Capacity of nonlinear bosonic systems

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by  Vittorio Giovannetti

Capacity of nonlinear bosonic systems
Vittorio Giovannetti1, Seth Lloyd1,2, and Lorenzo Maccone1
1Massachusetts Institute of Technology – Research Laboratory of Electronics
2Massachusetts Institute of Technology – Department of Mechanical Engineering
77 Massachusetts Ave., Cambridge, MA 02139, USA
WeanalyzetheroleofnonlinearHamiltoniansinbosonicchannels. Weshowthattheinformation
capacity as a function of the channel energy is increased with respect to the corresponding linear
case, although only when the energy used for driving the nonlinearity is not considered as part of
theenergetic cost and when dispersive effects are negligible.
PACSnumbers: 42.50.-p,03.67.-a,42.65.-k,89.70.+c
4
0 Noninteractingmasslessbosonicsystemshavebeenthe information, then one cannot do better than the bound
0 objectofextensiveanalysis[1,2]. Theirmaximumcapac- obtained in the noninteracting case [6, 7]. However, the
2 ityintransmittinginformationwasderivedforthenoise- enhancement discussed here is not to be underestimated

n lesscasebothinthenarrow-bandregime(whereonlyfew sinceinmostsituations manydegreesoffreedomarenot
a frequency modes are employed) and in the broad-band usable to encode information, but can still be employed
J
regime. However, it is still an open question whether to augment the capacity of other degrees of freedom. A

9 nonlinearities in the system may increase these bounds: typical example is when the sender is not able to modu-
2
linearityappearstobethemostimportantassumptionin late the signals sufficiently fast to employ the full band-

all previous derivations [2]. Up to now nonlinear effects width supported by the channel: an external pumping
2
have been used in fiber optics communications to over- (suchastheoneinvolvedintheparametricdownconver-
v
1 come practicallimitations, suchas using solitonsto beat sion case) may allow to increase the energy devoted to
0 dispersion or traveling wave amplifiers to beat loss [3]. the transmission modes.
0 Theapproachadoptedinthispaperisfundamentallydif- We start by describing a general procedure to evalu-
9 ferent from these and from others where nonlinearities ate the capacity of a system and we apply it to a lin-
0
and squeezing are employed at the coding stage when ear bosonic system to reobtain some known results (see
3
using linear channels [4]. We follow the cue of a recent Sec. I). Such a procedure is instructive since it empha-
0
/ proposal[5]whereinteractionswereexploitedinincreas- sizesthe roleofthe systemspectruminthe capacitycal-
h ing the capacity of a qubit-chain communicationline. In culation. We then analyze a collection of examples of
p
the case of linear bosonic systems, the information stor- nonlinear bosonic systems in the narrow-bandand wide-
-
t agecapacityofasignaldividedbythetimeittakesforit band regimes (see Sec. II): for each case we describe
n
topropagatethroughthemedium givesthe transmission the capacityenhancementsoverthecorrespondinglinear
a
u capacity of the channel. In the presence of nonlineari- systems. General considerations on the energy balance
q ties, dispersion can affect the propagation of the signal in the information storage and on the information prop-
: complicating the analysis, but an increase in the capac- agation conclude the paper (see Sec. III).
v
ity canbe shown,atleastwhen the dispersive effects are
i
X negligible. Acompleteanalysisofdispersioninnonlinear
r materials is impossible at this stage, since the quantiza-
a I. INFORMATION CAPACITY
tionofthesesystemshasbeensolvedonlyperturbatively.
The basic idea behind the enhancement we find is that
themodificationofthesystemspectrumduetononlinear The information capacity of a noiseless channel is de-
Hamiltonians may allow one to better employ the avail- finedasthemaximumnumberofbitsthatcanbereliably
able energy in storing the information: we will present sentper channeluse. Fromthe Holevo bound (provenin
some examples that exhibit such effect. Excluding the Ref. [1] for the infinite dimensional case), we know that
down conversion channel (a model sufficiently accurate itisgivenbythemaximumoftheVonNeumannentropy
to include the propagationissue –see Sec. IIC), allthese S(̺) = −Tr[̺log2̺] over all the possible input states
examplesarehighlyidealizedsystemsbutarestillindica- ̺ of the channel. In our case, since ̺ is the state of a
tive of the possible nonlinearity-induced enhancements masslessbosonic field, the associatedHilbert space is in-
in the communication rates. An important caveat is in finite dimensional and the maximum entropy is infinite.
order. In the physical implementations that we have an- However,forallrealisticscenariosacut-offmustbeintro-
alyzed, there is no capacity enhancement if we include duced by constrainingthe energyrequiredinthe storage
in the energy balance also the energy required to create orinthe transmission,e.g. requiringtheentropyS(̺)to
the nonlinear Hamiltonians. This is a general charac- bemaximizedonlyoverthosestatesthathaveanaverage
teristic of any system: if one considers the possibility of energy E, i.e.
employing all the available degrees of freedom to encode
E =Tr[̺H], (1)
2
where H is the system Hamiltonian. The constrained which is valid in the limit ~δω E. When applied
≪
maximization of S(̺) can be solved by standard varia- to communication channels, this last equation is usually
tionalmethods (see[8, 9]for examples),whichentailthe expressed in terms of the rate R (bits transmitted per
solution of unit time) and power P (energy transmitted per unit
time) as
λ λ′
δ S(̺) Tr[H̺] Tr[̺] =0, (2)
− ln2 − ln2 1 πP
(cid:26) (cid:27) R= , (10)
where λ and λ′ are Lagrange multipliers that take into ln2r3~
account the energy constraint (1) and the normalization by identifying the transmission time with 2π/δω. In the
constraint Tr[̺] = 1, and where the ln2 factor is in- next section we analyze how the capacities C and C
nb wb
troduced so that all subsequent calculations can be per- are modified by introducing nonlinear terms in the sys-
formedusingnaturallogarithms. Eq.(2)issolvedbythe tem Hamiltonian.
density matrix ̺=exp[ λH]/Z(λ) where
−
Z(λ) Tr[e−λH] (3)
II. NONLINEAR HAMILTONIANS
≡
is the partition function of the system and λ is deter-
mined from the constraint (1) by solving the equation Nonlinear terms in the Hamiltonian of the electro-
magnetic field derive from the interactions between the
∂ photons and the medium in which they propagate. In
E = lnZ(λ). (4)
−∂λ this section we will employ the techniques described
above to derive the narrow-band and wide-band ca-
The corresponding capacity is thus given by
pacities when quadratic nonlinearities are present. In
C =S[exp( λH)/Z(λ)]=[λE+lnZ(λ)]/ln2, (5) Secs. IIA, IIB and IIC we discuss parametric down-
− conversion type Hamiltonians in the narrow-band and
which means that we can evaluate the system capacity broad-band regimes. In Secs. IID and IIE we discuss a
only from its partition function Z(λ). modeswappinginteraction. Allthesenonlinearitiesarise
In general an explicit expression for Z(λ) is difficult inreal-worldsystemsfromχ(2) typecouplings,whenone
to derive, but it proves quite simple for noninteracting of the three fields involved in these kinds of interactions
bosonic systems, such as the free modes of the electro- is a strong pump field that can be considered classical
magnetic field. In fact, in this case the Hamiltonian is [10]. For all the cases analyzed we present the capac-
given by ity enhancement over the corresponding non interacting
Hamiltonian.
H = ~ω a†a , (6)
k k k
k
X
A. Squeezing Hamiltonian
where ω is the frequency of the kth mode and the
k
mode operators a satisfy the usual commutation rela-
k
tions [ak,a†k′]=δkk′. Hence, the partition function is ConsiderthesinglemodedescribedbytheHamiltonian
∞ H =~ωa†a+~ξ (a†)2+a2 /2+~Ω(ξ), (11)
1
Z(λ)= e−λ~ωknk = . (7)
1 e−λ~ωk where ξ is the squeezin(cid:2)g paramete(cid:3)r. We employ ξ < ω
Yk nXk=0 Yk − toavoidHamiltonianswhichareunboundedfrom|b|elow.
From Eqs. (5) and (7) it is clear that the capacity C In Eq. (11) the frequency Ω(ξ) 1(ω ω2 ξ2) has
≡ 2 − −
will be ultimately determined by the spectrum ω of the beenintroducedsothatthegroundstateofthesystemis
k p
system. In particular, in the narrow-band case (where null: withthischoice,theaverageenergyE istheenergy
onlyamodeoffrequencyω isemployed)Eq.(5)gives[2, associated to the mode a in the nonlinear medium. By
4, 8] applying the canonical transformation
1 ω+ξ
E a=A coshθ A† sinhθ , θ ln , (12)
C =g , (8)
nb (cid:18)~ω(cid:19) − ≡ 4 (cid:20)ω−ξ(cid:21)
the Hamiltonian H is transformed to the free field form
where g(x) (1 + x)log (1 + x) xlog x for x = 0
≡ 2 − 2 6 ~ ω2 ξ2A†A, so that the derivation of the previous
and g(0) = 0. On the other hand, in the homogeneous −
section can be employed to calculate the capacity. It is
wide-bandcase(whereaninfinitecollectionofequispaced p
frequencies ω = kδω is employed for k N) Eq. (5) thus immediate [see Eq. (8)] to find the capacity of this
k
∈ system as
gives [1, 2, 8, 9]
E
π 2E C =g . (13)
Cwb ≃ ln2 3~δω, (9) ~ ω2 ξ2!
r −
p
3
This quantity measures the amount of information that which is higher than the capacity of two independent
can be encoded when E is the total energy associated single-mode bosonic systems, given by Eq. (16) for ξ =
with the mode a, and a nonlinear squeezing-generating 0. [The factors 2 in (16) derive from the presence of
term is present in the Hamiltonian: this result is quite the two modes A and B.] The enhancement is again a
different from the one obtained by using squeezed states consequenceofthe factthatthe nonlinearityreducesthe
as inputs to a linear system (see for example [4]). The effective frequency of the two modes: ω ω2 ξ2.
−→ −
capacityC ofEq.(13)is higherthanthe capacityC of
nb
p
the linear case ξ =0 since g(x) is an increasing function
(seeFig.1). Thereasonbehindthisenhancementisthat C. Broadband parametric down-conversion
the nonlinearity reduces the effective frequencies of the
modes (from ω to ω2 ξ2), so that more energy levels Here we apply the above results to the case of two
−
can now be populated with the same energy. wide-band modes (signal and idler) coupled by a para-
p
metric down conversion interaction. The interaction is
mediated by a nonlinear crystal with second order sus-
ceptibility χ(2) pumped with an intense coherent field
of amplitude at frequency ω . Assuming undepleted
p p
E
pumping and perfect phase matching, the Hamiltonian
at the first order in the interaction is given by
H = ~ω a†a +~(ω ω )b†b
k k k p− k k k
Xk h
+~ξ (a†b† +a b )+~Ω , (17)
k k k k k k
i
wherea andb arethe modeoperatorsofthe downcon-
k k
verted modes of frequency ω and ω ω respectively.
k p k
−
Their interactionis described by the coupling parameter
FIG. 1: Capacity increase of the squeezing Hamiltonian of
Eq. (11) as a function of the energy parameter E/~ω and χ(2)π~ω
k p
of the squeezing ratio ξ/ω. The increase is evident from ξ = E Φ(ω ), (18)
k k
the positivity of C −C , where C is given in Eq. (13) and cǫ0 na(ωk)nb(ωp ωk)
nb −
C is the free space narrow-band capacity of Eq. (8). No-
nb with n and n the refractive indices of the signal and
tice that for high energy, the increase tends asymptotically a b
to −log2 1−ξ2/ω2 /2. idler, and Φ(ωk) the phase matching function that takes
into account the spatial matching of the modes in the
(cid:2) (cid:3) crystal [10]. As usual, the frequency Ω = [ω
k p
−
ω2 4ξ2]/2 has been introduced in the Hamiltonian
p− k
B. Two mode parametric down conversion tqo appropriately rescale the ground state energy. Notice
that, in contrast to the case described in Sec. IIB, the
Considerthetwointeractingmodesaandbevolvedby Hamiltonian (17) couples nondegenerate modes whose
the Hamiltonian frequenciessumuptothepumpfrequencyω . Canonical
p
transformationsanalogousto(15)allowustorewritethe
H =~ω(a†a+b†b)+~ξ(a†b†+ab)+~Ω(ξ), (14)
Hamiltonian in the free field form
where ξ < ω is the coupling constant and Ω(ξ) ω
| | ≡ − H = ~(ω Ω )A†A +~(ω ω Ω )B†B (.19)
ω2 ξ2 has again been introduced to ensure that the k− k k k p− k− k k k
energ−y ground state is null. [This Hamiltonian, as the Xk h i
p
previousone,possessesthestructureofthealgebraofthe WiththisHamiltonian,thepartitionfunctionisgivenby
SU(1,1)group,intheHolstein-Primakoffrealization]. We
can again morph the Hamiltonian to the free field form 1
lnZ(λ) = ln
using the two-field canonical transformation 1 e−λ~(ωk−Ωk)
k (cid:20) − (cid:21)
X
A=a coshθ+b† sinhθ 1 ω+ξ 1
, θ ln ,(15) + ln , (20)
B =a† sinhθ+b coshθ ≡ 4 ω ξ 1 e−λ~(ωp−ωk−Ωk)
(cid:26) (cid:20) − (cid:21) k (cid:20) − (cid:21)
X
which transforms the Hamiltonian (14) to the form
where the two contributions are due to the signal and
~ ω2 ξ2(A†A+B†B). The capacity in this case is
idlermodesrespectivelyandthesumoverk isperformed
−
on all the frequencies up to ω . For ease of calculation,
p p
E
C =2g , (16) we will assume that the coupling ξk is acting only over
2~ ω2 ξ2! a frequency band ζωp (ζ < 1) centered around ωp/2,
−
p
4
whereitassumestheconstantvalueξ. Thischoicegivesa process (amplification plus input energy), no gain is ob-
roughapproximationofthecrystalphasematchingfunc- tained. Thecorrectattitude is,ofcourse,toconsiderthe
tion Φ(ω ) that prevents the coupling of signal and idler signals alone.
k
photons when their frequencies are too mismatched [10].
InthehighenergyregimeE ~δω,thesumsinEq.(20)
≫
can be replaced by frequency integrals, obtaining a) b)
2Æ!
Casym
!p
2 ωp(1−ζ)/2 1
lnZ(λ) = dω ln
δω 1 e−λ~ω
0((cid:13))+(cid:15)
1((cid:13);(cid:16))
Z0 (cid:20) − (cid:21)
0((cid:13))
2 ωp(1+ζ)/2 1
+ dω ln
δω 1 e−λ~(ω−Ω)
Zωp(1−ζ)/2 (cid:20) − (cid:21)
2 ωp 1
1((cid:13);(cid:16))
+ dω ln , (21)
δω 1 e−λ~ω
Zωp(1+ζ)/2 (cid:20) − (cid:21) (cid:13) (cid:13)
whereΩ [ω ω2 4ξ2]/2. TheintegralsinEq.(21)
≡ p− p− FIG. 2: a) Capacity function c0(γ) (continuous line) and its
have no simple qanalytical solution, but we can give a correction c1(γ,ζ) (dashed line) of Eq. (22) with fractional
perturbative expansion in the low-interactionregime i.e. couplingbandwidthζ =.5andγ ≡Eδω/(2~ωp2). Thecapac-
ǫ 4ξ2/ω2 1. In this limit, the result is derived in ity increase accomplished by thenonlinearity is evidentfrom
Ap≡p. A anpd≪is given by thepositivityof thetermc1. b)Comparison between theca-
pacity C of Eq. (22) with ǫ≡4ξ2/ω2 =0.1 (continuous line)
p
C = 2δωωp c0 E~ωδω2 +ǫc1 E~ωδω2,ζ +O(ǫ2) ,(22) oabntdaitnheedaussyimngptaontiicnfitnwiotemfroedqeuernactye bCaansdym(d≡ottle2ndπ2lωδiωnpep).2γ/3,
(cid:20) (cid:18) p (cid:19) (cid:18) p (cid:19) (cid:21)
where the functions c and c are plotted in Fig. 2. The
0 1
zeroth order term c in (22) gives the capacity of two
0
D. Swapping Hamiltonian
broad-bandnon-interactingmodes with cutoff frequency
ω : in the limit of infinite bandwidth (ω ), this
p p
→ ∞
function reaches the asymptotic behavior C C Consider N modes that are pairwise coupled through
asym
→ ≡
2π E/(3~δω) that corresponds to C of Eq. (9) for the Hamiltonian
ln2 wb
two noninteracting wide-band bosonic systems. To first
p N
oξridsegrivinenǫ,btyhethinecvraelauseeoinf cca,pwachiitcyhdiuseatpootshiteivientqeuraacnttiiotny H = ~ωa†jaj + Λjj′a†jaj′
1 j=1 j6=j′
(see App. A). It can be shown that c 0 in the limit X X
ω ,sothatintheinfinite bandwi1d→thregimenoim- = ~~a† (ω11+Λ) ~a, (23)
p · ·
→∞
provement is obtained from the interaction. In fact, an
where~a is the column vectorcontaining the annihilation
infinitecontinuousspectrumisinvariantunderthetrans-
operators a of the N modes and Λ is an N N sym-
formation ωk −→ ωk2−ξk2. As in the non-interacting metric realjmatrix with null diagonal. This Ha×miltonian
broad-band case of Eqs. (9) and (10), the transmission
p describes N modes a whose photons have a probabil-
time τ of the signal can be estimated as 2π/δω. The j
validity of this assumption (which implies negligible dis- ity amplitude Λjj′ to be swapped into the mode aj′. By
performing a canonical transformation on all the mode
persion) rests on the fact that the Hamiltonian (17) is
operators, it is possible to rewrite Eq. (23) in the free
valid to first order in the interaction term ξ and the
k
field form
small dispersion which derives from such term does not
play any role in the capacity [8, 11]. N
In this calculation E is the energy devoted to the sig- H = ~(ω+λ )A†A , (24)
j j j
nalandidlermodes: itdoesnotincludetheenergyspent
j=1
X
to pump the crystal. This last quantity is proportional
to p 2 (apart from corrections of order ǫ which derive where the λjs are the N eigenvalues of Λ. Two condi-
|E |
from the coupling). In the limit of undepleted pump- tions must be satisfied: the positivity of H requires that
ing we have considered, this is a very large contribution λj 6ω forallj,and,sincethe diagonalizationofΛmust
which overshadows E and that is not directly used to preserve its trace, jλj = 0. The capacity of this sys-
encode information. In this respect, the system is very tem can now be easily computed as [1, 2, 8]
P
inefficient in using the total available energy. However,
thisisnotthecorrectattitudetoevaluatesuchasystem: N
e
j
our process is analogous to the amplification of signals C = max g , (25)
where, if one considers the total energy employed in the e1,e2,···,eNj=1 (cid:18)~(ω+λj)(cid:19)
X
 
5
where the maximum must be evaluated under the re- complicated,weshowanincreaseincapacitybyconsider-
quirement that e =E. In the simple case of N = 2 ing the following choice ofthe coupling constants (which
j j
(whereλ = λ ξ),themaximization(25)canbeeas- may well not be the optimal one):
1 2
− P≡
ily performed numerically: the increase in capacity over
λ = ω r for j =1, ,N 1 (28)
the twomode noninteractingcase is presentedin Fig.3. jk k
− ··· −
In this case, in the strong coupling regime (ξ ω) the λNk = ωk(N 1)r , (29)
capacitydivergesaslog [E/(~(ω ξ ))]: this|co|r→responds −
to employing for the in2formation−s|to|rage only the lowest where 0 6 r 6 1. The choices (28) and (29) auto-
frequency mode among A and A . matically guarantee both the positivity condition on the
1 2
Hamiltonian andthe trace preservationcondition on the
Λ(k) matrices. With this choice, the Hamiltonian (27)
becomes
N−1
H = (1 r) ~ω A† A
− k jk jk
k j=1
X X
+[1+(N 1)r] ~ω A† A . (30)
− k Nk Nk
k
X
Essentiallywehavechosenacouplingthatcontractsbya
factor1 rthefrequenciesofthefirstN 1normal-modes
− −
andstretches by a factor[1+(N 1)r] the frequency of
−
theNthone. Choosingr 1,wecanincreasethecapac-
ityad libitumoverthecas→eofN non-interactingsystems.
FIG. 3: Capacity increase of the swapping Hamiltonian as a In fact, in this limit a straightforwardapplication of the
fruanticotiξo/nωo,fgitvheenebnyerpglyottpianrgam∆eCte≡r EC/−~ω2ga(End/(o2f~ωth)e)(ctohuepfilirnsgt TwhidisebraensdultcamlcuusltatbioencoofmEpaqr.ed(9)w,itghivethseCca≃se iN1n−−rw1Chwicbh.
term being the capacity C of Eq. (25) and the second the the N wideband modes are independent where the ca-
capacity of two non interacting modes that share an energy pacityscalesas√NC . Clearlyanarbitraryincreasein
wb
E). capacity is gained as r 1.
→
III. GENERAL CONSIDERATIONS
E. Broadband swapping Hamiltonian
Intheprevioussectionswehavederivedthecapacityof
A generalization of the Hamiltonian (23) given in the varioustypesofnonlinearbosonicsystems. Thecommon
previous subsection can be obtained by considering N feature of all these results is that the nonlinearities were
parallelbroadbandmodes in whichthe coupling joins all used to reshape the spectrum by compressingit to lower
the modes with the same frequency, namely, frequencies, where it is energetically cheaper to encode
information. A couple of remarks are in order. First of
N all,inourcalculationsthe meanenergyE representsthe
H = ~ωka†jkajk energy of the modes in the system, whereas customarily
k j=1 one considers the input energy. Our choice is motivated
XX
+ Λ(jkj′)a†jkaj′k , (26) binyfotrhmeafaticotnthbaetarninonglminoeadress,yssotetmhastdtihspeeinnspeuetneenregrygytodtohees
k j6=j′
XX not necessarily coincide with the amount of energy that
where a is the annihilation operator of the jth system the medium where information is encoded needs to sus-
jk
with frequency ω . This Hamiltonian can be taken to a tain. This last quantity is a practically relevant one for
k
free field form using the procedure of the previous sub- the casesinwhichthe degreesoffreedomusedtoencode
section, so that information cannot handle high energies (as for example
optical fibers [12]). Notice that, in some of the cases we
N havestudied (the exampleof Sec.IIC in particular),the
H = ~(ω +λ )A† A , (27) nonlinearityis achievedby supplyingthe systemwithan
k jk jk jk
k j=1 externalenergysourceintheformofanintensecoherent
XX
beam. Ifoneweretotakeintoaccountalsothiscontribu-
where λ is the jth eigenvalue of the matrix Λ(k). Ex- tionintheenergybalance,thennocapacityenhancement
jk
pressed in terms of the normal modes A , the Hamil- would be evident, since the pumping energy is used only
jk
tonian (27) describes N independent wideband modes. indirectlytostoretheinformation. However,itisnotun-
Since in this case the maximization of the form (25) is warrantedtoexcludethe pumpfromthe energybalance,
6
since it is only used to set up the required Hamiltonian band modes (signaland idler) with cut-off frequency ω .
p
andnotdirectlyemployedinthe informationprocessing. ReplacingEq.(A1)intotheenergyconstraint(4)wecan
Finally, our analysis is limited to the noiseless case. In find the value of the Lagrange multiplier λ, contained in
the presence of noise in place of the von Neumann en- the parameter β, by solving the equation
tropy in Eq. (2), one would have to consider the Holevo
information [13]. This is a highly demanding problem
∂f (β) ∂f (β,ζ)
0 1
because of the yet unknown additivity properties of this +ǫ = γ , (A4)
∂β ∂β −
quantity. At least in the case of linear bosonic systems,
thecapacityinthepresenceofnoisewasstudiedin[4,14].
This work was funded by the ARDA, NRO, NSF, and where γ Eδω/(2~ω2) is a dimensionless quantity. By
≡ p
by ARO under a MURI program. expanding the solution β for small ǫ as β = β +ǫβ , it
0 1
follows that
APPENDIX A
∂f (β )
0 0
= γ , (A5)
∂β −
In this appendix we derive formula (22) for the broad-
band parametric down-conversion Hamiltonian. In the ∂f1(β0,ζ) ∂2f0(β0)
β = . (A6)
low-interaction regime ǫ 1, Eq. (21) becomes 1 − ∂β , ∂β2
≪
2ω
lnZ(λ)= p f (β)+ǫf (β,ζ)+ (ǫ2) , (A1)
δω 0 1 O These two equations can be numerically solved for any
(cid:2) (cid:3) value of γ. Replacing the solution in (A1) and using
where β λ~ωp and the capacity formula (5) we can evaluate the parametric
≡
down-conversioncapacity as reported in Eq. (22) where
1 β 1
f (β) dx ln (A2)
0 ≡ β 1 e−x
Z0 (cid:20) − (cid:21) c0(γ) [β0γ+f0(β0)]/ln2, (A7)
1 1 e−β(1+ζ)/2 ≡
f1(β,ζ) ≡ 4ln 1−e−β(1−ζ)/2 . (A3) c1(γ,ζ)≡f1(β0,ζ)/ln2. (A8)
(cid:20) − (cid:21)
Notice that the zeroth order term f in the expansion Both these functions depend on the system energy only
0
(A1) correspondsto the partition function of two broad- through the quantity γ.
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