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Journal of Luminescence 128 (2008) 978–981

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Quantum hydrodynamic modes in one-dimensional polaron system

Satoshi Tanakaa,�, Kazuki Kankia, Tomio Petroskyb

aDepartment of Physical Science, Osaka Prefecture University, Gakuen-cho 1-1, Sakai, Osaka 599-8531, JapanbCenter for Quantum Complex Systems, The University of Texas at Austin, Austin, TX 78712, USA

Available online 15 December 2007

Abstract

Dynamical relaxation process of one-dimensional polaron system weakly coupled with a thermal phonon field is theoretically

investigated. In addition to the diffusion relaxation, we have found that there appears a new macroscopic quantum sound mode which

stabilizes the wave packet of the quantum particle even under the random collision with the thermal phonon. This coherent sound mode

is a new hydrodynamic mode obeying a macroscopic linear wave equation for the density of the particle, instead of wave function.

r 2008 Elsevier B.V. All rights reserved.

Keywords: One-dimensional polaron; Kinetic theory; Relaxation dynamics; Hydrodynamics; Quantum sound mode

One-dimensional (1D) polaron system exhibits charac-teristic relaxation dynamics, such as polyacetylene, a-helix,halogen bridged mixed-valent compounds, which has beentheoretically and experimentally investigated [1–3]. Aquantum particle weakly coupled with a phonon fieldbehaves as a quantum Brownian particle accompanied withirreversible processes. Time evolution of the Brownianparticle has been studied in most cases by phenomenolo-gical dissipative equations based on Markov approxima-tion [4], but the origin of the irreversibility has not yet beenanalyzed on the level of microscopic dynamics. On theother hand, Prigogine et al. have clarified that the Poincareresonance associated in microscopic dynamics is respon-sible for the time symmetry breaking [5,6]. In the presentwork, we reveal that in a 1D polaron system this Poincareresonance plays a crucial role to the characteristicrelaxation process in which a new macroscopic coherentmotion arises as a hydrodynamic quantum sound mode.

We consider the 1D polaron Hamiltonian given by [7]:

H ¼X

p

epaypap þX

q

_oqbyqbq

þ

ffiffiffiffiffiffi2pL

r Xp;q

gqay

pþ_qapðbq þ by�qÞ, ð1Þ

e front matter r 2008 Elsevier B.V. All rights reserved.

min.2007.12.022

ing author. Tel./fax: +81 72 254 9710.

ess: stanaka@p.s.osakafu-u.ac.jp (S. Tanaka).

where ap is an annihilation operator for the fermionicparticle with energy ep and momentum p. In this article weconsider a free particle ep ¼ p2=2m coupled with a singleacoustic phonon mode _oq ¼ cjqj. The annihilationoperator of the phonon system is bq. The length of the1D chain is given by L, and we impose a periodic boundarycondition leading to the discrete wave numbers withp=_; q ¼ 2pj=L, with integer j. We consider the caseL!1, later.The time evolution of the total system obeys the

quantum Liouville equation: iqrðtÞ=qt ¼LrðtÞ, where theLiouvillian L is defined by L� � ½H ; ��=_. We focus onthe time evolution of the normalized reduced densitymatrix of the particle defined by f ðtÞ � Trph½rðtÞ�, where thephonon system is initially in thermal equilibrium repre-sented by rph ¼ exp½�

Pq _oqbyqbq=kBT �=Zph.

We can express the reduced density operator in themomentum space as: f kðP; tÞ � hPþ _k=2jf ðtÞjP� _k=2i.Fourier transform for k gives the Wigner function in phasespace:

f WðX ;P; tÞ �

1

2p

Z 1�1

f kðP; tÞeikX dk. (2)

Note that k ¼ 0 component of f k¼0ðP; tÞ is a momentumdistribution function, while ka0 component represents aninhomogeneity in space.The equation of motion of the reduced distribution

function f kðP; tÞ is obtained by using the complex spectral

ARTICLE IN PRESS

Fig. 1. The spectrum of K for the temperatures, kBT=mc2 ¼ 1; 10, and100.

S. Tanaka et al. / Journal of Luminescence 128 (2008) 978–981 979

representation of the Liouvillian [6]. Since we consider theweakly coupling case in which the so-called Van Hove limitcan be taken [5], the momentum distribution functionfollows the kinetic equation:

qqt

f 0ðP; tÞ ¼ �Kf 0ðP; tÞ, (3)

where K is the collision operator which is given by

K ¼ iPð0ÞLepQð0Þ1

i0þ �L0

Qð0ÞLepPð0Þ, (4)

where L0 is the unperturbed Liouvillian and Lep is theinteraction part of the Liouvillian. The operator Pð0Þ

denotes the projection operator for the density operatoronto the k ¼ 0 subspace in Eq. (2) and Qð0Þ is the onecomplement to Pð0Þ. Since the spectrum of L0 hascontinuous spectrum in the thermodynamical limitL!1, the Poincare resonance occurs in Eq. (4), lettingthe system non-integrable. This resonance effect is the veryreason for the emergence of the irreversibility [5].

The explicit form of the collision operator K is given by

K ¼2p

_2

Zdqjgqj

2 deP � ePþ_q

_þ oq

� �nq

n

þdeP�_q � eP

_þ oq

� �ðnq þ 1Þ

o

�2p

_2

Zdqjgqj

2 deP�_q � eP

_þ oq

� �nq exp½�_qq=qP�

n

þdeP � ePþ_q

_þ oq

� �ðnq þ 1Þ exp½_qq=qP�

o, ð5Þ

where nq � 1=ðexp½_oq=kBT � � 1Þ. It should be emphasizedthat K vanishes in the classical limit of _! 0 in Eq. (5).This means that the dissipation in the present weaklycoupled system is a pure quantum effect in the 1D case.

Once the eigenvalue problem for the collision operatorK is solved as

Kfð0Þj ðPÞ ¼ lð0Þj fð0Þj ðPÞ, (6)

then the time evolution of the momentum distributionfunction is obtained by f 0ðP; tÞ ¼

Pj exp½�l

ð0Þj t�fð0Þj ðPÞc

ð0Þj ,

where cð0Þj depends on the initial value of f 0ðP; 0Þ. We can

prove that all the eigenvalues of lð0Þj take non-negativediscrete values, which guarantees the monotonous ap-proach to the steady state (the H-theorem). Indeed, withuse of Eqs. (5) and (6), we have

lð0Þj ¼ 2

ZdP

Zdqjgqj

2e�P=kBT nqdð�P � �Pþ_q þ _oqÞ

�jfð0Þj ðPÞ � e_oq=kBTfð0Þj ðPþ _qÞj2X0. ð7Þ

In Eq. (7) the equality holds when the following tworelations are simultaneously satisfied: (A) detailed balance,fð0Þ0 ðPÞ ¼ exp½_oq=kBT �fð0Þ0 ðPþ _qÞ, and (B) resonancerelation, �P þ _oq ¼ �Pþ_q.

In Fig. 1 we display the spectrum of K obtainednumerically for several temperatures, where we specificallyadopt the deformation potential interaction for thecoupling gq in Eq. (1): gq ¼ ajqj= ffiffiffiffiffiffioq

p. It should be noted

that there is a finite gap lð0Þ1 between the lowest and the firstexcited states. The inverse of the gap 1=lð0Þ1 determines therelaxation time taken for the system to relax into the steadystate by the random collision with the thermal phononfield. It is found that the gap is proportional to a2 and1=

ffiffiffiffiTp

.A striking feature may be deduced from the above two

relations (A) and (B). The distribution function corre-sponding to the steady state, i.e. collision invariant, is notunique contrary to the case of a quantum Brownianparticle; the collision invariant is infinitely degenerated. Ifwe put fðPÞ ¼ feqðPÞwðPÞ, where feqðPÞ is the Maxwelliandistribution, and substitute it into the relations (A) and (B),we find that any functions which possess a property ofwðPÞ ¼ wð�P� 2mcÞ fulfill the condition of the collisioninvariant. The origin of this periodicity is two holds: (1) theone is due to 1D nature of the system and (2) the other is aquantum effect that is expressed by the shift operatorexp½�_qq=qP� in Eq. (5), which is in contrast to themomentum derivative operators in classical systems.It is an established knowledge in non-equilibrium

statistical mechanics that phenomenological concepts inhydrodynamics, such as sound modes, diffusion modes,etc., have a microscopic dynamical justification that thesemodes are obtained by removing the degeneracy in thecollision invariance by introduction of the flow term in thekinetic equation as a perturbation [8]. With the flow termthe kinetic equation reads

qqt

f kðP; tÞ ¼ �ðikP=mþ KÞf kðP; tÞ. (8)

In classical gas systems, it is well known that hydro-dynamics modes are obtained for a physical situation thatsatisfies the so-called hydrodynamics condition which

ARTICLE IN PRESSS. Tanaka et al. / Journal of Luminescence 128 (2008) 978–981980

corresponds to the case where the wavelength 1=k is muchlonger than the mean-free-length that is proportional to theinverse of the smallest positive eigenvalue of the homo-geneous collision operator with k ¼ 0. In our polaronsystem, this corresponds to the physical situation jkjc5lð0Þ1 .In other words, this is a physical situation that the lengthscale of the inhomogeneity of the electron distribution inspace is much longer than the length scale defined by c

times the momentum relaxation time to local-equilibrium.For this physical situation the flow term kP=m in Eq. (8)can be treated as a small perturbation. The degeneracy of

Fig. 2. Calculated sound velocity as a function of P for temperatures of

kBT=mc2 ¼ 0:1 (solid), 1.0 (dashed), and 3.0 (dotted).

Fig. 3. Propagation of the wave packet for the polaron in (a)–(c). Tha

the collision invariants are lifted up by the first order in k,giving the sound velocity sðP;TÞ as

sðP;TÞ

¼

P1l¼�1 ð�1Þ

lððP=mÞ þ 2clÞ exp½�ðPþ 2mclÞ2=2mkBT �P1

l¼�1 exp½�ðPþ 2mclÞ2=2mkBT �.

ð9Þ

We show in Fig. 2 the calculated sound velocity as afunction of P. It is found that the sound velocity is aperiodic function of P with the periodicity 4mc and the P

dependence becomes small as T increases.Thus we have found out the emergence of the quantum

sound mode in macroscopic sense in the exactly samemanner as in the classical gas system [8]. In Fig. 3, thepropagation of the sound wave of the quantum particle isdepicted. A minimum uncertainty wave packet is preparedas a initial distribution f W

ðX ;P; 0Þ as shown in Fig. 3(a).After the relaxation time �1=lð0Þ1 , the local equilibrium hasbeen achieved, where the momentum distribution relaxesinto the distribution of the steady state leaving the spatialdistribution unchanged (Fig. 3(b)). Due to the flow term,the wave propagates following a macroscopic waveequation q2f W=qt2 ¼ s2ðP;TÞq2f W=qX 2, with the soundvelocity sðP;TÞ as shown in Fig. 3(c). Since the soundvelocity sðP;TÞ becomes less dependent on P for highertemperatures, the collapse of the wave packet due to thephase mixing is more suppressed, and the wave packetbecomes more stable. This is quite intriguing and counter-intuitive because the random collision of the quantumparticle with phonons becomes the reason for stabilizing

t for the free particle with the same initial packet is shown in (d).

ARTICLE IN PRESSS. Tanaka et al. / Journal of Luminescence 128 (2008) 978–981 981

the macroscopic coherent motion of the particle. When wecompare the sound wave propagation Fig. 3(c) with a wavepacket propagation of a free quantum particle Fig. 3(d) forthe same initial condition, it is clearly seen that ourcoherent motion is far more stable than the free particlemotion. Nonetheless, the quantum sound mode eventuallydisappears due to the diffusion process that comes fromthe higher order contribution of order k2, as in the caseof the classical sound mode. We note that the quantumsound modes do not appear for the system with higherdimension more than one because of the lack of theperiodicity in collision invariants. Furthermore, the soundmodes do not appear for the case of the coupling withoptical phonon because of the symmetry of the collisioninvariants.

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