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experimental realization of trapped one - dimensional ( 1d ) cold atom systems @xcite are triggering more and more theoretical efforts to study the 1d many - body physics beyond the mean - field theory . for the ultracold quantum gases tightly confined in waveguides , the dynamics are effectively described by a 1d model due to the radial degrees of freedom are frozen @xcite . further , the ability of tuning the effective 1d interactions by feshbach resonance allows experimental access to the very strongly interacting regime where correlation effects are greatly enhanced @xcite . in the limit of the tonks - girardeau ( tg ) @xcite gas with effective coupling constant @xmath2 , the many - body problem of a tg gas can be mapped to that of a free fermi gas by the bose - fermi mapping , which has been verified by two experimental groups @xcite . this bose - fermi duality was generalized to show the equivalence between a 1d fermionic system and a bosonic one with the reversed role of strong and weak couplings @xcite . recently , the exact ground state of the fermionic tg ( ftg ) gas , defined as a 1d spin - polarized fermionic gas with infinitely strong attractive @xmath0-wave interactions , has been determined by inversely fermi - bose mapping to the ideal bose gas @xcite .    a key experimental challenge is to obtain superfluidity with pairs in nonzero orbital angular momentum states by using @xmath0-wave , or maybe even @xmath3-wave feshbach resonances . in general , the @xmath0-wave interaction is very weak comparing with the @xmath4 -wave interaction . however , for a spin - polarized fermionic gas , the @xmath4-wave scattering is forbidden due to the pauli exclusion principle and thus the @xmath0 -wave interaction is dominant . furthermore , the @xmath0-wave interaction can be greatly enhanced by the feshbach resonances @xcite and using a @xmath0-wave feshbach resonance between @xmath5 atoms jin s group at jila have successfully produced and detected molecules with lifetimes on the order of milliseconds on both the bec and the bcs side of the resonance @xcite . for a 1d gas , the additional confinement induced resonance permits one to tune the 1d effective interaction via a 3d feshbach resonance @xcite .    in this paper , we report on a detailed study of the 1d fermi gases in the infinitely deep square potential well . we will show that the model of fermionic gases with attractive @xmath0-wave interactions in such a one - dimensional hard - wall trap is exactly solvable by the bethe - ansatz method . the experimental efforts in trapping ultracold gases near micro - fabricated surfaces , the so - called `` atom chips '' @xcite , and various innovative features in designing the optical box trap @xcite , are specifically aimed at studying the surprisingly rich variety of physical regimes predicted for the 1d bose gas and have stimulated many theoretical studies on the physics in a box trap @xcite . different from the harmonic trap , the interacting model in a hard - wall trap is integrable and thus could provide us some exact pictures for understanding the trapped many - body systems . so far , there has been a growing interest in the exactly solved models in the hard - wall trap @xcite , but most of them focus on the bose gas and the fermi gas with odd - wave interactions is not addressed . while the theoretical understanding of the correlation effect of bosonic system has been investigated extensively @xcite , the fermionic system is not well understood except in the so - called ftg limit @xcite . we consider an @xmath6-particle system with finite , attractive @xmath0-wave interaction in a one - dimensional box of length @xmath7 , which obviously fills the gap between the ftg limit and free fermions . the schrdinger equation can be formulated as @xmath8 \psi = e\psi , \ ] ] where @xmath9 is the pseudo - potential describing the @xmath0 -wave scattering . it has been shown that the @xmath0-wave scattering of two spin - polarized fermions in a tightly confined waveguide can be well described by the contact condition @xcite @xmath10 where @xmath11 ^{-1}\ ] ] is the effective 1d scattering length with @xmath12 the @xmath0-wave scattering length and @xmath13 the transverse oscillator length @xcite . the contact condition can be reproduced by using the following pseudo - potential @xcite @xmath14 where @xmath15 and @xmath16 the scattering length can be tuned readily from @xmath17 to @xmath18 by sweeping an external magnetic field - the feshbach resonance , or by changing the geometry of the trapping potential - the confinement induced resonance and in this paper the full physical regimes @xmath19 will be studied . similar to the case of bose gas , the important parameter characterizing the different physical regimes of the 1d fermi gas is @xmath20 , where @xmath21 and @xmath22 . a standard rescaling procedure brings the schrdinger equation into a dimensionless one ( for simplicity we keep the original notations ) @xmath23 with @xmath24 where in the dimensionless interaction constant @xmath25 we intentionally keep the factor 2 in accordance with the bosonic case @xcite . the wave function takes the general form @xmath26 where we have used @xmath27 to label the region @xmath28 . the wave function of fermions should follow the antisymmetry of exchange , so our model is simplified into the solution of @xmath29 in the region @xmath27 with the open boundary condition @xmath30    using the bethe ansatz method we obtain the wavefunction parameterized by the set of quantum number @xmath31 known as quasi - momenta or rapidities @xcite @xmath32 \sin \left ( k_{p_1}x_{q_1}\right ) \\ & & \times \prod_{1<j < n}\sin \left ( k_{p_j}x_{q_j}-\sum_{l < j}\omega _ { p_lp_j}\right ) \\ & & \times \exp \left ( ik_{p_n}l\right ) \sin \left ( k_{p_n}\left ( l - x_{q_n}\right ) \right)\end{aligned}\ ] ] with @xmath33 and @xmath34 . here @xmath35 and @xmath36 denote sign factors associated with even / odd permutations of @xmath37 and @xmath38 , respectively . the quasi - momenta @xmath39 can be easily determined from the bethe ansatz equations @xmath40 with @xmath41 . these quasi - momenta lead us immediately to important physical quantities for our system . for example , the energy eigenvalue is given by @xmath42 and the total momentum by @xmath43 . it is clear that these bethe ansatz equations are the same as those in the case of bose gas if we simply make a substitution @xmath44 @xcite . in the regime @xmath27 , there is one - to - one correspondence between the quasi - momentum solution of attractive @xmath0-wave fermi gas and that of repulsive bose gas , but the total wave functions @xmath45 take different forms because of their distinct exchange symmetries . this difference can be easily seen from the one - body density matrix and the momentum distribution . taking the logarithm of bethe ansatz equations , we have @xmath46 for the ground state the set of integer @xmath47 ( @xmath48 ) . for simplicity , in the following evaluation the length @xmath7 will be taken as one . it is obvious that the solutions of @xmath49 is only relevant to @xmath50 for different @xmath7 . the subsequent procedure is standard . by numerically solving the transcendental equations eqs.([bae ] ) , we obtain the quasi - momenta @xmath51 and thus the ground state wave function . in principle , all necessary information about the system can be inferred , including the one - body density matrix , the momentum distribution , and the excitation spectrum . furthermore , one can apply the thermodynamic formalism for dealing with the one dimensional interacting systems developed by yang and yang @xcite . -wave attractive interactions as a function of the dimensionless coordinates @xmath52 and @xmath53 for @xmath54 . ( a ) @xmath55 ; ( b ) @xmath56 ; ( c ) @xmath57 ; ( d ) @xmath58.,width=307 ]    -wave attractive interactions for @xmath54.,width=336 ]    before we proceed to the general case with intermediate interaction strength , we d like to take a look at the situation in the two limiting cases . in the limit of free fermion case , we have @xmath59 ( @xmath60 ) . in this exceptional case of @xmath55 , the quasi - momentum acquires the genuine physical meaning of momentum and the atoms occupy the single particle momentum states according to pauli exclusion principle . we thus have the total wave function @xmath61 with @xmath62 the normalization constant , which is nothing but the slater determinant of the lowest @xmath6 eigenstates @xmath63 of the system of single particle in the hard wall potential . in the other limit of strongly attractive interaction , i.e. , the ftg limit , all quasi - momenta take the same value @xmath64 ( @xmath60 ) and @xmath65 . thus we have the total wave function @xmath66 on the other hand , the fermi - bose mapping method has been used to give exactly the ground state of the ftg gas in @xcite @xmath67 with the antisymmetric function @xmath68sgn@xmath69 . it is obvious that the above two wave functions match each other . we now turn to the system with finite interaction strength @xmath70 . in this case the quasi - momenta are decided by numerically solving the bethe ansatz equations and the total wave function is obtained by eq.([wf ] ) through @xmath71 under the restriction of exchange antisymmetry . for the one dimensional interacting system , a quantity of fundamental importance in many - body physics is the one - body density matrix , which , in terms of the ground state wave function @xmath72 , is given by @xmath73 this quantity furnishes the expectation values of single particle observables such as the position density distribution @xmath74 , and the momentum distribution which is simply the fourier transformation of @xmath75 , @xmath76    we display the one - body density matrix and the position density distribution in fig . 1 and in fig . 2 for different @xmath0-wave attractive interactions . the one - body density matrix expresses the self correlation and it means the probability that two successive measurements , one immediately following the other , will find the particle at the point @xmath52 and @xmath77 , respectively . we notice that for all interacting strengthes there exists a strong enhancement of the diagonal contribution @xmath78 along the line @xmath79 . to see this more clearly the position density distribution @xmath80 for @xmath54 particles is shown in fig . 2 with more variable interacting strengthes . for noninteracting system the atoms behave as ideal fermions and the density profiles show obvious spatial oscillation structure . increasing the strength of the attractive @xmath0-wave interaction , as seen in fig . 2 , first leads to the depression of the amplitude of the oscillations of the density profile , followed by the emergence of the typical gaussian - like bosonic behavior and contraction of the half - width of the density . in the limit of infinitely strong attractive interaction between the fermions , the system enters into the ftg regime , and the density shows the same smooth profile as that of noninteracting bosons . particularly , the density distribution in fig . 2 for ftg is identical to that of the free bose gas and there exists one - to - one correspondence for the density distribution between the attractive fermionic gas and the repulsive bosonic gas with the interacting strength related by @xmath81 @xcite . -wave attractive interactions for @xmath54.,width=336 ]    .,width=336 ]    . ( a ) @xmath82 ; ( b ) @xmath83 ; ( c ) @xmath84 ; ( d ) @xmath85 ; ( e ) @xmath86 ; ( f ) @xmath87 . , width=336 ]    it is worth to emphasize that the density distribution is identical for tg bosons and noninteracting fermions , and also for ftg fermions and noninteracting bosons . nevertheless , their momentum distributions show remarkable differences as shown in fig . 3 for fermions with different attractive interaction strengths and in fig . 4 for bosons with the corresponding repulsive interaction strengths . the momentum distribution for fermions oscillate in the full regime and the number of oscillation peaks remains equal to the number of atoms in the system . it becomes more and more nonuniform in the momentum space and and two sharp spikes indicating that there is high probability of finding the atom in momentum states around @xmath88 . two other peaks at higher momentum diminish with increasing interaction but remain there even in the ftg limit due to the fermionic statistics . the half - width of the profiles become larger as the attractive interaction increases .    as a comparison , the momentum distributions of bosons with repulsive interactions are given in fig . 4 which are obtained by the fourier transformation of the one - body density matrix shown in fig . obviously , the momentum distribution for the bosons exhibits quite different behaviors . as shown in fig . 4 , there is an obvious peak around the zero momentum point and the height of the peak shrinks with the increase of the repulsive interaction . furthermore we find no oscillation at all in the momentum distribution for bosons . even in the tonks limit , the momentum distribution of bosons does not show shell structure like free fermions . the largest probability of distribution appears around the zero point of momentum and decreases rapidly for finite momentum values . stronger repulsive interaction between the bosonic atoms tends to spread out the distribution into higher momentum space . although the momentum distribution for the interacting bosonnic gases has been studied by different numerical methods @xcite , an exact result has never been given except in the tg limit @xcite .    from the eqs . ( [ energy ] ) and ( [ bae ] ) , we see that the energy level structure of our fermionic model is exactly the same as the corresponding bosonic model with interaction strength related by @xmath44 . therefore the thermodynamic properties of the fermionic atoms with attractive @xmath0-wave interactions are the same as the well known thermodynamic properties of the 1d boson gas @xcite with inverse coupling @xmath44 . this implies that there exists a bose - fermi duality between the @xmath0-wave fermionic model and bosonic one with arbitrary interactions which can not be distinguished by the thermodynamic properties . however , due to the different exchange symmetry of the wave functions , the observables associated with the wave functions rather than the square of wave functions ( density distribution ) should display different behaviors . as we have shown , the off - diagonal density matrix and the momentum distributions are different greatly , which result from the different statistics properties followed by bosons and fermions . in summary , we have investigated the ground - state properties of fermionic gases with attractive @xmath0-wave interactions in a one - dimensional hard - wall trap . with the bethe ansatz method , the explicit wave function of the ground state and therefore the one - body density and momentum distributions are obtained . it turns out that the density distributions show one - to - one correspondence between fermions with attractive @xmath0-wave interactions and bosons with repulsive interactions . for weak attractive interaction the density distributions of fermions display shell structures and the boson - like distributions appear as the interactions become stronger . in the ftg limit , the fermi gas exhibits the same distribution as that of the ideal bose gas . this again confirms the bose - fermi duality : strongly interacting bosons behave like fermions , and vice versa . nevertheless , from the viewpoint of momentum distribution , the conclusion is rather different . in the full interacting regime the momentum distributions of fermions show typical oscillations , which is in sharp contrast with the case of bosonic atoms . in the tonks limit of infinite interaction , although the density profiles of one dimensional bosons display the fermion - like distribution , the momentum distribution is still boson - like . is supported by nsf of china under grant no . 10574150 and programs of chinese academy of sciences . y.z . is supported by 973 program under grant no . 2006cb921102 and shanxi province youth science foundation under grant no . 20051001 . the authors would like to acknowledge the hospitality of kitpc in beijing .
we investigate the ground state of the one - dimensional fermionic system enclosed in a hard - wall trap with attractive contact @xmath0-wave interactions . 
 based on the bethe ansatz method , the explicit wave function is derived by numerically solving the bethe ansatz equations for the full physical regimes ( @xmath1 ) . with the exact wave function some quantities which are important in many - body physics 
 are obtained , including the one - body density matrix and the momentum distribution of the ground state for finite system . 
 it is shown that the shell structure of the density profiles disappears with the increase of the interaction and in the fermionic tonks - girardeau ( ftg ) limit the density distribution shows the same behavior as that of an ideal bose gas . 
 however the one - body density matrix and the momentum distribution exhibit completely different structures compared with their bosonic counterparts .