Reply to the Comment on "General Non-Markovian Dynamics of Open Quantum System"

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by  Wei-Min Zhang

Reply to the Comment on
“General Non-Markovian Dynamics of Open Quantum System”
Wei-Min Zhang1, Ping-Yuan Lo1, Heng-Na Xiong1, Matisse Wei-Yuan Tu1 and Franco Nori2,3
1Department of Physics, National Cheng Kung University, Tainan, 70101 Taiwan
2Center for Emergent Matter Science, RIKEN, Saitama 351-0198, Japan
3Physics Department, The University of Michigan, Ann Arbor, Michigan, 48109-1040, USA
(Dated: July 28, 2014)
The letter [1] presents three examples. For the steady- To be more specific, let us begin with the valid initial
5 statesolutionofthefirstexample,i.e.,thedissipationless state: |ψ(t0)i=a†|0,{0k}i, in which the system initially
1 part of Eq.(12) in [1], the 2nd version of the Comment contains one particle, and the environment is in its vac-
0
[2]claimedthat“this[dissipationless]regimeexistsifand uum, i.e., the system and the environment are initially
2
only if the total Hamiltonian is unbounded from below, decoupled[7]. Thecorrespondingenergyofthe totalsys-

r castingseriousdoubtsontheusefulnessofthisresult.”In temisjusttheenergycarriedbytheparticleintheinitial
a
M the following, we shall show that this Comment is again state, i.e., Etot=ωs>0. Solving exactly the Schr¨odinger
incorrect. equationwiththisinitialstate,thesteadystateoftheto-
5 The total Hamiltonian used in the first example in [1] talsystemis|ψ(t→∞)i=(cid:2)e−iωbtZ(cid:0)a†+P Vk b†(cid:1)+
k ωb−ωk k
2 is Htot= ωsa†a+Pkωkb†kbk +PkVk(a†bk +b†ka). Di- Pke−iωkt[ωk−ωs−∆(k)+iγ(k)]−1b†k(cid:3)|0,{0k}iwhich is
] agonalizing Htot leads to Htot= ωbC†C +Pkωk′Dk†Dk, a superposition of the renormalized mode ωb of the sys-
ph where ωb= ωs−R0∞dωωJ−(ωω)b is the renormalized mode of wtehmerpelutshealfilrostthteerrpmosgsiivbelsetmhoeddeisssωipkaotfiotnhleesesnpvairrotnomfetnhte,
- the system, and ω <ω′ <ω . The operators {C†,D†}
t k k k+1 k system in [1]. The derivation of this result is given in
n are all normal modes of the total system after a Bogoli-
[3]. In the strong-coupling regime, ω is negative, as
a ubovtransformationfromthebasis{a†,b†}[3],andC† is b
u k shown in [1], but the energy of the total Hamiltonian
just the single-excitation given in [2]. In the continuous
q is positive, E = ω > 0, because the total Hamilto-
[ limit,ωk′ =ωk. Thus,thefirstmistakemadein[2]isthat nian and the ttoottal pasrticle number are conserved during
2 at operator level, Htot is not given only by C†. the time evolution. Let us now extend the above solu-
v Secondly, Arai and Hirokawa proved [4] that the spec- tion to the initial states |ψ (t )i ∝ (a†)n|0,{0 }i, where
n 0 k
2 trum of the above Hamiltonian in the strong-coupling n = 1,2,3,···, can be any arbitrary integer. The cor-
1 regime is unbound from below when the particle num-
responding steady state of the total system is |ψ (t →
n
20 dbuereintoththeerecnoonrsmeravlaizteiodnmoofdethωebtiostaulnbpoaurntidc.leHnouwmebveerr,, ∞)i∝(cid:2)e−iωbtZ(cid:0)a†+Pk ωbV−kωkb†k(cid:1)+Pke−iωkt[ωk−ωs−
01. [HHatmoti,lNtotnoita]n=c0a,nwbheerweriNttteont =asaa†ad+irePctksbu†kmbk,otfhdeectoomta-l t∆h(eks)e+staiγte(sk)E]−to1tb,n†k(cid:3)=n|n0,ω{s0>k}0i.(pToshietivteo-tdaelfienniteer)g.ieIsnoffacatll,
4 for all valid initial states and the corresponding exact
posedHamiltonians. EachdecomposedHamiltonianwith
1 solutions of the master equation given in [1], the total
fixed total particle number always has a lower bound for
:
v arbitrary coupling V [5]. A similar “unbound” ground Hamiltonian is always positive-definite. This is consis-
k
i tent with the fact that the total Hamiltonian with fixed
X state energy also exists for the total Hamiltonian of a
total particle numbers has a lower bound, due to the to-
r Dirac particle in QED, where the possible trouble from
a the unbound ground-state energy in QED is avoideddue talparticlenumberconservation. Thus,theabove-quoted
criticism in [2] is obviously incorrect. The Comment [2]
tothetotalmomentumconservation,i.e.,thedecomposed
ignoredthe validity of the exactmaster equation derived
Hamiltonianwithfixedtotalmomentahasalowerbound
from the Feynman-Vernon influence functional to reach
forarbitraryQEDcoupling[6]. Thus,thesecondmistake
an incorrect conclusion.
made in [2] was not to consider the important role of the
particle number conservation. The only correct part in the Comment is the last part
Furthermore, because H is not simply given by the of [2], where they pointed out that the problem exists in
tot
single-excitationC†,thethirdmistakemadein[2]isthat their own works, i.e., Refs. 3 and 4 in [2] for the quan-
the possible energy divergence claimed in [2] is not ap- tum Brownianmotion(QBM). This is because the QBM
plicable to the non-Markovian dynamics studied in [1]. used a system-environment coupling Hint = Pkckxqk =
Non-Markovian dynamics relies on the initial states of Pkc′k(a†bk +ab†k +a†b†k +abk), which breaks the con-
the total system. The exact master equation formalism servation of the total particle number, [H ,N ] 6= 0.
int tot
given in [1] requires that the initial states of the total Because of this, the QBM will cause both dynamical
system must be a direct product state between the sys- and thermodynamic instabilities in the strong-coupling
tem and its environment. These states always carry a regime, as claimed in their Comment [2]. The authors of
positive-definite total energy. This decoupling condition [2]didnotrealizethattheinstabilitiesoftheQBMinthe
must be obeyed for any exact master equation derived strong-coupling regime come from the breakdown of the
from the Feynman-Vernon influence functional [7]. Oth- total particle number conservation. Our studies focus on
erwise one cannot carry out the influence functional and systemspreservingthetotalparticlenumberconservation
therebywouldbeunabletoderivetheexactmasterequa- andthereforedonothavesuchproblem(seemorediscus-
tion. Thus,duetothetotalenergyconservation,thetotal sionsin[8]). They mistakenlybelieve thatthe instability
2
other Hamiltonians. [4] A.AraiandM.Hirokawa, Rev. Math. Phys.12, 10851135
(2000), see Theorem 6.3.
[5] SeeProposition 2.5 in theReport byM. Hirokawa [URL].
[6] I. Sasaki, J. Math. Phys. 46, 102307 (2005).
[1] W.M.Zhang,et al.,Phys. Rev. Lett. 109, 170402 (2012). [7] R.P.FeynmanandF.L.Vernon,Ann.Phys.24,118(1963).
[2] D. P. S.McCutcheon, et al.,Comment (unpublished).
[8] W. M. Zhang, et al., arXiv:1401.2012v1 (2014).
[3] See derivations given in the
SupplementalMaterials [URL].