A new model for quantum dot light emitting-absorbing devices: proofs and supplements

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A new model for quantum dot light emitting-absorbing devices: proofs and supplements

H. Neidhardt, L. Wilhelm, V.A. Zagrebnov

To cite this version:

H. Neidhardt, L. Wilhelm, V.A. Zagrebnov. A new model for quantum dot light emitting-absorbing devices: proofs and supplements. Nanosystems: Physics, Chemistry, Mathematics, St. Petersburg National Research University of Information Technologies, Mechanics and Optics., 2015, 6 (1), pp.6- 45. �10.17586/2220-8054-2015-6-1-6-45�. �hal-00919509�

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A new model for quantum dot light emitting-absorbing devices

Dedicated to the memory of Pierre Duclos Hagen Neidhardt

Lukas Wilhelm

WIAS Berlin, Mohrenstr. 39, 10117 Berlin, Germany E-mail: hagen.neidhardt@wias-berlin.de

Valentin A. Zagrebnov

Laboratoire d’Analyse, Topologie, Probabilités - UMR 7353 CMI - Technopôle Château-Gombert

39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France E-mail: Valentin.Zagrebnov@latp.univ-mrs.fr

December 16, 2013

Abstract

Motivated by the Jaynes-Cummings (JC) model, we consider here a quantum dot coupled simultaneously to a reservoir of photons and to two electric leads (free- fermion reservoirs). This Jaynes-Cummings-Leads (JCL) model makes possible that the fermion current through the dot creates a photon flux, which describes a light-emitting device. The same model is also describe a transformation of the photon flux into current of fermions, i.e. a quantum dot light-absorbing device.

The key tool to obtain these results is an abstract Landauer-Büttiker formula.

Keywords: Landauer-Büttiker formula, Jaynes-Cummings model, coupling to leads, light emission, solar cells

Mathematics Subject Classification 2000:47A40, 47A55, 81Q37, 81V80

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1 Introduction

The Landauer-Büttiker formula is widely used for the analysis of the steady state current flowing trough a quantum device. It goes back to [18] and [7] and was initially developed based on phenomenological arguments for non-interacting electrons (free- fermions). The essential idea was to describe a quantum system as an inner or sample system (dot) with left and right leads attached to it, i.e. free-fermion reservoirs with two different electro-chemical potentials. The goal was to calculate the steady electron current going from one lead through the dot to another one.

It was Landauer and Büttiker who found that this current is directly related to the transmission coefficientsof some natural scattering system related to this particle transport problem. The phenomenological approach of Landauer and Büttiker later has been justified in several papers by deriving the formula from fundamental concepts of the Quantum Mechanics, see the series of papers [1, 5, 8, 9, 10, 11, 12, 13, 14] and [19].

Note that this quantum mechanical approach is possible since for the case of free- fermion reservoirs the corresponding transport problem reduces to study the Hamilto- nian dynamics of extended “one-particle” system. During last decade there has been an

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important progress in rigorous development of the Quantum Statistical Mechanics of Open Systems [2, 3, 4]. This is a many-body approach adapted for interacting systems.

It also allows, besides the Hamiltonian [2], to develop a Markovian description of ef- fective microscopic dynamics of the sample system (dot) connected to environment of external reservoirs [3]. Then evolution the sample system is governed by a quantum Master Equation. Although powerful and useful the Markovian approach needs a microscopic Hamiltonian justification, which is a nontrivial problem [3].

In the present paper we follow the one-particle quantum mechanical Hamiltonian approach. Motivated by the quantum optics Jaynes-Cummings (JC) model, we consider here atwo-levelquantum dot coupled simultaneouslyto environment of three external reservoirs. The first is the standard JC one-mode photon resonator, which makes the JC quantum dot anopen system[16]. Two others arefree-fermion reservoirs coupled to the quantum dot. They mimic two electric leads. This new Jaynes- Cummings-Leads (JCL-) model makes possible that the fermion current through the dot creates a photon flux into the resonator, i.e. it describes alight-emittingdevice. The same model is also able to describe a transformation of the external photon flux into a current of fermions, which corresponds to a quantum dotlight-absorbingdevice.

The aim of the paper is to analyze the fermion current going through the dot as a function of electro-chemical potentials on leads and the contact with the photon reservoir. Although the latter is the canonicalJC-interaction, the coupling of the JC model with leads needs certain precautions, if we like to stay in the framework of one-particle quantum mechanical Hamiltonian approach and the scattering theory.

We discuss the construction of ourJCL-model in Sections 2.2-2.7. For simplicity, we choose for the leads Hamiltonians the one-particle discrete Schrödinger operators with constant one-site (electric) potentials on each of leads. Notice that these Hamil- tonians are one band bounded self-adjoint operators. The advantage is that one can easily adjust the leads band spectra positions (and consequently the dot-leads transmission coefficients) shifting them with respect to the two-point quantum dot spectrum by varying the one-site electric potentials (voltage). In Section 2.5 we show that the our model fits into framework oftrace-class scattering and in Section 2.7 we verify the important property that the coupled Hamiltonian has no singular continuous spectrum.

Our main tool is an abstract Landauer-Büttiker-type formula applied in Sections 3.1 and 3.2 to the case of theJCL-model. Note that this abstract formula allows to calculate not only the electron current but also fluxes for other quantities, such as photon or energy/entropy currents. In particular, we calculate the outgoing flux of photons induced by electric current via leads. This corresponds to alight-emittingdevice. We also found that pumping the JCL quantum dot by photon flux from resonator may in- duce current of fermions into leads. This reversing imitates a quantumlight-absorbing cell device. These are the main properties of our model and the main application of the Landauer-Büttiker-type formula of Sections 3.1 and 3.2. They are presented in Sections 4 and 5, where we distinguishcontact-inducedandphoton-inducedfermion currents.

To describe the results of Sections 4 and 5 note that in our setup the sample Hamil- tonian is atwo-levelquantum dotdecoupledfrom the one-mode resonator. Then the unperturbed HamiltonianH0 describes is a collection of four totally decoupled subsystems: the sample, the resonator and the two leads. The perturbed HamiltonianHis

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a fully coupled system and the feature of our model is that it istotally(i.e. including the leads)embeddedinto the external electromagnetic field of resonator. This allows a systematic application of the abstract Landauer-Büttiker-type formula, c.f. Sections 3.1 and 3.2.

As we see there is a variety of possibilities to switch on interactions between subsystems, i.e. to produce intermediate Hamiltonians. We distinguish the following two of them:

(a) First to switch on the coupling between sample and resonator: the standard JC modelHJC, see e.g. [16]. Then to connect it to leads, which gives the Hamilto- nianHJCL:=Hof the fully coupled system.

(b) First to couple the sample to leads: the corresponding Hamiltonian HSL is a standard “Black Box”SL-model for free-fermion current, see [1], [4]. Then to embed it into resonator and to couple the sample with electromagnetic field by theJC-interaction. This again produces ourJCL-model withHJCL=H.

Similar to theSL-model{HSL, H0}, it turns out that theJCL-model also fits into the framework of the abstract Landauer-Büttiker formula, and in particular, is a trace- class scattering system{HJCL = H, HSL}. The current in theSL-model is called thecontact-inducedcurrentJ_el^c. It was a subject of numerous papers, see e.g. [1, 5], or [4] and references quoted there. Note that the currentJel is due to the difference of electro-chemical potentials between two leads, but it may bezeroeven if this difference is not null [12, 13].

The fermion current in the JCL-model, takes into account the effect of the electron-photon interaction under the assumption that the leads are already coupled.

It is called thephoton-inducedcomponentJ_el^ph of the total current. Up to our knowl- edge the present paper is the first, where it is studied rigorously. We show that the total free-fermion currentJ in theJCL-model decomposes into a sum of the contact- and the photon-induced currents:Jel:=J_el^c +J_el^ph. An extremal case is, when the contact- induced current is zero, but the photon-induced component is not, c.f. Section 5.1. In this case the flux of photonsJphout of the quantum dot (sample) is also non-zero, i.e.

the dot serves as the light emitting device, c.f. Section 5.2. In general theJph6= 0only when the photon-induced componentJ_el^ph6= 0.

In this paper we derive explicit formulas for these currents in the following three cases which are important for the understanding of theJCL-model:

(i) The electro-chemical potentials of fermions in the left and right leads are equal.

Note that in this case the (contact-induced) current in theJCL-model is zero.

(ii) The spectrum of the left and right lead Hamiltonians do not overlap. Again, in this case the contact-induced electron currentJ_el^c of the current in theJCL- model is zero, and only the photon-induced electron currentJ_el^ph of the total current is possible.

(iii) The leads are coupled to the Jaynes-Cummings model such that left and right leads interact only by virtue of the photon interaction in the Jaynes-Cummings model. Then the contact-induced electron currentJ_el^c is also zero.

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For these cases we find that the photon induced electron currentJ_α,el^ph entering the left (α=l) or right (α=r) lead is given by

J_α,el^ph =− X

m,n∈N₀ κ∈{l,r}

e 2π

Z

R

dλ bσ_n^ph

κmα(λ)×

ρ^ph(n)fF D(λ−µα−nω)−ρ^ph(m)fF D(λ−µκ−mω) . where bσ_n^ph

κmα(λ) ≥ 0 is a partial scattering cross-section between the left channel withm-photons and theκ-channel withn-photons at energyλ ∈ R. Bye > 0 the magnitude of the electron charge is denoted. The photon current is given by

Jph= X

m,n∈N₀ α,κ∈{l,r}

(n−m)ρ^ph(m) 1 2π

Z

R

dλ fF D(λ−µα−mω)bσ_n^ph

κmα(λ).

Both formulas become simpler if it is assumed that theJCL-model is time reversible symmetric. In this case we get

J_l,el^ph =− X

m,n∈N₀

e 2π

Z

R

dλ σb^ph_n_r_m_l(λ)×

ρ^ph(n)fF D(λ−µl−nω)−ρ^ph(m)fF D(λ−µr−mω) , and

Jph= X

m,n∈N₀,n>m κ,α∈{l,r}

1 2π

Z

R

dλ bσ_n^ph

κmα(λ)×

(n−m) ρ^ph(m)fF D(λ−µα−mω)−ρ^ph(n)fF D(λ−µκ−nω) . It turns out that choosing the parameters of the model in an suitable manner one gets either a photon emitting or a photon absorbing system. HenceJCL-model can be used either as a light emission device or as a light-cell. Proofs of explicit formulas for fermion and photon currentsJ_l,el^ph ,Jphis the contents of Sections 4 and 5.

Note that theJCL-model is calledmirror symmetricif (roughly speaking) one can interchange left and right leads and theJCL-model remains unchanged. In Section 5 we discuss a surprising example of a mirror symmetricJCL-model such that the free-fermion current iszerobut the model is photon emitting. This peculiarity is due to a specific choice of thephoton-fermioninteraction in our model.

2 Jaynes-Cummings quantum dot coupled to leads

2.1 Jaynes-Cummings model

The starting point for construction of ourJCL-model is the quantum optics Jaynes- Cummings HamiltonianH^JC. Its simplest version is atwo-levelsystem (quantum dot)

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with the energy spacingε, defined by HamiltonianhS on the Hilbert spaceh_S =C², see e.g. [16]. It is assumed that this system is “open” and interacts with the one-mode ω photon resonator with Hamiltonianh^ph.

Since mathematicallyh^phcoincides with quantum harmonic oscillator, the Hilbert space of the resonator is the boson Fock spaceh^ph=F+(C)overCand

h^ph=ω b^∗b . (2.1)

Hereb^∗andbare verifying the Canonical Commutation Relations (CCR) creation and annihilation operators with domains in F₊(C)≃ℓ²(N₀). Operator (2.1) is self-adjoint on its domain

dom(h^ph) = (

(k0, k1, k2, . . .)∈ℓ²(N₀) : X

n∈N0

n²|kn|²<∞ )

.

Note that canonical basis{φn:= (0,0, . . . , kn= 1,0, . . .)}ⁿ∈N₀inℓ²(N₀)consists of eigenvectors of operator (2.1):h^phφn=nω φn.

To model the two-levelsystem with the energy spacing ε, one fixes in C² two ortho-normal vectors{e^S₀, e^S₁}, for example

e^S₀ :=

0 1

and e^S₁ :=

1 0

, (2.2)

which are eigenvectors of HamiltonianhS with eigenvalues{λ^S₀ = 0, λ^S₁ = ε}. To this end we put

hS:=ε 1 0

0 0

, (2.3)

and we introduce twoladderoperators:

σ⁺:=

0 1 0 0

, σ⁻:=

0 0 1 0

. (2.4)

Then one getshS =ε σ⁺σ⁻as well as

e^S₁ =σ⁺e^S₀ , e^S₀ =σ⁻e^S₁ and σ⁻e^S₀ = 0

0

. (2.5)

So, e^S₀ is the ground state of Hamiltonian hS. Note that non-interacting Jaynes- Cummings HamiltonianH₀^JC lives in the spaceH^JC = h_S ⊗h^ph = C²⊗F₊(C) and it is defined as thematrixoperator

H₀^JC :=hS⊗Ih^ph+IhS ⊗h^ph. (2.6) HereIh^ph denotes the unit operator in the Fock spaceh^ph, whereasIhS stays for the unit matrix in the spacehS.

With operators (2.4) the interactionVSbbetween quantum dot and photons (bosons) in the resonator is defined (in the rotating-wave approximation [16]) by the operator

VSb:=gSb(σ⁺⊗b+σ⁻⊗b^∗). (2.7)

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Operators (2.6) and (2.7) define the Jaynes-Cummings model Hamiltonian

HJC :=H₀^JC+VSb, (2.8)

which is self-adjoint operator on the common domain dom(H₀^JC)∩dom(VSb). The standard interpretation ofHJC is that (2.8) describes an “open” two-level system interacting with external one-mode electromagnetic field [16].

Since the one-mode resonator is able to absorbinfinitelymany bosons this interpretation sounds reasonable, but one can see that the spectrumσ(H^JC)of the Jaynes- Cummings model isdiscrete. To this end note that the so-called number operator

N_JC:=σ⁺σ⁻⊗Ih^ph+Ih_S ⊗b^∗b commutes withHJC. Then, since for anyn≥0

H^JC_n>0:={ζ0e^S₀ ⊗φn+ζ1e^S₁ ⊗φn−1}^ζ^0,1∈C, H^JC_n=0:={ζ0e^S₀ ⊗φ0}^ζ⁰∈C, (2.9) are eigenspaces of operatorNJC, they reduceHJC, i.e. HJC : H^JC_n → H^JC_n . Note thatH^JC = L

n≥0H^JC_n , where each H^JC_n is invariant subspace of operator (2.8).

Therefore, it has the representation HJC = M

n∈N0

H_JC⁽ⁿ⁾, n >1, H_JC⁽⁰⁾ = 0. (2.10)

Here operatorsH_JC⁽ⁿ⁾are the restrictions ofHJC, which act in eachH^JC_n as

H_JC⁽ⁿ⁾(ζ0e^S₀ ⊗φn+ζ1e^S₁ ⊗φn−1) = (2.11) [ζ0nω+ζ1gSb√

n]e^S0 ⊗φn+ [ζ1(ε+ (n−1)ω) +ζ0gSb√

n]e^S1 ⊗φn−1. Hence, the spectrumσ(HJC) = S

n≥0σ(H_JC⁽ⁿ⁾). By virtue of (2.11) the spectrum σ(H_JC⁽ⁿ⁾)is defined forn≥1by eigenvaluesE(n)of two-by-two matrixHb_JC⁽ⁿ⁾acting on the coefficient space{ζ0, ζ1}:

Hb_JC⁽ⁿ⁾ ζ1

ζ0

=

ε+ (n−1)ω gSb√n gSb√n nω

ζ1

ζ0

=E(n) ζ1

ζ0

. (2.12)

Then (2.10) and (2.12) imply that the spectrum of the Jaynes-Cummings model Hamil- tonianHJC ispure point:

σ(HJC) =σp.p.(HJC) = (2.13)

{0} ∪ [

n∈N

nω+1

2(ε−ω)± q

(ε−ω)²/4 +g²_Sbn

.

This property is evidently persists for any system HamiltonianhS with discrete spectrum and linear interaction (2.7) with a finite mode photon resonator [16].

We resume the above observations concerning the Jaynes-Cummings model, which is our starting point, by following remarks:

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(a) The standard Hamiltonian (2.8) describes instead of flux only oscillations of photons between resonator and quantum dot, i.e. the systemhS is not “open”

enough.

(b) Since one our aim is to model alight-emittingdevice, the systemhS needs an externalsource of energy to pump it into dot, which then be transformed by interaction (2.7) into the outgoingphoton currentpumping the resonator.

(c) To reach this aim we extend the standard Jaynes-Cummings model to ourJCL- model by attaching to the quantum dothS (2.3) twoleads, which are (infinite) reservoirs offree fermions. Manipulating with electro-chemicalpotentials of fermions in these reservoirs we can force one of them to inject fermions in the quantum dot, whereas another one to absorb the fermions out the quantum dot with the same rate. This current of fermions throughout the dot would pump it and produce the photon current according scenario (b).

(d) The most subtle point is to invent aleads-dotinteractionVlS, which ensures the above mechanism and which is simple enough that one still be able to treat this JCL-model using our extension of the Landauer-Büttiker formalism.

2.2 The JCL-model

First let us make some general remarks and formulate certain conditions indispensable when one follows the modeling (d).

(1) Note that since the Landauer-Büttiker formalism [13] is essentially a scattering theory on a contact between two subsystems, it is developed only on a

“one-particle” level. This allows to study with this formalism only ideal (non- interacting) many-body systems. This condition we impose on many-body fermion systems (electrons) in two leads. Thus, only direct interaction between different components of the system: dot-photonsVSb and electron-dotVlS are allowed.

(2) It is well-known that fermion reservoirs are technically simpler to treat then boson ones [13]. Moreover, in the framework of our model it is also very natural since we study electric current although produced by “non-interacting electrons”.

So, below we use fermions/electrons as synonymous.

(3) In spite of precautions formulated above, the first difficulty to consider an ideal many-body system interacting with quantized electromagnetic field (photons) is inducedindirect interaction. If electrons can emit and absorb photons, it is possible for one electron to emit a photon that another electron absorbs, thus creating the indirect photon-mediated electron-electron interaction. This interaction makes impossible to develop the Landauer-Büttiker formula, which requires non-interacting framework.

Assumption 2.1 To solve this difficulty we forbid in our model the photon-mediated interaction. To this end we suppose that every electron (in leads and in dot) interacts

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with itsowndistinct copy of the electromagnetic field. So, to consider electrons to- gether with its photon fields as non-interacting “composed particles”, which allows to apply the Landauer-Büttiker approach. Formally it corresponds to the “one-electron”

Hilbert spaceh^el⊗h^ph, whereh^phis the Hilbert space of the individual photon field.

The fermion description of composed-particlesh^el⊗h^ph corresponds to the antisym- metric Fock spaceF₋(h^el⊗h^ph).

The composed-particle assumption 2.1 allows us to use the Landauer-Büttiker formalism developed for ideal many-body fermion systems. Now we come closer to the formal description of our JCL-model with two (infinite) leads and a one-mode quantum resonator.

Recall that the Hilbert space of the Jaynes-Cummings Hamiltonian with two energy levels is H^JC = C² ⊗F+(C). The boson Fock space is constructed from a one- dimensional Hilbert space since we consider only photons of a single fixed frequency.

We model the electrons in the leads as free fermions living on a discrete semi-infinite lattices. Thus

hêl =ℓ²(N)⊕C²⊕ℓ²(N) =hêl_l ⊕h_S⊕hêl_r (2.14) is the one-particle Hilbert space for electrons and for the dot. Here,hêl_α,α∈ {l, r}, are the Hilbert spaces of theleftrespectivelyrightlead andh_S =C²is the Hilbert space of the quantum dot. We denote by

{δ_n^α}ⁿ∈N, {δ^S_n}¹j=0

the canonical basis consisting of individual lattice sites ofh^el_α,α∈ {l, r}, and ofh_S, respectively. With the Hilbert space for photons,h^ph =F₊(C) ≃ℓ²(N₀), we define the Hilbert space of thefullsystem, i.e. quantum dot with leads and with the photon field, as

H=h^el⊗h^ph= ℓ²(N)⊕C²⊕ℓ²(N)

⊗ℓ²(N₀). (2.15) Remark 2.2 Note that the structure of full space (2.15) takes into account the condition 2.1 and produces composed fermions via the last tensor product. It also manifests that electronsin the dotas well as thosein the leadsare composed with photons. This makes difference with the picture imposed by the the Jaynes-Cummings model, when only dotis composed with photons:

H=ℓ²(N)⊕C²⊗ℓ²(N₀)⊕ℓ²(N) , H^JC=C²⊗ℓ²(N₀), (2.16) see (2.6), (2.7) and (2.8), whereH^JC = h_S ⊗h^ph. The next step is a choice of interactions between subsystems: dot-resonator-leads.

According to (2.14) the decoupled leads-dot Hamiltonian is the matrix operator h^el₀ =



h^el_l 0 0 0 hS 0 0 0 h^el_r



 on u=



ul

uS

ur



 , {uα∈ℓ²(N)}α∈{l,r}, uS ∈C²,

whereh^el_α =−∆^D+vαwith a constant potential biasvα∈R,α∈ {l, r}, andhS can be any self-adjoint two-by-two matrix with eigenvalues{λ^S₀, λ^S₁ :=λ^S₀ +ε},ε > 0,

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and eigenvectors{e^S₀, e^S₁}, cf (2.3). Here,∆^Ddenotes the discrete Laplacian onℓ²(N) with homogeneous Dirichlet boundary conditions given by

(∆^Df)(x) := f(x+ 1)−2f(x) +f(x−1), x∈N, dom(∆^D) := {f∈ℓ²(N₀) :f(0) := 0},

which is obviously a bounded self-adjoint operator. Notice thatσ(∆^D) = [0,4].

We define the lead-dot interactionfor couplinggel ∈ Rby the matrix operator acting in (2.14) as

vel=gel



 0 h·, δ0^Siδ^l1 0 h·, δ₁^liδ₀^S 0 h·, δ^r₁iδ^S₁

0 h·, δ1^Siδ^r1 0



 , (2.17)

where non-trivial off-diagonal entries are projection operators in the Hilbert space (2.14) with the scalar productu, v 7→ hu, viforu, v ∈ h^el. Here{δ₀^S, δ₁^S}is ortho- normal basis inh^el_S, which in general may be different from{e^S₀, e^S₁}. Hence, interaction (2.17) describes quantumtunnelingbetween leads and the dot via contact sites of the leads, which are supports ofδ₁^l andδ^r₁.

Then Hamiltonian for the system of interacting leads and dot we define ash^el :=

hêl₀ +vel. Here bothhêl₀ andhêlare bounded self-adjoint operators onhêl.

Recall that photon Hamiltonian in the one-mode resonator is defined by operator h^ph = ωb^∗bwith domain in the Fock spaceF+(C) ≃ℓ²(N₀), (2.1). We denote the canonical basis inℓ²(N₀)by{Υn}ⁿ∈N₀. Then for the spectrum ofh^phone obviously gets

σ(h^ph) =σpp(h^ph) = [

n∈N₀

{nω}. (2.18)

We introduce the following decoupled HamiltonianH0, which describes the system when the leads are decoupled from the quantum dot and the electron does not interact with the photon field.

H0:=H₀^el+H^ph, (2.19)

where

H₀êl:=hêl₀ ⊗Ih^ph and H^ph:=Ihêl⊗h^ph.

The operatorH0is self-adjoint on dom(H0) =dom(Ihêl⊗h^ph). Recall thathêl₀ and h^ph are bounded self-adjoint operators. HenceH₀êl andHêl are semi-bounded from below which yields thatH0is semi-bounded from below.

The interaction of the photons and the electrons in the quantum dot is given by the coupling of the dipole moment of the electrons to the electromagnetic field in the rotating wave approximation. Namely,

Vph=gph (·, e^S₀)e^S₁ ⊗b+ (·, e^S₁)e^S₀ ⊗b^∗

(2.20) for some coupling constantgph∈R. The total Hamiltonian is given by

H:=H^el+H^ph+Vph=H0+Vel+Vph, (2.21)

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whereH^el:=h^el⊗Ih^phandVel:=vel⊗Ih^ph.

In the following we callS={H, H0}the Jaynes-Cummings-leads system, in short JCL-model, which we are going to analyze. In particular, we are interested in the electron and photon currents for that system. The analysis will be based on the abstract Landauer-Büttiker formula, cf. [1, 13].

Lemma 2.3 H is bounded from below self-adjoint such thatdom(H) =dom(H0).

Proof.Letc≥2. Then

kbΥnk²≤ kb^∗Υnk²=n+ 1≤c⁻¹n²+c, n∈N₀. Consider elementsf ∈h_S⊗h^ph∩dom(Ih^el⊗h^ph)with

f =X

j,l

βjlej⊗Υl, j∈ {0,1}, l∈N₀,

which are dense inH^JC :=h^el_S⊗h^ph. Thenkfk²=P

j,l|βjl|²andk(Ih^el⊗b^∗b)fk²= P

j,l=1|βjl|²l². We obtain

k((·, e^S₁)e^S₀ ⊗b)fk²≤X

j,l

|βjl|²kbΥlk²≤ X

j,l

|βjl|²(c⁻¹l²+c) =c⁻¹k(Ih^el⊗b^∗b)fk²+ckfk²

Similarly,

k((·, e^S₁)e^S₀ ⊗b^∗)fk²≤c⁻¹k(Ih^el⊗b^∗b)fk²+ckfk².

Ifc ≥2is large enough, then we obtain thatVph is dominated byH^ph with relative bound less than one. HenceH is self-adjoint and dom(H0) = dom(H). SinceH₀^el andVel are bounded andH^phis self-adjoint and bounded from below, it follows that H =H₀^el+H^ph+Vel+Vphis bounded from below [17, Thm. V.4.1].

2.3 Time reversible symmetric systems

A system described by the HamiltonianH is called time reversible symmetric if there is a conjugationΓdefined onHsuch thatΓH =HΓ. Recall thatΓis a conjugation if the conditionsΓ²=Iand(Γf,Γg) = (f, g),f, g∈H.

Leth^ph_n ,n∈N₀, the subspace spanned by the eigenvectorΥninh^ph. We set H_n_α:=h^el_α⊗h^ph_n , n∈N₀, α∈ {l, r}. (2.22) Notice that

H= M

n∈N₀,α∈{l,r}

H_n_α

A new model for quantum dot light emitting-absorbing devices: proofs and supplements

HAL Id: hal-00919509

https://hal.archives-ouvertes.fr/hal-00919509

A new model for quantum dot light emitting-absorbing devices: proofs and supplements

H. Neidhardt, L. Wilhelm, V.A. Zagrebnov

To cite this version:

A new model for quantum dot light emitting-absorbing devices

Contents

1 Introduction

2 Jaynes-Cummings quantum dot coupled to leads