Nonzero temperature effects on antibunched photons emitted by a quantum point contact out of equilibrium

Nonzero temperature effects on antibunched photons emitted by a quantum point contact out of equilibrium

Fulga, I.C.; Hassler, F.; Beenakker, C.W.J.

Citation

Fulga, I. C., Hassler, F., & Beenakker, C. W. J. (2010). Nonzero temperature effects on

antibunched photons emitted by a quantum point contact out of equilibrium. Physical Review B, 81(11), 115331. doi:10.1103/PhysRevB.81.115331

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License: Leiden University Non-exclusive license Downloaded from: https://hdl.handle.net/1887/51731

Note: To cite this publication please use the final published version (if applicable).

Nonzero temperature effects on antibunched photons emitted by a quantum point contact out of equilibrium

I. C. Fulga, F. Hassler, and C. W. J. Beenakker

Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 共Received 25 January 2010; revised manuscript received 3 March 2010; published 26 March 2010兲 Electrical current fluctuations in a single-channel quantum point contact can produce photons共at frequency

␻ close to the applied voltage V⫻e/ប兲 which inherit the sub-Poissonian statistics of the electrons. We extend the existing zero-temperature theory of the photostatistics to nonzero temperature T. The Fano factorF 共the ratio of the variance and the average photocount兲 is ⬍1 for T⬍Tc 共antibunched photons兲 and ⬎1 for T⬎Tc

共bunched photons兲. The crossover temperature Tc⯝⌬␻⫻ប/kBis set by the bandwidth⌬␻ of the detector, even ifប⌬␻ⰆeV. This implies that narrow-band detection of photon antibunching is hindered by thermal fluctua- tions even in the low-temperature regime where thermal electron noise is negligible relative to shot noise.

DOI:10.1103/PhysRevB.81.115331 PACS number共s兲: 73.50.Td, 42.50.Ar, 42.50.Lc, 73.23.⫺b

I. INTRODUCTION

It is a celebrated result of Glauber that classical fluctua- tions of the electrical current I共t兲 produce photons with Pois- son statistics.¹The variance Var n of the number n of pho- tons detected in a time t_detis then equal to the mean具n典. The photons produced by a classical current thus behave as inde- pendent classical particles. Quantum fluctuations of the cur- rent共with I共t兲 and I共t⬘兲 noncommuting operators兲 change the photostatistics. The bosonic nature of the photons would naturally lead to photon bunching, with Var n⬎具n典. Photon antibunching, with Var⬍具n典, is also possible, if the photons can somehow inherit the sub-Poissonian statistics of the electrons.²One then speaks of nonclassical light.³

While nonclassical light emitted by a quantum mechani- cal current has not yet been observed, the theory is well developed^4–6 and there is an active experimental search.^7,8 Nonclassical light emitted by a quantum conductor such as a quantum point contact⁹would occur in a continuous range of GHz frequencies, in contrast to the discrete frequencies pro- duced by electronic transitions in quantum dots or quantum wells.^10,11Various methods of measuring the photostatistics have been developed, such as detection by means of photo- assisted tunneling,^12–14or by means of the Hanbury-Brown- Twiss effect.^8,15

The theoretical prediction⁵ is that photons emitted by a single-channel quantum point contact should have a Fano factor F=Var n/具n典 smaller than unity at zero temperature, for frequencies␻close to the applied voltage V⫻e/ប. More specifically,

F = 1 −²₃共␥0⌬␻兲␶共1 −␶兲, 共1.1兲 for photodetection with efficiency␥0in the frequency inter- val 共eV/ប−⌬␻^{, eV}/ប兲. The transmission probability ␶ through the quantum point contact is assumed to be energy independent on the scale of eV. Equation共1.1兲 is derived in the limit of weak coupling 共␥0⌬␻Ⰶ1兲 of electrons to pho- tons, so that the deviations from Poisson statistics remain small. It is also assumed that the photons can be detected individually, see Ref.6for an alternative detection scheme.

It is the purpose of the present paper to extend the theory of Ref. 5 to nonzero temperatures, in order to identify the conditions on the temperature needed to observe the photon antibunching. Clearly, photon bunching should take over when the electrical shot noise drops below the thermal noise, which happens when kBT becomes larger than eV. While k_BT⬍eV is the condition for photon antibunching in the case of wide-band detection, a more stringent condition k_BT

⬍ប⌬␻ holds for narrow-band detection.

More precisely, we obtain a crossover temperature Tc

⬇ប⌬␻/4kB at which F=1 for ⌬␻ⰆeV. In this low- temperature regime shot noise still dominates over thermal noise, yet the photon antibunching is lost. One qualitative way to understand this is, is to compare the coherence time t_coh⯝1/⌬␻of the detected radiation with the coherence time tT⯝ប/kBT of thermally excited electron-hole pairs. For t_coh

⬎tT the detected photons result from many uncorrelated electron-hole recombination events, and the one-to-one rela- tionship between electron and photon statistics is lost.

In the next section, we give the nonzero temperature gen- eralization of the theory of Ref. 5, and then in Sec. III, we specialize to the shot-noise regime k_BTⰆeV. General results in both the shot noise and thermal noise regimes are pre- sented in Sec. IV. Technical details are summarized in Ap- pendixes A and B.

II. GENERATING FUNCTION AT NONZERO TEMPERATURE

We seek the non-zero-temperature generalization of the formula⁵

F共␰兲 =

冓

^m=1

^兿

^{N Det共1 + Tm关eZeZ†− 1兴兲}

冔

^{, 共2.1兲}

for the factorial-moment generating function F共␰兲 of the pho- tocount. We first introduce the notation and then present the required generalization.

The photons are produced by time-dependent current fluc- tuations in a quantum point contact, characterized by trans-

mission eigenvalues T₁, T₂, . . . T_N, with N the number of propagating electronic modes共counting both orbital and spin degrees of freedom兲. The current flows between two reser- voirs, with Fermi functions

fL共␧兲 = 共1 + exp关共␧ − eV − EF兲/kBT兴兲⁻¹, 共2.2兲

fR共␧兲 = 共1 + exp关共␧ − EF兲/kBT兴兲⁻¹. 共2.3兲

The current fluctuations can be due to thermal noise共at tem- perature T兲 or due to shot noise 共at a voltage V applied over the point contact兲. We take the transmission eigenvalues Tm

as energy independent in the range max共eV,kBT兲 near the Fermi energy EF.

The photons are detected during a time t_det in a narrow frequency interval⌬␻around frequency⍀, as determined by the detection efficiency ␥共␻兲. Antibunching of the photons requires that⍀ is tuned to the applied voltage, ⍀⯝eV/ប. 共In the following we setប and e equal to unity.兲

The average 具¯典 in Eq. 共2.1兲 indicates a Gaussian inte- gration over the complex numbers zp,

具 ¯ 典 =

兿

_p ^␥_␲^p

冕

^{d2zpe−␥p兩zp兩2. . . . 共2.4兲}

The matrix Z has elements Z_pp⬘=␰^1/2zp−p⬘␥p−p⬘, depending only on the difference of the indices p and p⬘. This differ- ence represents the discretized frequency ␻p−p⬘=共p−p⬘^兲

⫻2␲/tdet of a photon emitted by an electronic transition from energy ␧p to ␧p⬘ and detected with efficiency ␥p−p⬘

=共2␲/tdet兲␥共␻p−p⬘兲. Since␥共␻兲⬅0 for␻ⱕ0, the matrix Z is a lower-triangular matrix. The discretization of frequency and energy is eliminated at the end of the calculation, by taking the limit t_det→⬁.

The expansion

F共␰兲 =

兺

k=0

⬁ ␰^k

k!具n^k典f 共2.5兲

of F共␰兲 in powers of ␰ gives the factorial moments 具n^k典f

=具n共n−1兲共n−2兲¯共n−k+1兲典 of the number of detected pho- tons. Antibunching means that the variance of the photocount Var n =具n²典−具n典²is smaller than the average, or equivalently that the Fano factorF=Var n/具n典⬍1.

As outlined in Appendix A, at nonzero temperature we have instead of Eq. 共2.1兲 the generating function

F共␰兲 =

冓

^m=1

^兿

^{N Det}

^{冉 冑}

^{Tm1 + T共1 − TmfmL共e兲fRZe共eZ−Z†− 1兲− eZ†兲}

^冑

^{T1 + Tm共1 − TmfRm共e兲f−ZLe共e−Z−Z††− 1兲− eZ兲}

^冊 冔

^{. 共2.6兲}

The Fermi function fL共␧兲 in the left electronic reservoir is contained in the diagonal matrix f_L, with elements 共fL兲pp⬘

=␦pp⬘fL共␧p兲, ␧p= p⫻2␲/tdet. Similarly, the Fermi function f_R共␧兲 in the right reservoir is contained in the diagonal ma- trix fR.

Following the steps in Appendix A, the expression 共2.6兲 can be reduced to the more compact form

F共␰兲 =

冓

^m=1

^兿

^{N Det共1 + Tm关f¯ReZ†fL− fRe−Zf¯L兴M兲}

冔

^,

共2.7兲 with the definitions f¯_L= 1 − f_L, f¯_R= 1 − f_R, M=e^Z− e^−Z†. The zero-temperature limit关Eq. 共2.1兲兴 follows from Eq. 共2.7兲 by setting fL= 1, fR= 0 in the energy interval EF⬍␧⬍EF+ V.

共There are no current fluctuations outside of this energy in- terval for T = 0.兲

III. SHOT NOISE REGIME

The result关Eq. 共2.7兲兴 holds for any temperature, provided that the energy dependence of the transmission eigenvalues may be neglected. In particular, it describes both thermal noise and shot noise. A simpler formula is obtained in the shot noise regime kBTⰆV. Thermal noise can then be ne-

glected and only the finite temperature effects on the shot noise are retained. We assume ⌬␻Ⰶ⍀⯝V, so even if kBT ⰆV, the relative magnitude of ⌬␻^{and k}BT is still arbitrary.

A. Generating function

The first simplification in this regime is that we may set fRe^−Zf¯L→0, since fR共␧兲f¯L共␧⬘^{兲→0 for ␧}⬘ⱕ␧. Equation 共2.7兲 reduces to

F共␰兲 =

冓

^m=1

^兿

^{N Det共e−Z†+ TmfLMf¯R兲}

冔

^{, 共3.1兲}

where we have multiplied by Det e^−Z†= 1.

The second simplification is that we can ignore energies separated by pV with pⱖ2, because V is the largest energy scale in the problem. Since Z^pand Z^†pconnect energies sepa- rated by p⍀⯝pV, we may set Z^p, Z^†p→0 for pⱖ2. From Eq. 共3.1兲 we arrive at

F共␰兲 =

冓

^m=1

^兿

^{N Det共1 − Z†+ TmfL共Z + Z†兲f¯R兲}

冔

^{. 共3.2兲}

Following the steps in Appendix B, the determinant may be rewritten in the more convenient form 共bilinear in Z,Z^†兲,

FULGA, HASSLER, AND BEENAKKER PHYSICAL REVIEW B 81, 115331共2010兲

115331-2

F共␰兲 =

冓

^m=1

^兿

^{N Det共1 + Tm共1 − Tm兲fLZf¯RZ†兲}

冔

^{. 共3.3兲}

B. Moment expansion

The generating function 共3.3兲 is of the form F共␰兲

=兿mDet共1+Xm兲 with Xmof order␰. An expansion in powers of ␰can be obtained by starting from the identity

兿

^{mDet共1 + Xm兲 = exp}

关 ^兺

^mTr ln共1 + Xm兲

兴

^{, 共3.4兲}

and expanding in turn, the logarithm and the exponential. Up to second order in ␰we have the expansion

F共␰兲 = 1 +

具 ^兺

^{mTr Xm}

典

⁻¹2

具 ^兺

^{mTr Xm2}

典

⁺¹2

具共 ^兺

^{mTr Xm}

兲

²

典

+O共␰³兲, 共3.5兲

from which we can extract the first two factorial moments,

F共␰兲 = 1 +␰具n典 +1

2␰²共具n²典 − 具n典兲 + O共␰³兲. 共3.6兲 We perform the Gaussian averages and obtain the average photocount具n典 and the variance Var n=具n²典−具n典²in the shot noise regime,

具n典 =t_det

2␲^S1

冕

^{d␻␥共␻兲}

冕

^{d␧fL共␧ +␻兲f¯R共␧兲, 共3.7兲}

Var n =具n典 +t_det 2␲^S1

2

冕

^d␻

冋

^␥共␻兲

冕

^{d␧fL共␧ +␻兲f¯R共␧兲}

册

²

−t_det

2␲^S2

冕

^d␧

冋

^fL共␧兲

冕

^{d␻␥共␻兲f¯R共␧ −␻兲}

册

²

−t_det

2␲^S2

冕

^d␧

冋

^{f¯R共␧兲}

^冕

^{d␻␥共␻兲fL共␧ +␻兲}

册

^{2. 共3.8兲}

We have defined

S_p=

兺

_m ^{关Tm共1 − Tm兲兴p. 共3.9兲}

Since the two reservoirs are at the same temperature, we can write f_L共␧兲= f共␧−V−EF兲 and f¯R= f共EF−␧兲 in terms of a single Fermi function

f共␧兲 = 共1 + e^␧/kBT兲⁻¹. 共3.10兲

We abbreviate ⌫共␧,␻兲=␥共␻兲f共␧兲f共␻⁻␧−V兲 and can then write Eqs.共3.7兲 and 共3.8兲 in the compact form

具n典 =t_det

2␲^S1

冕

^d␻

冕

^{d␧⌫共␧,␻兲, 共3.11兲}

Var n =具n典 +t_det

2␲

冕

^d␻

冕

^{d␧⌫共␧,␻兲}

⫻

冋

^S12

冕

^{d␧⬘⌫共␧⬘,␻兲 − 2S2}

冕

^{d␻⬘⌫共␧,␻⬘兲}

册

^.

共3.12兲 The difference Var n −具n典 contains a positive term ⬀S₁²and a negative term ⬀S2. The sign of this difference determines whether there is bunching or antibunching of the detected photons.

C. Crossover from antibunching to bunching

To investigate the crossover from antibunching to bunch- ing with increasing temperature, we take a block-shaped re- sponse function

␥共␻兲 =

再

^{␥00 if V −otherwise.⌬␻⬍␻⬍ V}

冎

^{, 共3.13兲}

In the low-temperature regime kBTⰆ⌬␻ the function

⌫共␧,␻兲 then has a block shape as well and we recover the results

具n典 =t_det⌬␻ 2␲ ␥0⌬␻¹

2S₁, 共3.14兲

Var n −具n典 =t_det⌬␻

2␲ ^共␥0⌬␻兲²1

3共S12− 2S₂兲 共3.15兲 of Ref.5. These correspond to a Fano factor

F = 1 +²3␥0⌬␻共S1− 2S₂/S1兲. 共3.16兲 For a single-channel conductor S₂= S₁², so there is antibunch- ing共F⬍1兲 at low temperatures.

At high temperatures kBTⰇ⌬␻, but still in the shot-noise regime k_BTⰆV, we may substitute ⌫共␧,␻兲→

−␥共␻兲kBTdf共␧兲/d␧ into Eqs. 共3.11兲 and 共3.12兲, which gives 具n典 =t_det⌬␻

2␲ ␥0kBTS₁, 共3.17兲

Var n −具n典 =t_det⌬␻

2␲ ^共␥0k_BT兲²S₁². 共3.18兲 The Fano factor

F = 1 +␥0k_BTS₁ 共3.19兲 is now⬎1—hence, there is photon bunching.

The crossover temperature Tc, at whichF=1, can be cal- culated numerically from Eqs. 共3.11兲 and 共3.12兲. In the single-channel case, when S₂= S₁², we find

k_BT_c⬇ 0.25⌬␻^. 共3.20兲 The crossover is shown graphically in Fig. 1.

IV. BEYOND THE SHOT-NOISE REGIME

In the previous section we assumed kBTⰆV 共shot noise regime兲. For arbitrary relative magnitude of kBT and V, the general formula共2.7兲 can be used. With the help of Eq. 共3.4兲, this general expression of the form Det共1+X兲 was expanded to second order in powers of ␰. In this case however, since X =O共

冑

_␰兲, terms up to order X⁴had to be retained. The first two moments of n are obtained as integrals over energy and frequency, similar to Eqs. 共3.11兲 and 共3.12兲 but containing many more terms in the integrands. The results shown in Figs.2and3are for the case N = 1, T₁=␶of a single channel, and for the box-shaped response function共3.13兲.

As expected, all curves converge to the shot-noise results when kBTⰆV 共shown dashed兲. At higher temperatures, the Fano factor lies above the shot noise limit due to the appear- ance of thermal noise. The temperature T_c at which anti- bunching crosses over into bunching, so whenF=1, follows the shot-noise limit 关Eq. 共3.20兲兴 for narrow-band detection

共⌬␻ⰆV兲. With increasing bandwidth, Tc drops below the shot noise limit, in particular for small transmission probabil- ity ␶^{. For}␶= 0.5 the shot-noise limit remains accurate even for bandwidths ⌬␻as large as V/2.

V. CONCLUSION

In conclusion, we have investigated the effects of a non- zero temperature on the degree of antibunching of photons produced by current fluctuations in a quantum point contact.

Antibunching crosses over into bunching as a result of ther- mal noise in the point contact, but this is not the dominant effect in the case of narrow-band detection. In that case, the finite coherence time of electron-hole pairs governs the tran- sition from photon antibunching to photon bunching, which occurs at a temperature kBTc⯝⌬␻^{even if k}BTcⰆV 共so even if thermal noise is negligible relative to shot noise兲.

The optimal conditions for the observation of antibunched photons are reached for a bandwidth ⌬␻⬇V/2 and a trans- mission probability␶⬇1/2 through a 共spin-resolved兲 single- channel quantum point contact. In that case k_BT_c⬇V/8 has the largest value at any given applied voltage. The currently available⁸ detection range 共4 GHz⬍␻⬍8 GHz⇒⌬␻

= 4 GHz= V/2兲, should make it possible to detect antibunch- ing at temperatures below 1 GHz⬵50 mK.

ACKNOWLEDGMENTS

We thank D. C. Glattli for a discussion which motivated this work and for correspondence on the experimental pa- rameters. Our research was supported by the Dutch Science Foundation NWO/FOM and by the EU Network NanoCTM.

APPENDIX A: DERIVATION OF THE GENERATING FUNCTION AT NONZERO TEMPERATURE We briefly describe how the analysis of Ref. 5 can be generalized to nonzero temperatures, in order to arrive at Eq.

共2.6兲. Referring to the equations in that paper, the first equa- tion which changes is Eq. 共5兲 of Ref.5, which now reads FIG. 1. Crossover with increasing temperature from antibunch-

ing 共Fano factor F⬍1兲 to bunching 共F⬎1兲 of the photons pro- duced by a single-channel quantum point contact in the shot noise regime 共kBTⰆV兲. The solid curve is calculated from Eqs.

共3.11兲–共3.13兲. The dashed line is the asymptote 关Eq. 共3.19兲兴. The crossover temperature T_cfrom Eq.共3.20兲 is indicated.

FIG. 2. Same as Fig.1, but now without making the restriction to the shot noise regime共so without assuming kBTⰆV兲. The two solid curves are calculated from Eq.共2.7兲 for two values of ⌬␻/kBT 共both for the single-channel case with transmission probability ␶

= 0.5兲. Both curves converge to the shot noise result at low tempera- tures共shown dashed兲.

FIG. 3. 共Color online兲 Dependence of the crossover temperature T_c共at which F=1兲 on the bandwidth ⌬␻. The points are calculated from Eq.共2.7兲 for three values of the single-channel transmission probability␶. For ⌬␻ⰆV all points converge to the shot noise limit 关Eq. 共3.20兲兴 共dashed line兲.

FULGA, HASSLER, AND BEENAKKER PHYSICAL REVIEW B 81, 115331共2010兲

115331-4

F共␰兲 = 具e^−a†DZae^b†DZbe^b†DZ†be^{−a†DZ†a}典. 共A1兲 The four factors correspond to the four current operators that need to be taken into account: I_in^†, I_out^† , I_out, and I_in.

The operator a^† creates an incoming electron, while b^† creates an outgoing electron. The matrix D projects on the right lead, where the current is evaluated. 共Since D com- mutes with Z, we can write DZ instead of DZD.兲 One can relate b = Sa, with S the unitary scattering matrix, so one can write the entire generating function in terms of the operators a. The expectation value 具¯典 is both an expectation value over the fermion operators a, as well as the average over the Gaussian variables Z , Z^†.

Following the steps of Ref. 5, we calculate the expecta- tion value of the fermion operators by means of the identity

冓 ^兿

^{n ea†Ana}

冔

= Det共1 + AB兲, 共A2兲 A =

冉 ^兿

^{n eAn}

冊

^{− 1, Bij=具a†jai典. 共A3兲}

We have B_ij=␦ijf_i, with f_ithe Fermi occupation number in channel i. The matrix A is given by A = e^Xe^Ye^Y†e^X†− 1, with X = −DZ and Y = S^†DZS. Notice that X^p= D共−Z兲^p and Y^p

= S^†DZ^pS.

We now make the assumption of an energy independent scattering matrix, so S , S^†commute with Z , Z^†. The determi- nant is invariant under a change of basis, and by working in the eigenchannel basis we can reduce S to a 2⫻2 matrix Sm

for each eigenchannel,

Sm=

冉 ^冑

^{1 − T}

^冑

^Tm^m −

冑 ^冑

1 − T^Tm m

冊

^{, 共A4兲}

with Tm, m = 1 , 2 , . . . N the transmission eigenvalue. The ma- trix structure of f, D, and Z in this basis is

f =

冉

^{f0L f0}R

冊

^{, D =}

冉

^{0 00 1}

冊

^{, Z =}

冉

^{Z0 Z0}

冊

^{. 共A5兲}

Substitution of Eqs. 共A2兲–共A5兲 into Eq. 共A1兲 leads after some algebraic manipulations to the result 关Eq. 共2.6兲兴.

The determinant in Eq.共2.6兲 can be reduced by means of the folding identity

Det

冉

^MM1121 ^MM¹²₂₂

冊

^{= Det M11Det共M22− M21M11−1M12兲,}

共A6兲 leading to

F共␰兲 =

冓

^m=1

^兿

^{N Det关1 + TmfL共eZeZ†}− 1兲兴Det共1 + TmfR共e^−Ze^−Z†− 1兲

− Tm共1 − Tm兲fR共e^−Z− e^Z†兲关1 + TmfL共e^Ze^Z†− 1兲兴⁻¹fL共e^−Z†− e^Z兲兲

冔

^{. 共A7兲}

We continue the reduction of the determinant, using first the identity

关1 + TmfL共e^Ze^Z†− 1兲兴⁻¹fL共e^−Z†− e^Z兲 = − fL共e^Ze^Z†− 1兲关1 + TmfL共e^Ze^Z†− 1兲兴⁻¹e^−Z†, 共A8兲 then multiplying the determinant by Det e^Z†= 1, and finally combining the product of three determinants into a single deter- minant. In this way we eliminate the matrix inversion, arriving at

F共␰兲 =

冓

^m=1

^兿

^{N Det共关1 + TmfR共e−Ze−Z†− 1兲兴eZ†关1 + TmfL共eZeZ†− 1兲兴 + Tm共1 − Tm兲fR共e−Z− eZ†兲fL共eZeZ†− 1兲兲}

冔

=

冓

^m=1

^兿

^{N Det共1 + Tm关共1 − fR兲eZ†fL− fRe−Z共1 − fL兲兴共eZ− e−Z†兲兲}

冔

^{. 共A9兲}

This is Eq.共2.7兲 in the main text.

APPENDIX B: DERIVATION OF THE GENERATING FUNCTION IN THE SHOT NOISE REGIME Starting from the expression共3.2兲 for the generating func- tion in the shot noise regime kBTⰆV, we give the steps required to arrive at the bilinear form共3.3兲. We group terms with Z and with Z^† in the matrices A_m= T_mf_LZf¯_R and B_m

= T_mf_LZ^†f¯_R− Z^†, so that Eq.共3.2兲 can be written as

F共␰兲 =

冓

^m=1

^兿

^{N Det共1 + Am+ Bm兲}

冔

^{. 共B1兲}

Because energies separated by V^p with pⱖ2 can be dis- carded, we may set A_m²→0, Bm

2→0. For any pair of matrices A , B which square to zero, one has the identity

Det共1 + A + B兲 = Det共1 − AB兲. 共B2兲 This leads to

F共␰兲 =

冓

^m=1

^兿

^{N Det共1 + TmZf¯RZ†fL− Tm2Zf¯RfLZ†f¯RfL兲}

冔

^{. 共B3兲}

Eq. 共3.3兲 follows by noting that Zf¯Rf_L→Zf¯R for k_BT Ⰶ⍀⯝V, since the Fermi function fLin this term is evaluated at energies near E_F, where it can be replaced by unity. Simi- larly Z^†f¯_Rf_L→Z^†f_L, since f¯_R is evaluated at energies near E_F+ V where it can be replaced by unity.

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