Enantiodiscrimination of chiral molecules via quantum correlation function

We propose a method to realize enantiodiscrimination of chiral molecules based on quantum correlation function in a driven cavity-molecule system, where the chiral molecule is coupled with a quantized cavity field and two classical light fields to form a cyclic three-level model. According to the inherent properties of electric-dipole transition moments of chiral molecules, there is a $\pi$-phase difference in the overall phase of the cyclic three-level model for the left- and right-handed chiral molecules. Thus, the correlation function depends on this overall phase and is chirality-dependent. The analytical and numerical results indicate that the left- and right-handed chiral molecules can be discriminated by detecting quantum correlation function. Our work opens up a promising route to discriminate molecular chirality, which is an extremely important task in pharmacology and biochemistry.

In this paper, we propose a method for the discrimination of the left-and right-handed chiral molecules based on quantum correlation function (i.e., equal-time second-order correlation function) of the cavity field. The second-order correlation function is a quantum physical quantity and can reveal the quantum properties of the field. By analysing the second-order correlation function in cyclic three-level models of atom [59,60], the photon (or magnon) blockade effect has been investigated. Here, based on the similar cyclic three-level models of chiral molecules, we study enantiodiscrimination of chiral molecules by detecting the equal-time second-order correlation function. In the weak-driving case, we derive the analytical expression of the equal-time second-order correlation function by using the probability amplitude method [61][62][63][64][65], and find that the correlation function is chirality-dependent. According to the theoretical analysis, we further demonstrate that the left-and right-handed chiral molecules can be distinguished by detecting the correlation function. Therefore, our method provides a feasible way to discriminate molecular chirality.
The rest of this paper is organized as follows. In Sec. 2, we introduce the physical model of the cavity-molecule system and present the system Hamiltonian. In Sec. 3, we derive the analytical expression of the equal-time second-order correlation function of the cavity field in the weak-driving case. In Sec. 4, we investigate the dependence of the equal-time second-order correlation function for the left-and right-handed chiral molecules on the parameters (e.g. the detunings and the driving strengths). Finally, A summary is given in Sec. 5.

Model
We consider a cavity-molecule system consisting of a cavity and a chiral molecule, as shown in Fig. 1. The chiral molecule is coupled with a quantized cavity field and two classical light fields to form the cyclic three-level model, and the cavity is continuously driven by a monochromatic weak field with the driving strength and driving frequency . For convenience, we adopt the subscripts " " and " " to mark the left-and right-handed chiral molecules, respectively. | and | ( = 1, 2, 3) are, respectively, the th eigen-states of the left-and right-handed chiral molecules with the same eigen-energy ℏ . Under the dipole approximation and rotating-wave approximation, the Hamiltonian of the cavity-molecule system for the left-or right-handed chiral molecules reads (ℏ = 1) where the superscript ( = or ) is introduced to represent the molecular chirality. Here † ( ) is the creation (annihilation) operator of the cavity field with resonance frequency , and = | | ( , = 1, 2, 3) are the molecular raising and lowering operators for ≠ and the molecular population operators for = . The parameter denotes the coupling strength between the molecular transition |1 ↔ |2 and the quantized cavity field, Ω 31 (Ω 32 ) and 31 ( 32 ) are the driving strength and driving frequency of the classical light field acting on the molecular transition |1 ↔ |3 (|2 ↔ |3 ), and is overall phase of the cyclic three-level model. For simplicity but without loss of generality, we have taken these parameters ( , , Ω 31 , and Ω 32 ) as positive real numbers. The molecular chirality is reflected by choosing the overall phases of the left-and right-handed chiral molecules as [16,39] = , = + .

Analytical results of the correlation function
In order to obtain the analytical results of the correlation function in the cavity field, we consider a weak-driving case ({Ω 31 , } ). In this case, the two driving terms Ω 31 ( 13 + 31 ) and ( + † ) can be considered as perturbative terms. For the Hamiltonian ) in the absence of the two driving terms, the total excitation number operatorˆ= † + 22 + 33 is a conserved quantity due to the commutative relation [ˆ, ] = 0. The subspaces corresponding to the total excitation number = 0, 1, 2, . . . , , . . . are spanned over the basis states To include the influence of the dissipations of the cavity field and the chiral molecule on the quantum statistics, we phenomenologically add the imaginary dissipation terms to Hamiltonian (3) as [59,60] with and ( , = 1, 2, 3) being the decay rates of the cavity field and the chiral molecule, respectively. Here the non-Hermitian Hamiltonian non is obtained based on the quantum-jump approach [73] and the pure dephasing of the system is neglected.
In Fig. 2, we show the equal-time second-order correlation function (2) (0) ( = , ) as a function of the detuning Δ . Concretely, we consider the cases of = 0 in Fig. 2(a) and = /2 in Fig. 2(b). Here the solid curves correspond to the numerical results obtained by solving Eq. (4), while the dashed curves correspond to the analytical results given in Eq. (12). We find that there is a slight discrepancy between the analytical and numerical results, but the main physical results are the same. Such a discrepancy can be understood as follows: (i) We have mentioned in Sec. 3 that the quantum jump terms are ignored in the analytical calculation compared with the numerical result. (ii) In the derivation of the analytical result, we have made the perturbation approximation. In the numerical calculation, the quantum master equation (4)  truncation dimension of the cavity field needs to be chosen such that the numerical result of the correlation function is convergent. Hence, here we choose the truncation dimension of the cavity field as = 8. In the case of = 0, we can see from Fig. 2(a) that the correlation function (2) (0) for the left-handed chiral molecule is larger than 1 around Δ = 0 (the light orange area), which corresponds to the super-Poissonian distribution of the photons. However, the correlation function for the right-handed chiral molecule is (2) (0) < 1 around Δ = 0 corresponding to the sub-Poissonian distribution. For the case of = /2, it can be seen in Fig. 2(b) that, in the red-detuning (blue-detuning) regime Δ > 0 (Δ < 0), a bunching peak of the correlation function (2) can be observed for the left-handed (right-handed) chiral molecule. In addition, we find that the correlation function for the left-handed chiral molecule is (2) and for the right-handed chiral molecule is (2) in the red (blue) area. This indicates that the left-and right-handed chiral molecules can be discriminated by detecting the equal-time second-order correlation function of the cavity field.
To analyze the influence of the driving strength Ω 32 of the classical light field acting on the molecular transition |2 ↔ |3 on the correlation function, we display in Figs. 3(a) and 3(b) the correlation functions (2) (0) for the left-and right-handed chiral molecules versus the detuning Δ at various values of Ω 32 . All results in Fig. 3 are obtained by solving Eq. (4), and we consider only the case of = /2. It can be seen that the detuning locations of the bunching peaks in the correlation function for the left-and right-handed chiral molecules correspond to the red detuning Δ ≈ Ω 32 and the blue detuning Δ ≈ −Ω 32 , respectively. In particular, we find that the curve of the correlation function for the left-handed chiral molecule is mutually symmetric with the right-handed chiral molecule at = /2. To further understand the reason of the bunching peak generation, we also analyze the steady-state occupations |1,1 = 1, 1| ss |1, 1 and |1,2 = 1, 2| ss |1, 2 for the left-and right-handed chiral molecules versus the detuning Δ at Ω 32 / = 0.5, as shown in Figs. 3(c) and 3(d). We find that the location of the bunching peak in the correlation function (2) (0) corresponds to the location of the dip in |1,1 . For the cavity-molecule system, there are two different transition paths from the state |1, 0 to |1, 1 . Here the direct transition path is |1, 0 −→ |1, 1 , and the indirect transition path is When a perfect destructive quantum interference happens between these two transition paths, the value of the steady-state occupation |1,1 is zero. However, the value of |1,1 decreases for an imperfect destructive quantum interference, corresponding to the location of the dip in |1,1 . This indicates that the generation of the bunching peak is based on the quantum interference effect between the two different transition paths [65].
We also analyze how the correlation function depends on the driving strength Ω 31 of the classical light field acting on the molecular transition |1 ↔ |3 . The correlation functions log 10 ( In the above discussions, we neglect the effect of the pure dephasing rates in the system. Below, we will discuss the influence of the pure dephasing rates on the correlation function. Finally, we present a discussion on the implementation of the scheme. Concretely, we consider 1,2-propanediol molecules [79][80][81] as an example to realize the single-loop three-level models. We choose three working states of the cyclic three-level model as |1 = | |0 000 , |2 = | |1 110 , and |3 = | (|1 101 + |1 10−1 )/ √ 2, where | and | are, respectively, the vibrational ground state and first-excited state with transition frequency vib = 2 ×100.950 THz [79]. The rotational state is marked as | [19,82] with the angular moment quantum number , the magnetic quantum number , and ( ) runs form (0) to 0 ( ) in unit step with decreasing energy. According to the rotational constants for 1,2-propanediol molecules = 2 × 8524.405 MHz, = 2 × 3635.492 MHz, and = 2 × 2788.699 MHz [80], we can obtain the transition frequencies between the three working states as 21 ≡ 2 − 1 = 2 × 100.961 THz, 31 ≡ 3 − 1 = 2 × 100.962 THz, and 32 ≡ 3 − 2 = 2 × 0.847 GHz [82]. Hence, the single-loop three-level models can be realized by choosing three linearly polarized electromagnetic fields.
Here the state |1 is coupled to the state |2 by the -polarized quantized light field, and the state |1 (|2 ) is coupled to the state |3 by the -polarized ( -polarized) classical light field based on electric-dipole interaction. In particular, the frequencies of the three electromagnetic fields can be all microwave fields [48][49][50][51][52][53][54], or one microwave and two infrared fields [12,14,27]. For the case of the microwave (infrared) cavity with quantized field, it can be a transmission line resonator (Fabry-Pérot cavity).

Conclusion
In conclusion, we have proposed a feasible scheme to discriminate molecular chirality by detecting the equal-time second-order correlation function of the cavity field in a driven cavity-molecule system. By analytically and numerically calculating the correlation function, we find that this quantum correlation function for the left-handed (right-handed) chiral molecule is (2) (0) > 1 [ (2) (0) < 1] around Δ = 0 for the overall phase = 0. In the case of = /2, we find that the bunching peak of the second-order correlation function for the left-handed (right-handed) chiral molecule can be observed in the red-detuning (blue-detuning) regime. The obtained results indicate that the left-and right-handed chiral molecules can be discriminated by detecting the equal-time second-order correlation function. Our work provides a promising method to discriminate molecular chirality and has the following features: (i) For the case of single molecule, when the single-molecule coupling is weak (e.g. the coupling strength between the single molecule and the quantum light field is less than the decay rate of cavity field), the left-and right-handed chiral molecules can also be discriminated by measuring the correlation function of the cavity field. For the single-molecule case with weak coupling, the previous scheme (e.g. Refs. [12,14]) would not work well. (ii) Our scheme of enantiodiscrimination is robust against the fluctuation of the parameters under the appropriate parameter conditions.