Table Of ContentQuantum Equilibration under Constraints and Transport Balance
Gernot Schaller∗
Institut fu¨r Theoretische Physik, Technische Universita¨t Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
For open quantumsystems coupled to a thermal bath at inverse temperatureβ, it is well known
thatundertheBorn-,Markov-,andsecularapproximationsthesystemdensitymatrixwillapproach
thethermal Gibbsstate with the bath inversetemperature β. Wegeneralize thisto systems where
there exists a conserved quantity (e.g., the total particle number), where for a bath characterized
by inverse temperature β and chemical potential µ we find equilibration of both temperature and
chemical potential. For couplings to multiple baths held at different temperatures and different
chemicalpotentials,weidentifyaclassofsystemsthatequilibratesaccordingtoasinglehypothetical
average but in general non-thermal bath, which may be exploited to generate desired non-thermal
states. Under special circumstances the stationary state may be again be described by a unique
Boltzmann factor. These results are illustrated by several examples.
PACSnumbers: 05.60.Gg, 03.65.Yz
1
1 Thermalization is a classical phenomenon: Coupling situation for transport [13] from one reservoir through
0
two materials at different temperature will lead to equi- the system to another reservoir, which may be used to
2
libration at some intermediate temperature – depending generate interesting non-equilibrium stationary states in
n on the heat capacities of the constituents. Especially the system.
a
when one piece is significantly larger than the other, the This paper is organized as follows: Having introduced
J
temperature of the larger piece will hardly change, such the terminologyinSec. I we showhowconservedquanti-
1
that it may be understood as a heat bath. In contrast, ties lead to additional properties of the dampening coef-
3
thetemperatureofthesmallerpiecewillsimplyapproach ficients in Sec. II. The case of a single bath is discussed
] the bath temperature in that limit. in Sec. IIIA, followed by a discussion of multiple baths
h inSec.IIIB. Wederivegeneralstatementsontheresult-
The dynamics of open quantum systems that are cou-
p
ingnon-equilibriumstationarystateformasterequations
- pled to a thermal bath is howevermore difficile [1, 2]. A
nt powerful tool to describe the evolution of such systems that are tridiagonal in the system energy eigenbasis in
Sec. IIIC. The conditionsunderwhichsucha stationary
a in various limits is the quantum master equation [3, 4]:
u A first order differential equation – typically with con- statemaystillappearthermalarediscussedinSec.IIID.
q stant coefficients – describing the evolution of the sys- Finally, the results are demonstrated with a number of
[ examples in Sec. IV.
tem part of the density matrix. As the derivation of an
3 exact master equation is impossible in most cases, one
v has to rely on perturbative schemes. In such schemes, it
I. PRELIMINARIES
4 is often already a challenge to preserve the fundamental
9
propertiesofthedensitymatrixsuchasitstrace,itsself-
0 adjointness, and its positive semidefiniteness. Starting We will consider a large closed quantum system with
3
from microscopic models, especially the last property is the total Hamiltonian
.
0 often hard to fulfill, as for master equations with con-
1 H =H +H +H , (1)
stant coefficients, preservation of positivity requires the S B SB
0
dissipator to be of Lindblad [5] form. Such Lindblad
1 whereH andH actonlyonthesystemandbathparts,
S B
: form dissipators are generically derived in the singular respectively, and H mediates a coupling. The latter
v SB
coupling limit [6], the weak-coupling limit – also termed
i may generally be decomposed as [3]
X Born-Markov-secular [7] (BMS) approximation – and in
r coarse-graining schemes [8]. Within the BMS approxi- M
a mation, thermalization of the system and equilibration H = A B (2)
SB α α
of the systems temperature with that of the bath have ⊗
α=1
X
been proven [9]. However, some baths are not only de-
scribedbyatemperature,butmayalsoequilibrateunder with M hermitian system (Aα = A†α) and bath (Bα =
further side constraints – typically modeled by a chem- B†) coupling operators. By convention, the system cou-
α
ical potential. When we consider couplings to multiple plingoperatorsmaybechosentracelessandorthonormal
baths held at different temperatures [7, 10] and/or dif- Tr AαAβ = δαβ. For example, for an N-dimensional
ferent chemical potentials [11, 12], we have the generic sys{tem Hi}lbert space one may use the M =N2 1 gen-
−
erators of the symmetry group SU(N) for the system
coupling operators.
Under the Born, Markov, and secular (BMS) approx-
∗Electronicaddress: [email protected] imations [3], and assuming that the bath is kept in an
2
equilibriumstateρ¯ withthepropertiesTr B ρ¯ =0 onecaneasilyverify[3]theKubo-Martin-Schwinger[17–
B B α B
{ }
as well as [H ,ρ¯ ]=0, one derives a master equation of 19] (KMS) condition (τ) = ( τ iβ). Since the
B B αβ βα
C C − −
Lindblad form for the system density matrix ρ . In the bath correlationfunctions are analytic in the lower com-
S
system energy eigenbasis H a E a it assumes the plex half plane, the Fourier transform of the KMS con-
S a
| i ≡ | i
form [14] dition reads
γ ( ω)=e−βωγ (+ω), (8)
αβ βα
ρ˙ = i H + σ˜ a b ,ρ (t) −
S S ab S
− " | ih | #
ab and can be used to prove [9] that the equilibrated Gibbs
X
† state
+ γ˜ a b ρ (t) c d
ab,cd S
| ih | | ih |
aXbcd h(cid:16) (cid:17) (cid:16) (cid:17) ρ¯ = e−βHS (9)
1 c d † a b ,ρS(t) , (3) S Tr{e−βHS}
−2 | ih | | ih |
(cid:26)(cid:16) (cid:17) (cid:16) (cid:17) (cid:27)i is a stationary state of Eq. (3).
where σ˜ = σ˜∗ defines the unitary action of decoher-
ab ba
ence (also denoted Lamb-shift [3] or exchange field [15])
and the dampening coefficients γ˜ab,cd describe the non- II. CONSERVED QUANTITIES
unitary (dissipative) terms due to the interaction with
the reservoir. The net effect of the secular approxima-
Now assume that there exists a conserved quantity
tion mentioned above is that these coefficients may van-
N =N +N ,whereN andN actonlyonsystemand
S B S B
ish when some transition frequencies are not matched
bath,respectively. Thatis,weassumethat[H ,N ]=0,
S S
1 [HB,NB] = 0, and [HSB,NS +NB] = 0, such that non-
σ˜ab = 2i σαβ(Ea−Ec)δEb,Ea × trivial evolution arises via [HSB,NS]6=06=[HSB,NB].
c αβ The conservation laws imply the identity
XX
c Aα a ∗ c Aβ b , HSB =e−κ(NS+NB)HSBe+κ(NS+NB). Acting on this
×h | | i h | | i expression with e+κNS[...]e−κNS yields with Eq. (2) the
γ˜ = γ (E E )δ
ab,cd αβ b− a Ed−Ec,Eb−Ea × identity
αβ
X
×ha|Aβ|bihc|Aα|di∗ , (4) e+κNSAαe−κNS ⊗Bα =
α
which is formally expressed by the Kronecker-δ symbols X(cid:0) (cid:1)
(but see [14, 16] for a Lindblad-form coarse-grainingap- Aα e−κNBBαe+κNB , (10)
⊗
proach circumventing the secular approximation): The α
X (cid:0) (cid:1)
Lamb-shiftHamiltonianfor examplewill only actwithin
such that effectively a transformation of the form
the subspace of energetically degenerate states. Note e+κNS[...]e−κNS on a system coupling opera-
that when the spectrum of the system Hamiltonian is
tor is mapped into a transformation of the form
non-degenerate, Eq. (3) may be simplified into a rate e−κNB[...]e−κNB on the corresponding bath coupling
equation system (which is independent on the Lamb-
operator. We will show in the following that this
shift) for the diagonals of ρ in the energy eigenbasis.
S identity leads to additional properties of the dampening
In the dampening coefficients, the functions
coefficients in Eq. (3), when the bath density matrix is
+∞ assumed to be in the grand-canonical equilibrium state
γ (ω) C (τ)e+iωτdτ, with chemical potential µ
αβ αβ
≡
−Z∞ e−β(HB−µNB)
ρ¯ = . (11)
+∞ B TrB e−β(HB−µNB)
σ (ω) C (τ)sgn(τ)e+iωτdτ (5)
αβ αβ
≡ −Z∞ Note that – depending o(cid:8)n the spectrum(cid:9) ofHB – normal-
izability of ρ¯ may impose constraints on the chemical
B
are even (γαβ) and odd (σαβ) Fourier transforms of the potential, compare Sec. IVB, Sec. IVC, and Sec. IVD.
bath correlationfunctions Evidently, as [H ,N ] = 0, we may and will in the
S S
following choose a to be the common eigenbasis of the
Cαβ(τ)≡TrB e+iτHBBαe−iτHBBβρ¯B . (6) two operators wit|hiHS a Ea a and NS a Na a .
| i≡ | i | i≡ | i
When we multiply the Lamb-shift coefficients σ˜ by a
Thebathcorrelation(cid:8)functionshavemanyint(cid:9)erestingan- ij
alytic properties [3]. For example, when the bath is held factoroftheforme+βµ(Ni−Nj),wemayuseEq.(4)tore-
at a thermal equilibrium state (canonical ensemble) place the eigenvalues by operators,such that the system
operators in Eq. (4) are rotated. Then, the identity (10)
e−βHB with Eq. (5) can be used to transfer the pseudo-rotation
ρ¯ = , (7)
B TrB e−βHB to the bath correlation functions. Finally, we may use
{ }
3
the invariance of the trace over the bath degrees of free- c. Fordegenerateenergylevelswehavetheadditional
dom under cyclic permutations and [N ,ρ¯ ] = 0 to see possibility that i=j but E =E . Cancellation of
B B i j
6
that the Lamb-shift terms in (16) results fromEq.(12).
Showing that the remaining dissipative terms also
σ˜ e+βµ(Ni−Nj) =σ˜ , (12)
ij ij vanish amounts to
i.e., the LambshiftHamiltonianonly acts onstates with 0 = γ˜ e−β(Ea−Ei)e+βµ(Na−Nj)
ia,ja
both degenerate energy and particle number. An analog
a
calculation for the dissipative coefficients γ˜ reveals X
ab,cd 1
the identity γaj,ai e+βµ(Ni−Nj)+1 , (18)
−2
γ˜ e+βµ(Ni−Nj) =γ˜ . (13) Xa h i
aj,ai aj,ai
which is directly evident from relations (13)
WhenweconsideradditionalthermalBoltzmannfactors, and (14).
one needs to change the integration path in the Fourier
transform(5)–using thatthe bathcorrelationfunctions Tosummarize,wehaveshownthatthestate(15)isasta-
are analytic – to show that the balance relation tionary state of the quantum master equation (3), when
the reservoir density matrix is of the form (11). Gener-
γ˜ia,jae−β(Ea−Ei)e+βµ(Na−Nj) =γ˜aj,ai (14) allyofcourse,theexistenceoffurtherstationarystatesis
possible,butforanergodic[3]evolutiontheBMSapprox-
holds. Such relations are termed fluctuation theo-
imationschemeforasinglereservoirleadstoequilibration
rems [20]. Specifically, relations (12), (13), and (14)
of both temperature and chemical potential. This equili-
generalize the KMS condition (µ = 0) for the quantum
bration has been noted earlier for specific examples [21–
master equation in Eq. (8) to systems with a conserved
23]andhas been usedquite generallyfor systems ofrate
quantity.
equations [24]. Here however, we have a rigorous proof
for the quantum master equation.
III. STATIONARY STATE
B. Multiple Reservoirs
A. Single Reservoir
When the system of interest is is not only coupled to
The matrix elements of the grand-canonical Gibbs a single, but multiple (K) reservoirs
state (11) read
K K
ρ¯ij = hi|e−β(HS−µNS)|ji = δije−β(Ei−µNi) , (15) HSB = Aα⊗ Bα(k), HB = HB(k), (19)
Z Z α k=1 k=1
X X X
whereZ =Tr e−β(HS−µNS) denotesthenormalization. where varying coupling strengths are absorbed in the
For such a diagonal density matrix, the time-evolution B(k) operators and the independent reservoirs are char-
α
(cid:8) (cid:9)
of off-diagonal matrix elements may in principle still be acterized by different inverse temperatures β(k) and dif-
influenced by the diagonals, as Eq. (3) reduces to ferent chemical potentials µ(k)
ρ¯˙ij = −iσ˜ij(ρ¯jj −ρ¯ii)+ γ˜ia,jaρ¯aa K e−β(k)(cid:16)HB(k)−µ(k)NB(k)(cid:17)
1 Xa ρ¯B = k=1 Tr(k) e−β(k)(cid:16)HB(k)−µ(k)NB(k)(cid:17) , (20)
γ˜ (ρ¯ +ρ¯ ) . (16) O B
aj,ai ii jj
−2 (cid:26) (cid:27)
a
X
much less is known about the resulting stationary
To show stationarity of the Gibbs state (15), it is conve-
state [25]. A decomposition of the interaction Hamil-
nient to distinguish different cases:
tonian in the form of Eq. (19) with identical system
coupling operators for each bath is always possible, as
a. Trivially,when i=j and also E =E in Eq. (16),
i j
wehaveρ¯˙ =0,s6inceallcoefficien6 tssimplyvanish, we have chosen the Aα operators to form a complete
ij
basis set for hermitian operators in the system Hilbert
cf. Eq. (4).
space. We assume that some interaction Hamiltonians
b. Wredhuecnesit=o tjhaenrdateeveidqeunattliyonalssyostEeim= Ej, Eq. (16) mayobeyaconservedquantityN(k) ≡NS+NB(k),where
N(k),H(k) = 0 = N +N(k), A B(k) . Evi-
B B S B α α⊗ α
ρ¯˙ii = + γ˜ia,iaρ¯aa− γ˜ai,aiρ¯ii =0, (17) hdently, the fiorm of Eqh. (3) remainsPinvariant withi
a a
X X K K
which vanishes due to the detailed balance rela- γ˜ = γ˜(k) , σ˜ = σ˜(k), (21)
ab,cd ab,cd ab ab
tion (14), evaluated for i=j.
k=1 k=1
X X
4
whereγ˜(k) andσ˜(k) describethedissipationandLamb- or bosonic
ab,cd ab
shift, respectively, due to the k-th bath only. Accord-
1
ingly, each bath yields separate detailed balance condi- F(k)(ω) = n(k)(ω) (26)
tionsoftheformofEqns.(12),(13),and(14). Ingeneral, + eβ(k)(ω−µ(k)) 1 ≡
−
this will lead to a non-equilibrium stationary state.
baths. It is obvious from Eq. (23) that for coupling to
a single bath, neither g nor the energy dependence of
m
the tunneling rate G(k)(ω) do affect the stationary state
C. Rate Equations for Ladder Spectra
–onlythetransientrelaxationdynamicswillbechanged.
However, when the system is coupled to multiple baths
Quite general statements on the resulting non- obeyingEqns.(24),itiseasytoshowthatthestationary
equilibrium stationary state may be obtained for mas- state – as characterized by Eq. (23)
ter equations that assume tri-diagonal form in an
NH-dmimen=sioEnalmenearngdyNandmnu=mNber meig)e,nwbhaseirse ρ(with ρ¯m+1 = kG(k)(ωm+1,m)F±(k)(ωm+1,m)
hmS||ρSi|mi amnd| wie assumSe|ordieringmw|ithirespect tomth≡e ρ¯m kPG(k)(ωm+1,m) 1±F±(k)(ωm+1,m)
(quasi-)particlenumbersinthesystem(Nm+1−Nm =1) PF¯±(ωm+1,m) h i
= (27)
1 F¯ (ω )
ρ˙ = γ˜ ρ +γ˜ ρ ± m+1,m
m m,m−1 m−1 m,m+1 m+1 ±
[γ˜ +γ˜ ]ρ . (22)
m−1,m m+1,m m is the same as one for a hypothetical single structured
−
bath with the weighted averageoccupation function
In this basis, the populations evolve independently from
the coherences (which we assume to decay), and tran- G(k)(ω)F(k)(ω)
sitions between populations are mediated by single par- F¯ (ω) k ± . (28)
± ≡ G(k)(ω)
ticle tunneling processes. Note however, that even an P k
effective rate equation system of the form (22) may
For a single (K = 1) bathPof either fermionic or bosonic
keep genuine quantum properties as the eigenstates m
| i naturetheratioin(27)reducestotheconventionalBoltz-
themselves may e.g. be entangled between different sub-
parts of the system. At the boundaries m and m of mann factor ρ¯m+1/ρ¯m = e−β(1)(ωm+1,m−µ(1)), such that
1 2
we will term it generalized Boltzmann factor further-
the spectrum (where N = 0 or N = N) the un-
m1 m2 on. Evidently, the energy dependence of the tunnel-
physical tunneling rates vanish, for example we have
ing rates enters the average occupation function (28)
ρ˙ =+γ˜ ρ γ˜ ρ . Computing the
m1 m1,m1+1 m1+1− m1+1,m1 m1 and may therefore be used to tune the resulting non-
stationarystateandusingtheresultsfromtheboundary,
equilibrium stationary state. For example, whereas the
this implies that for all m, it has to satisfy
canonical Gibbs state (9) will for finite temperature al-
ρ¯ γ˜ ways favor the ground state and the grand-canonical
m+1 m+1,m
= . (23)
Gibbs state (11) may favor states with a certain particle
ρ¯ γ˜
m m,m+1
number, it is here possible e.g. to select multiple states
of interest. Note however, that with using only bosonic
Together with the trace condition ρ¯ =1 this com-
m m
baths (with n(ω) 0) it is not possible to achieve e.g.
pletely defines the stationary state. In addition, we as-
≥
P ρ¯ >ρ¯ ,whichisinstarkcontrasttofermionicbaths,
sumethat(compareSec.IVforexamples)thedampening m+1 m
coefficientsassociatedwiththekth bathhavethedecom- where this only requires F¯−(ωm+1,m)>1/2.
position Toillustratethisidealetusforsimplicityparameterize
the tunneling rates phenomenologically(but see e.g. [26,
γ˜(k) = g G(k)(ω ) 1 F(k)(ω ) , 27]foramicroscopicjustification)byaLorentzianshape
m,m+1 m m+1,m ± ± m+1,m
γ˜m(k+)1,m = gmG(k)(ωm+1,m)Fh±(k)(ωm+1,m), i(24) Γ(k)(ω)= (ω Γω¯kδ)k22+δ2 (29)
− k k
where ω E E , and g , G(k)(ω) and
F±(k)(ω) mco+n1,tmain≡themd+e1ta−ils mof the symstem, the cou- wWithhenmtahxeimauvmerargaeteoΓcckupatatfiroenqufeunnccytioω¯nk aatndthweidttrhanδski-.
pling, and the thermal properties of the bath, re- tion frequency ω between states m and m+1
m+1,m
spectively. Using this factorization in the local bal- exceeds 1/2, the population of the sta|tesiwill i|ncreasei
ance condition (14) leads with Nm+1 − Nm = 1 to ρm+1 >ρm,whereastheoppositeistruefor¯f(ωm+1,m)<
γ˜(k) e−β(k)ωm+1,me+β(k)µ(k) = γ˜(k) , which is auto- 1/2, see Fig. 1. Depending on the number and the
m,m+1 m+1,m
matically fulfilled for fermionic thermal properties used baths, the resulting hypotheti-
calaveragedistribution (28) may assume quite arbitrary
F(k)(ω) = 1 f(k)(ω) (25) shapes, such that more sophisticated statistical station-
− eβ(k)(ω−µ(k))+1 ≡ ary mixtures than in Fig. 1 are conceivable.
5
0 1 2 3 4 5 6 7 8 910 fix the parameters β¯ and µ¯. Note however, that then in
stationary occupation
1 1100--11 general β¯ < 0 is possible, which corresponds to popula-
1100--22 tion inversion (i.e., favoring for µ¯ = 0 the most excited
state) Evidently, this forbids the interpretation of the
0.8 quantum number m 1100--33 parameter β¯ as inverse temperature.
n
o Alternatively, for N > 2, Eqns. (27) may still be lin-
ati0.6
p early dependent, as is the case when the system Hamil-
u
c tonianprovidesonlyasingletransitionfrequency(e.g.,a
h oc0.4 harmonicoscillator). Inanapproximatesense,linearde-
at pendence may also be generated when the averageoccu-
b
e 0.2 pationfunction F¯±(ω) is essentially flatin the transition
v
cti frequencywindow probedby the system. Assuming only
effe 0 arastiensgaleretrdaensscitriiobnedfrbeqyuaensciyngΩle, snuucmhbtehratGt(hk)e(Ωtu)nnelΓing,
-4 -2 tran0sition2 frequ4ency6 (8units1 0of 1 2 ) 14 and vanishing chemical potentials µ(k) = 0 = µ¯ i≡n thke
low-energy limit β(k)Ω 1, Eqn. (27) defines average
FIG.1: (ColorOnline)Effectivebathoccupation(solidblack) temperatures for bosons≪and fermions
inEq.(28)generatedbytwofermionicbaths(dottedredand
dashed green curves, chemical potentials µ(k) and temper- Γ
atures β(k) −1 are given by the center and width of the T¯Bose ≈ ΓkT(k),
h i k
X
shaded boxes, respectively) with Lorentzian tunneling rates 1 Γ 1
k
of form (29). The detuning of the tunneling rates is ad- (31)
T¯ ≈ Γ T(k)
justed such that the average hypothetical occupation func- Fermi k
tion ¯f(ω) is not monotonically decaying. Then, the thresh- X
old 1/2 is passed several times (vertical solid blue lines) with Γ Γ , that correspond to the weighted arith-
≡ ℓ ℓ
withinthetransitionfrequencywindowprobedbythesystem. metic (bosons) and weighted harmonic (fermions) mean.
This will lead to an increasing population ρm+1 > ρm when The bosonPic averagetemperature does well resemble the
¯f(ωm+1,m)>1/2 and to a decreasing population ρm+1 <ρm Richmann mixing formula and has been found previ-
when ¯f(ωm+1,m) < 1/2 (compare the inset). Such multi- ously [10]. However, it should be noted that this is
modaldistributionsareclearlynon-thermal,i.e.,theycannot
different from the so-called classical (high-energy) limit
begenerated byasingle thermal bathby adjustingtempera-
β(k)Ω 1, for which one obtains an arithmetic mean of
tureandchemicalpotential. Parametershavebeenchosenas ≫
∆ǫ β(1) = ∆ǫ β(2) = 2, µ(1) = ∆ǫ , µ(2) = 9∆ǫ , Γ = Γ , the Boltzmann factors from Eq. (27)
0 0 0 0 1 2
δ =δ =0.1∆ǫ ,andω¯ =∆ǫ withω¯ =9∆ǫ . Theshown
1 2 0 1 0 2 0
equidistant transition frequencies (vertical solid and dashed e−β¯Ω Γke−β(k)Ω (32)
bluelines) correspond to eigenvalues that scale quadratically ≈ Γ
k
with m and may be realized by the example in Sec. IVA by X
using N =10, ǫ=∆ǫ , U =∆ǫ , and T =0.
0 0 for – as expected – both fermionic and bosonic baths.
D. Trivial dynamic equilibria
IV. EXAMPLES
It has been observed for special systems [10] that the
For a single reservoir the equilibration of both tem-
equilibrium state had a thermal form with some average
perature and chemical potential has been observed for
temperatureeventhoughthe systemofinterestwascou-
interacting double dots [21] in the BMS approximation.
pled to more than just a single thermal bath with differ-
Therefore,weonly giveexamples toillustrate the results
ent temperatures. For the ladder-like systems discussed
inSec.IIIB,Sec.IIIC,andSec.IIID. Naturally,incase
in the previous section, it is obvious that to describe the
stationary state with just two parameters β¯ and µ¯ re- of only a single coupling bath, the case of Sec. IIIA is
also reproduced here.
quires Eqns. (27) to be compatible with
ρ¯m+1 = F¯±(ωm+1,m) =! e−β¯(ωm+1,m−µ¯), (30)
ρ¯ 1 F¯ (ω ) A. Homogeneous Electronic Nanostructure
m ± m+1,m
±
where the effective average occupation is defined in Consider a nanostructure with N homogenous elec-
Eq. (28). This can be achieved in severalpossible ways:
tronic sites (we neglect the spin)
Firstly, for a three-dimensional system Hilbert space
(N = 2 in the previous section), we will only have two N
U
transitionfrequenciesandaccordinglyonlytwoequations H =ǫ d†d + d†d d†d +T d†d (33)
S i i 2 i i j j i j
of the above form. These two equations would uniquely
i=1 i6=j i6=j
X X X
6
with single-particle energy ǫ, Coulomb interaction U, and chemical potentials µ(k) entering the rates as in
and hopping term T that are all assumed com- Eq. (34). The general non-equilibrium steady state of
pletely isotropic. The permutational symmetry sug- Eq. (22) fulfills
geststoreducetheproblemtothesymmetrizedsubspace
withthebasis N,m+1 1 D† N,m with ρ¯ Γ(k)(ω )f(k)(ω )
| i≡ √(N−m)(m+1) | i m+1 = k m+1,m m+1,m (35)
0te≤m,mw≤heNre dDenotingNthednaunmdbeNr o,0f el=ectr0o,n.s..i,n0threepsyres-- ρ¯m kPΓ(k)(ωm+1,m) 1−f(k)(ωm+1,m)
≡ i=1 i | i | i and is thus Pidentical with the(cid:2) non-equilibrium(cid:3)steady
sents the N particle Fock space vacuum. Clearly, both
H and N Pd†d are diagonal in this basis. For state for coupling to a single hypothetical non-
S S ≡ i i i equilibrium bath with average non-thermal distribution
N = 1 we recover the single resonant level [28], but for
P of the form (28), compare also Fig. 1.
for N > 1, the spectrum of the system Hamiltonian be-
comes nontrivial. The eigenvalues of the symmetric sub-
space read E = mǫ + m(m 1)U/2 + m(N m)T,
m
− − B. Coupled oscillators
such that the spectrum may become equidistant when
U = 2T with ω ǫ+(N 1)T – which still ad-
m+1,m
mits a strongly interac→ting model−. We consider a harmonic oscillator H = Ωb†b cou-
S
At first we assume that the nanostructureis only cou- pled to many others H = ω b†b with positive
B k k k k
βpleadndtochaemsinicgallepleoatdentHiaBl µ=viaktǫhkec†ktcuknnaetlitnegmHpearmaitlutore- eHigen=frebquenciehs ωb†k+>b† 0 viaPhq∗uba.si-parTtihcele ctounnsneerlvinedg
nian [33] HSB =D⊗ kgkc†kP+D†⊗ kgk∗ck, where gk quSaBntity⊗isPcomk pkoskedfrom⊗NPSk=kb†kbandNB = kb†kbk.
representsafrequency-dependentcouplingconstant,and Rewriting the interaction Hamiltonian in terms of her-
ck arefermionicannihPilationoperatorsPactingonthelead mitianoperators,weobtainA1 =(b†+b),A2 =Pi(b† b),
−
Hilbert space. The conservedquantity is composedfrom B = (h b†+h∗b )/2,andB =i (h b† h∗b )/2.
NS and NB = kc†kck. We may write the interaction Th1e matkrixkelekmenktskofthe Four2iertranksforkmks−(5)kokfthe
Hamiltonian alsPo as HSB =A1⊗B1+A2⊗B2, with bath cPorrelation functions equate toPγ11(ω) = γ22(ω) =
the hermitian and trace-orthogonal (not-normalized) 1/4[+Θ(+ω)[1+n(+ω)]Γ(+ω)+Θ( ω)n( ω)Γ( ω)]
system coupling operators A1 = D†+D and for the diagonals and γ2∗1(ω) − = −γ12(ω−) =
A = i D† D , and the associated hermitian i/4[+Θ(+ω)[1+n(+ω)]Γ(+ω) Θ( ω)n( ω)Γ( ω)]
2 − (cid:0) (cid:1) − − − −
bath coupli(cid:0)ng opera(cid:1)tors B1 = k gkc†k+gk∗ck /2 and fsoterpthfeunocfft-iodnia,gΓon(ωal)s, wh2eπre Θ(hω)2dδe(nωotesωth)etHheeaqvuisaisdie-
B2 =i k gkc†k−gk∗ck /2. PFor(cid:16)a bath in(cid:17)thermal particle tunneling ra≡te, anPdkn|(ωk|) den−oteskthe bosonic
equilibrium(cid:16) with inver(cid:17)se temperature β and chemi- occupationnumberasdefinedinEq.(26). Thecondition
P
cal potential µ, such that its density matrix is given that µ < mink(ωk) grants positivity of all bath occupa-
by Eq. (11), we obtain for Fourier transforms (5) of tions. In addition, it implies that Γ(Ω)|Ω<µ = 0, such
the bath correlation functions γ (ω) = γ (ω) = that we assume also Ω > µ throughout. Accordingly,
11 22
Γ(+ω)[1 f(+ω)]/4 + Γ( ω)f( ω)/4 and γ (ω) = the Fourier transform matrix of the bath correlation
12
γ∗ (ω)=−iΓ(+ω)[1 f(+ω)−]/4 −iΓ( ω)f( ω)/4, where functions is positive semidefinite at all frequencies. In
Γ2(1ω) 2π g 2δ−(ω ǫ )ist−hetun−neling−rateandthe the Fock space basis (where b† n = √n+1 n+1 ),
≡ k| k| − k we obtain a rate equation of |thie form (22)|, wheire
Fermi function f(ω) encodes the bath properties β and
P 0 n< . However, since even the original eigenstates
µ, compare Eq. (25). Obviously, the Fourier-transform
≤ ∞
are non-degenerate, the dampening coefficients equate
matrix of bath correlation functions has non-negative
(with Ω>0) to
eigenvalues– a consequence ofBochners theorem[3, 29].
These lead to the non-vanishing dampening coefficients
γ˜ = (n+1)Γ(Ω)[1+n(Ω)] ,
n,n+1
γ˜m,m+1 = (N −m)(m+1)Γ(ωm+1,m)[1−f(ωm+1,m)] , γ˜n+1,n = (n+1)Γ(Ω)n(Ω), (36)
γ˜ = (N m)(m+1)Γ(ω )f(ω ), (34)
m+1,m m+1,m m+1,m
− which is also compatible with assumption (24). The re-
where ω E E and the factoring condi- sultingsystemisinfinitelylarge,onemayhowever,intro-
m+1,m m+1 m
≡ −
tion(24)isobviouslyfulfilled. Weobtainarateequation duce a cutoff size N and solve for the stationary state
cut
of the form (22) with ρ N,m ρ N,m , where the oftherateequationsforfiniteN . Theratioρ /ρ of
m S cut n+1 n
≡ h | | i
Lamb-shifttermsareirrelevant,sincewehavebyexploit- two successive populations yields the desired Boltzmann
ing the permutational symmetry mapped our system to factor, which even happens to be independent of N .
cut
a nondegenerate one. For multiple baths, we obtain equilibration in a ther-
Now we consider tunnel couplings to multiple baths mal state with the unique generalized Boltzmann factor
with factorizing density matrices as in Eq. (20). The
form of Eq. (22) remains invariant, and we simply have ρ Γ(k)(Ω)n(k)(Ω)
n+1 = k , (37)
γ˜m,m±1 = kγ˜m(k,)m±1, with different temperatures β(k) ρn kPΓ(k)(Ω) 1+n(k)(Ω)
P P (cid:2) (cid:3)
7
consistent with Eq. (27). This generalized Boltzmann but such a model may represent scattering processes
factoristhesamethatonewouldobtainforcontactwith with a further fermionic bath that omitted from the
asingle hypotheticalnon-thermalbathatanaverageoc- description. As before, we require that µ(1) < Ω and
cupation compatible with Eq. (28), which in the high- µ(1) < min (ω ). Choosing the system coupling op-
k k
temperature and µ(k) 0 limit reduces to the bosonic erators as A = σx and A = σy, we obtain B(1) =
→ 1 2 1
averagetemperatureinEq.(31). Thiscoincideswellwith
1/2 h b† +h∗b , B(1) = i/2 h b† h∗b ,
thewell-knowntemperaturemixingformulaandprevious k k k k k 2 k k k− k k
results [10]. andPB1((cid:16)2) = 1/2(cid:17) k gkc†k+gk∗ckP (cid:16)with B2(2) (cid:17)=
i/2 k gkc†k−gk∗ck P. Th(cid:16)eFouriertra(cid:17)nsformofthebath
C. Spin-Boson Model correlat(cid:16)ion function(cid:17)for the first bath corresponds to
P
Sec. IVC, whereas the Fourier transform matrix for the
A variant of the spin-boson model coupled to a sin- second bath is identical to that of Sec. IVA. Accord-
gle bath has already been provided in Ref. [30], such ingly, we obtain in the σz-eigenbasis σz a = ( 1)a a
| i − | i
that we here only generalize to K 2 baths and non- with a 0,1 and ρ a ρ a the master equation
a S
≥ ∈{ } ≡h | | i
vanishing chemical potentials. We consider a large spin
system H = Ω/2Jz with Ω > 0, coupled to a bath ρ = γ˜(1)+γ(2) ρ + γ˜(1)+γ(2) ρ ,
of harmonSic oscillators H = ω b†b with ω > 0 0 − 1,0 1,0 0 0,1 0,1 1
via HSB = J+⊗ khkb†kB+J−P⊗k kkhkkbkk, wherekJα ≡ ρ1 = +hγ˜1(1,0)+γ1(2,0)iρ0−hγ˜0(1,1)+γ0(2,1)iρ1 (39)
N σα and σ± = (σx iσy)/2. The conserved quan- h i h i
tityi=i1s giiven by NP+N±= Jz/P2+ b†b . We im- with the dampening coefficients γ˜0(1,1) =Γ1(Ω)[1+n(Ω)],
paPsosbeetfohree:saµme<comnSidniti(oωnBs)oann−dthµe c<hemΩP.ickaClkhpokootseinntgialt(hse) γΓ˜1(1,(0)Ω=)f(ΩΓ1),(Ωw)nh(eΩre), fγ˜(0(ω2,1)) =anΓd2(nΩ()ω[)1−haf(vΩe)]b,eaennddγ˜e1(fi2,0)ne=d
k k 2
system coupling operators as A = Jx and A = Jy, in Eqns. (25) and (26), respectively, and Γ (ω) =
1 2 1
the Fourier transforms (5) of the bath correlation func- 2π h 2δ(ω ω ) with Γ (ω)=2π g 2δ(ω ǫ )
k| k| − k 2 k| k| − k
tion are identical to the previous section. In order representthecouplingstrengthstothetwobaths,respec-
to calculate the dampening coefficients, permutational tivePly. The stationary state of Eq. (39P) is characterized
symmetry suggests to use the angular momentum ba- by the generalized Boltzmann factor
sis N/2,m with N/2 m +N/2. Using that
J± |j,m =i j(j+−1) m≤(m 1≤) j,m 1 , we obtain ρ1 = Γ1(Ω)n(Ω)+Γ2(Ω)f(Ω) , (40)
a ra|te eqiuation of the f−orm (22±) wit|h the±coiefficients ρ0 Γ1(Ω)[1+n(Ω)]+Γ2(Ω)[1 f(Ω)]
p −
N N which is consistent with Eq. (27).
γ˜ = +1 m(m+1) Γ(Ω)[1+n(Ω)] ,
m,m+1
2 2 −
(cid:20) (cid:18) (cid:19) (cid:21)
N N V. CONCLUSIONS
γ˜ = +1 m(m+1) Γ(Ω)n(Ω). (38)
m+1,m
2 2 −
(cid:20) (cid:18) (cid:19) (cid:21)
Under the Born, Markov,and secularapproximations,
Solving this rate equation with multiple baths for its
quantum systems coupled to a single bath described by
steady state yields a thermal bath with the same unique
inverse temperature β and chemical potential µ relax –
generalized Boltzmann factor as Eq. (37), as predicted
when the total Hamiltonian conserves a (quasi-)particle
by Eq. (27).
number – into a stationary equilibrium ensemble that
is described by the same inverse temperature and the
samechemicalpotential. As longasonly a singlebathis
D. Mixed Spin Model
involved,thisalsoholdswhenthespectrumofthesystem
Hamiltonian is not equidistant.
We consider a spin-1/2 system H = Ω/2σz that is
S For coupling to multiple thermal baths and tridiago-
firstly coupled to a bosonic bath HB(1) = kωkb†kbk nal rate equations, a hypothetical non-thermal average
via the dissipative coupling H(1) = σ+ h b† + bath is effectively felt by the system, which will in gen-
SB ⊗ Pk k k
σ− h∗b , and secondly to a fermionic bath H(2) = eral lead to a non-thermal stationary state. However,
⊗ k k k P B when there exists only a single transition frequency as
ǫ c†c via the coupling H(2) = σ+ g c† +
σ−k kPk kg∗c . Note that inSBcontrast t⊗o thke kprkevi- e.g. in two-level systems, the resulting stationary state
P ⊗ k k k P may be well characterized by a single Boltzmann factor
ous examples we do now consider two different baths withtwoparametersβ¯andµ¯. Thepossibilityofcreating
P
from the beginning. The interaction Hamiltonians ex- level inversion however demonstrates that then β¯ does
plicitly obeys the conserved quantity constructed from
not always define an inverse temperature.
N = σz/2, N(1) = b†b , and N(2) = c†c .
S − B k k k B k k k There are several interesting consequences: Firstly,
NotethatH(2)doesnotconservethenumberoffermions, by using equilibration under side constraints one should
SB P P
8
be able to prepare not only the ground state of a sys- terest are often the stationary current through the sys-
tem Hamiltonian by dissipative means but also an en- tem and its fluctuations (noise), see e.g. Refs. [31, 32].
ergetically sufficiently isolated energy eigenstate with a The calculation of these quantities heavily depends on
desired particle number when temperature and chemi- the knowledge of the stationary state and may therefore
cal potential of a single grand-canonical bath are tuned strongly benefit from its analytic knowledge.
accordingly. Secondly, in order to generate interesting
non-equilibrium stationary states via coupling to multi-
ple baths, fermionic baths appear more favorable. Be-
VI. ACKNOWLEDGMENTS
yond this, it is necessary (though not sufficient) to con-
sider system Hamiltonians with multiple allowed transi-
tion frequencies. Energy-dependent tunneling rates can Financial support by the DFG (Project
then significantly enhance the non-thermal signatures of SCHA 1646/2-1) is gratefully acknowledged. The
the resulting stationary state. Finally, when the focus author has benefited from discussions with T. Brandes,
is on particle or thermal transport, the quantities of in- M. Esposito, G. Kießlich, and M. Vogl.
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