- Email: [email protected]
Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field
Journal Pre-proof Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field C.A. Jiménez-Orjuela, H. Vinck-Posada, José M. Villas-Bôas
PII: DOI: Reference:
S0921-4526(20)30082-X https://doi.org/10.1016/j.physb.2020.412070 PHYSB 412070
To appear in:
Physica B: Physics of Condensed Matter
Received date : 5 August 2019 Revised date : 5 November 2019 Accepted date : 4 February 2020 Please cite this article as: C.A. Jiménez-Orjuela, H. Vinck-Posada and J.M. Villas-Bôas, Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field, Physica B: Physics of Condensed Matter (2020), doi: https://doi.org/10.1016/j.physb.2020.412070. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Journal Pre-proof *Manuscript Click here to view linked References
Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field C. A. Jim´enez-Orjuela∗ and H. Vinck-Posada
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Departamento de F´ısica, Universidad Nacional de Colombia, 111321, Bogota, Colombia
Jos´e M. Villas-Bˆ oas
Instituto de F´ısica, Universidade Federal de Uberlˆ andia, 38400-902 Uberlˆ andia, MG, Brazil (Dated: November 5, 2019) Including an external magnetic field B in a Tavis-Cummings model enable us to consider four exciton states in each quantum dot, two brights and two darks. With a continuous and incoherent pump to a cavity, we explore the conditions of occupation and emission as a function of B. We have found that although the dark excitons are optically inactive, these show a trace of emission as a result of both the strong coupling between light and matter and the interaction between bright and dark exciton states mediated by the magnetic field. Further, tuning one of the quantum dots at resonance with the cavity, we report the set of parameters that allows us to change the resonance condition between exciton states from different quantum dots, namely, we use the magnetic field as a control parameter to select which exciton state and which quantum dot will be mostly coupled to the cavity. I.
INTRODUCTION
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Cavity quantum electrodynamics CQED in the last years has witnessed amazing experimental and theoretical progress because of its application in quantum information and quantum networks technologies [1–4]. In this sense, the control of the spins for the exciton states in the quantum dot (QD) and the coupling with the polarized photons inside a microcavity is essential, this control may be done by the implementation of electric fields [5], changes in temperature or as we implement in this manuscript, with a magnetic field B [6]. The magnetic field has two main directions, parallel to the growth direction of the heterostructure which is called the Faraday configuration, and perpendicular to growth direction, which is known as Voigt configuration[7]. The inclusion of B forces us to consider the spin configuration in the exciton states, where it rises four exciton states, two bright which interact with photons of different polarization according to selection rules and two dark which are optically inactive [8, 9]. The magnetic field has shown to be a good control parameter for the exciton spin [10, 11] and its coupling with the cavity states [12], additionally, it may be used to overcome an experimental difficulty in reach the strong coupling between the cavity and many quantum dots with close energy to the cavity [2]. For this purpose, the magnetic field allows to tune the energy of the quantum dot, and reach the strong coupling with the cavity [13]. In this paper, we simulate two quantum dots strongly coupled to a cavity but no between them, this is called the Tavis-Cummings model. Moreover, we include an incoherent and continuous pump to the cavity and an ex-
∗
[email protected]
ternal tilted magnetic field which rises four exciton states in the quantum dot, two bright and two dark. We use the master equation in the Lindblad form with the BornMarkov approach to solving the density matrix in the steady state, then, we calculate the occupation as the mean value of the exciton operators. Further, we use the quantum regression theorem to calculate the energy emission spectrum of photons which escape from the cavity. We identify the correspondence between the occupation and the emission for the exciton states in each quantum dot and report the set of parameters which allow us reaching a maximum of occupation in the exciton states for the quantum dot. II.
THEORY
Inside a quantum dot, an exciton rises by the interaction of an electron in the conduction band with spin projection S = ±1/2 and a hole in the valence band with spin projection J = ±3/2. In this way, the exciton state |S, Ji have four possibilities according to the addition of angular momentum. The bright state |1i = |−1/2, 3/2i with spin projection +1, which may be recombined emitting a photon with right polarization. The bright state |2i = |1/2, −3/2i with spin projection −1, which may be recombined emitting a photon with left polarization. And the dark states |3i = |1/2, 3/2i with spin projection +2 and |4i = |−1/2, −3/2i with spin projection −2, which are optically inactive. Finally, the ground state |0i result by the recombination of the electron and the hole. Although the radiative recombination of dark states is forbidden, these may decay by another process such like the inclusion of additional carriers or spin-flip processes which we does not consider because we will take very low temperatures and our main purpose is to establish the effect of the resonant properties of two quantum dots mediated only
Journal Pre-proof 2 by the magnetic field [14, 15].
Hqd =
X i i i i ωxi σ11 + ωxi σ22 + ωdi σ33 + ωdi σ44
i=1,2
+
δ2 i δ1 i i i σ12 + σ21 σ34 + σ43 + , 2 2
Hcav = ωc a† a + ga
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i=1,2
i i β+ + αB 2 σ11 + −β+ + αB 2 σ22
i i + −β− + αB 2 σ33 + β− + αB 2 σ44 i i i i + βe (σ13 + σ31 + σ24 + σ42 )
i i i i i + βh (σ14 + σ41 + σ32 + σ23 ) ,
+ ωc b b + gb
(1)
The magnetic field has two main components, Bz parallel to growth direction known as Faraday configuration and Bx perpendicular to growth direction known as Voigt configuration. In Faraday configuration, the magnetic field lifts the degeneration between the two BE and the two DE, splitting the energies between them but without mixing the BE with the DE. In Voigt configuration the magnetic field generates a mix between the BE and the DE states, in both cases, there is a diamagnetic effect which causes a blue shift in the energy emission. We take the magnetic field as B = (Bx , Bz ) = (B cosθ, B sinθ), where θ is the tilted angle from the plane such that θ = 0 define the Voigt configuration and θ = π/2 define the Faraday configuration[7]. We include the magnetic field like a correction Hmag in the quantum dot hamiltonian[17], Hmag =
X
i i (a† σ01 + aσ10 )
i=1,2
†
here, ωxi is the bright exciton frequency for the i-QD, ωdi = ωxi − δ0 is the dark excitons frequency of the i-QD, δ0 = 0.1 meV is the energy splitting between the bright and dark excitons, δ1 = 50µeV is the exchange interaction which lift the degeneration between the two bright exciton and it leads to the observation of two linearly polarized transitions even in absence of the magnetic field [7]. Finally, δ2 = 1µeV is the exchange interaction between the dark excitons [16]
X h
The light is quantized inside the microcavity and may be described by the ladder operators a† (b† ) which create a photon with right (left) polarization. The hamiltonian including the interaction with the exciton states is (~ = 1),
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In this paper we consider two quantum dots, defining i the j-state for the i-QD like |ji , here, j = 0, 1, 2, 3, 4 label the exciton state and i = 1, 2 label the QD. The hamiltonian for the i-QD is set up with the projection i i i operators σjk = |ji hk| , such that the hamiltonian with ~ = 1 for the two quantum dots is,
Bohr magneton, gez = −0.8, ghz = −2.2, gex = −0.65 and ghx = −0.35 are the electron and hole effective g-factor in the z and x directions, respectively, and α = 20µeV /T 2 is the diamagnetic factor[12].
(2)
here we have the β-factors, where β+ = µB B sin θ(gez + ghz )/2 and β− = µB B sin θ(gez − ghz )/2 are Faraday-like factors which open the energies for the BEs and the DEs, respectively. βe = µB B cos θgex /2 and βh = 12 µB Bghx cos θ, are Voigt-like factors which mix the BEs with the DEs. Also, µB = 57.9 µeV/T is the
X
i i (b† σ02 + bσ20 ),
(3)
i=1,2
here, ωc is the transition frequency of the cavity and ga = gb = 50 µeV are the cavity-QD coupling strength. We define the cavity states with right (left) polarization as |ni (|mi) such that, a† a |ni = n |ni
b† b |mi = m |mi .
(4)
The whole Hamiltonian including the two quantum dots and the magnetic effect is H = Hqd + Hmag + Hcav .
(5)
In order to find the density matrix which contains all the information of the system, we make use of the master equation in the Lindblad form with the Born-Markov approach, dρ = −ı[H, ρ] + κ (D[a] + D[b]) dt X i i +γ D[σ01 ] + D[σ02 ] + Pc D[a† ] + D[b† ] , i=1,2
(6)
where, D[L] = LρL† − 12 (L† Lρ + ρL† L) is the Lindblad superoperator, κ = 0.15 meV is the rate of photons which escape from the cavity, γ = 0.7 µeV the photons rate that escape from the QD, and Pc = 10 µeV a continuous and incoherent pump to cavity. To solve this equation let the basis, 1
2
|i, j, n, mi = |ii ⊗ |ji ⊗ |ni ⊗ |mi , 1
2
(7)
where |ii and |ji are the exciton states for the first and second quantum dot, respectively (i, j = 0, 1, 2, 3, 4). n (m) represents the number of photons inside the cavity with right (left) polarization. The excitation operator is defined as,
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ˆ = a† a + b† b + N
4 X i=1
1
|ii hi| +
4 X j=1
2
|ji hj| ,
(8)
1 1 hσii i = T r σii ρ
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and allow us to define the manifold number of a state ˆ |si, and this represents the |si = |i, j, n, mi by, N = hs| N excitation in the system. The master equation (6) is a set of several differentials equations that grow quadratically with the basis size. The low pumping regime allows us to consider each QD as a two-level system and a low occupation of photons inside the cavity. We cut off the number of equations considering a maximum manifold number of N = 2. With this basis, we solve the master equation in the steady-state and calculate the mean value of the exciton operator for the two quantum dots, 2 2 hσii i = T r σii ρ
(9)
which represent the occupation of the system in the exciton states, i = 1, 2 for the bright excitons, and i = 3, 4 for the dark excitons. Finally, the connection of the theory with the experiment is done through the emission of photons which may be measured by photoluminescence. Whereby, we make use of the quantum regression theorem QRT to calculate the energy emission spectrum of photons which escape from the cavity [18]. III.
RESULTS
ωx2
1 = ωc + δ12 + δ, 2
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1 ωx1 = ωc − δ12 + δ 2
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All the parameters have been defined previously except the strength of the magnetic field B, the tilted angle θ and the energies of the cavity and the two quantum dots. For InAs/GaAs-self-assembled quantum dots embedded in a microcavity of a GaAs/AlGaAs-micropillar, the typical energies are ∼ 1.3 eV [12]. In our simulation, we define the cavity energy fix at ωc = 1 eV and the energy for each quantum dot as,
(10)
where δ12 = 0.4 meV is the split energy between the two quantum dots and δ is the detuning between the cavity and the central frequency of the QDs. First of all, we consider the absence of magnetic field and explore the effect of the detuning between the cavity and the QDs. With B = 0, the Fig.1 show the occupation for the bright excitons in each quantum dot (lower panel) and the energy emission spectrum from the cavity (upper panel). In the plots of Fig.1, we show the correspondence of the maximum occupation of the bright exciton with the emission of each quantum dot in resonance with the
0.10
0.00
- 0.4
0.0
0.4
FIG. 1. (Color online) Occupation of the excitonic states for the two QDs as a function of the detuning δ (lower panel). Energy emission spectrum of the cavity as a function of the detuning δ (upper panel). The dark zone in 1000 meV is the cavity energy emission and the two white diagonal-lines represent the emission of the QDs, also, the arrows display the correspondence between the emission and the maximum occupation with each QD. It has been taken B = 0.
cavity. The emission from the cavity is higher by two main reasons, the first one is because the pumping is carried through the cavity mode and the second one is because the cavity loss rate κ is higher than exciton loss rate γ. Besides, the emission of the two quantum dots is smaller far from resonance, but near the resonance it shows the typical anticrossing for systems in strong coupling regime and the bright zone is the vacuum Rabi splitting which is a signature of the polariton emission [6]. Further, the BE in each QD is degenerate and the DE are not populated, because the magnetic field is zero. Here we have shown that the detuning between the cavity and the two QDs allow us to choose which QD is populated. With δ = −0.2 meV the cavity is resonant with the second QD (ωx2 = ωc ) and far from resonance with the first QD (ωx1 = ωc − 0.4 meV). With this choice, we take the Faraday configuration (θ = π/2) and calculate the occupation and emission as a function of the magnetic field strength B. In the plots of Fig.2 for B = 0, the second QD has a maximum occupation because of the strong coupling with the cavity. When B increases, the magnetic field open the energy emission of the bright excitons for the two QDs due to the Zeeman effect, then one of the bright exciton of the first QD get the maximum occupation. Finally, for higher values in B the diamagnetic effect is relevant and let the occupation for the others bright excitons, such that all the bright excitons states for the
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4
FIG. 3. (Color online) Occupation of the excitonic states for the first QD (lower panel) and the second QD (upper panel) as a function of the magnetic field B. Energy emission spectrum of the cavity as a function of the magnetic field B (middle panel). The arrows display the correspondence between the emission and the maximum occupation with each QD. It has been taken θ = 0, Voigt Configuration.
two QDs may be populated just tuning the magnetic field intensity.
1 of the second QD to the exciton state hσii i of the first QD as,
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FIG. 2. (Color online) Occupation of the excitonic states for the first QD (lower panel) and the second QD (upper panel) as a function of the magnetic field B. Energy emission spectrum of the cavity as a function of the magnetic field B (middle panel). The arrows display the correspondence between the emission and the maximum occupation with each QD. It has been taken θ = π/2, Faraday configuration.
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In order to populate dark excitons we choice the Voigt configuration (θ = 0) and repeat the previous calculation. In the Fig.3 for B = 0 the cavity is resonant with the second QD and there is a maximum occupation for its BEs. When the magnetic field increases the DEs appear in both QDs due to the mix between bright and dark excitons through the β-factors in equation (2). However, the BEs and the DEs remain degenerate because there is not Zeeman effect. For B high enough, the diamagnetic effect allows to get a maximum occupation for all the exciton states in the first QD. Currently we have shown the effect of magnetic field in both configuration, where tuning one of the QDs in resonance with the cavity, we may get occupation in any of the exciton states for the other QD, this effect allow us to get resonance of the cavity with any state of any QD given the resonance of the cavity with a particular state. Let δ = −0.2 meV which defines the cavity resonant with the second QD and far from resonance with the first QD. To quantify how efficient is the mechanism to get resonance of the cavity with a particular state, we 2 define the resonant function from the bright exciton hσ11 i
Ti =
1 2 hσii i − hσ11 i 1 2 hσii i + hσ11 i
i = 1, 2, 3, 4,
(11)
where we have taken as reference one of the BE of the second QD because is one of the most populated for B = 0. Ti → −1 if the system is maximally occupied in 2 the state hσ11 i, and Ti → 1 if the system is maximally 1 occupied in the state hσii i, namely, the resonant function is maximum. The Fig.4 shows the Ti as a function of the intensity B and tilted angle θ. Naturally, in the low magnetic field regime, Ti → −1 where the second QD is maximally occupied. When B increases, there is different values in θ that let populate the different exciton states which are 1 the dark regions in these plots. The bright exciton hσ11 i may be maximally populated for any angle and B around 6 T. The other states are maximally populated near the Voigt zone (θ ∼ 0 or θ ∼ π) and B ∼ 5 T. In the Faraday region (θ ∼ π/2), the dark excitons states are totally 1 forbidden and the bright exciton hσ22 i is some populated for B ∼ 3 T .
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FIG. 4. (Color online) Resonant function Ti between quantum dots states as a function of the magnetic field intensity B and tilted angle θ.
IV.
CONCLUSIONS
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Including an external and tilted magnetic field in the Tavis-Cummings model enlarge the number of exciton states which are relevant in the system. Further, with the two quantum dots far from the resonance between
[1] J. P. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, A. Forchel, Strong coupling in a single quantum dot–semiconductor microcavity system, Nature 432 (7014) (2004) 197 (2004). [2] P. Lodahl, S. Mahmoodian, S. Stobbe, Interfacing single photons and single quantum dots with photonic nanostructures, Reviews of Modern Physics 87 (2) (2015) 347 (2015). [3] C.-P. Yang, S.-I. Chu, S. Han, Possible realization of entanglement, logical gates, and quantuminformation transfer with superconducting-quantuminterference-device qubits in cavity qed, Physical Review A 67 (4) (2003) 042311 (2003). [4] A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, A. Small, et al., Quantum information processing using quantum dot spins and cavity qed, Physical review letters 83 (20) (1999) 4204 (1999). [5] A. Laucht, N. Hauke, J. Villas-Bˆ oas, F. Hofbauer, G. B¨ ohm, M. Kaniber, J. Finley, Dephasing of exciton polaritons in photoexcited ingaas quantum dots in gaas nanocavities, Physical review letters 103 (8) (2009)
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them, we explore the condition where each QD is individually and collective coupled to the cavity in terms of the magnetic field see Fig.1 and Fig.2. Also, with one of the quantum dot in resonance with the cavity, we show the effect in the occupation and emission spectrum of the magnetic field in both configurations. A remarkable phenomenon reported in this paper is the trace shown by the dark exciton in the emission spectrum see Fig.3, it is unexpected for zero magnetic field because the dark excitons are optically inactive, but the trace shown in the emission is a signature of the coupling between dark and bright excitons mediated by the magnetic field. Finally, the magnetic field allows us to transfer occupation between exciton states of different quantum dots, and we report the set of parameters which reach a maximum resonance of population in each state see Fig.4. The change in resonance states reached by the magnetic field is significant because it allows us to select the quantum dot and the exciton state which will be resonant with the cavity and this is useful when there are many QDs with close energy to the cavity as actually occurs in the experiments. V.
ACKNOWLEDGMENTS
The authors gratefully acknowledge funding by COLCIENCIAS projects “Emisi´ on en sistemas de Qubits Superconductores acoplados a la radiaci´ on. C´odigo 110171249692, CT 293-2016, HERMES 31361” and “Exploraci´ on y modelaci´ on de la iridiscencia en especies Colombianas. C´ odigo 110156933525, CT 026-2013, HERMES 17432”. CAJ also, acknowledge to “fundaci´on mazda para el arte y la ciencia” and to “Beca de Doctorados Nacionales de COLCIENCIAS 727”.
087405 (2009). [6] H. Kim, T. C. Shen, D. Sridharan, G. S. Solomon, E. Waks, Magnetic field tuning of a quantum dot strongly coupled to a photonic crystal cavity, Applied Physics Letters 98 (9) (2011) 091102 (2011). [7] M. Bayer, G. Ortner, O. Stern, A. Kuther, A. Gorbunov, A. Forchel, P. Hawrylak, S. Fafard, K. Hinzer, T. Reinecke, et al., Fine structure of neutral and charged excitons in self-assembled in (ga) as/(al) gaas quantum dots, Physical Review B 65 (19) (2002) 195315 (2002). [8] R. J. Warburton, Single spins in self-assembled quantum dots, Nature materials 12 (6) (2013) 483 (2013). [9] P. Chen, C. Piermarocchi, L. Sham, D. Gammon, D. Steel, Theory of quantum optical control of a single spin in a quantum dot, Physical Review B 69 (7) (2004) 075320 (2004). [10] C. Emary, X. Xu, D. Steel, S. Saikin, L. Sham, Fast initialization of the spin state of an electron in a quantum dot in the voigt configuration, Physical review letters 98 (4) (2007) 047401 (2007). [11] M. Atat¨ ure, J. Dreiser, A. Badolato, A. Imamoglu, Observation of faraday rotation from a single confined spin, Nature Physics 3 (2) (2007) 101 (2007).
Journal Pre-proof 6 [15] J. Johansen, B. Julsgaard, S. Stobbe, J. M. Hvam, P. Lodahl, Probing long-lived dark excitons in self-assembled quantum dots, Physical Review B 81 (8) (2010) 081304 (2010). [16] S. L¨ uker, T. Kuhn, D. Reiter, Direct optical state preparation of the dark exciton in a quantum dot, Physical Review B 92 (20) (2015) 201305 (2015). [17] C. Jim´enez-Orjuela, H. Vinck-Posada, J. M. Villas-Bˆ oas, Dark excitons in a quantum-dot–cavity system under a tilted magnetic field, Physical Review B 96 (12) (2017) 125303 (2017). [18] J. Perea, D. Porras, C. Tejedor, Dynamics of the excitations of a quantum dot in a microcavity, Physical Review B 70 (11) (2004) 115304 (2004).
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[12] S. Reitzenstein, S. M¨ unch, P. Franeck, A. Rahimi-Iman, A. L¨ offler, S. H¨ ofling, L. Worschech, A. Forchel, Control of the strong light-matter interaction between an elongated in 0.3 ga 0.7 as quantum dot and a micropillar cavity using external magnetic fields, Physical review letters 103 (12) (2009) 127401 (2009). [13] H. Kim, D. Sridharan, T. C. Shen, G. S. Solomon, E. Waks, Strong coupling between two quantum dots and a photonic crystal cavity using magnetic field tuning, Optics express 19 (3) (2011) 2589–2598 (2011). [14] M. Korkusinski, P. Hawrylak, Atomistic theory of emission from dark excitons in self-assembled quantum dots, Physical Review B 87 (11) (2013) 115310 (2013).
Highlights
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Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field
Highlights
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Two quantum dot inside a microcavity can be described with the TavisCummings model, this system behaves like two qubits interacting with photons but not between them. The dark exciton states, although are optically inactive, show a trace of emission in the spectrum due to the magnetic field interaction. The magnetic field allows driving the spins in the electron and hole in each quantum dot and in this way behaves like a tuning parameter to select the exciton state. An external magnetic field used to drive the occupation in the system is a remarkable process because it is relatively easy to implement experimentally
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*Conflict of Interest form
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Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field conflccts of lnctrersct
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The authors have no conflcts oo lnterest ln our research.
PII: DOI: Reference:
S0921-4526(20)30082-X https://doi.org/10.1016/j.physb.2020.412070 PHYSB 412070
To appear in:
Physica B: Physics of Condensed Matter
Received date : 5 August 2019 Revised date : 5 November 2019 Accepted date : 4 February 2020 Please cite this article as: C.A. Jiménez-Orjuela, H. Vinck-Posada and J.M. Villas-Bôas, Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field, Physica B: Physics of Condensed Matter (2020), doi: https://doi.org/10.1016/j.physb.2020.412070. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
Journal Pre-proof *Manuscript Click here to view linked References
Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field C. A. Jim´enez-Orjuela∗ and H. Vinck-Posada
lP repro of
Departamento de F´ısica, Universidad Nacional de Colombia, 111321, Bogota, Colombia
Jos´e M. Villas-Bˆ oas
Instituto de F´ısica, Universidade Federal de Uberlˆ andia, 38400-902 Uberlˆ andia, MG, Brazil (Dated: November 5, 2019) Including an external magnetic field B in a Tavis-Cummings model enable us to consider four exciton states in each quantum dot, two brights and two darks. With a continuous and incoherent pump to a cavity, we explore the conditions of occupation and emission as a function of B. We have found that although the dark excitons are optically inactive, these show a trace of emission as a result of both the strong coupling between light and matter and the interaction between bright and dark exciton states mediated by the magnetic field. Further, tuning one of the quantum dots at resonance with the cavity, we report the set of parameters that allows us to change the resonance condition between exciton states from different quantum dots, namely, we use the magnetic field as a control parameter to select which exciton state and which quantum dot will be mostly coupled to the cavity. I.
INTRODUCTION
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Cavity quantum electrodynamics CQED in the last years has witnessed amazing experimental and theoretical progress because of its application in quantum information and quantum networks technologies [1–4]. In this sense, the control of the spins for the exciton states in the quantum dot (QD) and the coupling with the polarized photons inside a microcavity is essential, this control may be done by the implementation of electric fields [5], changes in temperature or as we implement in this manuscript, with a magnetic field B [6]. The magnetic field has two main directions, parallel to the growth direction of the heterostructure which is called the Faraday configuration, and perpendicular to growth direction, which is known as Voigt configuration[7]. The inclusion of B forces us to consider the spin configuration in the exciton states, where it rises four exciton states, two bright which interact with photons of different polarization according to selection rules and two dark which are optically inactive [8, 9]. The magnetic field has shown to be a good control parameter for the exciton spin [10, 11] and its coupling with the cavity states [12], additionally, it may be used to overcome an experimental difficulty in reach the strong coupling between the cavity and many quantum dots with close energy to the cavity [2]. For this purpose, the magnetic field allows to tune the energy of the quantum dot, and reach the strong coupling with the cavity [13]. In this paper, we simulate two quantum dots strongly coupled to a cavity but no between them, this is called the Tavis-Cummings model. Moreover, we include an incoherent and continuous pump to the cavity and an ex-
∗
[email protected]
ternal tilted magnetic field which rises four exciton states in the quantum dot, two bright and two dark. We use the master equation in the Lindblad form with the BornMarkov approach to solving the density matrix in the steady state, then, we calculate the occupation as the mean value of the exciton operators. Further, we use the quantum regression theorem to calculate the energy emission spectrum of photons which escape from the cavity. We identify the correspondence between the occupation and the emission for the exciton states in each quantum dot and report the set of parameters which allow us reaching a maximum of occupation in the exciton states for the quantum dot. II.
THEORY
Inside a quantum dot, an exciton rises by the interaction of an electron in the conduction band with spin projection S = ±1/2 and a hole in the valence band with spin projection J = ±3/2. In this way, the exciton state |S, Ji have four possibilities according to the addition of angular momentum. The bright state |1i = |−1/2, 3/2i with spin projection +1, which may be recombined emitting a photon with right polarization. The bright state |2i = |1/2, −3/2i with spin projection −1, which may be recombined emitting a photon with left polarization. And the dark states |3i = |1/2, 3/2i with spin projection +2 and |4i = |−1/2, −3/2i with spin projection −2, which are optically inactive. Finally, the ground state |0i result by the recombination of the electron and the hole. Although the radiative recombination of dark states is forbidden, these may decay by another process such like the inclusion of additional carriers or spin-flip processes which we does not consider because we will take very low temperatures and our main purpose is to establish the effect of the resonant properties of two quantum dots mediated only
Journal Pre-proof 2 by the magnetic field [14, 15].
Hqd =
X i i i i ωxi σ11 + ωxi σ22 + ωdi σ33 + ωdi σ44
i=1,2
+
δ2 i δ1 i i i σ12 + σ21 σ34 + σ43 + , 2 2
Hcav = ωc a† a + ga
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i=1,2
i i β+ + αB 2 σ11 + −β+ + αB 2 σ22
i i + −β− + αB 2 σ33 + β− + αB 2 σ44 i i i i + βe (σ13 + σ31 + σ24 + σ42 )
i i i i i + βh (σ14 + σ41 + σ32 + σ23 ) ,
+ ωc b b + gb
(1)
The magnetic field has two main components, Bz parallel to growth direction known as Faraday configuration and Bx perpendicular to growth direction known as Voigt configuration. In Faraday configuration, the magnetic field lifts the degeneration between the two BE and the two DE, splitting the energies between them but without mixing the BE with the DE. In Voigt configuration the magnetic field generates a mix between the BE and the DE states, in both cases, there is a diamagnetic effect which causes a blue shift in the energy emission. We take the magnetic field as B = (Bx , Bz ) = (B cosθ, B sinθ), where θ is the tilted angle from the plane such that θ = 0 define the Voigt configuration and θ = π/2 define the Faraday configuration[7]. We include the magnetic field like a correction Hmag in the quantum dot hamiltonian[17], Hmag =
X
i i (a† σ01 + aσ10 )
i=1,2
†
here, ωxi is the bright exciton frequency for the i-QD, ωdi = ωxi − δ0 is the dark excitons frequency of the i-QD, δ0 = 0.1 meV is the energy splitting between the bright and dark excitons, δ1 = 50µeV is the exchange interaction which lift the degeneration between the two bright exciton and it leads to the observation of two linearly polarized transitions even in absence of the magnetic field [7]. Finally, δ2 = 1µeV is the exchange interaction between the dark excitons [16]
X h
The light is quantized inside the microcavity and may be described by the ladder operators a† (b† ) which create a photon with right (left) polarization. The hamiltonian including the interaction with the exciton states is (~ = 1),
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In this paper we consider two quantum dots, defining i the j-state for the i-QD like |ji , here, j = 0, 1, 2, 3, 4 label the exciton state and i = 1, 2 label the QD. The hamiltonian for the i-QD is set up with the projection i i i operators σjk = |ji hk| , such that the hamiltonian with ~ = 1 for the two quantum dots is,
Bohr magneton, gez = −0.8, ghz = −2.2, gex = −0.65 and ghx = −0.35 are the electron and hole effective g-factor in the z and x directions, respectively, and α = 20µeV /T 2 is the diamagnetic factor[12].
(2)
here we have the β-factors, where β+ = µB B sin θ(gez + ghz )/2 and β− = µB B sin θ(gez − ghz )/2 are Faraday-like factors which open the energies for the BEs and the DEs, respectively. βe = µB B cos θgex /2 and βh = 12 µB Bghx cos θ, are Voigt-like factors which mix the BEs with the DEs. Also, µB = 57.9 µeV/T is the
X
i i (b† σ02 + bσ20 ),
(3)
i=1,2
here, ωc is the transition frequency of the cavity and ga = gb = 50 µeV are the cavity-QD coupling strength. We define the cavity states with right (left) polarization as |ni (|mi) such that, a† a |ni = n |ni
b† b |mi = m |mi .
(4)
The whole Hamiltonian including the two quantum dots and the magnetic effect is H = Hqd + Hmag + Hcav .
(5)
In order to find the density matrix which contains all the information of the system, we make use of the master equation in the Lindblad form with the Born-Markov approach, dρ = −ı[H, ρ] + κ (D[a] + D[b]) dt X i i +γ D[σ01 ] + D[σ02 ] + Pc D[a† ] + D[b† ] , i=1,2
(6)
where, D[L] = LρL† − 12 (L† Lρ + ρL† L) is the Lindblad superoperator, κ = 0.15 meV is the rate of photons which escape from the cavity, γ = 0.7 µeV the photons rate that escape from the QD, and Pc = 10 µeV a continuous and incoherent pump to cavity. To solve this equation let the basis, 1
2
|i, j, n, mi = |ii ⊗ |ji ⊗ |ni ⊗ |mi , 1
2
(7)
where |ii and |ji are the exciton states for the first and second quantum dot, respectively (i, j = 0, 1, 2, 3, 4). n (m) represents the number of photons inside the cavity with right (left) polarization. The excitation operator is defined as,
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ˆ = a† a + b† b + N
4 X i=1
1
|ii hi| +
4 X j=1
2
|ji hj| ,
(8)
1 1 hσii i = T r σii ρ
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and allow us to define the manifold number of a state ˆ |si, and this represents the |si = |i, j, n, mi by, N = hs| N excitation in the system. The master equation (6) is a set of several differentials equations that grow quadratically with the basis size. The low pumping regime allows us to consider each QD as a two-level system and a low occupation of photons inside the cavity. We cut off the number of equations considering a maximum manifold number of N = 2. With this basis, we solve the master equation in the steady-state and calculate the mean value of the exciton operator for the two quantum dots, 2 2 hσii i = T r σii ρ
(9)
which represent the occupation of the system in the exciton states, i = 1, 2 for the bright excitons, and i = 3, 4 for the dark excitons. Finally, the connection of the theory with the experiment is done through the emission of photons which may be measured by photoluminescence. Whereby, we make use of the quantum regression theorem QRT to calculate the energy emission spectrum of photons which escape from the cavity [18]. III.
RESULTS
ωx2
1 = ωc + δ12 + δ, 2
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1 ωx1 = ωc − δ12 + δ 2
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All the parameters have been defined previously except the strength of the magnetic field B, the tilted angle θ and the energies of the cavity and the two quantum dots. For InAs/GaAs-self-assembled quantum dots embedded in a microcavity of a GaAs/AlGaAs-micropillar, the typical energies are ∼ 1.3 eV [12]. In our simulation, we define the cavity energy fix at ωc = 1 eV and the energy for each quantum dot as,
(10)
where δ12 = 0.4 meV is the split energy between the two quantum dots and δ is the detuning between the cavity and the central frequency of the QDs. First of all, we consider the absence of magnetic field and explore the effect of the detuning between the cavity and the QDs. With B = 0, the Fig.1 show the occupation for the bright excitons in each quantum dot (lower panel) and the energy emission spectrum from the cavity (upper panel). In the plots of Fig.1, we show the correspondence of the maximum occupation of the bright exciton with the emission of each quantum dot in resonance with the
0.10
0.00
- 0.4
0.0
0.4
FIG. 1. (Color online) Occupation of the excitonic states for the two QDs as a function of the detuning δ (lower panel). Energy emission spectrum of the cavity as a function of the detuning δ (upper panel). The dark zone in 1000 meV is the cavity energy emission and the two white diagonal-lines represent the emission of the QDs, also, the arrows display the correspondence between the emission and the maximum occupation with each QD. It has been taken B = 0.
cavity. The emission from the cavity is higher by two main reasons, the first one is because the pumping is carried through the cavity mode and the second one is because the cavity loss rate κ is higher than exciton loss rate γ. Besides, the emission of the two quantum dots is smaller far from resonance, but near the resonance it shows the typical anticrossing for systems in strong coupling regime and the bright zone is the vacuum Rabi splitting which is a signature of the polariton emission [6]. Further, the BE in each QD is degenerate and the DE are not populated, because the magnetic field is zero. Here we have shown that the detuning between the cavity and the two QDs allow us to choose which QD is populated. With δ = −0.2 meV the cavity is resonant with the second QD (ωx2 = ωc ) and far from resonance with the first QD (ωx1 = ωc − 0.4 meV). With this choice, we take the Faraday configuration (θ = π/2) and calculate the occupation and emission as a function of the magnetic field strength B. In the plots of Fig.2 for B = 0, the second QD has a maximum occupation because of the strong coupling with the cavity. When B increases, the magnetic field open the energy emission of the bright excitons for the two QDs due to the Zeeman effect, then one of the bright exciton of the first QD get the maximum occupation. Finally, for higher values in B the diamagnetic effect is relevant and let the occupation for the others bright excitons, such that all the bright excitons states for the
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FIG. 3. (Color online) Occupation of the excitonic states for the first QD (lower panel) and the second QD (upper panel) as a function of the magnetic field B. Energy emission spectrum of the cavity as a function of the magnetic field B (middle panel). The arrows display the correspondence between the emission and the maximum occupation with each QD. It has been taken θ = 0, Voigt Configuration.
two QDs may be populated just tuning the magnetic field intensity.
1 of the second QD to the exciton state hσii i of the first QD as,
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FIG. 2. (Color online) Occupation of the excitonic states for the first QD (lower panel) and the second QD (upper panel) as a function of the magnetic field B. Energy emission spectrum of the cavity as a function of the magnetic field B (middle panel). The arrows display the correspondence between the emission and the maximum occupation with each QD. It has been taken θ = π/2, Faraday configuration.
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In order to populate dark excitons we choice the Voigt configuration (θ = 0) and repeat the previous calculation. In the Fig.3 for B = 0 the cavity is resonant with the second QD and there is a maximum occupation for its BEs. When the magnetic field increases the DEs appear in both QDs due to the mix between bright and dark excitons through the β-factors in equation (2). However, the BEs and the DEs remain degenerate because there is not Zeeman effect. For B high enough, the diamagnetic effect allows to get a maximum occupation for all the exciton states in the first QD. Currently we have shown the effect of magnetic field in both configuration, where tuning one of the QDs in resonance with the cavity, we may get occupation in any of the exciton states for the other QD, this effect allow us to get resonance of the cavity with any state of any QD given the resonance of the cavity with a particular state. Let δ = −0.2 meV which defines the cavity resonant with the second QD and far from resonance with the first QD. To quantify how efficient is the mechanism to get resonance of the cavity with a particular state, we 2 define the resonant function from the bright exciton hσ11 i
Ti =
1 2 hσii i − hσ11 i 1 2 hσii i + hσ11 i
i = 1, 2, 3, 4,
(11)
where we have taken as reference one of the BE of the second QD because is one of the most populated for B = 0. Ti → −1 if the system is maximally occupied in 2 the state hσ11 i, and Ti → 1 if the system is maximally 1 occupied in the state hσii i, namely, the resonant function is maximum. The Fig.4 shows the Ti as a function of the intensity B and tilted angle θ. Naturally, in the low magnetic field regime, Ti → −1 where the second QD is maximally occupied. When B increases, there is different values in θ that let populate the different exciton states which are 1 the dark regions in these plots. The bright exciton hσ11 i may be maximally populated for any angle and B around 6 T. The other states are maximally populated near the Voigt zone (θ ∼ 0 or θ ∼ π) and B ∼ 5 T. In the Faraday region (θ ∼ π/2), the dark excitons states are totally 1 forbidden and the bright exciton hσ22 i is some populated for B ∼ 3 T .
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FIG. 4. (Color online) Resonant function Ti between quantum dots states as a function of the magnetic field intensity B and tilted angle θ.
IV.
CONCLUSIONS
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Including an external and tilted magnetic field in the Tavis-Cummings model enlarge the number of exciton states which are relevant in the system. Further, with the two quantum dots far from the resonance between
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them, we explore the condition where each QD is individually and collective coupled to the cavity in terms of the magnetic field see Fig.1 and Fig.2. Also, with one of the quantum dot in resonance with the cavity, we show the effect in the occupation and emission spectrum of the magnetic field in both configurations. A remarkable phenomenon reported in this paper is the trace shown by the dark exciton in the emission spectrum see Fig.3, it is unexpected for zero magnetic field because the dark excitons are optically inactive, but the trace shown in the emission is a signature of the coupling between dark and bright excitons mediated by the magnetic field. Finally, the magnetic field allows us to transfer occupation between exciton states of different quantum dots, and we report the set of parameters which reach a maximum resonance of population in each state see Fig.4. The change in resonance states reached by the magnetic field is significant because it allows us to select the quantum dot and the exciton state which will be resonant with the cavity and this is useful when there are many QDs with close energy to the cavity as actually occurs in the experiments. V.
ACKNOWLEDGMENTS
The authors gratefully acknowledge funding by COLCIENCIAS projects “Emisi´ on en sistemas de Qubits Superconductores acoplados a la radiaci´ on. C´odigo 110171249692, CT 293-2016, HERMES 31361” and “Exploraci´ on y modelaci´ on de la iridiscencia en especies Colombianas. C´ odigo 110156933525, CT 026-2013, HERMES 17432”. CAJ also, acknowledge to “fundaci´on mazda para el arte y la ciencia” and to “Beca de Doctorados Nacionales de COLCIENCIAS 727”.
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Highlights
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Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field
Highlights
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Two quantum dot inside a microcavity can be described with the TavisCummings model, this system behaves like two qubits interacting with photons but not between them. The dark exciton states, although are optically inactive, show a trace of emission in the spectrum due to the magnetic field interaction. The magnetic field allows driving the spins in the electron and hole in each quantum dot and in this way behaves like a tuning parameter to select the exciton state. An external magnetic field used to drive the occupation in the system is a remarkable process because it is relatively easy to implement experimentally
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*Conflict of Interest form
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Strong coupling of two quantum dots with a microcavity in the presence of an external and tilted magnetic field conflccts of lnctrersct
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The authors have no conflcts oo lnterest ln our research.