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In this paper, we explore the graphical representation of two-qubit entanglement on two Bloch Spheres via stabilizer formalism. We relate the density matrix to the graphical representation on two Bloch Spheres by showing how both may be derived from the stabilizer group structure of the state. Then we use the representation to explore the symmetries present in maximally entangled states., Comment: 5 figures, 10 pages

We extend Bloch Sphere formalism to pure two qubit systems. Combining insights from Geometric Algebra and analysis of entanglement in different conjugate bases we identify Two Bloch Sphere geometry that is suitable for representing maximally entangled states. It turns out that relative direction of coordinate axes of the two Bloch Spheres may be used to describe the states. Moreover, coordinate axes of one Bloch sphere should be rignt-handed and of the other one - left-handed. We describe and depict separable and maximally entangled states as well as entangling and non-entangling rotations. We also offer graphical representation of workings of a CNOT gate for different inputs. Finally we provide a way to also represent partially entangled states and describe entanglement measure related to the surface area of the sphere enclosing the state representation., Comment: 36 pages, 16 Figures

We investigated the absence of certain bright peaks in Autler-Townes laser excitation spectra of alkali metal atoms. Our research revealed that these dips in the spectra are caused by a specific architecture of adiabatic (or ``laser-dressed'') states in hyperfine (HF) components. The dressed states' analysis pinpointed several cases where constructive and destructive interference between HF excitation pathways in a two-photon excitation scheme limits the available two-photon transitions. This results in a reduction of the conventional two-photon selection rule for the total angular momentum $F$, from $\Delta F= 0,\pm 1$ to $\Delta F\equiv 0$. Our discovery presents practical methods for selectively controlling the populations of unresolvable HF $F$-components of $ns_{1/2}$ Rydberg states in alkali metal atoms. Using numerical simulations with sodium and rubidium atoms, we demonstrate that by blocking the effects of HF interaction with a specially tuned auxiliary control laser field, the deviations from the ideal selectivity of the HF components population can be lower than $0.01\%$ for Na and $0.001\%$ for Rb atoms., Comment: 20 pages, 14 figures, to be submitted to PRA. Changelog: * added missing ORCID iDs; * Notation used in figures 4,6 is now consistent with the rest of the manuscript: Pump paser -> S-laser; * corrected typos, more precise language througout the paper; * improved legibility of labels in figures 4-8

Efficient initialization and manipulation of quantum states is important for numerous applications and it usually requires the ability to perform high fidelity and robust swapping of the populations of quantum states. Stimulated Raman adiabatic passage (STIRAP) has been known to perform efficient and robust inversion of the ground states populations of a three-level system. However, its performance is sensitive to the initial state of the system. In this contribution we demonstrate that a slight modification of STIRAP, where we introduce a non-zero single-photon detuning, allows for efficient and robust population swapping for any initial state. The results of our work could be useful for efficient and robust state preparation, dynamical decoupling and design of quantum gates in ground state qubits via two-photon interactions.

We show a method of describing processes with indefinite causal order (ICO) by a definite causal order. We do so by relabeling the processes that take place in the circuit in accordance with the basis of measurement of control qubit. Causal nonseparability is alleviated at a cost of nonlocality of the acting processes. This result highlights the key role of superposition in creating the paradox of ICO. We also draw attention to the issue of growing incompatibility of language in its current form (especially the logical structures it embodies) with the quantum logic., Comment: 5 pages

In this study we explored the possibilities for observing the angular momentum alignment-to-orientation conversion (AOC) in the ground state of various alkali metals: K, Rb, Cs. For theoretical analysis we used a model that is based on the Optical Bloch equations for the density matrix. Our model includes the interaction of all neighboring hyperfine levels with laser radiation, the mixing of magnetic sublevels in an external magnetic field, the coherence properties of the exciting laser radiation, and the Doppler effect. Additionally we simulated signals where the ground- or the excited-state coherent processes were numerically switched off in order to determine the origins of the features of the obtained signals. We also performed experiments on Cs atoms with two laser beams: a linearly polarized Cs $D_1$ pump and circularly polarized Cs $D_2$ probe. We used the pump beam to create angular momentum alignment in the ground state and observed the transmission signal of the probe beam as we changed the magnetic field. A detailed analysis of the experimentally obtained transmission signal from a single circularly polarized probe laser component is provided. Finally, prospects for observing AOC conversion experimentally are discussed, as well as experiments were even a weak AOC signal could lead to systematic errors., Comment: arXiv admin note: text overlap with arXiv:2006.15019 Revised Sections: Introduction, Results & Discussion, Conclusions Minor corrections in Sections: Theoretical model, Experiment

We studied the dynamic nuclear spin polarization of nitrogen in negatively charged nitrogen-vacancy (NV) centers in diamond both experimentally and theoretically over a wide range of magnetic fields from 0 to 1100 G covering both the excited-state level anti-crossing and the ground-state level anti-crossing magnetic field regions. Special attention was paid to the less studied ground-state level anti-crossing region. The nuclear spin polarization was inferred from measurements of the optically detected magnetic resonance signal. These measurements show that a very large (up to $96 \pm 2\%$) nuclear spin polarization of nitrogen can be achieved over a very broad range of magnetic field starting from around 400 G up to magnetic field values substantially exceeding the ground-state level anti-crossing at 1024 G. We measured the influence of angular deviations of the magnetic field from the NV axis on the nuclear spin polarization efficiency and found that, in the vicinity of the ground-state level anti-crossing, the nuclear spin polarization is more sensitive to this angle than in the vicinity of the excited-state level anti-crossing. Indeed, an angle as small as a tenth of a degree of arc can destroy almost completely the spin polarization of a nitrogen nucleus. In addition, we investigated theoretically the influence of strain and optical excitation power on the nuclear spin polarization., Comment: 9 pages, 7 figures

In this paper cross-relaxation between nitrogen-vacancy (NV) centers and substitutional nitrogen in a diamond crystal was studied. It was demonstrated that optically detected magnetic resonance signals (ODMR) can be used to measure these signals successfully. The ODMR were detected at axial magnetic field values around 51.2~mT in a diamond sample with a relatively high (200~ppm) nitrogen concentration. We observed transitions that involve magnetic sublevels that are split by the hyperfine interaction. Microwaves in the frequency ranges from 1.3 GHz to 1.6 GHz ($m_S=0\longrightarrow m_S=-1$ NV transitions) and from 4.1 to 4.6 GHz ($m_S=0\longrightarrow m_S=+1$ NV transitions) were used. To understand the cross-relaxation process in more detail and, as a result, reproduce measured signals more accurately, a model was developed that describes the microwave-initiated transitions between hyperfine levels of the NV center that are undergoing anti-crossing and are strongly mixed in the applied magnetic field. Additionally, we simulated the extent to which the microwave radiation used to induce ODMR in the NV center also induced transitions in the substitutional nitrogen via cross-relaxation. The improved understanding of the NV processes in the presence of a magnetic field will be useful for designing NV-diamond-based devices for a wide range of applications from implementation of q-bits to hyperpolarization of large molecules to various quantum technological applications such as field sensors., Comment: 12 pages, 10 figures

We proposed utilizing a medium with a high optical depth (OD) and a Rydberg state of low principal quantum number, $n$, to create a weakly-interacting many-body system of Rydberg polaritons, based on the effect of electromagnetically induced transparency (EIT). We experimentally verified the mean field approach to weakly-interacting Rydberg polaritons, and observed the phase shift and attenuation induced by the dipole-dipole interaction (DDI). The DDI-induced phase shift or attenuation can be viewed as a consequence of the elastic or inelastic collisions among the Rydberg polaritons. Using a weakly-interacting system, we further observed that a larger DDI strength caused a width of the momentum distribution of Rydberg polaritons at the exit of the system to become notably smaller as compared with that at the entrance. In this study, we took $n =32$ and the atomic (or polariton) density of 5$\times10^{10}$ (or 2$\times10^{9}$) cm$^{-3}$. The observations demonstrate that the elastic collisions are sufficient to drive the thermalization process in this weakly-interacting many-body system. The combination of the $\mu$s-long interaction time due to the high-OD EIT medium and the $\mu$m$^2$-size collision cross section due to the DDI suggests a new and feasible platform for the Bose-Einstein condensation of the Rydberg polaritons.

Hereby we inspect two-partite entanglement using thought experiment that relates properties of incoherently mixed states to the impossibility of faster-than-light (FTL) signalling. We show that if there appears a way to distinguish ensembles of particles that are described by the same density matrix, but are generated using different pure states - properties of entanglement (namely, non-classical correlations) could be employed to create an FTL signalling device. We do not claim FTL signalling is possible, rather, we establish the logical connection between the aforementioned properties of current physical theory which has not so far been evident., Comment: 13 pages, 3 figures

STIRAP (Stimulated Raman Adiabatic Passage) is a powerful laser-based method, usually involving two photons, for efficient and selective transfer of population between quantum states. A particularly interesting feature is the fact that the coupling between the initial and the final quantum states is via an intermediate state even though the lifetime of the latter can be much shorter than the interaction time with the laser radiation. Nevertheless, spontaneous emission from the intermediate state is prevented by quantum interference. Maintaining the coherence between the initial and final state throughout the transfer process is crucial. STIRAP was initially developed with applications in chemical dynamics in mind. That is why the original paper of 1990 was published in The Journal of Chemical Physics. However, as of about the year 2000, the unique capabilities of STIRAP and its robustness with respect to small variations of some experimental parameters stimulated many researchers to apply the scheme in a variety of other fields of physics. The successes of these efforts are documented in this collection of articles., Comment: accepted for publication in Journal of Physics B: Atomic, Molecular and Optical Physics

In this paper we examine Interaction-free measurement (IFM) where both the probe and the object are quantum particles. We argue that in this case the description of the measurement procedure must by symmetrical with respect to interchange of the roles of probe and object. A thought experiment is being suggested that helps to determine what does and what doesn't happen to the state of the particles in such a setup. It seems that unlike the case of classical object, here the state of both the probe and the object must change. A possible explanation of this might be that the probe and the object form an entangled pair as a result of non-interaction., Comment: 15 pages, 5 fugures, Editorial changes in comparison with the previous version (indicated in color)

Energy levels of nitrogen-vacancy centers in diamond were investigated using optically detected magnetic-resonance spectroscopy near the electronic ground-state level anticrossing (GSLAC) at an axial magnetic field around 102.4~mT in diamond samples with a nitrogen concentration of 1~ppm and 200~ppm. By applying radiowaves in the frequency ranges from 0 to 40 MHz and from 5.6 to 5.9 GHz, we observed transitions that involve energy levels mixed by the hyperfine interaction. We developed a theoretical model that describes the level mixing, transition energies, and transition strengths between the ground-state sublevels, including the coupling to the nuclear spin of the NV center\textquotesingle s $^{14}$N and $^{13}$C atoms. The calculations were combined with the experimental results by fitting the ODMR spectral lines based on a theoretical model, which yielded information about the polarization of nuclear spins. This study is important for the optimization of experimental conditions in GSLAC-based applications, e.g., microwave-free magnetometry and microwave-free nuclear-magnetic-resonance probes., Comment: Significant changes from v2: 1) Reinserted treatment of high-density (200ppm) sample; 2) updated paper to version accepted for publication

Recent developments in magnetic field sensing with negatively charged nitrogen-vacancy centers (NV) in diamond employ magnetic-field (MF) dependent features in the photoluminescence (PL) and eliminate the need for microwaves (MW). Here, we study two approaches towards improving the magnetometric sensitivity using the ground-state level anti-crossing (GSLAC) feature of the NV center at a background MF of 102.4\,mT. Following the first approach, we investigate the feature parameters for precise alignment in a dilute diamond sample; the second approach extends the sensing protocol into absorption via detection of the GSLAC in the diamond transmission of a 1042\,nm laser beam. This leads to an increase of GSLAC contrast and results in a magnetometer with a sensitivity of 0.45\,nT/$\sqrt{\text{Hz}}$ and a photon shot-noise limited sensitivity of 12.2 pT/$\sqrt{\rm{Hz}}$.

We propose methods of optical pumping that are applicable to open, high-angular-momentum transitions in atoms and molecules, for which conventional optical pumping would lead to significant population loss. Instead of applying circularly polarized cw light, as in conventional optical pumping, we propose to use techniques for coherent population transfer (e.g., adiabatic fast passage) to arrange the atoms so as to increase the entropy removed from the system with each spontaneous decay from the upper state. This minimizes the number of spontaneous-emission events required to produce a stretched state, thus reducing the population loss due to decay to other states. To produce a stretched state in a manifold with angular momentum J, conventional optical pumping requires about 2J spontaneous decays per atom, one of our proposed methods reduces this to about log_2(2J), while another of the methods reduces it to about one spontaneous decay, independent of J., Comment: 13 pages, 12 figures

We have shown that it is possible to model accurately optical phenomena in intense laser fields by taking into account the intensity distribution over the laser beam. We developed a theoretical model that divided an intense laser beam into concentric regions, each with a Rabi frequency that corresponds to the intensity in that region, and solved a set of coupled optical Bloch equations for the density matrix in each region. Experimentally obtained magneto-optical resonance curves for the $F_g=2\longrightarrow F_e=1$ transition of the $D_1$ line of $^{87}$Rb agreed very well with the theoretical model up to a laser intensity of around 200 mW/cm$^2$ for a transition whose saturation intensity is around 4.5 mW/cm$^2$. We have studied the spatial dependence of the fluorescence intensity in an intense laser beam experimentally and theoretically. An experiment was conducted whereby a broad, intense pump laser excited the $F_g=4\longrightarrow F_e=3$ transition of the $D_2$ line of cesium while a weak, narrow probe beam scanned the atoms within the pump beam and excited the $D_1$ line of cesium, whose fluorescence was recorded as a function of probe beam position. Experimentally obtained spatial profiles of the fluorescence intensity agreed qualitatively with the predictions of the model., Comment: 8 pages, 9 figures

We present the results of an investigation of the different physical processes that influence the shape of the nonlinear magneto-optical signals both at small magnetic field values (~ 100 mG) and at large magnetic field values (several tens of Gauss). We used a theoretical model that provided an accurate description of experimental signals for a wide range of experimental parameters. By turning various effects "on" or "off" inside this model, we investigated the origin of different features of the measured signals. We confirmed that the narrowest structures, with widths on the order of 100 mG, are related mostly to coherences among ground-state magnetic sublevels. The shape of the curves at other scales could be explained by taking into account the different velocity groups of atoms that come into and out of resonance with the exciting laser field. Coherent effects in the excited state can also play a role, although they mostly affect the polarization components of the fluorescence. The results of theoretical calculations are compared with experimental measurements of laser induced fluorescence from the D2 line of atomic rubidium as a function of magnetic field., Comment: 8 pages, 6 figures

We studied magneto-optical resonances caused by excited-state level crossings in a nonzero magnetic field. Experimental measurements were performed on the transitions of the $D_2$ line of rubidium. These measured signals were described by a theoretical model that takes into account all neighboring hyperfine transitions, the mixing of magnetic sublevels in an external magnetic field, the coherence properties of the exciting laser radiation, and the Doppler effect. Good agreement between the experimental measurements and the theoretical model could be achieved over a wide range of laser power densities. We further showed that the contrasts of the level-crossing peaks can be sensitive to changes in the frequency of the exciting laser radiation as small as several tens of megahertz when the hyperfine splitting of the exciting state is larger than the Doppler broadening., Comment: 10 pages, 10 figures

Nonlinear magneto-optical resonances on the hyperfine transitions belonging to the D2 line of rubidium were changed from bright to dark resonances by changing the laser power density of the single exciting laser field or by changing the vapor temperature in the cell. In one set of experiments atoms were excited by linearly polarized light from an extended cavity diode laser with polarization vector perpendicular to the light's propagation direction and magnetic field, and laser induced fluorescence (LIF) was observed along the direction of the magnetic field, which was scanned. A low-contrast bright resonance was observed at low laser power densities when the laser was tuned to the Fg=2 --> Fe=3 transition of Rb-87 and near to the Fg=3 --> Fe=4 transition of Rb-85. The bright resonance became dark as the laser power density was increased above 0.6mW/cm2 or 0.8 mW/cm2, respectively. When the Fg=2 --> Fe=3 transition of Rb-87 was excited with circularly polarized light in a second set of experiments, a bright resonance was observed, which became dark when the temperature was increased to around 50C. The experimental observations at room temperature could be reproduced with good agreement by calculations based on a theoretical model, although the theoretical model was not able to describe measurements at elevated temperatures, where reabsorption was thought to play a decisive role. The model was derived from the optical Bloch equations and included all nearby hyperfine components, averaging over the Doppler profile, mixing of magnetic sublevels in the external magnetic field, and a treatment of the coherence properties of the exciting radiation field., Comment: 9 pages, 7 figures

We present and experimental and theoretical study of nonlinear magneto-optical resonances observed in the fluorescence to the ground state from the 7P_{3/2} state of cesium, which was populated directly by laser radiation at 455 nm, and from the 6P_{1/2} and 6P_{3/2} states, which were populated via cascade transitions that started from the 7P_{3/2} state and passed through various intermediate states. The laser-induced fluorescence (LIF) was observed as the magnetic field was scanned through zero. Signals were recorded for the two orthogonal, linearly polarized components of the LIF. We compared the measured signals with the results of calculations from a model that was based on the optical Bloch equations and averaged over the Doppler profile. This model was adapted from a model that had been developed for D_1 and D_2 excitation of alkali metal atoms. The calculations agree quite well with the measurements, especially when taking into account the fact that some experimental parameters were only estimated in the model., Comment: small changes to text of previous version; 12 pages, 8 figures

The formation of dark states under interaction of degenerate atomic states with incoherent broadband radiation (white light) is discussed. A simple coupling scheme in a three level Lambda-system, which allows both qualitative and quantitative analysis is discussed. We found a stationary solution of the optical Bloch equations in a broad excitation line approximation that describes the dynamics of the atom-white light interaction and demonstrated its identity to a conventional dark state created with coherent laser fields. We than examine the efficiency of the population transfer induced by broadband radiation in a model Lambda-system and revealed that high efficiency (attaining 100 percent) of stimulated Raman adiabatic passage-like processes can be achieved with certain temporal control of light polarization. The corresponding criterion of adiabaticity was formulated and justified by means of numerical simulations., Comment: 11 pages, 7 figures

We describe the so-called "Lambda-Zeeman method" to investigate individual hyperfine transitions between Zeeman sublevels of atoms in an external magnetic field of 0.1 mT - 0.25 T. Atoms are confined in a nanocell with thickness L = Lambda, where Lambda is the resonant wavelength (794 nm or 780 nm for D1 or D2 line of Rb). Narrow resonances in the transmission spectrum of the nanocell are split into several components in a magnetic field; their frequency positions and probabilities depend on the B-field. Possible applications are described, such as magnetometers with nanometric spatial resolution and tunable atomic frequency references.

Bright and dark nonlinear magneto-optical resonances associated with the ground state Hanle effect have been studied experimentally and theoretically for D1 excitation of atomic cesium. This system offers the advantage that the separation between the different hyperfine levels exceeds the Doppler width, and, hence, transitions between individual levels can be studied separately. Experimental measurements for various laser power densities and transit relaxation times are compared with a model based on the optical Bloch equations, which averages over the Doppler contour of the absorption line and simultaneously takes into account all hyperfine levels, as well as mixing of magnetic sublevels in an external magnetic field. In contrast to previous studies, which could not resolve the hyperfine transitions because of Doppler broadening, in this study there is excellent agreement between experiment and theory regarding the sign (bright or dark), contrast, and width of the resonance. The results support the traditional theoretical interpretation, according to which these effects are related to the relative strengths of transition probabilities between different magnetic sublevels in a given hyperfine transition., Comment: 9 pages, 6 figures

We report the experimental observation of alignment to orientation conversion in the 7D_3/2 and 9D_3/2 states of Cs in the presence of an external dc electric field, and without the influence of magnetic fields or atomic collisions. Initial alignment of angular momentum states was created by two-step excitation with linearly polarized laser radiation. The appearance of transverse orientation of angular momentum was confirmed by the observation of circularly polarized light. We present experimentally measured signals and compare them with the results of a detailed theoretical model based on the optical Bloch equations., Comment: 4 pages with 4 figures (6 postscript files), submitted to Physical Review Letters

In this paper we, to our knowledge, for the first time obtain the rate equations for Zeeman coherences in the broad line approximation and steady-state balance equations directly from optical Bloch equations without the use of the perturbation theory. The broad line approximation allows us to use the adiabatic elimination procedure in order to eliminate the optical coherences from the optical Bloch equations, but the steady-state condition allows us to derive the balance equations straightforward. We compare our approach with the perturbation theory approach as given previously and show that our approach is more flexible in analyzing various experiments. Meanwhile we also show the validity and limitations of the application of the rate equations in experiments with coherent atomic excitation, when either broad line approximation or steady-state conditions hold. Thus we have shown the basis for modeling the coherent atomic excitation experiments by using the relatively simple rate equations, provided that certain experimental conditions hold.

We report the first use of an extremely thin vapor cell (thickness ~ 400 nm) to study the magnetic-field dependence of laser-induced-fluorescence excitation spectra of alkali atoms. This thin cell allows for sub-Doppler resolution without the complexity of atomic beam or laser cooling techniques. This technique is used to study the laser-induced-fluorescence excitation spectra of Rb in a 50 G magnetic field. At this field strength the electronic angular momentum J and nuclear angular momentum I are only partially decoupled. As a result of the mixing of wavefunctions of different hyperfine states, we observe a nonlinear Zeeman effect for each sublevel, a substantial modification of the transition probabilities between different magnetic sublevels, and the appearance of transitions that are strictly forbidden in the absence of the magnetic field. For the case of right- and left- handed circularly polarized laser excitation, the fluorescence spectra differs qualitatively. Well pronounced magnetic field induced circular dichroism is observed. These observations are explained with a standard approach that describes the partial decoupling of I and J states.

The fluorescence spectra of a sub-micron atomic cesium vapor layer observable under resonant excitation on D2 line have been studied in the presence of an external magnetic field. Substantial changes in amplitudes and frequency positions of the individual (resolved) hyperfine transitions have been recorded in moderate magnetic fields (up to ~ 50 Gauss). These features are caused by mixing of the hyperfine states of the upper level resulting from comparable values of the hyperfine splitting of the 62^P_{3/2} manifold and Larmor frequencies of the magnetic sublevels. The results of simulation show a good agreement with the experimental spectra. Possible application of the results for high spatial resolution magnetometry is discussed.

Excited state angular momentum alignment -- orientation conversion for atoms with hyperfine structure in presence of an external magnetic field is investigated. Transversal orientation in these conditions is reported for the first time. This phenomenon occurs under Paschen Back conditions at intermediate magnetic field strength. Weak radiation from a linearly polarized diode laser is used to excite Rb atoms in a cell. The laser beam is polarized at an angle of pi/4 with respect to the external magnetic field direction. Ground state hyperfine levels of the 5S_1/2 state are resolved using laser-induced fluorescence spectroscopy under conditions for which all excited 5P_3/2 state hyperfine components are excited simultaneously. Circularly polarized fluorescence is observed to be emitted in the direction perpendicular to both to the direction of the magnetic field B and direction of the light polarization E. The obtained circularity is shown to be in quantitative agreement with theoretical predictions., Comment: Accepted for publication in Phys. Rev. A

Origin of recently discovered reversed (opposite sign) dark resonances was explained theoretically and verified experimentally. It is shown that the reason for these resonances is a specific optical pumping of ground state level in a transition when ground state angular momentum is smaller than the excited state momentum., Comment: 4 ps figures