Spin-spin interaction and spin-squeezing in an optical lattice

a r X i v :h e p -t h /9810045v 1 7 O c t 1998TPI-MINN -98/04Quasi-Exactly Solvable Models with Spin-Orbital Interaction 1Alexander UshveridzeTheoretical Physics Institute,University of MinnesotaandDepartment of Theoretical Physics,University of Lodz,Pomorska 149/153,90-236Lodz,Poland 2Abstract First examples of quasi-exactly solvable models describing spin-orbital interaction are constructed.In contrast with other examples of matrix quasi-exactly solvable models discussed in the literature up to now,our models admit infinite (but still incomplete)sets of exact (algebraic)solutions.The hamiltonians of these models are hermitian operators of the form H =−∆2+V 1(r )+(s ·l )V 2(r )+(s +l )·h V 3(r )where V 1(r ),V 2(r )and V 3(r )are scalar functions,l is a vector of the angular momentum operator,s is a matrix-valued vector spin-operator and h is an external (constant)vector magnetic field.1Introduction Quasi-exactly solvable (QES)problems are distinguished by the fact that only some of their energy levels and corresponding wavefunctions admit explicit construction.In the last decade a big progress has been acieved in elaborating the concepts of the phenomenon of quasi-exact solvability and formulating methods of constructing and solving QES models of a variety oftypes (for more detail see e.g.the reviews [[4],[7],[3],[5]]and book [[8]]).In this paper we undertake a new step in this direction and construct new classes of QES models which (up to now)have never been discussed in the QES-literature.These are QES models with a spin-orbital interaction.The potentials of such models have the following general formV (x,y,z )=V 1(r )+(s ·l )V 2(r )+(j ·h )V 3(r )where r =1This work was supported by the grant No DOE/DE-FG02-94ER408232E-mail address:alexush@mvii.uni.lodz.pl and alexush@krysia.uni.lodz.plenergy levels and corresponding eigenvalues.They are however quasi-exactly solvable because the set of their exact solutions is still incomplete and does notfill all the spectrum of a model. First examples of such models were presented in our recent work[[1]]where they have been called”infinite QES models”.Another unusual feature of these models is that they(in contrast with models discussed in paper[[1]])are matrix models with physically realistic hermitian hamiltonians.Everybody who has some experience with quasi-exact solvability in the matrix(multi-channel)case knows how difficult is to satisfy the condition of hermiticity when constructing such models.It is hardly neccessary to remind the reader that up to now only a couple of hermitian matrix QES models have been constructed(see e.g.[[4],[2]]).2Starting pointTo demonstrate how does our construction procedure work we start with the simplest one-dimensional QES model with hamiltonianH=−∂2r2+[b2−2a(2m+c+1)]r2+2abr4+a2r6(1)acting in Hilbert space of functions defined on the positive half axis r∈[0,∞]and vanishing sufficiently fast at its ends r=0and r=∞.Here a,b,c are real parameters satisfying the conditions(a>0,c>0)and m is a non-negative integer.As it was demonstrated in[[8]], for anyfixed m the Schroedinger equationHψ(r)=Eψ(r)(2) for model1admits algebraic solutions whose general form is given by the formulasψ(r)=r c−1/2mi=1 r2/2−ξi exp −ar42 (3)E=2b(2m+c)+8ami=1ξi(4)The m complex numbersξi in expressions3and4satisfy the system of m algebraic equationsmk=1,k=i12ξi−b−2aξi=0,i=1,...,m.(5)It turns out that system5has only m+1permutationally invariant solutions for any given m which are represented by the sets of real pointsξi.Each solution is completely characterized by a(quantum)number k=0,1,...,m which indicates the number of positiveξi-points. According to formula3,the number of positiveξi-points determines the number of(real) wavefunction zeros,which,in turn,determines the ordinal number of an excitation(oscillation theorem).This means that model1has m+1exactly constructable solutions describing the ground state and mfirst excited states.A more detailed exposition of properties of model1 and its algebraic solutions can be found in the book[[8]].3The modified equationIt is not difficult to see that the transformationψ(r)=rϕ(r)(6)reduces the equation2to the form−∂2r∂r2+V(r,l,m) ϕ(r)=Eϕ(r)(7) in which we used the notationV(r,l,m)=[b2−2a(2m+5/2+l)]r2+2abr4+a2r6(8) andl=c−3/2(9)Hereafter we shall consider l as a new independent parameter taking(by agreement)only non-negative integer values.The form of thefirst three terms in the equation7coincides with the form of the radial part of a tree-dimensional Laplace operator.For this reason it seems quite natural to interpret l as the3-dimensional angular momentum and try to relate the equation7to a certain3-dimensional quantum problem.In the following three sections we show that there are three such possibilities leading to three different kinds of quasi-exactly solvable problems in the3-dimensional space.4Thefirst possibilityOne of the simplest possibilities of interpreting equation7is based on the assumption that function8entering into7is l independent:V(r,l,m)=V0(r,N)=[b2−2a(N+5/2)]r2+2abr4+a2r6(10) For this the numberN=l+2m(11)must befixed.In this case,equation7takes the form of a typical radial Schroedinger equation for a spherically symmetric3-dimensional equation(−∆+V0(r,N))Ψ(x,y,z)=EΨ(x,y,z).(12) Since both m and l are assumed to be positive,the condition11leads to afinite number of possibilities with m=0,1,...,[N/2],and l=N,N−2,...,N−2[N/2],respectively.For this reason,for any given N,the model12is quasi-exactly solvable and has(as usually)only a finite([N/2]([N/2]+1)/2)number of explicit solutions.The model of such a form and even its more complicated spherically non-symmetric versions were considered many years ago in papers[[6],[7]].5The second possibilityAnother possibility of interpreting equation7is to consider m as afixed number not restricting the value of l.In this case the function8becomes linearly dependent on l and can be represented in the formV(r,l,m)=V1(r,m)−l·V2(r)={[b2−2a(2m+5/2)]r2+2abr4+a2r6}−l·{2ar2}(13)It is quite obvious that,in order to associate the equation13with a certain3-dimensional Schroedinger equation,we mustfind a proper3-dimensional source for the term which is linear in the momentum l.Thefirst think which comes in ones head is to look for the3-dimensional scalar operators O which wuld commute with both the Laplace operator and r and would have the eigenvalues linear in l.In this case we could consider7as a reduction of a3-dimensional problem(−∆+V1(r,m)−O·V2(r))Ψ(x,y,z)=EΨ(x,y,z)(14)The linearity in l means that the operator must be proportional to the operator of the angular momentuml=(l x,l y,l z)=(i(y∂z−z∂y),i(z∂x−x∂z),i(x∂y−y∂x)).(15) But this is a3-dimensional vector while the operator we are looking for must be a scalar. The only possibility to construct a scalar from15is to take a scalar product of l with another vector operator.It is quite obvious that there is no such operator if we restrict ourselves to the single-channel problems.However,if we admit the consideration of multi-channel problems, then a good candidate for the second operator can immediately be found.This is obviously the spin operator s!Rerstricting ourselves(for the sake of simplicity)to the1/2-spin case(2 by2matrices),we can easily check that the spectrum of the operatorO=2·s·l(16)(which,obviously commutes with both∆and r)is linear in l.Indeed,representing operator 16in the formO=j2−l2−s2(17) (where j=l+s is a total momentum)and taking for concreteness a particular case with j=l+1/2we easilyfind the corresponding branch of the spectrumo=j(j+1)−l(l+1)−s(s+1)=(l+1/2)(l+3/2)−l(l+1)−3/4=l.(18)Thisfinally leads us to a3-dimensional matrix QES models−∆+{[b2−2a(2m+5/2)]r2+2abr4+a2r6}−2(s·l)·{2ar2} Ψ(x,y,z)=EΨ(x,y,z)(19) describing spin-orbital interaction.It is a time to ask ourselves of what kind of models did we obtain?First of all,one should stress again that these models are actually quasi-exactly solvable.This follows from the fact that for any given m and l they have an infinite number of normalizable solutions,but only m+1of them are exactly(algebraically)constructable.Second,and this is may be the most important thing,despite the fact that the set of exactly constructable solutions is incomplete,this set is infinitely large.This is so because the number l is notfixed by the3-dimensional model19.It appears as a solution of the eigenvalue problem for operators O and may take arbitrary non-negative integer values.In conclusion of this section note that the hamiltonians of models we obtained are hermi-tian by construction.6The third possibilityThe last interesting possibility of reducing the equation7to a3-dimensional form appears when the function8depends on both parameters l and m.In this case the function8becomes linearly dependent on both l and m and can be represented in the formV(r,l,m)=V1(r)−l·V2(r)−m·V3(r)={(b2−5a)r2+2abr4+a2r6}−l·{2ar2}−m·{4ar2}(20) By analogy with the previous section we can consider the numbers l as the eigenvalues of the operator of spin-orbital interaction,and the only thing which remains to do is to interpret m as an independent quantum number appearing in equation7as an eigenvalue of a certain operator M commuting with the variable r,Laplasian∆and the spin-orbital operator s·l.A good candidate for such an operator is the z-projection of the total momentum s+l.In fact,it should not neccessarily be a z-projection.Because of the spherical symmetry,it could be equally weel a x-or y-projection,or any other projection.We can therefore represent this operator in a covariant formM=2(s+l)·h(21) where h is a unit magneticfield.We introduced an additional factor2to make the eigenvalues of operator M integer rather than half integer.Of course,the negative integers are not interesting for us,because,as we remember,only for non-negative integer values of m the system admits algebraic solution.Summarizing,we can consider7as a reduction of a3-dimensional problem(−∆+V1(r)−(s·l)·V2(r)−2(s+l)·h·V3(r))Ψ(x,y,z)=EΨ(x,y,z)(22) which can be treated as a spectral problem for a matrix quantum model describing spin-orbital interaction together with the interaction of a total momentum with an external magneticfield. It is remarkable,that the model22does not contain any integer parameters anylonger.All these parameters appear dynamically as solutions of the eigenvalue problems for additionally introduced symmetry operators.At the same time,the model22remains quasi-exactly solvable,because for any particular values of these eigenvalues the equation7has only a certain incomplete set of solutions.7ConclusionThe method of construction infinite(matrix)QES models exposed in this paper is,obviously, quite general and can easily be used for building other spin-orbital models with more com-plicated potentials and higher matrix dimensions.For this it is sufficient to start with other known one-dimensional QES modelsfirst rewritting them in the form of a radial Schroedinger equation and then interpreting the l-dependent terms appearing in their potential as the eigenvalues of a spin-orbital operators.8AcknowledgementsI would like to express my sincere gratitude to my colleagues from the Theoretical Physics Institute of the University of Minnesota(where this work has been written)for their kind hos-pitality.I am especially grateful to Professor M.Shifman for very intersting and fruitful dis-cussions during my visit.This work was supported by the grant DOE/DE-FG02-94ER40823. 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Quantum Computingdd,1F.Jelezko,flamme,3,4Y.Nakamura,5,6C.Monroe,7and J.L.O’Brien 81EdwardL.Ginzton Laboratory,Stanford University,Stanford,California 94305-4088,USA2Physikalisches Institut,Universit¨a t Stuttgart,Pfaffenwaldring 57,D-70550,Germany3Institute for Quantum Computing and Department of Physics and Astronomy,University of Waterloo,200University Avenue West,Waterloo,ON,N2L 3G1,Canada 4Perimeter Institute,31Caroline Street North,Waterloo,ON,N2L 2Y5,Canada5Nano Electronics Research Laboratories,NEC Corporation,Tsukuba,Ibaraki 305-8501,Japan6Frontier Research System,The Institute of Physical and Chemical Research (RIKEN),Wako,Saitama 351-0198,Japan7Joint Quantum Institute,University of Maryland Department of Physicsand National Institute of Standards and Technology,College Park,MD 20742,USA8Centre for Quantum Photonics,H.H.Wills Physics Laboratory &Department of Electrical and Electronic Engineering,University of Bristol,Merchant Venturers Building,Woodland Road,Bristol,BS81UB,UK(Dated:June 15,2009)Quantum mechanics—the theory describing the fundamental workings of nature—is famously counterintuitive:it predicts that a particle can be in two places at the same time,and that two re-mote particles can be inextricably and instantaneously linked.These predictions have been the topic of intense metaphysical debate ever since the theory’s inception early last century.However,supreme predictive power combined with direct experimental observation of some of these unusual phenom-ena leave little doubt as to its fundamental correctness.In fact,without quantum mechanics we could not explain the workings of a laser,nor indeed how a fridge magnet operates.Over the last several decades quantum information science has emerged to seek answers to the question:can we gain some advantage by storing,transmitting and processing information encoded in systems that exhibit these unique quantum properties?Today it is understood that the answer is yes.Many research groups around the world are working towards one of the most ambitious goals humankind has ever em-barked upon:a quantum computer that promises to exponentially improve computational power for particular tasks.A number of physical systems,spanning much of modern physics,are being devel-oped for this task—ranging from single particles of light to superconducting circuits—and it is not yet clear which,if any,will ultimately prove successful.Here we describe the latest developments for each of the leading approaches and explain what the major challenges are for the future.I.INTRODUCTIONOne of the most bizarre and fascinating predictions of the theory of quantum mechanics is that the information processing capability of the universe is much larger than it seems.As the theory goes,a collection of quantum objects inside a closed box will in general proceed to do everything they are physically capable of,all at the same time.This closed system is described by a “wave function”,which for more than a few particles is an incredibly large mathemati-cal entity describing states of matter and energy far beyond experience and intuition.The wave function,however,is only maintained until the box is opened and the system “collapses”randomly into one particular “classical”out-come.Erwin Schr¨odinger attempted to reduce these notions to absurdity by connecting the known quantum behavior of an atomic nucleus to a cat in a box that becomes simultane-ously alive and dead before the box is opened.Schr¨odinger intended for the difficulty of imagining a cat in a “superpo-sition”of alive and dead to make us question whether this quantum theory could possibly be correct.And yet,nearly a century later,quantum theory has yet to fail in predicting an experiment.Although observing an actual “alive and dead”cat is still beyond experimental ca-pabilities,a number of useful technologies have arisen from the counterintuitive quantum world.The quantum com-puter,a device which uses the full complexity of a many-particle wavefunction to solve a computational problem,may soon be one of these technologies.The nature and purpose of quantum computation are of-ten misunderstood.The context for the development of quantum computers may be clarified by comparison to a more familiar quantum technology:the laser.Before the invention of the laser we had the sun,and fire,and the lantern,and then the lightbulb.Despite these advances in making light,until the laser this light was always “incoher-ent”,meaning that the many electromagnetic waves gener-ated by the source were emitted at completely random times with respect to each other.One possibility allowed by quan-tum mechanics,however,is for these waves to be generated in phase,and by engineering and ingenuity methods were discovered for doing so,and hence came about the laser.But lasers do not replace light bulbs for most applications;instead,they produce a different kind of light—coherent light—which is useful for thousands of applications from eye surgery to cat toys,most of which were unimagined by the first laser physicists.Likewise,a quantum computer will not necessarily be faster,bigger,or smaller than an ordinary computer.Rather,it will be a different kind of computer,engineered to control coherent quantum mechanical waves for different applica-tions.The result will be a “closed box”,designed to simul-taneously perform everything it is physically capable of,all at once,with all of those possibilities focused toward a com-putational problem whose solution will be observable aftera r X i v :1009.2267v 1 [q u a n t -p h ] 12 S e p 20102the box is opened.So what will be in the box,and what will it be able to do? Both questions are currently subjects of ongoing research. Thefirst question will be addressed in ensuing sections;the second is worthy of a review of comparable size,and inter-ested readers are advised to see Ref.1.For now,we provide only a brief synopsis of quantum computer“software”. One example of a task for a quantum computer is the quantum fourier transform,which continues the exponen-tial increase in computational efficiency begun by the fast fourier transform2.This subroutine is at the core of Pe-ter Shor’s seminal quantum algorithm for factoring large numbers3,which is one among several quantum algorithms that would allow modestly sized quantum computers to outperform the largest classical supercomputers in solving the specific problems required for decrypting encoded in-formation.Although these algorithms have done much to spur the development of quantum computers,another ap-plication is likely to be far more important in the long term. This application is thefirst envisioned for quantum com-puters,by Richard Feynman in the early1980s4:the effi-cient simulation of that large quantum universe underlying all matter.Such simulations may seem to lie in the esoteric domain of research physics,but these same quantum laws govern the behavior of the many emerging forms of nan-otechnology,including nature’s nanomachinery of biologi-cal molecules.The engineering of the ultra-small will con-tinue to advance and change our world in coming decades, and as this happens we will likely use quantum computers to understand and engineer such technology at the atomic level.Quantum information research promises more than com-puters,as well.Similar technology allows quantum com-munication,which enables the sharing of secrets with secu-rity guaranteed by the laws of physics.It also allows quan-tum metrology,in which distance and time are measured with higher precision than would be possible otherwise. The full gamut of potential technologies have probably not yet been imagined,nor will it be until actual quantum in-formation hardware is available for future generations of quantum engineers.This brings us to the central question of this review:what form will quantum hardware take?Here there are no easy answers.Quantum computers are often imagined to be con-structed by controlling the smallest form of matter,isolated atoms,as in ion traps and optical lattices,but they may like-wise be made from electrical components far larger than routine electronic components,as in superconducting phase qubits,or even from a vial of liquid,as in Nuclear Magnetic Resonance(NMR).Of course it would be convenient if a quantum computer can be made out of the same material that current computers are made out of,i.e.silicon,but it may be that they will be made out of some other material entirely,such as InAs quantum dots or microchips made of diamond.In fact,very little ties together the different implementa-tions of quantum computers currently under consideration. We provide a few general statements about requirements in the next section,and then describe the diverse technological approaches for satisfying these requirements.II.REQUIREMENTS FOR QUANTUM COMPUTING Perhaps the most critical,universal aspect of the various implementations of quantum computers is the“closed box”requirement:a quantum computer’s internal operation, while under the programmer’s control,must otherwise be out of contact with the rest of the universe.Small amounts of information-exchange into and out of the box can dis-turb the fragile,quantum mechanical waves that the quan-tum computer depends on,causing the quantum mechani-cally destructive process known as decoherence,discussed further in Sec.III.Unfortunately no system is fully free of decoherence,but a critical development in quantum com-puter theory is the ability to correct for small amounts of it through various techniques under the name of Quantum Error Correction(QEC).In QEC,entropy introduced from the outside world isflushed from the computer through the discrete processes of measuring and re-initializing qubits, much as digital information today protects against the noise sources problematic to analog technology.Of course,the correction of errors may be useless if the act of correcting them creates more errors.The ability to correct errors us-ing error-prone resources is called fault-tolerance5.Fault-tolerance has been shown to be theoretically possible for er-ror rates beneath a critical threshold that depends on the computer hardware,the sources of error,and the protocols used for QEC.Realistically,most of the resources a fault-tolerant quantum computer will use will be in place to cor-rect its own errors.If computational resources are uncon-strained,the fault-tolerant threshold can be as high as3%6. An early characterization of the physical requirements for an implementation of a fault-tolerant quantum computer was carried out by David DiVincenzo7.However,since that time the ideas for implementing quantum computing have diversified,and the DiVincenzo criteria as originally stated are difficult to apply to many emerging concepts.Here,we rephrase DiVincenzo’s original considerations into three, more abstract criteria,and in so doing introduce a number of critical concepts common to most quantum technologies.1.Scalability:the computer must operate in a Hilbert space whose dimensions may be grown exponentially without an exponential cost in resources(such as time, space or energy.The standard way to achieve this follows thefirst Di-Vincenzo criterion:one may simply add well-characterized qubits to a system.A qubit is a quantum system with two states,and1,such as a quantum spin with S=1/2. The logic space available on a quantum system of N qubits is described by a very large group[known as SU(2N)], which is much larger than the comparable group[SU(2)⊗N] for N unentangled spins or for N classical bits.Ultimately, it is this large space that provides a quantum computer its power.For qubits,the size and energy of a quantum com-puter generally grows linearly with N.3Although qubits are a convenient way to envision a quan-tum computer,they are not a prerequisite.One could use quantum d-state systems(qudits)instead,or even the con-tinuous degrees of freedom available in laser-light.In all cases,however,an exponentially large space of accessible quantum states must be available.In principle,there is an exponentially large Hilbert space in the bound states a single hydrogen atom,a system which is clearly bounded by the Rydberg energy of13.6eV and consists of only two particles!However,the states of a hydrogen atom in any realistic experiment have afinite width due to decoherence,limiting the useful Hilbert space (for which DiVincenzo introduced his third criterion;see Sec.III).Further,access to an exponentially large set of a hy-drogen atom’s states comes at the exponentially large cost in the size of that atom and the time required to excite it to any arbitrary state8.While it is straightforward to see why a single-atom quantum computer is“unscalable”,declaring a technology “scalable”is a tricky business,since the resources used to define and control a qubit are diverse.They may include space on a microchip,classical microwave electronics, dedicated lasers,cryogenic refrigerators,etc.For a system to be scalable,these“classical”resources must be made scalable as well,which tie into complex engineering issues and the infrastructure available for large-scale technologies.2.Universal Logic:the large Hilbert space must be acces-sible using afinite set of control operations;the resources for this set must also not grow exponentially.In the most standard picture of computing,this criterion (DiVincenzo’s fourth)means that a system must have avail-able a universal set of quantum logic gates.In the case of qubits,it is sufficient to have available any“analog”single-qubit gate(e.g.an arbitrary rotation of a spin-qubit),and almost any“digital”two-qubit logic operation,such as the controlled-NOT gate.But quantum computers need not be made with gates. In adiabatic quantum computation9,one defines the answer to a computational problem as the ground state of a com-plex network of interactions between qubits,and then one adiabatically evolves those qubits into that ground state by slowly turning on the interactions.In this case,eval-uation of this second criterion requires that one must ask whether the available set of interactions is complex enough, how long it takes to turn on those interactions,and how cold the system must be maintained.As another example, in cluster-state quantum computation10,one particular quan-tum state(the cluster state)is generated in the computer through a very small set of non-universal quantum gates, and then computation is performed by changing the way in which the resulting wave function is measured.Here,the measurements provide the“analog”component that com-pletes the universal logic.Adiabatic and cluster-state quan-tum computers are provably equivalent in power to gate-based quantum computers11,but their implementation may be simpler for some technologies.One theoretical issue in the design of fault-tolerant quan-tum computers is that for most QEC protocols,“digital”quantum gates(or,more precisely,those in the Clifford group)are relatively easy to perform fault-tolerantly on en-coded qubits,while the“analog”(non-Clifford)quantum gates are substantially more challenging.In other protocols, the analog gates may become easy,and then the digital ones become difficult.The modern design of fault-tolerant proto-cols centers around maintaining universality and balancing the difficulties between the two types of operations.No matter what scheme is used,however,QEC funda-mentally requires the third abstract criterion:3.Correctability:It must be possible to extract the en-tropy of the computer to maintain the computer’s quan-tum state.Regardless of QEC protocol,this will require some com-bination of efficient initialization(DiVincenzo’s second crite-rion)and measurement(DiVincenzo’sfifth criterion).Initial-ization refers to the ability to quickly cool a quantum system into a low-entropy state;for example,the polarization of a spin into its ground state.Measurement refers to the abil-ity to quickly determine the state of a quantum system with the accuracy allowed by quantum mechanics.It is possible that these two abilities are the same.For example,a quan-tum non-demolition(QND)measurement alters the quantum state by projecting to the measured state,which remains the same even after repeated measurements.Clearly,perform-ing a QND measurement also initializes the quantum system into the state measured.Some QND measurements also al-low quantum logic;they are therefore quite powerful for quantum computing.The relationship between the need for initialization and measurement is complex;depending on the scheme used for fault-tolerance,one may generally be replaced by the other.Of course,some form of mea-surement is always needed to read out the state of the com-puter at the end of a computation.Notably,the amount of required physical initialization is not obvious,as schemes have been developed to quantum compute with states of high entropy12.Quantum computation is difficult because the three basic criteria we have discussed appear to be conflicted.For ex-ample,those parts of the system in place to achieve rapid measurement must be turned strongly“on”for error cor-rection and read-out,but must be turned strongly“off”to preserve the coherences in the large Hilbert space.Gener-ally,neither the“on”state nor the“off”state are as difficult to implement as the ability to switch between the two! DiVincenzo introduced extra criteria related to the abil-ity to communicate quantum information between distant qubits,for example by converting stationary qubits to“fly-ing qubits”such as photons.This ability is important for other applications of quantum processors such as quantum repeaters13,but the ability to add non-local quantum com-munication also substantially aids the scalability of a quan-tum computer technology.Quantum communication al-lows small quantum computers to be“wired together”to make larger ones,it allows specialized measurement hard-ware to be located distant from sensitive quantum mem-ories,and it makes it easier to achieve the strong qubit-connectivity required by most schemes for fault-tolerance.4Evaluating the resources required to make a quantum technology truly scalable is an emerging field of quantum computer research,known as quantum computer architec-ture.Successful development of quantum computers will require not only further hardware development,but also the continued theoretical development of algorithms and QEC ,and the architecture connections between the theory and the hardware.These efforts strive to find ways to main-tain the simultaneous abilities to control quantum systems,to measure them,and to preserve their strong isolation from uncontrolled parts of their environment.The simultane-ity of these aspects forms the central challenge in actually building quantum computers,and in the ensuing sections,we introduce the various technologies researchers are cur-rently employing to solve this challenge.III.QUANTIFYING NOISE IN QUANTUM SYSTEMSA key challenge in quantum computation is handling noise.For a single qubit,noise processes lead to two types of relaxation.First,the energy of a qubit may be changed by its environment in a random way which,on-average,brings the qubit to thermal equilibrium with its environment.The timescale for this equilibration is T 1.Typically,systems used for qubits have long T 1timescales,which means that T 1can usually be ignored as a computation error.However,in many experimental systems,T 1sets the timescale for ini-tialization.More dangerous for quantum computing are processes which randomly change the phase of a qubit;i.e.pro-cesses that scatter a superposition such as 0 + 1 into+exp (i φ) 1 ,for an unknown value of φ.This is known as decoherence,and the timescale for phase randomization by decoherence is called T 2.The processes leading to T 1also contribute to T 2,resulting in T 2being upper bounded by 2T 1.But T 2processes cost no energy,and as a result may be much more frequent than T 1processes.In studying noise,one must average over a large ensem-ble of measurements.It is frequently the case that in this ensemble of measurements,the energy of a qubit is slightly different in each measurement.As a result,superpositions again develop unknown phases,and as a result effects ap-pear which resemble those contributing to T 2.This pro-cess is known as dephasing,and it occurs on a timescale T ∗2≤T 2.However,the phase evolution that contributes to T ∗2is constant for each member of the ensemble,and may therefore be reversed.The standard method for doing so is known as the spin-echo,following the NMR technique de-veloped in 195014.By unconditionally flipping the state of a qubit after a time τ,and then allowing evolution for an-other time τ,any static phase evolution is reversed,leading to an apparent “rephasing.”Through spin-echo techniques,the effects of decoherence (T 2)can be distinguished fromthose of dephasing (T ∗2).The value of T 2is used as an initial characterization of many qubits,since,at a bare minimum,qubits need to be operated much faster than T 2to allow fault-tolerant quan-tum computation.This is the third DiVincenzo criterion.However,T 2is not the timescale in which an entire compu-tation takes place,since QEC may correct for phase errors.Also,the measured values of T 2are not fundamental to a material and a technology.Generally,T 2can be extended by a variety of means,such as defining qubits with decoher-ence free subspaces 15which are less sensitive to noise;apply-ing dynamic decoupling techniques 16–21,such as the spin-echo itself,to periodically reverse the effects of environmental noise;or simply improving those aspects of the apparatus or material that leads to the T 2noise process in the first place.Other noise processes exist besides T 1and T rge-dimensional systems,such as multiple-coupled qubits,may be hurt by noise processes distinct from single-qubit T 1and T 2processes.Also,some qubits suffer noise processes that effectively remove the qubit from the computer,such as loss of a photon in a photonic com-puter or the scattering of an atom into a state other than a qubit state.These processes may also be handled by error correction techniques.In practice,once relaxation times are long enough to al-low fault-tolerant operation,imperfections in the coherent control of qubits are more likely to limit a computer’s per-formance.As devices are scaled up to a dozen of qubits,the use of state and process tomography,useful to fully un-derstand the evolution of very small quantum systems,be-comes impractical.For this reason,protocols that assess the quality of control in larger quantum processors have been developed.These enable a characterisation of gate fidelity that can be used to benchmark various technologies.The table below gives measured T 2decoherence times and the results of one-qubit and multi-qubit benchmarking or tomography for several technologies.5Type of Matter QubitCoherence Benchmarking ω0/2πT 2Q1qbit 2qbit A M OTrapped Optical Ion 22,23(40Ca +)400THz 1ms 10120.1%0.7%∗Trapped Microwave Ion 24–26(9Be +)300MHz 10sec 10100.48%†3%Trapped Neutral Atoms 27(87Rb)7GHz 3sec10115%Liquid Molecule Nuclear Spins 28500MHz 2sec1090.01%†0.47%†S o l i d -S t a t ee −Spin in GaAs Quantum Dot 29–3110GHz 3µs 1055%e −Spins Bound to 31P:28Si 32,3310GHz 60ms 1095%10%Nuclear Spins in Si 3460MHz 25sec1095%NV −Center in Diamond 35–373GHz 2ms1072%5%Superconducting Phase Qubit 38–4010GHz 350ns 1042%∗24%∗Superconducting Charge Qubit 41–4310GHz 2µs 105 1.1%†10%∗Superconducting Flux Qubit 44,4510GHz 4µs1053%60%Table comparing the current per-formance of various matter qubits.The approximate resonant fre-quency of each qubit is listed as ω0/2π;this is not necessarily the speed of operation,but sets a limit for defining the phase of a single qubit.Therefore,Q =ω0T 2is a very rough quality factor.Bench-marking values show approximate error rates for single or multi-qubit gates.Values marked with *are found by state tomography,and give the departure of the fidelity from 100%.Values marked with †are found with randomized bench-marking.Other values are rough experimental gate error estimates.IV .CA VITY QUANTUM ELECTRODYNAMICSMany concepts for scalable quantum computer architec-tures involve wiring distant qubits via communication us-ing the electromagnetic field,e.g.infrared photons in fiber-optic waveguides or microwave photons in superconduct-ing transmission lines.Unfortunately,the interaction be-tween a single qubit and the electromagnetic field is gen-erally very weak.For applications such as measurement,in which quantum coherence is deliberately discarded,us-ing more and more photons in the electromagnetic field can sometimes be enough.However,photons easily get lost into the environment,which causes decoherence,and this happens more quickly with stronger fields.Coherent oper-ation requires coupling qubits to weak,single-photon fields with very low optical loss.Such coupling becomes available when discrete,atom-like systems are placed between mir-rors that form a high-quality cavity,introducing the physics known as cavity quantum electrodynamics (c QED )46.Cav-ity QED has been an important topic of fundamental re-search for many years 47–50,and was employed for one of the earliest proposals for quantum computing 51.A cavity enables quantum information processes for sev-eral reasons.First,one may imagine that a photon in a cav-ity bounces between its mirrors a large number of times be-fore leaking out;this number is called the quality factor Q .If Q is high,one single photon may interact Q times with a sin-gle atom,and if each interaction accomplishes a weak,QND measurement (see Sec.II),then the measurement strength is enhanced by Q .But a cavity does more than this.It also confines the electromagnetic field into a small volume.One manifesta-tion of this is evident in the spontaneous emission of atoms.Spontaneous emission can be considered as the simultane-ous coupling of an atom to an infinite continuum of modes of the electromagnetic field.A cavity makes the coupling to one particular mode —the cavity mode —substantially stronger than other,free space modes.This mode is emit-ted from the cavity at a rate κ=ω0/Q ,where ω0is theresonant frequency of the cavity.The coupling of the atom to the cavity mode,g ,is proportional tof /V .Here f is the oscillator strength of the atom,a measure of its general coupling to electromagnetic fields irrespective of the cavity,which depends on details such as the size and resonant fre-quency of the atom.The mode-volume of the cavity,V ,is a critical parameter to minimize for strong interactions.If the energy levels of the atom are matched to the cavity pho-ton energy ¯h ω0,the rate at which the combined atom/cavity system emits photons is approximately 4g 2/κ.It is possible for this rate to be much larger than the rate of emission into non-cavity modes,γ,leading to a very large resonant Purcell factor :Purcell factor =4g 2κγ=34π2 λn 3QV ,(1)where λ/n is the wavelength of the emitted photons in thematerial of refractive index n .A large Purcell factor roughly means that when an atom emits a photon,it is very likely that the emitted photon enters the cavity mode.This cav-ity mode may then be well coupled to a waveguide,which strongly directs that photon to an engineered destination.This parameter is critical for a large variety of proposals us-ing cQED,even those not involving Purcell-enhanced spon-taneous emission of the atom.The Purcell factor for a res-onant atom/cavity system is also known as the coopera-tivity factor ,and its inverse is known as the critical atom number 47,i.e.the number of atoms in a cavity needed to have a profound effect on its optical characteristics.Large Purcell factors are generally observed in cavities in the weak or intermediate coupling regime ,also known as the bad cavity limit ,in which κ>g .This regime is use-ful for applications such as single photon sources,in which the cavity increases the speed,coherence,and directional-ity of emitted photons.It is also the appropriate regime for schemes in which distant qubits are probabilistically entan-gled by heralded photon scattering 52–55(as opposed to pho-ton absorption/emission 56).However,a variety of schemes are enabled by the strong coupling limit,in which g κ,γ,。

a r X i v :0705.3936v 2 [n u c l -t h ] 11 N o v 2007Λ∗-hypernuclei in phenomenological nuclear forcesA.Arai,M.Oka and S.YasuiDepartment of Physics,Tokyo Institute of Technology,Tokyo 152-8551,JapanFebruary 1,2008Abstract The Λ∗-hypernuclei,which are bound states of Λ(1405)and nuclei,are discussed as a possible interpretation of the ¯K -nuclei.The Bonn and Nijmegen potentials are extended and used as a phenomenological potential between Λ∗and N .The K -exchange potential is also considered in the Λ∗and N interaction.The two-body (Λ∗N )and three-body (Λ∗NN )systems are solved by a variational method.It is shown that the spin and isospin of the ground states are assigned as Λ∗N (S =1,I =1/2)and Λ∗NN (S =3/2,I =0),respectively.The binding energies of the Λ∗-hypernuclei are discussed in comparison with experiment.1Introduction Possibility of kaon-nuclear bound states is of great interest in hadron and nuclear physics.A deeply bound kaonic nuclear state with a relatively small decay width was recently predicted and further studied in literatures.[1,2,3,4]Possible high density matter caused by the strong kaon attractive force is also a subject of heated discussion.[1,2,3,4]In order to find such exotic states,several experimental searches have been carried out.[5,6]FINUDA collaboration has reported an observation of bound ppK −state at the binding energy,115MeV.[6]Interpretation of the peak found in this experiment is yet under discussion,while some advanced calculations,such as a 3-body calculation `a la Fadeev equation with ¯KN−πΣchannel coupling,have been performed.[7,8]Other approaches include an interpretation as a nine-quark state studied in the MIT bag model [9],and a kaon absorption process between two nucleons [10].While the coming J-PARC facility will certainly answer to the question whether such deeply bound kaonic nuclear states exist,we need to study such a system in more extensive views.The purpose of this paper is to give a new interpretation to the “kaon-nucleus”states.We consider Λ∗-hypernuclear states.Λ∗is the lowest negative parity baryon with mass around 1405MeV and strangeness −1.This baryon is in many senses unique.For instance,it is below any other non-strange baryons with negative parity and is isolated1not forming an octet in SU(3).If it is assumed to be a p-wave baryon with spin1/2,then it requires a large spin-orbit splitting.Its uniqueness has attracted a lot of attention and various exotic views ofΛ∗have been proposed.We here do not consider specific composition ofΛ∗,but simply assume that it belongs to aflavor singlet representation,and thus isolated.Our claim is that the so-called kaon-nuclear bound states can be interpreted as a bound state ofΛ∗in a nucleus,or Λ∗-hypernucleus.Then the FINUDA observation is regarded as a two-body bound state ofΛ∗and N,whose binding energy is rger systems,such as strange tribaryon can be aΛ∗NN system and so on.In this paper,we construct a model ofΛ∗N interaction according to the one-boson exchange model and consider the two-body(Λ∗N)and three-body(Λ∗NN)systems.As the NN interaction,we choose the same one-boson exchange model,the Bonn potential [11]and the Nijmegen soft-core potential models[12,13].We extend these models to the Λ∗N systems and the K-exchange is also included in the same context betweenΛ∗and N.We then solve the two-body and three-body Schr¨o dinger equations using a variational method.The content of the paper is as follows.In Section2,we construct the potential model forΛ∗N system.The features of the model in the context of possible quantum num-bers of theΛ∗-nuclear states are also given.In Section3,the numerical results for the binding energies of two-body and three-body bound states are shown.In Section4,some discussion on the present results are given.The conclusion is given in Section5.2ModelThe phenomenological nuclear force is quite successful in studying nuclear systems.In order to setup the model for theΛ∗-hypernuclei,we discuss the phenomenologicalΛ∗N interaction.The nuclear forces among hyperons are not yet established,although a great amount of studies have been performed experimentally and theoretically so far.In general, it is considered that the interaction between baryons is described by exchange of mesons supplemented by phenomenological short-range repulsion.In the present study,based on the one-boson exchange picture,we extend the phenomenological nuclear forces between the NN pair to theΛ∗N pair interaction.In the Bonn[11]and Nijmegen potentials[12, 13],the scalar(σ,a0),pseudoscalar(π,η)and vector(ω,ρ)bosons are exchanged between the NN pair.The explicit equations and the parameter sets for the Bonn potential are shown in Appendix.By considering the SU(3)symmetry,we assume that these potentials are also applied to theΛ∗N pair.Here the isovector mesons(a0,π,ρ)are irrelevant to theΛ∗N interaction,since theΛ∗is isosinglet.Instead the K(and¯K)meson comes into the game in the exchange processΛ∗N→NΛ∗.Then,theΛ∗N potential is given by the sumVΛ∗N=Vσ+Vη+Vω+V K.(1) Theσ-exchange potential,Vσ,is the most attractive force,and theω-exchange potential, Vω,plays an essential role for the spin determination,as we see below.In the following,2we investigate several possibilities of theΛ∗N potential.In the extended phenomenological nuclear forces in theΛ∗N pair,we have some unknown parameters;the coupling constants gΛ∗NM(for M meson)and the momen-tum cutoffs,which appear in the form factors.Among them,theΛ∗N¯K vertex con-stant,gΛ∗N¯K,is determined by the SU(3)relation from the observed decay width of the Λ∗→Σπchannel.TheΛ∗Σπcoupling g2Λ∗Σπ/4π=0.064is obtained from the decay widthΓ(Λ∗→Σπ)=50.0±2.0MeV.Assuming thatΛ∗is a purely SU(3)singlet state, the SU(3)isoscalar factor,(Λ∗)=(N¯KΣπΛηΞK)=18(23−1−2)1/2,(2)predicts the coupling constant g2Λ∗NK/4π=g2Λ∗Σπ/4π=0.064.It should be noted that this coupling constant is much smaller than the NNσcoupling constant g2NNσ/4π=7.78. Therefore,the K meson plays only a minor role in theΛ∗N bound state as seen in the numerical calculation later.Here,the K-exchange has to be treated carefully.BecauseΛ∗lies close to the¯KN threshold,the kaon may have four momentum,qµ,near the on-mass-shell values.Then the kaon propagator can be enhanced strongly.In order to include this effect,we replace the kaon mass by an effective mass.Assuming that the baryons are static,we obtain the finite energy transfer,q0≃MΛ∗−M N,and the effective mass,˜m K≡ m2K−(MΛ∗−M N)2=171MeV/c2.(3) This replacement indeed enhances theΛ∗N potential more strongly than the case ofΛN orΣN potential.This is because that theΛ∗N¯K coupling is a scalar coupling and the effect of˜m K comes only in the range of the potential.In contrast,a p-wave coupling gives rise to an extra˜m2K factor,which significantly suppresses its effect on theΛN orΣN potential.As a result,the K-exchange potential forΛ∗N pair is given by the coupling constant g2Λ∗NK/4π=0.064and the effective mass˜m K in Eq.(10)in Appendix.Yet the other parameters in theΛ∗N interaction are notfixed due to lack of experi-mental information.In the present study,we treat the gΛ∗Λ∗σas a free parameter,since the results happen to be most sensitive to theσ-exchange.We further assume that the other parameters are the same as the NN interaction for simplicity.Here it is important for the later discussions to prospect the properties of theΛ∗N interaction.Let us see the spin-dependence of theΛ∗N interaction.The spin-spin inter-action is induced by theη,K andωmesons.We consider theω-exchange force,since theωmeson coupling is much stronger than theηand K meson couplings.The static ω-exchange potential is written in the momentum space asVω( k2)=−C( σ1· σ2) k2k2+m2ω( σ1· σ2),(4)3with a positive constant C,and theωmeson mass mω.In the real space,thefirst term is a delta function type,while the second term is the standard Yukawa potential.With a form factor introduced,the delta type potential becomes afinite range potential.For the Yukawa type potential,the spin singlet is more attractive than the spin triplet.On the other hand,for the delta type potential,the spin triplet is more attractive than the spin ually the delta type potential at short distance plays a minor role due to strong repulsive core of the NN force,hence the Yukawa potential is much more important.However,this is not the case for theΛ∗N potential.There,at short distances, the delta function is overwhelming to the Yukawa type potential.Consequently,in the Λ∗N potential,the spin triplet potential is more attractive at short distance than the spin singlet one.Now we look at the possible quantum states of theΛ∗-hypernuclei,which are obtained in the spin and isospin combinations.It is assumed that all the particles occupy the lowest-energy s-orbit.SinceΛ∗has spin S=1/2and isospin I=0,theΛ∗N system has two states of spin and isospin;(S=0,I=1/2)and(S=1,I=1/2).On the other hand,theΛ∗NN system takes the isosinglet and isotriplet states.The isosinglet state has the spin S=1/2and S=3/2states,where the NN pair is isosinglet and spin triplet.The isotriplet state has spin S=1/2,where the NN pair is isotriplet and spin singlet.Therefore the possible quantum numbers of the s-waveΛ∗NN system are (S=1/2,I=0),(S=3/2,I=0)and(S=1/2,I=1).Here,we prospect the quantum numbers of the ground states of theΛ∗-hypernuclei by using the spin dependence of theω-exchange at short distance.It is expected that the ground state of the two body system(Λ∗N)is spin triplet.In order to understand the ground state of the three body system(Λ∗NN),the three possible states,(S=1/2,I=0), (S=3/2,I=0)and(S=1/2,I=1),are expanded in terms of theΛ∗N pair;|Λ∗(NN)S=1;S=1/2,I=0 =1332|(Λ∗N)S=0N ,(7) for the isotriplet state.Among these states,the spin triplet interaction is contained most strongly in the(S=3/2,I=0)state due to the largest value of coefficients of the (Λ∗N)S=1component.Therefore,it is expected that the(S=3/2,I=0)is the ground state for theΛ∗NN state.In the next section,it will be shown that the above analysis is consistent with the numerical results.The above conclusion of the spin of the two-body and three-body states is in strong contrast with the kaon bound state approach.There,the spin dependence is induced by the isospin dependence of¯KN interaction.For instance,in the¯KNN system,the (¯KN)I=0interaction is the driving force for the bound state.It is easy to see that the ¯K(NN)state contains more(¯KN)I=0component than¯K(NN)I=0,S=1state.Thus, I=1,S=0the ground state of¯KNN is expected to have S=0.4-400-2002000 0.2 0.4 0.60.8 1 1.2 1.4V (r ) [M e V ]r [fm]Figure 1:The Bonn potentials applied for Λ∗N pair in the S =0(dashed line)and S =1(solid line)channels for the coupling constants g Λ∗Λ∗σ/g NNσ=0.39.3Numerical ResultsIn this section,we discuss the numerical results for the two-body (Λ∗N )and three-body (Λ∗NN )systems by solving the Schr¨o dinger equation with the Bonn [11]and Nijmegen (SC89[12]and ESC04[13])potentials for the Λ∗N and NN interactions.For simplicity,derivative terms are not considered.We mainly show the results for the Bonn potential.The results do not differ qualitatively for the Nijmegen potentials with SC89and ESC04.3.1Two-body systemThe two-body system (Λ∗N )has the spin singlet (S =0,I =1/2)and triplet (S =1,I =1/2)states.In Fig.1,the Bonn potentials are plotted as functions of the relative distance r between Λ∗and N in the S =0and S =1channels for the coupling constant g Λ∗Λ∗σ/g NNσ=0.39.The S =1potential is strongly attractive at short distance (r <∼0.4fm),while the S =0one is repulsive.This difference is caused by the delta type interaction in the ωpotential (4),which is attractive for S =1and repulsive for S =0.It should be noted that the delta function is smeared to a finite range potential due to the form factor in the phenomenological nuclear forces.In order to see the contributions from individual mesons in the S =1Λ∗N potential,the components from the σ,ωand η-exchange potentials and the additional K -exchange potentials are plotted in Fig.2.The σ-exchange potential is attractive at medium range (r >∼0.4fm),while the ωpotential is repulsive.However,the ωpotential is strongly attractive at shorter distances (r <∼0.4fm)due to the delta function in (4).Hence,the sum of them employs a deeply attractive potential at short distance,which is one of the characteristic properties in the Λ∗N potential.The contributions from the ηand K mesons are relatively weak.Here we investigate the on-shell effect of the K -exchange.As we have already dis-5-400-2002000 0.2 0.4 0.60.8 1 1.2 1.4V (r ) [M e V ]r [fm]ωηKFigure 2:The components from the σ,ω,ηand K mesons in the Bonn potential between Λ∗N pair for S =1for g Λ∗Λ∗σ/g NNσ=0.39.The Bonn potential is indicated by the solid line.cussed,the K meson propagating between the Λ∗and N may have large energy and thus can be close to the on-mass-shell kinematics.Such K -exchange might be largely enhanced.We treat this effect as an effective mass ˜m K defined in Eq.(3).In Fig.3,the K -exchange potential is plotted for the bare mass m K =495MeV (solid line)and for the reduced mass ˜m K =171MeV (dashed line).As the effective K meson mass gets smaller,both the potential strength and the range increase.However,this enhancement has little significance to the binding energy of the Λ∗N ,since the K -exchange potential is still weaker than the σand ω-exchange potentials.Solving the Schr¨o dinger equation for the Λ∗N with the Bonn potential,we obtain the binding energy and the wave function of the Λ∗N bound states.As a set of trial wave functions,we use a linear combination of Gaussian functions;ψ(r )=n i =1a i e −αi r 2,(8)where a i are the normalization constants,αi the width parameters,and r the relative distance between Λ∗and N .As a result,the binding energy strongly depends on the coupling constant g Λ∗Λ∗σ/g NNσas shown in Fig.4.The σpotential plays an essential role for the formation of the bound state.Indeed,without the Λ∗Λ∗σcoupling,the bound state does not exist.For S =1,the Λ∗N is bound for the coupling constant g Λ∗Λ∗σ/g NNσ>∼0.37.At g Λ∗Λ∗σ/g NNσ=0.39,the binding energy reaches 88MeV.The observed binding energy 115MeV of ppK −state reported by the FINUDA collaboration [6]is interpreted as 88MeV for the binding energy of the Λ∗N state.For S =0,a stronger coupling g Λ∗Λ∗σ/g NNσ>∼0.98is required for the bound state.The binding energy 88MeV is obtained at g Λ∗Λ∗σ/g NNσ=1.01.In the current experimental status,the quantum number of the bound state is not yet known.Our result suggests an S =1bound state rather than S =0.The probability density of the Λ∗N with S =1is plotted in Fig.5.The wave function is very compact as compared with the nucleon size,suggesting that the60 510152025 303540450 0.5 11.5 2V (r ) [M e V ]r [fm]m K =495MeV m K =171MeV Figure 3:The K -exchange component in the Λ∗N potential.The solid line is for the bare mass m K =495MeV and the dashed line is for the reduced mass ˜m K =171MeV.See the text.obtained bound state is produced mainly by the short-range ωattraction assisted by the medium-range σ-exchange that cancels the medium range ωrepulsion.So far,we have discussed the results of the Bonn potential.In order to check the model dependence,we also apply different types of the nuclear force.In particular,as the short-range part of the interaction is important,the mechanisms of short-range NN repulsion is in question.The Bonn potential acquires the short-range repulsion mainly from the ω-exchange potential,while the other models,such as the Nijmegen potential,introduce a new component for the repulsion.The pomeron exchange is represented in the Nijmegen potential by a gaussian potential of short range.In the case of NN interaction,it is strong enough to expel the wave functions away from the center.When we apply the same repulsion to Λ∗N system,we find that the short-range attraction from ω-exchange may still make a bound state,if the σ-exchange attraction at the medium range is strong enough.However,compared with the Bonn potential,the extra pomeronic repulsion for Λ∗N makes the system less bound.That means that the required g Λ∗Λ∗σcoupling constant becomes larger.We employ the Nijmegen potential of the versions SC89and ESC04.It is found that SC89requires the minimal coupling constants g Λ∗Λ∗σ/g NNσ=0.77and g Λ∗Λ∗σ/g NNσ=0.98to form a Λ∗N bound state for S =1and S =0,respectively.The binding energy (88MeV)reported by the FINUDA group is obtained by setting g Λ∗Λ∗σ/g NNσ=0.832for S =1and g Λ∗Λ∗σ/g NNσ=1.128for S =0.The probability density is pushed out from the center as compared with the Bonn potential.This is because the range of the Λ∗N potential is longer than that from the Bonn potential.The result of ESC04is qualitatively the same as that of SC89.Only difference is that absolute values of the σ,ωand pomeron exchange potentials in ESC04are smaller than those in SC89.Then,the minimum coupling constants of Λ∗Λ∗σbecome larger,and the experimental binding energy is reproduced at g Λ∗Λ∗σ/g NNσ=1.118for S =1and g Λ∗Λ∗σ/g NNσ=1.398for S =0.70.360.370.380.390.4g Λ∗Λ∗σ/g NN σ020406080100120B i n d i n g E n e r g y [M e V ](S=1, I=1/2)1 1.1 1.2g Λ∗Λ∗σ/g NN σ(S=0, I=1/2)Figure 4:The binding energy of the Λ∗N as functions of the coupling constant g Λ∗Λ∗σ/g NNσfor the Bonn potential.The solid line is for S =1and the dashed line for S =0.0 0.2 0.4 0.60.8 1 1.2 1.4D i s t r i b u t i o nr [fm]Figure 5:The probability density of the bound Λ∗N system with S =1and g Λ∗Λ∗σ/g NNσ=0.39.The binding energy is 88MeV.The horizontal axis is the relative distance between Λ∗and N .8Table1:The ratios of the coupling constants gΛ∗Λ∗σ/g NNσrequired to obtain the binding energy,88MeV,of theΛ∗N two-body system.Λ∗N Nijmegen(SC89) B.E.[MeV]0.39 1.1181.1 1.398The results above discussed are summarized in Table1.The ratios of coupling con-stants gΛ∗Λ∗σ/g NNσthat correspond to the experimental observation in the FINUDA collaboration are listed.The smaller value indicates that the system can form a bound state more easily.Wefind that in all cases that the coupling constant in S=1bound state is smaller than that in S=0one in the Bonn and Nijmegen potentials.Therefore, we expect that the lowest-energy bound state is the S=1state.3.2Three-body systemNow we discuss the three-body system(Λ∗NN).The binding energies and wave functions are obtained variationally by solving the Schr¨o dinger equation by using the phenomeno-logical NN andΛ∗N potentials.The trial wave function is given byψ(r,r′)= i,j a ij e−αi r2e−βi r′2(9)with a ij are the normalization constants,αi andβi the width parameters,and r and r′the Jacobi coordinates forΛ∗NN.Then we obtain the binding energies as func-tions of the coupling constant gΛ∗Λ∗σ/g NNσfor the(S=3/2,I=0),(S=1/2,I=1) and(S=1/2,I=0)states indicated by solid,long-dashed,short-dashed lines,respec-tively,in Fig.6.Wefind that a bound(S=3/2,I=0)state can be formed only for gΛ∗Λ∗σ/g NNσ>∼0.406,while the(S=1/2,I=1)and(S=1/2,I=0)bound states re-quire stronger coupling constant.Therefore our analysis shows the(S=3/2,I=0)state as the ground state of theΛ∗NN system.Note that forΛ∗N pair in the S=1channel is more attractive than the S=0channel.From theΛ∗NN states written as combinations of(Λ∗N)S=1N and(Λ∗N)S=0N as Eq.(6)and(7),one sees that the(S=3/2,I=0) state contains more the(Λ∗N)S=1component.Through the discussions given above,it is clear that the short-distance behavior of the nuclear potential plays an important role.We recall that at short distance the smeared delta interaction in theω-exchange inΛ∗N interaction overwhelms the Yukawa type potential,and it induces a strong attractive potential for S=1.This is contrary to the long range behavior of theω-exchange potential;repulsive for S=1and attractive for S=0.The strong attractive potential at short distance is also seen in the three-body system.It is directly observed from the probability densities ofΛ∗N andΛ∗NN,which concentrate at short distances.90.40.410.420.43g Λ∗Λ∗σ/g NN σ020406080100120140160B i n d i n g E n e r g y [M e V ](S=3/2, I=0)0.640.650.660.67 g Λ∗Λ∗σ/g NN σ(S=1/2, I=1)0.90.920.94g Λ∗Λ∗σ/g NN σ(S=1/2, I=0)Figure 6:The binding energy of the Λ∗NN as functions of the coupling constant g Λ∗Λ∗σ/g NNσfor the Bonn potential.The solid,long-dashed and short-dashed lines indicate the (S =3/2,I =0),(S =1/2,I =1)and (S =1/2,I =0)states,respectively.Table 2:The ratios of the coupling constants g Λ∗Λ∗σ/g NNσrequired to obtain the binding energy,167MeV,of the Λ∗NN three-body system.Λ∗NNNijmegen (SC89) B.E.[MeV]0.4241.2170.64 1.325104DiscussionWe here compare our results with some other studies for the same system.In the original Akaishi-Yamazaki picture[1,2,3],the binding energy of ppnK−is shown to be larger than ppK−.Furthermore,it is indicated that¯K-nuclei can be stable for larger baryon numbers.However,this is not the case for theΛ∗-hypernuclei.Let us compare the minimum coupling constants for the two-body(Λ∗N)and three-body(Λ∗NN)bound systems.For example,for the Bonn potential,the coupling gΛ∗Λ∗σ/g NNσ=0.37gives a bound state ofΛ∗N with(S=1,I=1/2),while gΛ∗Λ∗σ/g NNσ=0.406is required for Λ∗NN with(S=3/2,I=0).Therefore,the present framework leads to the conclusion that theΛ∗N two-body state is bound more easily than theΛ∗NN three-body state.This is qualitatively the same for the Nijmegen potentials with SC89and ESC04.Furthermore, it may be expected that the theΛ∗-hypernuclei becomes unbound as the baryon number increases.This picture is in opposite direction as compared with the¯K-nucleus bound states.[1,2,3]A recent proposal[15]of interpreting the binding of¯K in terms of“migration”of¯K is in fact quite similar to the K-exchange part of our picture.The difference between their analysis and ours is mainly in the coupling strengths.We determine the coupling constant so as to reproduce the relatively narrow width ofΛ∗,while Ref.[15]employs the coupling to reproduce the binding energy ofΛ∗as a¯KN bound state.A further analysis on this difference will be needed.A comment is in order for the dependence of the result on the coupling constant in the phenomenological nuclear potentials forΛ∗N.The coupling constant gΛ∗Λ∗σis taken as a free parameter,and the couplings for the other mesons arefixed to the same value as the NN interaction.Theσpotential is the most attractive force among them,and theωpotential is the second strongest force.Therefore,as a further step,it is interesting to alter the coupling strength ofΛ∗Λ∗ω.However,concerning the spin of the bound state,the result will not be changed as long as theω-exchange potential is attractive at short-range.This is because that the spin of theΛ∗N pair is almost determined by the ω-exchange potential,not by the other mesons.Lastly,we note that in the present framework only the meson exchange potential has been considered as interaction betweenΛ∗and N.However,the resulting bound states are compact where the substructure of the baryons should be important.In the present analysis,we have introduced the form factors representing the baryon structure.It may be important to consider the quark structure of baryons explicitly so that the short range baryon-baryon interactions are correctly taken into account.[14]This subject is left for a future work.5ConclusionThe possibility of theΛ∗-hypernuclei is discussed by considering theΛ∗as a compound state in the kaonic-nuclei.The two-body(Λ∗N)and three-body(Λ∗NN)systems are investigated by using phenomenological nuclear forces.Based on the one-boson exchange11picture,the Bonn and Nijmegen(SC89and ESC04)potentials are extended to theΛ∗N interaction.The K-exchange is also included in the extended nuclear forces.The gΛ∗N¯K coupling is determined from the experimental value of the decay width of theΛ∗.Only Λ∗Λ∗σcoupling constant is left as a free parameter with the other parametersfixed.As a result,for the two-body system(Λ∗N),an appropriateΛ∗Λ∗σcoupling constant reproduces the binding energy of the ppK−which is comparable to the value reported by the FINUDA collaboration.The most stable state in theΛ∗N bound state is the S=1state.For the three-body system(Λ∗NN),the(S=3/2,I=0)state is the most stable state.These results are understood by the fact that theω-exchange potential is strongly attractive at short distances for S=1channel rather than in S=0.There the K-exchange plays only a minor pared with theΛ∗N andΛ∗NN,the minimum coupling constant gΛ∗Λ∗σin the former is smaller than that of the latter.Therefore,the Λ∗N state is more easily produced in comparison with theΛ∗NN state.Our conclusion is qualitatively the same for the Bonn and Nijmegen(SC89and ESC04)potentials.In the present study,Λ∗is considered as a stable particle.In reality,however,multi-channel decays are open forΛ∗.The decay toΣπis the main source ofΛ∗free decay, while the in-medium decaysΛ∗N→ΛN,ΣN are new and interesting.In our discussion, the boundΛ∗N andΛ∗NN states are very compact objects.Therefore,the conversion width toΛN orΣN may be modified.These subjects are closely related to experimental researches in DAΦNE and J-PARC.Further studied are left for future works. AcknowledgementThis work is supported by a Grant-in-Aid for Scientific Research for Priority Areas,MEXT (Ministry of Education,Culture,Sports,Science and Technology)with No.17070002. AThe Bonn potential used in the present paper is explicitly shown in a coordinate formalism [11].In the followings,m is the mass of the meson,g ij and f ij are coupling constants in the reaction process,1+2→3+4with the baryon mass M1and M2.The parameter set is summarized in Table3.The pseudoscalar type potential is given byV ps(m,r)=g13g244M1M2m 14π 1−m24π g13g24 1+m24MM1+f13g24m216M2M1M2 φ(mr) 12Table3:The parameter set of the Bonn potential.(The superscript a and b denote the NN pairs with isospin T=1and T=0,respectively.)π138.0314.9 1.3η548.83 1.5ρ7690.95 5.8 1.3ω782.620 1.5a0983 2.6713 2.0σ550a7.7823a 2.0715b16.2061b 2.04M1M2 g13g24+g13f24M2M+f13f24M1M28M1M22x,(13)χ(x)= m x+3r2− σ1· σ2,(15) with the propagating meson mass m.When the exchanged meson has isospin,τ1·τ2is multiplied.The form factor is introduced for each meson by replacing asVα(m,r)→Vα(m,r)−Λ22−m2Λ22−Λ12Vα(Λ2,r),(16)whereΛ1=Λ+ǫ,Λ2=Λ−ǫ,(17) withǫ/Λ≪1,such asǫ≈10MeV.References[1]Y.Akaishi and T.Yamazaki,Phys.Rev.C65(2002),044005.[2]A.Dot´e,H.Horiuchi,Y.Akaishi and T.Yamazaki,Phys.Lett.B590(2004),51.13[3]A.Dot´e,H.Horiuchi,Y.Akaishi and T.Yamazaki,Phys.Rev.C70(2004),044313.[4]A.Dot´e and W.Weise,arXiv:nucl-th/0701050.[5]T.Suzuki et al.,Phys.Lett.B597(2004),263.[6]M.Agnello et al.(FINUDA Collaboration),Phys.Rev.Lett.94(2005),212303.[7]N.V.Shevchenko,A.Gal and J.Mareˇs,Phys.Rev.Lett.98(2007),082301.[8]Y.Ikeda and T.Sato,arXiv:nucl-th/0701001and arXiv:0704.1978[nucl-th].[9]Y.Maezawa,T.Hatsuda and S.Sasaki,Prog.Theor.Phys.114(2005),317.[10]V.K.Magas,E.Oset,A.Ramos and H.Toki,Phys.Rev.C74(2006),025206.[11]R.Machleidt,K.Holinde and Ch.Elster,Phys.Rep.149(1987),1.[12]M.M.Nagels,T.A.Rijken and J.J.de Swart,Phys.Rev.D17(1978),768.[13]Th.A.Rijken,Phys.Rev.C73(2006),044007.[14]M.Oka and K.Yazaki,Phys.Lett.B90,41(1980).[15]T.Yamazaki and Y.Akaishi,Phys.Rev.C76(2007),045201.14。