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The most important problem of spectroscopic chiral analysis is the inherently weak chiral signals are easily overwhelmed by the environment noises. Enormous efforts had been spent to overcome this problem by enhancing the symmetry break in the light-molecule interactions or reducing the environment noises. Here, we propose an alternative way to solve this problem by using frequency-entangled photons as probe signals and detecting them in coincidence, i.e., using quantum chiral spectroscopy. For this purpose, we develop the theory of entanglement-assisted quantum chiral spectroscopy. Our results show that the signals of left- and right-handed molecules in the quantum spectrum are always distinguishable by suitably configuring the entangled probe photons. In construct, the classical spectrum of the two enantiomers become indistinguishable when the symmetry break in the interactions is overwhelmed by the environment noises. This offers our quantum chiral spectroscopy a great advantage over all classical chiral spectroscopy. Our work opens up an exciting area that exploring profound advantages of quantum spectroscopy in chiral analysis.

Chiral discrimination that is efficient to tiny amounts of chiral substances, especially at the single-molecule level, is highly demanded. Here, we propose a single-shot nondestructive quantum sensing method addressing such an issue. Our scheme consists of two steps. In the first step, the two enantiomers are prepared in different rotational states via microwave enantio-specific state transfer. Then, the chiral discrimination is transferred to quantum hypothesis testing. In the second step, we for the first time introduce a non-destructive quantum-state detection technique assisted with a microwave resonator to chiral discrimination, through which the molecular chirality is determined by the sign of the output signals. Using a typical chiral molecule, 1,2-propanediol, and an experimentally feasible model based on spherical Fabry-P\'{e}rot cavity, we show that the molecular chirality of slowly moving enantiopure gaseous samples with $10^2 - 10^3$ molecules can be highly credibly distinguished in a single-shot detection. By further trapping chiral molecules, it is promising to achieve chiral discrimination at the single molecule level by using our approach.

Multiplicity fluctuations play a crucial role in relativistic heavy-ion collisions. In this work, we explore how the multiplicity fluctuations can be effectively suppressed in the measurement of particle correlations. In particular, through proper normalization, particle correlations can be evaluated in a manner irrelevant to multiplicity. When the multiplicity fluctuations are adequately extracted, Monte Carlo simulations show that the remaining correlations possess distinct features buried in the otherwise overwhelming fluctuations. Moreover, we argue that such a normalization scheme naturally agrees with the multi-particle correlator, which can be evaluated using the Q-vectors. The implications of the present study in the data analysis are also addressed., Comment: 15 pages, 3 figures

Semi-quantum private comparison (SQPC) enables two classical users with limited quantum capabilities to compare confidential information using a semi-honest third party (TP) with full quantum power. However, entanglement swapping, as an important property of quantum mechanics in previously proposed SQPC protocols is usually neglected. In this paper, we propose a feasible SQPC protocol based on the entanglement swapping of Bell states, where two classical users do not require additional implementation of the semi-quantum key distribution protocol to ensure the security of their private data. Security analysis shows that our protocol is resilient to both external and internal attacks. To verify the feasibility and correctness of the proposed SQPC protocol, we design and simulate the corresponding quantum circuits using IBM Qiskit. Finally, we compare and discuss the proposed protocol with previous similar work. The results reveal that our protocol maintains high qubit efficiency, even when entanglement swapping is employed. Consequently, our proposed approach showcases the potential applications of entanglement swapping in the field of semi-quantum cryptography., Comment: 17 pages, 6 figures. arXiv admin note: text overlap with arXiv:2210.03421

Developing effective methods to measure the enantiomeric excess of the chiral mixture is one of the major topics in chiral molecular researches, yet remains challenging. Enantiodetection method via two-dimensional (2D) spectroscopy based on a four level model, containing a cyclic three-level system (CTLS), of chiral molecules was recently proposed and demonstrated, yet with a strict condition of the one-photon resonance (where three driving fields are exactly resonantly coupled to the three electric-dipole transitions, respectively) in the CTLS and narrowband probe pulse assumption. Here, we extend the 2D spectroscopy method to more general experimental conditions, with three-photon resonance (where the sum of the two smaller frequencies among the three driving fields equals to the third one) and broadband probe pulse. Our method remains effective on enantiodetection with the help of experimental techniques, such as the chop detection method, which is used to eliminate the influence of the other redundant levels existing in the real system of chiral molecules. Under these more general conditions, the enantiomeric excess of the chiral mixture is estimated by taking an easily available standard sample (usually the racemic mixture) as the reference., Comment: 9 pages, 4 figures

In relativistic heavy-ion collisions, event-by-event fluctuations are known to have non-trivial implications. Even though the probability distribution is geometrically isotropic for the initial conditions, the anisotropic $\varepsilon_n$ still differs from zero owing to the statistical fluctuations in the energy profile. On the other hand, the flow harmonics extracted from the hadron spectrum using the multi-particle correlators are inevitably subjected to non-vanishing variance due to the finite number of hadrons emitted in individual events. As one aims to extract information on the fluctuations in the initial conditions via flow harmonics and their fluctuations, finite multiplicity may play a role in interfering with such an effort. In this study, we explore the properties and impacts of such fluctuations in the initial and final states, which both notably appear to be statistical ones originating from the finite number of quanta of the underlying system. We elaborate on the properties of the initial-state eccentricities for the smooth and event-by-event fluctuating initial conditions and their distinct impacts on the resulting flow harmonics. Numerical simulations are performed. The possible implications of the present study are also addressed., Comment: 11 pages, 2 figures

We explore the possibility of evaluating flow harmonics by employing the maximum likelihood estimator (MLE). For a given finite multiplicity, the MLE simultaneously furnishes estimations for all the parameters of the underlying distribution function while efficiently suppressing the variance of measures. Also, the method provides a means to assess a specific class of mixed harmonics, which is not straightforwardly feasible by the approaches primarily based on particle correlations. The results are analyzed using the Wald, likelihood ratio, and score tests of hypotheses. Besides, the resultant flow harmonics obtained using MLE are compared with those derived using particle correlations and event plane methods. The dependencies of extracted flow harmonics on the multiplicity of individual events and the total number of events are analyzed. It is shown that the proposed approach works efficiently to deal with the deficiency in detector acceptability. Moreover, we elaborate on a fictitious scenario where the event plane is not a well-defined quantity in the distribution function. For the latter case, the MLE is shown to largely perform better than the two-particle correlation estimator. In this regard, one concludes that the MLE furnishes a meaningful alternative to the existing approaches for flow analysis., Comment: 15 pages, 5 figures, and 8 tables

Metric learning plays an essential role in image analysis and classification, and it has attracted more and more attention. In this paper, we propose a quantum adversarial metric learning (QAML) model based on the triplet loss function, where samples are embedded into the high-dimensional Hilbert space and the optimal metric is obtained by minimizing the triplet loss function. The QAML model employs entanglement and interference to build superposition states for triplet samples so that only one parameterized quantum circuit is needed to calculate sample distances, which reduces the demand for quantum resources. Considering the QAML model is fragile to adversarial attacks, an adversarial sample generation strategy is designed based on the quantum gradient ascent method, effectively improving the robustness against the functional adversarial attack. Simulation results show that the QAML model can effectively distinguish samples of MNIST and Iris datasets and has higher robustness accuracy over the general quantum metric learning. The QAML model is a fundamental research problem of machine learning. As a subroutine of classification and clustering tasks, the QAML model opens an avenue for exploring quantum advantages in machine learning., Comment: arXiv admin note: substantial text overlap with arXiv:2303.07906

Label propagation is an essential semi-supervised learning method based on graphs, which has a broad spectrum of applications in pattern recognition and data mining. This paper proposes a quantum semi-supervised classifier based on label propagation. Considering the difficulty of graph construction, we develop a variational quantum label propagation (VQLP) method. In this method, a locally parameterized quantum circuit is created to reduce the parameters required in the optimization. Furthermore, we design a quantum semi-supervised binary classifier based on hybrid Bell and $Z$ bases measurement, which has shallower circuit depth and is more suitable for implementation on near-term quantum devices. We demonstrate the performance of the quantum semi-supervised classifier on the Iris data set, and the simulation results show that the quantum semi-supervised classifier has higher classification accuracy than the swap test classifier. This work opens a new path to quantum machine learning based on graphs.

Enantio-conversion of chiral mixtures, converting the mixtures composed of left- and right-handed chiral molecules into the homochiral ensembles, has become an important research topic in chemical and biological fields. In previous studies on enantio-conversion, the tunneling interaction between the left- and right-handed chiral states was often neglected. However, for certain chiral molecules, this tunneling interaction is significant and cannot be ignored. Here we propose a scheme for enantio-conversion of chiral mixtures through optical pumping based on a four-level model of chiral molecules, comprising two chiral ground states and two achiral excited states, with a tunneling interaction between the chiral states. Under one-photon large detuning and two-photon resonance conditions, one of the achiral excited states is eliminated adiabatically. By well designing the detuning and coupling strengths of the electromagnetic fields, the tunneling interaction between two chiral states and the interaction between one of the chiral states and the remaining achiral excited state can be eliminated. Consequently, one chiral state remains unchanged, while the other can be excited to an achiral excited state, establishing chiral-state-selective excitations. By numerically calculating the populations of two chiral ground states and the enantiomeric excess, we observe that high-efficiency enantio-conversion is achieved under the combined effects of system dissipation and chiral-state-selective excitations., Comment: 14 pages, 6 figures

An interesting method of fast enantio-specific state transfer is proposed for cyclic three-level systems of chiral molecules. We show that the fast population transfer via shortcut to adiabaticity can be accomplished for the cyclic three-level system of a general (chiral) molecule with invariant-based inverse engineering of the coupling strengths. By choosing appropriate parameters, the two enantiomers, which are initially prepared in their ground states in the three-level systems, will evolve respectively along their enantio-selective shortcut-to-adiabaticity paths to different-energy final states simultaneously, namely achieving the fast enantio-specific state transfer., Comment: 8 pages, 4 figures

Enantiodetection is an important and challenging task across natural science. Nowadays, some chiroptical methods of enantiodetection based on decoherence-free cyclic three-level models of chiral molecules can reach the ultimate limit of the enantioselectivities in the molecular responses. They are thus more efficient than traditional chiroptical methods. However, decoherence is inevitable and can severely reduce enantioselectivities in these advanced chiroptical methods, so they only work well in the weak decoherence region. Here, we propose an enantioselective switch on the radiation of dissipative chiral molecules and develop a novel chiroptical method of enantiodetection working well in all decoherence regions. In our scheme, radiation is turned on for the selected enantiomer and simultaneously turned off for its mirror image by designing the electromagnetic fields well based on dissipative cyclic three-level models. The enantiomeric excess of a chiral mixture is determined by comparing its emissions in two cases, where the radiations of two enantiomers are turned off respectively. The corresponding enantioselectivities reach the ultimate limit in all decoherence regions, offering our scheme advantages over other chiroptical methods in enantiodetection. Our work potentially constitutes the starting point for developing more efficient chiroptical techniques for enantiodection in all decoherence regions.

Locally chiral light is an emerging tool for probing and controlling molecular chirality. It can generate large and freely adjustable enantioselectivities in purely electric-dipole effects, offering its major advantages over traditional chiral light. However, the existing types of locally chiral light are phase-mismatched, and thus the global efficiencies are greatly reduced compared with the maximum single-point efficiencies or even vanish. Here, we propose a scheme to generate phase-matched locally chiral light. To confirm this advantage, we numerically show the robust highly efficient global control of enantiospecific electronic state transfer of methyloxirane at nanoseconds. Our work potentially constitutes the starting point for developing more efficient chiroptical techniques for the studies of chiral molecules., Comment: 5 pages, 3figures, 1 supplment document

Semi-quantum protocols construct connections between quantum users and ``classical'' users who can only perform certain ``classical'' operations. In this paper, we present a new semi-quantum private comparison protocol based on entangled states and single particles, which does not require pre-shared keys between the ``classical'' users to guarantee the security of their private data. By utilizing multi-particle entangled states and single particles, our protocol can be easily extended to multi-party scenarios to meet the requirements of multiple ``classical'' users who want to compare their private data. The security analysis shows that the protocol can effectively prevent attacks from outside eavesdroppers and adversarial participants. Besides, we generalize the proposed protocol to other semi-quantum protocols such as semi-quantum key agreement, semi-quantum summation, and semi-quantum anonymous ranking protocols. We compare and discuss the proposed protocols with previous similar protocols. The results show that our protocols satisfy the demands of their respective counterparts separately. Therefore, our protocols have a wide range of application scenarios., Comment: 19 pages 5 tables

In this paper, a novel multi-party quantum private comparison (MQPC) protocol for equality comparison with n-level single-particle states is constructed, where the encoded particles are transmitted in a circular way. Here, n parties employ the qudit shifting operation to encode their private secrets and can compare the equality of their private secrets within one time execution of protocol. The proposed MQPC protocol can overcome both the outside attack and the participant attack. Specially, each party's secret can be kept unknown to other parties and the third party (TP)., Comment: 11 pages, 1 table. arXiv admin note: substantial text overlap with arXiv:2205.06406

Mediated semi-quantum key distribution (M-SQKD) permits two limited "semi-quantum" or "classical" users to establish a secret key with the help of a third party (TP), in which TP has fully quantum power and may be untrusted. Several protocols have been studied recently for two-party scenarios, but no one has considered M-SQKD for multi-party scenarios. In this paper, we design a circular M-SQKD protocol based on Bell states, which offers an approach to realizing multiple "classical" users' key distribution. Then, we prove the protocol is unconditional security in the asymptotic scenario. The protocol's key rate and noise tolerance can be derived by utilizing the parameters observed in the channel. The results show that our protocol may hold similar security to a fully quantum one. We also compare the proposed protocol with similar protocols in terms of noise tolerance, qubit efficiency, communication cost, and scalability. Finally, the security proof method of this paper may contribute to studying the security of other circular semi-quantum cryptography protocols., Comment: 16pages, 4figures

In this paper, we successfully design the semi-quantum private comparison (SQPC) protocol with the measure-resend characteristic by using two-particle product states as the initial prepared quantum resource which allows two classical users to compare the equality of their private secrets under the help of a quantum third party (TP). The quantum TP is semi-honest in the sense that he is allowed to misbehave on his own but cannot conspire with either of users. Both the output correctness and the security against the outside attack and the participant attack can be guaranteed. Compared with the previous SQPC protocols, the advantage of our protocol lies in that it only employs two-particle product states as the initial prepared quantum resource, only requires TP to perform single-photon measurements and does not need quantum entanglement swapping. Our protocol can be realized with current quantum technologies., Comment: 11 pages, 1 figure, 3 tables

In this paper, by using d-level single-particle states, two novel multi-party quantum private comparison protocols for size relation comparison with two semi-honest third parties and one semi-honest third party are constructed, respectively. Here, each protocol can compare the size relation of secret integers from n parties rather than just the equality within one time execution. In each protocol, every third party is assumed to be semi-honest in the sense that she may misbehave on her own but is not allowed to collude with anyone else; and each party employs the qudit shifting operation to encode her secret integer. Each protocol can resist both the outside attack and the participant attack. Specially, each party's secret integer can be kept unknown to other parties and the third parties. The proposed protocol with two third parties is workable in a stranger environment, as there are no communication and no pre-shared key between each pair of party. The proposed protocol with one third party is workable in an acquaintance environment, as all parties need to share a common private key beforehand., Comment: 12 pages, 2 figures, 2 tables

In this paper, we propose two semi-quantum dialogue (SQD) protocols by using single photons as the quantum carriers, where one requires the classical party to possess the measurement capability and the other does not have this requirement. The security towards active attacks from an outside Eve in the first SQD protocol is guaranteed by the complete robustness of present semi-quantum key distribution (SQKD) protocols, the classical one-time pad encryption, the classical party's randomization operation and the decoy photon technology. The information leakage problem of the first SQD protocol is overcome by the classical party' classical basis measurements on the single photons carrying messages which makes him share their initial states with the quantum party. The security towards active attacks from Eve in the second SQD protocol is guaranteed by the classical party's randomization operation, the complete robustness of present SQKD protocol and the classical one-time pad encryption. The information leakage problem of the second SQD protocol is overcome by the quantum party' classical basis measurements on each two adjacent single photons carrying messages which makes her share their initial states with the classical party. Compared with the traditional information leakage resistant QD protocols, the advantage of the proposed SQD protocols lies in that they only require one party to have quantum capabilities. Compared with the existing SQD protocol, the advantage of the proposed SQD protocols lies in that they only employ single photons rather than two-photon entangled states as the quantum carriers. The proposed SQD protocols can be implemented with present quantum technologies., Comment: 11 pages, 2 figures, 2 tables

Semi-quantum secret sharing (SQSS) is an important branch of semi-quantum cryptography, and differs from quantum secret sharing (QSS) in that not all parties are required to possess quantum capabilities. All previous SQSS protocols have three common features: (1) they adopt product states or entangled states as initial quantum resource; (2) the particles prepared by quantum party are transmitted in a tree-type way; and (3) they require the classical parties to possess the measurement capability. In this paper, two circular SQSS protocols with single particles are suggested, where the first one requires the classical parties to possess the measurement capability while the second one does not have this requirement. Compared with the previous SQSS protocols, the proposed SQSS protocols have some distinct features: (1) they adopt single particles rather than product states or entangled states as initial quantum resource; (2) the particles prepared by quantum party are transmitted in a circular way; and (3) the second protocol releases the classical parties from the measurement capability. The proposed SQSS protocols are robust against some famous attacks from an eavesdropper, such as the measure-resend attack, the intercept-resend attack and the entangle-measure attack, and are feasible with present quantum technologies in reality., Comment: 15 pages, 4 figures, 2 tables

Enantiodetection of chiral molecules is important to pharmaceutical drug production, chemical reaction control, and biological function designs. Traditional optical methods of enantiodetection rely on the weak magnetic-dipole or electric-quadrupole interactions, and in turn suffer from the weak signal and low sensitivity. We propose a new optical enantiodetection method to determine the enantiomeric excess via two-dimensional (2D) spectroscopy of the chiral mixture driven by three electromagnetic fields. The quantities of left- and right- handed chiral molecules are reflected by the intensities of different peaks on the 2D spectrum, separated by the chirality-dependent frequency shifts resulting from the relative strong electric-dipole interactions between the chiral molecules and the driving fields. Thus, the enantiomeric excess can be determined via the intensity ratio of the peaks for the two enantiomers., Comment: 6 pages, 4 figures

We propose an enantio-detection method of chiral molecules in a cavity with external drive. The chiral molecules are coupled with a quantized cavity field and two classical light fields to form the cyclic three-level systems. The chirality-dependent cavity-assisted three-photon process in the three-level systems leads to the generation of intracavity photons. Simultaneously, the drive field also results in the chirality-independent process of the generation of intracavity photons. Based on the interference between the intracavity photons generated from these two processes, one can detect the enantiomeric excess of chiral mixture via monitoring the transmission rate of the drive field., Comment: 8 pages, 5 figures

We study the enantio-specific state transfer in a four-level model for symmetric-top chiral molecules. Such a model is formed by coupling the electric-dipole transitions among four appropriate working states with three electromagnetic fields. It includes two cyclic three-level substructures, where the overall phases differ by $\pi$ with enantiomers and reflect the chirality dependence of the molecule. Based on this four-level model, two dynamic ways are proposed to achieve the approximately perfect enantio-specific state transfer for symmetric-top chiral molecules., Comment: 7 pages, 3 figures

We explore the spatial enantioseparation of gaseous chiral molecules for the cyclic three-level systems coupled with three electromagnetic fields. Due to molecular rotations, the specific requirements of the polarization directions of the three electromagnetic fields lead to the space-dependent part of the overall phase of the coupling strengths. Thus, the overall phase of the coupling strengths, which differs with $\pi$ for the enantiomers in the cyclic three-level model of chiral molecules, varies intensely in the length scale of the typical wavelength of the applied electromagnetic fields. Under the induced gauge potentials resulting from the space-dependent part of the overall phase and the space-dependent intensities of coupling strengths, we further show spatial enantioseparation for typical parameters of gaseous chiral molecules., Comment: 8 pages, 4 figures

We propose a method for enantio-detection of chiral molecules based on a cavity-molecule system, where the left- and right-handed molecules are coupled with a cavity and two classical light fields to form cyclic three-level models. Via the cavity-assisted three-photon processes based on the cyclic three-level model, photons are generated continuously in the cavity even in the absence of external driving to the cavity. However, the photonic fields generated from the three-photon processes of left- and right-handed molecules differ with the phase difference {\pi} according to the inherent properties of electric-dipole transition moments of enantiomers. This provides a potential way to detect the enantiomeric excess of chiral mixture by monitoring the output field of the cavity., Comment: 13 pages, 5 figures

Enantio-conversion with the help of electromagnetic fields is an essential issue due to the chirality-dependence of many chemical, biological, and pharmaceutical processes. Here, we propose a method for this issue based on a five-level double-$\Delta$ model of chiral molecules. By utilizing the breaking of left-right symmetry in the two $\Delta$-type sub-structures, we can establish the chiral-state-selective excitation with one chiral ground state being excited to an achiral excited state and the other one being undisturbed. In the meanwhile, the achiral excited state will relax to the two chiral ground states. The two effects simultaneously acting on the chiral mixtures can convert molecules of different chiralities to the ones of the same chirality, i.e., the enantio-conversion via optical pumping. With typical parameters in gas-phase experiments, we numerically show that highly efficient enantio-conversion can be achieved. Our method works in the appearance of decoherences and without the precise control of pulse-durations (pulse-areas) and/or pulse-shapes. These advantages offer it promising features in promoting the future exploring of enantio-conversion., Comment: 7 pages, 3 figures

Laser-assisted enantio-conversion is an ambitious issue related to chiral molecules in the atomic, molecular, and optical physics. The theoretical laser-distillation method had been proposed to realize enantio-conversion based on a four-level double-$\Delta$ model of two degenerated chiral ground states and two achiral excited states. Here, we re-investigate and improve the laser-distillation method so that a chiral mixture can be converted to an enantiopure sample of the desired chirality, i.e., complete enantio-conversion, which has not been discussed in the previous theoretical works of the laser-distillation method. Since the undesired chirality may be inefficient or even cause serious side effects in pharmacology, our work plays as an important improvement to the original theoretical works of the laser-assisted enantio-conversion.

Optical methods of enantiomeric-specific state transfer had been proposed theoretically based on a cyclic three-level system of chiral molecule. According to these theoretical methods, recently the breakthrough progress has been reported in experiments [S. Eibenberger et al., Phys. Rev. Lett. 118, 123002 (2017); C. P\'{e}rez et al., Angew. Chem. Int. Ed. 56, 12512 (2017)] for cold gaseous chiral molecules but with low state-specific enantiomeric enrichment. One of the limiting factors is the influence of the thermal population in the selected three purely rotational states in experiment. Here, we theoretically propose an improved optical method of enantiomeric-specific state transfer to effectively evade such an adverse impact of thermal population by introducing ro-vibrational transitions for the cyclic three-level system of chiral molecules. Then, at the typical temperature in experiments approximately only the lowest state in the chosen three-level system is thermally occupied and the optical method of enantiomeric-specific state transfer works well. Comparing with the case of purely rotational transitions where all the three states are thermally occupied, this modification will remarkably increase the obtained state-specific enantiomeric enrichment with enantiomeric excess approximately 100%., Comment: 6 pages, 4 figures

The spontaneous emission spectrum of a multi-level atom or molecule with nonreciprocal transition is investigated. It is shown that the nonreciprocal transition can lead to the elimination of a spectral line in the spontaneous emission spectrum. As an application, we show that nonreciprocal transition arises from the phase-related driving fields in chiral molecules with cyclic three-level transitions, and the elimination of a spectral line induced by nonreciprocal transition provides us a method to determine the enantiomeric excess for the chiral molecules without requiring the enantio-pure samples., Comment: 8 pages, 6 figures

Based on the four-level double-$\Delta$ model composed of two degenerated (left- and right-handed) chiral ground states and two achiral excited states, we propose a purely optical method for enantio-conversion of chiral mixture. By choosing appropriate parameters, the original four-level model will be simplified to two effective two-level sub-systems with each of them involving one chiral ground state. Then, with the help of well-designed optical operations, the initial unwanted and wanted chiral ground states are converted, respectively, to the wanted chiral ground state and an auxiliary chiral excited state with the wanted chirality, i.e., achieving the enantioconversion of the chiral mixture. Comparing with the original work of enantioconversion based on the four-level double-$\Delta$ model with the requirement of the time-consuming relaxation step and repeated operations, our method can be three orders of magnitude faster since we use only purely optical operations. Thus, it offers a promising candidate for fast enantioconversion when the total operation time is limited due to the racemization and/or experimental conditions.

We propose a theoretical method for enantio-discrimination based on the light deflection effect in four-level models of chiral molecules. This four-level model consists of a cyclic three-level subsystem coupled by three strong driving fields and an auxiliary level connected to the cyclic three-level subsystem by a weak probe field. It is shown that the induced refractive index for the weak probe field is chirality-dependent. Thus it will lead to chirality-dependent light deflection when the intensities of two of the three strong driving fields are spatially inhomogeneous. As a result, the deflection angle of the weak probe light can be utilized to detect the chirality of pure enantiomers and enantiomeric excess of chiral mixture. Therefore, our method may act as a tool for enantio-discrimination., Comment: 9 pages, 6 figures

In this work, we explore various relevant aspects of the Smoothed Particle Hydrodynamics regarding Burger's equation. The stability, precision, and efficiency of the algorithm are investigated in terms of different implementations. In particular, we argue that the boundary condition plays an essential role in the stability of numerical implementation. Besides, the issue is shown to be closely associated with the initial particle distribution and the interpolation scheme. Among others, we introduce an interpolation scheme termed symmetrized finite particle method. The main advantage of the scheme is that its implementation does not involve any derivative of the kernel function. Concerning the equation of motion, the calculations are carried out using two distinct scenarios where the particles are chosen to be either stationary or dynamically evolved. The obtained results are compared with those obtained by using the standard finite difference method for spatial derivatives. Our numerical results indicate subtle differences between different schemes regarding the choice of boundary condition. In particular, a novel type of instability is observed where the regular distribution is compromised as the particles start to traverse each other. Implications and further discussions of the present study are also addressed., Comment: 18 pages, 5 figures