Instituut-Lorentz

Leiden University

Master Thesis

Anti-bunched Photons Emitted by aQuantum Point Contact

Author:Ion Cosma Fulga

Supervisor:C. W. J. Beenakker

April 7, 2010

1

Contents

1 Introduction 3

2 Current operators and dynamical properties 42.1 Electrical Current Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Thermal and Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Elements of Photodetection Theory 9

4 Anti-bunched Photons 144.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 From boson to fermion operators . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Factorial moment generating function . . . . . . . . . . . . . . . . . . . . . . . 17

5 Results 225.1 Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Non-zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2.1 Shot noise regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2.2 Anti-bunching conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Experimental efforts 30

7 Conclusion 33

8 Acknowledgments 34

A Performing Gaussian averages 35

2

1 Introduction

One of the most interesting subfields of condensed matter physics that has been developedin the past few decades is that of mesoscopic physics [1]. The miniaturization of devices to thepoint in which quantum effects become important has given rise to a wealth of fascinating,never before seen phenomena.

One of the topics of this field which has seen a lot of development lately is that of dynamicalfluctuations in mesoscopic conductors. An active subject of research is to study a mesoscopicconductor by taking into account the quantum optical properties of the radiation which isproduced by the current fluctuations. This problem has received a great deal of interest inthe physical community, both from an experimental [2] - [7], as well as a theoretical [8] - [10]viewpoint.

It is the purpose of this thesis to present findings which were made in this area, namely onnon-zero temperature effects on the statistics of photons emitted by a quantum point contactout of equilibrium. It will be explained how, under certain conditions, photons can inherit thefermionic, sub-Poissonian statistics of the electrons which produced them, leading to photonanti-bunching.

In contrast to the more familiar setups which generate photon anti-bunching, also referredto as non-classical light, namely electron hole recombinations in the discrete levels of a quan-tum dot or quantum well [11], [12], in the case of a quantum point contact electron transitionscan occur over a continuous energy range in the Fermi sea. Though a zero temperature solu-tion of this problem, taking into account all moments of the photon distribution had alreadybeen developed as early as 2001 [8], recent advances in experimental physics [5] and the abil-ity to probe previously unaccessible parameter ranges have motivated the development of atheory which takes into account the effect of non-zero temperature.

This work has been submitted for publication in Phys. Rev. B, and is in the mean timeaccessible through arXiv:1001.4389. This thesis represents an in-detail version providing morebackground on the theory involved, containing explicit derivations, as well as an overview ofsome experimental efforts towards the detection of anti-bunched photons.

3

2 Current operators and dynamical properties

As mentioned in the introduction, current fluctuations provide a means of probing meso-scopic conductors. In this section, the so called scattering approach, or Landauer approachwill be employed in order to deduce expressions for the quantum operators describing theelectrical currents. The fluctuations of these currents will be classified by introducing the con-cepts of thermal, and shot noise. This description assumes a noninteracting electron model.Much of this section follows the lines of the review by Blanter and Buttiker [1].

2.1 Electrical Current Operators

A seminal idea in mesoscopic physics, the so called Landauer approach, consists of relatingthe various transport properties of a system to its scattering properties. These properties couldthen, in principle, be determined by performing a quantum-mechanical calculation.

An abstract representation of the setup considered in this work, the so called two terminalsetup, is shown in Fig. 1.

sampleμL μ

R

TL T

R

aL

^aR

^

bL

^bR

^

L R

Figure 1: Two terminal scattering problem for a single transverse channel [1].

The sample, a mesoscopic conductor, is connected to two electron reservoirs, labeled hereL for left, and R for right. The large size of the reservoirs means that the sample can beconsidered a small perturbation, thus allowing their characterization in terms of equilibriumproperties, such as temperature T and chemical potential µ. They act as sources of electrons,and absorb the electrons coming out of the sample. Given their equilibrium properties, one candescribe the distribution of electrons within the reservoir by means of the Fermi distributionfunction,

fα(ε) =1

exp [(ε− EF − µα)/kBTα] + 1, (2.1)

with α ∈ {L,R} and kB the Boltzmann constant. The sample is biased at a voltage differenceV , meaning that one can set µL = eV and µR = 0 for the left and right chemical potentials.

4

Away from the sample, in the two leads that connect it to the reservoirs, the motion ofelectrons may be separated into a longitudinal motion characterized by the continuous wavevector kl, having an energy El = h2k2

l /2m, and a quantized transverse motion described by atotal number N of transverse channels, each having an energy ER,L;n. Note that, in principle,the number of channels in the left and right leads may be different.

The fermionic operators a and b create and annihilate electrons in the so called scatteringstates of the left and right lead. As such, a†Ln and b†Ln create electrons in the nth channelof the left lead which are heading towards the sample, and away from it, respectively. Theconnection between the incoming and outgoing states of the sample is given by the sampleproperties themselves, and can be expressed by means of the unitary scattering matrix, suchthat

b =

bL1

. . .

bLNL

bR1

. . .

bRNR

= Sa = S

aL1

. . .aLNL

aR1

. . .aRNR

. (2.2)

The scattering matrix S has dimensions N×N , or equivalently (NL+NR)×(NL+NR), andis constrained to be unitary, meaning S†S = I, by the requirement of probability conservation.It can be written in a form containing blocks which describe the transmission, t, and reflection,r, through and from the sample, as

S =(r′ t′

t r

). (2.3)

The prime symbol was used to indicate the fact that in general, the sample may be asym-metric, in the sense that scattering from the left and from the right may occur with differentamplitudes. Note that throughout this paper the energy dependence of the transmission andreflection amplitudes that enter in the matrix S will be neglected.

It is always possible to use a basis such that each block of the scattering matrix is diagonal.In this so called eigenchannel basis, the scattering matrix of a symmetric sample can be reducedto a set of 2× 2 matrices, one for each eigenchannel

Sn =( √

1− Tn√Tn√

Tn −√

1− Tn

)(2.4)

where Tn is the transmission probability through eigenchannel n.Making use of the general definition (2.3), one can write the current operator in a usual

way [13],

5

IR(z, t) =he

2im

∫dr⊥

[ψ†R(r, t)

∂

∂zψR(r, t)−

(∂

∂zψ†R(r, t)

)ψR(r, t)

], (2.5)

where the current in the right lead was arbitrarily chosen, and the ψ operators are defined as:

ψR =∫dE e−iEt/h

NR∑n=1

χRn(r⊥)√2πhvRn(E)

[aRne

−ikRnz + bRneikRnz

]. (2.6)

In the above expressions, r⊥ denotes the set of transverse coordinates, while z denotes thecoordinates along the leads. The χRn are the complex, normalized transverse wave functions,

and the wave vector kRn =√

2m(E − ERn)/h2. The Fermi velocity in the nth transversechannel of the right lead has been defined as vRn(E) = hkRn/m. Using (2.6) in (2.5) leads to:

IR(z, t) =e

4πh

∑n

∫∫dEdE′ ei(E−E

′)t/h 1√vRn(E)vRn(E′)

{[vRn(E) + vRn(E)]

×[exp

[i(kRn(E′)− kRn(E))z

]b†Rn(E)bRn(E′)

− exp[i(kRn(E)− kRn(E′))z

]a†Rn(E)aRn(E′)

]+ [vRn(E)− vRn(E)]×

[exp

[−i(kRn(E′) + kRn(E))z

]b†Rn(E)aRn(E′)

− exp[i(kRn(E′) + kRn(E))z

]a†Rn(E)bRn(E′)

]}. (2.7)

This lengthy expression may be simplified by making two assumptions [1]: first of all,that the energies E and E′ are identical, or very close to each other, and second, that theFermi velocity varies slowly with energy, meaning that one can effectively neglect its energydependence. This enables the current operator expression to be cast into a more elegant, zindependent form [8],

I(t) =e

2πh

∫∫dεdε′ ei(ε−ε

′)t/ha†(ε)Ma(ε′) (2.8)

where the R index has been dropped for ease of notation. In the above expression, for conve-nience, the energy is measured relative to the Fermi energy, with ε = E −EF . The quantitiesa and a† are column vectors of the annihilation and creation operators in all channels of boththe left and right lead, and M = S†DS −D, with

D =(

0 00 1

), (2.9)

a matrix which selects the current in the right lead.

6

2.2 Thermal and Shot noise

Since one of the main concerns of this paper is that of current fluctuations, it is convenientto offer a brief enumeration of the two main types of fluctuations which are considered, fromnow on referred to as noise: thermal noise, and shot noise. To do this, it is useful to definethe correlation function

S(t− t′) =12

⟨∆I(t)∆I(t′) + ∆I(t′)∆I(t)

⟩, (2.10)

in terms of ∆I(t) = I(t) − 〈I〉. Notice that if there are no time dependent external fieldsthen the only dependence is on the time difference between the current operators, namelyt − t′. Computing averages of such products of current operators boils down to determiningthe averages of products of fermion creation and annihilation operators. This can be done bytaking into account that⟨

a†αn(E)aβm(E′)⟩

= δαβδmnδ(E − E′)fα(E), with α, β ∈ {L,R}, (2.11)

where fα(E) is the distribution function in reservoir α. Averages of products of multipleoperators can then be performed by means of Wick contractions [14].

The Fourier transform of Eq. (2.10), also called the spectral density of noise, or noisepower reads

2πδ(ω + ω′)S(ω) =⟨

∆I(ω)∆I(ω′) + ∆I(ω′)∆I(ω)⟩. (2.12)

To differentiate between the two kinds of noise the zero-frequency spectral density functionis analyzed in two cases.

Firstly, in the equilibrium case where the reservoirs are at a temperature T , one obtains

S = 4kBTG, (2.13)

with G being the conductivity. This equilibrium noise, also called thermal noise or Nyquist-Johnson noise, is the result of the thermally induced variations in the occupation numbers ofthe two reservoirs.

On the other hand, if one considers a zero temperature transport system, the Fermi dis-tribution functions become step functions. In this scenario, the spectral density functionbecomes

S =e3|V |πh

∑n

Tn(1− Tn). (2.14)

This type of noise, also referred to as shot noise, is the result of the discrete nature of thecharge transported through the system, and vanishes for channels which are either completely

7

open, Tn = 1, or completely closed, Tn = 0. Its maximal value corresponds to a transmissionprobability of 1/2.

8

3 Elements of Photodetection Theory

Having discussed electric current fluctuations, it is instructive to present some elementsof photodetection theory, in order to be later able to link the two and show how currentfluctuations couple to the electromagnetic field. In the following discussion, which followsclosely the one of [15], Planck’s constant as well as the elementary charge were set to one,h = e = 1, for ease of notation.

The natural starting point is the electric field operator [15], [8]

E(t) ∝∫ ∞

0dω(a†(ω)eiωt + a(ω)e−iωt

), (3.1)

where a different font was used in order to make a distinction between the bosonic creation andannihilation operators of the above equation, and the fermionic ones present in the expressionfor the current operator of Eq. (2.8). Other quantities, such as prefactors, polarization vectors,position dependence, and mode indices were deliberately omitted from Eq. (3.1) given thatthey bring little to the following conceptual discussion.

Given its expression, it is natural to decompose the electric field operator into a positivefrequency part proportional to a, and a negative frequency part proportional to the creationoperator a†, such that E(t) = E(+)(t) + E(−)(t). Since the act of detecting a photon removesthat photon from the system, one can consider that the amplitude for detecting a photon attime t1, within the small time interval ∆t1 is given by 〈f |E(+)(t1)|i〉, where |i〉 and |f〉 denotesome initial and final state of the field [16].

Thus, by taking the squared modulus of this amplitude, and summing over all possiblestates one obtains the probability

P1(t1)∆t1 = α1∆t1∑f

〈i|E(−)(t1)|f〉〈f |E(+)(t1)|i〉, (3.2)

which, by making use of the completeness relation for the final states (∑

f |f〉〈f | = 1), becomes

P1(t1)∆t1 = α1∆t1∑f

〈i|E(−)(t1)E(+)(t1)|i〉, (3.3)

where α1 is a constant characterizing the detector, containing information pertaining to, forinstance, detector sensitivity. It is convenient to define then an intensity operator, I(t) =E(−)(t)E(+)(t), where a different font is used so that there is no confusion between this quantityand the current operator of Eq. (2.8).

As a next step, one can consider the joint probability of detecting n photons, the first oneat time t1 in a small interval ∆t1, the second photon at time t2 > t1 in the interval ∆t2, andso on. By a similar argument, the detection amplitude becomes

9

〈f |E(+)(tn) . . . E(+)(t2)E(+)(t1)|i〉, (3.4)

leading to a probability

Pn(t1, t2, . . . , tn)∆t1∆t2 . . .∆tn = (3.5)

= αn〈i|E(−)(t1)E(−)(t2) . . . E(−)(tn)E(+)(tn) . . . E(+)(t2)E(+)(t1)|i〉∆t1∆t2 . . .∆tn.

Notice how in this formulation both normal ordering, meaning creation operators to theleft of annihilation operators, as well as time ordering arise naturally. All negative frequencyparts of the electric field operator are ordered with later times to the right, while the positivefrequency parts have later times to the left. Thus, by defining an ordering symbol such that: aa† := a†a, as well as a time ordering symbol T , one can make use of the intensity operatorI(t) to write the joint probability as [15]

Pn(t1, t2, . . . , tn)∆t1∆t2 . . .∆tn = αn〈i|T : I(t1)I(t2) . . . I(tn) : |i〉∆t1∆t2 . . .∆tn. (3.6)

One immediate use of this expression involves the use of the two photon joint detectionprobability, P2(t, t+τ), in order to define a so called normalized intensity correlation function,

λ(τ) =〈T : I(t)I(t+ τ) :〉〈I(t)〉〈I(t+ τ)〉

− 1, (3.7)

which can be used to define bunching and anti-bunching.First of all, notice that if all photons are emitted and detected independently of each

other, then the average in the numerator of Eq. (3.7) factors out, and one obtains λ(0) = 0.The usual case, λ(0) > 0 is referred to as photon bunching, while λ(0) < 0 is called photonanti-bunching.

It is easy to relate the sign of the intensity correlation function to the relative magnitude ofvariance and average of the resulting photon distribution. By making use of the commutationrelations between the bosonic creation and annihilation operators, [a†,a] = 1, [a†,a†] = 0, and[a,a] = 0, one can express the variance as

Varn = 〈n2〉 − 〈n〉2 = 〈(a†a)2〉 − 〈a†a〉2 (3.8)

= 〈(a†)2(a)2〉+ 〈a†a〉 − 〈a†a〉2.

This means that the difference between the variance and the average is proportional to theintensity correlation function (3.7),

10

1〈n〉2

(Varn− 〈n〉) =〈(a†)2(a)2〉〈a†a〉2

− 1 = λ(0). (3.9)

In order to make contact with experiments, one should however use the integral countingprobability, namely the probability to detect n photons in the detection time tdet, between atime t and a time t+tdet. In order to relate this quantity to the joint detection probability of Eq.(3.6), one considers that the in the time tdet measurement can be seen as a sequence of manysuccessive measurements, each occurring in a short time interval ∆t. One can therefore dividethe detection time into tdet/δt short time intervals of duration δt, labeled as t1, t2, . . . , ttdet/δt.The joint probability of making n photon detections in the intervals ti1, ti2, . . . , tin is given byEq. (3.6)

Pn(ti1, ti2, . . . , tin) = 〈T : [αI(ti1)δt] . . . [αI(tin)δt] :〉 . (3.10)

The probability of making n detections in the intervals ti1, ti2, . . . , tin and no detectionsin the intervals tj1, tj2, . . . , tjn, Pn(ti1, ti2, . . . , tin; tj1, tj2, . . . , tjn), can then be computed bymaking use of probability theory. For example, by summing the probability of making onedetection at ti together with zero, one, two, etc. detections at times tj , one arrives at P1(ti):

P1(ti) = P1(ti; tj) + P2(ti, tj) + P3(ti, tj , tj) + . . . (3.11)

This means that

P1(ti; tj) = P1(ti)− P2(ti, tj)− P3(ti, tj , tj)− . . .=⟨T : [αI(ti)δt]

[1− (αI(tj)δt)− (αI(tj)δt)2 − . . .

]:⟩

(3.12)

In the general case, Pn(ti1, ti2, . . . , tin; tj1, tj2, . . . , tjn) will have a factor of [αI(ti)δt] foreach interval in which a detection occurs, and a factor of

[1− (αI(tj)δt)− (αI(tj)δt)2 − . . .

]for each interval in which no detection occurs:

Pn(ti1, ti2, . . . , tin; all other intervals)

=

⟨T :

n∏s=1

[αI(tis)δt]tdet/δt∏r=1︸ ︷︷ ︸

r 6=i1,i2,...,in

[1− (αI(tr)δt)− (αI(tr)δt)2 − . . .

]:

⟩. (3.13)

The integral counting probability, P (n, t, t + tdet), can then be computed by summingover all possible time intervals when the n detections can occur, and then letting δt → 0, or

11

equivalently tdet → ∞. Note that when doing so, each set of intervals is repeated n! times,meaning that the result should be divided by this quantity in order to avoid multiple counting.

P (n, t, t+ tdet) = limδt→0

1n!

tdet/δt∑i1,...,in=1

Pn(ti1, ti2, . . . , tin; all other intervals)

= limδt→0

1n!

⟨T :

tdet/δt∑i1,...,in=1

[αI(ti1)δt] . . . [αI(tin)δt]

×tdet/δt∏r=1

[1− (αI(tr)δt)− (αI(tr)δt)2 − . . .

]:

⟩. (3.14)

In the above expression, the order of averaging and summation was interchanged, and thefactors for which r = i1, . . . , in were added. Note that adding these factors does not lead toa different result, given that n is finite, and each of the factors tends to 1 in the limit whenδt→ 0. The resulting product can be approximately expressed to leading order in δt as

tdet/δt∏r=1

[1− (αI(tr)δt)− (αI(tr)δt)2 − . . .

]= 1−

tdet/δt∑i=1

[(αI(ti)δt) + (αI(ti)δt)2 + . . .

]+

12!

tdet/δt∑i,j=1︸ ︷︷ ︸i 6=j

[(αI(ti)δt) + (αI(ti)δt)2 + . . .

] [(αI(tj)δt) + (αI(tj)δt)2 + . . .

]

− 13!

tdet/δt∑i,j,k=1︸ ︷︷ ︸i 6=j 6=k 6=i

[(αI(ti)δt) + (αI(ti)δt)2 + . . .

]

×[(αI(tj)δt) + (αI(tj)δt)2 + . . .

] [(αI(tk)δt) + (αI(tk)δt)2 + . . .

]+ . . .+O(δt)

≈ exp

− tdet/δt∑i=1

αI(ti)δt

. (3.15)

By neglecting terms of order δt or smaller one obtains

12

P (n, t, t+ tdet) = limδt→0

1n!

⟨T :

tdet/δt∑i=1

αI(ti)δt

n exp

− tdet/δt∑i=1

αI(ti)δt

:

⟩. (3.16)

After taking the limit, the sums effectively become integrals, the final result being

P (n, t, t+ tdet) =

⟨T :

Wn exp(−W )n!

:

⟩(3.17)

where W = α

∫ t+tdet

tI(t′)dt′. (3.18)

Notice that this is just the expectation value of a time ordered, normal ordered Poissondistribution. This is consistent with the prediction that a classical current produces photonswith Poisson statistics [17].

13

4 Anti-bunched Photons

Having described both fluctuations of the electrical currents, as well as the probabilitiesof detecting photons, it is time to link the two and derive a counting probability similar toEq. (3.17) in terms of the fermionic creation and annihilation operators that enter in theexpression for the current, (2.8).

4.1 Experimental Setup

Before attempting this however, it is good to make contact with experiment and presenta particularized picture of the two terminal setup of section 2.

The simplest mesoscopic conductor, a quantum point contact, was chosen. The left (L)and right (R) reservoirs are in thermal equilibrium at a temperature T , and are biased witha voltage difference V , which enables the writing of the associated chemical potentials asµL = V , µR = 0 (still in units where h = e = 1). Electrons injected through this constrictionwill decay, emitting photons. Conversely, the holes left behind will absorb photons, though inthis treatment only detection through emission of photons is considered. Figure 2 shows thissetup [8], [9].

eV

0

VFigure 2: Illustration of the scattering geometry. Electrons that tunnel through the biasedquantum point contact will decay, emitting photons which are absorbed by a nearby detector,[8],[9].

The statistics of the transfered charge is sub-Poissonian, meaning that electrons are anti-

14

bunched, which is an immediate consequence of the Pauli principle. This is however notgenerally the case with photons. A photon of energy ω may be populated by electrons decayingfrom a range V −ω of initial energies. As such, photon statistics is generally super-Poissonian.In order to obtain anti-bunched photons, these must somehow inherit the electron statistics.This can be achieved by ensuring a one-to-one correspondence between electron tunnelingevents and photon emission events [9]. Conditions which lead to such a correspondence, bothin the zero and non-zero temperature cases will be discussed in detail in Section 5.

4.2 From boson to fermion operators

In order to be able to perform a quantitative description of the physical processes portrayedin Fig. 2, one needs to express the counting probability (3.17) in terms of the current operators(2.8). Note that in this setup, in the zero temperature case the energy integrals of Eq. (2.8)should be carried out in the interval [0, V ], while if T 6= 0 the range should extend to (−∞,∞).This derivation closely follows the lines of [8], the starting point being a slightly differentversion of the counting probability, namely

P (n, 0, tdet) =

⟨T±

Wne−W

n!

⟩(4.1)

W =∫ ∞

0dω α(ω)

∫∫ tdet

0dt−dt+ e

iω(t+−t−)E(t−)E(t+). (4.2)

Here, α(ω) is the frequency dependent detector sensitivity, and the symbol T± indicatesKeldysh time-ordering. This means that all t− are placed to the left of all t+, earlier t− to theleft of later t−, earlier t+ to the right of later t+. This ordering reduces to the time ordering andnormal ordering of Eq. (3.17) when inserting the electric field form of Eq. (3.1) and makingthe rotating wave approximation, meaning neglecting all terms of the form exp(i(ω + ω′)t)while keeping terms ∝ exp(i(ω − ω′)t).

The electrons and photons are coupled via the interaction Hamiltonian −∫dr j(r, t) ·

A(r, t), where j is the electron current density and A the electromagnetic vector potential.Due to this coupling, one can relate the current operator of Eq. (2.8) to the electric fieldoperator via

E(t)→ E(t) +∫ ∞−∞

dt′ g(t− t′)I(t′) (4.3)

where g(t− t′) is a propagator obeying causality, meaning that g(t < 0) = 0.It has been shown that the effect of E(t) in the right hand side of Eq. (4.3) may be

neglected if the Keldysh ordering is applied to I(t) [18], leading to an expression for W of theform

15

W =∫ ∞

0dω α(ω)

∫∫ tdet

0dt′dt′′ eiω(t′′−t′)

∫∫ ∞−∞

dt−dt+ g(t′ − t−)g(t′′ − t+)I(t−)I(t+). (4.4)

It is convenient to express the ordering of the current operators in a simpler form, byconsidering that the total current of Eq. (2.8) is made up of two parts, one coming out of thesample, and one going into it. As such, I(t) = Iout(t)− Iin(t) with

Iout(t) =1

2π

∫∫dεdε′ ei(ε−ε

′)ta†(ε)S†DSa(ε′), (4.5)

and

Iin(t) =1

2π

∫∫dεdε′ ei(ε−ε

′)ta†(ε)Da(ε′). (4.6)

The main reason for this separation is the fact that the commutation relations betweenthese components leads to a simpler expression of the ordering. Iout(t) commutes with Iout(t′),as do Iin(t) and Iin(t′). On the other hand Iout(t) commutes with Iin(t′) as long as t < t′.This means that the Keldysh ordering may be realized by simply placing all I(t−) operatorsto the left of all I(t+), irrespective of the values of the time arguments. Having performedthis ordering, one can now consider the Fourier transforms,

P (n, 0, tdet) =

⟨OW

ne−W

n!

⟩(4.7)

with

W =∫ ∞

0dω α(ω)U †(ω)U(ω), (4.8)

and

U(ω) =∫ ∞−∞

dω′

2π

∫ tdet

0dt′ ei(ω−ω

′)t′g(ω′)[Iout(ω′)− Iin(ω′)

]. (4.9)

The ordering symbol O arranges the current operators in the order I†in, I†out, Iout, Iin, andthe current operators in Fourier space are given by

Iout(ω) =∫dε a†(ε)S†DSa(ε+ ω), (4.10)

and

Iin(ω) =∫dε a†(ε)Da(ε+ ω). (4.11)

16

In this way, the relation between the current fluctuations in terms of fermion creation andannihilation operators and the statistics of the observed photons has been realized. In theexperimentally relevant limit of a long detection time tdet, one may discretize the frequenciesωn = 2πn/tdet, leading to

∫ tdet

0 dt′ ei(ωn−ωm)t′ = tdetδn,m and U(ωn) = g(ωn)I(ωn).The quantities of interest in this case will be the factorial moments of the photon distri-

bution. They are defined as

〈np〉f = n(n− 1)(n− 2) . . . (n− p+ 1) = 〈OW p〉 (4.12)

or,

〈np〉f =⟨O[∫

dω γ(ω)I†(ω)I(ω)]p⟩

(4.13)

where γ(ω) = α(ω)|g(ω)|2 is a real valued response function proportional to the detectorsensitivity, and the integral sign was used, even though a discrete sum over frequencies isimplied.

In particular, for the first moment of the distribution which is just the average, 〈n1〉f = 〈n〉,the ordering operator O may be omitted, with the result that

〈n〉 =∫ ∞

0dω γ(ω)〈I†(ω)I(ω)〉 (4.14)

and

〈I†(ω)I(ω)〉 = tdet

∫ ∞−∞

dt eiωt〈I(0)I(t)〉 (4.15)

which is just the unsymmetrized version of the noise power (see (2.10)), in agreement with [3].The higher moments of the distribution however are much harder to determine, given thatthe second moment of the photon distribution relates to the fourth moment of the currentfluctuations [5],[9]. The advances of recent experimental techniques have enabled the probingof a parameter range which approaches the conditions required to detect anti-bunching [5].

4.3 Factorial moment generating function

The quantities needed to classify the photon distribution as either sub- or super-Poissonian,the variance and average, can be computed from the first and second factorial moments

〈n〉 = 〈n1〉f , Varn = 〈n2〉f − 〈n1〉2f + 〈n1〉f . (4.16)

It is more elegant however to use the approach of [9] and define a so called factorial momentgenerating function which enables the expression of all factorial moments of the probabilitydistribution:

17

P (n, 0, τ) =1n!

limξ→−1

dn

dξnF (ξ), (4.17)

with

F (ξ) =∞∑k=0

ξk

k!〈nk〉f =

⟨O exp

[ξ

∫ ∞0

dω γ(ω)I†(ω)I(ω)]⟩

. (4.18)

Again, even though the integral sign is used, it can be interpreted as a sum over discreteenergies and frequencies. This means ωn = 2πn/tdet, εn = 2πn/tdet with integer n, andγn = γ(ωn)2π/tdet such that

∫dω γ(ω) →

∑n γn. The detection time tdet will be set to

infinity at the end of the calculation. The following derivations closely follow the lines of [9].In order to simplify Eq. (4.18), a Hubbard-Stratonovich transformation is performed by

introducing a complex Gaussian field zp = z(ωp) with zero mean and variance 〈|zp|2〉 = 1/γp:

F (ξ) =⟨O exp

[√ξ

∫ ∞0

dω γ(ω)[z(ω)I†(ω) + z∗(ω)I(ω)

]]⟩. (4.19)

Now the product of current operators has been expressed as a sum, making it easier toperform the quantum average. The trade off however is that now the brackets 〈. . .〉 indicateboth a quantum mechanical average, as well as an average over the Gaussian variables zp andz∗p ,. The latter is defined as

〈· · · 〉gaussian =∏p

γpπ

∫d2zp e

−γp|zp|2 . . . . (4.20)

Products of current operators have been eliminated from the exponent, allowing for thedecomposition of the current operator in its outward and inward components as in (4.10) and(4.11). Taking into account the ordering O leads to:

F (ξ) =⟨e−√ξ∫∞0 dω I†in(ω)z(ω)e

√ξ∫∞0 dω I†out(ω)z(ω)e

√ξ∫∞0 dω Iout(ω)z∗(ω)e−

√ξ∫∞0 dω Iin(ω)z∗(ω)

⟩(4.21)

By expressing the integrals as discrete sums one obtains

F (ξ) =⟨e−a

†DZaea†S†DZSaea

†S†DZ†Sae−a†DZ†a

⟩(4.22)

where the vectors a have elements apn =√

2π/τ an(εp) and the matrix Z is defined as

Zpn,p′n′ =√ξ δnn′zp−p′γp−p′ , (4.23)

being diagonal in the channel indices n, n′. This means that, since the scattering matrix Swas assumed to be diagonal in the energy indices p, p′, it commutes with Z, [Z, S] = 0. For

18

the same reasons, [Z,D] = 0, but notice that the scattering and detection matrices do notcommute [S,D] 6= 0.

By further considering only photodetection by emission, which amounts to setting γ(ω ≤0) = 0, or equivalently γp≤0 = 0, one obtains the additional condition that the matrix Z islower triangular. This will prove very helpful later on when computing traces, given that thetrace of a lower triangular matrix is zero, tr(Z) = 0.

In order to bring the generating function as written in Eq. (4.22) back to the form of asingle exponential, the relation

∏i

ea†Aia = N exp

[a†

(∏i

eAi − 1

)a

], (4.24)

is used, where Ai denote any set of matrices, and the normal ordering symbol N means puttingall annihilation operators to the right of all creation operators [8]. On the other hand, thequantum mechanical expectation value of a normally ordered exponential is a determinant[19],

〈N ea†Aa〉 = det(1 +AB). (4.25)

The matrix B is given by

Bnp,n′p′ = 〈a†npan′p′〉 = δnn′δpp′fnεp , (4.26)

where fnp is the Fermi occupation number in channel n at energy εp (see (2.1)). It is consideredthat the two reservoirs are in thermal equilibrium at a temperature T , and µn = µL for thosechannels of the left lead, while µn = µR for the channels in the right lead. The total numberof channels is denoted by N . The matrix A has the form

A = eXeY eY†eX† − 1, (4.27)

with X = −DZ and Y = S†DSZ.In order to simplify relation (4.25), it would be helpful to express the matrices A and B,

D, S and Z, such that the determinant is written as a product over all channels, and onlyenergy indices are left. To this end, the eigenchannel basis is chosen, in which the scatteringmatrix and the detection matrix are given by (2.4) and (2.9). The B matrix then takes theblock form

B =(fL 00 fR

), (4.28)

with diagonal blocks (fL)pp′ = δpp′fL(εp) and (fR)pp′ = δpp′fR(εp) in terms of the Fermifunctions defined in (2.1). The matrix Z in this basis is given by

19

Z =(Z 00 Z

). (4.29)

The determinant (4.25) can be further simplified by making use of the fact that Dq = Dand that S†S = I, as well as the commutation relations [Z, S] = [Z,D] = 0. This implies thatXq = D(−Z)q, and Y q = S†DSZq, allowing for the manipulations

e−DZ = (e−DZ − 1) + 1 = D(e−Z − 1) + 1, (4.30)

eS†DSZ = (eS

†DSZ − 1) + 1 = S†DS(eZ − 1) + 1. (4.31)

By applying the relations (4.24) to (4.31), one arrives at a generating function having theform

F (ξ) =

⟨N∏m=1

Det

(1 + TmfL(eZeZ

† − 1)√Tm(1− Tm) fL(e−Z

† − eZ)√Tm(1− Tm) fR(e−Z − eZ†) 1 + TmfR(e−Ze−Z

† − 1)

)⟩. (4.32)

where the brackets 〈. . .〉 now only indicate Gaussian averaging.By making use of the determinant relation

Det(M11 M12

M21 M22

)= DetM11 Det (M22 −M21M

−111 M12), (4.33)

one arrives at

F (ξ) =⟨ N∏m=1

Det[1 + TmfL(eZeZ

† − 1)]Det

(1 + TmfR(e−Ze−Z

† − 1)

− Tm(1− Tm)fR(e−Z − eZ†)[1 + TmfL(eZeZ† − 1)]−1fL(e−Z

† − eZ))⟩

. (4.34)

Additionally, by using that

[1+TmfL(eZeZ†−1)]−1fL(e−Z

†−eZ) = −fL(eZeZ†−1)[1+TmfL(eZeZ

†−1)]−1e−Z†, (4.35)

and then multiplying by Det eZ†

= 1, one arrives at

20

F (ξ) =⟨ N∏m=1

Det([

1 + TmfR(e−Ze−Z† − 1)

]eZ†[

1 + TmfL(eZeZ† − 1)

]+ Tm(1− Tm)fR(e−Z − eZ†)fL(eZeZ

† − 1))⟩

=⟨ N∏m=1

Det(

1 + Tm[(1− fR)eZ†fL − fRe−Z(1− fL)](eZ − e−Z†)

)⟩. (4.36)

This expression may be written in the compact form

F (ξ) =⟨ N∏m=1

Det(

1 + Tm[fReZ†fL − fRe−Z fL]M

)⟩, (4.37)

by defining fL = 1 − fL, fR = 1 − fR, M = eZ − e−Z† . This result is consistent with thezero temperature one obtained from a very similar derivation in Ref. [9], which is obtainedby setting fL = 1, fR = 0 and the energy interval [0, V ].

By expanding this function in powers of ξ and performing the Gaussian average, one candetermine the factorial moments of the photon distribution.

21

5 Results

As mentioned in the previous section, observing photon anti-bunching requires a one-to-one correspondence between electron tunneling events and photon emission events. In the zerotemperature case, it is sufficient to restrict the bandwidth of the detector to frequencies closeto the applied voltage. This amounts to saying that for each tunneling event, one can haveat most one emitted photon. One would naively assume that such a constraint would lead toanti-bunching also in the case of a small non-zero temperature. The results obtained howeverindicate that another more stringent condition is required, as will be outlined in this section.

In order to simplify the problem, the real response function γ(ω) is assumed to have thebox form

γ(ω) ={γ0 if V −∆ω < ω < V,0 otherwise.

. (5.1)

This allows for the evaluation of the energy and frequency integrals that enter in thegenerating function.

5.1 Zero Temperature

The case in which T = 0 had been already solved as early as 2001 [8], and the full photoncounting statistics were later derived starting from the factorial moment generating function[9]. The anti-bunching condition in this case, is that ∆ω < V/2, where the detector bandwidth∆ω is defined in Eq. (5.1). The variance and average of the detected photon distribution werefound to be

〈n〉 =τ∆ω2π

γ0∆ω12S1, (5.2)

Varn− 〈n〉 =τ∆ω2π

(γ0∆ω)2 13

(S21 − 2S2), (5.3)

where Sp =∑

n[Tn(1− Tn)]p. The anti-bunching condition in this case is S21 < 2S2.

Note that in the case of a quantum point contact (see Fig. 2) all channels are either open,Tn = 1, or closed Tn = 0 except one, meaning that the condition S2

1 < 2S2 is always satisfied.Another way of expressing anti-bunching is in terms of the so called Fano factor, F ,

defined as the ratio between the variance and the average of the distribution. As such, F > 1corresponds to bunching, while F < 1 corresponds to anti-bunching. In the zero temperaturecase

F =Varn〈n〉

= 1 +23γ0∆ω(S1 − 2S2/S1). (5.4)

22

5.2 Non-zero temperature

In the case of a small, non-zero temperature, the generating function in the form (4.37)should be expanded in orders of ξ, each term being proportional to one factorial moment.This can be achieved by making use of the identity

∏m

Det(1 + Ξm) = exp

[∑m

tr(log(1 + Ξm))

], (5.5)

when in this particular case Ξm is given by Eq. (4.37)

Ξm = Tm[fReZ†fL − fRe−Z fL]M. (5.6)

By expanding in turn the logarithm and the exponential, one can express the generatingfunction such that the powers of ξ can be read off. This however leads to a lengthy expression,given that Ξ is of order

√ξ, meaning that in order to extract the second moment one has

to go to fourth order terms. Luckily, several simplifications are brought about by the factthat Gaussian averaging requires an equal number of Z and Z† terms in order to producenon-zero results, and also by the fact that since the matrix Z is triangular, its trace is zero,tr(Z) = tr(Z†) = 0. The result is

F (ξ) = 1− 14

⟨∑m

tr(Ξ4m)

⟩+

13

⟨∑m

tr(Ξ3m)

⟩− 1

2

⟨∑m

tr(Ξ2m)

⟩

+18

⟨(∑m

tr(Ξ2m)

)2⟩+

⟨∑m

tr(Ξm)

⟩− 1

2

⟨(∑m

tr(Ξm)

)(∑m

tr(Ξ2m)

)⟩

+12

⟨(∑m

tr(Ξm)

)2⟩+O(ξ5/2). (5.7)

By computing every trace and every Gaussian average, then expressing the resulting sumsin terms of integrals over energy and frequency (see Appendix), one can compute the varianceand average of the photon distribution. However, due to the many terms which result, theobtained expressions were only evaluated numerically, as will be shown in the following.

5.2.1 Shot noise regime

In order to bring these results to a simpler form, the limiting case of low temperature hasbeen considered, kBT � V . This approximation, which shall be referred to as the shot noiseregime, amounts to neglecting thermal noise and keeping just the temperature dependence of

23

the shot noise. In order not to impose restrictions on the relative magnitude of kBT and ∆ω,the bandwidth of the detector was set such that ∆ω � V .

The first simplification to the generating function (4.37) is made by setting fRe−Z fL → 0in the low temperature limit, given that fR(ε)fL(ε′)→ 0 when ε′ ≤ ε. Afterwards, multiplyingwith Det e−Z

†= 1 leads to

F (ξ) =⟨ N∏m=1

Det(e−Z

†+ TmfLMfR

)⟩. (5.8)

Secondly, given that in this limit the largest energy scale of the system is the appliedvoltage V , one can ignore energies separated by at least 2V . Since Z2 and Z†

2 connectenergies separated by 2V one may set Z2, Z†

2 → 0, and obtain the result

F (ξ) =⟨ N∏m=1

Det(

1− Z† + TmfL(Z + Z†)fR

)⟩. (5.9)

As a further simplification, this expression is brought to a form bilinear in Z and Z†.Denoting Am = TmfLZfR and Bm = TmfLZ

†fR − Z†, one can re-express Eq. (5.9) as

F (ξ) =⟨ N∏m=1

Det(1 +Am +Bm)⟩. (5.10)

Of course, setting Z2, Z†2 → 0 is equivalent to saying that A2

m → 0 and B2m → 0, and

under these conditions the determinant obeys the condition

Det (1 +A+B) = Det(1−AB), (5.11)

leading to

F (ξ) =⟨ N∏m=1

Det(1 + TmZfRZ

†fL − T 2mZfRfLZ

†fRfL)⟩

=⟨ N∏m=1

Det(

1 + Tm(1− Tm)fLZfRZ†)⟩

. (5.12)

The last equality sign comes by replacing ZfRfL → ZfR and Z†fRfL → Z†fL given thatkBT � V and the Fermi function fL in the first term is evaluated at energies near EF , whilefR in the second term is evaluated at energies near EF + V , where they can both be replacedby unity.

24

This simplification turns out to be of tremendous use in determining the moments, giventhat now

Ξm = Tm(1− Tm)fLZfRZ† (5.13)

is of order O(ξ), so that only expansions up to second order need to be retained. By applyingrelation (5.5) one obtains

F (ξ) = 1 +⟨∑m

Tr Ξm⟩− 1

2⟨∑m

Tr Ξ2m

⟩+

12⟨(∑

m

Tr Ξm)2⟩+O(ξ3). (5.14)

From this expression one can read off the variance and average, given that

F (ξ) = 1 + ξ〈n〉+ 12ξ

2(〈n2〉 − 〈n〉

)+O(ξ3). (5.15)

By performing the traces and the Gaussian averages, the variance and average in this lowtemperature limit are found to be

〈n〉 =tdet

2πS1

∫dω γ(ω)

∫dε fL(ε+ ω)fR(ε), (5.16)

Varn = 〈n〉+tdet

2πS2

1

∫dω

[γ(ω)

∫dε fL(ε+ ω)fR(ε)

]2

− tdet

2πS2

∫dε

[fL(ε)

∫dω γ(ω)fR(ε− ω)

]2

− tdet

2πS2

∫dε

[fR(ε)

∫dω γ(ω)fL(ε+ ω)

]2

, (5.17)

again in terms of

Sp =∑m

[Tm(1− Tm)]p. (5.18)

Since the two reservoirs are at the same temperature T and the applied voltage differenceenables the setting of chemical potentials to µL = V and µR = 0, one can express the twoFermi functions in terms of a single function such that fL(ε) = f(ε − V ) and fR = f(−ε)where ε is the energy measured with respect to the Fermi energy and

f(x) =1

1 + exp(x/KbT ). (5.19)

25

In order to write the expressions (5.16) and (5.17) into an even more compact form, onecan define Γ(ε, ω) = γ(ω)f(ε)f(ω − ε− V ), such that

〈n〉 =tdet

2πS1

∫dω

∫dεΓ(ε, ω), (5.20)

Varn = 〈n〉+tdet

2π

∫dω

∫dεΓ(ε, ω)

×[S2

1

∫dε′ Γ(ε′, ω)− 2S2

∫dω′ Γ(ε, ω′)

]. (5.21)

As can be seen from the above expression, the difference between the variance and the av-erage of the photon distribution is made up of two terms, a positive one, which is proportionalto S2

1 , and a negative one ∝ S2. It is the difference between these terms which determineswhether anti-bunching is realized or not.

When kBT � ∆ω, temperature becomes the smallest energy scale of the system, andthe function Γ(ε, ω) acquires a block form. This leads to the zero temperature results of [9],namely

〈n〉 =tdet∆ω

2πγ0∆ω

12S1, (5.22)

Varn− 〈n〉 =tdet∆ω

2π(γ0∆ω)2 1

3(S2

1 − 2S2) (5.23)

with

F = 1 + 23γ0∆ω(S1 − 2S2/S1). (5.24)

On the other hand, when kBT � ∆ω but still in the shot noise regime such that kBT � Vthe function Γ(ε, ω) becomes Γ(ε, ω)→ −γ(ω)kBTdf(ε)/dε, meaning that relations (5.20) and(5.21) become

〈n〉 =tdet∆ω

2πγ0kBTS1, (5.25)

Varn− 〈n〉 =tdet∆ω

2π(γ0kBT )2S2

1 , (5.26)

with the corresponding Fano factor given by

F = 1 + γ0kBTS1. (5.27)

The expression (5.27) clearly shows that anti-bunching is lost, F > 1, even in the shotnoise regime where kBT � V as long as ∆ω � kBT . This is not an obvious statement, as

26

one would have expected that in the shot noise regime neglecting thermal noise would alwayslead to anti-bunching. This means that sub-Poissonian statistics is destroyed not only bythe overlapping tails of the non-zero temperature Fermi functions, since they were neglectedin this limit, but rather by the rounding of the Fermi function itself, as sketched in Fig. 3.Therefore, the smaller the bandwidth the more nonuniform the Fermi function in that energyinterval is, leading to photon bunching.

0 . 0

0 . 5

1 . 0

E n e r g y

V

Figure 3: Rounding of the Fermi function for non-zero temperatures shown together withtwo bandwidths of different sizes (shown with different shades, marked by arrows). For alarge bandwidth the distribution function appears to be flat, while for small ∆ω it variessignificantly within the energy interval dictated by the bandwidth.

Another way of getting a qualitative explanation of this fact is comparing the coherencetime of the detected radiation, tcoh ' 1/∆ω, with the coherence time of thermally excitedelectron hole pairs tT ' 1/kBT . For tcoh > tT , the detected photons are the result of manyuncorrelated electron hole recombination acts, meaning that the one-to-one correspondencebetween electron tunneling events and photon detection events, and hence also anti-bunching,are lost.

5.2.2 Anti-bunching conditions

In order to establish the conditions under which anti-bunching is observed, the crossovertemperature Tc for which F = 1 was numerically determined in the shot noise regime, startingfrom equations (5.20) and (5.21). The obtained result is

27

kBTc ≈ 0.25 ∆ω. (5.28)

Fig. 4 illustrates the difference between the variance and average of the detected photondistribution in the shot noise regime, together with the asymptotic high-temperature case.

Figure 4: Reduced, normalized Fano factor as a function of temperature in the shot noiseregime (full line). Dashed line indicates the high temperature limit. The crossover temperatureTc is shown, and a single channel is considered.

As a next step the validity of this approximation is tested by direct comparison with thedistribution moments obtained numerically from the general generating function of Eq. (4.37),in the case of a single channel quantum point contact having the transmission probability τ ,and using the box shaped response function (5.1). It can be seen in Fig. 5 that the curvesare in good agreement at low temperatures, especially in the anti-bunching region, convergingtowards the shot noise result as temperature becomes smaller than bandwidth.

Finally, the crossover temperature result in the shot noise regime, Eq. (5.28), was comparedto the general result, numerically obtained from the generating function (4.37).

It can be seen from Fig. 6 that the shot noise estimate is accurate in the low bandwidthlimit. With increasing ∆ω, the crossover temperature Tc drops below this estimate, thedecrease being enhanced for small transmission probabilities. This can be understood bynoting the fact that shot noise has a maximum when the transmission τ = 0.5, as can also beseen from the expression (2.14). Setting τ away from this value increases the effect of thermalnoise with respect to that of shot noise, and one has to decrease the temperature in orderto observe anti-bunching. When τ = 0.5 however, the shot noise regime estimate remainsaccurate even for bandwidths as large as V/2.

28

Figure 5: Reduced, normalized Fano factor as a function of temperature for various bandwidths(full lines). At low temperatures, the curves fall on top of the shot noise estimate (showndashed), while for higher temperatures they separate, indicating that thermal noise becomesimportant. A single channel is considered, having the transmission probability τ = 0.5.

Figure 6: Dependence of the crossover temperature on bandwidth, for several single channeltransmission probabilities (shown on plot). All curves converge towards the shot noise estimate(dashed line) in the low bandwidth limit.

29

6 Experimental efforts

Many experiments have addressed the detection of radiation emitted by mesoscopic conduc-tors [2] - [7], as well as the problem of the resulting photon statistics. This section representsan overview of recent experimental efforts in this direction, and provides a short descriptionof the systems considered and the methods employed.

One such method is detecting photons emitted by a mesoscopic conductor such as a quan-tum point contact by means of a tunable two level system [3], [6], [7]. One explicit realizationof such a system is a double quantum dot (DQD), as depicted in Fig. 7.

Figure 7: Detecting photons emitted by a quantum point contact by means of a doublequantum dot (reproduced from [7]). Panel (a) represents a picture of the setup indicating thequantum point contact (QPC), the two quantum dots (circles) coupled in series between asource (S) and a drain (D). The gates used to control the occupation number in the quantumdots, as well as the height of the tunnel barrier between the two dots are labeled G1, G2, Uand L. Panel (b) depicts the scheme of the DQD occupation in the single photon detectionsetup, while panel (c) is the current measured in the quantum point contact in the regimeshown in (b). For regions I-III, the current stays the same, while upon entering the region IV,a spike in the current which signifies the absorption of a single photon.

Unlike atoms or molecules, which too absorb light via electronic transitions, a DQD hasthe advantage that the difference between the energy levels can be easily tuned by means ofgate voltages, and thus makes a good candidate for single photon detection. As depicted inFig. (7)b, in the beginning of the cycle (step I), the current through the double quantum dotis suppressed because of Coulomb blockade. If an electron in the first dot absorbs a photonwith frequency equal to the energy difference between the levels of the dots, δ, it will tunnel

30

to the second dot (step II). Afterwards it can either leave the system with a rate ΓD, or relaxback to the energy level of the first dot with a rate Γrel. After existing the DQD, there willbe a period of time dictated by ΓS in which the considered energy levels of the two dots willbe unoccupied (step IV).

In the experiment of [7], ΓD � Γrel such that an electron entering the DQD will suffermany excitation and relaxation processed before exiting the system. However, given that thissystem is coupled to that of the quantum point contact, there will be a measurable differencein the QPC current when the DQD contains one extra electron (steps I-III), as opposed towhen it contains no extra electrons (step IV). The resulting spikes in the measured current,Fig. 7c can be directly related to processes in which an electron absorbs a photon of energyδ, and afterwards exits the DQD.

It is expected that in the near future, using an array of double quantum dots will make itpossible to probe the statistics of the emitted photons.

One method of determining the higher moments of the statistics of photons produced bymesoscopic conductors is by measuring Hanbury-Brown–Twiss (HBT) correlations [2], [4], [5].The experiment proposed by Hanbury Brown and Twiss [20], originally in the field of astron-omy, consisted in using a beam splitter and two detectors to measure the power fluctuationsof a single electromagnetic source (see Fig. 8). The sign of those correlations can give insightinto the nature of the photon statistics of the source [4]. Positive correlations for instance area trademark of photon bunching, and have been observed for thermal photons, while photonswith a Poisson distribution have vanishing HBT correlations. Measurements on sources ofnon-classical light lead to negative correlations.

Figure 8: Schematic depiction of the Hanbury-Brown–Twiss experiment (taken from [2]).

In a recent experiment by Zakka-Bajjani et al. [5], such a method was applied in order todetermine the statistics of photons emitted by a tunnel junction. In their experimental setup,they have also managed to measure the unsymmetrized noise power, Eq. (4.15), directly, asshown in Fig 9.

Given that a tunnel junction consists of many weakly transmitting channels, one expects tofind photon bunching even in the zero temperature regime, S2

1 > 2S2 in Eq. (5.3), and indeed

31

Figure 9: Cross-correlation spectrum of power fluctuations (taken from [5]). Open squaresrepresent experimental data, while lines represent theoretical predictions. It can be seen thatthe data agrees with the unsymmetrized shot noise prediction of [8] (full line; see also Eq.(4.15)). The dashed line is obtained by plotting the symmetrized version of the shot noisepower (see Eq. (2.10)).

this is observed my means of positive HBT correlations. Considering the wide frequency rangeavailable in the experiment, 4 − 8 GHz, one hopes that replacing the tunnel junction by aquantum point contact, in which only one channel is neither completely open not completelyclosed, will lead to the experimental observation of anti-bunched photons.

32

7 Conclusion

In conclusion, the effects of non-zero temperature on the statistics of the photons emittedby a quantum point contact out of equilibrium were investigated. The crossover temperaturefrom sub- to super-Poissonian statistics is set by the bandwidth of the detector, even in thecase of a small bandwidth, ∆ω � V , as shown in Eq. (5.28). This means that there exists aneven more stringent condition for observing photon anti-bunching at non-zero temperaturesapart from kBT < V , namely kBT < ∆ω. Although thermal noise is responsible for thecrossing from bunching to anti-bunching, it is not the dominant effect in the case of narrowbandwidth detection. In that case, the crossover is instead governed by the coherence time ofthe electron hole pairs. Finally, it is concluded that the optimal conditions for the experimentalobservation of anti-bunched photons in this quantum point contact setup are achieved whenthe detector bandwidth is set to ∆ω ≈ V/2, and the single channel transmission probabilityto τ ≈ 1/2. In this case kBTc ≈ V/8 has the largest value at any applied voltage. Thecurrently available detection range (4 GHz < ω < 8 GHz ⇒ ∆ω = 4 GHz = V/2) [5], opensthe possibility of detecting anti-bunching at temperatures below 1 GHz ∼= 50 mK.

33

8 Acknowledgments

I would like to thank my supervisor, C.W.J Beenakker, for his input and help throughout thisproject. Additionally, I would like to thank F. Hassler for helpful discussions and pointing outkey facts, and P.J.H. Denteneer for a second evaluation of this thesis.

34

A Performing Gaussian averages

In this section, the explicit computation of terms in the generating function given by (4.37)or (5.12) is outlined. Making use of the definition (4.20), one can immediately compute that

〈zp〉 = 0, 〈z∗pzp′〉 =δpp′

γp, (A.1)

and all other averages containing unequal numbers of zp and z∗p , or equivalently of Z and Z†,are zero. Averages over multiple zp and z∗p can be performed by pairwise averaging, meaningthat

⟨zp1zp2z

∗p3z∗p4

⟩=⟨zp1z

∗p3

⟩ ⟨zp2z

∗p4

⟩+⟨zp1z

∗p4

⟩ ⟨zp2z

∗p3

⟩=δp1p3γp1

δp2p4γp2

+δp1p4γp1

δp2p3γp2

(A.2)

Because of the fact that the expansion is performed only up to second order in ξ and thematrix Z defined in Eq. (4.23) is proportional to

√ξ, the maximum number of Z matrices that

will be averaged over is 4. Cyclic permutation within the trace further reduces the number ofpossible averages over a single trace to a total of three:

1. ⟨tr(ZZ†)

⟩=⟨

tr(Z†Z)⟩

(A.3)

2. ⟨tr(ZZZ†Z†)

⟩=⟨

tr(Z†ZZZ†)⟩

=⟨

tr(Z†Z†ZZ)⟩

=⟨

tr(ZZ†Z†Z)⟩

(A.4)

3. ⟨tr(ZZ†ZZ†)

⟩=⟨

tr(Z†ZZ†Z)⟩

(A.5)

There is however the possibility of a Gaussian average containing multiple traces, as inEq. (5.7). However, because of the fact that Z and Z† are triangular, tr(Z) = tr(Z†) = 0,and as such any non-zero Gaussian average can be made up of at most two traces. The onlysolution is a term of the form ⟨

tr(ZZ†)tr(ZZ†)⟩. (A.6)

Of course, the traces may also contain the diagonal Fermi matrices fL, fR, or fL, fR,which do not commute with the Z and Z† matrices. The result is that there are four differentterms

35

1. ⟨tr(AZBZ†)

⟩(A.7)

2. ⟨tr(AZBZCZ†DZ†)

⟩(A.8)

3. ⟨tr(AZBZ†CZDZ†)

⟩(A.9)

4. ⟨tr(AZBZ†)tr(CZDZ†)

⟩(A.10)

where the matrices A, B, C, D can be either any of the diagonal Fermi matrices, or the unitmatrix. Each of these terms will be computed separately and expressed in terms of frequencyand energy integrals in the following.

The first term is⟨tr(AZBZ†)

⟩=∑pq

AppBqqγp−q =

∫dω γ(ω)

∫dεA(ε+ ω)B(ε) (A.11)

upon relabeling the variables as ω = p− q, ε = q.The second term takes the form

⟨tr(AZBZCZ†DZ†)

⟩=∑pqrs

γs−rγp−s (δqsAppBqqCrrDss + δp−q,s−rAppBqqCrrDss)

=∑pqr

γq−rγp−qAppBqqCrrDqq +∑pqr

γp−qγq−rAppBqqCrrDp−q+r,p−q+r

=∫dωdω′ γ(ω)γ(ω′)

∫dεA(ε+ ω + ω′)B(ε+ ω)C(ε)D(ε+ ω)

+∫dωdω′ γ(ω)γ(ω′)

∫dεA(ε+ ω + ω′)B(ε+ ω)C(ε)D(ε+ ω′) (A.12)

by relabeling ω = q − r, ω′ = p− q, and ε = r.The third term becomes

36

⟨tr(AZBZ†CZDZ†)

⟩=∑pqrs

γp−sγr−q (δprAppBqqCrrDss + δqsAppBqqCrrDss)

=∑pqr

γp−rγp−qAppBqqCppDrr +∑pqr

γr−qγp−qAppBqqCrrDqq

=∫dωdω′ γ(ω)γ(ω′)

∫dεA(ε+ ω + ω′)B(ε+ ω)C(ε+ ω + ω′)D(ε+ ω)

+∫dωdω′ γ(ω)γ(ω′)

∫dεA(ε+ ω)B(ε)C(ε+ ω′)D(ε) (A.13)

when the index substitution ω = p− r, ω′ = p− q, ε = q+ r− p is performed in the first sum,and ω = p− q, ω′ = r − q, ε = q in the second sum.

Finally, the term containing a product of two traces can be expressed as

⟨tr(AZBZ†)tr(CZDZ†)

⟩=

=∑pqp′q′

γp−qγp′−q′δp−q,p′−q′AppBqqCp′p′Dq′q′ +∑pqp′q′

γp−qγp′−q′AppBqqCp′p′Dq′q′

=[∫

dω γ(ω)∫dεA(ε+ ω)B(ε)

] [∫dω′ γ(ω′)

∫dε′C(ε′ + ω′)D(ε′)

]+∫dω γ(ω)2

[∫dεA(ε+ ω)B(ε)

] [∫dε′C(ε+ ω′)D(ε)

](A.14)

when denoting ω = p− q, ε = q, and ε′ = q′.

37

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