Decoherence dynamics of interacting qubits coupled to a bath of local optical phonons

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by  Muzaffar Qadir Lone

Decoherence dynamics of interacting qubits coupled to a bath of local optical phonons
Muzaffar Qadir Lone
∗
Department of Physics, University of Kashmir, Srinagar-190006,India and
TCMP Division, 1/AF Salt Lake, Saha Institute of Nuclear physics, Kolkata, India.
S. Yarlagadda
†
TCMP Division, 1/AF Salt Lake, Saha Institute of Nuclear physics, Kolkata, India.
(Dated: January 21, 2016)
6
1 We study decoherence in an interacting qubit system described by infinite range Heisenberg
0 model(IRHM)inasituation wherethesystemiscoupledtoabathoflocal opticalphonons. Using
2 perturbation theory in polaron frame of reference, we derive an effective Hamiltonian that is valid
in the regime of strong spin-phonon coupling under non-adiabatic conditions. It is shown that the
n
effectiveHamiltoniancommuteswiththeIRHMuptoleadingordersofperturbationandthushasthe
a
same eigenstates as the IRHM. Using a quantum master equation with Markovian approximation
J
of dynamical evolution, we show that the off diagonal elements of the density matrix donot decay
0
in the energy eigen basis of IRHM.
2

PACSnumbers: 03.65.Yz,03.67.Pp
]
h
p
- I. INTRODUCTION (i.e.,many-spin)systemcanhavedistancedependentin-
t teraction. The two limiting casesforinteractionarespin
n
a interactions that are independent of distance and spin
A closed isolated quantum system will always follow
u chains with nearest-neighbor interactions only. In this
unitary quantum dynamics given by Schrodinger equa-
q work we consider the extreme case of distance indepen-
[ tion. However every quantum system that we try to
dent interaction among the spins, i.e., the IRHM.
study ormodel is inevitably coupledto some formof en-
1
vironmentandhenceanopenquantumsystem[1,2]. The
v
3 couplingofaquantumsystemwithitsenvironmentleads Inthispaper,weemploytheanalyticallysimplerframe
6 to decoherence, the process by which information is de- ofreferenceofhard-core-bosons(HCBs)ratherthanthat
1 graded. Decoherence is the fundamental mechanism by of spins so that the single particle excitation spectrum
5 which fragile superpositions are destroyed thereby pro- can be easily obtained and exploited; we show that the
0 ducing a quantum to classical transition [3, 4]. In fact, effective Hamiltonian even in higher order (i.e., greater
.
1 decoherence is one of the main obstacles for the prepa- than secondorder)perturbation theory retains the same
0 ration, observation, and implementation of multi-qubit eigenstates as the IRHM when the spins are coupled to
6 entangled states. The intensive work on quantum infor- local phonons. Furthermore, decoherence is studied us-
1 mation and computing in recent years has tremendously ing the quantum master equation approach [32]. Our
:
v increased the interest in exploring and controlling deco- analysis based on Markovian quantum master equation
Xi herence effects [5–23]. shows that the off diagonal matrix elements of density
Thedynamicsofanopenquantumsystemcoupledtoa matrix in eigen-basis of IRHM do not decay.
r
a bath can be either Markovian or non-Markovian [1, 24–
29]. However in this paper we are concerned with the
Markoviandyanmicsofthe systeminfinite rangeHeisen- The rest of the paper is organized as follows: In sec-
berg model (IRHM) coupled to a bath of local optical tionII,weintroducetheIRHMHamiltonianandmapthe
phonons. In case of Markovian processes, the environ- Hamiltonian to a HCB model. Also in the same section
ment acts as a sink for the system information; the sys- we transform IRHM to a polaron frame using a canon-
tem of interest loses information into the environment ical transformation. In section III, we use second order
perturbation theory and with the help of Schrieffer-Wolf
and this lost information plays no role in the dynamics
of the system [1, 30]. Although the theory of decoher- (SW)transformation,wederiveaneffectiveHamiltonian
that commutes with H and thus have same set of
encehasundergonemajoradvances[3,4],yet,thereexist IRHM
many definitions of decoherence [31]. For the analysis in eigenstates. In section IV, we use the master equation
approach and show that the system does not decohere
thispaper,wechoosethemostcommonlyuseddefinition
of decoherence: Loss of off-diagonal elements in the sys- under Markovianapproximation. Finally we conclude in
section V and make some general remarks regarding the
tem’s reduced density matrix. In general, a many-qubit
wider context of our results. The paper also contains
an Appendix A, where we derive the third order pertur-
bation contribution to our effective Hamiltonian (H )
eff
∗email: [email protected] and show that the eigenstates ofthe IRHM Hamiltonian
†email: [email protected] are retained by our Heff.
2
II. MODEL HAMILTONIAN FOR THE SYSTEM given by
COUPLED TO A BATH OF PHONONS
H = J [(0.5b†ibj +H.c.)+∆(ni−0.5)(nj −0.5)]
We consider a system of spin-1 particles interacting i,j>i
2 X
with each other through a infinite range anisotropic 1
Heisenberg antiferromagnetic exchange interaction i.e. +ω a†jaj +gω (nj − 2)(aj +a†j), (3)
IRHM: Xj Xj
where nj ≡ b†jbj. Subsequently, we perform the well-
H =J S~.S~ +(∆ 1)SzSz (1) known Lang-Firsov (LF) transformation [39, 40] on this
IRHM i j − i j Hamiltonian. Under the LF transformation given by
iX,j>ih i eSHe−S =H0+HI withS =−g i(ni−12)(ai−a†i),the
where J >0, ∆ 0,andS = 1σ , i=x, y, z. We note operators b and a transform like fermions and bosons;
that H com≥mutes witih bo2thiSz ( Sz) and this is due jto the ijnteresting comPmutation properties of
IRHM Total ≡ i i
2 HCB given below:
iS~i (≡ ST2otal). In equation (1), it isPunderstood
(cid:16)thPat J(cid:17)= J⋆/(N − 1) (where J⋆ is a finite quantity) [bi,bj] = [bi,b†j]=0, for i6=j,
so that the energy per site remains finite as N .
The eigenstates of HIRHM are characterized by ST→(i.∞e., {bi,b†i} = 1. (4)
the total spin eigenvalue) and Sz (or the eigenvalue of
T Next, the unperturbed Hamiltonian H is expressed as
the z-component of the total spin Sz ). The ground 0
Total [40]
state corresponds to Sz = 0 and S = 0 which is SU(2)
T T
invariant.
H =H +H , (5)
0 s env
The IRHM has relevance to many physical problems.
The Lipkin-Meshkov-Glick (LMG) model [33] HLMG = where we identify Hs as the system Hamiltonian
2h( Sz) 2λ[( Sx)2+γ( Sy)2]/N wellstudied
i−n nuPclejarjma−ny boPdyj pjroblem P(fojr hj= 0 and γ = 1) is Hs = J [(0.5e−g2b†ibj +H.c.)
a special case of the above mentioned long-range model i,j>i
X
forcertiansetofparamters. IthasbeenshownbyEzawa +∆(n 0.5)(n 0.5)], (6)
i j
that the long-range ferromagnetic Heisenberg model de- − −
scribes well a zigzag graphene nanodisc [34]. For spin and H as the Hamiltonian of the environment
env
systems with spins defined on the corners of a regular
tetrahedron can be realized (from a Hubbard model) as Henv =ω a†jaj. (7)
exact special cases of the above long-range model [35].
j
X
In solid state quantum computation using semiconduc-
torquantumdots,spinstatesareprepared,manipulated, Ontheotherhand,theinteractionHI whichwewilltreat
andmeasuredusingrapidcontrolofHeisenbergexchange as perturbation is given by
interaction [36].
Therealquantumcomputerwillnotbefreefromnoise HI =J [0.5e−g2b†ibj]{S+ij†S−ij −1}+H.c., (8)
andthustheentangledstateshaveatendencytoundergo i,j>i
X
decoherence. To study decoherence due to phonons, we
consider interaction with optical phonons such as would where ij =exp[ g(ai aj)]. Inthetransformedframe,
beencounteredwhenconsideringtransitionmetaloxides. the sysSt±em Ham±iltonia−n depicts that all the HCBs are
We write the total Hamiltonian H as coupled to the same phononic mean-field. Thus, the
T
unperturbed Hamiltonian H comprises of the system
0
HT =HIRHM+gω Siz(a†i +ai)+ω a†iai, (2) Hamiltonian Hs representing HCBs with the reduced
Xi Xi hopping term 0.5Je−g2 and the environment Hamilto-
nian H involving displaced bath oscillators corre-
env
where a is the phonondestructionoperator[37], ω is the
sponding to local distortions. Here it should be pointed
opticalphononfrequency,andg isthecouplingstrength.
outthatboththeinteractionoftheHCBwiththemean-
In order to make the calculations simple from spin exci-
fieldaswellasthe localpolaronicdistortionsinthe bath
tations toparticle excitationsinour model,we makethe
oscillators involve controlled degrees of freedom. Now,
mapping of spin operators for spin-1 particles on HCBs.
2 the system Hamiltonian Hs can be expressed as
HCBs are defined on lattice sites i = 1,...,N with re-
stricted occupation numbers, ni = 0,1 [38]. The con- Hs =HIRHM+(Hs HIRHM) (9)
strainedcreationanddestructionoperatorsb and b,are −
†
definedasb =S+, b=S ,andb b=Sz+0.5. Wethen When we changethe Hamiltonian fromH to H by
† − † IRHM s
observe that conservation of Sz implies conservation adiabaticallyturningontheperturbation(H H ),
Total s− IRHM
of total number of HCB. The total Hamiltonian is then the resulting state of the system is still obtainable from
3
that of H by using unitary Hamiltonian dynamics obtainedbythecanonicalSWtransformation[44,45]de-
IRHM
andis thus predictable basedon a knowledgeofthe cou- coupling the low-energy and the high-energy subspaces;
plingparameterg [41]. Thusnoirreversibilityisinvolved this decoupling is a consequence of J⋆e g2 ω. We
−
in going from HIRHM to Hs. On the other hand, pertur- now make the important observation that t≪he effective
bation HI pertains to the interaction of HCBs with local Hamiltonian Hs + H(2), when expressed in terms of
deviations from the phononic mean-field; the interaction spins, has the following form:
term H represents numerous or uncontrolled environ-
I
mental degrees of freedom and thus has the potential for J (SxSx+SySy)+J SzSz , (14)
tr i j i j lng i j
producing decoherence. Furthermore, it is of interest to
i,j>i
notethattheinteractiontermisweakinthetransformed X (cid:2) (cid:3)
frame compared to the interaction in the original frame; and thus has eigenstates identical to those of the
thusonecanperformperturbationtheorywiththeinter- original Hamiltonian H in equation (1) because
IRHM
action term. (SzSz) and H commute. On carrying out higher
i,j>i i j s
order(i.e., beyondsecondorder)perturbationtheory(as
P
discussed in Appendix A), and expressing the results in
III. EFFECTIVE HAMILTONIAN FROM
the spin language, we still get an effective Hamiltonian
SECOND ORDER PERTURBATION THEORY
H ofthe followingformthathasthe sameeigenstates
eff
as the s.
In this section we derive an effective Hamiltonian us-
ingsecondorderperturbationtheoryandSchrieffer-Wolff
(SW)transformation. Werepresenttheeigenstatesofthe Heff = Jxy( Skz)(SixSjx+SiySjy)
" #
unperturbed Hamiltonian H as n,m n m i,j>i k
0 s ph X X
| i ≡ | i ⊗| i
wnithitshtehecoerirgeesnpsotnadteinogfetihgeenseynsteermgiewsiEthn,emig=enEennser+gyEEmphs; + Jz( Skz)Siz, (15)
|whisile m is the eigenstate for the environment withn Xi Xk
ph
eigenen|erigy Eph. Henceforth, for brevity, we will use where J and J are functions of the Sz (= Sz )
m xy z Total k k
tωhme ≡groEumpnhd. sOtanteoebxsperevcitnagtiotnhavtahlu0,e0o|Hf It|h0e,0diev=ia0tio(nis.ei.s, othpaetraetnoarb.lIetsitshteheeiignefinnstitaetersanogfethoeftshyestHemeisteonbreemrPgaminoudne-l
zero), we obtain the next relevant second-order pertur- changed. Next, we study decoherence in a dynamical
bation term [40] context and gain more insight into how the states of our
H can be decoherence free.
s
0,0H n,m n,mH 0,0
E(2) = h | I| ih | I| i. (10)
E E
0,0 n,m
n,m −
X
IV. MARKOVIAN DYNAMICS
EmployingtheSWtransformation(seeAppendix Aof
Ref [43]) with the conditions of strong coupling (g > 1)
In this section, we study the markovian dynamics of
and non-adiabaticity (J⋆/ω 1), we get the following
second-order term H(2) [42] ≤ our system in polaron frame of reference. The dynamics
of the system, described by the reduced density matrix
ρ (t)attimet,isobtainedfromthedensitymatrixρ (t)
0H m mH 0 s T
H(2) = phh | I| iph phh | I| iph of the total system by taking the partial trace over the
− ω
m m degrees of freedom of the environment:
X
= (0.5J(2)b†ibj +H.c.) ρ (t)=Tr [ρ (t)]=Tr U(t)ρ (0)U (t) , (16)
⊥ s R T R T †
iX,j>ih
0.5J(2) n (1 n )+n (1 n ) ,(11) where U(t) representsthe time-(cid:2)evolutionoperat(cid:3)orof the
i j j i
− { − − }
k total system. Now it is evident from the above equation
i
where that we need first to determine the dynamics of the to-
tal system which is a difficult task in most of the cases.
J(2) (N 2)f (g)J2e−2g2 (N 2)J2e−g2, (12) By contrast,master equationapproachconvenientlyand
≡− − 1 2ω ∼− − 2g2ω directly yields the time evolution of the reduced density
⊥
matrix of the system interacting with an environment.
Thisapproachrelievesusfromtheneedofhavingtofirst
J(2) [2f (g)+f (g)]J2e−2g2 J2 , (13) determine the dynamicsofthe totalsystem-environment
≡ 1 2 2ω ∼ 4g2ω combinationandthentotraceoutthedegreesoffreedom
k
of the environment.
with f1(g) ≡ ∞n=1g2n/(n!n) and f2(g) ≡ We consider the following Hamiltonian:
∞ ∞ g2(n+m)/[n!m!(n + m)]. The effective
n=1 m=1 P
Hamiltonian H + H(2) is a low energy Hamiltonian H =H +H , (17)
s 0 I
P P
4
where H is the system-environment Hamiltonian given which the system changes is τ 1 and the bath
0 s ∼ J⋆e−g2
btoyneiaqnuagtiivoenn(b5)yaenqduaHtiIonrep(8r)e.seDntesfinthineginatneroapcteiroantoHraOmiiln- cporrorxeilmataitoinontiims emosctaivleateisdτbcy∼theω1c.onTdhiteioMnaJr⋆keovgia2n apω-
interaction picture as O˜ = eiH0tOe−iH0t, we write the already mentioned in section III. The Markov−app≪roxi-
quantum master equation in Born approximation [1]
mation (τ τ ) allows us to set the upper limit of the
c s
≪
integral to in equation (18). Thus we obtain the sec-
∞
dρ˜ (t) ondordertime-convolutionlessMarkovianquantummas-
s = i TrR[H˜I(t),ρs(0) Ro] ter equation
dt − ⊗
t
dτTr [H˜ (t),[H˜ (t τ),ρ˜ (t) R ]](18) dρ˜ (t)
− R I I − s ⊗ 0 s = i Tr [H˜ (t),ρ (0) R ]
Z0 dt − R I s ⊗ o
where R0 = n|niphphhn|e−βωn/Z is the bath density ∞dτTrR[H˜I(t),[H˜I(t τ),ρ˜s(t) R0]].(19)
matrixwithZ asthepartitionfunction. Inordertostudy − Z0 − ⊗
P
the Markovian dynamics of the system, we assume that
thecorrelationtimescaleτ fortheenvironmentalfluctu- Defining n as the basis set for phonons, therefore,
c ph
{| i }
ationsisnegligiblysmallcomparedtotherelaxationtime wecanwritethemasterequationas(SeeAppendixBfor
scaleτ forthe system,i.e., τ τ . The time scaleover details):
s c s
≪
dρ˜s(t) = ∞dτ ph 0HI m ph 2 ρ˜s(t)e−iωmτ +ρ˜s(t) ph 0HI m ph 2eiωmτ
dt − | h | | i | | h | | i |
m Z0
X (cid:2) (cid:3)
+ ∞dτ nH 0 ρ˜ (t) 0H n eiωnτ
ph I ph s ph I ph
h | | i h | | i
n Z0
X (cid:2)
+ph nHI 0 phρ˜s(t)ph 0HI n phe−iωnτ
h | | i h | | i
= ∞dτ e−i(ωn−iη)τ ph 0HI n ph 2 ρ˜s(t) (cid:3)
− | h | | i |
n (cid:20)Z0
X
+ ∞dτ ei(ωn+iη)τ ρ˜ (t) 0H n 2
s ph I ph
| h | | i |
Z0
∞ dτ eiωnτ nH 0 ρ˜ (t) 0H n .
ph I ph s ph I ph
− h | | i h | | i
Z−∞ (cid:21)
(20)
Now, we know ∞ dτeiωnτ δ(ωn). Therefore, on which implies that
∝
using this relation −an∞d the fact that 0H 0 = 0,
R phh | I| iph
the third term in equation (20) vanishes; hence, we get s nρs(t)m s = e−i(En−Em)t s nρs(0)m s, (23)
h | | i h | | i
dρ˜s(t) ph 0HI n ph 2 ph 0HI n ph 2 whereE =Es+E(2). Thusweseefromtheaboveequa-
=i | h | | i | ρ˜s(t) ρ˜s(t)| h | | i | n n n
dt n (cid:20) ωn − ωn (cid:21) tion that there is only a phase shift but no decoherence!
X Since the matrix elements of an operator are invariant
(.21)
under canonical transformation, thus under Markovain
dynamics, it should be clear that no loss in off-diagonal
The term |phh0|HI|niph|2 corresponds to the effec- density matrix elements (i.e., no decoherence) in the LF
n ωn
tive Hamiltoniahn H(2) in secoind order perturbation and transformed frame of reference implies no loss in off-
P
diagonal density matrix elements (i.e., no decoherence)
commutes with H (see section III). Let n be the si-
0 s
multaneous eigenstate for H(2) and H wi|thieigenvalues in the original untransformed frame of reference. Al-
s
thoughtheHCB’sintheoriginalframeofreferenceform
E(2) and Es, respectively. Then, from the above equa-
n n polarons and are thus entangled with the environment,
tion we get:
nevertheless no decoherence results. For greater clarity,
wetaketheexampleoftwoqubitstateofIRHMi.eN=2.
s nρ˜s(t)m s = e−i(En(2)−Em(2))t s nρ˜s(0)m s, (22) Fromequation(23),thematrixelements nρs(t)m scan
h | | i h | | i h | | i
5
be written as where Xi =e2g(ai−a†i). Using the above, we obtain
nρ (t)m = n nρ (t)n m
s s s s ph T ph s
h | | i h | " h | | i # | i
n
X
= n neSρo(t)e S n m ,
sh | phh | T − | iph | is
n
X
(24)
1
where ρoT(t) is the total density matrix in the original e−S√2(|10i±|01i)|m1,m2iph
frame of reference . Now, we illustrate this quantity by
considering the simple two-spin (i.e., N=2) case of the =[X1X2†|10i±X2X1†|01i]|m1,m2iph(.26)
IRHM.The singletstate 1 ( )andthe triplet
√2 |↑↓i−|↓↑i
state 1 ( + ) are the Sz = 0 eigenstates of the
√2 | ↑↓i | ↓↑i T
two-qubit IRHM Hamiltonian; in HCB language, these
statesareexpressedas 1 (10 01 )and 1 (10 +01 ),
√2 | i−| i √2 | i | i
respectively. Now, the operator e S can be expressed as
−
e−S = egPi=1,2(ni−21)(ai−a†i) wherem andm correspondtophononoccupationnum-
= eg(ni−21)(ai−a†i) bers at s1ite 1 an2d site 2 respectively. Therefore, from
i=1,2 equation (24) we can write the density matrix element
Y
betweensingletandtripletstatesintheoriginalframeof
= "niXi+(1−ni)Xi†#, (25) reference as
i=1,2
Y
1
( 10 01) ρ (t)(10 + 01 )
s
2 h |−h | | i | i
1
= 2 phhm1,m2| h10|X2X1†−h01|X1X2† ρoT(t) X1X2†|10i+X2X1†|01i |m1,m2iph.
mX1,m2 (cid:16) (cid:17) (cid:16) (cid:17)
(27)
Depending upon the presence or absence of HCB, ap- V. CONCLUSIONS
propriate deformation will be produced at each site and
X1X2†|10i±X2X1†|01i |m1,m2iph represents pola- In conclusion, we have shown that the eigenstates of
hro(cid:16)nic states. Furthermor(cid:17)e, in equatioin (27), no loss in Heff are the same as those of HIRHM upto the lead-
the off-diagonalmatrix elementonthe left handsideim- ing order of perturbation. Also we have shown that
plies no loss in the off-diagonal matrix element on the the off-diagonal elements of the reduced density ma-
righthandside (i.e., nolossinthe measureddensityma- trix donot deacy in polaron frame of reference, thus
trix elements in the originalframe of reference) which in no decoherence results. More specifically, for local
turn means no decoherence results. phonons,s nρs(t)m sdiffersfroms nρs(0)m sonlyby
h | | i h | | i
a phase factor and nρ (0)m can be obtained from
s s s
h | | i
nρ m (density matrix element of IRHM) by an
s IRHM s
h | | i
exact unitary evolution [41]. It would be of cosiderable
Thus,uptosecondorderinperturbation,theassump- interest to analyze the non-Markovain decoherence dy-
tion J⋆e−g2 << ω, the infinite range of the Heisenberg namics in the system and is left as future exercise.
model, and the Markov approximation (τ τ )
c ≪ s Next, our decoherence analysis for local optical
together have ensured that the system, with a fixed Sz,
T phonons will continue to be valid even for the more gen-
does not decohere. While the above analysis is valid in
eral optical phonon terms given below:
the regime k T/ω << 1, the finite temperature case
B
k T/ω & 1 needs additional extensive considerations
B 1
and will be dealt seperately. N1/2 Siz[ωk(gka†k,i+gk⋆ak,i)]+ ωka†k,iak,i. (28)
i,k k,i
X X
6
FIG. 1: Open loop hopping processes contributing to ef- FIG. 2: Closed-loop hopping processes contributing to effec-
fective hopping term Tnli in third-order perturbation theory. tive interaction term Vni in third-order perturbation theory.
Hereemptycirclescorrespondtositeswithnoparticleswhile Herefilled (empty)circles correspond to sites with (without)
filled circles correspond to sites with hard-core-bosons. The hard-core-bosons. Thenumbers1,2,and3representhopping
numbers1, 2, and 3 indicate theorder of hopping. sequence.
We also must mention that our approachcannot accom- Hamiltonian:
modate the acoustic phonon case as here the condition
J⋆e g2 <<ω cannotbesatisfiedinthelongwavelength
− k 0H m mH n nH 0
limit. H(3)= phh | I| iph phh | I| iph phh | I| iph.
∆Eph∆Eph
Next, we make a remarkon applicability of our model m=0,n=0 m n
6 X6
andthedecoherenceinsomerealprocesses. Understand- (A1)
ingthehighlyefficienttransportofabsorbedlight-energy
through molecules in photosynthesis is of significant sci-
Here ∆Eph = ω ω . Evaluation of H(3) leads to
entific interestand also key to designing light-harvesting m m − 0
various hopping terms and interaction terms.
technology [46–48]. The model that is used for the
study of the excitationenergy in Fenna-Matthews-Olson
(FMO) complexes is an extreme long range interaction 6 3 3
H(3) = t Tli+ t Tli + v Vi,
model [48] for excitons with uniform hopping strength n n cn Cn n n
" #
between any pair of chromophores in FMO complexes. iX,l6=i nX=1 nX=1 Xi nX=1
Thephononfluctuationsatvariouschromophoresareun- (A2)
correlatedto eachother [47] i.e., localphononeffects are
significantin such complexes. The system-bath coupling
where t (J3e g2)/(g2ω)2, t J3e g2/(gω)2, and
in photosynthetic complexes is thought to be not weak v Jn3/∼(g2ω)2−(as will be excpnla∼ined l−ater). We will
buttobeatleastintheintermediateregime[47];instead n ∼
demonstrate below that H(3) is of the following form
of employing the usual quantum master equation tech-
niques validfor the weak-couplinglimit, LF transformed
master equation can be used.
H(3) = T( nk)b†lbi+H.c. + V( nk)ni,
" #
i,l>i k i k
X X X X
(A3)
Appendix A
whereT andV arefunctionsofthetotalnumberoperator
In this appendix, we will show that the third-order
n . SincetheIRHMcommuteswiththetotalnumber
perturbation theory also produces a term that has the k k
operator, H(3) has the same eigenstates as IRHM!
same eigenstates as IRHM. To this end, we obtain the
P
following third-order perturbation term in the effective Therearesixopen-loophoppingprocessesTlidepicted
n
7
by
T2li = b†jbib†lbkb†kbj
k=i,l,jj=i,l
6X X6
= (1−b†kbk) b†jbjb†lbi
k=i,l,j j=i,l
6X X6
= (1−b†kbk) b†jbjb†lbi
k=i,l j=i,l
X6 X6
= (1−b†kbk) b†jbj −1b†lbi
k=i,l j
X6 X
 
=  b†jbj −1"(N −1)− b†kbk)#b†lbi.(A5)
j k
X X
 
The hopping process Tli in figure 1 (c) is expressed as
FIG.3: Hoppingprocesses(involvingclosedloops)contribut- 3
ingtoeffectivehoppingtermTli inthird-orderperturbation T3li = k=i,l,j j=i,lb†lbkb†jbib†kbj = T2li. The fourth
theory. Filled (empty) circlesCnrepresent occupied (unoccu- hoppingpr6ocessTli6 infigure1(d)isobtainedasfollows.
P P4
pied) sites.
T4li = b†kbjb†jbib†lbk
j=i,l,kk=i,l
6X X6
in figure 1. We analyze them sequentially below. = (1−b†jbj) b†kbkb†lbi
j=i,l,k k=i,l
6X X6
= Tli. (A6)
2
T1li = b†lbkb†kbjb†jbi The hopping process Tli in figure 1 (e) yields Tli =
k=i,l,jj=i,l 5 5
= 6X (X16 −b†kbk) (1−b†jbj)b†lbi lasjt6=hi,ol,pkpinkg6=pi,rlobc†kebsjsb†lTbklib†jinbifi=guTr4eli.1W(fe).analyzebelowthe
k=i,l,j j=i,l P P 6
6X X6
=  (1−b†kbk)−1 (1−b†jbj)b†lbi T6li = b†jbib†kbjb†lbk
k=i,l j=i,l
X6 X6 k=i,l,jj=i,l
   6X X6
=  (1−b†kbk)−1(N −2)− b†jbjb†lbi = k=i,l,jb†kbkj=i,lb†jbjb†lbi
k=i,l j=l 6X X6
X6 X6
  
=  (1−b†kbk)−1(N −1)− b†jbjb†lbi = k=i,lb†kbk−1j=i,lb†jbjb†lbi
k=i,l j X6 X6
X6 X  
  
= (N −1)− b†jbj (1−b†kbk)−1b†lbi = k=i,lb†kbk−1 j b†jbj −1b†lbi
j k=i,l X6 X
X X6   
  
= (N −1)− b†jbj"(N −2)− b†kbk#b†lbi. = Xj b†jbj −1"Xk b†kbk−2#b†lbi. (A7)
j k
X X  
  (A4)
We will now deal with closed-loop hopping processes
such as those in figure 2. These lead to effective inter-
The second hopping process Tli in figure 1 (b) is given actions. The process Vi in figure 2 (a), obtained from
2 1
8
figure 1 (a) by setting l=i, is given as follows. To obtain the effective hopping term Tli corresponding
C2
to figure 3 (b), we take j =l in figure 1 (a):
V1i = b†ibkb†kbjb†jbi
kX6=i,jXj6=i TCli2 = b†lbkb†kblb†lbi
= (1−b†kbk) (1−b†jbj)b†ibi kX6=i,l
kX6=i,j Xj6=i = (1−b†kbk)b†lbi
k=i,l
X6
=  (1−b†kbk)−1 (1−b†jbj)b†ibi
Xk6=i Xj6=i = (N −2)− b†kbkb†lbi
   k=l
X6
= (N)−Xj b†jbj"(N −1)−Xk b†kbk#b†ibi.(A8) = "(N −1)− b†kbk#b†lbi
  Xk
Next, the hopping process V2i corresponding to closed = TCli1. (A12)
loopinfigure2(b)isobtainedfromfigure1(c)bytaking
l=i. The effective hopping term Tli depicted in figure 3 (c)
C3
[upon setting k =i and j =l in figure 1 (a)] is given by
V2i = b†ibkb†jbib†kbj
kX6=i,jXj6=i TCli3 =b†lbib†iblb†lbi =b†lbi. (A13)
= (1−b†kbk) b†jbjb†ibi Thus we have shown that H(3) contains effective hop-
k=i,j j=i
= XkX66=i(1−b†kbk)Xj6=X6ib†jbjb†ibi +pSiiHnngc.cet.T]eramanndsdPeVffie,acl>rteiiv[fTeun(inPctteikornancskt)oibof†lntbhiteertmotsal(nuimVb(erokpnekra)ntoi)r.,
H(3) and IRHM have the same eigenstaPtes. TPhese argu-
= (1−b†kbk) b†jbj −1b†ibi mentscanbeextendedtoevenhigher-orderperturbation
k=i j theoryto showthat the effective Hamiltonian(after tak-
X6 X
  ing all orders of perturbation into account) will give the
same eigenstates as IRHM!
=  b†jbj −1"(N)− b†kbk)#b†ibi. (A9) Wewillnowexplaintheexpressionsforthecoefficients
j k
X X t , v , and t in equation (A2), obtained from third-
n n cn
 
Lastly, the hopping Vi [depicted by the closed loop in order perturbation theory, using typical schematic dia-
3
grams shown in figure 4 [for details of corresponding di-
figure2(c)]isobtainedfromfigure1(e)bysetting l=i.
agrams and analysis in second order perturbation, see
V3i = b†kbjb†ibkb†jbi reference [43]]. We consider two distinct time scales as-
sociatedwith hoppingprocessesbetweentwo sites: (i)
jX6=i,kXk6=i 1/(Je g2)correspondingtoeitherfulldistortionatasit∼e
−
= (1−b†jbj) b†kbkb†ibi toformasmallpolaronicpotentialwell(ofenergy g2ω)
jX6=i,k Xk6=i orfullrelaxationfromthe smallpolaronicdistortio−nand
= Vi. (A10) (ii) 1/J relatedto negligibledistortion/relaxationata
2 ∼
site. The coefficient t corresponds to the typical dom-
n
Finally, we consider figures 3 (a), (b), and (c) which
inant distortion processes shown schematically in figure
deal with effective hopping terms Tli involving closed
Cn 4 (a) with the pertinent typical hopping processes being
loops. The effective hopping term Tli , corresponding to
C1 depicted in figure 1 (a). In figure 4 (a), after the HCB
figure 3 (a), is obtained by setting k=i in figure 1 (a):
hops away from the initial site, the intermediate states
havethe same distortionas the initial state. Next, when
TCli1 = b†lbib†ibjb†jbi the HCB hops to its final site there is a distortion at
jX6=i,l thisfinalsitewithaconcomitantrelaxationattheinitial
= (1−b†jbj)b†lbi sJi/te(2.gH2ωen)cetJh/e(c2ogn2ωtr)ibutJioengt2otheJc3oeeffig2c/ie(gn2tωtn)2b.eAcosmrees-
jX6=i,l gardscoeffi×cientv ,it×canb−ede∼duced−basedonthetypi-
n
cal dominant hopping-cum-distortion processes depicted
= (N −2)− b†jbjb†lbi in figure 4 (b) which typifies the hopping processes in
Xj6=l figure 2 (a). In figure 4 (b), when the particle hops to
 
different sites and reaches finally the initial site, there
= (N −1)− j b†jbjb†lbi. (A11) iesstnimoactheadngtoe ibnedJi/st(o2rgt2iωon) atJa/n(y2gs2iωte). HJenceJv3n/(cga2nω)b2e.
X × × ∼
 
9
Lastly, we obtain the coefficient t by considering the
cn
i j k l typical dominant diagram in figure 4 (c) corresponding
tothetypicalprocessinfigure3(a). Infigure4(c),where
the first intermediate state depicts the particle hopping
but leaving the distortion unchanged, we get a contribu-
j k l tionJ/(2g2ω);forthenextintermediatestate,wherethe
i
HCB returns to the initial site, the initial site has to un-
dergo a slight relaxation (involving absorbing a phonon
soastoyieldanon-zerodenominatorintheperturbation
theory)leadingtothecontributionJ/ω;andlastly,when
theHCBhopstothefinalsite,thereisadistortionatthe
i j k l
finalsitewithasimultaneousrelaxationattheinitialsite
thereby producing a contribution Je g2. Thus we calcu-
−
late t to be J/(2g2ω) J/ω Je g2 J3e g2/(gω)2.
cn − −
× × ∼
i j k l
(a)
j j
i
i
k k
j j
i i
k k
(b)
i j l i j l
i j l i j l
(c) Appendix B
FIG. 4: Schematic diagrams (a), (b), and (c), correspond-
ing to the hopping processes depicted in figure 1 (a), figure
2 (a), and figure 3 (a), respectively, yield coefficients tn, vn,
and tcn, respectively. The intermediate states give the typ-
ical dominant contributions. Here empty circles correspond
to empty sites, while filled circles indicate particle positions.
Parabolic curve at a site depicts full distortion at that site In this appendix we will evaluate the various terms in
with corresponding energy −g2ω (+g2ω) if the hard-core- master equation (19). Defining n as the basis set
ph
boson is present (absent) at that site. forphonons,therefore,we canwr{i|teithe}masterequation
(equation (19)) as:
10
dρ˜ (t)
s = i n[H˜ (t),ρ (0) R ]n
ph I s o ph
dt − h | ⊗ | i
n
X
∞dτ nH˜ (t)H˜ (t τ)ρ˜ (t) R n
ph I I s o ph
− h | − ⊗ | i
Xn Z0 h
nH˜ (t)ρ˜ (t) R H˜ (t τ)n
ph I s o I ph
− h | ⊗ − | i
nH˜ (t τ)ρ˜ (t) R H˜ (t)n
ph I s o I ph
− h | − ⊗ | i
+ nρ˜ (t) R H˜ (t τ)H˜ (t)n . (B1)
ph s o I I ph
h | ⊗ − | i
i
In order to simplify the abovemaster equation,we need to evaluate the time evolutionofthe operatorsinvolvedin
H . Considering the second term in the equation (B1), yields
I
e−βωn
phhn|H˜I(t)H˜I(t−τ)ρ˜s(t)⊗Ro|niph =XeiHstphhn|HI|miphe−iHst eiHs(t−τ)phhm|HI|niphe−iHs(t−τ)ρ˜s(t) Z ei(ωn−ωm)τ.
m
(B2)
In momentum space, we express HCB operators as: b†j = √1N keikrj b†k and bj = √1N ke−ikrj bk; then, it is
important to note that the hopping term in the system Hamiltonian can be written as:
P P
0.5J (e−g2b†ibj +H.c.) = 0.5Je−g2 b†ibj − b†ibi
i,j>i i,j i
X X X
N 
= 0.5J⋆( )e−g2nˆ0 0.5Je−g2Nˆp
N 1 −
−
= ǫkb†kbk, (B3)
k
X
where we used J = J⋆/(N −1), Nˆp ≡ kb†kbk and nˆ0 ≡ b†0b0 (i.e., the particle number in momentum k = 0 state).
Here it should be mentioned that using HCBs instead of spins has enabled us to obtain (with ease) the excitation
P
spectrum ǫ which is crucial for the analysis given below. Let q denote the complete set of energy eigenstates
k s
(with eigenenergies Es) of the system Hamiltonian H ; then we{c|ani w}rite:
q s
eiHstHIe−iHst =0.5Je−g2 |qisshq|eiHstN1 b†kbpei(krl−prj)e−iHst|q′isshq′|{S+lj†S−lj −1}
l,j>lq,q′ k,p
X X X
  +H.c.,
(B4)
which implies
eiHst ph nHI m phe−iHst = q ssq ph nHI m ph q′ ssq′ ei(Eqs−Eqs′)t,
h | | i | i h | h | | i | i h |
q,q′
X
(B5)
where Es Es =0.5J⋆( N )e g2 or 0 . Here we have takenthe totalnumber of HCBs to be conserved;then, only
| q − q′| N 1 −
thehoppingterminH will−contributetotheparticleexcitationenergy. Substituting equation(B5)inequation(B2),
s
we get
nH˜ (t)H˜ (t τ)ρ˜ (t) R n = [ q q nH m q q mH n q q
ph I I s o ph ss ph I ph ′ ss ′ ph I ph ′′ ss ′′
h | − ⊗ | i {| i h | h | | i | i h | h | | i | i h |}
m q,q′,q′′
X X
ei[(Eqs−Eqs′)t+(Eqs′−Eqs′′)(t−τ)]ρ˜s(t)e−βωnei(ωn−ωm)τ . (B6)
× Z
(cid:21)