Influence of a Mesoscopic Bath on Quantum Coherence

a r X i v :c o n d -m a t /0409630v 1 [c o n d -m a t .m e s -h a l l ] 23 S e p 2004

In?uence of a Mesoscopic Bath on Quantum Coherence

Onuttom Narayan 1and Harsh Mathur 2

1

Department of Physics,University of California,Santa Cruz,CA 95064

2

Physics Department,Case Western Reserve University,10900Euclid Avenue,Cleveland OH 44106-7079

For a quantum double well system interacting with a mesoscopic bath,it is shown that a single particle in the bath is su?cient to substantially reduce tunneling between the two wells.This is demonstrated by considering an ammonia molecule in the center of a ring;in addition to halving the maser line frequency,there is an increase in intensity by four orders of magnitude.The tunneling varies non-monotonically with the number N of electrons in the ring,re?ecting the changing elec-tronic correlations.Although the tunneling is reduced for small N ,it turns around and grows to its free value for large N.This is shown to not violate Anderson’s orthogonality theorem.Experimental implementations are discussed.

The importance of decoherence for the transition from quantum to classical behavior is well known.Var-ious authors have shown that when a system is coupled to a bath consisting of an in?nite number of degrees of freedom,irreversibility—and therefore,decoherence—is a consequence;this has given rise to the ?eld of dis-sipative quantum mechanics[1,2].Studies in this ?eld have been further stimulated by the experimental real-isation of quantum coherent mesoscopic systems that are small on the macroscopic scale but large on the mi-croscopic,atomic scale[3].Control of coupling to the environment is also critical to the success of quantum computing,a fast-expanding area of research[4].

In this paper we consider a microscopic quantum system that is predominantly coupled to a mesoscopic bath.In dissipative quantum mechanics,it is gener-ally presumed that the quantum system is coupled to a macroscopic bath with an essentially in?nite number of degrees of freedom.The general belief is that “more is worse”:the greater the coupling to the environment,the more quantum e?ects are destroyed.Here we show that,for a mesoscopic bath,it is possible for the tun-neling between the two wells of a quantum system in a double well potential to be highly non-monotonic as a function of the number of particles in the bath.There is a substantial reduction in the tunneling with a sin-gle particle.As the number of particles is increased,the e?ect of the mesoscopic bath at ?rst sharply de-creases,then grows,and ?nally decreases again,essen-tially vanishing for su?ciently large particle number.We will show that the last assertion does not contradict Anderson’s orthogonality catastrophe [5].

We demonstrate these results by considering an often studied double well system,the ammonia molecule[6],at the center of a mesoscopic ring.The mesoscopic bath is taken to be a one dimensional ring;qualita-tively similar results would be obtained for a multi-channel ring or for a singly connected geometry such as a disk.If the ring has just one electron,the tunneling

between the two con?gurations for the ammonia dipole is reduced by as much as 50%,reducing the frequency of the transition between the symmetric and antisym-metric con?gurations by the same amount.In addi-tion,the coupling to the electron in the ring increases the dipole moment approximately hundredfold.This dramatically enhances the intensity of the spectral line for the transition,by ～104.As the number of elec-trons on the ring is increased,the rich non-monotonic behaviour noted above unfolds.Possible experimental realizations are discussed towards the end of the paper.Parenthetically we note that it is known the cou-pling of a molecule to light is enhanced by proximity to a metallic nanoparticle,an e?ect that is the ba-sis of sensitive optical detection of molecules [7].The physical origin of this essentially classical e?ect is ?eld enhancement due to the coupling of optical radiation to metallic plasmons.In contrast the enhancement in the spectral line identi?ed here is quantum in origin.To establish these results,?rst consider an ammonia molecule in the center of a ring of radius R.Assume for the moment that the dipole moment of the molecule is in the plane of the ring,and that there is only one elec-tron in the ring.In the absence of the ring,for every orientation of the hydrogen plane,there are two pos-sible positions of the nitrogen atom.The ground and ?rst excited states are the symmetric and antisymmet-ric combinations of these.Transitions between the two states are used in the ammonia maser.

In the presence of the ring,the two possible posi-tions of the nitrogen atom,which have opposite dipole moments,perturb the electron in the ring,changing its ground state to two di?erent ground states.The possible states of the combined system of the ammonia molecule and the electron in the ring are then

|↑ ?|E n ,

|↓ ?|E ′

n

(1)

where |E n is the n th electronic eigenstate when the

ammonia electric dipole moment is in one orienta-

2 tion,and|E′n is the corresponding electronic eigen-

state when the dipole moment is in the other orien-

tation.The fact that|E n and|E′n are di?erent—

partially orthogonal—states reduces the tunneling be-

tween the two dipole orientations,since the tunneling

part of the Hamiltonian connects|↑ ?|ψ to|↓ ?|ψ

with the same|ψ for both states.Thus the splitting

between the symmetric and antisymmetric state is re-

duced.The di?erent orientations of the hydrogen tri-

angle do not a?ect this calculation,since there is no

tunneling between the di?erent orientations,only be-

tween the two di?erent positions of the nitrogen atom

for a?xed orientation of the hydrogen triangle.

The potential of the electron in the ring due to the

ammonia dipole moment is V(θ)=ep cosθ/(4π?0R2),

causing matrix elements between the unperturbed elec-

tronic eigenstates.The unperturbed eigenstates of the

electron areψ0=1/

√π

for n=0,with energiesˉh2n2/(2mR2).We neglect the

sin(nθ)wavefunctions,since they are not mixed into

the ground state by V(θ).The Hamiltonian for the

electron in the presence of the ammonia molecule is

H ij=ˉh2

2?1)ρ(δi,1δj,2+δi,2δj,1)]

(2)

whereρ=mep/(4π?0ˉh2)is the dimensionless pertur-bation https://www.sodocs.net/doc/023006532.html,ingρ≈0.6137D for ammonia,H can be truncated to a n×n matrix,and its eigenvalues obtained numerically.Changing n from4to5changes the?rst four eigenvalues by less than1%,so we use a 4×4truncation.The eigenvalues are

[E0,E1,E2,E3]=

ˉh2

2mR2

.(4)

Since the tunneling Hamiltonian does not act on the electron in the ring,the tunneling matrix element be-tween|↑ ?|E n and|↓ ?|E′m is t E′m|E n ,which can be evaluated numerically.

If t is su?ciently small,the degenerate ground state in the8×8

Hamiltonian is split by2t E0|E′0 . Numerically,with Eq.(4),i.e.a ring of radius10 nm,the splitting between the lowest two eigenvalues is0.122ˉh2/(2m),to a good approximation equal to 2|t| E0|E′0 .The frequency of the radiation for tran-sition between these two states is thus less than for the free ammonia molecule by a factor of approxi-mately0.5.If the temperature is large compared to (E1?E0),one has to consider transitions between ex-cited states as well.For a10nm ring,the temperature T has to be much less than8K to ignore the excited states.For R<30nm and T<<8K,to a good ap-proximation the Hamiltonian can thus be truncated to a symmetric2×2matrix,for which the ground and ?rst excited states are the symmetric and antisymmet-ric combinations|I ,|II =1

2

[|↑;E0 ±|↓;E′0 ].In order to excite transitions between|I and|II ,one has to apply a time dependent electric?eld,which couples to the dipole moment of the states.With P=P amm+P el,one can verify that I|P|II = ↑|P amm|↑ + E0|P el|E0 .Since the electronic ground state has a dipole moment opposed to the direction of the ammonia moment,this changes the intensity of the spectral line.Numerically,one obtains

I|P|II =0.325e?A?0.65eR.(5) For R=10nm, I|P|II is changed by a factor of -200,increasing the line intensity by4×104.(Recall that R<30nm for the2×2truncation to be valid.) We now reexamine the approximations made so far. For a general orientation of the ammonia molecule, only the component of its dipole moment in the plane of the ring couples to the electron in the ring,reduc-ing the e?ects discussed above.For randomly oriented molecules,a broad band will be seen.Also,if the molecule is not in the center of the ring,the symme-try between the two orientations of its dipole moment is destroyed:the dipole moment prefers to align itself towards the closest point on the ring.

We now consider the e?ect of increasing the number of electrons in the ring.First,we consider the two electron case in detail.Ifθ1and?θ2are the angular positions of the electrons with respect to the dipole moment of the ammonia,changing toθ+=(θ1+θ2)/2 andθ?=(θ1?θ2),the electronic Hamiltonian is

H=

ˉh2

R2

cosθ+cos(

θ?

2R cosθ?

.

(6) In the absence of the ammonia dipole,the Hamiltonian is separable.Since the kinetic energy is small compared to the potential energy,θ?≈π.Expandingθ?asπ?

3

2δ,Hδ≈(ˉh2/mR2)?2/?2δ+e2δ2/4R,so that

δ2 =ˉh(mRe2)?1/2.(7) The wavefunction is uniform in theθ+coordinate.

With the dipole,the Hamiltonian of Eq.(6)is no longer separable,so we perform an approximate anal-

ysis.From classical arguments it is clear that δ(θ+)

is greatest whenθ+=0,for which case by mini-mizing?2(ep/R2)sinδ+e2/(2R cosδ)we?nd that

δ2 ≈(4p/eR)2.By comparing with Eq.(7),we see that the change in the ground state wavefunction is

negligible for theδcoordinate.For theθ+coordinate,

withδset at its optimal value(δ=4p/eR forθ+=0,δ=0forθ+=±π/2),there is an e?ective potential energy?(4p2/R3)cos2θ+.Since this is negligible com-pared toˉh2/2mR2,the kinetic energy still forces the wavefunction to be essentially uniform inθ+.Thus the overlap in the ground state wavefunction of the two electrons with and without the ammonia dipole poten-tial is almost unity,as is therefore the overlap E′0|E0 . If more electrons are added to the ring, E′0|E0 decreases:for N electrons at low density the elec-tronic state is essentially a Wigner crystal and the dipole potential shifts each electron from its lattice position by an angle O(p/eRN),whereas the angular spread of each electron wavefunction is O(1/N).Thus E0′|E0 =[1?O(p/eR)2]N,which tends to zero for large N.But in reality,as N increases,the Wigner crystal melts[8].The overlap E′0|E0 then reverses its N dependence,approaching unity as N→∞.Phys-ically this is because in the liquid state the electrons e?ciently screen the dipole potential so that it does not signi?cantly perturb the electronic ground state. To make this plausible,let us consider the more tractable problem of N electrons in a spherical shell of inner radius R,perturbed by a point charge q placed at the center of the shell.Within the Thomas Fermi approximation,the point charge potential is screened within a distance～d s from R.If d s<

Alternatively,we could consider N electrons in a spherical box of radius R perturbed by a point charge?e placed at the centre of the box.Ander-

son’s bound on the overlap can also be expressed as exp ?k2σtot ln N/(12π3) whereσtot is the cross sec-tion for an electron at the Fermi surface to scatter from

the screened potential of the perturbing point charge. If we take the screened potential to be of the Thomas-Fermi form and use the Born approximation,both justi?ed in the high density limit k f a B?1(where a B=4π?0ˉh2/me2is the Bohr radius),we?nd

E′0|E0 ≤exp ?19π 1/31a B ln N (8)

which tends to1as N→∞if R is held constant. Again,this does not con?ict with Anderson’s result[5]. The case of a strictly one-dimensional ring deserves special consideration.For spinless electrons with short-range interactions at high density,the problem can be analysed by bosonisation,whereby the electron liq-uid is described in terms of non-interacting bosonic density-wave oscillators[9].In this description,the smooth ammonia dipole potential perturbs just one bosonic oscillator via a linear coupling.Thus,in con-trast to the case of a sharp Kane-Fisher impurity[10], the problem remains trivially soluble.A simple cal-culation reveals that the overlap of the electronic ground states corresponding to the two con?gurations of the ammonia molecule is exp[?4ρ2g/v2N2],essen-tially perfect as N→∞.Here v and g are Luttinger liquid interaction parameters[9].Physically,the rea-son for this dependence is that the single oscillator to which the ammonia molecule couples becomes sti?er as N(and therefore the Fermi velocity)increases.

In three dimensions,the single particle level spacing scales as1/N1/3.Thus as N continues to grow,the bath will ultimately cross over to a macroscopic regime when the single particle level spacing falls well below the tunneling scale t.In this regime,the ammonia molecule can couple to an enormous number of possi-ble particle-hole excitations of the bath,and its tunnel-ing behaviour should therefore be well-described in the conventional framework of the spin-boson model[2,11]. In contrast to the mesoscopic bath,the imperfect over-lap between the ground states of a macroscopic bath can renormalize the tunneling frequency of the two level system all the way down to zero,leading to a complete suppression of tunneling,or localization[2]. Also,there is damping for the macroscopic bath,but not the mesoscopic.Another distinction is the meso-scopic bath-induced ampli?cation of the dipole cou-pling of ammonia to external radiation.This has,to our knowledge,no counterpart in the physical realisa-tions of the spin-boson model studied so far.

4

We now brie?y consider possible experimental real-isations.Metallic rings are unsuitable because of the high density of electrons in metals:rings of the requi-site size would contain far too many electrons.Con-ventional semiconductor devices such as silicon MOS-FETs and GaAs MODFETs can have the required low electron density but su?er from the di?culty that the electron gas is buried deep inside the device below a dielectric layer of oxide or semiconductor.However it may be possible to circumvent this di?culty by fab-ricating devices in which the electron gas lies on the outer surface of the semiconductor and in which the vacuum plays the role of the dielectric layer.GaAs is unsuitable for such devices because of a high den-sity of surface traps,but Kane and co-workers have recently succeeded in passivating the 111 surface of silicon and making working devices of this type[12]. By suitably gating these devices,in principle,it should be possible to laterally con?ne the electrons to rings. Other systems in which the electron gas has low density and is not deeply buried include electrons deposited on liquid helium[13]and newer semiconductor structures that are grown by bottom-up techniques and that may be injected with electrons by photo-excitation of a dye with which the structures have been coated[14].In the former system too electrons could be shepherded into rings by suitable gating[15];in the latter,the electrons would be con?ned to nanoscale semiconductor particles that are typically spherical in shape.

Apart from fabricating the rings,the other experi-mental task is to position ammonia molecules in their vicinity.A simple extension of the calculation above shows that a free ammonia molecule placed at the cen-ter of a ring experiences a force that tends to de?ect it o?-center where the coupling to the ring is much weaker.However this should not be an insurmountable problem at least in the solid state systems mentioned above.For example,with silicon devices,the ammonia molecules could be deposited at random on the silicon surface,to which they would stick.Although ammonia sticks to a bare silicon surface by covalent bonding of the nitrogen atom[16]that renders dipolar oscillations impossible,on a hydrogen passivated Si surface such as used in the devices of Ref.[12]it is expected that ammonia would stick without such covalent bonding. In summary,we have shown that a quantum two level system coupled to a mesoscopic bath responds to the bath in an extremely non-monotonic manner as the number of particles in the bath is changed,and has a strong response even with one particle in the bath. Apart from its intrinsic interest,such an interaction could be relevant to a quantum computing architecture integrating mesoscopic solid state and?ying atomic or ionic qubits[17],a potential application deserving fur-ther exploration.As a prototype,we have considered an ammonia molecule in the center of a ring with a small number of electrons.We have discussed possible experimental techniques to make such devices with a single electron in a ring.The experimental signatures of the e?ects studied here are:(i)a shift in the am-monia spectral line due to the suppressed tunneling, (ii)a factor of104increase in the strength of the line due to the enhanced e?ective dipole moment and(iii) strong inhomogeneous broadening of the line due to variations in the rings and the couplings between the rings and ammonia molecules.A key role in our analy-sis is played by the sensitivity of the electronic ground state to external perturbation.It is desirable to deter-mine the dependence of this“wave function sti?ness”on N and N/R more rigorously,e.g.numerically.

It is a pleasure to acknowledge helpful discussions with Mike Crommie,Arnie Dahm,Josh Deutsch, Carine Edder,Anupam Garg,Bruce Kane,Peter Lit-tlewood and Jie Shan.

[1]A.O.Caldeira and A.J.Leggett,Ann.Phys.(NY)149,

374(1983).

[2]A.J.Leggett,S.Chakravarty, A.T.Dorsey,M.P.A.

Fisher,A.Garg and W.Zwerger,Rev.Mod.Phys.59, 1(1987).

[3]Y.Imry,Introduction to Mesoscopic Physics(Oxford

University Press,New York,2002).

[4]I.L.Chuang and M.A.Nielsen,Quantum Computation

and Quantum Information(Cambridge Univ Press, New York,2000).

[5]P.W.Anderson,Phys.Rev.Lett.18,1049(1967).

[6]R.P.Feynman,R.B.Leighton and M.Sands,The Feyn-

man Lectures on Physics,vol III(Addison-Wesley, 1965).

[7]A.J.Haes,D.McFarland,and R.P.Van Duyne,Pro-

ceedings of SPIE5223,197(2003);G.C.Schatz and R.P.Van Duyne,in Handbook of Vibrational Spec-troscopy,edited by J.M.Chalmers and P.R.Gri?ths (New York,Wiley,2002).

[8]D.M.Ceperley and B.J.Alder,Phys.Rev.Lett.45,

566(1980).

[9]For a pedagogical introduction see,e.g.,J.von Delft

and H.Schoeller,Ann Phys7,225(1998).

[10]C.L.Kane and M.P.A.Fisher,Phys.Rev.Lett.68,

1220(1992).

[11]If single particle spacings are treated as uniform over

the energy range of interest,the particle-hole excita-tions of the electronic system may be considered an as-sembly of bosons via bosonisation,as shown by R.Den-ton,B.M¨u hlschlegel and D.J.Scalapino,Phys Rev B7, 3589(1973).If the matrix element of the dipole poten-tial between two single particle levels is also constant (or depends only on the di?erence in the energy)the ammonia-bath interaction is of the spin-boson form.

5

[12]K.Eng,R.McFarland and B.Kane,Bulletin,APS

March Meeting(2004);private communication.

[13]M.W.Cole,Rev.Mod.Phys.46,451(1994).

[14]C.B.Murray,C.R.Kagan and M.G.Bawendi,Ann Rev

Mater Sci30(2000);G.M.Turner,M.C.Beard and

C.A.Schmuttenmaer,J.Phys.Chem.B106,11716

(2002).

[15]A.Dahm,private communication.

[16]R.J.Hamers,Ph.Avouris and F.Bozso,Phys.Rev.

Lett.59,2071(1987).

[17]A.Steane,Nature422,387(2003),and refs therein.

从实践的角度探讨在日语教学中多媒体课件的应用

从实践的角度探讨在日语教学中多媒体课件的应用 在今天中国的许多大学，为适应现代化，信息化的要求，建立了设备完善的适应多媒体教学的教室。许多学科的研究者及现场教员也积极致力于多媒体软件的开发和利用。在大学日语专业的教学工作中，教科书、磁带、粉笔为主流的传统教学方式差不多悄然向先进的教学手段而进展。 一、多媒体课件和精品课程的进展现状 然而，目前在专业日语教学中能够利用的教学软件并不多见。比如在中国大学日语的专业、第二外語用教科书常见的有《新编日语》（上海外语教育出版社）、《中日交流标准日本語》（初级、中级）（人民教育出版社）、《新编基础日语（初級、高級）》（上海译文出版社）、《大学日本语》（四川大学出版社）、《初级日语》《中级日语》（北京大学出版社）、《新世纪大学日语》（外语教学与研究出版社）、《综合日语》（北京大学出版社）、《新编日语教程》（华东理工大学出版社）《新编初级（中级）日本语》（吉林教育出版社）、《新大学日本语》（大连理工大学出版社）、《新大学日语》（高等教育出版社）、《现代日本语》（上海外语教育出版社）、《基础日语》（复旦大学出版社）等等。配套教材以录音磁带、教学参考、习题集为主。只有《中日交流標準日本語（初級上）》、《初級日语》、《新编日语教程》等少数教科书配备了多媒体DVD视听教材。 然而这些试听教材，有的内容为日语普及读物，并不适合专业外语课堂教学。比如《新版中日交流标准日本语（初级上）》，有的尽管DVD视听教材中有丰富的动画画面和语音练习。然而，课堂操作则花费时刻长，不利于教师重点指导，更加适合学生的课余练习。比如北京大学的《初级日语》等。在这种情形下，许多大学的日语专业致力于教材的自主开发。 其中，有些大学的还推出精品课程，取得了专门大成绩。比如天津外国语学院的《新编日语》多媒体精品课程为2007年被评为“国家级精品课”。目前已被南开大学外国语学院、成都理工大学日语系等全国40余所大学推广使用。

新视野大学英语全部课文原文

Unit1 Americans believe no one stands still. If you are not moving ahead, you are falling behind. This attitude results in a nation of people committed to researching, experimenting and exploring. Time is one of the two elements that Americans save carefully, the other being labor. "We are slaves to nothing but the clock,” it has been said. Time is treated as if it were something almost real. We budget it, save it, waste it, steal it, kill it, cut it, account for it; we also charge for it. It is a precious resource. Many people have a rather acute sense of the shortness of each lifetime. Once the sands have run out of a person’s hourglass, they cannot be replaced. We want every minute to count. A foreigner’s first impression of the U.S. is li kely to be that everyone is in a rush -- often under pressure. City people always appear to be hurrying to get where they are going, restlessly seeking attention in a store, or elbowing others as they try to complete their shopping. Racing through daytime meals is part of the pace

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新视野大学英语第三版第二册课文语法讲解 Unit4

新视野三版读写B2U4Text A College sweethearts 1I smile at my two lovely daughters and they seem so much more mature than we,their parents,when we were college sweethearts.Linda,who's21,had a boyfriend in her freshman year she thought she would marry,but they're not together anymore.Melissa,who's19,hasn't had a steady boyfriend yet.My daughters wonder when they will meet"The One",their great love.They think their father and I had a classic fairy-tale romance heading for marriage from the outset.Perhaps,they're right but it didn't seem so at the time.In a way, love just happens when you least expect it.Who would have thought that Butch and I would end up getting married to each other?He became my boyfriend because of my shallow agenda:I wanted a cute boyfriend! 2We met through my college roommate at the university cafeteria.That fateful night,I was merely curious,but for him I think it was love at first sight."You have beautiful eyes",he said as he gazed at my face.He kept staring at me all night long.I really wasn't that interested for two reasons.First,he looked like he was a really wild boy,maybe even dangerous.Second,although he was very cute,he seemed a little weird. 3Riding on his bicycle,he'd ride past my dorm as if"by accident"and pretend to be surprised to see me.I liked the attention but was cautious about his wild,dynamic personality.He had a charming way with words which would charm any girl.Fear came over me when I started to fall in love.His exciting"bad boy image"was just too tempting to resist.What was it that attracted me?I always had an excellent reputation.My concentration was solely on my studies to get superior grades.But for what?College is supposed to be a time of great learning and also some fun.I had nearly achieved a great education,and graduation was just one semester away.But I hadn't had any fun;my life was stale with no component of fun!I needed a boyfriend.Not just any boyfriend.He had to be cute.My goal that semester became: Be ambitious and grab the cutest boyfriend I can find. 4I worried what he'd think of me.True,we lived in a time when a dramatic shift in sexual attitudes was taking place,but I was a traditional girl who wasn't ready for the new ways that seemed common on campus.Butch looked superb!I was not immune to his personality,but I was scared.The night when he announced to the world that I was his girlfriend,I went along

英语造句

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新大学日语简明教程课文翻译

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新视野大学英语第一册Unit 1课文翻译

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舌尖上的中国分集简介

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