光子带隙热库中原子的发光特性研究
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摘要
自发辐射和共振荧光辐射作为光与物质相互作用的基本过程一直就是物理学的一个重要研究课题。到了二十世纪七十年代,人们开始研究如何在量子体系中实现对原子发光特性的控制和修正。光子带隙热库因为其独特的控制光子运动的能力被加以利用。近年来,原子在光子带隙热库中的辐射在理论研究、相关实验及其应用开发等领域都得到了迅速发展。
本论文对光子带隙热库中原子的自发辐射和共振荧光辐射进行了深入的研究。通过比较原子与外相干场耦合的不同方式,得出光子带隙热库中原子自发辐射的量子干涉效应,以及不同光子晶体的态密度对位相依赖现象的影响。并讨论了在弱激发的情况下,光子带隙热库中原子的共振荧光谱不同于真空情况下的谱线特征。当在光子晶体中引入缺陷形成纳米腔时,利用腔量子电动力学得到了单量子和双量子点在腔中的发光特性,并在缀饰态表象下揭示其物理本质,为研究多原子系统打下基础。
光子带隙热库中原子的发光特性研究对光信息存储、量子信息与量子计算、光群速减慢、光通讯、光谱识别、高效率发光材料等领域的理论和实验研究均具有一定的指导意义。
In this thesis for doctorate we study the effects of quantum interference on sponta-neous emission spectrum (SES) and resonance fluorescence (RF) spectrum of differentatomic systems, which are embedded in isotropic Photonic Band Gap (PBG) reservoirs,anisotropic PBG reservoirs and vacuum reservoirs, respectively. This thesis consists ofthree parts:
1. Comparing the“lower level coupling”and“upper level coupling”, theinfluence of the external field on the quantum interference and the phase dependencecharacter of the spontaneous emission are analyzed.
(1) The influence of the external field on the quantum interference in the sponta-neous emission
A two-level atom embedded in Photonic Crystals (PCs) is coupled to a third level|3> by a external field. The coupling between |3> and |1> is named as the“lower levelcoupling”; the coupling between |3> and |2> is named“upper level coupling”. Throughcomparing the“lower level coupling”and“upper level coupling”, the quantuminterference is analyzed.
For the“lower level coupling”, the coherence caused by the driving field separatesthe lower level of the atom into two close dressed states. The particle drop into the
dressed states both from the upper level. This means that the interference plays no rolein the decay ways. It presents an incoherent superposition of two Lorentzian lines asshown in the figure 1. And the two side lobes are separated farther asΔ_0 becomesstronger. Due to the singularities of the density of state (DOS) of the isotropic PBGreservoirs, four holes appear in the SES as shown in the figure 1(a).
For the“upper level coupling”, the coherence caused by the driving field separatesthe upper level into two close dressed states. When the two dipole moments of theatomic transition from the two sole dressed states to the ground level they are orthogonalimplying that there is no quantum interference denoted by p = 0. When the dipolemoments of the two transitions are parallel or anti-parallel, it implies that the quantum interference is maximal denoted by p =±1. But in the“upper level coupling”, theupper splitting states are phased together by the driving field, so that the interference isunavoidable. The key signature of the interference effect in the emission spectrum of theisotropic PBG and the anisotropic PBG is the appearance of a dark line, the same as thatin vacuum. The dark line in the fluorescent spectrum locates at .It is different from the vacuum situation which locates atδ_k =Δ_0. So the dark line isnot only respect to the detuning but also depend on the location of the PBG. Whenδ_(21c1) <Δ_0 orδ_(21c1) >δ_(c2c1) +Δ_0, an additional peak appears between the band edgeand the dark line as shown the solid line in figure 2 and dash line in figure 3. WhenΔ_0≤δ_(21c1)≤δ_(c2c1) +Δ_0, the dark line disappears as shown the solid line in figure 3. Thisphenomenon should not be confused with the situation that the driving field approacheszero(as the dash line in figure 2). Although the dark line disappears in both situations,the quantum dynamic is absolutely different: the former is induced cut-off effect of thePBG and the latter is induced by the no destructive quantum interference.
(2) The phase dependence character in the SES
The two models researched are introduced as follows. The first one is“lower level coupling”model, whose two close lower levels in aΛ-type atom are coupled by amicrowave field. The second one is the“upper level coupling”model whose two closeupper levels in a V-type atom are coupled by a microwave field. Through comparingthe two model, we can obtain the physical dynamics of the phase dependence characterfrom the three-level atom embedded in PBG reservoirs.
We consider the situation thatδ= 0 and the center of the band gap locates in themiddle of the transition frequencies of the atom. Two dressed states |α> and |β> aregenerated by the quantum coherence between the atom and the microwave field. In the“lower level coupling”, the SES is shown in figure 4. The spectrum corresponding to|α> is comprised of two linesα_1 andα_2, centered atω_(31)-|Ω_0| andω_(32)-|Ω_0|. And the |β>is composed of two linesβ_1 andβ_2, centered atω_(31) + |Ω_0| andω_(32) + |Ω_0|. In the case ofthe“upper level coupling”, the SES is shown in figure 5. The structures of two linesα_1andα_2 are centered atω_(20)-|Ω_0| andω_(10)-|Ω_0|. Whileβ_1 andβ_2 are centered atω_(20)+|Ω_0|andω_(10) + |Ω_0|. In these models, we only consider the situation thatω_(21) < 2|Ω_0| due tothe small interval between two close levels |1 and |2 . Then the two spectral structuresα_(1,2) andβ_(1,2) locate in each side of the band gap.The phase termη_(1)2e~(±iφ) plays an important role in our models. Whenη_(12) = 0, thephase term is equal to 0, then the spectrum is not changed with the phase. So the phasedependence occurs only ifη_(12)≠0. Namely, the phase sensitive phenomenon is theconsequence of the quantum interference between the transitions |3>→|1> and |3>→|2>in the“lower level coupling”and between the transitions |1>→|0 and |2>→|0> in the“upper level coupling”. The DOS in different reservoirs only influence the the relativeheight and width of peaks which not play a role in the quantum interference.When the phase is changed from 0 toπ, the SES from the two models can varyas the phase in different tendency. Comparing to the the figure4 and 5 we found thatthe variety of the phase can bring oppositive influence on the two models. And the twoSES of each system are almost symmetrical with each other when the phase is equal to0 andπ.
2. RF spectrum from double-band PBG reservoirs
It is well-known that in vacuum the asymmetrical RF can be obtained only in theoff-resonance case. But in PBG reservoirs the RF appears different characters. Whenall the dressed-states lie in a region away from the band gap of the PCs, one can seethat the RF profiles represented by the dash curves in figures 6 are characterized bythree asymmetrical peaks. This is induced by the different DOS, and the intensity ratioamong the three peaks has different value in different PCs. As the transition frequencymoves to the center of the band gap, the RF spectrum are displayed by the solid linesin figure 6. The central peak disappears and the symmetrical character is obtained. Thedisappearance of the central peak can be explained that the elastic part is inhibited bythe band structure of the PCs. Due to the strong DOS in the edges of the band gap, whenthe transition frequency moves to the center of the band gap, the edges of the band gapincrease obviously. More precisely, the character of the RF spectrum directly dependson the location of the resonance energy with respect to the band gap of the PCs.
The off-resonance case are shown in figure 7. We compare it with the solid linein figure 6 which is in the resonance case. They are both in the case that the frequencyof the external field lies at the center of the band gap, the RF spectra changes from asymmetrical structure to two asymmetrical peaks. This means that the asymmetricalcharacter is absolutely induced by the detuning. And when two detunings are inverse toeach other, two mirror RF spectra are denoted by the solid lines and dash lines in figure7. The spectral features are similar to the ordinary vacuum case regime except for theinhibition of the central peak.
3. Spontaneous emission of Quantum Dots coupled with a single-mode cavity inPCs
In the Cavity Quantum Electrodynamics (CQED), we research the florescenceemission in the side and along the axis cavity from one single Quantum Dot(QD) andtwo QDs which are coupled with a single-mode nanocavity field in thermal equilibrium.In the CQED strong-coupling regime, the emission process of this system is reversibleand a photon emitted by the atom can be coherently reabsorbed before it is emitted outof the cavity.
First we obtain the coupling dynamic between a QD and a single mode cavity.Both side and axis spectra components whose emission probabilities are different fromeach other, approach the so-called vacuum Rabi spectrum. However, in generally, thetwo spectra have different shapes which can be shown in the two QDs model. Threeimportant parameters are used to characterize the system: the QD-cavity coupling con-stant, the QD decay rate, and the cavity decay rate. So two detunings are obtained tocontrol the emission in the side and along the axis cavity: One detuning is between thedipole moments (ω_(AB) =ω_A -ω_B), the other detuning is between two transition frequen-cies of two quantum dots (Δd_(21) = d_(32)~A- d_(21)~B). While the cavity decay rateκwhich onlydepends on the cavity mode is small all the time. Finally, the spontaneous emissionspectra will be controlled byω_(AB) andΔd_(21).
To explore the origins of the unusual spectral features produced by quantum in-terference the dressed atomic-cavity-state representation is employed. In the resonantcase, each energy eigenstate decays to the ground state via two output channels: thetransition operator (sideways channel).and the transition operator (axial channel).
In the axis channel as shown in equation (3), firstly we consider the case of twoidentical QDs, due to the same coupling constants, the contribution of the decay from|2_A,1_B,0> and |1_A,2_B,0> at zero detuning is mutually counteracted, inducing a slippycurve and two symmetric peaks in each side of the zero detuning shown in figure 8(b).When the detuning between dipole moments is unequal to 0, the singularity (peak ordip) will appear at the frequency of the coherent field. So when the dipole moment of A-QD is smaller than B-QD as shown in figure 9(b), there is a peak in the middle of thespectrum. Contrarily there occurs a hole in figure 10(b). And the singularity is obviousas the increase of the detuning value. In the side emission, the photon can decay fromall of the three dressed states as shown in equations (1) and (2). And no matter how thedipole moments are changed, the triple-peak spectrum always exists as shown in figure9(a) and 10(a).
In the off-resonance situation we can obtain the decay via the side pathwayand via the axial pathway
The above equations (4), (5), (6) indicate that the photon can be emitted from|φ_a> , |φ_b> and |φ_c> both in the side direction and the axis direction. The population on the bare stateσ_(22)~A(0) can not be symmetrically distributed on the initial dressedstates. And there are not any other external fields acting on the system, so the SESdisplays a asymmetrical shape withω_c. By changing the detunning of the two transitionfrequencies, we can see the influence ofω_(AB) on the SES. Shown as the solid and dashlines in the figure 11, when the two detunings between the transition frequencies ofthe two QDs are inverse to each other, their spectra have a mirror relation. Withω_(AB)increases, the spectrum will change in different ways as shown by the dot and dash linesin figure 11.
The purpose of this thesis are:
(1) We research the quantum interference and the phase dependence character ofthe spontaneous emission, which is induced by the external driving field, from an ex-cited atom embedded in a double-band PCs .
(2) We research the RF spectrum when the atom is embedded in double-band PCswho is the non-Markvo reservoirs. Through the zero-th order Liouville operator ex-pansion approximation in optical Bloch equation and the noise operators, we obtainthe emission property which is different from the vacuum case in the resonance andoff-resonance case.
(3) In the Cavity Quantum Electrodynamics(CQED), we research the florescenceemission in the side and along the axis cavity from one single QDs and two QDs whichare strongly coupled with a single-mode nanocavity field in PCs. And we clearly ex-plain the strange emission property of the florescence emission spectrum in the dressedstate picture.
本论文对光子带隙热库中原子的自发辐射和共振荧光辐射进行了深入的研究。通过比较原子与外相干场耦合的不同方式,得出光子带隙热库中原子自发辐射的量子干涉效应,以及不同光子晶体的态密度对位相依赖现象的影响。并讨论了在弱激发的情况下,光子带隙热库中原子的共振荧光谱不同于真空情况下的谱线特征。当在光子晶体中引入缺陷形成纳米腔时,利用腔量子电动力学得到了单量子和双量子点在腔中的发光特性,并在缀饰态表象下揭示其物理本质,为研究多原子系统打下基础。
光子带隙热库中原子的发光特性研究对光信息存储、量子信息与量子计算、光群速减慢、光通讯、光谱识别、高效率发光材料等领域的理论和实验研究均具有一定的指导意义。
In this thesis for doctorate we study the effects of quantum interference on sponta-neous emission spectrum (SES) and resonance fluorescence (RF) spectrum of differentatomic systems, which are embedded in isotropic Photonic Band Gap (PBG) reservoirs,anisotropic PBG reservoirs and vacuum reservoirs, respectively. This thesis consists ofthree parts:
1. Comparing the“lower level coupling”and“upper level coupling”, theinfluence of the external field on the quantum interference and the phase dependencecharacter of the spontaneous emission are analyzed.
(1) The influence of the external field on the quantum interference in the sponta-neous emission
A two-level atom embedded in Photonic Crystals (PCs) is coupled to a third level|3> by a external field. The coupling between |3> and |1> is named as the“lower levelcoupling”; the coupling between |3> and |2> is named“upper level coupling”. Throughcomparing the“lower level coupling”and“upper level coupling”, the quantuminterference is analyzed.
For the“lower level coupling”, the coherence caused by the driving field separatesthe lower level of the atom into two close dressed states. The particle drop into the
dressed states both from the upper level. This means that the interference plays no rolein the decay ways. It presents an incoherent superposition of two Lorentzian lines asshown in the figure 1. And the two side lobes are separated farther asΔ_0 becomesstronger. Due to the singularities of the density of state (DOS) of the isotropic PBGreservoirs, four holes appear in the SES as shown in the figure 1(a).
For the“upper level coupling”, the coherence caused by the driving field separatesthe upper level into two close dressed states. When the two dipole moments of theatomic transition from the two sole dressed states to the ground level they are orthogonalimplying that there is no quantum interference denoted by p = 0. When the dipolemoments of the two transitions are parallel or anti-parallel, it implies that the quantum interference is maximal denoted by p =±1. But in the“upper level coupling”, theupper splitting states are phased together by the driving field, so that the interference isunavoidable. The key signature of the interference effect in the emission spectrum of theisotropic PBG and the anisotropic PBG is the appearance of a dark line, the same as thatin vacuum. The dark line in the fluorescent spectrum locates at .It is different from the vacuum situation which locates atδ_k =Δ_0. So the dark line isnot only respect to the detuning but also depend on the location of the PBG. Whenδ_(21c1) <Δ_0 orδ_(21c1) >δ_(c2c1) +Δ_0, an additional peak appears between the band edgeand the dark line as shown the solid line in figure 2 and dash line in figure 3. WhenΔ_0≤δ_(21c1)≤δ_(c2c1) +Δ_0, the dark line disappears as shown the solid line in figure 3. Thisphenomenon should not be confused with the situation that the driving field approacheszero(as the dash line in figure 2). Although the dark line disappears in both situations,the quantum dynamic is absolutely different: the former is induced cut-off effect of thePBG and the latter is induced by the no destructive quantum interference.
(2) The phase dependence character in the SES
The two models researched are introduced as follows. The first one is“lower level coupling”model, whose two close lower levels in aΛ-type atom are coupled by amicrowave field. The second one is the“upper level coupling”model whose two closeupper levels in a V-type atom are coupled by a microwave field. Through comparingthe two model, we can obtain the physical dynamics of the phase dependence characterfrom the three-level atom embedded in PBG reservoirs.
We consider the situation thatδ= 0 and the center of the band gap locates in themiddle of the transition frequencies of the atom. Two dressed states |α> and |β> aregenerated by the quantum coherence between the atom and the microwave field. In the“lower level coupling”, the SES is shown in figure 4. The spectrum corresponding to|α> is comprised of two linesα_1 andα_2, centered atω_(31)-|Ω_0| andω_(32)-|Ω_0|. And the |β>is composed of two linesβ_1 andβ_2, centered atω_(31) + |Ω_0| andω_(32) + |Ω_0|. In the case ofthe“upper level coupling”, the SES is shown in figure 5. The structures of two linesα_1andα_2 are centered atω_(20)-|Ω_0| andω_(10)-|Ω_0|. Whileβ_1 andβ_2 are centered atω_(20)+|Ω_0|andω_(10) + |Ω_0|. In these models, we only consider the situation thatω_(21) < 2|Ω_0| due tothe small interval between two close levels |1 and |2 . Then the two spectral structuresα_(1,2) andβ_(1,2) locate in each side of the band gap.The phase termη_(1)2e~(±iφ) plays an important role in our models. Whenη_(12) = 0, thephase term is equal to 0, then the spectrum is not changed with the phase. So the phasedependence occurs only ifη_(12)≠0. Namely, the phase sensitive phenomenon is theconsequence of the quantum interference between the transitions |3>→|1> and |3>→|2>in the“lower level coupling”and between the transitions |1>→|0 and |2>→|0> in the“upper level coupling”. The DOS in different reservoirs only influence the the relativeheight and width of peaks which not play a role in the quantum interference.When the phase is changed from 0 toπ, the SES from the two models can varyas the phase in different tendency. Comparing to the the figure4 and 5 we found thatthe variety of the phase can bring oppositive influence on the two models. And the twoSES of each system are almost symmetrical with each other when the phase is equal to0 andπ.
2. RF spectrum from double-band PBG reservoirs
It is well-known that in vacuum the asymmetrical RF can be obtained only in theoff-resonance case. But in PBG reservoirs the RF appears different characters. Whenall the dressed-states lie in a region away from the band gap of the PCs, one can seethat the RF profiles represented by the dash curves in figures 6 are characterized bythree asymmetrical peaks. This is induced by the different DOS, and the intensity ratioamong the three peaks has different value in different PCs. As the transition frequencymoves to the center of the band gap, the RF spectrum are displayed by the solid linesin figure 6. The central peak disappears and the symmetrical character is obtained. Thedisappearance of the central peak can be explained that the elastic part is inhibited bythe band structure of the PCs. Due to the strong DOS in the edges of the band gap, whenthe transition frequency moves to the center of the band gap, the edges of the band gapincrease obviously. More precisely, the character of the RF spectrum directly dependson the location of the resonance energy with respect to the band gap of the PCs.
The off-resonance case are shown in figure 7. We compare it with the solid linein figure 6 which is in the resonance case. They are both in the case that the frequencyof the external field lies at the center of the band gap, the RF spectra changes from asymmetrical structure to two asymmetrical peaks. This means that the asymmetricalcharacter is absolutely induced by the detuning. And when two detunings are inverse toeach other, two mirror RF spectra are denoted by the solid lines and dash lines in figure7. The spectral features are similar to the ordinary vacuum case regime except for theinhibition of the central peak.
3. Spontaneous emission of Quantum Dots coupled with a single-mode cavity inPCs
In the Cavity Quantum Electrodynamics (CQED), we research the florescenceemission in the side and along the axis cavity from one single Quantum Dot(QD) andtwo QDs which are coupled with a single-mode nanocavity field in thermal equilibrium.In the CQED strong-coupling regime, the emission process of this system is reversibleand a photon emitted by the atom can be coherently reabsorbed before it is emitted outof the cavity.
First we obtain the coupling dynamic between a QD and a single mode cavity.Both side and axis spectra components whose emission probabilities are different fromeach other, approach the so-called vacuum Rabi spectrum. However, in generally, thetwo spectra have different shapes which can be shown in the two QDs model. Threeimportant parameters are used to characterize the system: the QD-cavity coupling con-stant, the QD decay rate, and the cavity decay rate. So two detunings are obtained tocontrol the emission in the side and along the axis cavity: One detuning is between thedipole moments (ω_(AB) =ω_A -ω_B), the other detuning is between two transition frequen-cies of two quantum dots (Δd_(21) = d_(32)~A- d_(21)~B). While the cavity decay rateκwhich onlydepends on the cavity mode is small all the time. Finally, the spontaneous emissionspectra will be controlled byω_(AB) andΔd_(21).
To explore the origins of the unusual spectral features produced by quantum in-terference the dressed atomic-cavity-state representation is employed. In the resonantcase, each energy eigenstate decays to the ground state via two output channels: thetransition operator (sideways channel).and the transition operator (axial channel).
In the axis channel as shown in equation (3), firstly we consider the case of twoidentical QDs, due to the same coupling constants, the contribution of the decay from|2_A,1_B,0> and |1_A,2_B,0> at zero detuning is mutually counteracted, inducing a slippycurve and two symmetric peaks in each side of the zero detuning shown in figure 8(b).When the detuning between dipole moments is unequal to 0, the singularity (peak ordip) will appear at the frequency of the coherent field. So when the dipole moment of A-QD is smaller than B-QD as shown in figure 9(b), there is a peak in the middle of thespectrum. Contrarily there occurs a hole in figure 10(b). And the singularity is obviousas the increase of the detuning value. In the side emission, the photon can decay fromall of the three dressed states as shown in equations (1) and (2). And no matter how thedipole moments are changed, the triple-peak spectrum always exists as shown in figure9(a) and 10(a).
In the off-resonance situation we can obtain the decay via the side pathwayand via the axial pathway
The above equations (4), (5), (6) indicate that the photon can be emitted from|φ_a> , |φ_b> and |φ_c> both in the side direction and the axis direction. The population on the bare stateσ_(22)~A(0) can not be symmetrically distributed on the initial dressedstates. And there are not any other external fields acting on the system, so the SESdisplays a asymmetrical shape withω_c. By changing the detunning of the two transitionfrequencies, we can see the influence ofω_(AB) on the SES. Shown as the solid and dashlines in the figure 11, when the two detunings between the transition frequencies ofthe two QDs are inverse to each other, their spectra have a mirror relation. Withω_(AB)increases, the spectrum will change in different ways as shown by the dot and dash linesin figure 11.
The purpose of this thesis are:
(1) We research the quantum interference and the phase dependence character ofthe spontaneous emission, which is induced by the external driving field, from an ex-cited atom embedded in a double-band PCs .
(2) We research the RF spectrum when the atom is embedded in double-band PCswho is the non-Markvo reservoirs. Through the zero-th order Liouville operator ex-pansion approximation in optical Bloch equation and the noise operators, we obtainthe emission property which is different from the vacuum case in the resonance andoff-resonance case.
(3) In the Cavity Quantum Electrodynamics(CQED), we research the florescenceemission in the side and along the axis cavity from one single QDs and two QDs whichare strongly coupled with a single-mode nanocavity field in PCs. And we clearly ex-plain the strange emission property of the florescence emission spectrum in the dressedstate picture.
引文
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