Statistics and fluctuation theorem for boson and fermion transport through mesoscopic junctions

Statistics and fluctuation theorem for boson and fermion transport through mesoscopic junctions

Upendra Harbola,¹ Massimiliano Esposito,^1,2and Shaul Mukamel¹

1Department of Chemistry, University of California, Irvine, California 92697-2025, USA

2Center for Nonlinear Phenomena and Complex Systems, Campus Plaine, Code Postal 231, Universitibre de Bruxelles, B-1050 Brussels, Belgium

共Received 14 February 2007; published 8 August 2007兲

The statistical properties of quantum particles moving between two heat reservoirs at different temperatures are studied by solving the quantum master equation using a generating function technique. Bosons and fermi- ons satisfy the same fluctuation theorem: Logarithm of the ratio of the probabilities of forward and backward transports reaches a constant value at long times, with an asymptotic⬃1/t correction. Non-Poissonian transfer statistics 共bunching for bosons and antibunching for fermions兲 are examined using the Mandel parameter.

These come primarily from the tail of the distribution corresponding to transfer of large number of particles, kⰇ具k典.

DOI:10.1103/PhysRevB.76.085408 PACS number共s兲: 05.60.⫺k, 44.10.⫹i, 05.30.⫺d

I. INTRODUCTION

With the recent advances in technology, it is now possible to construct submicron共mesoscopic兲 electronic devices with fairly controlled physical parameters. Fluctuations play a fundamental role in transport through such small devices.

Quantum effects and the statistical properties of the current carrying particles dominate the transport properties. This has led to several recent experimental^1–6and theoretical^7–11stud- ies devoted to quantum transport statistics of current carrying particles 共fermions and bosons兲 through mesoscopic junc- tions. Most experimental studies employ the Hanbury, Brown, and Twiss¹²共HBT兲 configuration where an incoming particle current共beam兲 is split into two beams and the cross- covariance between the two output beams provides informa- tion about the quantum nature of the source.

In Ref. 3, a HBT-type experiment was conducted on a thermal source of two-dimensional electrons and a negative cross-covariance 共antibunching兲 was directly observed be- tween the two electron beams. This was analyzed later theo- retically by Texier and Büttiker.⁸They predicted that, within certain parameter range, a positive cross-covariance共bunch- ing兲 can also be observed due to inelastic scattering between different current carrying states. This has recently been con- firmed experimentally.²

Theoretical analysis of the photon counting statistics of a laser-driven molecule has been done earlier.^13–17 HBT ex- periments were also carried out on ultracold bosonic atoms.

Full counting statistics of a monoenergetic atom laser ob- tained from the Bose-Einstein condensate of⁸⁷Rb atoms has been carried out.¹⁸Jeltes et al.¹⁹have reported a comparison of the fermionic and the bosonic HBT effect on thermal³He 共fermion兲 and ⁴He 共boson兲 atoms. Most HBT experimental studies focus only on the second-order moment of the par- ticle counting statistics which can be more easily sampled. In order to have a full understanding of transport properties of a quantum device, one needs to understand the full probability distribution, which involves all moments. Öttl et al.¹⁸ have reported the full counting statistics of the atom laser which was found to be Poissonian. However, a systematic compara- tive study共experimental or theoretical兲 of full counting sta-

tistics of bosons and fermions is still lacking. This is the motivation for the present work.

We shall use a quantum master equation^21,20 共QME兲 to compare the statistics of boson and fermion transfers through a mesoscopic system attached to two baths with different temperatures. We compare the full counting statistics of the forward and the backward transfer processes in a nonequilib- rium steady state. We observe super共sub兲-Poissonian behav- ior in the transport statistics of bosons共fermions兲 across the junction. Deviations of the probability distribution functions 共PDFs兲 for the bosons and fermions from Poisson form arise mainly from large共rare兲 fluctuations 共kⰇ具k典, where k is the number of particles transferred in a given time兲. We also compared the steady-state fluctuation theorem^22,23for bosons and fermions and found that bosons with a temperature driv- ing force satisfy a fluctuation theorem very similar to fermi- ons driven by a chemical-potential gradient.²⁴ For finite times, deviations from the fluctuation theorem 共FT兲 decay asymptotically as 1 / t, irrespective of particle statistics.

In the next section, we briefly discuss the QME for a boson system attached to two harmonic baths and obtain the PDF using a generating function technique. We compute the Mandel parameter M共or the Fano factor, F=M +1兲 which is directly obtained from the HBT-type experiments. The fluc- tuation theorem for fermions and bosons is also presented. In Sec. IV, we numerically compare the two statistics. We con- clude in Sec. V.

II. QME FOR BOSON TRANSPORT

We consider a model of noninteracting bosons which can be transferred between two baths held at two different tem- peratures, Tl 共left兲 and Tr 共right兲, through a single site 共the system兲 with energy␻0共ប=1兲. The model can represent, for example, phonons and the system could be a single harmonic vibration. The total Hamiltonian of the system is

H =␻0b^†b +

兺

_i

苸␯␻ia_i^†ai+

兺

_i

␯ ␬i␯共ai†␯

+ ai␯兲共b^†+ b兲, 共1兲

where b^†共b兲 are the boson creation 共annihilation兲 operators at the site. The other two terms represent the free baths and the bath-system coupling, respectively. ␬i␯ is the coupling between the system and the ith oscillator of the ␯^th 共␯

= l , r兲 bath. a^†共a兲 are the bath boson creation 共annihilation兲 operators. These operators satisfy commutation relation

␺i␺j

†−␺j

†␺i=␦ij, ␺i= b,ai␯. 共2兲 The two baths have different temperatures and the system population dynamics will be described by a master equation.

We assume that the system-bath coupling is weak so that it can be treated by second-order perturbation and the bath cor- relation time scale is small compared to the time scale of evolution of the system density matrix. The time dependence of the system density matrix can be described by the QME of the Lindblad form^25,26

⳵

⳵^t␳ⁿ共t兲 = − i␻0关b^†b,␳ⁿ共t兲兴 + 关k¯ub^†␳ⁿ⁻¹共t兲b + k¯db␳ⁿ⁺¹共t兲b^†

− k¯

d␳ⁿ共t兲b^†b − k¯

ubb^†␳ⁿ共t兲 + H.c.兴, 共3兲 where␳ⁿrepresents the system density matrix with n number of quanta on the system. Note that in Eq. 共3兲, coherences between system many-body states with different numbers of quanta do not contribute.²⁷ ¯k

d 共k¯u兲 is the transfer rate of quanta out of共into兲 the system and is given in terms of the bath correlation functions

¯k_d=

兺

_i␯ ^{兩␬i␯兩2}

冕

0

⬁

d␶^ei␻0␶具Bi␯B_i^†_␯共␶兲典, 共4兲

¯k

u=

兺

_i␯ ^{兩␬i␯兩2}

冕

0

⬁

d␶^e−i␻0␶具B†i␯B_i␯共␶兲典, 共5兲

where Bi␯= a_i^†_␯+ ai␯ and 具¯典 represents trace over the bath equilibrium density matrix. The contributions of the two baths to the rates are additive and we have k¯

d= k¯

d l+ k¯

d r and

¯k

u= k¯

u l+ k¯

d r.

By plugging the nth eigenstate 兩n典 from the left and the right to Eq. 共3兲 and using b兩n典=

冑

n兩n−1典 and b^†兩n典

=

冑

n + 1兩n+1典, we obtain²⁰

⳵Pn

⳵^{t = nkuPn−1共t兲 − 关nkd+共n + 1兲ku兴Pn}共t兲 + 共n + 1兲kdPn+1共t兲, 共6兲 wherePnis the probability of having n共=0,1, ... ,⬁兲 quanta in the system, with k_d= k¯

d+ k¯

d

*and k_u= k¯

u+ k¯

u

*. By calculating the bath correlation functions, we obtain kd=⌫共␻0兲关1 + n¯共␻0兲兴 and ku=⌫共␻0兲n¯共␻0兲, where n¯_␯共⍀0兲=共e^␤␯␻0− 1兲⁻¹ is the Bose-Einstein distribution for the bath with ␤␯

= 1 / k_BT_␯,␯= l , r and⌫共␻0兲=兺i␯兩␬i␯兩²共␻0/⍀i␯兲␦共⍀i␯−␻0兲.

Note that for each bath the two rates are related as k_d^␯= e^␻0␤␯k_u^␯, ␯^{= l,r} 共7兲 and therefore

k_d^lk_u^r

k_u^lk_d^r= e^{−␻0⌬␤}, 共8兲

where⌬␤=␤r−␤l. Equation 共8兲 will be crucial for proving the FT for boson transfer.

III. GENERATING FUNCTION FOR BOSON COUNTING STATISTICS

The statistics of thermal fluctuations is related to the num- ber共k兲 of quanta transferred between the baths and the sys- tem. The probability P共k,t兲 that k quanta are transferred dur- ing time t is

P共k,t兲 = 具Iˆ兩␳^ˆ共k兲共t兲典, 共9兲

where Iˆ is the identity matrix and ␳^ˆ共k兲共t兲 is the system density-matrix conditional that k quanta transferred between the bath and the system共see Appendix A兲. In the following, we shall denote a matrix, a vector, and a scalar by Bˆ , 兩Bˆ典, and B, respectively. The generating共characteristic兲 function 共GF兲 for the probability P共k,t兲 is defined as

G共␭,t兲 =

兺

_k ^{P共k,t兲e␭·k. 共10兲}

Since there are four possible elementary processes through which the number of particles 共heat quanta兲 in the system can change 共either “in” or “out” from left and right junc- tions兲, ␭·k=兺_i=1⁴ ␭iki, where k =兵ki其, i=1,4, and ␭=兵␭i其 are four-component vectors with k₂共k1兲 关k4共k3兲兴 representing the number of quanta transferred from 共to兲 the left 共right兲 bath to共from兲 the system at time t.

All observables of interest can be calculated by taking the derivative of the GF with respect to ␭i. For example, the average number of quanta transferred through process i at time t,

具ki典共t兲 =

兺

k

kiP共k,t兲 =

冏

⳵^⳵␭i

具Iˆ兩Gˆ共␭,t兲典

冏

_␭=0^{. 共11兲}

The GF will be calculated for the model of Eq. 共6兲 by solving the equation of motion共see Appendix A for deriva- tion兲

兩Gˆ˙ 共␭,t兲典 = Mˆ共␭兲兩Gˆ共␭,t兲典, 共12兲 where兩Gˆ共␭,t兲典 is an infinite dimensional vector and Mˆ is a tridiagonal matrix

Mˆ =

冢

^{klue␭− k2¯¯+ k0uure␭4 2共kk−ldule关ke␭␭1d2¯¯+ k+ 2k+ krdureue␭兴␭34兲 2−共kld关2ke␭¯1¯d0+ k+ 3kdreu␭兴3兲 3共klde␭1¯¯¯0+ krde␭3兲 0 ¯ ¯¯ ¯ ¯¯ ¯ ¯¯ ¯ ¯¯ ¯ ¯}

冣

^{, 共13兲}

with兩Gˆ共␭,t=0兲典=兩␳^ˆ共0兲典. The formal solution of Eq. 共12兲 is 兩Gˆ共␭,t兲典 = e^{Mˆ 共␭兲t}兩␳^ˆ共0兲典. 共14兲 It can be expanded in the left and right eigenvectors具Lˆ_␰共␭兲兩 and兩Rˆ_␰共␭兲典 of Mˆ 共␭兲 corresponding to the eigenvalue␥_␰共␭兲,

兩Gˆ共␭,t兲典 =

兺

m=0

e^{␥␰共␭兲t}兩Rˆ_␰共␭兲典具Lˆ_␰共␭兲兩␳^ˆ共0兲典. 共15兲 Note that in order for the system to reach a steady state, the real part of all the eigenvalues ␥_␰共␭=0兲, 共␰⫽0兲 must be negative and ␥0共␭=0兲=0. The standard method to solve tridiagonal matrix equations, such as Eq.共12兲, is to use the continued fraction method.^14,28 Instead, we shall solve Eq.

共15兲 numerically.

A. Average current and the Mandel parameter We first consider the lowest two moments of the current.

The average number of bosons transferred between the sys- tem and the baths through a process 共i=1,2,3,4兲 during time t is given by the derivative of the GF with respect to the corresponding parameter␭iat␭=0, as given in Eq. 共11兲. The average heat Q_avtransferred between the system and the bath is therefore Q_avⁱ 共t兲=␻0具ki典共t兲. The corresponding current Ii共t兲 is given by the rate of change of the average number of quanta transferred,

I_i共t兲 = ⳵

⳵^{t具ki典共t兲. 共16兲}

At steady state,具ki典 grows linearly with time. By differ- entiating Eq.共14兲 with respect to ␭i, we get

冏

⳵^⳵␭i

兩Gˆ共␭,t兲典

冏

_␭=0⁼

冏

^t⳵M⳵^{ˆ 共␭兲}␭i

e^{Mˆ 共␭兲t}

冏

␭=0兩␳^ˆ共0兲典. 共17兲 In the long-time limit, e^{Mˆ 共␭=0兲t}兩␳^ˆ共0兲典=兩␳^ˆst典. This vector cor- responds to the stationary-state solution of Eq.共6兲. Equation 共17兲 with Eq. 共11兲 then gives

具ki典共t兲 = 具Iˆ兩⌫ˆi兩␳^ˆst典t, 共18兲 where⌫ˆi=兩⳵Mˆ 共␭兲/⳵␭i兩_␭=0is known as the resetting matrix²⁹ for the ith process. The stationary solution of Eq.共6兲 is²⁰

␳st

n=共kd− k_u兲

k_d

冉

^kkud

冊

^{n. 共19兲}

The steady-state current I_i^sis

I_i^s= lim

t→⬁

1

t具ki典共t兲 =

冏

^limt→⬁

1 t

⳵

⳵␭i

G共␭,t兲

冏

_␭=0^{=具Iˆ兩⌫ˆi兩␳ˆst典.}

共20兲 Substituting Eq.共19兲 into Eq. 共20兲, we obtain

I₁^␯_共3兲= k_d^␯ku

kd− ku

, ␯^{= l,r,}

I₂^␯_共4兲= k_u^␯kd

kd− ku

. 共21兲

The total steady-state current from the left共right兲 bath to the system is I^s= I_2共4兲^␯ − I_1共3兲^␯ .

All other moments and cumulants of the probability dis- tribution function can be obtained from the GF by taking higher-order derivatives with respect to␭. The second cumu- lant for the ith transport process,C_i^共2兲=具ki2典−具ki典², can be ob- tained as

Ci共2兲共t兲 =

冏

⳵^⳵␭²i

2ln G共␭,t兲

冏

_␭=0^{. 共22兲}

The Mandel parameter³⁰ for the ith process Miis defined as M_i共t兲 ⬅Ci共2兲共t兲

具ki典共t兲− 1. 共23兲

For Poisson statistics, M共t兲=0. M ⬎0 and M ⬍0 correspond to super-Poissonian 共bunching兲 and sub-Poissonian 共anti- bunching兲 statistics. The Mandel parameter has been widely used^3,31as a convenient measure of non-Poissonian statistics.

The Fano factor,³² Fi⬅Mi+ 1, is sometimes used^3,33,34 in- stead. Assuming that the initial state ␳共0兲 共when counting starts兲 corresponds to the steady state, the Mandel parameter can be expressed^14,24in terms of the eigenvalues共␥_␰兲 and the left and right eigenvectors,具Lˆ_␰兩 and 兩Rˆ_␰典, of M共␭=0兲.

Mi共t兲 = 2

兺

␰ ⬘

冉

^e␥␥^␰t_␰^{− 1}t + 1

␥_␰

冊

^{具Lˆ0兩⌫ˆ}_具Lˆ^{i兩Rˆ␰典具Lˆ␰兩⌫ˆi兩Rˆ0典}

0兩⌫ˆi兩Rˆ₀典 , 共24兲 where 具Lˆ0兩 and 兩Rˆ0典 are the eigenvectors corresponding to zero eigenvalue共␥0= 0兲 and the prime indicates that the sum over␰excludes the zero eigenvalue.

B. Steady-state fluctuation theorem for bosons The net heat current flowing through the left and the right system-bath interfaces can be calculated by substituting ␭1

= −␭2⬅␭L and ␭4= −␭3⬅␭R in the generating function G共␭,t兲. The average current through each junction is then computed by taking the derivative of the current GF with respect to␭Lor␭R. At steady state, the two currents are the same and given by the sum of the currents defined in Eqs.

共21兲.

The steady-state FT for currents quantifies the probability of the forward and backward transfer processes. It follows from Eq.共8兲, as shown in Appendix B, that

lim

t→⬁

P共k,t兲

P共− k,t兲= e^␻0⌬␤k, 共25兲

where k is the net number of transfer events between the␯^th 共␯= l , r兲 bath and the system over a time t. According to Eq.

共25兲, the probability of transferring bosons against the heat gradient is exponentially small compared to that of the re- verse process. The fraction of reverse processes is sup- pressed exponentially as the system frequency, temperature gradient, or the number of transfers k is increased.

C. Comparison with fluctuation theorem of fermions Equation 共3兲 has the same structure as for a Fermi system.²⁷ The difference arises from the commutation rules 共boson statistics兲 of the system and bath operators 共see Ap- pendix C兲.

We consider a system with a single orbital with energy⑀0

attached to two metal leads. We assume that the leads are at equilibrium with their respective temperatures T_land T_rand chemical potential ␮0. The statistics of electron transport through a system of coupled quantum dots has recently been studied²⁴ using the QME. Here, we extend that model by incorporating the temperature gradient between the two leads. Following Ref.24, the steady-state FT for this model reads

lim

t→⬁

P共k,t兲

P共− k,t兲= e^{共⑀0−␮0兲k⌬␤}. 共26兲 Note that when⑀0=␮0, the ratio of the two PDFs is unity and the system is always at equilibrium irrespective of⌬␤^.

The fluctuation theorems共25兲 and 共26兲 only hold for large time. In Ref.24, it was shown that for finite times and small nonequilibrium constraints, when the PDF can be well ap- proximated by a Gaussian function, the FT modifies to

P共k,t兲

P共− k,t兲= e^{共S−␣共t兲兲k}, 共26a兲 and the GF satisfies the symmetry

G共␭,t兲 = G共S − ␭ −␣共t兲,t兲, 共27兲 where S=␻0⌬␤ for bosons, S=共⑀0−␮0兲⌬␤ for fermions, and␣共t兲 is the finite time correction 共FTC兲 which␣共t兲→0 as t→⬁. Since G共␭=0,t兲=1, Eq. 共27兲 reduces to

G共S −␣共t兲,t兲 = 1. 共28兲 At sufficiently long times共after few tens of transfer events兲,

␣共t兲=␣X/ t and Eq.共28兲 can be solved²⁴to compute the FTC for bosons共␣X=␣b兲 and fermions 共␣X=␣f兲. For the transfer statistics between the system and left bath, we obtained

␣b= 1

I^slog

冋

^{共krd− k共kdru− k兲共kd2u兲k2ruk− krukrdu2kdr兲}

册

^{, 共29兲}

␣f= 1

I_f^slog

冋

^{共vdr+vrdvvurru兲共v共vd2dv+ruv+u兲v22uvrd兲}

册

^{, 共30兲}

where I^sand I^s_fare the net steady-state currents for the boson 关Eqs. 共20兲兴 and fermion²⁷ systems, respectively. vu=兺_␯v_u^␯ andvd=兺_␯v_d^␯are the rates corresponding to in and out trans- fers of fermions from the system.^24,27The FTC for the sta- tistics between the right bath and the system is obtained by

7.5
5
2.5 0 2.5 5 7.5 10

k

0 0.05 0.1 0.15 0.2 0.25 0.3

PDF

7.5
5
2.5 0 2.5 5 7.5 10

k

0 0.05 0.1 0.15 0.2

PDF

15
10
5 0 5 10 15 20

k

0 0.02 0.04 0.06 0.08

PDF

20
10 0 10 20 30

k

0 0.01 0.02 0.03 0.04 0.05 0.06

PDF

(a)

(b)

(c)

(d)

FIG. 1. Probability distribution function for boson共cross兲 and fer- mion 共triangle兲 transfers between left lead and the system at four different binning times t:共a兲 500, 共b兲 1000, 共c兲 5000, and 共d兲 10 000.

␻0=⑀0= 2.0, temperature of two baths is T_l= 0.7 and T_r= 0.6, and the coupling to the left and the right baths is ␬l= 0.1 and ␬r

= 0.05.

0 25 50 75 100 125 150 175 t

0 0.1 0.2 0.3 0.4

MandelParameter
M1

Β8.75

Β8.34

Β7.50

Β5.00

0 25 50 75 100 125 150 175

t

0.14

0.12

0.1

0.08

0.06

0.04

0.02 0

MandelParameter
M1 ^Β5.00

Β7.50

Β8.34

Β8.75

0 25 50 75 100 125 150 175

t 0

0.1 0.2 0.3 0.4 0.5 0.6

MandelParameter
M2

Β8.75

Β8.34

Β7.50

Β5.00

0 25 50 75 100 125 150 175

t

0.25

0.2

0.15

0.1

0.05 0

MandelParameter
M2 ^Β5.00

Β7.50

Β8.34

Β8.75

0 25 50 75 100 125 150 175

t 0

0.025 0.05 0.075 0.1 0.125 0.15 0.175

MandelParameter
M3

Β8.75

Β8.34

Β7.50

Β5.00

0 25 50 75 100 125 150 175

t

0.08

0.06

0.04

0.02 0

MandelParameter
M3 ^Β5.00

Β7.50

Β8.34

Β8.75

0 25 50 75 100 125 150 175

t 0

5 10
⁶ 0.00001 0.000015 0.00002 0.000025 0.00003 0.000035

MandelParameter
M4

Β8.75

Β8.34

Β7.50

Β5.00

0 25 50 75 100 125 150 175

t

0.000028

0.000026

0.000024

0.000022

0.00002

0.000018

MandelParameter
M4

Β5.00

Β7.50

Β8.34

Β8.75

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

FIG. 2. 共Color online兲 The time-dependent Mandel parameter corresponding to the statistics of 共a,c,e,g兲 boson and 共b,d,f,h兲 fermion transfers. M₁and M₃ correspond to the particle transfer from the system to the left and the right baths while M₂ and M₄ correspond to transfer from the left and right baths to the system, respectivel.␻0=⑀0= 1.0, T_r= 0.1.

replacing 兵ku

r, k_d^r其↔兵ku

l, k_d^l其 and 兵vu

r,v_d^r其↔兵vu

l,v_d^l其 in Eqs.

共29兲 and 共30兲.

Closed expressions for the current and the Mandel param- eter in a fermionic system were given in Ref. 24. For the present model 共with different bath temperatures兲, we only need to replace␤^by␤land␤rin the left and right rates关Eqs.

共3兲 in Ref.24兴. The numerical results for fermion and boson statistics are presented in the next section.

IV. NUMERICAL SIMULATIONS

We used the following parameters in our simulations. The coupling of the system with the left and right baths is fixed at

␬l= 0.1 and␬r= 0.05. For the system energy, we consider two different values, ␻0= 1 and ␻0= 2. The right bath has the lower temperature共Tl⬎Tr兲. The current GF is obtained from Eq.共15兲 for boson and fermion transfers, and the correspond- ing PDFs are obtained using the inverse Fourier transform of Eq. 共10兲. The QME for bosons is an infinite hierarchy in many-body space. In order to solve the QME for bosons关or Eq. 共12兲兴, we therefore need to truncate it at some suitable order 共n兲. This is done by analyzing the exact, analytical solution for the steady-state population, Eq.共19兲. The many- body states with steady-state population less than 10⁻⁷ have been neglected. Note that the population falls off almost ex- ponentially with the many-body states共n兲. So, the contribu- tion to statistics coming from many-body states with large n can be safely ignored. In the numerical calculations, n was varied between 5 and 15, depending on the parameter values 兵␻0, Tl, Tr其. In Fig. 1, we show the probability distribution functions for boson and fermion transfers between the left bath and the system, respectively. Positive共negative兲 k cor- respond to a net transfer from 共to兲 the left bath. The peak position of the PDF moves to higher values as the observa- tion共binning兲 time is increased. Equation 共18兲 shows that the average number of particles transferred between the left bath and the system grows linearly in time. The slope that gives the average current, which for the parameter values consid- ered here, is always found to be larger for fermions than for bosons. Therefore, the peak of PDF, which is roughly equivalent to the average number of particles transferred具k典 for fermions, is always at larger k.

In Fig. 2, we compare the Mandel parameter for boson and fermion transfer of different processes as function of time and the inverse temperature difference of baths, ⌬␤^. The distribution is non-Poissonian in both cases, bunched 共Mi⬎0兲 for bosons and antibunched 共Mi⬍0兲 for fermions.

For bosons, all the transfer processes become more bunched as ⌬␤ is increased. For fermions, however, the statistics of the process related to particle transfer from the right lead to the system becomes more Poissonian共see Mandel parameter M₄兲 as ⌬␤is increased. In Fig.3, we show the PDFs for the transfer from the left bath to the system共corresponding Man- del parameter is M₂兲 for bosons and fermions. The corre- sponding Poisson distributions, f共k兲=具k典^ke^具k典/共k!兲, are shown as well. The non-Poissonian features共Mi⫽0兲 mainly comes from the rare events of transfer of a large number of particles kⰇ具k典. Moreover, we notice that for kⰇ具k典, in the case of

bosons 共fermions兲, the Poisson distribution underestimates 共overestimates兲 the actual probability. This is due to the fact that in the case of fermions, due to Pauli exclusion, the sys- tem cannot be occupied by more than one particle and the transfer processes are therefore negatively correlated.³ This effect shows up more clearly as a large number of particles is transferred. This is why we see a clear deviation from Pois- son only for large k. For bosons, on the other hand, two or more particles can occupy the same quantum state, enhanc- ing successive transfer events.

By computing the full PDFs for particle transfer statistics, we can compare the fluctuation theorem for bosons and fer- mions which quantifies the rare events. In the present model, these events correspond to the particle transfer processes against the nonequilibrium constraints. In order to test the fluctuation theorem, we have computed the logarithm of the ratios of the PDF for boson transfer, Eq. 共25兲. The straight line in Fig. 4 represents limt→⬁log关P共k,t兲/ P共−k,t兲兴 as a function of k. At finite times, this ratio deviates from the FT, Eq. 共26a兲, which is shown as dots at different times. The deviation from the straight line decreases as the binning time is increased. In Fig.5, we plot关S−␣共t兲兴/S as a function of the binning time t. The horizontal line represents the FT. In order to satisfy the FT, more bosons need to be transferred than fermions, i.e., the required binning time for bosons is larger than the fermions. Continuous curves show a fit to

0 20 40 60 80 100 120

k

12

10

8

6

4

2 0

Log
P
k 

t1000

t5000 t10000

0 20 40 60 80 100 120 140

k

12

10

8

6

4

2 0

Log
P
k 

t1000

t5000 t10000

(a)

(b)

FIG. 3. PDFs corresponding to the共a兲 boson and 共b兲 fermion transfer processes from the left bath to the system at three different times. Lines represent the Poisson fits共see text兲. Deviations respon- sible for a nonzero Mandel parameter are more pronounced at large kⰇ具k典.

1 −关␣b共f兲/S兴t⁻¹, with ␣b= 8.34 and ␣f= 3.97 for bosons and fermions as obtained by Eqs.共29兲 and 共30兲, respectively.

A. Interplay of chemical potential and temperature gradient for fermions

So far, we assumed that the chemical potential of the two baths共leads兲 is equal. When a finite 共dc兲 bias 共V兲 is applied across the junction such that the two baths are in equilibrium at chemical potentials␮_␯^,␯^{= l , r}共␮r=␮0兲, Eq. 共26兲 is modi- fied as

lim

t→⬁

P共k,t兲

P共− k,t兲= e^{共⑀0−␮0兲k⌬␤}e^e␤l⌬␮k, 共31兲

where⌬␮=␮l−␮r is the applied bias. Note that when ⌬␤

= 0 or⑀0=␮0, Eq.共26兲 reduces to the result obtained in Ref.

24. The first exponential on the right-hand side of Eq.共26兲 is due to the heat gradient while the second accounts for the chemical-potential gradient.

According to Eq.共31兲, when the chemical-potential gra- dient is opposite 共␮l⬍␮r兲 to the temperature gradient 共␤l

⬍␤r兲 between two leads, for each bias value, there exists a temperature difference⌬T=Tl− Tr,

⌬T T_r = eV

␮0−⑀0

, 共32兲

where the ratio of the probabilities in Eq.共26兲 is unity; the ratio of the rates v_d^␯/v_u^␯ for ␯= l , r becomes equal and the system is at equilibrium. When condition共32兲 is satisfied, the probability distribution for the current共steady state兲 is sym- metric around k = 0 and the Mandel parameters correspond- ing to the transfer process related to the same bath are equal, as shown in Fig.6. This implies that the two currents I₁^␯_共3兲 and I_2共4兲^␯ become equal.

V. CONCLUSIONS

We have studied the thermal transport properties of bosons and fermions through a mesoscopic junction. The generating function approach was used to calculate the full counting statistics of the transport. Numerical calculations show that the Mandel parameter for bosons is positive 共bunching兲 while that for fermions, it is negative 共antibunch- ing兲. By computing the full PDFs, we notice that the non- Poissonian character of the statistics mainly comes from the transfer of the large number 共Ⰷk兲 of particles. We showed that both fermion and boson statistics satisfy a steady-state fluctuation theorem. Irrespective of particle statistics, we found that for finite times, the deviation from the FT, Eqs.

共25兲 and 共26兲, vanishes asymptotically as 1/t. This is due to the fact that the leading-order correction to the large devia- tion function共LDF兲 defined in Eq. 共B6兲 vanishes as 1/t. We further observed that the mean number of particles trans- ferred between the bath and the system grows linearly in time with the value for fermions being larger than that for bosons.

0 1 2 3 4

k

0 0.5 1 1.5

LogP
K,tP

k,t

FT t
10000 t
5000 t
1000 t
500 t
200 t
100

FIG. 4. Fluctuation theorem at finite times for boson statistics.

Straight line represents the steady-state FT共25兲 with slope␻0⌬␤

= 0.476. Dots represent the deviations共slope different from 0.476兲 for FT at finite times.

0 1000 2000 3000 4000 5000 6000

t

0.9 0.92 0.94 0.96 0.98 1

LogP
KP

k
kΩ0Β

FIG. 5. Time dependence of the finite time correction to FT:

log关P共k兲/ P共−k兲兴/共k␻0⌬␤兲 is shown at different times. The FT is satisfied asymptotically. Continuous curve is the fit, 1 −␣X/St⁻¹ with␣b= 8.34 and␣f= 3.97 for bosons and fermions, respectively, as predicted by Eqs.共29兲 and 共30兲.

100
50 0 50 100

k

0.01 0.02 0.03 0.04

PDF

1
0.8
0.6
0.4
0.2

V

0.014

0.012

0.01

0.008

0.006

0.004

0.002

M4

M3

M2

M1

FIG. 6. 共Color online兲 The PDFs for fermions are shown at t

= 100 for three sets of parameters 共upper: Tl= 0.5, T_r= 0.4, and

⌬␮=0.5; middle: Tl= 0.8, T_r= 0.5, and ⌬␮=1.2; and lower: Tl

= 0.7, T_r= 0.6, and⌬␮=0.34兲 when condition 共32兲 is satisfied. Inset shows the Mandel parameters at t = 100 as a function of the bias with T_l= 0.5, T_r= 0.4, and ⑀0= 2. Note that M₁= M₂ and M₃= M₄ when Eq.共32兲 holds.

ACKNOWLEDGMENTS

The financial support from the National Science Founda- tion共Grant No. CHE-0446555兲 and NIRT 共Grant No. EEC- 0406750兲 is gratefully acknowledged. M.E. is partially sup- ported by the FNRS Belgium共Collaborateur Scientifique兲.

APPENDIX A: DERIVATION OF EQUATION (12) In Liouville space, the QME共3兲 can be written as

兩␳^ˆ˙共t兲典 = Wˆ 兩␳^ˆ共t兲典, 共A1兲 where兩␳^ˆ典 is a vector comprising populations␳nnand coher- ences␳nm, n⫽m. Since populations are decoupled from co- herences, the matrix Liouville operatorWˆ can be factorized in two parts,Wˆ_candWˆ_p, describing the evolution of coher- ences and the populations, respectively. The off-diagonal el- ements,⌫ˆp, inWˆ

p represent the rate of transfer of particles between the bath and the system and are responsible for the population change due to the particle transfer.

In order to compute the system density-matrix condition l on the transfer of a fixed number of electrons, we transform Eq. 共A1兲 into the interaction picture with respect to Mˆ

0

⬅Wˆ −⌫ˆp.

兩␳^ˆ˙I共t兲典 = ⌫ˆp共t兲兩␳^ˆI共t兲典, 共A2兲 where兩␳^ˆI共t兲典=Uˆ共0,t兲兩␳^ˆ共t兲典 and ⌫ˆp共t兲=U共0,t兲⌫ˆpU共t,0兲, with Uˆ 共0,t兲=exp兵−Mˆ

0t其, is the time evolution operator in the absence of electron transfer.

The formal solution of Eq.共A1兲 can be written as 兩␳^ˆI共t兲典 = exp+

冋 ^冕

^{0t⌫ˆ共␶兲d␶}

册

^{兩␳ˆI共0兲典. 共A3兲}

Expanding the 共time-ordered兲 exponential in Eq. 共A3兲, we can write兩␳^ˆI共t兲典=兺k兩␳^ˆ_I^共k兲共t兲典, where

兩␳^ˆI共k兲共t兲典 =

冕

0 t

d␶k

冕

0

␶k

d␶k−1¯

⫻

冕

0

␶2

d␶1⌫ˆp共␶k兲⌫ˆp共␶k−1兲 ¯ ⌫ˆp共␶1兲兩␳^ˆI共0兲典.

共A4兲 In Eq. 共A4兲, each interaction with ⌫ˆp at times t =␶n 共n

= 1 , k兲 records the transfer of a particle between baths and the system. Multiplying Eq. 共A4兲 from the left by Uˆ共t,0兲, the conditional density matrix for transferring k particles in the Schrödinger picture is obtained as

兩␳^ˆ共k兲共t兲典 =

冕

0 t

d␶k

冕

0

␶k

d␶k−1¯

⫻

冕

0

␶2

d␶1Uˆ 共t,␶k兲⌫ˆpUˆ 共␶k,␶k−1兲⌫ˆp¯

⫻Uˆ共␶2,␶1兲⌫ˆpUˆ 共0,␶1兲兩␳^ˆ共0兲典. 共A5兲

According to Eq.共A5兲 the system 共conditional兲 density ma- trix evolves from time t = 0 to t =␶1 without any particle transfer. At t =␶1,⌫ˆpacts on the density matrix and a particle is transferred. The density matrix again evolves without any particle transfer until a second particle is transferred at t

=␶2. This process is repeated until the k particle has been transferred at t =␶k and the density matrix finally evolves from t =␶k to t = t without additional particle transfer. Note that all particle transfers are instantaneous so that the density matrix does not evolve during particle transfer. There are four different processes through which particles can be trans- ferred between the bath and the system, we write k

= k₁, k₂, k₃, k₄, such that k =兺_i=1⁴ ki.

The probability of transferring k particles at time t is then obtained by tracing the conditional density matrix

P共k,t兲 = 具Iˆ兩␳^ˆ共k兲共t兲典 =

兺

_i

冕

0 t

d␶k具Iˆ兩⌫ˆp i共␶k兲兩␳^ˆIk−Iˆ

i共␶k兲典, 共A6兲 whereIˆiis a vector with 1 at the position i and the rest of the elements zero. From Eqs.共10兲 and 共A6兲, the GF in the inter- action picture is then obtained as

Gˆ

I共␭,t兲 =

兺

_k

i,i

冕

0 t

d␶^e␭iki具Iˆ兩⌫ˆp i共␶兲兩␳^ˆI

k−Iˆ i共␶兲典

=

兺

_i

冕

0 t

d␶

冓

^Iˆ

^冏

^{e␭i⌫ˆpi共␶兲}

^兺

^{ki e␭i共ki−1兲}

^冏

^{␳ˆIki−1共␶兲}

冔

=

兺

i

冕

0 t

d␶具Iˆ兩e^␭i⌫ˆp

i共␶兲兩GˆI共␭,␶兲典. 共A7兲

Taking the time derivative and transforming back to the Schrödinger picture, we get

兩Gˆ˙ 共␭,t兲典 =

冉

^{Mˆ 0+}

^兺

^{i e␭i⌫ˆpi}

冊

^{兩Gˆ共␭,t兲典. 共A8兲}

The matricesMˆ

0and⌫ˆp

i are defined by the QME共6兲. When substituting into Eq.共A8兲, we obtain Eq. 共12兲.

APPENDIX B: DERIVATION OF EQUATION (25) Following Ref. 24, Eq. 共25兲 can be derived by defining the LDF for the particle number fluctuations as

R共␨兲 ⬅ − lim

t→⬁

1

t ln P共␨,t兲, 共B1兲 where␨= k / t. Equation共B1兲 uses the ansatz

P共␨^,t兲 = C共␨^,t兲e^{−R共␨兲t}, 共B2兲 with limt→⬁C共␨, t兲/t=0. Using Eqs. 共10兲 and 共B2兲, the cur- rent GF can be expressed as

G共␭,t兲 =

冕

^{d␨C共␨,t兲e}−关␭␨+R共␨兲兴t. 共B3兲 In long time, the main contribution to the integral in Eq.共B3兲 comes from the␨⁼␨^*value where dR共␨兲/d␨= −␭. Expanding

R共␨兲 around ␨^* and keeping up to second order and substi- tuting into Eq.共B3兲, we get

G共␭兲 = e^{−关␭␨*+R共␨*兲兴t}

冕

^{d␨C共␨,t兲e}共1/2兲R¨共␨兲共␨ − ␨^*兲²t

. 共B4兲 Now consider the generatorM共␭兲 of the current generat- ing operator which is obtained by setting ␭1=␭=−␭2 and

␭3=␭4= 0 in Eq.共13兲. We notice that the ␭-dependent terms in the characteristic polynomial of the current generator ap- pear only as a product,A共␭兲B共␭兲=共kd

le^␭+ k_d^r兲共ku

le^−␭+ k_u^r兲. Us- ing Eq. 共8兲, we see that A共␭兲B共␭兲=A共−␭+␻0⌬␤兲B共−␭

+␻0⌬␤兲. Therefore, the characteristic polynomial and hence the eigenvalues gn共␭兲, n=0,1,2,... of the current GF show the symmetry property

gn共␭兲 = gn共− ␭ +␻0⌬␤兲. 共B5兲 We define a LDF, S共␭兲, corresponding to the generating function

S共␭兲 = − lim

t→⬁

1

t ln G共␭,t兲. 共B6兲

Since the eigenvalues of the current GF satisfy the symmetry property 共B5兲, it follows that S共␭兲 also satisfies the same property,

S共␭兲 = S共␻0⌬␤⁻␭兲. 共B7兲 Using Eqs.共B4兲 and 共B6兲, we obtain

S共␭兲 = ␭␨^*+ R共␨^*兲. 共B8兲 Equation共B8兲 shows that S共␭兲 is the Legendre transform of the LDF R共␨^*兲. The inverse Legendre transform gives R共␨兲

= S共␭^*兲−␭^*␨. Substituting ␭^*=␻0⌬␤ into the inverse Leg- endre transform and using Eq.共B7兲, we obtain

R共␨兲 = R共−␨兲 −␻0␨⌬␤^. 共B9兲 Using Eqs. 共B1兲 and 共B2兲, it is straightforward to verify Eq.共25兲.

APPENDIX C: QME FOR FERMIONS

Here, we present the QME for a fermionic system. The model is a system site coupled to two fermionic共noninter- acting兲 leads held at two different temperatures. The details of the calculations are given in Ref.27. The Hamiltonian for the system is

H =⑀0c₀^†c₀+

兺

_␯ ^{⑀␯c␯†c␯+}

兺

_0␯ ^{共␬0␯c0†c␯+ H.c.兲, 共C1兲}

where c₀^†共c0兲 are the Fermi creation 共annihilation兲 operators for the system site and satisfy the commutation relations c₀^†c₀+ c₀c₀^†= 1. The first two terms in Eq.共C1兲 represent the noninteracting system and leads; the last term is the coupling between the system and two leads. Direct coupling between the two leads is neglected.

When the coupling J is turned on, the system density matrix evolves according to the QME analogous to Eq.共3兲.

⳵

⳵^t␳ⁿ共t兲 = − i⑀0关c0†

c₀,␳ⁿ共t兲兴 + 关vuc₀^†␳ⁿ⁻¹共t兲c + vdc₀␳ⁿ⁺¹共t兲c0†

−vd␳ⁿ共t兲c₀^†c₀−vuc₀c₀^†␳ⁿ共t兲 + H.c.兴, 共C2兲 wherevd=v_d^l+v_d^r andvu=v_u^l+v_u^r are the particle rates

v_d^␯=␲ⁿ␯共⑀0兲兩␬0␯兩²关1 − f_␯共⑀0兲兴, vu␯=␲ⁿ␯共⑀0兲兩␬0␯兩²f_␯共⑀0兲, 共C3兲 where f_␯共⑀0兲=关1+e^{␤␯共⑀0−␮␯兲}兴 is the Fermi function for the␯^th lead at chemical potential ␮_␯ and inverse temperature ␤_␯^. When traced over two system states兩0典 共empty兲 and 兩1典 共oc- cupied兲 in Fock space, and using the relations

c₀兩0典 = 0, c^†兩0典 = 兩1典,

c₀兩1典 = 兩0典, c^†兩1典 = 0, 共C4兲 we obtain the coupled equations

␳^˙0共t兲 = − 2vu␳0共t兲 + 2vd␳1共t兲,

␳^˙1共t兲 = − 2vd␳1共t兲 + 2vu␳0共t兲, 共C5兲 where ␳n=具n兩␳^ˆ兩n典, n=0,1. Equations 共C2兲 and 共C5兲 are analogous to the boson关Eqs. 共3兲 and 共6兲兴, respectively.

1T. Rom, Th. Best, D. van Oosten, U. Schneider, S. Folling, B.

Paredes, and I. Bloch, Nature共London兲 444, 733 共2006兲.

2S. Oberholzer, E. Bieri, C. Schönenberger, M. Giovannini, and J.

Faist, Phys. Rev. Lett. 96, 046804共2006兲.

3W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, Science 284, 299共1999兲.

4H. Kiesel, A. Renz, and F. Asselbach, Nature共London兲 418, 392 共2002兲.

5M. Iannuzzi, A. Orecchini, F. Sacchetti, P. Facchi, and S.

Pascazio, Phys. Rev. Lett. 96, 080402共2006兲.

6T. Fujisawa, T. Hayashi, R. Tomita, and Y. Hirayama, Science 312, 1634共2006兲.

7T. L. Schmidt, A. Komnik, and A. O. Gogolin, Phys. Rev. Lett.

98, 056603共2007兲.

8C. Texier and M. Büttiker, Phys. Rev. B 62, 7454共2000兲.

9L. S. Levitov and M. Reznikov, Phys. Rev. B 70, 115305共2004兲.

10L. Basano, P. Ottonello, and B. Torre, J. Opt. Soc. Am. B 22, 1314共2005兲.

11M. Büttiker, Science 313, 1587 共2006兲; B. Cleuren and C.

Van den Broeck, Europhys. Lett. 79, 30001共2007兲.

12R. H. Brown and R. Q. Twiss, Nature共London兲 177, 27 共1956兲.

13R. J. Glauber, Phys. Rev. 131, 2766共1963兲.

14F. Sanda and S. Mukamel, J. Chem. Phys. 124, 124103共2006兲.

15S. Mukamel, Phys. Rev. A 68, 063821共2003兲.

16L. Mandel, Phys. Rev. Lett. 49, 136共1982兲.

17F. Sanda and S. Mukamel, Phys. Rev. A 71, 033807共2005兲.