Rapid and highly sensitive detection of DNA molecules contributes to ultrasensitive and automated biological assays such as sensing, imaging, immunoassay, and other diagnostics applications 123. Conventional approaches have focused on inorganic/organic hybrid DNA biomedicine sensors for biological labels, cell tracking, and monitoring response to therapeutic agents 456. Among numerical hybrid components, the unique size-dependent, narrow, symmetric, bright, and stable fluorescence of quantum dots (QDs) have made themselves powerful tools for investigating a wide range of biological problems 7. This is a difficult task with standard fluorophores because their relatively narrow excitation and broad emission spectra often result in spectra overlap. Besides, the optical behaviors of quantum dots are typically unaffected while they are conjugating to bio-molecules, which make them highly stable and bright probes, especially suitable for photon-limited in vivo studies and continuous tracking experiments over extended time periods 7. Recently, the coherent optical spectroscopy of a strongly driven quantum dot has been experimentally investigated by Xu et al. 89. They have shown that, like single atom two- and three-level quantum systems, single QD can also exhibit interference phenomena including Autler-Townes splitting and gain without population inversion when driven simultaneously by two optical fields. In this case, researchers are indulged in quantum dots and DNA conjugates to study biological activities and medical diagnosis 1011, which have applications in biomolecule targets exploitation 1213. But for the research of coherent optical spectrum in such coupled DNAs-QD, no study has ever been undertaken, neither in experiment nor in theory.

Furthermore, there is another question. The metallic quantum dots used in biological assays always have toxicity, which may limit the capabilities of biomedicine assays and bring in some unnecessary troubles. So the search for cadmium-free quantum dots has therefore becomes another major research area. Most recently, Amdursky et al. 1415 have experimentally demonstrated that the peptide quantum dots represent one of the simplest forms of quantum dot and the most important feature of these quantum dots is the nontoxicity to the environment and to human body. These quantum dots will become new labeling materials in biological and biomedical assays. However, the coherent optical properties of such QDs coupled to DNAs are still lacking.

In the present study, we theoretically investigate the coherent optical spectroscopy for a peptide quantum dot (QD) coupled to DNA molecules, with pump-probe technique. Recently, this two-laser technique has been realized by several groups 1617181920 while investigating the optomechanical system. Here we show that this hybrid peptide QD-DNA system will become transparent due to the DNA's vibrations when applying a strong control laser. Under some conditions the output signal laser even be enhanced significantly. Furthermore, the vibrational frequency of DNA molecule and the coupling strength between peptide QD and DNA can be measured due to the absorption splitting peaks in all-optical domain.

We consider a system composed of a biological semiconductor neutral quantum dot and DNA molecules in the simultaneous presence of a strong control field and a weak signal field. The physical situation is illustrated in Figure 1a. Figure 1b shows the energy levels of peptide QD when dressing the vibrational modes of DNA molecules. The energy levels of peptide quantum dot can be modeled as traditional quantum dot, which consists of two energy states, the ground state |g〉 and the first excited state (single exciton) |ex〉. Usually, this two-level exciton can be characterized by the pseudospin -1/2, operators S^± and S^z. The coherent optical spectroscopy of a strongly driven quantum dot without DNA molecules has been experimentally investigated by Xu et al. 89. Then the Hamiltonian of exciton in a QD can be described by

H Q D = ℏ ω e x S z ,

where ω_ex is the exciton frequency of peptide quantum dot.

We use the following Hamiltonian to describe DNA molecules, which is modeled as a harmonic oscillator and described by the position and momentum operators q and p, which have a commutation relation [q, p] = iħ 21, and then

H D = ∑ i = 1 n p i 2 2 m i + 1 2 m i ω i 2 q i 2 ,

where m_i and ω_i are the mass and vibrational frequency of DNA molecule, respectively.

In the unlimited or large volume of aqueous solution, all the vibrational modes of DNA molecules are strongly attenuated. But in a small volume of aqueous solution, the DNA longitudinal vibrational modes decayed slowly, while the other vibrational modes remain strong attenuation 22. Therefore, for small volume aqueous solution of DNA molecules coupled to a quantum dot system, only the longitudinal modes of DNA molecules can be taken into account. Such vibrational mode of DNA can be simulated as harmonic vibration of spring oscillator. Since the flexion of DNA molecules induces extensions and compressions in the structure of Figure 1a, the longitudinal strain will modify the energy of the electronic states of QD through deformation potential coupling 2324. Then the Hamiltonian of the vibrational modes of DNA molecules coupled to the peptide QD can be described by

H Q D - D = ℏ S z ∑ i = 1 n M i q i .

where M_i is the coupling strength between the peptide QD and the ith DNA. It is should be noted that due to the dilute aqueous solution of DNA molecules, here we do not consider the effect of the coupling between the DNA molecules although it may be significant in the dense aqueous solutions 22.

The Hamiltonian of the peptide quantum dot coupled to the strong control field and weak signal field is described as

H Q D - f = - μ ( E c S + e - i ω c t + E c * S - e i ω c t ) - μ ( E s S + e - i ω s t + E s * S - e i ω s t ) ,

where μ is the electric dipole moment of the exciton, ω_c (ω_s) is the frequency of the control field (signal field), and E_c (E_s) is the slowly varying envelope of the control field (signal field). Therefore, we obtain the total Hamiltonian of the coupled peptide QD-DNA in the presence of two optical fields as follows

H = H Q D + H D + H Q D - D + H Q D - f = ℏ ω e x S z + ∑ i = 1 n p i 2 2 m i + 1 2 m i ω i 2 q i 2 + ℏ S z ∑ i = 1 n M i q i - μ ( E c S + e - i ω c t + E c * S - e - i ω c t ) - μ ( E s S + e - i ω s t + E s * S - e i ω s t ) .

In a rotating frame at the control field ω_c, the total Hamiltonian of the system reads as

H = ℏ Δ c S z + ∑ i = 1 n p i 2 2 m i + 1 2 m i ω i 2 q i 2 + ℏ S z Q - ℏ ( Ω c S + + Ω c * S - ) - μ ( E s S + e - i δ t + E S * S - e - i δ t ) ,

where Δ_c = ω_ex - ω_c, Q = ∑ i = 1 n M i q i , Ω_c is the Rabi frequency of the control field, and δ = ω_s - ω_c is the detuning between the signal field and the control field.

Furthermore, we may consider the decoherence and relaxation of exciton and DNA mode in combination with their interaction to external environments into the Hamiltonian 25262728. In general, the environments can be described as independent ensembles of harmonic oscillators with spectral densities. We also assume that DNA molecules interact bilinearly with external environment via its position, and the exciton interacts with the environment through S^x operator and S^z operator. The S^x coupling to the environment models the relaxation process of the exciton, while the S^z coupling to the environment models the pure dephasing process of the exciton 25262728. On the other hand, because ω_ex is much larger than ω_i, it is reasonable to use the rotating-wave approximation to the exciton-environment coupling term, but not to the DNA-environment coupling term in the system-environment coupling Hamiltonian.

In accordance with standard procedure 25262728, we can obtain the Born-Markovian master equation of the reduced density matrix of the coupled system, ρ (t), through tracing out the environmental degrees of freedom as

d ρ d t = - i ℏ [ H , ρ ] + A { [ S - , [ S + , ρ ] ] + h . c . } + B { [ S - , { S + , ρ } ] + h . c . } + E [ S z , [ S z , ρ ] ] + D [ Q , [ Q , ρ ] ] + G [ Q , [ p , p ] ] + i ℏ L [ Q , { p , ρ } ] ,

where the coefficients A, B, E, D, G, and L correspond to the characteristics of the coupling, and to the structure and properties of the environments. Their explicit form can be written as A = - 1 2 ℏ { γ 1 2 ( 1 + 2 N ( ω e x ) ) } , B = 1 2 ℏ γ 1 2 , E = - 1 ℏ γ 2 2 ( 1 + 2 N ( 0 ) ) , D = - 1 4 ℏ γ 3 ( 1 + 2 N ( ω ) ) , G = - ∑ i = 1 n Δ 3 2 ℏ m i ω i , L = - ∑ i = 1 n γ 3 4 m i ω i , where γ₁ = 2π J_x ( ω_ex), γ₂ = 2π J_z(0), γ₃ = 2_π J_c(ω_i) Δ 3 = P ∫ 0 ∞ d ω J c ( ω ) ω - ω i ( 1 + 2 N ( ω ) ) . N ( ω ) = ∑ i = 1 n 1 / [ exp ( ℏ ω / k B T ) - 1 ] is the Boltzman-Einstein distribution of the thermal equilibrium environments. J_x, J_z, and J_c describe the spectral densities of the respective environments coupled through S^x and S^z to the exciton, and through Q to the DNA molecule, respectively. P denotes the principal value of the argument.

According to the master equation (7), we can obtain the equation of motion for the expectation value of any physical operator O of the coupled system by calculating Ȯ ( t ) = Tr [ O ρ · ( t ) ] . We thus have

d d t S z = - Γ 1 S z + 1 2 + i Ω c S + - S - + i μ E s e - i δ t ℏ S + - i μ E s * e i δ t ℏ S - ,

d d t S - = - [ i ( Δ c + Q ) + Γ 2 ] S - - 2 i Ω c S z - 2 i μ E s e - i δ t ℏ S z ,

d 2 d t 2 Q + 1 τ D d d t Q + ω D 2 Q = - λ ω D 2 S z ,

where λ = ∑ i = 1 n ℏ M i 2 / ( m i ω D 2 ) corresponds to the peptide QD-DNA coupling strength, ω_D is the DNA longitudinal vibrational modes, Γ₁ and Γ₂ are the exciton relaxation rate and dephasing rate, respectively, τ_D is the vibrational lifetime of DNA molecule 29. They are derived microscopically as

Γ 1 = 2 ℏ γ 1 2 ( 1 + 2 N ( ω e x ) ) ,

Γ 2 = 1 ℏ γ 1 2 ( 1 + 2 N ( ω e x ) ) + 4 ℏ γ 2 2 ( 1 + 2 N ( 0 ) ) ,

τ D = ∑ i = 1 n 2 m i ω i γ 3 .

Note that if the pure dephasing coupling is neglected, i.e., γ₂ = 0, then Γ₁ = 2Γ₂. In order to solve these equations, we first take the semiclassical approach by factorizing the DNA molecule and exciton degrees of freedom, i.e., 〈QS^z〉 = 〈Q〉〈S^z〉, in which any entanglement between these systems should be ignored. And then we make the following ansatz 30

S - ( t ) = S 0 + S + e - i δ t + S - e i δ t ,

S z ( t ) = S 0 z + S + z e - i δ t + S - z e i δ t ,

Q ( t ) = Q 0 + Q + e - i δ t + Q - e i δ t .

Upon substituting these equations to Equations (8)-(10) and working to the lowest order in E_s, but to all orders in E_c, we finally obtain the linear optical susceptibility S₊ in the steady state as the following solution

χ ( ω s ) e f f ( 1 ) = μ S + E s = μ 2 ℏ Γ 2 χ ( ω s ) ,

where the dimensionless susceptibility χ(ω_s) is given by

χ ( ω s ) = w 0 f ( δ 0 ) × { e 1 e 2 [ 2 i + δ 0 ) ( e 1 + δ 0 ) − 2 Ω c o 2 ] − e 2 Ω c o 2 λ 0 η w 0             + Ω c o 2 ( 2 e 2 − λ 0 η w 0 ) ( e 1 + δ 0 ) } ,

where the function f(δ₀) and auxiliary function η(ω_s) are given by

f ( δ 0 ) = ( e 2 + δ 0 ) { e 1 e 2 [ ( 2 i + δ 0 ) ( e 1 + δ 0 ) − 2 Ω c o 2 ] − e 2 Ω c o 2 λ 0 η w 0 }     − e 1 Ω c o 2 ( 2 e 2 − λ 0 η w 0 ) e 1 + δ 0 ) ,

η ( ω s ) = ω D 0 2 ω D 0 2 - δ 0 2 - i δ 0 / τ D 0 ,

where e₁ = i + Δ_c0-λ₀/(2w₀), e₂ = i-Δ_c0 + λ₀/(2w₀), δ₀ = δ/Γ₂, Ω_c0 = Ω_c/Γ₂, λ₀ = λ/Γ₂ ω_D0 = ω_D/Γ₂, τ_D0 = τ_DΓ₂, Δ_c0 = Δ_c/Γ₂, w 0 = 2 S 0 z , and Γ₁ = 2Γ₂.

And the population inversion (w₀) of the exciton is determined by the following equation

( w 0 + 1 ) Δ c 0 - λ 0 w 0 2 2 + 1 + 2 Ω c 0 2 w 0 = 0 .

For illustration of the numerical results, we choose the realistic coupled system of a peptide QD linked to the DNA molecules in the simultaneous presence of a strong control beam and a weak signal beam as shown in Figure 1. In such coupled system, many DNA molecules linked with one QD. These DNA molecules in solution form may be distorted in mess, but one can extend these molecules into linear form by applying electromagnetic field or fluid force 31. In addition, the longitudinal vibrational frequency can be determined by the length of DNA molecules. In the theoretical calculation, we select the vibrational frequency and the lifetime of DNA molecule are ω_D = 32 GHz and τ_D = 3 ns, respectively 24323334. The decay time of peptide quantum dot is 6 fs 15, which corresponds to Γ₁ = 160 THz.

Figure 2a shows the signal absorption spectrum as a function of control-signal detuning. In the middle of the figure, we can see that the features appear in the signal absorption spectrum as those in atomic two-level systems 30. However, the new features which are different from those in atomic systems without DNA molecules also appear at the both sides of the spectrum. Figure 2b gives the origin of these new features. The leftmost (1) of Figure 2b shows the dressed states of exciton when coupling with DNA molecules (|n〉 denotes the number states of the DNA molecules). Part (2) shows the origin of DNA vibrational mode induced three-photon resonance. Here the electron makes a transition from the lowest dressed level |g, n〉 to the highest dressed level |ex, n + 1〉 by the simultaneous absorption of two control photons and emission of a photon at ω_c - ω_D. This process can amplify a wave at δ = -ω_D, as indicated by the region of negative absorption in Figure 2a. The part (3) in Figure 2b shows the origin of DNA stimulated Rayleigh resonance. The Rayleigh resonance corresponds to a transition from the lowest dressed level |g, n〉 to the dressed level |ex, n〉. Each of these transitions is centered on the frequency of the control laser. The rightmost part (4) corresponds to the usual absorption resonance as modified by the ac Stark effect.

In this article, we theoretically investigated the coherent optical spectroscopy in a coupled DNA-peptide quantum dot system in the presence of two optical fields. Theoretical analysis shows that the vibrational frequency of DNA and the coupling strength between peptide QD and DNA can be measured effectively and precisely in all-optical domain. Finally, we hope that our predictions in the present study can be testified by experiments in the near future.

The authors declare that they have no competing interests.

JJL finished the main work of this article, including deducing the formulas, plotting the figures, and drafting the manuscript. KDZ conceived of the idea, participated in its writing and provided some useful suggestions. All authors read and approved the final manuscript.