## Abstract

We investigate the Berry phase in the nanocrystal complex made of a metal nanoparticle and a slowly rotating semiconductor quantum dot under the radiation of a circularly polarized light. The Berry phase in the dynamic system is found to be more effective to manifest the interaction between the plasmon in the metal nanoparticle and the exciton in the quantum dot. The dependences of the Berry phase on the interparticle distance and the relative position are studied in the weak field condition. The methods to observe the Berry phase are also given.

© 2017 Optical Society of America

## 1. Introduction

Recently much attention has been paid to the nanocrystal complexes, composed of various building blocks such as metal nanoparticles (MNPs) [1–3], semiconductor quantum dots (SQDs) [4, 5], dielectric nanostructures [6], biomolecules [7], colloid quantum dots [8], semiconductor and metal nanowires [9, 10]. One of the most popular system is undoubtedly the MNP-SQD complex, whose optical properties have been investigated intensively, such as but definitely not limited to, the Fano effect [11–17], the Rabi oscillation [18–22], the spin-plasmon coupling [23–25], the optical bistability [14, 26–31], the population dynamics [32–35], the quantum coherence [36], the resonance fluorescence [37–41], the Förster energy transfer [42], the photoluminescence [43], the spontaneous emission [44], and the enhanced difference frequency generation [45]. Later the system has been modified to some more complex composites [16, 17, 31, 46, 47] by including more MNPs or SQDs, which further extends the scope of the exciton-plasmon interactions in this kind of nanocrystals and possible optical applications.

In generally speaking, most of studies relevant to this kind of composites focused on the optical properties and usually in static conditions [48–70]. The strong field enhancement effect is a characteristics of the coupled systems. However, the field enhanced optical processes become outstanding only when the interdistance between the MNP and the two-level system decreases down to order of nanometers [71–73]. Even so, the improvement is still demanding for detection in experiments. For example, the theoretical results showed the typical red-shift of the steady absorption in the MNP-SQD system is about 0.05 meV [74] and could only be improved to about 0.5 meV even if the multipole effect considered [75]. Besides, the concerned optical properties maybe spoiled as the interdistance further decreasing [20,45] due to the rapid increasing dephasing rate. Thus a more effective way to manifest the dipole-dipole interaction is necessary.

In the paper, we will investigate the evolution of the quantum states in the SQD when cycling around the MNP, which is a dynamic process rather than a static one. In contrast to the optical properties, the highlight is put on the phase of quantum states rather than the energy level shift or inter-level transition. Particularly the Berry phase obtained for each energy level after the SQD finishing the cycling will be studied [76–82]. As we expected, the quantum state of SQD does get a nonzero Berry phase under the circularly polarized optical field, similar to the spin of electrons does under the precessed magnetic field [83, 84]. Moreover, the Berry phase in the system has some new characteristics: it not only has a geometric dependence on the path but also changes sensitively with the interparitcle distance when compared with the energy-level shift in the same condition. The results are obtained almost by analytical derivations and the different contributions to the Berry phase are studied in detail. The sensitive and obvious dependence on the configuration of the system means we could observe the dipole-dipole interaction easily via the Berry phase.

The paper is organized as follows. In Section II, the system and its interaction with the external field are introduced. Detailed derivations for the external field, the dipole induced in the SQD and the Berry phase are given orderly in Subsection A, B, C, respectively. In Section III, the numerical results are shown. The contributions from different parts to the Berry phase and its dependences on different parameters are studied. In Section IV, we will discuss how to observe the Berry phase by two different methods. The assumed measurements are demonstrated. Finally the conclusion is given in Section V.

## 2. The model and Berry phase

The studied system is the nanocrystal complexes composed of an MNP of radius *R*_{0} and an SQD of radius *R _{s}* (

*R*≪

_{s}*R*

_{0}), as shown in Fig. 1. The SQD is assumed to be a two-level system whose polarization is along the

*z*direction, i.e., the interlevel optical transition matrix element

**=**

*μ**μ*

**. In contrast to other similar systems, the SQD is not fixed, but designed to rotate around the MNP slowly. With the same origin as the Cartesian coordinates, the spherical coordinates of the SQD is the (**

*ẑ**R*,

_{d}*θ*,

*φ*), where

*R*is the interdistance,

_{d}*θ*and

*φ*are polar angle and azimuthal angle of the SQD at the position, respectively. The dynamic path is just the trajectory by changing

*φ*continuously. A circularly polarized light propagating along

*z*direction is employed as the excitation. The external field is considered to be spatially uniform due to the small size of the system. Once the plasmon in the MNP and the exciton in the SQD are excited, the exciton-plasmon interaction takes effect through the modifications of electric fields felt by each other. As the adiabatical condition is satisfied, the system gets the steady state at each moment in the cycle. So let us begin with the exact solution of the steady electric fields.

#### 2.1. The electric field

The incident light is a circularly polarized light propagating along *z* direction, whose electric field can be written as

At a given position, the electric field felt by the SQD can be split into three parts:

which will be explained in detail as follows. This classification is proposed by us before [75], but note that the center-to-center line here is not always parallel or perpendicular to the polarization of external field so that the electric field felt by the SQD is different.The permittivities of the MNP and the SQD are denoted as *ε _{m}* and

*ε*, respectively, while that of the environment around the nanocrystal complex is

_{s}*ε*. The first part

_{e}

*E*_{1}is the electric field exerted by the light directly,

*ε*= (

_{eff}*ε*+ 2

_{s}*ε*)/3. This part of the electric field would not excite the exciton in the SQD here because

_{e}**·**

*μ***= 0.**

*E*The second part is the one produced by the polarized MNP. In the paper we will adopt the dipole approximation, which is sound enough to demonstrate the qualitative physics if the interdistance *R _{d}* is not so small compared with the radius of the MNP

*R*

_{0}. The polarization inside the MNP induced directly by the external field is

*γ*

_{1}= [

*ε*(

_{m}*ω*) −

*ε*]/[

_{e}*ε*(

_{m}*ω*) + 2

*ε*]. The polarized MNP will created an additional electric field around and therefore has an influence on the SQD. Since that the first part has no contribution to the exciton in the SQD, this part has the leading contribution. We only need the

_{e}*z*component of the electric field in the SQD, which is found to be

The third part comes from the effective electric field produced by the polarization in the MNP induced by the effective dipole of the SQD (*ε _{e}/ε_{eff}*)

*p**(*

_{s}*t*). Technically, we can decompose the dipole vector into two parts, parallel and perpendicular to the center-to-center line, calculate the induced electric field by each part, and then combine them. By careful calculation, the

*z*component of this part is found as

*φ*as the dipole of the SQD

*p**(*

_{s}*t*) has a rotational symmetry. The relationship of $~{R}_{d}^{6}$ debases its contribution compared with that of the second part.

To sum up, the interaction item between the SQD and the field is then

#### 2.2. The dipole in the SQD

Now we want to calculate the dipole of the SQD *p** _{s}* (

*t*) analytically. The Hamiltonian of the SQD coupled to the external optical field reads

*a*and ${a}_{i}^{\u2020}$ are the annihilation and the creation operators of the level

_{i}*i*(

*i*=1,2), respectively. The upper and lower levels have the energies of

*ħω*

_{0}/2 and −

*ħω*

_{0}/2, respectively. As the time dependence of the external field is ∼

*e*

^{±i(ωt+φ)}, the induced polarization inside the SQD also has the same time dependence ∼

*e*

^{±i(ωt+φ)}. Therefore we can assume the dipole of the SQD has the form

*p̃*is time-independent. With the help of rotating wave approximation, the Hamiltonian can be written as

_{s}

*p**(*

_{s}*t*) =

**(**

*μ**p*+

*p*), where the interlevel polarization

^{*}*p*satisfies Dot above the quantity represents the time derivation in the paper.

#### 2.3. The Berry phase

Now we calculate the Berry phase of the excited energy level when the SQD completes a cycle adiabatically. For convenience, we express the Hamiltonian by Pauli operator ** σ**,

The above Hamiltonian is better to be transformed to a frame rotating at the frequency *ω* to ignore the high frequency part with the help of the following unitary transformation,

*U*= exp(

*iωtσ*/2). The transformed Hamiltonian takes the form of

_{z}*δ*=

*ω*

_{0}−

*ω*is the detuning between the transition energy and the excitation.

The adiabatical condition means the cycling time period *T* should satisfy the condition $T\cdot \left(\frac{\delta}{2\pi}\right)\gg 1$. During the cycle, each quantum state |*ψ*〉 will pick up a geometric phase as well as the dynamic phase, i.e., *e*^{iψD} *e ^{iγ}* |

*ψ*〉, where

*γ*is just the well-known Berry phase. Now we give the explicit expression of the Berry phase in the system. The instantaneous eigenvalue of the excited state for Hamiltonian (15) is

*χ*-parameter space. By straight calculation, the Berry phase of the excited state can be expressed as where

*λ*is the instantaneous eigenvalue given above. Note that a linear-polarization light will not bring about Berry phase, because the left and right circularly polarized light produce the opposite Berry phase in the system, which can be easily checked following the above derivations.

In the model of spin-1/2 particle under an adiabatically precessing magnetic field, the excited state acquires the Berry phase $\gamma =\pi \left(1+\text{cos}\frac{\alpha}{2}\right)$, where *α* is the apex angle corresponding to the solid angle. The magnitude of the precessing magnetic field is fixed. In comparison, the Berry phase in our system is caused by the processing of the induced electric field accompanying the SQD. As the magnitude of the electric field has a strong dependence on the interdistance, the Berry phase is not only determined by the solid angle but also by the interdistance.

## 3. Numerical results

Now we present the numerical results of the Berry phase based on the obtained formulae. The typical parameters used to do the calculations are as follows: *R*_{0} = 10*nm*, *ε _{e}* = 12

*ε*

_{0}and

*ε*

_{0}is the permittivity of vacuum, the transition matrix element

*μ*=

*er*

_{0}and

*r*

_{0}= 0.6

*nm*, the detuning

*ħδ*= 0.5

*meV*and

*ħω*

_{0}= 1.55

*eV*. The material of the MNP is gold (Au) whose permittivity is (−29 + 2

*i*)

*ε*

_{0}[85] at this frequency. The permittivity of the SQD is 12

*ε*

_{0}.

We first analyze the dependence of Berry phase on the polar angle *θ*. In the weak field case, the quantity Δ ≈ 1. Here the intensity of the external field is set as *E*_{0} = 10^{5}*V/m* so that Δ ≈ 1 is checked to be satisfied. In Fig. 2, the Berry phases for a certain interparticle distance such as *R _{d}* = 12

*nm*are shown. To see the contributions made by different parts of electric fields, the results when only

*E*

_{2}part is considered only (hollowed symbols) and when both parts

*E*

_{2}+

*E*

_{3}are included (solid symbols) are calculated, respectively. For case of only

*E*

_{2}considered, the Berry phase increases firstly and then decreases as the angle

*θ*increasing, with the extremum locating at

*θ*=

*π*/4. The reason is obvious because of the sin 2

*θ*factor in the expression of electric field

*E*

_{2}. Note that the Berry phases corresponding to 2

*π*and 0 are equivalent and thus the extremum is actually a maximal change of the phase. The rule is also applied for the other interparticle distances. However, the electric field

*E*

_{3}has a big influence on the Berry phase especially when the interparticle distance is small. As it shows, when both parts are considered, the Berry phase has a complex relation with the polar angle

*θ*, which is determined not only by the unidirectional influence from the MNP to the SQD, but also by the interaction between them. The obvious dependence of the interaction on the angle gives a new way to observe the strong coupling in the system.

The difference brought about by *E*_{3} will disappear only when the SQD and the MNP are separated by a big interparticle distance as it takes effect according to the relation of ${R}_{d}^{-6}$. In this case the contribution of *E*_{3} can be neglected.

The dependence of the Berry phase on the interparticle distance can be seen from Fig. 3, where the calculated Berry phases with and without *E*_{3} part considered are drawn for several different interparticle distances. For a relatively big distance, the results in two cases are almost same, because the electric field of *E*_{3} has a decreasing contribution with the increasing distance. When two nanoparticles are close, the Berry phase for these two cases are totally different. The difference is caused by the non-negligible *E*_{3}. The obvious sensitivity of the Berry phase on the *E*_{3} provides an effective way to study or observe the dipole-dipole interaction in the complex.

## 4. Observation of the Berry phase

The Berry phase is usually observed by interference. Here we design two schemes to observe the Berry phase by adding another SQD in the system. All the building blocks in the complex could be imbedded in a dielectric slab, which is radiated by the excitation light propagating along the normal direction. Rotating the slab around the light can realize the slow variation of this dynamic system. The principle is to measure the interference result of two produced lights polarized along *z* direction by two SQDs with different Berry phases.

#### 4.1. By different positions

In this method, we place one SQD above and the other below the MNP with the same interparticle distance and supplementary polar angle. We only need to rotate the sample around *z* direction along which a circularly polarized light propagates, as shown in Fig. 4. We assume these two SQDs have the identical energy level. If the incident light is right circularly polarized light for one SQD, the other feels the left circularly polarized light according to the symmetry, therefore the Berry phases produced in these two SQDs are exactly opposite.

The detected light is determined by the polarizations in the SQDs. Based on the above analysis, the polarizations after finishing a cycle obtain the Berry phases *γ _{a}* and

*γ*, respectively, which have the relation

_{b}*γ*= −

_{a}*γ*. The intensity of polarization can be expressed as

_{b}*P*∝

*e*

^{iφD}

*e*

^{iγi}

*E*

_{0}, where

*φ*is the dynamical phase and

_{D}*γ*is the Berry phase [86]. Then intensity of the detected light is

_{i}*θ*=

*π*/4 and

*R*= 15

_{d}*nm*, stated in caption of Fig. 4, the corresponding Berry phase through a circle is about

*γ*= 1.6

_{a}*π*. The intensity of the detected light after a cycle according changes the factor cos 2

*γ*, as shown in right part of Fig. 4. By measuring the strength changing, the Berry’s phase in the nanocrystal complexes can be obtained.

_{a}#### 4.2. By different detunings

As the Berry phase is also determined by the detuning *δ*, we can use the sign of detuning to produce the opposite Berry phases. Now we add another SQD in the same pathway with respect to the original one, and rotate the whole sample around *z* axis, as shown in Fig. 5. In this case, the energy level of these two SQDs are different, satisfying (*ω*_{1} + *ω*_{2})/2 = *ω*_{0}. Based on the theory, the polarizations of these two SQDs acquire the opposite Berry phases and consequently the intensity of the detected light is determined by

*z*direction by the time period of

*T*=

*π/δ*, then the intensity is just ${E}_{0}^{2}\text{cos}2{\gamma}_{a}$, which changes in the same way as the first method. Similarly, by measuring the strength of the detected light, the Berry phase in the nanocrystal complex can be obtained.

## 5. Conclusion

We investigate the Berry phase in the rotating MNP-SQD system radiated by a circularly polarized light. Exact solutions to the electric fields, the dipole in the SQDs and the Berry phase of the quantum states are presented analytically. The Berry phase of the quantum states in such dynamic system are found to be more sensitive than the shift of eigen energies and therefore is easy to be detected. We also design two methods to observe the Berry phase by interference and present the theoretical analyses. The properties of the Berry phase are beneficial to study the dipole-dipole interaction in the system.

## Funding

National Key Research and Development Program of China (Grant 2016YFA0301300); National Natural Science Foundation of China (NSFC) (Grant 11004015).

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