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the gottesmann - knill theorem for continuous variables ( cv ) states that quantum computing based only on components described by one- or two - mode quadratic hamiltonians , gaussian states input , and measurements on canonical variables can be efficiently simulated by a classical computation . that means , although some of the algorithms are of fundamentally quantum nature , they do not provide any speedup over classical processes  @xcite . it has also been proved that construction of a cv universal quantum computer for transformations that are polynomial in those variables requires cubic or higher non - linear operations  @xcite . therefore , investigation of the non - gaussian transformations and states generation is crucial for a nontrivial quantum computation . more recently it has been noted that kerr - like nonlinearities , in a variety of systems , enable high precision quantum metrology that would otherwise require entanglement to achieve it  @xcite .    over the last decade ion trap experiments have led the emerging technologies of coherent quantum control , especially in quantum information theory  @xcite and quantum computation  @xcite . those applications require efficient creation and precise manipulation of both electronic and motional trapped ion state . various theoretical proposals on how to produce nonclassical arbitrary states of ion motion have been discussed . in experiment , fock number states  @xcite , coherent states  @xcite , vacuum squeezed states @xcite , and schrdinger cat states  @xcite have been realized . in this latter case , the state is an entangled state of the vibrational and electronic degrees of freedom . on the contrary , in this paper we give a deterministic way to prepare the vibrational degree of freedom in a non - gaussian state that is not entangled with the electronic states . according to our knowledge , neither non - gaussian state ( other than an entangled cat state ) nor a superposition of more than two coherent states has been observed so far . in the case of photons there is no practical method of non - gaussian state generation so far . efforts have been made to explore a class of the @xmath0 non - gaussian states produced using a photon coherent state @xmath1 interacting with @xmath0 kerr nonlinearity in an optical fiber @xmath2 this class of non - gaussian states , parametrized by the unitless evolution parameter @xmath3  @xcite , is the most popular one . the state ( [ eq : kerr ] ) is known also as the kerr state . in general , this is a highly nonclassical state and after a certain time of evolution @xmath3 in the fiber its wigner function would take negative values in the phase space @xcite . however , the nonlinearity in a fiber , or any other experimentally achievable kerr medium , is too small , @xmath4 , to reach a highly nonlinear regime and thus produce the negativity in an experimentally reasonable time , before it is destroyed by dissipation @xcite . although microstructured fibers seem to be more promising with @xmath5 , their length does not exceed @xmath6 with current technology . the most known examples of the kerr state are the cat states @xmath7 corresponding to @xmath8 , which have been found useful for studies of quantum decoherence and quantum - classical boundary  @xcite . the larger cat states for which the two components @xmath9 and @xmath10 are almost orthogonal ( @xmath11 ) find their application in quantum information processing  @xcite and quantum computation  @xcite . recently , there has been introduced a probabilistic method of non - gaussian state generation relying on a conditional photon subtraction from a squeezed vacuum state  @xcite . such a state is a good approximation for the cat state if the amplitude is small @xmath12 . however , it is neither practical to subtract more than two photons in the experiment nor to produce a state with @xmath13  @xcite . moreover , the state is not pure . another approach is suggested by recent achievements in the the polzik group  @xcite based on their demonstration of teleportation of a quantum state from optical to matter degrees of freedom . if this process was reversed , so that a non - gaussian state is either teleported or mapped from matter to light , optical kerr states might be robustly generated and thus overcome obstacles which are met with in trying to generate optical non - gaussian states directly . of course this requires us to demonstrate a way to make non - gaussian states in matter degrees of freedom . the proposals for transferring an arbitrary motional quantum state of an atom to a cavity field already exist @xcite . the effort to integrate ion traps and optical fields  @xcite might offer a path based on the extraordinary level of coherent quantum control one has over the vibrational degrees of freedom for trapped ions .    in this paper we discuss two efficient and deterministic methods of @xmath0 non - gaussian state generation for light using a single trapped ion . both methods require ion motional state transfer to the cavity mode . the first method is based on the well - known protocol  @xcite for an ion finite motional superposition state generation . it allows to produce the @xmath0 non - gaussian states with arbitrary good approximation . we give criteria , based on wigner function comparison and its measurement precision , which quantify the error resulting from the approximation . the second method is novel and it enables an exact non - gaussian state generation using one laser pulse only . we point out that a wigner function measurement of ion motional state can be performed using currently available technology and already existing experimental schemas . we also suggest a quantum metrology application , based on the work of caves and co workers  @xcite . this paper is organized as follows . in section [ sec1 ] we show that , applying a well - known protocol , one can produce an approximated @xmath0 non - gaussian state of motion for a trapped ion . the method relies on using a series of laser pulses to couple electronic and vibrational degrees of freedom to effect the desired state preparation for the motional degree of freedom . we give the criteria for quantifying the extent to which the prepared state approximates the desired non - gaussian state , and discuss the technical limitations of the method . in section [ sec2 ] we present an alternative , and novel , method for generation of an exact @xmath0 non - gaussian state . we also discuss the range of application of this method . we finish the paper with conclusions and a brief discussion of possible applications to quantum metrology . an ion in a paul trap  @xcite may be prepared in an arbitrary state of the form @xmath14 using the method proposed in @xcite , where @xmath15 are finite superpositions of ion motional states ( @xmath16 ) , @xmath17 is a fock state of a harmonic oscillator potential in the trap , @xmath18 and @xmath19 are the ion electronic ground and excited states respectively . parameters @xmath20 and @xmath21 are complex numbers obeying the normalization constraint @xmath22 and are set at the start of the experiment . the method is based on applying a series of alternating laser pulses tuned , first to the carrier , and then the red sideband , transition of the trapped ion . it works both within and beyond the lamb - dicke regime . the ion is initially prepared in its electronic and vibronic ( motional ) ground state @xmath23 . adjusting the rabi frequencies and duration of each pulse properly , one could achieve @xmath24 equal to @xmath25 these coefficients correspond to the coefficients of a quantum non - gaussian state resulting from the unitary evolution of a self - kerr interaction ( [ eq : kerr ] ) decomposed in the fock basis , up to the normalization factor ( which results from the fact that we cut off the infinite sum in ( [ eq : kerr ] ) and take into account only the first @xmath26 terms ) . this method produces an approximated @xmath0 non - gaussian state @xmath27 corresponding to the state reached via a kerr interaction with an arbitrary value of evolution parameter @xmath3 . furthermore the effective value of @xmath3 can be much larger than can be achieved via unitary interaction under a realistic optical kerr interaction were the bosonic degree of freedom an optical field mode . a proper choice of the cut - off value @xmath26 enables one to approximate the state ( [ eq : kerr ] ) arbitrarily closely . if we set @xmath28 for simplicity , the ion state is given by @xmath29 the number of required pulses for preparing the state ( [ eq : approx_kerr ] ) is equal to @xmath30 latexmath:[\[|\psi^{(m)}(\alpha,\tau)\rangle = r_m c_{m-1 } \cdot ... \cdot r_1 c_0    carrier and a red sideband laser pulse respectively . therefore @xmath26 should be as small as possible @xcite . the hint for existence of a good non - gaussian state approximation comes from a simple observation that only a finite amount of coefficients @xmath32 contribute to the sum ( [ eq : kerr ] ) significantly and the exact number of them is strongly @xmath33 dependent . the coefficients , evaluated for exemplary values of the amplitude , @xmath34 and @xmath35 , are depicted on fig . [ fig : significant_coeff_a1 ] .    for a given value of an amplitude @xmath33 the choice of the cut - off value @xmath26 is based on three criteria . all criteria rely on comparing the wigner function @xmath36 of an approximated state ( [ eq : approx_kerr ] ) with the untruncated one @xmath37 evaluated for ( [ eq : kerr ] ) . please note that the comparison is made for @xmath38 . we chose this particular value of the evolution parameter because for this value the original wigner function is given by a simple analytic formula , which can be computed directly . the first criterion derives from an investigation of the approximated wigner function isolines . the smaller the value of @xmath26 the more the function deviates from an ideal gaussian function for @xmath38 : the isolines are no longer circles . the isolines of interest can be chosen arbitrarily . we however chose @xmath39 , @xmath40 , @xmath41 and calculate , for each isoline , the ratio between the most and the least distant point with respect to the point @xmath42 separately . for an ideal circle the ratio is always equal to @xmath43 .    in order to take into account the interference resulting from the approximate state , which takes the values around zero , we calculate the maximal and average ratio of the difference @xmath44 to the value of the ideal wigner function , for all points @xmath45 in a phase space . therefore , the second criterion gives the maximal and average relative error of the approximation respectively . the third criterion reveals the percentage of points @xmath45 in a phase space for which @xmath46 for a given precision . we assume a precision of order of either @xmath47 or @xmath48 to be good enough , since it relates to the accuracy of both the wigner function computer visualisation using the density plots and the wigner function reconstruction using quantum tomography  @xcite . the above criteria turn out to be more subtle than the well - known fidelity @xmath49 which measures the overlap of the approximated and the original state in the phase space @xmath50 obviously , the fidelity approaches unity in the limit of @xmath51 . we would like to point out that having the ion already prepared in state ( [ eq : approx_kerr ] ) one could measure the wigner function of its vibronic state using either the standard method of quantum tomography @xcite or the method of direct measurement developed in  @xcite . this would allow for detailed investigation of @xmath0 non - gaussian states , which has never been verified experimentally so far . the direct wigner function measurement method is especially interesting . it relies on the fact that the wigner function of a displaced vibronic state of an ion is related to the probability of finding the ion in the ground and the excited state . the probabilities are measured by detecting a fluorescence signal . such a measurement has been done for a light in a cavity with accuracy of @xmath52  @xcite . estimation of the minimal number of laser pulses for @xmath53 generation , which approximates the original state @xmath54 for any value of the evolution parameter @xmath3 well , requires comparison of the approximated wigner functions @xmath36 evaluated for a few different cut - off values @xmath26 , with the original wigner function @xmath37 . the selection of the best possible cut - off values is based on analysis of significant coefficients @xmath32 . for @xmath34 only the coefficients ranging from @xmath55 to @xmath56 are greater than @xmath57 . the coefficients @xmath58 to @xmath59 are of order of @xmath60 . therefore , we test @xmath61 . we next investigate the isolines . table [ tabela_a2 ] shows the ratios of the most to the least distant point , with respect to the point of @xmath62 for the isolines ( @xmath39 , @xmath40 , @xmath41 ) of the approximated wigner functions @xmath36 . the minimal value of @xmath26 for which the ratios evaluated for @xmath36 are equal to the ratios evaluated for the original gauss function using the grid ( discretized phase space ) with a step equal to @xmath63 is equal to @xmath64 . it means that for the given grid and greater values of @xmath26 , the numerical simulations of the wigner function will not differ . therefore , the value of @xmath65 could be regarded as an appropriate cut - off value .    [ cols="^,^,^,^,^,^,^,^,^",options="header " , ]     these quantities were evaluated for the following rabi frequencies @xmath66 , @xmath67 corresponding to carrier transition and red sideband respectively , and lamb - dicke parameter @xmath68 . the total time of state build - up is equal to @xmath69 . the wigner function of the approximated compass state for @xmath70 is is presented on fig . [ fig : wigner_a2_t0.5pi_comparem ] . we also present the wigner functions for @xmath71 and the original one for @xmath72 for comparison . +   +   +      at the end of this section we estimate the maximum value of the amplitude @xmath33 for which the method works . the maximum value seems to be @xmath73 . this limitation does not result from the decoherence of the ion state . both the time of single pulse duration and the total duration of state built - up remain within the coherence of the ion for the vibronic state @xmath74 . the coherence time for the electronic state is equal to @xmath75 . the number of laser pulses is limited due to the finite binding energy in the trap . the cut - off value @xmath26 corresponds to the maximal ion excitation @xmath76 . the @xmath77 is the upper limit for trapping the ion so far  @xcite . the one laser pulse method of @xmath0 non - gaussian state generation requires one carrier resonance pulse applied to the ion cooled down to a lamb - dicke regime . the vibronic state of the ion is initially prepared in a coherent state @xmath1 . the advantage of this method over the first one is that it allows for generation of the original kerr state ( [ eq : kerr ] ) without any approximations . however , not all values of the evolution parameter are accessible : the pulse duration is limited due to the ion decoherence . the interaction between an ion and a laser pulse of rabi frequency @xmath78 is governed by the following hamiltonian @xmath79 where @xmath80 is an electronic state rising operator , @xmath81 is a vibronic state annihilation operator , @xmath82 is a trapping frequency , and @xmath83 is a lamb - dicke parameter . using the expansion of the exponens function into a taylor series @xmath84 in the rotating wave approximation and in an interaction picture the hamiltonian is approximated by the first five terms of the expansion @xmath85 the terms in the first line in the above formula govern the free ion evolution . the last term in ( [ rozwiniecie ] ) we associate with a self - kerr interaction hamiltonian , up to the electronic state operators . therefore , the unitary evolution operator resulting from the nonlinear part of the hamiltonian is given by @xmath86 it allows for reading out the effective value of the evolution parameter @xmath87 in the further discussion we neglect the electronic state evolution since it may be made to take no part in the dynamics . to do this one first needs to prepare the electronic state in an eigenstate of @xmath88 using a @xmath89 pulse tuned to the carrier transition , but the pulse must not excite the vibrational degree of freedom in any way . this can be done by making the pulse propagate orthogonal to the vibrational axis . please note that by way of contrast to the method described in the previous section this method is independent of the initial amplitude @xmath33 . the laser pulse duration required for the kerr state generation is the same for all values of the amplitude . in other words , the time required for generation e.g. @xmath90 and @xmath91 is the same . however , accessibility of the evolution parameter @xmath92 is limited instead . the greater rabi frequency @xmath78 and lamb - dicke parameter value are the shorter pulse duration @xmath93 must be to obtain @xmath92 . on the other hand , the ion has to remain within the lamb - dicke regime in order to hold the expansion ( [ rozwiniecie ] ) true . figure [ fig : one_pulse_duration ] shows the time of the pulse duration assuming its rabi frequency @xmath94 and the lamb - dicke parameters @xmath95 , @xmath96 , @xmath68 , and @xmath97 . for the first two lamb - dicke parameters it is less than @xmath98 , which is within the coherence time of vibronic ion state ( @xmath99 ) , to obtain @xmath100 and therefore produce the cat state . for the two remaining @xmath83 values the duration of the pulse is of order of tenth of second . the duration of the pulse required for @xmath101 generation is equal to @xmath102 for @xmath95 and @xmath103 for @xmath96 ( @xmath104 for @xmath68 and @xmath105 for @xmath97 ) . for @xmath106 the first negativities in the wigner function become visible  @xcite . this method allows for obtaining also the other coherent state superpositions easily . for example , for @xmath107 one achieves superposition of six states and if @xmath108 one achieves superposition of four states ( the compass state ) @xcite . for @xmath96 and @xmath109 the pulse duration is equal to @xmath110 , and for @xmath111 it is equal to @xmath112 . these values are experimentally feasible . +    the one laser pulse method of @xmath0 non - gaussian state generation is limited due to introduced cut - off in the expansion ( [ eq : rozwiniecie ] ) . it is valid only if we take the expansion for a small parameter . to estimate the small parameter we approximate the quantum operators by the classical amplitudes @xmath113 @xmath114 and read out the small parameter in the expansion to be @xmath115 . the above series is decreasing if @xmath116 . this constraint is enough to ensure that the contribution from the higher order terms is negligible . for example , for @xmath35 and @xmath95 , obeying @xmath117 ( @xmath118 and @xmath119 ) the coefficients @xmath120 are equal to : @xmath41 ( @xmath121 ) for @xmath122 , @xmath123 ( @xmath124 ) for @xmath125 , @xmath126 ( @xmath127 ) for @xmath128 . for coefficients obeying @xmath119 the sixth order coefficient is of two orders of magnitude smaller than the forth order coefficient . we now show how the cat - state weak force detection protocol proposed in munro et al @xcite can be implemented in an ion trap given the ability to engineer an approximate cat state as we have described . it must be admitted that there is no compelling case to use the vibrational motion of a trapped ion as a weak force detector . however , the method we propose here would be a nice demonstration of how non - gaussian states can beat the standard quantum limit for weak force detection . suppose one was able to prepare the vibrational degree of freedom in the non - gaussian state @xmath129 with the amplitude @xmath33 real . when a weak force acts it can be described by the action of the unitary displacement operator @xmath130 acting on the initial state @xmath131 to give the output state @xmath132 . using the result that @xmath133 we find that for @xmath134 , and @xmath33 real that @xmath135 where the even and odd parity states are given by @xmath136 in other words , the weak force is well approximated by a rotation in the two dimensional parity subspace . as in @xcite it then follows that the minimum detectable force is then given by @xmath137 which beats the standard quantum limit by a factor of @xmath138 . ion trap provide a simple way to reach this lower bound . suppose that after the weak force has acted , the total vibrational and electronic state is given by eq . ( [ roatated ] ) . the first step is to make a @xmath89 rotation of the electronic state , followed by the conditional rotation @xmath139 which can easily be done @xcite . finally another @xmath89 rotation gives the state @xmath140 the electronic state can now be readout , and the probability to find the ion in , say , the excited state is @xmath141 sampling this distribution gives the estimation for the force with a minimum detectable force that is inversely proportional to @xmath33 . for example , the minimum force required to shift the interference distribution by one fringe is @xmath142 . in the paper we have presented two experimentally feasible methods of @xmath0 non - gaussian states generation for light using a trapped ion . the first method is based on a well - known protocol ( a series of laser pulses ) and allows for generation of an arbitrarily well approximated non - gaussian state . the approximation is quantified by three criteria based on wigner function analysis . this method is only limited by technical parameters of the trap : the binding energy . the second method enables an exact non - gaussian state generation using one laser pulse . it is limited by the decoherence time of the motional ion state . however , adjusting the laser pulse and trap parameters properly one is able to produce a cat state . based on the proposed protocols @xcite we believe that non - gaussian state transfer from ion motion to a light beam is possible in a foreseeable future . it will enable application of a nontrivial quantum computing protocols based for example on coherent states . this is also a step towards exploring so far unknown branch of quantum optics such as non - gaussian states . m. s. acknowledges support by the alexander von humboldt foundation and by men grant no . n202 021 32/0700 . m. s. also thanks i. cirac for the discussion . m. keller , b. lange , k. hayasaka , w. lange , and h. walther , appl . b * 76 * , 125 ( 2003 ) ; a. b. mundt , a. kreuter , c. russo , c. becher , d. leibfried , j. eschner , f. schmidt - kaler , and r. blatt , appl . phys . b * 76 * , 117 ( 2003 ) .
according to the gottesmann - knill theorem the non - gaussian states are necessary component for a nontrivial quantum computation . 
 we show two efficient and deterministic methods of @xmath0 non - gaussian state generation for a cavity mode using a single trapped ion . both require ion motional state transfer to the cavity field . 
 the methods are experimentally feasible . 
 the first is based on the well - known protocol for an ion finite motional superposition state generation . 
 it allows for an arbitrary good approximation of @xmath0 non - gaussian states . 
 we give criteria based on the wigner function which quantify the error resulting from the approximation . 
 the second and novel method enables an exact non - gaussian state generation using one laser pulse only .