Self-induction and magnetic effects in electron transport through a photon cavity

Vidar Gudmundsson,1, ∗ Nzar Rauf Abdullah,2, 3 Chi-Shung

Tang,4, † Andrei Manolescu,5, ‡ and Valeriu Moldoveanu6, §

1Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Iceland2Physics Department, College of Science, University of Sulaimani, Kurdistan Region, Iraq

3Computer Engineering Department, College of Engineering,Komar University of Science and Technology, Sulaimani 46001, Kurdistan Region, Iraq

4Department of Mechanical Engineering, National United University, Miaoli 36003, Taiwan5School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland

6National Institute of Materials Physics, PO Box MG-7, Bucharest-Magurele, Romania

We explore higher order dynamical effects in the transport through a two-dimensional nanoscaleelectron system embedded in a three-dimensional far-infrared photon cavity. The nanoscale systemis considered to be a short quantum wire with a single circular quantum dot defined in a GaAsheterostructure. The whole system, the external leads and the central system are placed in a constantperpendicular magnetic field. The Coulomb interaction of the electrons, the para- and diamagneticelectron-photon interactions are all treated by a numerically exact diagonalization using step-wisetruncations of the appropriate many-body Fock spaces. We focus on the difference in transportproperties between a description within an electric dipole approximation and a description includingall higher order terms in a single photon mode model. We find small effects mostly caused by anelectrical quadrupole and a magnetic dipole terms that depend strongly on the polarization of thecavity field with respect to the transport direction and the photon energy. When the polarizationis aligned along the transport direction we find indications of a weak self-induction that we analyzeand compare to the classical counterpart, and the self-energy contribution of high-order interactionterms to the states the electrons cascade through on their way through the system. Like expectedthe electron-photon interaction is well described in the dipole approximation when it is augmentedby the lowest order diamagnetic part for a nanoscale system in a cavity in an external magneticfield.

I. INTRODUCTION

The non-perturbative coupling of two-dimensional(2D) electrons in a magnetic field in a heterostructurewith high-quality-factor terahertz or far-infrared photonshas been achieved [1]. The coupling of electronic systemsin circuit QED planar microwave cavities to externalleads gives the hope that this will also be accomplishedfor the terahertz systems [2–6]. Several groups have mod-eled various aspects of the transport of electrons throughnanoscale systems placed in photon cavities [7–11], justto mention few. Like for most physical phenomena, dif-ferent modeling approaches have been applied rangingfrom non-equilibrium Green functions [12–14] to masterequations of various types [15–17]. The modeling effortshave not been straight forward, and some authors haveemphasized the role of the geometrical shape of the sys-tems [9], while other have explored how appropriate theuse of a single cavity mode is [18], with respect to causal-ity, or if relativistic corrections have to be taken into ac-count [19], besides the fundamental issues on the deriva-tion of an appropriate description when using a masterequation approach [20, 21].

∗ vidar@hi.is† cstang@nuu.edu.tw‡ manoles@ru.is§ valim@infim.ro

Various approaches have been used for the matter-photon interactions in different systems of spins, elec-trons, or atoms placed in a cavity. Naturally, most canbe referred back to an electrical dipole interaction, ex-pressed either in terms of a space integral over the innerproduct of the vector potential A and the charge cur-rent density j, or the integral of the inner product of theelectrical field and the position operator [22, 23]. Somemodels have included the much weaker diamagnetic in-teraction, that is often referred to as the A-square term[24–27].

Analytical or high-order numerical methods lead to re-sults of high accuracy within the electrical dipole interac-tion approximation, relying or not [28, 29] on the rotatingwave approximation [22, 23].

The dipole approximation is known to be valid if thewavelength of the electromagnetic field is usually muchlarger than the size of the matter system. Magnetic ef-fects in the interaction will only be evident as the varia-tion of the electromagnetic vector potential is taken intoaccount within the matter or the electron system. Theyare thus bound to be tiny in most cases.

Earlier, we have shown that high order transitions canbe important when describing the approach of systemsto a steady state in which otherwise low order forbiddentransitions play a role [30]. This has lead us to the ques-tions: What about the higher order electric and magneticterms of the para- and diamagnetic electron-photon in-teractions? Do they lead to effects, that among others,can be understood as self-induction in time-dependent

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electron transport through a system with a nonzero spa-tial extension in a photon cavity?

To explain our ideas, let us consider the following set-up: Initially, the external leads are coupled to an emptycentral system, i. e. neither an electron nor a photon arepresent, the initial system is in its vacuum state. Thecoupling to the external leads offers only electrons to thesystem, no photons. We consider an external photonreservoir coupled to the cavity to be at zero tempera-ture, so no photons enter the system from the photonreservoir. The electron-photon coupling in the centralsystems turns all many-body states into either pure pho-ton states, or electrons states dressed with cavity pho-tons. Thus, after the coupling to the leads the photonexpectation value in the cavity increases and can becomelarge depending on which dressed electron states are inthe bias window defined by the external leads. In this set-up the photon content of the central system will reach amaximum at some intermediate time and decrease as thesystem reaches a steady state.

In a classical system this emergence of an electromag-netic component in transport could lead to the consid-eration of inductive effects if higher order terms wereconsidered for the electron-photon interaction. For a“few-body” quantum system one expects the identifica-tion of such inductive effects probably to be obscured bythe quantum details of countably few transport paths ortransitions through the system. Comparison with clas-sical circuits is further hindered by the simplicity of theapproach to view few-body high frequency quantum cir-cuits using classical terms as “lumped elements”, withseparable capacitive and inductive elements.

In quantum electrodynamics (QED) all higher orderterms of an electron-photon interaction, describing pro-cesses during which an electron interacts with its ownelectromagnetic field, are included in the self-energy ofthe electron. Here, in a confined central system the dis-crete electron states all acquire a higher-order self-energythat depends on the geometry of both the electron statesand the photon field in the cavity in addition to otherproperties the states have.

The paper is organized as follows: Section II gives ashort description of the model with the time-independentproperties in the Subsection II A, but the time-dependenttransport in Subsection II B. Results are presented in Sec-tion III, with general quantities outlined in subsectionIII A, while differences between the models with differentelectron-photon interactions are displayed in SubsectionIII B. Conclusions are drawn in Section IV.

II. MODEL

We model a short quantum wire of length Lx = 180nm in the x-direction with a parabolic confinement withcharacteristic energy ~Ω0 = 2.0 meV in the y-direction.We assume a hard wall confinement at the ends in the x-direction and a constant external magnetic field B = Bz

with strength B = 1.0 T perpendicular to the two-dimensional quantum wire assumed to be formed in aGaAs heterostructure with effective mass m∗ = 0.067me

dielectric constant κe = 12.4, and g∗ = −0.44. Embed-ded in the short wire is a quantum dot potential as isshown in Fig. 1 The short quantum wire with the dot is

-4

-2

0

2

4

x/aw

-4-2

0 2

4

y/aw

-5

0

5

10

V (meV)

0

5

10

V (

meV

)

FIG. 1. The potential, V , defining the shape of the centralsystem, a circular quantum dot embedded in a short quan-tum wire. The length of the wire is 180 nm, extending fromx/aw ≈ −4.3 to x/aw ≈ +4.3, where aw ≈ 20.75 nm is theeffective magnetic length for the parabolic confinement in they-direction with characteristic energy ~Ωw = 2.0 meV. Thetiny gaps at either end of the short wire indicate the onset ofthe external leads with same confinement in the y-direction.

defined by the potential

V (x, y) =θ

(Lx2− |x|

)[1

2m∗Ω2

0y2 − eVg+ (1)

× Vd

2∑i=1

exp−β2x2 − β2(y − y0i)2

],

formed by two strongly overlapping Gaussian dips in or-der to get an almost circular shape in the parabolic con-finement. The parameters are Vd = −6.0 meV, β = 0.018nm−1, defining their depth and extent, y01 = −48 nm,and y02 = +48 nm defining their location. Vg is a plungergate voltage used to raise or lower certain states into thebias window. Here, it will be set to Vg = 2.47 meV.

A. Time-independent properties

The Hamiltonian of the central system is

HS =

∫d2rψ†(r)

π2

2m∗+ V (r)

ψ(r)

+HEM +HCoul +HZ

+1

c

∫d2r j(r) ·Aγ +

e2

2m∗c2

∫d2r ρ(r)A2

γ , (2)

3

in terms of the fermionic field operators ψ and ψ† for elec-trons, the probability density ρ = ψ†ψ, and the chargecurrent density j = −eψ†πψ + π∗ψ†ψ/(2m∗), withπ = (p+eAext/c), where Aext = (−By, 0, 0) is the vectorpotential defining the external constant magnetic field B.The external magnetic field together with the parabolicconfinement energy ~Ω0 define the effective confinementenergy ~Ωw = ~(ω2

c + Ω20)1/2 and the effective magnetic

length aw = (~/(m∗Ωw))1/2, where ωc = (eBext)/(m∗c)

is the cyclotron frequency. We will explore properties ofthe central system for B = 1.0 T, such that aw ≈ 20.75nm and ~Ωw ≈ 2.642 meV. HEM = ~ωa†a is the Hamil-tonian for the single-mode cavity with energy ~ω, HZ

is the Zeeman term for the electrons, and HCoul is theCoulomb interaction of the electrons with a kernel

VCoul(r− r′) =e2

κe√|r− r′|2 + η2c

, (3)

where a small regularization parameter ηc/aw = 3×10−7

has been used. The electron-photon interactions termsin the 3rd line of Eq. (2) are the paramagnetic- and thediamagnetic parts, respectively.

We model a rectangular photon-cavity with dimensionsac × bc × dc, using the Coulomb gauge for the quantizedvector potential Aγ for the single-mode photon field ofthe cavity. The electric field of the cavity photons isaligned to the transport in the x-direction (with the unitvector ex) in the TE011 mode, and perpendicular to it(defined by the unit vector ey) in the TE101 mode. Thegeneral configuration of the electric and magnetic com-ponents of the cavity field are schematically displayed inFig. 2 for the two linear polarizations. The quantized

x

y

z

B

E

BE

FIG. 2. A schematic diagram indicating the directions of theelectric (E, red) and the magnetic (B, blue) fields of the cav-ity photon field for y-polarization (upper) and x-polarization(lower). The black horizontal strip represents the plane of thetwo-dimensional electron gas.

vector potential for the two polarizations for the cavityphoton field is expressed as [30, 31]

Aγi(r) = eiAa+ a†

ui(z) (4)

where i = x or y labels the direction of the polarization

and

ux(z) = cos

(πy

bc

)cos

(πz

dc

)and,

uy(z) = cos

(πx

ac

)cos

(πz

dc

). (5)

a is the annihilation operator for a cavity photon, and a†

is the corresponding creation operator. The magnitude ofthe vector potential, A, and the electron-photon couplingconstant are related by gEM = eAΩwaw/c. For the cavitywith photon energy ~ω = 2.63 meV the ratio of the sizeof the cavity to the length of the central system, ac/Lxor bc/Lx, is 26.40, and for the energy ~ω = 0.98 meV it is70.7, so as expected the higher-order terms will be small.Simplistically, one might expect the higher photon energyleads to increased higher order effects, but this view willbe challenged below.

For an easier comparison of the electron-photon inter-action used in other models we use the vector potential(4) and the field operators for the electrons to rewrite theHamiltonian for the electron-photon interactions as [30]

He−EM =gEM

∑ij

d†idjgpija+ a† (6)

+g2EM

~Ωw

∑ij

d†idjgdij

[(a†a+

1

2

)+

1

2(a†a† + aa)

],

where di is the annihilation operator for an electron in

the single-particle state labeled by i, and d†i is the cor-responding creation operator. The definition of the cou-pling matrices (in the z = 0 plane) [30] in terms of thewavefunctions of the original single-electron basis |a〉is

gdiab = 〈a|ui(0)2|b〉, (7)

with i = x or y for the diamagnetic electron-photon in-teraction. If the size of the cavity would be assumed toapproach infinity (compared to Lx) ui(0) → 1 and thengdab → δa,b. The coupling matrix for the paramagneticinteraction is written symbolically as

gpiab =aw2~〈a|(ei · π) ui(0)|b〉, (8)

where the basis |a〉 consists of the original single-electron states of the wire-dot system with no interac-tion to the photons. As the counter rotating terms inthe electron-photon interactions are important [32–34],especially for the self-energy of the dressed states [30]we do not apply the rotating wave approximation. Theeigenstates of the central system described by the Hamil-tonian (6) are denoted by |µ).

If in Eqs. (7-8) the spatial variation of the vector po-tential of the photon field, Aγ(r) over the electron sys-tem is neglected, we have a kind of a dipole interaction,but including the diamagnetic electron-photon interac-tion. Note, that in this case we still use the exact numer-ical diagonalization that thus includes higher order terms

4

of the dipole origin into the self-energies of the dressedelectron many-body states. We will notice below that atleast the lowest order of the diamagnetic interaction isnecessary due to the strong external magnetic field hav-ing orbital effects.

B. Transport description

The external leads have the same parabolic confine-ment as the short quantum wire and are subjects to thesame perpendicular magnetic field. Their coupling to thecentral system at t = 0 is described by the Hamiltonian

HT = θ(t)∑il

∫dq(T lqic

†qldi + (T lqi)

∗d†i cql

). (9)

Electrons in the short wire are created (annihilated) by

the operators d†i (di), and in the leads by the operators

c†ql (cql). The quantum number q stands both for thecontinuous momenta in the leads and the appropriatesubband index. Due to the shape of the central system itis essential to account for the fact that states in the leadsand the central system couple differently well or badlydepending on their energy and the shape of their proba-bility densities. Thus, the coupling matrix T lqi is calcu-lated using the probability density of each single-electronstate of the lead l and the central electron system in thecontact region that is defined to extend approximatelyone aw into each subsystem [35–37]. See also AppendixA in Ref. [30]. The coupling of the external leads to the

central system has an overall strength g0gLRa3/2w = 0.101

meV, when the overall dimensionless coupling constantg0 = 1.0.

The time evolution of the central system under theinfluence of the reservoirs, the external leads and thephoton reservoir, is calculated using a Markovian mas-ter equation for the reduced density operator ρS in theLiouville space of transitions [38–40], that has been de-rived from a generalized master equation [41, 42] in amany-body Fock space by applying a Markovian approx-imation and a vectorization of the matrices [30, 43, 44].The coupling of the cavity to the external photon reser-voir is κ = 1.0 × 10−5 meV, and the average photonnumber in the reservoir is set by the parameter nR [45].

From the generalized master equation we derive themean current [35–37, 45] into the central system, IL, fromthe left lead (L), and the mean current out of the centralsystem, IR, into the right lead (R). For comparison withsimple classical circuit it would be convenient to considerthe mean net current through the central system. Dueto the capacitance and charging of the central systemthis quantity is not well defined, and we use the meancurrent, I = IL + IR, as a measure of it. Conveniently, inthe regime when IL = IR, IL is double this net currentthrough the system, but the net current into the centralsystem, Ii = IL − IR, vanishes.

In addition to calculating the mean current, I, we eval-uate the mean number of electrons in the central system,Ne, the mean number of photons Nγ , the mean value ofthe z-component of the total spin Sz, and the Reniy-2entropy of the central system [46–48]

S = −kB ln [Tr(ρ2S)]. (10)

The mean entropy (10) serves as a sensitive measure ofchanges in the central system, and we only use it for thatpurpose here.

Information about details in the numerical computa-tions is found in Appendix A.

III. RESULTS

We first concentrate on properties of the system withthe full electron-photon interactions. Then we analyzethe differences of these quantities between a system withthe full electron-photon interactions and a system wherethe dipole approxmation is considered.

A. Transport results for numerically exactone-mode electron-photon interaction

The external homogeneous magnetic field B = 1.0 T.We select the chemical potentials of the left and rightlead to be µL = 1.7 meV and µL = 1.4 meV defining abias window ∆µ of 0.3 meV. The plunger gate voltageVg is set such that −eVg = 2.47 meV. Further more wewill explore the transport properties of the system fortwo values of the cavity photon energy, ~ω = 0.98, and2.63 meV. The one-electron ground state of the quantumdot (and the central system) is a typical circular sym-metric state with vanishing angular momentum, whilethe next one-electron states in energy are typical circularring states with unit angular momenta quantum num-bers. In the Fock-Darwin energy spectrum for a circularsymmetric parabolically confined quantum dot in a mag-netic field the radial wavefunctions are the same for theM = ±1 states [49]. In our case they are very similar,but the higher once (in the bias window) look slightlyellipical due to the opening of the dot into the short wirein the x-direction. In our quantum dot the energy differ-ence between these two states is close to the lower photonfrequency, leading to a very special Rabi-splitting as willbe seen below. We select these circular states of a quan-tum dot in an external magnetic field to enhance possibleself-induction in the system. At the same time the one-electron ground state is in resonance with the first excitedone-electron state.

As Figure 3 shows the one electron ground state |1) and

its partner with opposite electron spin |2) (correspondingto the M = 0 states of the Fock-Darwin spectrum) arewell below the bias window. For the case of ~ω = 2.63meV only the two spin partners of the second excited one-particle states |6) and |7) (corresponding to the higher

5

-1

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0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60

0

1

2

3

4⟨s

z⟩,

⟨N

e⟩,

⟨N

γ⟩

Eµ (

meV

)

µ

γe

SzEµ1.41.7

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60

0

1

2

3

4

5

⟨sz⟩,

⟨N

e⟩,

⟨N

γ⟩

Eµ (

meV

)

µ

γe

SzEµ1.41.7

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60

0

1

2

3

4

⟨sz⟩,

⟨N

e⟩,

⟨N

γ⟩

Eµ (

meV

)

µ

γe

SzEµ1.41.7

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60

0

1

2

3

4

5

⟨sz⟩,

⟨N

e⟩,

⟨N

γ⟩

Eµ (

meV

)

µ

γe

SzEµ1.41.7

FIG. 3. The energy (E), the electron (e), the photon (γ), expectation values for fully interacting state |µ), together with itsvalue of Sz, for ~ω = 0.98 meV (left), ~ω = 2.63 meV (right), x-polarization (upper), and y-polarization (lower). The twohorizontal yellow lines indicate the chemical potentials of the left µL and right µR leads defining the bias window. gEM = 0.1meV, B = 1.0 T, −eVg = 2.47 meV, and Lx = 180 nm.

M = ±1 Fock Darwin states, |4) and |5) correspond tothe lower ones) are within the bias window, but there is aRabi-resonance between two spin partners of the groundstate and states slightly above the bias window, such thatthe lower Rabi branch is either of the spin partners |9)

or |10), and the upper branch is |12) or |13). This canbest be seen by the mean photon content of these dressedone-electron states which is close to half a photon.

In this case the photon content of the dressed one-electron states in the bias window is small. When thephoton energy is ~ω = 0.98 meV there is, on the otherhand, a Rabi-resonance between the second and the thirdexcited one-electron states in the system on top of a res-onance between the ground state and the first excitedstate. We thus end up with six one-electron states inthe bias window, all with a noninteger photon content.For this case one would expect a faster charging of thesystem and it should likewise approach the steady stateas most transitions needed would be photon aided. Wealso notice from Fig. 3 that in the case of ~ω = 0.98 meV(left subfigures) the photon content of the dressed states

is more dependent on the polarization of the photon fieldthan for ~ω = 2.63 meV (right subfigures).

Indeed, in Fig. 4 we see this photon enhancement ofall transitions leading to a faster charge for ~ω = 0.98meV (upper) than for ~ω = 2.63 meV (lower). This canbe verified in Fig. 4 by the difference in the mean pho-ton content for either case with different photon energy.We notice that the steady states for these two cases aredifferent as can be seen by their difference in the meanspin z-component. We come back to this below.

Fig. 5 displays the mean currents for either photonenergy, the current from the left lead into the centralsystem, IL labeled (L), the current from the central sys-tem into the right lead, IR labeled (R), and the currentI = IL + IR labeled (T). As nR = 0 and there are sev-eral one-electron states below the bias window and thelowest two-electron states are above it, the system is ina Coulomb blockade in the steady state. If nR = 1, orhigher, the current will not vanish in the steady state.

Importantly, there is a time interval just when the sys-tem is getting fully charged where the left and right cur-rents are almost the same. During this time interval the

6

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0

0.5

1

1.5

2

2.5

3

1 10 100 1000 10000 100000 1x106

1x107

1x108

⟨Ne⟩,

⟨N

γ⟩, S

/kB

t (ps)

Ne

Nγ

tr(ρ)

S/kB

Sz

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1 10 100 1000 10000 100000 1x106

1x107

1x108

⟨Ne⟩,

⟨N

γ⟩, S

/kB

t (ps)

Ne

Nγ

tr(ρ)

S/kB

Sz

FIG. 4. The mean electron Ne and photon Nγ number, thetrace of the reduced density matrix, the mean entropy S, andz-component of the spin for photon energy ~ω = 0.98 meV(upper), and ~ω = 2.63 meV (lower). gEM = 0.1 meV, B =1.0 T, nR = 0, κ = 1.0 × 10−5 meV, −eVg = 2.47 meV,

Lx = 180 nm, and g0gLRa3/2w = 0.101 meV.

system is approaching the steady state through radiativeor nonradiative transitions [9] as can be seen by the timedependent occupations seen in Fig. 6 By comparing theinformation in Fig. 6 and 3 we see that on the way to thesteady state the system probes the two-electron groundstate, but then relaxes finally into a combination of one-electron states made up of contributions from both spincomponents of the one-electron ground state. The routetaken to the steady state depends strongly on the photonenergy as the electron-photon interaction makes differenttransitions available to the system.

The results shown in Figs. 4 and 5 allow us to estimatetwo time scales relevant for classical circuits. First, usingthe size of the bias window, ∆µ = 0.3 meV, and theaverage current outside the steady state, I = 0.1 nA, wearrive at the contact resistance R ≈ 3 MΩ. Secondly,from the geometry of the electron system we estimateits coefficient of inductance to be L ≈ 36 fH. Togetherthese give the time scale of the inductance to be τL ≈1.2×10−20 s = 1.2×10−8 ps. This shows plainly that we

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1 10 100 1000 10000 100000 1x106

1x107

1x108

Curr

ent (n

A)

t (ps)

L

R

T

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

1 10 100 1000 10000 100000 1x106

1x107

1x108

Curr

ent (n

A)

t (ps)

L

R

T

FIG. 5. The mean electron current from the left lead into thecentral system IL (L), from the central system to the rightlead IR (R), and the current I = IL + IR (T), for photonenergy ~ω = 0.98 meV (upper), and ~ω = 2.63 meV (lower).gEM = 0.1 meV, B = 1.0 T, nR = 0, κ = 1.0 × 10−5 meV,

−eVg = 2.47 meV, Lx = 180 nm, and g0gLRa3/2w = 0.101

meV.

can not expect to observe this τL in the present system.

From the charging time τC ≈ 106 ps (see Fig. 6) andthe contact resistance we estimate the capacity of theelectron system to be approximately C ≈ 1 pf. If oursystem could be compared to an RCL circuit this allowsus to extract the resonance frequency of oscillations tobe ω0 = 1/

√LC ≈ 5.3 (ps)−1, or their energy ~ω0 ≈ 3.5

meV. Interestingly, this resonance energy is inside thepart of the energy spectrum we are exploring, but as weare dealing with few particles in a system with discreteenergy levels we can not expect to find this classical res-onance energy in our quantum system.

Looking at Fig. 4 and 6 one has to wonder why thereare no Rabi-oscillations seen in the density or the occupa-tion of states like has been seen for different dot systemsat lower magnetic field [30, 31]. We would expect them,at least for the case of the photon energy ~ω = 0.98 meV,when there are Rabi-split states in the bias window. Theanswer lies in the combination of high external magnetic

7

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0.3

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0.6

0.7

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1

1 10 100 1000 10000 100000 1x106

1x107

1x108

Occupation

t (ps)

|01)|02)|03)|04)|05)|06)|07)|08)

|09)|10)|11)|12)|13)|14)|15)|16)

|17)|18)|19)|20)|21)|22)|23)|24)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000 100000 1x106

1x107

1x108

Occupation

t (ps)

|01)|02)|03)|04)|05)|06)|07)|08)

|09)|10)|11)|12)|13)|14)|15)|16)

|17)|18)|19)|20)|21)|22)|23)|24)

FIG. 6. The time-dependent occupation of selected many-body states |µ) for ~ω = 0.98 meV (upper), and ~ω = 2.63meV (lower). gEM = 0.1 meV, B = 1.0 T, nR = 0, −eVg =

2.47 meV, and Lx = 180 nm, and g0gLRa3/2w = 0.101 meV.

field (B = 1.0 T), and the simple circular dot potentialin the central system. Like, before we draw an analogywith the Fock-Darwin energy spectrum [49], mentionedabove, the lowest Rabi-resonance is between states withopposite angular momenta, but having the same radialwavefunctions. The resonance thus leads to circular os-cillations in the local current density in the central sys-tem. These divergence-free oscillations can not lead tooscillations in charge density. We have thus “transverse”Rabi-oscillations that will be clear in the current, butnot in the density. This behavior will even be clear on ahigher resolution scale introduced below.

This resonance between the lowest lying ring-formedstates corresponding to the lowest M = ±1 Fock-Darwinstates would not appear in the lowest order dipole ap-proximation, and at the same time there is a resonancebetween the one-electron ground state and the lower ofthe M = ±1 states that would be enabled in a dipoleapproximation. The Rabi-resonance caused by the dia-magnetic part of the electron-photon interactions occurseven for constant vector potential inside the electron sys-

tem, as it is facilitated by the external constant magneticfield, that leads to divergent-free rotating currents in thesystem [25].

B. Exploring dynamical differences

All the results described above have been calculatedusing a numerically exact diagonalization for the single-mode electron-photon interactions. We now look for dy-namic effects of the higher order terms in the results.To this end we assume the spatial variation of the vec-tor potential Aγ to vanish inside the electron system it-self. One might call that a dipole approximation, butwithin it we also keep the diamagnetic interaction, whichis usually not done in a dipole approximation, and weuse the exact numerical diagonalization for both cases.Within this approximation we calculate the same quan-tities as in subsection III A and then compare the twosets of results by defining ∆Ne = N exact

e − Ndipolee ,

∆Nγ = N exactγ −Ndipole

γ , and ∆I = Iexact − Idipole. Fig.7 displays ∆Ne and ∆Nγ in the central system for thewhole time range from the transient regime to the steadystate. The figure shows results for both photon energiesand polarizations, and for the photon energy ~ω = 2.63meV it shows results for three different values of the over-all system-lead coupling coefficient g0. We notice that es-pecially in the intermediate time range (ITR) the differ-ence in the electron and photon number becomes largerfor the x-polarization than the y-polarization.

Three of the subfigures of Fig. 7 show irregular oscilla-tions looking like noise. This is not the case here. Whena higher resolution is used for a narrower time interval (aswill be done below) the irregular oscillations are replacedby very regular Rabi oscillations.

In the lower panels of Fig. 7 for photon frequency~ = 2.63 meV, when no photon replicas or other elec-tron states with a high expectation value of photons isin the bias window we see that in the ITR both the ex-pectation values for electrons and photons is reduced forthe x-polarized cavity field. The time variation of both∆Ne and ∆Nγ can be correlated with changes in theoccupation of the states of the central system seen inthe lower panel of Fig. 6 if one considers at the sametime the change in the self-energies of the correspondingstates due to the higher-order electron-photon interac-tion shown in lower panel of Fig. 15 in Appendix B, andthe changes in the occupation of the states induced bythe same terms seen in the lower panel of Fig. 14 in thesame Appendix. The photon energy in the lower panelsin Fig. 7 ~ω = 2.63 meV is just below the effective con-finement energy of the short quantum wire, ~Ωw = 2.642meV favoring polarization of the electron charge densityin the x-direction, but at the same time allowing for asmall polarization in the y-direction. In the upper panelsthe photon energy is ~ = 0.98 meV, so polarization of thecharge density in the y-direction can only be very tiny,but as there is now a multiple Rabi resonance with the

8

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e

t (ps)

g0=0.9, x-polg0=1.0, x-polg0=1.1, x-polg0=0.9, y-polg0=1.0, y-polg0=1.1, y-pol

-5x10-6

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0

10 100 1000 10000 100000 1x106

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1x108

∆N

γ

t (ps)

g0=1.10, x-polg0=1.00, x-polg0=0.90, x-plog0=1.10, y-polg0=1.00, y-polg0=0.90, y-pol

FIG. 7. The differences in the mean electron ∆Ne (left) and photon ∆Nγ (right) numbers for a system with an exact, and asystem with a dipole approximation for the electron-photon interactions for a single cavity mode for ~ = 0.98 meV (upper),and for ~ = 2.63 meV (lower) with g0 = 1.0. nR = 0, gEM = 0.1 meV, B = 1.0 T, κ = 1.0× 10−5 meV, −eVg = 2.47 meV, andLx = 180 nm.

states in the bias window, the system can effectively bepolarized in the x-direction. Correspondingly, the timevariation of both ∆Ne and ∆Nγ in the upper panels ofFig. 7 is tiny for the y-polarized cavity field, but for thex-polarized field the effects are larger and ∆Nγ showsclear signs of Rabi-oscillations.

For the lower photon energy, ~ω = 0.98 meV, we see inAppendix B in the upper panel of Fig. 15 that, indeed,the self-energies of the relevant states due to the higher-order terms in the electron-photon interactions are van-ishingly small for the y-polarized cavity field, but for thex-polarized field the states in the bias window acquire noor relatively large positive or negative self-energy contri-bution. The early in time photon active transitions seenin the differences in the occupation in the upper panel ofFig. 14 in Appendix B lead both to the strong responsein ∆Ne in the upper left panel of Fig. 7 for the early andthe intermediate time.

All the results in Fig. 7 are for the case when no pho-tons flow from the photon reservoir to the cavity, i. e.the mean value of photons in the cavity is nR = 0,and as the there are several one-electron states below

the bias window the system ends up in a steady stateof a Coulomb-blockade. Figure 8 shows how ∆Ne looksfor a system with nR = 1, that does not end up in aCoulomb-blockade, but the current through the systemis maintained by the photons the reservoir supplies. Ini-tially, the ∆Ne shows similar behavior as in the upper leftpanel of Fig. 7, but in the late intermediate time thereare changes due to the nonvanishing current through thesystem.

Commonly, in classical circuits the current is analyzedin order to find traits of self-induction. In Fig. 9 we dis-play the change in the current I = IL + IR for the higherphoton energy, ~ω = 2.63 meV, for both polarizations ofthe cavity field, and four values of the overall couplingto the external leads g0. For the x-polarized cavity field(upper panel) the difference in the current caused by thehigher-order electron-photon interactions terms increaseswith growing overall coupling and is generally positive.The higher-order terms lead to more current through thecentral system. On the contrary, for a y-polarized cavityfield (lower panel) the current is weakly suppressed bythe higher-order terms, increasing with the overall cou-

9

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∆N

e

t (ps)

gEM=0.05 meV, y-polgEM=0.10 meV, y-polgEM=0.05 meV, x-polgEM=0.10 meV, x-pol

FIG. 8. The difference in the mean electron number for a sys-tem with an exact, and a system with a dipole approximationfor the electron-photon interactions for a single cavity mode.The figure compares ∆Ne for two values of the electron-photon coupling strength gEM, nR = 1, and the polarizationof the field. B = 1.0 T, ~ω = 0.98 meV, κ = 1.0× 10−5 meV,

−eVg = 2.47 meV, Lx = 180 nm, and g0gLRa3/2w = 0.101

meV.

pling g0. For nR = 1 (not shown here) the differencesin the current are very similar, except the higher orderterms do not suppress the current for the y-polarization.It is still of the same magnitude as for nR = 0.

In the ITR irregularly looking oscillations appear. Wefocus in on these oscillations in Fig. 10 for the overalldimensionless coupling g0 = 1.0, in order to show thatwith a higher resolution on a linear scale the oscillationsare totally regular. Below, we will show that the os-cillations are to the largest extent a beating pattern ofRabi-oscillations with other smaller oscillations superim-posed. It is important to remember here that the statesin the bias window are not directly participating in aRabi-resonance for ~ω = 2.63 meV.

In a simple classical serial RCL circuit we would expectthe appearance of induction always to lead to a reductionof the total current through it. Here, we see that thequantum counterparts, the higher order electron-photoninteraction terms, can either lead to a reduction or anincreased current through the system.

The change in the current ∆I for the lower photon en-ergy ~ω = 0.98 meV is displayed in Fig. 11. On the scaleappropriate for ∆I in the case of x-polarization of thecavity field ∆I for the y-polarization is vanishingly small.The mean change ∆I for the x-polarized field is posi-tive as before, but a complex pattern of Rabi-oscillationsemerges at the ITR, that will be analyzed below.

In the two remaining figures we show the Fourier powerspectrum for the oscillations in the difference of the meanphoton number, ∆Nγ for the two different photon ener-gies, starting with ~ω = 0.98 meV in Fig. 12. The Fouriertransform is taken using 40000 points equispaced in thetime interval from 65.82 – 8065.45 ps. In the upper panel

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1 100 10000 1x106

1x108

1x1010

∆I

t (ps)

g0=0.9g0=1.0g0=1.1g0=1.2

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1 100 10000 1x106

1x108

1x1010

∆I

t (ps)

g0=0.9g0=1.0g0=1.1g0=1.2

FIG. 9. The difference in the mean electron current I =IL + IR for x-polarized (upper), and y-polarized cavity field,for a system with an exact, and a system with a dipole approx-imation for the electron-photon interactions for a single cavitymode. The figure compares ∆I for three values of the dimen-sionless overall lead-system coupling strength g0. nR = 0.gEM = 0.1 meV, B = 1.0 T, ~ω = 2.63 meV, κ = 1.0 × 10−5

meV, −eVg = 2.47 meV, and Lx = 180 nm.

of Fig. 12 the whole energy range from 0 – 6.0 meV isseen and clearly there is an order of magnitude betweenthe two spectra for the x- and the y-polarization of thecavity field. Numerical noise is visible, specially for thehighest energies in the y-polarization. The lower panel ofFig. 12 presents the low energy range of the same spec-tra with three main peaks caused by the Rabi-splittingof the states in the bias window.

In the upper panel of Fig. 12 there are a small groupsof peaks just below 2.0 meV, and again just below 4.0meV. These peaks can all be correlated to active tran-sitions between one-electron states in the system. It isinteresting to note that from the energy spectra shown inthe left panels of Fig. 3 for ~ω = 0.98 meV there are cleardifferences in energy, and specially in the mean photonnumber of many states for the two different polarizationsof the cavity field. This difference is minimal for the spec-tra for ~ω = 2.63 meV seen in the right panels of Fig. 3,

10

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0 1000 2000 3000 4000 5000 6000 7000 8000

∆I (n

A)

t (ps)

x-poly-pol

FIG. 10. The difference in the mean electron current I =IL + IR for x- and y-polarized cavity fields, for a system withan exact, and a system with a dipole approximation for theelectron-photon interactions for a single cavity mode. Thefigure displays partial information from Fig. 9 on a linear t-scale with a higher resolution. nR = 0. gEM = 0.1 meV,g0 = 1.0, B = 1.0 T, ~ω = 2.63 meV, κ = 1.0 × 10−5 meV,−eVg = 2.47 meV, and Lx = 180 nm.

but the Fourier power spectra for ∆Nγ seen in Fig. 13again shows the order of magnitude between the spectrafor the two polarizations.

Here are no Rabi-split states in the bias window andthus no low-energy Rabi-resonances are seen, instead wenotice pairs of resonances emerging corresponding to theRabi-split states above the bias window, indicating thatthese states do participate, even though very weakly, inthe electron transport through the central system. Here,we have analyzed the difference in the mean photon num-ber ∆Nγ , similar analysis for the difference in the current∆I reveals corresponding information. The resonances in∆I are weaker, but in addition we then see peaks causedby transitions between dressed electron states with onlya minimal photon content.

IV. CONCLUSIONS

As expected, the dipole approximation for the electron-photon interaction together with the lowest order dia-magnetic terms, is a good approximation to describe theinteractions of electrons and photons in nanoscale sys-tems embedded in a cavity.

In our central system with discrete energy spectrumand few particles far away from the possibility of asemi-classical interpretation all correspondence to clas-sical self-induction in simple circuits is lost.

Another difficulty comes from the fact that we considera photon cavity with a single photon mode, one funda-mental frequency. Anybody using a Laplace transforma-tion to analyze a classical circuit driven by a step poten-tial has noticed that the results always imply a spread of

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A)

t (ps)

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A)

t (ps)

x-poly-pol

FIG. 11. The difference in the mean electron current I =IL + IR for x- and y-polarized cavity fields, for a system withan exact, and a system with a dipole approximation for theelectron-photon interactions for a single cavity mode. nR = 0.gEM = 0.1 meV, g0 = 1.0, B = 1.0 T, ~ω = 0.98 meV,κ = 1.0× 10−5 meV, −eVg = 2.47 meV, and Lx = 180 nm.

frequencies producing the inductance effects. Neverthe-less, the question about the self-induction is interestingin order to acquire a more complete understanding of thequantum transport of electrons through photon cavities,and a cavity leads to a dominance of their fundamen-tal frequency in the process, as only processes with thefundamental frequency of the cavity or integer multiplesthereof will form a standing wave (a cavity mode) withhigh intensity in the central system.

Our modeling of the electron transport through thecentral system, a nanoscale electron system embedded ina three-dimensional photon cavity with a single FIR pho-ton mode relies on a linear many-body space of photon-dressed electron states |µ). These states have beenconstructed using a step-wise numerically exact diago-nalization and truncations for all the interactions presentin the isolated central system [50]. By keeping this spacelarge enough we have used it to describe the dynamicalevolution of the system as it is coupled to the externalleads with different chemical potentials. This evolution

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γ

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1x10-2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

∆N

γ

E (meV)

x-poly-pol

FIG. 12. The Fourier power spectrum for the difference inthe mean photon number for a system with an exact, and asystem with a dipole approximation for the electron-photoninteractions for a single cavity mode. The two panels showdifferent energy intervals. ~ω = 0.98 meV, gEM = 0.1 meV,

g0 = 1.0, and g0gLRa3/2w = 0.101 meV.

takes place in the many-body Fock-space |µ), or sincewe consider only the Markovian evolution of the system(with respect to the coupling to the leads) is achievedby mapping the original non-Markovian master equationto a larger Liouville space of transitions [44]. As thecentral system has a certain geometry this approach in-cludes higher-order wavefunction effects, i. e. the statescontain the information of polarization of their charge orprobability densities, and in our experience, exactly thisrequirement puts the most difficult constrains on the sizeneeded for our basis used.

The higher order effects of the electron-photon inter-actions stemming form the interaction of the electronswith their own photon field are thus included in the self-energy of the many-body space and their compositionalchanges (by admixing of other states due to the interac-tions). These changes, in terms, lead to changes in thedynamical evolution of the system, i. e. when and how theelectrons transit between the states of the system havingdifferent self-energies.

1x10-10

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γ

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3.3 3.4 3.5 3.6 3.7 3.8 3.9

∆N

γ

E (meV)

x-poly-pol

FIG. 13. The Fourier power spectrum for the difference inthe mean photon number for a system with an exact, and asystem with a dipole approximation for the electron-photoninteractions for a single cavity mode. The two panels showdifferent energy intervals. ~ω = 2.63 meV, gEM = 0.1 meV,

g0 = 1.0, and g0gLRa3/2w = 0.101 meV.

The differences in the transport properties are causedby high-order terms in the electron-photon interactions.The strongest terms contributing the most are terms withelectrical quadrupole and magnetic dipole momenta, butwe can not separate the two contributions.

The approach using exact numerical diagonalizations,or configuration interactions, is essential here, as thisview is not easily extensible to formalisms built onmean-field theories, in which one has to guarantee self-consistence at each time step in the calculation. Thisview underlines the difference between the description ofa classical circuit with an inductance and the quantummechanical one. If the first reaction is that the quantummechanical view may seem more boring, then one has toappreciate what the quantum mechanical view is reallyoffering. Our calculations indicates that if the full geom-etry of the system is taken into count, we may not be ableto guess what small changes the higher-order effects maylead to, without a full calculation. Here, we have alreadyseen that even the simple question if the higher order

12

terms will increase or decrease the current through thesystem is not simple in the quantum mechanical sense.In a direct continuation we are not able to predict whatmay happen in systems with higher electron-photon cou-pling and different geometries, even though the effectsseen here are by no means large.

Our calculation have been performed for a relativelysmall system in order to conserve the RAM-memory sizeneeded, but the inductance of the electronic system canbe made larger by increasing its size, as there is spaceenough in the cavity. Increased system size would lead tothe need to include more electrons in the system, and theevolution of the results with the system size should pointout the path to the porperties of the classical system.

ACKNOWLEDGMENTS

This work was financially supported by the ResearchFund of the University of Iceland, and the Icelandic In-frastructure Fund. The computations were performedon resources provided by the Icelandic High PerformanceComputing Centre at the University of Iceland. V.M.acknowledges financial support from the Romanian CoreProgram PN19-03 (contract No. 21 N/08.02.2019).

Appendix A: Information about the numericalcomputations

We use a scheme of step-wise introduction of modelcomplexities and a step-wise truncation of the ensuingmany-body spaces to guarantee the accuracy of the cal-culations and to contain the RAM memory needed [50].Initially, a large single-electron basis |a〉 with 2688 el-ements is made for the central system, out of which thelowest 53 in energy are used to build a Fock space with1 0-electron, 52 1-electron, 1326 2-electron, and 16 3-electron states. All together 1395 Fock many-body eigen-states for the noninteracting central system in an exter-nal magnetic field. This eigenbasis is used to diagonalizethe many-body Hamiltonian of the electrons with theirmutual Coulomb interaction. The lowest in energy 512eigenstates in this basis are then tensor multiplied bythe 17 lowest eigenstates of the photon number opera-tor to form a basis with 8704 elements that is used todiagonalize the Hamilton matrix including the electron-photon interactions, Eq. 6. Of the resulting eigenstates|µ) for the fully interacting central system, the lowest inenergy 128 states are used to build the 16384-dimensionalLiouville basis of transitions in which the Markovian ver-sion of the master equation is solved. The rotating waveapproximation is not used for the electron-photon inter-action in the central system, but it is used when derivingthe dissipation terms for the coupling of the cavity tothe external photon reservoir. When constructing thesetherms special care has been taken as they have to betransformed from the basis of the non-interacting pho-

tons to the basis of interacting electrons and photons[20, 21, 51–53]. In the Schrodinger picture used here thisis implemented by dismissing all creation terms in thetransformed annihilation operators and all annihilationterms in the transformed creation operators. This guar-antees that an open system will evolve into the correctphysical steady state with respect to the photon decay[31]. The processing time is reduced by a general useof parallelism and a heavy use of off-loading large linearalgebra tasks to fast GPU cards [44].

Appendix B: Changes in the self-energy of statesand their occupation

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|01)|02)|03)|04)|05)|06)|07)|08)

|09)|10)|11)|12)|13)|14)|15)|16)

|17)|18)|19)|20)|21)|22)|23)|24)

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|01)|02)|03)|04)|05)|06)|07)|08)

|09)|10)|11)|12)|13)|14)|15)|16)

|17)|18)|19)|20)|21)|22)|23)|24)

FIG. 14. The difference in the time-dependent occupationof selected many-body states |µ) for a system with an exact,and a system with a dipole approximation for the electron-photon interactions for a single cavity mode, for ~ω = 0.98meV (upper), and ~ω = 2.63 meV (lower). gEM = 0.1 meV,B = 1.0 T, nR = 0, −eVg = 2.47 meV, and Lx = 180 nm,

and g0gLRa3/2w = 0.101 meV.

The electron transport through the central system ismodeled using a quantum many-body formalism, for theelectrons in the central system, the single-photon modeof the cavity electromagnetic field, and the electrons in

13

the external leads. The search for self-induction and sim-ilar higher-order effects where an electron interacts withits own electromagnetic field is carried out in a modelbased on linear many-body spaces, instead of grids, wethus always have to consider the self-energy of the cavityphoton dressed electron states. To aid this explorationwe present here Fig. 14 showing the changes in the dy-namical occupation of the lowest many-body states of thecentral system.

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eV

)

µ

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FIG. 15. The difference in the self-energy of the 32 lowestmany-body states |µ) for a system with an exact, and a systemwith a dipole approximation for the electron-photon interac-tions for a single cavity mode, for ~ω = 0.98 meV (upper),and ~ω = 2.63 meV (lower). The states within the two verti-cal black lines are the states in the bias window. gEM = 0.1meV, B = 1.0 T, nR = 0, −eVg = 2.47 meV, and Lx = 180

nm, and g0gLRa3/2w = 0.101 meV.

In addition, we find it necessary to display in Fig. 15the underlying changes in the self-energy of each many-body state |µ) caused by all higher-order terms of theelectron-photon interactions. In Fig. 15 we show whichstates are within the bias window for both energies of thecavity photons we have chosen. Note the differences inthe energy scales in Fig. 15.

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