of photon sources, performing emission analysis for silicon microcavities and laser diodes as well as examining quantum systems operating in the microwave regime.

Dr. habil. Magdalena Stobińska has also worked on light-matter interactions. In [4], the conditions necessary **for perfect excitation of a two-level atom with a photon in free space**, without a cavity, were derived. The computations were performed in the framework of nonrelativistic quantum electrodynamics. It was shown that the photon must be prepared in a state which is a time-reversed state resulting from a spontaneous emission (a time-reversed dipole wave). Any imperfection in preparation of this ideal state decreases the probability of the absorption. Additionally, Magdalena Stobińska examined the dynamics of the spontaneous emission from a two-level atom placed in the focus of a parabolic cavity. These results may lead to achieving ideal coupling between atom and photon qubits, which is necessary for establishing scalable quantum networks. They have already found applications in various branches of physics, e.g. in development of one-atom transistors, nanofibers and sensors employing ion traps.

Our group has experience in investigation of **multiphoton quantum states**. An important example of these states is the squeezed vacuum. It is produced with one of the most ubiquitous sources of quantum light based on the parametric down-conversion (PDC) process in a nonlinear crystal. Entanglement witnesses formulated in the spirit of the Duan criterion as well as an entanglement measure employing a generalized Fedorov ratio for quasicontinuous variables were proposed in [5]. It was shown that the entanglement between two multiphoton states can be easily produced by impinging a single photon with a coherent state on a beam splitter [6]. A possibility of violation of simple Bell inequalities of the CHSH (Clauser-Horne-Shimony-Holt) type for a singlet consisting of macroscopic qubits was investigated in [7]. This work was very inspiring because it allowed to thoroughly analyze possibilities of observing Bell nonlocality depending on the employed measurement type and its resolution. It encouraged the community to further work on improvement of the visibility of the states. Dr. habil. Stobińska developed a filter which processes two-mode states of light but preserves quantum superpositions. The filter passes only those components which feature the modulus of the photon number difference higher than a given threshold [8]. She proposed an experimental setup which well approximates this filter and showed that it implements a large family of non-Gaussian filters [9]. This work expanded the palette of quantum sources of light which are useful for quantum technologies, e.g. for enhanced optical phase estimation as well as generating entanglement between matter and light.

Recently, our attention was focused on **generation of single photons and their investigation in the time-frequency domain** with the newest integrated optics technology. Single-photon states are produced in a controlled PDC process realized in PPLN (periodically-poled lithium niobate) or PPKTP (periodically-poled potassium titanyl phosphate) nonlinear crystals for small values of the parametric gain. These crystals produce perfectly correlated narrow-band photon pairs and they can also act as heralded sources of single photons. The photons can be additionally shaped in the frequency domain and transmitted over optical fibers for further processing. In comparison with older sources of multiphoton quantum states, new equipment is characterized by simplicity and huge application potential.

[1] M. Stobińska, K. Wódkiewicz, *Witnessing Entanglement with Second-Order Interference*, Phys. Rev. A, **71**, 032304 (2005).

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[2] M. Stobińska, G. J. Milburn, K. Wódkiewicz, *Wigner function evolution of quantum states in presence of self-Kerr interaction*, Phys. Rev. A **78**, 013810 (2008).

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[3] N. B. Grosse, T. Symul, M. Stobińska, T. C. Ralph, P. K. Lam, *Measuring Photon Anti-Bunching from Continuous Variable Sideband Squeezing*, Phys. Rev. Lett. **98**, 153603 (2007).

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[4] M. Stobińska, G. Alber, G. Leuchs, *Perfect excitation of a matter qubit by a single photon in free space*, EPL **86**, 14007 (2009).

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[5] M. Stobińska, F. Töppel, P. Sekatski, M. V. Chekhova, *Entanglement witnesses and measures for bright squeezed vacuum*, Phys. Rev. A **86**, 022323 (2012).

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[6] P. Sekatski, N. Sangouard, M. Stobińska, F. Bussieres, M. Afzelius, N. Gisin, *Proposal for exploring macroscopic entanglement with a single photon and coherent states*, Phys. Rev. A **86**, 060301(R) (2012).

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[7] M. Stobińska, P. Sekatski, A. Buraczewski, N. Gisin, G. Leuchs, *Bell-inequality tests with macroscopic entangled states of light*, Phys. Rev. A **84**, 034104 (2011).

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[8] M. Stobińska, F. Töppel, P. Sekatski, A. Buraczewski, M. Żukowski, M. V. Chekhova, G. Leuchs, N. Gisin, *Filtering of the absolute value of photon-number difference for two-mode macroscopic quantum states*, Phys. Rev. A **86**, 063823 (2012).

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[9] M. Stobińska, *Feasible quantum engineering of quantum multi-photon superpositions*, Optics Commun. **337** (Special Issue on Macroscopic Quantumness), 83 (2015).

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