Antibunched photons emitted by a quantum point contact out of equilibrium

Antibunched photons emitted by a quantum point contact out of

equilibrium

Beenakker, C.W.J.; Schomerus, H.

Citation

Beenakker, C. W. J., & Schomerus, H. (2004). Antibunched photons emitted by a quantum

point contact out of equilibrium. Retrieved from https://hdl.handle.net/1887/1295

Version:

Not Applicable (or Unknown)

License:

Leiden University Non-exclusive license

Downloaded from:

https://hdl.handle.net/1887/1295

Antibunched Photons Emitted by a Quantum Point Contact out of Equilibrium

C.W. J. Beenakker1 and H. Schomerus2

Ilnstituut-Lorentz, Universiteil Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands ~Max-Planck-Institut für Physik komplexer Systeme, Nothmtier Slrasse 38, 01187 Dresden, Germany

(Received 2 May 2004; published 23 August 2004)

Motivated by t he experimcntal search for "GHz nonclassical light," we identify the conditions under which currcnt flucluations in a narrow constriction generate sub-Poissonian radiation. Antibunched elcctrons gencrically produce bunchcd pholons, because the same pholon mode can be populated by electrons decaying independently from a ränge of initial energies. Photon antibunching becomes possible at frequencies close to the applied voltage V X e/K, when the initial energy ränge of a decaying electron is restricted. The condition for photon antibunching in a narrow frcquency interval below eV/K reads [£„7",, (l — Γ,,)]2 < 2£,,[7"η(1 — 7"„)]2, with T„ an eigenvalue of the transmission

matrix. This condition is satisfied in a quanlum point contact, where only a single T„ differs from 0 or 1. The photon statistics is then a superposition of binomial distributions.

DOI: 10.1103/PhysRevLett.93.096801 PACS numbers: 73.50.Td, 42 50 Ar, 42.50 Lc, 73.23-b

In a recent experiment [1], Gabelli et cd. have measured the deviation from Poisson statistics of photons emitted by a resistor in equilibrium at m K temperatures. By cross correlating the power fluctuations they detected photon bunching, meaning that the variance Vam = (n-) — (n)2

in the number of detected photons exceeds the mean photon count (n). Their experiment is a Variation on the quanlum optics experiment of Hanbury Brown and Twiss [2], but now at GHz frequencies.

In the discussion of the implications of their novel experimental technique, Gabelli et a l. noticed that a general theory [3] for the radiation produced by a con-ductor out of equilibrium implies that the deviation from Poisson statistics can go either way: Super-Poissonian fluctuations (Varn > (n), signaling bunching) are the rule in conductors with a l arge number of scattering channels, while sub-Poissonian fluctuations (Varn < (n), signaling antibunching) become possible in few-channel conductors. They concluded that a quantum point contact could therefore produce GHz nonclassical light [4].

It is the purpose of this work to identify the conditions under which electronic shot noise in a quantum point contact can generate antibunched photons. The physical picture that emerges differs in one essential aspecl from eleclron-hole recombination in a quantum dot or quantum well, which is a familiär source of sub-Poissonian radia-tion [5-7]. In those Systems the radiaradia-tion is produced by transitions between a few discrele levels. In a quantum point contact the transitions cover a continuous ränge of energies in the Perm i sea. As we will see, this continuous spectrum generically prevents antibunching, except at frequencies close to the applied voltage.

Before presenting a quantitative analysis, we first dis-cuss Ihe mechanism in physical terms. As depicted in Fig. l , electrons are injected through a constriction in an energy ränge eV above the Per m i energy EF, leaving behind holes at the same energy. The statistics of the

Charge Q transferred in a time τ ϊΐ> Ιϊ/eV is binomial [8], with Vai'Q/e < (Q/e). This electron antibunching is a result of Ihe Pauli principle. Each scattering channel n = \, 2, . . . , N in the constriction and each energy inter-val 8E = Ιϊ/τ contributes independently to Ihe Charge statistics. The photons excited by the electrons would inherit Ihe antibunching if there would be a one-to-one correspondence between the transfer of an electron and the population of a photon mode. Generically, this is not what happens: A photon of frequency ω can be exciled by each scattering channel and by a ränge eV — Κω of initial energies. The resulling statislics of photocounts is negative-binomial [3], with Varn > (n). This is the same photon bunching äs in black-body radiation [9].

F.

VOLUME 93, NUMBER 9

P H Y S I C A L R E V I E W L E T T E R S

27 AUGUST 2004weck ending

In order to convert antibunched electrons into

anti-bunched photons, it is sufficient to ensuie a one-to-one

correspondence between electron modes and photon

modes. This can be reahzed by concentratmg the current

fluctuations in a smgle scattermg channel and by

restnct-mg the energy ränge eV — Κω. Indeed, in a

single-channel conductoi and m a narrow frequency lange ω S

eV/h we obtam sub-Poissonian photon statistics

regard-less of the value of the transmission probability. In the

more general multichannel case, photon antibunching is

found if [χ,,Γ,,Ο - T

n

)]

2

< 2£

n

[r„(l - Γ,,)]

2

(with T„

an eigenvalue of the transmission matnx product tfi)

The starting point of our quantitative analysis is the

general relationship of Ref. [3] between the photocount

distnbution P (n) and the expectation value of an ordered

exponential of the electncal current operator:

/>(«)=! lim

^-n\ ί— ι άξ" (1)

(2)

We summanze the notation. The function Γ(ξ) =

ΧΓ=ο(£

λ

Α')(«

Α

)ι is the generatmg function of the

facto-nal moments (n

k

)

i

= (n(n — \)(n — 2) · · · (n — k + I)}.

The current operator I = 7

out

— 7

U1

is the difference of

the outgoing current 7

out

(away from the constnclion) and

the incoming current 7

m

(toward the constuction). The

symbol 0 indicates ordermg of the cunent operatois from

left to nght in the order I

m

, 7^,,, 7

ollt

, 7

ln

The real

frequency-dependent response function γ(ω) is

propor-tional to the coupling strength of conductor and

photode-tectoi and proportional to the detector efficiency. Positive

(negative) ω corresponds to absorption (emission) of a

photon by the detector. We consider photodetection by

absoiption, hence γ(ω) = 0 for ω Ä 0. Integrals over

frequency should be interpreted äs sums over discrete

modes ω

ρ

= p Χ 2 π/τ, p = l, 2, 3, . . The detection

time τ is sent to infinity at the end of the calculation.

We denote γ

ρ

= γ(ω

ρ

) Χ 2ττ/τ, so that f αωγ(ω) —+

ΣρΎρ· F°

r ease

°f notation we set h = l, e = 1.

The exponent in Eq. (2) is quadiatic in the current

operators, which comphcates the calculation of the

ex-pectation value. We remove this complication by

mtroduc-mg a Gaussian field ζ(ω) and performing a

Hubbard-Stratonovich tiansformation,

F(i) =

άωγ(ω)(ζ(ω)Ιϊ(ω)

(3)

The angular brackets now indicate both a quantum

me-chanical expectation value of the current operators and a

classical average over independent complex Gaussian

variables z

p

= ζ(ω

ρ

) with zero mean and variance

(\z„\

2

) = i/r„.

We assume zero temperature, so that the incoming

current is noiseless. We may then replace 7 by 7

ollt

and

restnct ourselves to energies ε in the ränge (0, V) above

Ep Let &,ΐ(ε) be the operator that creates an outgomg

electron in scattermg channel n at energy ε. The outgoing

current is given in terms of the electron operatois by

= Γ ds\bl(s)b

l

,(e +ω)

/ n *--* (4)

Energy ε

ρ

= p Χ 2ττ/τ is discretized in the same way äs

frequency. The eneigy and channel indices p, n aie

col-lected m a vector b with elements b

pl

, =

(2ir/T)

l/2

b„(e ). Substitution of Eq. (4) into Eq (3) gives

(5)

The exponents contain the product of the vectors

b,b^ and a matrix Z with elements Z

pn p

/„/ =

ξ

ί

/

2

δ

ηιι

ιζ

ρ

_

ρ

ιγ

ρ

_

ρ

ι. Notice that Z is diagonal in the

chan-nel indices n, n' and Iowei-triangulär in the energy

m-dices p, p

1

.

Because of the ordermg O of the current operators, the

smgle exponential of Eq. (3) factonzes into the two

non-commuting exponentials of Eq. (5). In order to evaluate

the expectation value efficiently, we would like to bring

this back to a smgle exponential—but now with normal

ordermg N of the fermion creation and annihilation

operators. (Normal ordermg means b^ to the left of b,

with a minus sign for each permutation) This is

accom-phshed by means of the operatoi identity [10]

\e

A

<

- l M

(6)

vahd foi any set of matnces A, The quantum mechanical

expectation value of a normally ordered exponential is a

determmant [11],

{We

b>Ab

) = Det(l + AB), B„ = (b]b,) (7)

In oui case A = e

z

<?

zt

- l and B = trf, with t the N X N

transmission matnx of the constnction.

In the expenmentally lelevant case [1,12] the response

function γ(ω) is sharply peaked at a frequency Ω < V,

with a width Δ «: Ω. We assume that the energy

depen-dence of the transmission matnx may be disregarded on

the scale of Δ, so that we may choose an ε-mdependent

basis m which tfi is diagonal. The diagonal elements are

the transmission eigenvalues T\, T

2

,... T

N

& (0, 1).

Combining Eqs. (5)-(7) we arrive at

„z*

(8)

(In the second equahty we used that Detez1 = l, smce Z

is a lowei-tiiangulai matnx) The lemaimng average is over the Gaussian vanables zr contamed in the matiix Z

Smce the inteiesting new physics occuis when Ω is close to V, we simplify the analysis by assummg that

γ (ω) = 0 foi ω < V/2 Foi such a lesponse function one

has Z2 = 0 (This amounts to the Statement that no

elec-tion with excitaelec-tion eneigy ε < V can pioduce moie than a single photon of fiequency ω > V/2 ) We may theiefoie leplace e7· —>· l + Z and e~^ —> l - Zf in Eq (8) We

then apply the matiix identity

Det(l +A + B) = Det(l -AB) if A2 = 0 = B2, (9) and obtain

(10) We have defined ξΧ = ΖΖ^ and wi itten out the Gaussian aveiage The Heimitian matiix X has elements

PP / ζρ-<ιζ η!-α (Π)

point appioximation bieaks down when the detection fiequency fl appioaches the apphed voltage V Foi V —

Ω :£ Δ one has to calculate the mtegials in Eq (10)

exactly

We have evaluated the geneiatmg function (10) for a lesponse function of the block foi m

if V - Δ < ω if ω < V - Δ,

V,

(14) with Δ < V/2 The fiequency dependence foi ω > V is melevant In the case N = l of a single channel, with tiansmission piobability T] = Γ, we find [15]

InF(f) = ~ Γ άω 1η[1 + ξγ0Τ(1 ~ T)(V - ω)]

2.TT Ιν-Δ

τΔ (1 +λ)1η(1 + λ) - Α

2ττ Α (15)

with χ = ξγ0Τ(] — Τ1) Δ This is a supeiposition of

bino-mial distnbutions The factonal cumulants aie

(16) The second factonal cumulant is negative, so V am < (n) This is sub-Poissoman ladiation

We have not found such a simple closed foi m expies-sion m the moie geneial multichannel case, but it is stiaightfoiwaid to evaluate the low-oidei factonal cumu-lants fiom Eq (10) We find

o

2.ΊΤ — S\,Z (17)

The mtegeis p, p', q lange fiom l to ντ/2ττ

The Gaussian aveiage is easy if the dimensionless shot noise powei 5 = ]Γ;ΙΓ,,(1 — Tn) is » l We may then do the mtegials of Eq (10) in saddle-pomt appioximation, with theiesult [13]

2ττ - ξ5γ(ω)(Υ ~ ω)] (12)

The loganthm InF(^) is the geneiatmg function of the factonal cumulants {{nA))| [14] By expanding Eq (12) in

poweis of ξ we find

2ττ (13)

Equations (12) and (13) lepiesent the multimode supei-position of mdependent negative-bmomial distnbutions [9] All factonal cumulants aie positive, in paiticulai, the second, so V am > (n) This is supei-Poissoman ladiation

When S is not » l, eg , when only a smgle-channel contnbutes to the shot noise, the lesult (12) and (13) lemams valid if V — Ω » Δ This was the conclusion of Ref [3], that nanow-band detection leads geneucally to a negative-bmomial distiibution Howevei, the

saddle-6

- Tn)]p

5Ϊ-252), (18)

J

- 155,5, + 1553), (19)

Antibunching theiefoie ic-with Sp = Χ,,[

qunes S2 < 2S2

The condition on antibunchmg can be geneiahzed to aibitiaiy fiequency dependence of the lesponse function γ(ω) m the lange V — Δ < ω < ν ο ί detected fiequen-cies Foi Δ < V/2 we find

V am — (n) =

27Γ V - Δ

άω'γ(ω') j daj(V - ω)

VOLUME 93, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 27 AUGUST 2004weck ending

piovided that it mcieases monotonically m the lange (V — Δ, V) A steeply increasmg lesponse function in this lange is moie favoiable, but not by much Foi ex-ample, the powei law γ (ω) <* (ω — V + Δ)/; gives the

antibunchmg condition S1? < 2S2 X [l + p/(\ + p)],

which is only weakly dependent on the powei p

In conclusion, we have piesented both a qualitative physical pictuie and a quantitative analysis foi the con-veision of election to photon antibunchmg A simple cntenon, Eq (18), is obtamed foi sub-Poissoman photon statistics, in teims of the tiansmission eigenvalues T„ of the conductoi Smce an ./V-channel quantum pomt contact has only a single 7^ diffeient fiom 0 01 l, it should geneiate antibunched photons in a fiequency band (V — Δ, V)—legaidless of the value of TN The statistics

of these photons is the supeiposition (15) of binomial distubutions, inhented fiom the electionic binomial dis-tnbution Theie aie no stungent conditions on the band width Δ, äs long äs it is <V/2 (in oidei to pievent multi-photon excitations by a single electron [16]) This should make it feasible to use the cioss-conelation technique of Ref [1] to detect the emission of nonclassical miciowaves by a quantum pomt contact

We have benefited fiom conespondence with D C Glatth This woik was suppoited by the Dutch Science Foundation NWO/FOM

[1] J Gabelh, L - H Reydellet, G Feve, J M Berron, B Plagais, P Röche, and D C Glaltli, Phys Rev Leu 93, 056801 (2004)

[2] R H a n b u i y Brown and R Q Twiss, Natuie (London) 177, 27 (1956)

[3] C W J Beenakkci and H Schomcrus, Phys Rev Lctt 86, 700 (2001)

[4] Sub-Poissonian radiation is called "nonclassical" be-cause ils photocount stalistics cannot be interpreted m cla&sical teims äs a superposition of Poisson processes See L Mandel and E Wolf, Optical Coheience and Quantum Optics (Cambudgc Univcisity, Cambridge,

1995)

[5] J K i m , O Benson, H Kan, and Υ Yamamoto, Natuie (London) 397, 500 (1999), C Santon, M Pelton, G Solomon, Υ Dale, and Υ Yamamoto, Phys Rev Lett 86, 1502 (2001)

[6] P Michlei, A Imamoglu, M D Mason, P J Caison, G F Stiouse, and S K Buratto, Nature (London) 406, 968 (2000), P Michler, A Kiraz, C Bechei.W V Schocnfeld, P M Petioff, L Zhang, E Hu, and A Imamoglu, Science 290, 2282 (2000)

[7] Z L Yuan, B E Kaidynal, R M Stevenson, A J Shields, C J Lobo, K Coopcr, N S Beattie, D A Ritchie, and M Peppcr, Science 295, 102 (2002)

[8] L S Levitov and G B Lesovik, JETP Lett 58, 230 (1993)

[9] The negalive-binomial distnbulion P(n) °" (" + 'l~>) X

\_v/(n) + 1]~" counts Ihe number oi partitions of n bo-i,ons among v = τδω/2π states in a fiequency mterval δω The binomial distnbution P(ir) y vn\_i>/(n) — 1] "

counls the numbei öl paiUtions öl nfeimtons among v states

[10] Equation (6) is the multimaliix gcnerahzation oi the well-known identity exp(b^Ab) — JV"cxp[/71 (eA - \)b] [11] K E C a h i l l and R J Glaubei, Phys Rev A 59, 1538

(1999)

[12] R Aguado and L P Kouwenhoven, Phys Rev Lett 84, 1986 (2000)

[13] The saddlc pomt is at zp = 0, so to mtcgiate out the Gaussian iluctualions aiound the saddlc pomt we may hneanzc the detcimmant in Eq (10) ]~]„Det[l + T„(l — Τ,,)ξΧ] = exp[£STiX + O(X2)] The lesult is Eq (12)

[14] Factonal cumulants aie constructed fiom factonal mo-ments in the usual way The hisl two aie ((n})t — (n)

(("2))i = ("2)i ~ (n)~ ~ Vaiw — (n)

[15] Usmg computci algebra, wc find that ln(Det[l + ξΤ(\ — T)X]) = £^_, ln[l + ιηξγ0Τ(1 - Τ)(2π/τ)], foi each

matnx dimensionahly M that wc could check Wc are confident that this closed lorm holds for all M, but we have not yet found an analytical proof Equation (15) follows m the l i m i t M = τΔ/2π—> °ο upon conversion of the summation mto an Integration

[16] Multiphoton excitations do not contubute to Var/i if T„ e {0 1/2 1} for all n [cf Ref [3], Eq (19)] Foi a

quantum pomt contact, one fmds that antibunchmg pci-sists when Δ > V/2 piovided that TN(l - TN) > 1/6