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hep-ph-0405286 | {'hep-ph-0405286-1-0-0': 'In the context of the brane-world scenarios with compactified extra dimensions, we study the production of brane fluctuations (branons) in hadron colliders ([MATH], [MATH] and [MATH]) in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-1-0-1': 'From the absence of monojets events at HERA and Tevatron (run I), we set bounds on these parameters and we also study how such bounds could be improved at Tevatron (run II) and the future LHC.', 'hep-ph-0405286-1-0-2': 'The single photon channel is also analyzed for the two last colliders.', 'hep-ph-0405286-1-1-0': 'PACS: 11.25Mj, 11.10Lm, 11.15Ex', 'hep-ph-0405286-1-2-0': 'plain', 'hep-ph-0405286-1-3-0': '20true cm', 'hep-ph-0405286-1-4-0': '# Introduction', 'hep-ph-0405286-1-5-0': 'Since rigid objects do not exist in relativistic theories, it is clear that brane fluctuations must play a role in the so called brane world, proposed some years ago by Arkani-Hamed, Dimopoulos and Dvali (ADD scenarios [CITATION]), where the Standard Model (SM) particles are confined to live in the world brane and only gravitons are free to move along the [MATH] dimensional bulk space (see [CITATION] for recent reviews).', 'hep-ph-0405286-1-5-1': 'This fact turns out to be particularly true when the brane tension scale [MATH] being the brane tension) is much smaller than the [MATH] dimensional or fundamental gravitational scale [MATH], i.e., [MATH].', 'hep-ph-0405286-1-5-2': 'In this case the only relevant low-energy modes of the ADD scenarios are the SM particles and branons which are the quantized brane oscillations.', 'hep-ph-0405286-1-5-3': 'Branons can be understood as the (pseudo)Goldstone bosons corresponding to the spontaneous breaking of translational invariance in the bulk space produced by the presence of the brane.', 'hep-ph-0405286-1-5-4': 'It has been pointed out that branons could solve some of the problems of the brane-world scenarios such as the divergent virtual contributions from the Kaluza-Klein tower at the tree level or non-unitarity of the graviton production cross-sections [CITATION].', 'hep-ph-0405286-1-5-5': 'As Goldstone bosons, branons are in principle massless, but in the cases where the metric of the extra dimensions cannot be factorized, they can become massive [CITATION].', 'hep-ph-0405286-1-5-6': 'This is similar to the case of pions which, being the Goldstone bosons of the spontaneous breaking of chiral symmetry, acquire some mass due to the explicit breaking of the symmetry induced by the quark masses.', 'hep-ph-0405286-1-6-0': 'In previous works the different effective actions have been obtained, namely: the effective action for the SM fields on the brane, that for the branon self-interactions and finally that corresponding to the interaction between SM fields and branons [CITATION].', 'hep-ph-0405286-1-6-1': 'In general, this branon effective action can be parameterized by the number of branons [MATH], the tension scale [MATH] and the branon masses (for an explicit construction see [CITATION]).', 'hep-ph-0405286-1-6-2': 'Using the effective action it is possible to obtain the different Feynman rules, the amplitudes and finally the cross-sections for branon production from SM particles.', 'hep-ph-0405286-1-6-3': 'In [CITATION] the case of electron-positron colliders has been considered.', 'hep-ph-0405286-1-6-4': 'By using the Large Electron-Positron Collider (LEP) data it is possible to set important bounds on the tension scale and on the branon mass for a given branon number.', 'hep-ph-0405286-1-6-5': 'Other restrictions have also been set from astrophysical and cosmological considerations due to the fact that branon dark matter can present relevant abundances [CITATION].', 'hep-ph-0405286-1-7-0': 'In this work we study branon production in hadron colliders and also in electron-proton colliders such as HERA.', 'hep-ph-0405286-1-7-1': 'Most of these cross sections have been studied by Creminelli and Strumia for the massless branon case [CITATION].', 'hep-ph-0405286-1-7-2': 'We reproduce their results and extend the analysis for an arbitrary branon mass.', 'hep-ph-0405286-1-7-3': 'The paper is organized as follows: In Sec.II we shortly review the branon effective action.', 'hep-ph-0405286-1-7-4': 'In Sec.III we consider the case of proton-(anti)proton colliders like Tevatron or the future Large Hadron Collider (LHC).', 'hep-ph-0405286-1-7-5': 'In Sec.IV electron(positron)-proton colliders like HERA are studied.', 'hep-ph-0405286-1-7-6': 'In Sec.V we show the main results for the relevant examples and in Sec.VI we set the conclusions.', 'hep-ph-0405286-1-8-0': '# Effective action', 'hep-ph-0405286-1-9-0': 'The relevant effective action describing the low-energy interactions of SM particles and branons was derived in [CITATION], where the necessary vertices are detailed.', 'hep-ph-0405286-1-9-1': 'The branon effective action can be expanded according to the number of branon fields appearing in each term: [EQUATION] where the zeroth-order term is just a constant and the second-order is just the free action: [EQUATION] with [MATH] the branon fields where [MATH] and [MATH] is the squared mass matrix which, without loss of generality, can be assumed to be diagonal.', 'hep-ph-0405286-1-9-2': 'The effective action for the SM particles and their interactions with branons is given by [EQUATION] where [MATH] is the SM Lagrangian and [MATH] is the SM energy-momentum tensor defined as: [EQUATION] where [MATH] is some arbitrary metric on the world brane and [MATH] is the Minkowski metric.', 'hep-ph-0405286-1-10-0': 'In this work we are interested in the interactions between quarks and gluons or photons.', 'hep-ph-0405286-1-10-1': 'Thus, for Dirac fermions with masses [MATH] belonging to some representation of a gauge group, such as [MATH] or [MATH], with generators [MATH], the Lagrangian is [EQUATION] where the covariant derivative is defined as [MATH], [MATH] being the appropriate gauge coupling.', 'hep-ph-0405286-1-10-2': 'Thus the energy-momentum tensor is given by [EQUATION] from where it is possible to find vertices such as [MATH] and [MATH].', 'hep-ph-0405286-1-10-3': 'For gauge fields [MATH] the appropriate Lagrangian for perturbation theory is: [EQUATION] where as usual [MATH] and [MATH] is the Fadeev-Popov Lagrangian including the gauge fixing and the ghost terms.', 'hep-ph-0405286-1-10-4': 'The energy-momentum tensor is: [EQUATION] from where we can obtain the [MATH], [MATH] and [MATH] vertices.', 'hep-ph-0405286-1-11-0': 'Therefore, by using these energy-momentum tensors and the effective action above, it is possible to obtain the different Feynman rules involving branons.', 'hep-ph-0405286-1-11-1': 'One important observation is that in all the vertices obtained above, branons appear always by pairs.', 'hep-ph-0405286-1-11-2': 'In fact they interact in a way similar to gravitons since they couple to the energy momentum tensor.', 'hep-ph-0405286-1-11-3': 'This can be seen by making the formal identification of the graviton field [MATH] which appear in linearized gravity with [EQUATION] where [MATH] and [MATH] the Planck mass.', 'hep-ph-0405286-1-11-4': 'Of course the physical meaning is completely different for branons and gravitons.', 'hep-ph-0405286-1-11-5': 'In any case branons are expected to be weakly interacting and then they will scape to detection.', 'hep-ph-0405286-1-11-6': 'Hence their typical signature will be missing energy and momentum with a continuum spectrum.', 'hep-ph-0405286-1-11-7': 'In the following sections we will study the production mechanisms relevant for the different hadronic colliders.', 'hep-ph-0405286-1-12-0': '# Proton-(anti)proton colliders', 'hep-ph-0405286-1-13-0': 'For the case of proton-antiproton colliders like Tevatron, the most important processes for branon production are quark-antiquark annihilation or gluon fusion giving a gluon and a branon pair; and (anti)quark-gluon interaction giving an (anti)quark and a branon pair.', 'hep-ph-0405286-1-13-1': 'Therefore the expected experimental signal will be in both cases one monojet [MATH] and missing energy and momentum.', 'hep-ph-0405286-1-13-2': 'This is a very clear signature that in principle can be easily identified.', 'hep-ph-0405286-1-13-3': 'Another potentially interesting process is the quark-antiquark annihilation giving a photon and a branon pair.', 'hep-ph-0405286-1-13-4': 'In this case the signature is one single photon and missing energy and momentum.', 'hep-ph-0405286-1-14-0': 'The Feynman diagrams contributing to the main subprocesses [MATH], [MATH], [MATH] and [MATH] are shown in Fig. [REF], Fig. [REF] and Fig. [REF].', 'hep-ph-0405286-1-15-0': 'From these diagrams and the Feynman rules coming from the effective action of the previous section, it is possible to obtain the differential cross section: [EQUATION] where [MATH], [MATH], [MATH] and [MATH], [MATH] and [MATH] being the anti-quark and quark four-momenta respectively, [MATH] the gluon four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-1-15-1': 'We have assumed for the sake of simplicity that all the branons are degenerated with a common mass [MATH] and that all the quarks are massless.', 'hep-ph-0405286-1-15-2': 'We have also neglected the effects of the top quark.', 'hep-ph-0405286-1-15-3': 'In addition we have the well-known relation [MATH].', 'hep-ph-0405286-1-15-4': 'The contribution to the total cross section of the process [MATH] coming from this subprocess is given by [EQUATION] where [MATH] and [MATH] are the distribution functions of the anti-quark inside the antiproton and of the quark of flavor [MATH] inside the proton respectively, and [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and antiquark.', 'hep-ph-0405286-1-15-5': 'The different limits of the integrals can be written in terms of the cuts used to define the total cross-section.', 'hep-ph-0405286-1-15-6': 'For example, in order to be able to detect clearly the monojet one must impose a minimal value for its transverse energy [MATH] and a pseudorapidity range given by [MATH] and [MATH].', 'hep-ph-0405286-1-15-7': 'Then we have the limits [MATH], [MATH] and [MATH].', 'hep-ph-0405286-1-15-8': 'On the other hand [MATH] and [MATH] where [MATH] is the total center of mass energy squared of the process and [EQUATION]', 'hep-ph-0405286-1-15-9': 'In addition the dots in ([REF]) represent the contribution of the case in which the quark comes from the antiproton and the anti-quark comes from the proton.', 'hep-ph-0405286-1-16-0': 'The cross-section of the subprocess [MATH] is given by [EQUATION] where the Mandelstan variables are defined as in the previous case, with [MATH] and [MATH] being the initial gluon four-momenta, [MATH] the final gluon four-momentum and [MATH] the total branon four-momentum.', 'hep-ph-0405286-1-16-1': 'Then the contribution to the total cross section from the [MATH] reaction is [EQUATION]', 'hep-ph-0405286-1-16-2': 'Here [MATH] is the gluon distribution function of the (anti)proton, [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the initial gluons and the integration limits remain the same.', 'hep-ph-0405286-1-16-3': 'From the above equations, it is possible to compute the total cross-section [MATH] in terms of the cut in the gluon (monojet) transverse energy [MATH].', 'hep-ph-0405286-1-17-0': 'For the [MATH] process the cross-section is given by [EQUATION] with [MATH] and [MATH] being the quark and the gluon four-momenta respectively, [MATH] the final state quark four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-1-17-1': 'The Mandelstam variables are defined as in previous cases.', 'hep-ph-0405286-1-17-2': 'The cross-section for the conjugate process [MATH] is exactly the same.', 'hep-ph-0405286-1-17-3': 'Then the total cross section for the the reaction [MATH] is [EQUATION]', 'hep-ph-0405286-1-17-4': 'In this equation [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and gluon.', 'hep-ph-0405286-1-17-5': 'The different integration limits are defined as in the previous case in terms of the minimal transverse energy of the quark (monojet) [MATH] and the dots refer to the case where the initial gluon is coming from the proton and the quark is coming from the antiproton.', 'hep-ph-0405286-1-17-6': 'In addition we have the contribution from the conjugate case where we take an anti-quark from the proton and a gluon from the antiproton and conversely.', 'hep-ph-0405286-1-17-7': 'This amount just to a factor of two.', 'hep-ph-0405286-1-18-0': 'From all the above equations it is possible to compute the total cross-section [MATH] in terms of the cut in the jet transverse energy [MATH].', 'hep-ph-0405286-1-19-0': 'For the subprocess [MATH] the cross-section is given by [EQUATION]', 'hep-ph-0405286-1-19-1': 'Here the notation is similar to the [MATH] case with the obvious differences in couplings, color and charge factors.', 'hep-ph-0405286-1-19-2': 'Thus [EQUATION]', 'hep-ph-0405286-1-19-3': 'All the previous discussion about branon production in [MATH] reactions can be easily translated to the [MATH] case.', 'hep-ph-0405286-1-19-4': 'The only point is to change the antiproton distribution functions of the different partons by the corresponding proton ones.', 'hep-ph-0405286-1-20-0': '# Electron(positron)-proton colliders', 'hep-ph-0405286-1-21-0': 'For electron(positron)-proton colliders like HERA, the most interesting branon creating process is branon photoproduction, where a photon emitted by the electron(positron) interacts with a quark(antiquark) from the proton giving a quark (antiquark) and a branon pair.', 'hep-ph-0405286-1-21-1': 'Thus the experimental signature is again one monojet [MATH] plus missing energy and momentum.', 'hep-ph-0405286-1-21-2': 'The relevant Feynman diagrams are shown in Fig. [REF] and the corresponding differential cross-section for the subprocess [MATH] is [EQUATION] where [MATH] and [MATH], [MATH] being the photon, [MATH] the proton quark, [MATH] the final quark and [MATH] the total branon momenta respectively.', 'hep-ph-0405286-1-21-3': 'The total cross-section for the process [MATH] is given by [EQUATION] [MATH] and [MATH] are defined in this case as [MATH] and [MATH] with [MATH] and [MATH] being the proton and electron(positron) momenta respectively.', 'hep-ph-0405286-1-21-4': 'Thus at high energies compared with the proton mass [MATH] where [MATH].', 'hep-ph-0405286-1-21-5': 'The integral limits [MATH], [MATH], [MATH] and [MATH] are defined like in the proton-(anti)proton collider case.', 'hep-ph-0405286-1-22-0': 'The photon spectrum [MATH] can be obtained from the well-known Weizs[MATH]cker-Williams approximation [CITATION]: [EQUATION] with [MATH] and [MATH] being the electron mass.', 'hep-ph-0405286-1-23-0': 'The cross-section [MATH] can be obtained in a similar way.', 'hep-ph-0405286-1-23-1': 'Then the total contribution to monojet plus missing energy and momentum production for large enough [MATH] coming from branons can be written as the sum of [MATH] and [MATH].', 'hep-ph-0405286-1-24-0': '# Results', 'hep-ph-0405286-1-25-0': 'By using the cross-sections shown in the previous sections it is possible to compute the expected number of branon pairs produced in the different hadron colliders in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-1-25-1': 'To this end we have used the distribution functions which can be found in [CITATION].', 'hep-ph-0405286-1-25-2': 'The values of the electromagnetic and strong couplings have been taken at the electroweak boson masses, namely [MATH] and [MATH].', 'hep-ph-0405286-1-25-3': 'However our final results do not depend too much on the precise value of these couplings.', 'hep-ph-0405286-1-25-4': 'In fact our main source of error is the use of an effective action to describe the SM particles and branon interactions since, in principle, this is only guaranteed for energies well below [MATH].', 'hep-ph-0405286-1-26-0': 'As discussed in the introduction, our main goal in this work is to study the bounds that can be set on the [MATH], [MATH] and [MATH] parameters coming from hadron colliders.', 'hep-ph-0405286-1-26-1': 'We will present all our limits at the 95% confidence level.', 'hep-ph-0405286-1-26-2': 'In particular, for the electron(positron)-proton case, HERA is the most relevant experiment.', 'hep-ph-0405286-1-26-3': 'In fact, the ZEUS collaboration has studied the jet production in charged current deep inelastic [MATH] scattering.', 'hep-ph-0405286-1-26-4': 'Its results are perfectly compatible with the SM background and therefore, we can set some bounds on the branon production and hence on the [MATH], [MATH] and [MATH] parameters.', 'hep-ph-0405286-1-26-5': 'These data were taken from 1995 to 2000 at a maximum CM energy of [MATH] GeV.', 'hep-ph-0405286-1-26-6': 'The total integrated luminosity was [MATH] pb[MATH] and the cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for more details).', 'hep-ph-0405286-1-26-7': 'By using the same cuts with our cross-sections for monojet plus a branon pair production, we find the bound [MATH] GeV for massless branons.', 'hep-ph-0405286-1-26-8': 'For a branon mass larger than [MATH] GeV there is no restrictions on the [MATH] value because of kinematical reasons.', 'hep-ph-0405286-1-26-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] where we have assumed [MATH].', 'hep-ph-0405286-1-26-10': 'For other [MATH] values one just has to take into account that the bound scales like [MATH] since all the cross-sections are proportional to [MATH].', 'hep-ph-0405286-1-27-0': 'In the [MATH] case the most relevant experimental information so far is the one obtained at the Tevatron (Run I).', 'hep-ph-0405286-1-27-1': 'The [MATH] detector has studied the monojet channel and CDF the single photon one.', 'hep-ph-0405286-1-27-2': 'As far as the number of events found in both cases is compatible with the SM background, we can set new bounds on the branon theory parameters.', 'hep-ph-0405286-1-27-3': 'For light branons the most important bound comes from the [MATH] data.', 'hep-ph-0405286-1-27-4': 'These data were taken from 1994 to 1996 at a CM energy of [MATH] TeV and correspond to an integrated luminosity [MATH] pb[MATH].', 'hep-ph-0405286-1-27-5': 'The cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for the details of the analysis).', 'hep-ph-0405286-1-27-6': 'The total number of monojets observed was [MATH] and the expected number from the SM plus cosmic rays events was [MATH].', 'hep-ph-0405286-1-27-7': 'By using our cross sections for monojet plus a branon pair production with these cuts we get the bound [MATH] GeV for light branons.', 'hep-ph-0405286-1-27-8': 'The restrictions for [MATH] improve up to a branon mass of [MATH] GeV.', 'hep-ph-0405286-1-27-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] for [MATH].', 'hep-ph-0405286-1-28-0': 'In a similar way we can use the CDF data on single photon production.', 'hep-ph-0405286-1-28-1': 'In this case the total luminosity collected was [MATH] pb[MATH] and the pseudorapidity cut was [MATH].', 'hep-ph-0405286-1-28-2': 'For the transverse photon energy several cuts were considered (for example 55 GeV at the [MATH] efficiency).', 'hep-ph-0405286-1-28-3': 'The total expected background for this process was [MATH], without taking into account the QCD contribution (see [CITATION] for the details of the analysis), and the number of events found was [MATH].', 'hep-ph-0405286-1-28-4': 'Comparing this result with our computations for photon plus one branon pair production, we find the bound [MATH] GeV for massless branons and no bound for [MATH] larger than [MATH] GeV.', 'hep-ph-0405286-1-28-5': 'The bound obtained for the rest of the cases is shown also in Fig. [REF].', 'hep-ph-0405286-1-29-0': 'In addition to this analysis corresponding to the Tevatron data (Run I), it is also interesting to make some estimation about the bounds that could be set from future experiments such as Tevatron (Run II) and the LHC.', 'hep-ph-0405286-1-29-1': 'In the case of the Tevatron (Run II), which is already in progress, the main novelties are a CM energy which equals [MATH] TeV and an expected integrated luminosity [MATH] at the end of the run of about [MATH] pb[MATH].', 'hep-ph-0405286-1-29-2': 'The detectors are also improved so that the pseudorapidity cuts can be taken as [MATH] for [MATH] and [MATH] for CDF.', 'hep-ph-0405286-1-29-3': 'This would result in a factor of [MATH] on the statistical significance when compared to the Run I, with integrated luminosity [MATH], provided that the CM energy and the cuts were the same.', 'hep-ph-0405286-1-29-4': 'For massless branons, the bound on [MATH] scales as the CM energy [MATH].', 'hep-ph-0405286-1-29-5': 'Even more important is the possibility of exploring higher branon masses, since the kinematical limit is given by [MATH].', 'hep-ph-0405286-1-29-6': 'In Fig. [REF] we show the expected bounds from the Run II in the [MATH] plane, again for [MATH].', 'hep-ph-0405286-1-30-0': 'The LHC will produce [MATH] collisions at a CM energy of [MATH] TeV and the integrated luminosity will be something about [MATH] pb[MATH].', 'hep-ph-0405286-1-30-1': 'In order to estimate the bounds on the [MATH], [MATH] and [MATH] parameters that will be possible to obtain at the LHC, we have proceeded in a similar way as in the Tevatron case, with the obvious changes in the distribution functions due to the fact that now we are dealing with [MATH] instead of [MATH] collisions.', 'hep-ph-0405286-1-30-2': 'We have kept the same cuts except for the transverse energy which has been corrected in order to maintain the same proportion relative to the CM energy.', 'hep-ph-0405286-1-30-3': 'Again the best bounds for [MATH] come from monojet production, which for [MATH] turns out to be [MATH] GeV.', 'hep-ph-0405286-1-30-4': 'For low [MATH] the best bound for [MATH] is given by the single photon channel ([MATH]).', 'hep-ph-0405286-1-30-5': 'The LHC sensitivity for other values in the [MATH] plane can be found in Fig. [REF] for [MATH].', 'hep-ph-0405286-1-31-0': '# Conclusions', 'hep-ph-0405286-1-32-0': 'In this work we have studied the flexible brane-world scenario, where the brane tension scale [MATH] is much smaller than the fundamental [MATH]-dimensional gravitational scale [MATH].', 'hep-ph-0405286-1-32-1': 'In this case, the relevant low-energy degrees of freedom are the SM particles and the brane fluctuations or branons.', 'hep-ph-0405286-1-32-2': 'From the corresponding effective action, we have calculated the relevant cross-sections for different branon searches in hadronic colliders.', 'hep-ph-0405286-1-32-3': 'We have used the information coming from HERA and the first Tevatron Run in order to get different exclusion plots on the branon mass [MATH] and the tension scale [MATH] plane for a given branon number [MATH].', 'hep-ph-0405286-1-32-4': 'Monojet production turns out to be the most efficient process for light branons, whereas the single photon channel is the most important one for heavy branons.', 'hep-ph-0405286-1-33-0': 'We have also extended the analysis to future hadronic colliders.', 'hep-ph-0405286-1-33-1': 'The corresponding sensitivity regions for the second Tevatron run and the LHC have also been plotted (see Table [REF] for a summary of the analysis).', 'hep-ph-0405286-1-34-0': 'These analysis improve those already done for electron-positron colliders for heavy branons, whereas for light branons, the results are similar [CITATION].', 'hep-ph-0405286-1-34-1': 'The Tevatron (run I) limit [MATH] GeV can be compared to the analogous limit from LEP II [MATH] GeV [CITATION].', 'hep-ph-0405286-1-34-2': 'On the other hand, LHC could detect branons up to a mass of several TeV ([MATH] GeV) improving even the CLIC prospects ([MATH] GeV) [CITATION].', 'hep-ph-0405286-1-35-0': 'The study of branons in colliders can be complemented with other bounds coming from astrophysics and cosmology.', 'hep-ph-0405286-1-35-1': 'In fact, as shown in [CITATION], the branon relic abundance can have cosmological consequences.', 'hep-ph-0405286-1-35-2': 'Other issues related to branon phenomenology, such as their radiative corrections to the SM processes, or their distinctive signatures at colliders with respect to the KK gravitons will be analyzed elsewhere.'} | {'hep-ph-0405286-2-0-0': 'In the context of the brane-world scenarios with compactified extra dimensions, we study the production of brane fluctuations (branons) in hadron colliders ([MATH], [MATH] and [MATH]) in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-2-0-1': 'From the absence of monojets events at HERA and Tevatron (run I), we set bounds on these parameters and we also study how such bounds could be improved at Tevatron (run II) and the future LHC.', 'hep-ph-0405286-2-0-2': 'The single photon channel is also analyzed for the two last colliders.', 'hep-ph-0405286-2-1-0': 'PACS: 11.25Mj, 11.10Lm, 11.15Ex', 'hep-ph-0405286-2-2-0': 'plain', 'hep-ph-0405286-2-3-0': '20true cm', 'hep-ph-0405286-2-4-0': '# Introduction', 'hep-ph-0405286-2-5-0': 'Since rigid objects do not exist in relativistic theories, it is clear that brane fluctuations must play a role in the so called brane world, proposed some years ago by Arkani-Hamed, Dimopoulos and Dvali (ADD scenarios [CITATION]), where the Standard Model (SM) particles are confined to live in the world brane and only gravitons are free to move along the [MATH] dimensional bulk space (see [CITATION] for recent reviews).', 'hep-ph-0405286-2-5-1': 'This fact turns out to be particularly true when the brane tension scale [MATH] being the brane tension) is much smaller than the [MATH] dimensional or fundamental gravitational scale [MATH], i.e., [MATH].', 'hep-ph-0405286-2-5-2': 'In this case the only relevant low-energy modes of the ADD scenarios are the SM particles and branons which are the quantized brane oscillations.', 'hep-ph-0405286-2-5-3': 'Branons can be understood as the (pseudo)Goldstone bosons corresponding to the spontaneous breaking of translational invariance in the bulk space produced by the presence of the brane.', 'hep-ph-0405286-2-5-4': 'It has been pointed out that branons could solve some of the problems of the brane-world scenarios such as the divergent virtual contributions from the Kaluza-Klein tower at the tree level or non-unitarity of the graviton production cross-sections [CITATION].', 'hep-ph-0405286-2-5-5': 'As Goldstone bosons, branons are in principle massless, but in the cases where the metric of the extra dimensions cannot be factorized, they can become massive [CITATION].', 'hep-ph-0405286-2-5-6': 'This is similar to the case of pions which, being the Goldstone bosons of the spontaneous breaking of chiral symmetry, acquire some mass due to the explicit breaking of the symmetry induced by the quark masses.', 'hep-ph-0405286-2-6-0': 'In previous works the different effective actions have been obtained, namely: the effective action for the SM fields on the brane, that for the branon self-interactions and finally that corresponding to the interaction between SM fields and branons [CITATION].', 'hep-ph-0405286-2-6-1': 'In general, this branon effective action can be parameterized by the number of branons [MATH], the tension scale [MATH] and the branon masses (for an explicit construction see [CITATION]).', 'hep-ph-0405286-2-6-2': 'Using the effective action it is possible to obtain the different Feynman rules, the amplitudes and finally the cross-sections for branon production from SM particles.', 'hep-ph-0405286-2-6-3': 'In [CITATION] the case of electron-positron colliders has been considered.', 'hep-ph-0405286-2-6-4': 'By using the Large Electron-Positron Collider (LEP) data it is possible to set important bounds on the tension scale and on the branon mass for a given branon number.', 'hep-ph-0405286-2-6-5': 'Other restrictions have also been set from astrophysical and cosmological considerations due to the fact that branon dark matter can present relevant abundances [CITATION].', 'hep-ph-0405286-2-7-0': 'In this work we study branon production in hadron colliders and also in electron-proton colliders such as HERA.', 'hep-ph-0405286-2-7-1': 'Most of these cross sections have been studied by Creminelli and Strumia for the massless branon case [CITATION].', 'hep-ph-0405286-2-7-2': 'We reproduce their results and extend the analysis for an arbitrary branon mass.', 'hep-ph-0405286-2-7-3': 'The paper is organized as follows: In Sec.II we shortly review the branon effective action.', 'hep-ph-0405286-2-7-4': 'In Sec.III we consider the case of proton-(anti)proton colliders like Tevatron or the future Large Hadron Collider (LHC).', 'hep-ph-0405286-2-7-5': 'In Sec.IV electron(positron)-proton colliders like HERA are studied.', 'hep-ph-0405286-2-7-6': 'In Sec.V we show the main results for the relevant examples and in Sec.VI we set the conclusions.', 'hep-ph-0405286-2-8-0': '# Effective action', 'hep-ph-0405286-2-9-0': 'The relevant effective action describing the low-energy interactions of SM particles and branons was derived in [CITATION], where the necessary vertices are detailed.', 'hep-ph-0405286-2-9-1': 'The branon effective action can be expanded according to the number of branon fields appearing in each term: [EQUATION] where the zeroth-order term is just a constant and the second-order is just the free action: [EQUATION] with [MATH] the branon fields where [MATH] and [MATH] is the squared mass matrix which, without loss of generality, can be assumed to be diagonal.', 'hep-ph-0405286-2-9-2': 'The effective action for the SM particles and their interactions with branons is given by [EQUATION] where [MATH] is the SM Lagrangian and [MATH] is the SM energy-momentum tensor defined as: [EQUATION] where [MATH] is some arbitrary metric on the world brane and [MATH] is the Minkowski metric.', 'hep-ph-0405286-2-10-0': 'In this work we are interested in the interactions between quarks and gluons or photons.', 'hep-ph-0405286-2-10-1': 'Thus, for Dirac fermions with masses [MATH] belonging to some representation of a gauge group, such as [MATH] or [MATH], with generators [MATH], the Lagrangian is [EQUATION] where the covariant derivative is defined as [MATH], [MATH] being the appropriate gauge coupling.', 'hep-ph-0405286-2-10-2': 'Thus the energy-momentum tensor is given by [EQUATION] from where it is possible to find vertices such as [MATH] and [MATH].', 'hep-ph-0405286-2-10-3': 'For gauge fields [MATH] the appropriate Lagrangian for perturbation theory is: [EQUATION] where as usual [MATH] and [MATH] is the Fadeev-Popov Lagrangian including the gauge fixing and the ghost terms.', 'hep-ph-0405286-2-10-4': 'The energy-momentum tensor is: [EQUATION] from where we can obtain the [MATH], [MATH] and [MATH] vertices.', 'hep-ph-0405286-2-11-0': 'Therefore, by using these energy-momentum tensors and the effective action above, it is possible to obtain the different Feynman rules involving branons.', 'hep-ph-0405286-2-11-1': 'One important observation is that in all the vertices obtained above, branons appear always by pairs.', 'hep-ph-0405286-2-11-2': 'In fact they interact in a way similar to gravitons since they couple to the energy momentum tensor.', 'hep-ph-0405286-2-11-3': 'This can be seen by making the formal identification of the graviton field [MATH] which appear in linearized gravity with [EQUATION] where [MATH] and [MATH] the Planck mass.', 'hep-ph-0405286-2-11-4': 'Of course the physical meaning is completely different for branons and gravitons.', 'hep-ph-0405286-2-11-5': 'In any case branons are expected to be weakly interacting and then they will scape to detection.', 'hep-ph-0405286-2-11-6': 'Hence their typical signature will be missing energy and momentum.', 'hep-ph-0405286-2-11-7': 'Since branons are produced by pairs, the energy spectrum of any other particle present in the final state will be continuous.', 'hep-ph-0405286-2-11-8': 'In the following sections we will study the production mechanisms relevant for the different hadronic colliders.', 'hep-ph-0405286-2-12-0': '# Proton-(anti)proton colliders', 'hep-ph-0405286-2-13-0': 'For the case of proton-antiproton colliders like Tevatron, the most important processes for branon production are quark-antiquark annihilation or gluon fusion giving a gluon and a branon pair; and (anti)quark-gluon interaction giving an (anti)quark and a branon pair.', 'hep-ph-0405286-2-13-1': 'Therefore the expected experimental signal will be in both cases one monojet [MATH] and missing energy and momentum.', 'hep-ph-0405286-2-13-2': 'This is a very clear signature that in principle can be easily identified.', 'hep-ph-0405286-2-13-3': 'Another potentially interesting process is the quark-antiquark annihilation giving a photon and a branon pair.', 'hep-ph-0405286-2-13-4': 'In this case the signature is one single photon and missing energy and momentum.', 'hep-ph-0405286-2-14-0': 'The Feynman diagrams contributing to the main subprocesses [MATH], [MATH], [MATH] and [MATH] are shown in Fig. [REF], Fig. [REF] and Fig. [REF].', 'hep-ph-0405286-2-15-0': 'From these diagrams and the Feynman rules coming from the effective action of the previous section, it is possible to obtain the differential cross section: [EQUATION] where [MATH], [MATH], [MATH] and [MATH], [MATH] and [MATH] being the anti-quark and quark four-momenta respectively, [MATH] the gluon four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-2-15-1': 'We have assumed for the sake of simplicity that all the branons are degenerated with a common mass [MATH] and that all the quarks are massless.', 'hep-ph-0405286-2-15-2': 'We have also neglected the effects of the top quark.', 'hep-ph-0405286-2-15-3': 'In addition we have the well-known relation [MATH].', 'hep-ph-0405286-2-15-4': 'The contribution to the total cross section of the process [MATH] coming from this subprocess is given by [EQUATION] where [MATH] and [MATH] are the distribution functions of the anti-quark inside the antiproton and of the quark of flavor [MATH] inside the proton respectively, and [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and antiquark.', 'hep-ph-0405286-2-15-5': 'The different limits of the integrals can be written in terms of the cuts used to define the total cross-section.', 'hep-ph-0405286-2-15-6': 'For example, in order to be able to detect clearly the monojet one must impose a minimal value for its transverse energy [MATH] and a pseudorapidity range given by [MATH] and [MATH].', 'hep-ph-0405286-2-15-7': 'Then we have the limits [MATH], [MATH] and [MATH].', 'hep-ph-0405286-2-15-8': 'On the other hand [MATH] and [MATH] where [MATH] is the total center of mass energy squared of the process and [EQUATION]', 'hep-ph-0405286-2-15-9': 'In addition the dots in ([REF]) represent the contribution of the case in which the quark comes from the antiproton and the anti-quark comes from the proton.', 'hep-ph-0405286-2-16-0': 'The cross-section of the subprocess [MATH] is given by [EQUATION] where the Mandelstan variables are defined as in the previous case, with [MATH] and [MATH] being the initial gluon four-momenta, [MATH] the final gluon four-momentum and [MATH] the total branon four-momentum.', 'hep-ph-0405286-2-16-1': 'Then the contribution to the total cross section from the [MATH] reaction is [EQUATION]', 'hep-ph-0405286-2-16-2': 'Here [MATH] is the gluon distribution function of the (anti)proton, [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the initial gluons and the integration limits remain the same.', 'hep-ph-0405286-2-16-3': 'From the above equations, it is possible to compute the total cross-section [MATH] in terms of the cut in the gluon (monojet) transverse energy [MATH].', 'hep-ph-0405286-2-17-0': 'For the [MATH] process the cross-section is given by [EQUATION] with [MATH] and [MATH] being the quark and the gluon four-momenta respectively, [MATH] the final state quark four-momentum and [MATH] and [MATH] the branon four-momenta.', 'hep-ph-0405286-2-17-1': 'The Mandelstam variables are defined as in previous cases.', 'hep-ph-0405286-2-17-2': 'The cross-section for the conjugate process [MATH] is exactly the same.', 'hep-ph-0405286-2-17-3': 'Then the total cross section for the the reaction [MATH] is [EQUATION]', 'hep-ph-0405286-2-17-4': 'In this equation [MATH] and [MATH] are the fractions of the proton and antiproton energy carried by the subprocess quark and gluon.', 'hep-ph-0405286-2-17-5': 'The different integration limits are defined as in the previous case in terms of the minimal transverse energy of the quark (monojet) [MATH] and the dots refer to the case where the initial gluon is coming from the proton and the quark is coming from the antiproton.', 'hep-ph-0405286-2-17-6': 'In addition we have the contribution from the conjugate case where we take an anti-quark from the proton and a gluon from the antiproton and conversely.', 'hep-ph-0405286-2-17-7': 'This amount just to a factor of two.', 'hep-ph-0405286-2-18-0': 'From all the above equations it is possible to compute the total cross-section [MATH] in terms of the cut in the jet transverse energy [MATH].', 'hep-ph-0405286-2-19-0': 'For the subprocess [MATH] the cross-section is given by [EQUATION]', 'hep-ph-0405286-2-19-1': 'Here the notation is similar to the [MATH] case with the obvious differences in couplings, color and charge factors.', 'hep-ph-0405286-2-19-2': 'Thus [EQUATION]', 'hep-ph-0405286-2-19-3': 'All the previous discussion about branon production in [MATH] reactions can be easily translated to the [MATH] case.', 'hep-ph-0405286-2-19-4': 'The only point is to change the antiproton distribution functions of the different partons by the corresponding proton ones.', 'hep-ph-0405286-2-20-0': '# Electron(positron)-proton colliders', 'hep-ph-0405286-2-21-0': 'For electron(positron)-proton colliders like HERA, the most interesting branon creating process is branon photoproduction, where a photon emitted by the electron(positron) interacts with a quark(antiquark) from the proton giving a quark (antiquark) and a branon pair.', 'hep-ph-0405286-2-21-1': 'Thus the experimental signature is again one monojet [MATH] plus missing energy and momentum.', 'hep-ph-0405286-2-21-2': 'The relevant Feynman diagrams are shown in Fig. [REF] and the corresponding differential cross-section for the subprocess [MATH] is [EQUATION] where [MATH] and [MATH], [MATH] being the photon, [MATH] the proton quark, [MATH] the final quark and [MATH] the total branon momenta respectively.', 'hep-ph-0405286-2-21-3': 'The total cross-section for the process [MATH] is given by [EQUATION] [MATH] and [MATH] are defined in this case as [MATH] and [MATH] with [MATH] and [MATH] being the proton and electron(positron) momenta respectively.', 'hep-ph-0405286-2-21-4': 'Thus at high energies compared with the proton mass [MATH] where [MATH].', 'hep-ph-0405286-2-21-5': 'The integral limits [MATH], [MATH], [MATH] and [MATH] are defined like in the proton-(anti)proton collider case.', 'hep-ph-0405286-2-22-0': 'The photon spectrum [MATH] can be obtained from the well-known Weizs[MATH]cker-Williams approximation [CITATION]: [EQUATION] with [MATH] and [MATH] being the electron mass.', 'hep-ph-0405286-2-23-0': 'The cross-section [MATH] can be obtained in a similar way.', 'hep-ph-0405286-2-23-1': 'Then the total contribution to monojet plus missing energy and momentum production for large enough [MATH] coming from branons can be written as the sum of [MATH] and [MATH].', 'hep-ph-0405286-2-24-0': '# Results', 'hep-ph-0405286-2-25-0': 'By using the cross-sections shown in the previous sections it is possible to compute the expected number of branon pairs produced in the different hadron colliders in terms of the brane tension parameter [MATH], the branon mass [MATH] and the number of branons [MATH].', 'hep-ph-0405286-2-25-1': 'To this end we have used the distribution functions which can be found in [CITATION].', 'hep-ph-0405286-2-25-2': 'The values of the electromagnetic and strong couplings have been taken at the electroweak boson masses, namely [MATH] and [MATH].', 'hep-ph-0405286-2-25-3': 'However our final results do not depend too much on the precise value of these couplings.', 'hep-ph-0405286-2-25-4': 'In fact our main source of error is the use of an effective action to describe the SM particles and branon interactions since, in principle, this is only guaranteed for energies well below [MATH].', 'hep-ph-0405286-2-26-0': 'As discussed in the introduction, our main goal in this work is to study the bounds that can be set on the [MATH], [MATH] and [MATH] parameters coming from hadron colliders.', 'hep-ph-0405286-2-26-1': 'We will present all our limits at the 95% confidence level.', 'hep-ph-0405286-2-26-2': 'In particular, for the electron(positron)-proton case, HERA is the most relevant experiment.', 'hep-ph-0405286-2-26-3': 'In fact, the ZEUS collaboration has studied the jet production in charged current deep inelastic [MATH] scattering.', 'hep-ph-0405286-2-26-4': 'Its results are perfectly compatible with the SM background and therefore, we can set some bounds on the branon production and hence on the [MATH], [MATH] and [MATH] parameters.', 'hep-ph-0405286-2-26-5': 'These data were taken from 1995 to 2000 at a maximum CM energy of [MATH] GeV.', 'hep-ph-0405286-2-26-6': 'The total integrated luminosity was [MATH] pb[MATH] and the cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for more details).', 'hep-ph-0405286-2-26-7': 'By using the same cuts with our cross-sections for monojet plus a branon pair production, we find the bound [MATH] GeV for massless branons.', 'hep-ph-0405286-2-26-8': 'For a branon mass larger than [MATH] GeV there is no restrictions on the [MATH] value because of kinematical reasons.', 'hep-ph-0405286-2-26-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] where we have assumed [MATH].', 'hep-ph-0405286-2-26-10': 'For other [MATH] values one just has to take into account that the bound scales like [MATH] since all the cross-sections are proportional to [MATH].', 'hep-ph-0405286-2-27-0': 'In the [MATH] case the most relevant experimental information so far is the one obtained at the Tevatron (Run I).', 'hep-ph-0405286-2-27-1': 'The [MATH] detector has studied the monojet channel and CDF the single photon one.', 'hep-ph-0405286-2-27-2': 'As far as the number of events found in both cases is compatible with the SM background, we can set new bounds on the branon theory parameters.', 'hep-ph-0405286-2-27-3': 'For light branons the most important bound comes from the [MATH] data.', 'hep-ph-0405286-2-27-4': 'These data were taken from 1994 to 1996 at a CM energy of [MATH] TeV and correspond to an integrated luminosity [MATH] pb[MATH].', 'hep-ph-0405286-2-27-5': 'The cuts on the pseudorapidity and the transverse energy were [MATH] and [MATH] GeV (see [CITATION] for the details of the analysis).', 'hep-ph-0405286-2-27-6': 'The total number of monojets observed was [MATH] and the expected number from the SM plus cosmic rays events was [MATH].', 'hep-ph-0405286-2-27-7': 'By using our cross sections for monojet plus a branon pair production with these cuts we get the bound [MATH] GeV for light branons.', 'hep-ph-0405286-2-27-8': 'The restrictions for [MATH] improve up to a branon mass of [MATH] GeV.', 'hep-ph-0405286-2-27-9': 'For the intermediate [MATH] values the bounds obtained can be seen in Fig. [REF] for [MATH].', 'hep-ph-0405286-2-28-0': 'In a similar way we can use the CDF data on single photon production.', 'hep-ph-0405286-2-28-1': 'In this case the total luminosity collected was [MATH] pb[MATH] and the pseudorapidity cut was [MATH].', 'hep-ph-0405286-2-28-2': 'For the transverse photon energy several cuts were considered (for example 55 GeV at the [MATH] efficiency).', 'hep-ph-0405286-2-28-3': 'The total expected background for this process was [MATH], without taking into account the QCD contribution (see [CITATION] for the details of the analysis), and the number of events found was [MATH].', 'hep-ph-0405286-2-28-4': 'Comparing this result with our computations for photon plus one branon pair production, we find the bound [MATH] GeV for massless branons and no bound for [MATH] larger than [MATH] GeV.', 'hep-ph-0405286-2-28-5': 'The bound obtained for the rest of the cases is shown also in Fig. [REF].', 'hep-ph-0405286-2-29-0': 'In addition to this analysis corresponding to the Tevatron data (Run I), it is also interesting to make some estimation about the bounds that could be set from future experiments such as Tevatron (Run II) and the LHC.', 'hep-ph-0405286-2-29-1': 'In the case of the Tevatron (Run II), which is already in progress, the main novelties are a CM energy which equals [MATH] TeV and an expected integrated luminosity [MATH] at the end of the run of about [MATH] pb[MATH].', 'hep-ph-0405286-2-29-2': 'The detectors are also improved so that the pseudorapidity cuts can be taken as [MATH] for [MATH] and [MATH] for CDF.', 'hep-ph-0405286-2-29-3': 'This would result in a factor of [MATH] on the statistical significance when compared to the Run I, with integrated luminosity [MATH], provided that the CM energy and the cuts were the same.', 'hep-ph-0405286-2-29-4': 'For massless branons, the bound on [MATH] scales as the CM energy [MATH].', 'hep-ph-0405286-2-29-5': 'Even more important is the possibility of exploring higher branon masses, since the kinematical limit is given by [MATH].', 'hep-ph-0405286-2-29-6': 'In Fig. [REF] we show the expected bounds from the Run II in the [MATH] plane, again for [MATH].', 'hep-ph-0405286-2-30-0': 'The LHC will produce [MATH] collisions at a CM energy of [MATH] TeV and the integrated luminosity will be something about [MATH] pb[MATH].', 'hep-ph-0405286-2-30-1': 'In order to estimate the bounds on the [MATH], [MATH] and [MATH] parameters that will be possible to obtain at the LHC, we have proceeded in a similar way as in the Tevatron case, with the obvious changes in the distribution functions due to the fact that now we are dealing with [MATH] instead of [MATH] collisions.', 'hep-ph-0405286-2-30-2': 'We have kept the same cuts except for the transverse energy which has been corrected in order to maintain the same proportion relative to the CM energy.', 'hep-ph-0405286-2-30-3': 'Again the best bounds for [MATH] come from monojet production, which for [MATH] turns out to be [MATH] GeV.', 'hep-ph-0405286-2-30-4': 'For low [MATH] the best bound for [MATH] is given by the single photon channel ([MATH]).', 'hep-ph-0405286-2-30-5': 'The LHC sensitivity for other values in the [MATH] plane can be found in Fig. [REF] for [MATH].', 'hep-ph-0405286-2-31-0': '# Conclusions', 'hep-ph-0405286-2-32-0': 'In this work we have studied the flexible brane-world scenario, where the brane tension scale [MATH] is much smaller than the fundamental [MATH]-dimensional gravitational scale [MATH].', 'hep-ph-0405286-2-32-1': 'In this case, the relevant low-energy degrees of freedom are the SM particles and the brane fluctuations or branons.', 'hep-ph-0405286-2-32-2': 'From the corresponding effective action, we have calculated the relevant cross-sections for different branon searches in hadronic colliders.', 'hep-ph-0405286-2-32-3': 'We have used the information coming from HERA and the first Tevatron Run in order to get different exclusion plots on the branon mass [MATH] and the tension scale [MATH] plane for a given branon number [MATH].', 'hep-ph-0405286-2-32-4': 'Monojet production turns out to be the most efficient process for light branons, whereas the single photon channel is the most important one for heavy branons.', 'hep-ph-0405286-2-33-0': 'We have also extended the analysis to future hadronic colliders.', 'hep-ph-0405286-2-33-1': 'The corresponding sensitivity regions for the second Tevatron run and the LHC have also been plotted (see Table [REF] for a summary of the analysis).', 'hep-ph-0405286-2-34-0': 'These analysis improve those already done for electron-positron colliders for heavy branons, whereas for light branons, the results are similar [CITATION].', 'hep-ph-0405286-2-34-1': 'The Tevatron (run I) limit [MATH] GeV can be compared to the analogous limit from LEP II [MATH] GeV [CITATION].', 'hep-ph-0405286-2-34-2': 'According to the previous estimations, the Tevatron run II could also improve the bound [MATH] GeV obtained by LEP-II.', 'hep-ph-0405286-2-34-3': 'On the other hand, LHC could detect branons up to a mass of several TeV ([MATH] GeV) improving even the CLIC prospects ([MATH] GeV) [CITATION].', 'hep-ph-0405286-2-35-0': 'The study of branons in colliders can be complemented with other bounds coming from astrophysics and cosmology (see Fig. 10).', 'hep-ph-0405286-2-35-1': 'In fact, as shown in [CITATION], the branon relic abundance can have cosmological consequences.', 'hep-ph-0405286-2-35-2': 'Other issues related to branon phenomenology, such as their radiative corrections to the SM processes, or their distinctive signatures at colliders with respect to the KK gravitons will be analyzed elsewhere.', 'hep-ph-0405286-2-36-0': 'Note added: After this paper was completed, we were informed by M. Spiropulu that CDF collaboration had performed a monojet study [CITATION] which could improve the bounds in a more detailed analysis.'} | [['hep-ph-0405286-1-23-0', 'hep-ph-0405286-2-23-0'], ['hep-ph-0405286-1-23-1', 'hep-ph-0405286-2-23-1'], ['hep-ph-0405286-1-32-0', 'hep-ph-0405286-2-32-0'], ['hep-ph-0405286-1-32-1', 'hep-ph-0405286-2-32-1'], ['hep-ph-0405286-1-32-2', 'hep-ph-0405286-2-32-2'], ['hep-ph-0405286-1-32-3', 'hep-ph-0405286-2-32-3'], ['hep-ph-0405286-1-32-4', 'hep-ph-0405286-2-32-4'], ['hep-ph-0405286-1-13-0', 'hep-ph-0405286-2-13-0'], ['hep-ph-0405286-1-13-1', 'hep-ph-0405286-2-13-1'], ['hep-ph-0405286-1-13-2', 'hep-ph-0405286-2-13-2'], ['hep-ph-0405286-1-13-3', 'hep-ph-0405286-2-13-3'], ['hep-ph-0405286-1-13-4', 'hep-ph-0405286-2-13-4'], ['hep-ph-0405286-1-33-0', 'hep-ph-0405286-2-33-0'], ['hep-ph-0405286-1-33-1', 'hep-ph-0405286-2-33-1'], ['hep-ph-0405286-1-5-0', 'hep-ph-0405286-2-5-0'], ['hep-ph-0405286-1-5-1', 'hep-ph-0405286-2-5-1'], ['hep-ph-0405286-1-5-2', 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'hep-ph-0405286-2-29-3'], ['hep-ph-0405286-1-29-4', 'hep-ph-0405286-2-29-4'], ['hep-ph-0405286-1-29-5', 'hep-ph-0405286-2-29-5'], ['hep-ph-0405286-1-29-6', 'hep-ph-0405286-2-29-6'], ['hep-ph-0405286-1-18-0', 'hep-ph-0405286-2-18-0'], ['hep-ph-0405286-1-34-0', 'hep-ph-0405286-2-34-0'], ['hep-ph-0405286-1-34-1', 'hep-ph-0405286-2-34-1'], ['hep-ph-0405286-1-34-2', 'hep-ph-0405286-2-34-3'], ['hep-ph-0405286-1-27-0', 'hep-ph-0405286-2-27-0'], ['hep-ph-0405286-1-27-1', 'hep-ph-0405286-2-27-1'], ['hep-ph-0405286-1-27-2', 'hep-ph-0405286-2-27-2'], ['hep-ph-0405286-1-27-3', 'hep-ph-0405286-2-27-3'], ['hep-ph-0405286-1-27-4', 'hep-ph-0405286-2-27-4'], ['hep-ph-0405286-1-27-5', 'hep-ph-0405286-2-27-5'], ['hep-ph-0405286-1-27-6', 'hep-ph-0405286-2-27-6'], ['hep-ph-0405286-1-27-7', 'hep-ph-0405286-2-27-7'], ['hep-ph-0405286-1-27-8', 'hep-ph-0405286-2-27-8'], ['hep-ph-0405286-1-27-9', 'hep-ph-0405286-2-27-9'], ['hep-ph-0405286-1-15-0', 'hep-ph-0405286-2-15-0'], ['hep-ph-0405286-1-15-1', 'hep-ph-0405286-2-15-1'], ['hep-ph-0405286-1-15-2', 'hep-ph-0405286-2-15-2'], ['hep-ph-0405286-1-15-3', 'hep-ph-0405286-2-15-3'], ['hep-ph-0405286-1-15-4', 'hep-ph-0405286-2-15-4'], ['hep-ph-0405286-1-15-5', 'hep-ph-0405286-2-15-5'], ['hep-ph-0405286-1-15-6', 'hep-ph-0405286-2-15-6'], ['hep-ph-0405286-1-15-7', 'hep-ph-0405286-2-15-7'], ['hep-ph-0405286-1-15-8', 'hep-ph-0405286-2-15-8'], ['hep-ph-0405286-1-15-9', 'hep-ph-0405286-2-15-9']] | [['hep-ph-0405286-1-35-0', 'hep-ph-0405286-2-35-0']] | [] | [['hep-ph-0405286-1-11-6', 'hep-ph-0405286-2-11-6']] | [] | ['hep-ph-0405286-1-1-0', 'hep-ph-0405286-1-2-0', 'hep-ph-0405286-1-3-0', 'hep-ph-0405286-1-14-0', 'hep-ph-0405286-1-19-2', 'hep-ph-0405286-2-1-0', 'hep-ph-0405286-2-2-0', 'hep-ph-0405286-2-3-0', 'hep-ph-0405286-2-14-0', 'hep-ph-0405286-2-19-2'] | {'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'} | https://arxiv.org/abs/hep-ph/0405286 | null | null | null | null | null |
cond-mat-0305190 | {'cond-mat-0305190-1-0-0': 'We report on the effect of the back-action of a Single Cooper Pair Transistor electrometer (SCPT1) in measurements of charge [MATH] on the island of a second SCPT2.', 'cond-mat-0305190-1-0-1': 'The measurement of [MATH] is [MATH]-periodic in its gate charge [MATH] when the electrometer is biased at [MATH] or [MATH].', 'cond-mat-0305190-1-0-2': 'We show that this is due to quasiparticle poisoning of SCPT2 at a rate proportional to the number of quasiparticle tunneling events in SCPT1.', 'cond-mat-0305190-1-0-3': 'We are able to eliminate this back-action and recover a 2e-periodic [MATH] dependence using a new measurement method based on switching current modulation of SCPT1.', 'cond-mat-0305190-1-1-0': 'The superconducting box, e.g. a small island of Al film very weakly coupled to the outside circuitry by Josephson junctions, has shown considerable promise as a qubit for quantum information processing where the two states can be represented by superpositions of 0 or 1 excess Cooper pairs in the box [CITATION].', 'cond-mat-0305190-1-1-1': 'Measurement of the quantum state of this so-called charge qubit without inducing unwanted decoherence is a significant problem as is quasiparticle poisoning, i.e. the introduction of unpaired electrons (quasiparticles) into the box.', 'cond-mat-0305190-1-1-2': 'At temperatures of 10 mK or so, where experiments are commonly done, the number of quasiparticle should, in principle, be negligible.', 'cond-mat-0305190-1-1-3': 'However, such quasiparticle poisoning, due perhaps to the measurement process itself, is commonly observed.', 'cond-mat-0305190-1-1-4': 'A manifestation of this is seen in the so called Coulomb staircase.', 'cond-mat-0305190-1-1-5': 'When a charge, [MATH], is capacitively induced on the junctions of the box, one expects Cooper pairs to tunnel resonantly into or out of the box at [MATH], where [MATH] is an odd integer, to maintain the lowest energy charge state of the box.', 'cond-mat-0305190-1-1-6': 'This results in the Coulomb staircase of the charge in the box [MATH] with period 2[MATH] in [MATH].', 'cond-mat-0305190-1-1-7': 'On the other hand, if there are quasiparticles in the system, then maintaining the lowest charging energy state also leads to quasiparticle tunneling.', 'cond-mat-0305190-1-1-8': 'This gives rise to splitting of the steps in the Coulomb staircase, which shifts toward e-periodicity as the number of quasiparticles increases [CITATION].', 'cond-mat-0305190-1-1-9': 'As a result, the lowest energy state of the box at [MATH], no longer corresponds to a resonant state of the Cooper pair tunneling.', 'cond-mat-0305190-1-1-10': 'For the box qubit this means that relaxation does not bring the system back to its computational ground state at its operating point.', 'cond-mat-0305190-1-1-11': 'Since the ability to prepare the initial state of the qubit is an absolutely necessary condition for quantum computing, solving the problem of quasiparticle poisoning in charge qubits is essential.', 'cond-mat-0305190-1-2-0': '(a) Schematic of the measurement showing SCPT electrometer (E) coupled to the second SCPT (B).', 'cond-mat-0305190-1-2-1': 'Tunnel junctions are represented by double box symbols.', 'cond-mat-0305190-1-2-2': '(b) Average charge of B as a function of its gate voltage when electrometer operates in VM mode.', 'cond-mat-0305190-1-2-3': '(c) Switching current of B as a function of its gate voltage.', 'cond-mat-0305190-1-2-4': 'There is an undetermined shift in gate voltage between measurements in (b) and in (c) because of drift of background charge.', 'cond-mat-0305190-1-3-0': 'Quasiparticle poisoning has been a serious road block for groups working to build a charge qubit [CITATION].', 'cond-mat-0305190-1-3-1': 'The purpose of this paper is to investigate the effects of the measurement process on quasiparticle poisoning and to develop approaches to minimize these effects.', 'cond-mat-0305190-1-3-2': 'The Single Cooper Pair Transistor (SCPT) electrometer can be operated in several different modes when measuring the charge of the box [CITATION].', 'cond-mat-0305190-1-3-3': 'Commonly the charge measurements of the box are done by operating the SCPT electrometer in the voltage modulation mode (VM).', 'cond-mat-0305190-1-3-4': 'In this mode the SCPT is biased at a sufficiently high voltage that quasiparticles are generated, so it effectively functions as a SET where the source drain voltage is modulated by [MATH] with a period of [MATH].', 'cond-mat-0305190-1-3-5': 'However it is known that the switching current of SCPTs, i.e. the current at which it switches hystereticaly from the low voltage or phase-diffusion branch to [MATH] is also charge sensitive [CITATION] and can be used for charge measurement.', 'cond-mat-0305190-1-3-6': 'We refer to this as the switching current mode (SW).', 'cond-mat-0305190-1-3-7': 'The SW mode of operation has been analyzed [CITATION], but until now no measurements of the charge on a superconducting island or box using a SW mode electrometer have been reported.', 'cond-mat-0305190-1-3-8': 'This paper presents the results of the measurements of the island charge of a SCPT used in a box configuration (B) by a SCPT electrometer (E) operated in either the VM or SW mode.', 'cond-mat-0305190-1-3-9': 'The results demonstrate that measurement-induced poisoning, which leads to an e-periodic Coulomb staircase when using the electrometer in the voltage modulation mode, can be eliminated in the switching current mode.', 'cond-mat-0305190-1-4-0': 'The sample, shown in Fig. 1(a), consists of symmetric arraignment of the two nominally identical SCPTs, E and B, which are capacitively coupled.', 'cond-mat-0305190-1-4-1': 'Due to the symmetry of the circuit the roles of E and B are interchangeable.', 'cond-mat-0305190-1-4-2': 'The parameters of the circuit are as follows: E has a normal state resistance [MATH] k[MATH], charging energy [MATH]eV and Josephson energy [MATH]eV.', 'cond-mat-0305190-1-4-3': 'For B these parameters are [MATH] k[MATH], [MATH]eV and [MATH]eV.', 'cond-mat-0305190-1-4-4': '[MATH] is the average of the Josephson energies of two junctions as determined from the values of [MATH] and superconducting gap [MATH]eV by using the Ambakoekar-Baratoff formula [CITATION].', 'cond-mat-0305190-1-4-5': '[MATH] is determined from the amplitude of maximum voltage modulation of the devices.', 'cond-mat-0305190-1-4-6': 'The coupling capacitance between E and B is determined to be 80 aF from their measured coupling of 4.8%.', 'cond-mat-0305190-1-4-7': 'All devices are made using standard two angle shadow evaporation without having normal metal quasiparticle traps close to junctions.', 'cond-mat-0305190-1-4-8': 'The sample is placed in microwave tight copper can located on a temperature regulated stage of a dilution refrigerator having a base temperature of 6 mK.', 'cond-mat-0305190-1-4-9': 'All the measurement leads are filtered by low temperature microwave filters [CITATION] that are thermally anchored to the mixing chamber.', 'cond-mat-0305190-1-5-0': 'Figure 1(b) shows the charge on the island of B, [MATH], measured with E in the VM mode.', 'cond-mat-0305190-1-5-1': 'During this measurement the source and drain leads of B are at a common potential with respect to its gate.', 'cond-mat-0305190-1-5-2': 'Figure 1(c) presents the switching current modulation of B measured with the bias current through E, [MATH], set equal to zero.', 'cond-mat-0305190-1-5-3': 'As can be seen, the switching current modulation of B is 2e-periodic as expected at low temperature, but the charge of B, measured by the electrometer, is e-periodic.', 'cond-mat-0305190-1-5-4': 'Similar dependences of [MATH] on [MATH] were measured with E biased in either of its voltage sensitive regions, i.e. near the gap where [MATH] or near the Josephson-quasiparticle peak where [MATH].', 'cond-mat-0305190-1-5-5': 'To determine if the e-periodicity of [MATH] is due to the back-action of E on B, the quasiparticle poisoning rate of B, [MATH], was measured for a range of bias conditions of E.', 'cond-mat-0305190-1-5-6': 'In addition to this, we studied how the biases of two other SCPTs located on the same chip but coupled more weakly to B effected.', 'cond-mat-0305190-1-6-0': 'Switching current histograms of B measured for different bias conditions of E. Corresponding time delay from the beginning of the current ramp is shown in the upper horizontal axes.', 'cond-mat-0305190-1-6-1': 'The histograms are shifted on vertical axis for clarity.', 'cond-mat-0305190-1-6-2': '"odd" ("even") peaks in the histograms correspond to switching from the odd (even) parity states of B. Inset: IV-characteristic of E. Arrows indicate bias conditions at which switching current histograms of the main Figure were measured.', 'cond-mat-0305190-1-6-3': 'Diamonds mark positions where [MATH] has been measured for data in Fig. 3.', 'cond-mat-0305190-1-7-0': 'The details of the technique for determining [MATH] from switching current distributions have been reported previously [CITATION].', 'cond-mat-0305190-1-7-1': 'Briefly, as [MATH] is linearly ramped in time, the switching current histogram of B with its gate biased near [MATH] exhibits two peaks if the number of quasiparticles on the island changes during the measurement of histogram.', 'cond-mat-0305190-1-7-2': 'One peak, [MATH], which is close to the maximum of the [MATH] characteristic occurs when the island has an even number of electrons (even state) and the other -much lower current - peak, [MATH], is near the predicted switching current minimum at [MATH] when one quasiparticle occupies the island (odd state) [Fig. 2].', 'cond-mat-0305190-1-7-3': 'If [MATH] and B has not switched, it must be in the even state.', 'cond-mat-0305190-1-7-4': 'The entry of a quasiparticle onto the island effectively changes [MATH] to [MATH] which for [MATH] will cause B to switch rapidly to the running state.', 'cond-mat-0305190-1-7-5': 'This can be used to obtain the poisoning rate [MATH].', 'cond-mat-0305190-1-7-6': 'Previous studies have shown that [MATH] is independent of [MATH] in the region between the peaks giving an exponential decay of the probability of the even state.', 'cond-mat-0305190-1-7-7': 'Several of these histograms with increasing [MATH] as [MATH] and [MATH] increase are shown in Fig. 2.', 'cond-mat-0305190-1-7-8': 'The relatively low bandwidth of our filters limits these measurement to [MATH] ms[MATH].', 'cond-mat-0305190-1-8-0': 'First we determine when all the electrodes of electrometer are disconnected from the measurement circuitry and grounded.', 'cond-mat-0305190-1-8-1': 'In this case we still observe a small residual rate [MATH] ms[MATH].', 'cond-mat-0305190-1-8-2': 'This non-zero rate can be caused, e.g. , by the presence of impurity levels in the superconducting gap of Al [CITATION].', 'cond-mat-0305190-1-8-3': 'For the present discussion it is clear this small residual rate is not related to the back-action of E. [MATH] is unchanged if E is biased on its supercurrent branch or when E is biased on its return current branch at low voltage [MATH] as shown in Fig. 3.', 'cond-mat-0305190-1-8-4': 'For [MATH]V, [MATH] decreases rapidly, becoming too short to measure for slightly higher voltages.', 'cond-mat-0305190-1-8-5': 'The modulation characteristics of the current [MATH] with gate voltage also change at this point from being 2e-periodic for [MATH]V to e-periodic for [MATH]V.', 'cond-mat-0305190-1-8-6': 'A similar cross-over from e to 2e-periodicity at a voltage [MATH] has been seen in a similar system recently [CITATION].', 'cond-mat-0305190-1-9-0': 'To understand how the E generates quasiparticles in B when it is biased at the usual operating voltages of VM mode, i.e. [MATH] and [MATH], we study the rate as a function of bias current and voltage of two other SCPT devices fabricated on the same chip.', 'cond-mat-0305190-1-9-1': 'These two SCPT, which we call E1 and E2, are much more weakly coupled to B and therefore allow us to measure for bias currents, [MATH], that are several orders of magnitude higher than is possible using E.', 'cond-mat-0305190-1-9-2': 'The island of SCPT E1 is located 96 [MATH]m from the island of B and has [MATH] k[MATH], while the corresponding parameters for E2 are 143 [MATH]m and [MATH] k[MATH].', 'cond-mat-0305190-1-9-3': 'For these devices, we can measure the rate up to voltages [MATH].', 'cond-mat-0305190-1-9-4': 'Again, we see a small initial increase of [MATH] at [MATH]V and then sharp increases at voltages [MATH] and [MATH].', 'cond-mat-0305190-1-9-5': 'These voltages correspond approximately to the Josephson-quasiparticle tunneling and sequential quasiparticle tunneling thresholds in an SCPT and are accompanied by sharp increases in [MATH].', 'cond-mat-0305190-1-9-6': 'The inset of Fig. 3 shows the rate as a function of the currents in E1 and E2 above these two thresholds.', 'cond-mat-0305190-1-9-7': 'To test the hypothesis that [MATH] is proportional to the total number of quasiparticle tunneling events, the currents for electrometer voltages above [MATH] are multiplied by two.', 'cond-mat-0305190-1-9-8': 'This is because there are twice as many quasiparticle tunneling events for voltages [MATH] through the junctions of E than for [MATH] for a given current.', 'cond-mat-0305190-1-9-9': 'As one can see, this scaling collapses all of the data from each electrometer to a common line.', 'cond-mat-0305190-1-10-0': 'From these data one can conclude that the quasiparticle current of electrometer is the source of back-action noise leading to quasiparticle poisoning and e-periodic Coulomb staircase of B when measured by E. Further, the quasiparticle generation in B is proportional to the total number of quasiparticle tunneling events per second through E.', 'cond-mat-0305190-1-10-1': 'This relationship could indicate that the back-action of E results from the shot noise of tunneling quasiparticles.', 'cond-mat-0305190-1-10-2': 'On the other hand, this back-action could also be the result of the recombination of quasiparticles in E into pairs.', 'cond-mat-0305190-1-10-3': 'This quasiparticle recombination produces phonons and in smaller extent photons of energy [MATH].', 'cond-mat-0305190-1-10-4': 'These phonons/photons, which propagate from E to B without energy relaxation could generate quasiparticles in B. Determining the details of the interaction between E and B will require further work.', 'cond-mat-0305190-1-10-5': 'However, it is interesting to note that this sort of recombination noise would likely be suppressed by having normal metal leads close to the junctions.', 'cond-mat-0305190-1-10-6': 'This may provide an explanation of previous results [CITATION] in which it was possible to observe a 2e-periodic Coulomb staircase using a VM electrometer measuring a superconducting box, which had normal metal "quasiparticle traps" close to the junctions.', 'cond-mat-0305190-1-11-0': 'Average charge of B as a function of its gate charge when the electrometer operates in SW mode: solid line, quasiparticle flushing (see text) before each charge measurement; dashed line, no flushing.', 'cond-mat-0305190-1-11-1': 'Vertical dotted lines mark the positions at which the maxima of [MATH] appear.', 'cond-mat-0305190-1-11-2': 'Inset shows grey scale image of density of switching events of E as a function of gate charge and charge on the island of B.', 'cond-mat-0305190-1-11-3': 'The bright line shows the predicted average charge in B as a function of the gate charge for a coherent superposition of 0 and 1 Cooper pair states with [MATH].', 'cond-mat-0305190-1-11-4': 'Residual quasiparticle poisoning appears in the grey area pointed by the arrow.', 'cond-mat-0305190-1-12-0': 'In an attempt to eliminate the poisoning due the measurement electrometer, we have investigated using it in the switching current mode.', 'cond-mat-0305190-1-12-1': 'For this type of measurement, the electrometer is operated on its switching current (phase-diffusion) branch until after the measurement is made.', 'cond-mat-0305190-1-12-2': 'So, its voltage is well below that for which a significant increase in [MATH] is observed.', 'cond-mat-0305190-1-12-3': 'For these measurements, [MATH] is set near [MATH], where the transfer function of E, [MATH], is maximum, ([MATH]3.6 nA/e).', 'cond-mat-0305190-1-12-4': 'For each charge measurement [MATH] at fixed [MATH] several hundred switching events is measured at a ramp rate of [MATH]130 nA/s.', 'cond-mat-0305190-1-12-5': 'The switching current, which is related the charge on the island of B by non-linear transfer function [MATH], is then inverted to obtain [MATH].', 'cond-mat-0305190-1-12-6': 'The result of this measurement after correction for non-linearity of the electrometer and averaging over measurements at fixed [MATH] is shown with dashed line in Fig. 4.', 'cond-mat-0305190-1-12-7': 'The dotted vertical lines in Fig. 4 show the locations of the measured peaks in the switching current of [MATH].', 'cond-mat-0305190-1-12-8': 'These peaks are located at [MATH] and so serve to calibrate the scale and origin (modulo [MATH]) the abscissa.', 'cond-mat-0305190-1-13-0': 'The Coulomb staircase in Fig. 4 (dashed line) still shows the split steps at [MATH] characteristic of quasiparticles.', 'cond-mat-0305190-1-13-1': 'This is to be expected based on the measurements of [MATH] even if there is no quasiparticle generation by E before a measurement.', 'cond-mat-0305190-1-13-2': 'The residual rate, [MATH] ms[MATH], which is not related to the back-action of electrometer, leads to trapping of quasiparticles by the electrostatic potential of island for [MATH] near [MATH] with a probability approaching 100% for measurement times much greater than [MATH].', 'cond-mat-0305190-1-13-3': 'The observed difference in the length of the two steps implies that the escape rate for a quasiparticle near [MATH] is greater than [MATH].', 'cond-mat-0305190-1-13-4': 'One would expect this to happen if the density of quasiparticles in the leads of B is sufficiently low.', 'cond-mat-0305190-1-14-0': 'The probability of an even parity state of B for [MATH] can be greatly increased by flushing the quasiparticle from B before each measurement.', 'cond-mat-0305190-1-14-1': 'To prepare the even parity state, a voltage pulse [MATH] is applied across B just prior to each measurement of its charge by E.', 'cond-mat-0305190-1-14-2': 'The amplitude of [MATH] is chosen such that [MATH], in order to release quasiparticles trapped by the electrostatic potential of the island without generating new quasiparticles, which would increase [MATH].', 'cond-mat-0305190-1-14-3': '(Note that a similar flushing action occurs automatically on the return branch at the end of each switching current measurement in B or E).', 'cond-mat-0305190-1-14-4': 'Switching histograms show that this procedure prepares the even state with a probability of about 85%.', 'cond-mat-0305190-1-14-5': 'Immediately after B is reset to its even state, the measurement ramp of [MATH] begins.', 'cond-mat-0305190-1-14-6': 'The result of this procedure is shown in the top curve in Fig. 4.', 'cond-mat-0305190-1-14-7': 'As one can see, the splitting of the step at [MATH] has vanished.', 'cond-mat-0305190-1-14-8': 'The imperfect preparation of the initial state and the residual rate of [MATH] ms[MATH] along with the ramping time of the current [MATH] still lead to switching events of E ,which correspond to an unpaired electron on the island of B [inset of Fig.4].', 'cond-mat-0305190-1-14-9': 'At [MATH] , where the switching events of E corresponding to even and odd parity states of B can be well distinguished, the odd parity state of B accounts for 30 of the total switching events.', 'cond-mat-0305190-1-14-10': 'This is in good agreement with the estimate (28) based on [MATH], the ramping time of the current [MATH]3 ms) and an initial probability of the even state (85).', 'cond-mat-0305190-1-14-11': 'Neglecting the poisoned events curve [MATH] vs. [MATH] results from the mixing of 0 and 1 Cooper pair states and is determined by the ratio [MATH] [CITATION].', 'cond-mat-0305190-1-14-12': 'The solid curve in the inset of Fig. 4 is calculated using the independently measured value of [MATH] 0.44 and is in reasonable agreement with the data.', 'cond-mat-0305190-1-15-0': 'In conclusion, our measurements clearly show that operation of SCPT electrometer at voltages [MATH]V leads to substantial generation of quasiparticles on the island of second SCPT which the electrometer measures.', 'cond-mat-0305190-1-15-1': 'The rate of quasiparticle generation depends linearly on the total number of quasiparticle tunneling events per second through the junctions of the electrometer.', 'cond-mat-0305190-1-15-2': 'To overcome this back-action from the electrometer, we operate it in the mode which uses switching current modulation for charge detection.', 'cond-mat-0305190-1-15-3': 'Using this mode of operation, we are able to recover the 2e-periodic Cooper staircase of the SCPT, which is expected both theoretically and from the 2e-periodicity of its switching current.', 'cond-mat-0305190-1-16-0': 'The authors thank D. V. Averin, J. R. Friedman and K. K. Likharev for useful discussions and W. Chen and V. V. Kuznetsov for technical assistance.', 'cond-mat-0305190-1-16-1': 'Work is supported in part by AFOSR grant No. F49620010001.'} | {'cond-mat-0305190-2-0-0': 'We report on the effect of the back-action of a Single Cooper Pair Transistor electrometer (E) on the measurement of charge on the island of a superconducting box (B).', 'cond-mat-0305190-2-0-1': 'The charge is e-periodic in the gate bias of B when E is operated near voltages [MATH] or [MATH].', 'cond-mat-0305190-2-0-2': 'We show that this is due to quasiparticle poisoning of B at a rate proportional to the number of quasiparticle tunneling events in E per second.', 'cond-mat-0305190-2-0-3': 'We are able to eliminate this back action and recover 2e charge periodicity using a new measurement method based on switching current modulation of E.', 'cond-mat-0305190-2-1-0': 'The superconducting box, e.g. a small island of Al film very weakly coupled to the outside circuitry by Josephson junctions, has shown considerable promise as a qubit for quantum information processing where the two states can be represented by superpositions of 0 or 1 excess Cooper pairs in the box [CITATION].', 'cond-mat-0305190-2-1-1': 'Measurement of the quantum state of this so-called charge qubit without inducing unwanted decoherence is a significant problem as is quasiparticle poisoning, i.e. the introduction of an unpaired electron (quasiparticle) into the box.', 'cond-mat-0305190-2-1-2': 'At temperatures of 10 mK or so, where experiments are commonly done, the number of quasiparticles should, in principle, be negligible.', 'cond-mat-0305190-2-1-3': 'However, such quasiparticle poisoning, due perhaps to the measurement process itself, is commonly observed.', 'cond-mat-0305190-2-1-4': 'A manifestation of this is seen in the so called Coulomb staircase.', 'cond-mat-0305190-2-1-5': 'When a charge, [MATH], is capacitively induced on the box, one expects Cooper pairs to tunnel resonantly into or out of the box at [MATH], where [MATH] is an odd integer, to maintain the lowest energy charge state of the box.', 'cond-mat-0305190-2-1-6': 'This results in the Coulomb staircase of the charge in the box [MATH] with period [MATH] in [MATH].', 'cond-mat-0305190-2-1-7': 'On the other hand, if there are quasiparticles in the system, then maintaining the lowest charging energy state also leads to quasiparticle tunneling.', 'cond-mat-0305190-2-1-8': 'This gives rise to splitting of the steps in the Coulomb staircase, which shifts toward e-periodicity as the number of quasiparticles increases [CITATION].', 'cond-mat-0305190-2-1-9': 'As a result, the lowest energy state of the box at [MATH] no longer corresponds to a resonant state of the Cooper pair tunneling.', 'cond-mat-0305190-2-1-10': 'For the box qubit, this means that relaxation does not bring the system back to its computational ground state at its operating point.', 'cond-mat-0305190-2-1-11': 'Since the ability to prepare the initial state of the qubit is an absolutely necessary condition for quantum computing, quasiparticle poisoning has been a serious road block for groups working to build a charge qubit [CITATION].', 'cond-mat-0305190-2-1-12': 'Solving this problem in charge qubits is essential.', 'cond-mat-0305190-2-2-0': 'The purpose of this paper is to investigate the effects of measurement (i.e. back-action) on the measured charge in the box and to develop approaches to minimize these effects.', 'cond-mat-0305190-2-2-1': 'For this study, we use two capacitively coupled Single Cooper Pair Transistors (SCPT)s, one of which acts as an electrometer (E) and the other as a superconducting box (B).', 'cond-mat-0305190-2-2-2': 'The latter gives a good representation of the box and at the same time allows us to study quasiparticle poisoning effects without needing to operate E.', 'cond-mat-0305190-2-2-3': 'The SCPT electrometer can be operated in several different modes when measuring the charge of B [CITATION].', 'cond-mat-0305190-2-2-4': 'Commonly, the charge measurements of B are done by operating E in the voltage modulation mode (VM).', 'cond-mat-0305190-2-2-5': 'In this mode, E is biased at a sufficiently high voltage that quasiparticles are generated, so it effectively functions as a SET where the source drain voltage is modulated by [MATH] with a period of [MATH].', 'cond-mat-0305190-2-2-6': 'However, it is known that the switching current of a SCPT, i.e. the current at which it switches hystereticaly from the low voltage or phase-diffusion branch to [MATH], is also charge sensitive [CITATION] and can be used for charge measurement.', 'cond-mat-0305190-2-2-7': 'We refer to this as the switching current mode (SW).', 'cond-mat-0305190-2-2-8': 'The SW mode of operation has been analyzed [CITATION], but until now no measurements of the charge on the island of a box using a SW mode of the electrometer have been reported.', 'cond-mat-0305190-2-2-9': 'We present the results of the measurements of the island charge in B by E operated in either the VM or SW mode.', 'cond-mat-0305190-2-2-10': 'The results demonstrate that measurement-induced poisoning, which leads to an e-periodic Coulomb staircase when using E in the VM mode, can be eliminated in the SW mode.', 'cond-mat-0305190-2-3-0': 'The parameters of the sample (Fig. 1.', 'cond-mat-0305190-2-3-1': 'a) are as follows: E has a normal state resistance [MATH] k[MATH], charging energy [MATH]eV and Josephson energy [MATH]eV.', 'cond-mat-0305190-2-3-2': 'For B these parameters are [MATH] k[MATH], [MATH]eV and [MATH]eV.', 'cond-mat-0305190-2-3-3': '[MATH] is the average of the Josephson energies of two junctions as determined from the values of [MATH] and the superconducting gap [MATH]eV by using the Ambegaokar-Baratoff formula.', 'cond-mat-0305190-2-3-4': '[MATH] is determined from the amplitude of maximum voltage modulation of the devices.', 'cond-mat-0305190-2-3-5': 'The coupling capacitance between E and B is determined to be 80 aF from their measured coupling of 4.8%.', 'cond-mat-0305190-2-3-6': 'All devices are made using standard two angle shadow evaporations without having normal metal quasiparticle traps close to junctions.', 'cond-mat-0305190-2-3-7': 'The sample is placed in microwave tight copper can located on a temperature regulated stage of a dilution refrigerator having a base temperature of 6 mK.', 'cond-mat-0305190-2-3-8': 'All the measurement leads are filtered by low temperature microwave filters [CITATION] that are thermally anchored to the mixing chamber.', 'cond-mat-0305190-2-4-0': 'Figure 1.', 'cond-mat-0305190-2-4-1': 'b shows the charge on the island of B measured with E in the VM mode.', 'cond-mat-0305190-2-4-2': 'During this measurement, the source and drain leads of B are at a common potential with respect to its gate.', 'cond-mat-0305190-2-4-3': 'Figure 1.c presents the switching current modulation of B measured with the bias current through E, [MATH], set equal to zero.', 'cond-mat-0305190-2-4-4': 'In this and following measurements, the bias current of B, [MATH], is ramped at a rate of [MATH] nA/s.', 'cond-mat-0305190-2-4-5': 'As can be seen, the switching current modulation of B is 2e-periodic as expected at low temperature, but the charge of B, measured by the electrometer, is e-periodic.', 'cond-mat-0305190-2-4-6': 'Similar dependences of [MATH] on [MATH] were measured with E biased in either of its voltage sensitive regions, i.e. near the gap where [MATH] or near the Josephson-quasiparticle peak where [MATH].', 'cond-mat-0305190-2-4-7': 'To determine if the e-periodicity of [MATH] is due to the back-action of E on B, the quasiparticle poisoning rate of B, [MATH], was measured for a range of bias conditions of E.', 'cond-mat-0305190-2-4-8': 'In addition to this, we studied how the biases of two other SCPTs located on the same chip, but coupled more weakly to B, affected [MATH].', 'cond-mat-0305190-2-4-9': 'Switching current histograms of B measured for different bias conditions of E. Corresponding time delay from the beginning of the current ramp is shown in the upper horizontal axis.', 'cond-mat-0305190-2-4-10': 'The histograms are shifted on vertical axis for clarity.', 'cond-mat-0305190-2-4-11': 'The "odd" ("even") peaks in the histograms correspond to switching from the odd (even) parity states of B. Inset: IV-characteristic of E. Arrows indicate bias conditions at which switching current histograms of the main figure were measured.', 'cond-mat-0305190-2-4-12': 'Diamonds mark positions where [MATH] has been measured for data in Fig. 3.', 'cond-mat-0305190-2-5-0': 'The details of the technique for determining [MATH] from switching current distributions have been reported previously [CITATION].', 'cond-mat-0305190-2-5-1': 'Briefly, as [MATH] is linearly ramped in time, the switching current histogram of B, with its gate biased near [MATH], exhibits two peaks if the number of quasiparticles on the island changes during the measurement of histogram.', 'cond-mat-0305190-2-5-2': 'One peak, [MATH], which is close to the maximum of the switching-current characteristic [MATH], occurs when the island has an even number of electrons (even state) and the other -much lower current- peak, [MATH], is near the predicted switching current minimum at [MATH] when one quasiparticle occupies the island (odd state) (Fig. 2).', 'cond-mat-0305190-2-5-3': 'If [MATH] and B has not switched, it must be in the even state.', 'cond-mat-0305190-2-5-4': 'The entry of a quasiparticle onto the island effectively changes [MATH] to [MATH] which for [MATH] will cause B to switch rapidly to the running state, giving the time of the poisoning and thus the quasiparticle poisoning rate [MATH].', 'cond-mat-0305190-2-5-5': 'Previous studies have shown that [MATH] is independent of [MATH] in the region between the peaks, giving an exponential decay of the even state.', 'cond-mat-0305190-2-5-6': 'Several of these histograms with increasing [MATH] as [MATH] and [MATH] increase are shown in Fig. 2.', 'cond-mat-0305190-2-5-7': 'The relatively low bandwidth of our filters limits these measurement to [MATH] ms[MATH].', 'cond-mat-0305190-2-6-0': 'First we determine [MATH] when all the electrodes of the electrometer are disconnected from the measurement circuitry and grounded.', 'cond-mat-0305190-2-6-1': 'In this case we still observe a small residual rate [MATH] ms[MATH] for [MATH].', 'cond-mat-0305190-2-6-2': 'This non-zero rate can be caused, e.g., by the presence of impurity levels in the superconducting gap of Al [CITATION].', 'cond-mat-0305190-2-6-3': 'For the present discussion it is clear that this small residual rate is not related to the back-action of E. [MATH] is unchanged if E is biased on its supercurrent branch or when E is biased on its return current branch at low voltage [MATH]V [MATH] as shown in Fig. 3.', 'cond-mat-0305190-2-6-4': 'For [MATH]V, [MATH] decreases rapidly, becoming too short to measure for slightly higher voltages.', 'cond-mat-0305190-2-6-5': 'The modulation characteristics of the current [MATH] with gate voltage also changes at this point from being 2e-periodic for [MATH]V to e-periodic for [MATH]V. Similar cross-overs from e to 2e periodicity at a voltage [MATH] have previously been seen in related systems (see e.g. [CITATION]).', 'cond-mat-0305190-2-6-6': 'Quasiparticle poisoning rate [MATH] of B as a function of the voltage across E. Inset: [MATH] as a function of the total quasiparticle tunneling current in E1 (diamonds) and of E2 (triangles).', 'cond-mat-0305190-2-6-7': 'The open and filled symbols correspond to bias conditions [MATH] or [MATH], respectively.', 'cond-mat-0305190-2-6-8': '[MATH] or [MATH] for the filled or open symbols, respectively.', 'cond-mat-0305190-2-7-0': 'In order to study how [MATH] depends on the voltage and current of E through its entire operating range, and in particular near [MATH] and [MATH], we use two other SCPTs fabricated on the same chip but much more weakly coupled to B.', 'cond-mat-0305190-2-7-1': 'Since these two SCPTs, which we call E1 and E2, are much more weakly coupled to B, they allow us to measure [MATH] for bias currents, [MATH], that are several orders of magnitude higher than is possible using E.', 'cond-mat-0305190-2-7-2': 'The island of SCPT E1 is located 96 [MATH]m from the island of B and has [MATH] k[MATH], while the corresponding parameters for E2 are 143 [MATH]m and [MATH] k[MATH].', 'cond-mat-0305190-2-7-3': 'For these devices, we can measure the rate [MATH] up to voltages [MATH].', 'cond-mat-0305190-2-7-4': 'Again, we see a small initial increase of [MATH] at [MATH]V and then sharp increases at voltages [MATH] and [MATH].', 'cond-mat-0305190-2-7-5': 'These voltages correspond approximately to the Josephson-quasiparticle tunneling and sequential quasiparticle tunneling thresholds in a SCPT and are accompanied by sharp increases in [MATH].', 'cond-mat-0305190-2-7-6': 'The inset of Fig. 3 shows the rate [MATH] as a function of the currents in E1 and E2 above these two thresholds.', 'cond-mat-0305190-2-7-7': 'To test the hypothesis that [MATH] is proportional to the total number of quasiparticle tunneling events, the currents for the electrometer voltages [MATH] are multiplied by two.', 'cond-mat-0305190-2-7-8': 'This is done since, for a given current at voltages [MATH], there are twice as many quasiparticle tunneling events through the junctions of E as for [MATH].', 'cond-mat-0305190-2-7-9': 'As one can see, this scaling collapses all of the data from each electrometer to a common line.', 'cond-mat-0305190-2-8-0': 'From these data one can conclude that the quasiparticle current of electrometer is the source of back-action noise leading to quasiparticle poisoning and an e-periodic Coulomb staircase of B when measured by E. Further, the quasiparticle generation in B is proportional to the total number of quasiparticle tunneling events per second through E.', 'cond-mat-0305190-2-8-1': 'This relationship could indicate that the back-action of E results from the shot noise of tunneling quasiparticles.', 'cond-mat-0305190-2-8-2': 'On the other hand, this back-action could also be the result of the recombination of quasiparticles in E into pairs.', 'cond-mat-0305190-2-8-3': 'This quasiparticle recombination produces phonons and to a smaller extent photons of energy [MATH].', 'cond-mat-0305190-2-8-4': 'These phonons/photons, which propagate from E to B without energy relaxation could generate quasiparticles in B. Determining the details of the interaction between E and B will require further work.', 'cond-mat-0305190-2-8-5': 'However, it is interesting to note that this sort of recombination noise would likely be suppressed by having normal metal leads close to the junctions.', 'cond-mat-0305190-2-8-6': 'This may provide an explanation of previous results [CITATION] in which it was possible to observe a 2e-periodic Coulomb staircase using a VM electrometer measuring a superconducting box, which had normal metal "quasiparticle traps" close to the junctions.', 'cond-mat-0305190-2-8-7': 'Average charge of B as a function of its gate charge when the electrometer operates in SW mode: bottom curve, quasiparticle flushing (see text) before each charge measurement; top curve, no flushing.', 'cond-mat-0305190-2-8-8': 'The top curve is shifted in vertical direction for clarity.', 'cond-mat-0305190-2-8-9': 'The flushing pulse has amplitude 150 [MATH]V and duration 25 ms. The dotted lines mark the positions at which the maxima of [MATH] appear.', 'cond-mat-0305190-2-8-10': 'Inset shows the average switching current of E as a function of its gate charge.', 'cond-mat-0305190-2-8-11': 'The arrows marked by "0","1" and "P" correspond to switching currents of E when the island of B has 0 or 1 excess Cooper pairs or when it is in the poisoned state, respectively.', 'cond-mat-0305190-2-9-0': 'Quasiparticle poisoning of B due to the measurement of its charge can, in principle, be eliminated by the operation of E in the SW mode, where its voltage remains well below [MATH] until the measurement is made.', 'cond-mat-0305190-2-9-1': 'The SW mode of operation of the electrometer is illustrated in the inset of Fig. 4, which shows its switching current vs. gate charge transfer-function.', 'cond-mat-0305190-2-9-2': 'In general, [MATH] contains a component proportional to [MATH], in addition to the externally applied bias.', 'cond-mat-0305190-2-9-3': 'So, to measure changes in [MATH] as a function of [MATH], we bias E near point "0", where the switching current is very sensitive to variations of external charge ( inset of Fig. 4), and record several hundred switching events of E for each value of [MATH].', 'cond-mat-0305190-2-9-4': 'These switching data are then corrected for the measured nonlinearity in the transfer-function of E and averaged.', 'cond-mat-0305190-2-9-5': 'The top curve in Fig. 4.', 'cond-mat-0305190-2-9-6': 'shows the result obtained using this procedure.', 'cond-mat-0305190-2-10-0': 'While these data are no longer strictly e-periodic, as in Fig. 1b, the Coulomb staircase in Fig. 4 (top line) still shows the split steps around [MATH] characteristic of quasiparticle poisoning.', 'cond-mat-0305190-2-10-1': 'We can see that this is consistent with the measured residual rate, [MATH], as follows.', 'cond-mat-0305190-2-10-2': 'The edges of these steps should occur for an energy difference between the even and odd states, [MATH], such that the rates for even to odd ([MATH]) and odd to even ([MATH]) transitions of the island are equal.', 'cond-mat-0305190-2-10-3': 'This ratio is given by [MATH] [CITATION], where [MATH] is the sum of the quasiparticle densities on the leads and the island in the even state.', 'cond-mat-0305190-2-10-4': 'For the odd state, this sum is [MATH], where [MATH] is the volume of the island.', 'cond-mat-0305190-2-10-5': 'The second term in [MATH] accounts for the extra quasiparticle occupying the island in the odd state, which increases the density by [MATH]m[MATH].', 'cond-mat-0305190-2-10-6': '[MATH] can be estimated from the measured residual poisoning rate [MATH], the normal density of states of the Al film and the junction resistances, giving [MATH]m[MATH].', 'cond-mat-0305190-2-10-7': 'Taking the electron temperature of B to be 15 mK, which is reasonable since B is completely passive in these measurements, gives a length for the short step of [MATH] in agreement with the data shown in Fig. 4.', 'cond-mat-0305190-2-11-0': 'The effects which result from the residual rate [MATH] can be greatly reduced by flushing the quasiparticle from the island of B before each measurement.', 'cond-mat-0305190-2-11-1': 'As one possible approach to prepare the even parity state, we apply a voltage pulse [MATH] across B just prior to each measurement.', 'cond-mat-0305190-2-11-2': 'The amplitude of [MATH] is chosen such that [MATH], in order to release the quasiparticle from the electrostatic potential of the island but yet not to generate any new quasiparticles by the pulse [CITATION].', 'cond-mat-0305190-2-11-3': 'Switching histograms of B show that this procedure prepares the even state with a probability of about 85%.', 'cond-mat-0305190-2-11-4': 'Immediately after B is flushed, the measurement ramp of [MATH] begins.', 'cond-mat-0305190-2-11-5': 'The result of this procedure is shown in the bottom curve in Fig. 4.', 'cond-mat-0305190-2-11-6': 'As one can see, the quasiparticle-induced splitting of the step at [MATH] is no longer apparent.', 'cond-mat-0305190-2-11-7': 'However, individual histograms still show about 30% of the switching events in E near [MATH] are from the poisoned state of B.', 'cond-mat-0305190-2-11-8': 'This is consistent with the imperfect preparation of the initial state and additional poisoning with rate [MATH] in the finite time between flushing and the switching event.', 'cond-mat-0305190-2-11-9': 'Thus we see that, with the effects of residual poisoning greatly reduced by the flushing, the measurement of [MATH] by E in the SW mode gives results consistent with the 2e-periodic switching-current distribution of B.', 'cond-mat-0305190-2-12-0': 'In conclusion, our measurements clearly show that operation of a SCPT electrometer at voltages [MATH]V causes a substantial generation of quasiparticles in the circuit of the superconducting box leading to an e-periodic Coulomb staircase.', 'cond-mat-0305190-2-12-1': 'The rate of quasiparticle poisoning in the box depends linearly on the total number of quasiparticle tunneling events per second through the junctions of the electrometer.', 'cond-mat-0305190-2-12-2': 'To overcome this back-action from the electrometer, we operate it in a mode which uses switching-current modulation for charge detection.', 'cond-mat-0305190-2-12-3': 'Using this mode of operation, we are able to recover the 2e-periodic Coulomb staircase of the SCPT in the box configuration, which is expected both theoretically and from the 2e-periodicity of its switching current.', 'cond-mat-0305190-2-13-0': 'The authors thank D. V. Averin, J. R. Friedman and K. K. Likharev for useful discussions and W. Chen and V. V. Kuznetsov for technical assistance.', 'cond-mat-0305190-2-13-1': 'Work is supported in part by AFOSR grant No. F49620010001.'} | [['cond-mat-0305190-1-4-3', 'cond-mat-0305190-2-3-2'], ['cond-mat-0305190-1-4-5', 'cond-mat-0305190-2-3-4'], ['cond-mat-0305190-1-4-6', 'cond-mat-0305190-2-3-5'], ['cond-mat-0305190-1-4-8', 'cond-mat-0305190-2-3-7'], ['cond-mat-0305190-1-4-9', 'cond-mat-0305190-2-3-8'], ['cond-mat-0305190-1-7-0', 'cond-mat-0305190-2-5-0'], ['cond-mat-0305190-1-7-3', 'cond-mat-0305190-2-5-3'], ['cond-mat-0305190-1-7-7', 'cond-mat-0305190-2-5-6'], ['cond-mat-0305190-1-7-8', 'cond-mat-0305190-2-5-7'], ['cond-mat-0305190-1-9-2', 'cond-mat-0305190-2-7-2'], ['cond-mat-0305190-1-9-4', 'cond-mat-0305190-2-7-4'], ['cond-mat-0305190-1-9-9', 'cond-mat-0305190-2-7-9'], ['cond-mat-0305190-1-8-4', 'cond-mat-0305190-2-6-4'], ['cond-mat-0305190-1-16-0', 'cond-mat-0305190-2-13-0'], ['cond-mat-0305190-1-3-6', 'cond-mat-0305190-2-2-7'], ['cond-mat-0305190-1-1-0', 'cond-mat-0305190-2-1-0'], 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1804.01505 | {'1804.01505-1-0-0': 'The dynamic linear response of a quantum system is critical for understanding both the structure and dynamics of strongly-interacting quantum systems, including neutron scattering from materials, photon and electron scattering from atomic systems, and electron and neutrino scattering by nuclei.', '1804.01505-1-0-1': 'We present a general algorithm for quantum computers to calculate the dynamic linear response function with controlled errors and to obtain information about specific final states that can be directly compared to experimental observations.', '1804.01505-1-1-0': 'Quantum computers should enable dramatic new capabilities in simulating quantum many-body systems, particularly their dynamic properties [CITATION].', '1804.01505-1-1-1': 'Quantum dynamics is in general extremely difficult to treat on a classical computer except for a few special cases such as very low-energy scattering where spectral decomposition in finite volumes enable direct connections between spectra and phase shifts in the scattering of 2- or 3-clusters [CITATION] or very high-energy scattering that can be treated as nearly non-interacting final states, including y-scaling in neutron or electron scattering [CITATION] or inclusive deep inelastic scattering in QCD [CITATION].', '1804.01505-1-1-2': 'The general problem is essentially intractable because of quantum interference, the rapidly oscillating phases that arise in the relevant path integrals.', '1804.01505-1-2-0': 'Perhaps the simplest quantum dynamics problem is the dynamic linear response, framed as the response of a quantum system to a small perturbation.', '1804.01505-1-2-1': 'Examples are ubiquitous, including for example neutron scattering on materials, photon scattering in atomic systems, and electron and neutrino scattering from atomic nuclei.', '1804.01505-1-2-2': 'The response of the system can in principle tell us much about the structure of the system being probed as well as important properties of the dynamics.', '1804.01505-1-2-3': 'In the case of neutrinos scattered by nuclei it is also used to infer properties of the neutrino itself including masses, mixing angles, the mass hierarchy and CP violation in the neutrino sector (eg. [CITATION]).', '1804.01505-1-3-0': 'The ability to accurately calculate the dynamic response over a wide range of energy and momentum transfers, augmented by the possibility of determining specific features of the final states, would revolutionize our ability to extract information from many kinds of scattering experiments.', '1804.01505-1-3-1': 'Some information on quantum dynamics can be obtained using classical computers even for relatively large systems, typically by computing imaginary-time correlation functions [CITATION].', '1804.01505-1-3-2': 'Even for systems where the ground state or thermal ensembles can be simulated free of any sign problem, it is extremely difficult to invert these correlation functions to obtain the exact dynamic response.', '1804.01505-1-3-3': 'In this paper we discuss methods to determine the dynamic response on a quantum computer, as well as to detect important features of explicit final states that can be directly compared to experimental data.', '1804.01505-1-3-4': 'Our approach is particularly well suited for problems defined on a lattice, but these lattice methods can of course also be used to simulate systems in the continuum over a wide but finite range of energy and momenta, for example in lattice studies of cold atoms [CITATION] and nuclear systems [CITATION].', '1804.01505-1-3-5': 'Also we restrict ourselves to the response from the quantum ground state (T=0), generalizations to finite temperature are possible by preparing states in thermal equilibrium rather than the ground state [CITATION].', '1804.01505-1-4-0': 'We note that the similar problem of evaluating chemical reaction rates [CITATION] and time-dependent correlation functions [CITATION] have been already investigated in the quantum-chemistry literature.', '1804.01505-1-4-1': 'Our proposed algorithm improves on these earlier techniques in that our strategy is completely general, does not depend on simplifying assumptions on the excitation operator (for example, being able to diagonalize it as in [CITATION]) and requires only a polynomial number of measurements (instead of exponential like in [CITATION]).', '1804.01505-1-4-2': 'Also, working directly in frequency space allows us a direct access to the final states of a reaction which can be further analyzed.', '1804.01505-1-4-3': 'This is particularly important for neutrinos where the momentum and energy transfer are a priori unknown.', '1804.01505-1-5-0': 'Furthermore we are able to provide rigorous cost and error estimates of the computed dynamical properties.', '1804.01505-1-5-1': 'Available algorithms for evaluating energy spectra [CITATION] can in principle be adapted to compute response functions but they require resolution of individual excited states which grows exponentially in number for large systems.', '1804.01505-1-6-0': 'The paper is organized as follows, in Sec. [REF] we provide detailed definition of the Dynamical Response Function and describe the implementation of our method.', '1804.01505-1-6-1': 'In Sec. [REF] we provide an example of final state characterization by discussing the estimation of the one- and two-body momentum distribution and conclude in Sec. [REF].', '1804.01505-1-7-0': '# Method', '1804.01505-1-8-0': 'In the linear regime the response of a system of interacting particles due to a perturbative probe characterized by the excitation operator [MATH] can be fully characterized using the Dynamical Response Function, which can be expressed as [EQUATION] where [MATH] is the ground-state of the system with energy [MATH], [MATH] are the final states of the reaction with energies [MATH] and [MATH] is the energy transfer.', '1804.01505-1-8-1': "It is convenient to rescale the response function so that it's zero moment (the integral over frequencies) is [MATH]; this can be achieved by defining [EQUATION]", '1804.01505-1-8-2': 'The final normalization can be restored by either using the knowledge of one of the sum rules or by direct evaluation of the ground state expectation value [MATH].', '1804.01505-1-8-3': 'Understanding this, in the following we will drop the superscript [MATH].', '1804.01505-1-9-0': 'Our goal is to estimate the dynamical response function [MATH] with energy resolution [MATH] and a precision [MATH] with probability [MATH].', '1804.01505-1-9-1': 'We will indicate the difference between the largest eigenvalue of [MATH] and the ground state energy by: [MATH].', '1804.01505-1-9-2': 'Note that this quantity grows only polynomially with system size for most Hamiltonians of interest (see discussion below).', '1804.01505-1-10-0': 'In the following we will assume to have access to three black-box quantum procedures (oracles):', '1804.01505-1-11-0': 'Even though the oracle [MATH] may be impractical to implement for a general Hamiltonian, for most systems of interest many different algorithms are available in the literature ([CITATION]) and some have already be tested on simple nuclear systems [CITATION].', '1804.01505-1-11-1': 'Also, close to optimal strategies to implement the time-evolution operator for sparse Hamiltonians are known [CITATION] and for Hubbard-type Hamiltonians (like those derived within lattice-EFT [CITATION]) efficient implementations of Trotter steps with sub-linear circuit depth are available [CITATION].', '1804.01505-1-11-2': 'For the common case where [MATH] is a one-body operator the latter strategies can be used to implement [MATH] efficiently.', '1804.01505-1-12-0': 'Our scheme is composed of two quantum circuits', '1804.01505-1-13-0': 'For typical situations where the implementation of [MATH] requires considerable effort the success probability of the first routine can be amplified to [MATH] with additional [MATH] calls to the oracle [MATH].', '1804.01505-1-13-1': 'An alternative algorithm which removes the dependence of [MATH] on [MATH] but is more difficult to make deterministic is also presented in Sec. [REF].', '1804.01505-1-14-0': 'This whole circuit needs to be run a number of times given approximately by [EQUATION] independent of the target resolution [MATH].', '1804.01505-1-15-0': 'In summary, for a given choice of the excitation operator [MATH] our algorithm can be described by the following steps:', '1804.01505-1-16-0': 'iteration number less than [MATH] prepare the ground state using [MATH] run the first quantum algorithm (Sec. [REF]) algorithm succeeds we have prepared [MATH] run the second quantum algorithm (Sec. [REF]) store result for classical post-processing final state information needed measure final state (eg.', '1804.01505-1-16-1': 'Sec [REF])', '1804.01505-1-17-0': 'In the next sections we describe in detail the implementation of the two quantum routines introduced above.', '1804.01505-1-17-1': 'We also present examples obtained by classical simulation of a simple 2D fermionic system described by the Hubbard hamiltonian [EQUATION] where [MATH] indicates the nearest-neighbor lattice sites and [MATH] denotes the number operator.', '1804.01505-1-17-2': 'The results shown here were obtained for [MATH] "nucleons", [MATH] lattice sites and [MATH].', '1804.01505-1-17-3': 'These parameters are chosen to give a bound state considerably smaller than the lattice.', '1804.01505-1-18-0': '## State preparation algorithm', '1804.01505-1-19-0': 'The first problem we have to solve is the preparation of the state [MATH] given a quantum register initialized in the ground-state [MATH].', '1804.01505-1-19-1': "Let's start by adding an ancilla qubit and defining the unitary operator [EQUATION] where the Pauli [MATH] operator acts on the ancilla and the final matrix representation is on the basis spanned by the states [MATH] of the ancilla.", '1804.01505-1-19-2': 'Note that this unitary can be implemented efficiently with just [MATH] calls to a controlled version of the oracle [MATH] and additional [MATH] one-qubit gates.', '1804.01505-1-20-0': 'By initializing the ancilla register to [MATH], applying [MATH] and measuring the state [MATH] we have effectively produced [EQUATION] which differs from the wanted state by corrections of order [MATH].', '1804.01505-1-20-1': 'The error in the implementation of the unitary [MATH] needs to be at least of the same order, which means a simple single Trotter step will suffice.', '1804.01505-1-20-2': 'The state preparation has a success probability of [EQUATION]', '1804.01505-1-20-3': 'This approach for the application of a non-unitary transformation is similar in spirit to earlier work (see eg. [CITATION]) and it suffers from a possibly very low efficiency since we may need [MATH] trials to succeed.', '1804.01505-1-20-4': 'One option is to perform the algorithm at a few relatively large values of [MATH] and fit a quadratic function to extrapolate out the error from the final response function.', '1804.01505-1-20-5': 'This approach is however complicated if one is interested also in properties of the final states.', '1804.01505-1-20-6': 'A second approach, already proposed in [CITATION], is to repeat the application of the unitary [MATH] until success.', '1804.01505-1-20-7': 'This works because [MATH] is approximately the identity.', '1804.01505-1-20-8': 'In order to obtain a success probability [MATH] we will need [MATH] repetitions.', '1804.01505-1-20-9': "In addition, if the inverse [MATH] of the ground-state preparation circuit is available then it's possible to use Amplitude Amplification [CITATION] to gain a quadratic speedup over this .", '1804.01505-1-21-0': 'Note that by using the normalized state [MATH] we will compute the normalized response function Eq. [REF].', '1804.01505-1-21-1': 'If no sum-rules are known one can estimate the normalization constant by estimating the success probability Eq. [REF] at different values of [MATH] and extrapolating.', '1804.01505-1-22-0': 'Since the state preparation through the unitary [MATH] is only approximate, the parameter [MATH] would need to depend on the final target accuracy.', '1804.01505-1-22-1': 'As mentioned in the introduction an alternative scheme that avoids this problem by removing the error in Eq. [REF] can be obtained by representing the excitation operator [MATH] as a linear combination of [MATH] unitary matrices [EQUATION] which can be efficiently implemented employing additional [MATH] ancilla qubits using known techniques [CITATION].', '1804.01505-1-22-2': 'The success probability in this case is given by [EQUATION] which depending on the particular case may be larger than Eq. [REF].', '1804.01505-1-22-3': 'The main drawback of this approach is that Amplitude Amplification is the only process that can make the algorithm deterministic since upon failure the output state can in general be very different from the starting point.', '1804.01505-1-23-0': '## Response Function estimation', '1804.01505-1-24-0': 'We now present our strategy to obtain the response function trough the standard Phase Estimation Algorithm (PEA) [CITATION].', '1804.01505-1-24-1': 'It is convenient to shift and scale the original Hamiltonian: [EQUATION] so that we map the energy spectrum to [MATH].', '1804.01505-1-25-0': 'By direct calculation we see that the response function [MATH] obtained from [MATH] is related to the original one by [EQUATION] for a scaled frequency [MATH].', '1804.01505-1-26-0': 'Our goal is to estimate [MATH] efficiently.', '1804.01505-1-26-1': 'We do this by using PEA on an auxiliary register of [MATH] qubits with the evolution operators [EQUATION] for [MATH].', '1804.01505-1-26-2': 'The resulting circuit will have depth [MATH], where the first term comes from the inverse Quantum Fourier Transform [CITATION] and [MATH] is the gate count needed for a time evolution of [MATH] using the oracle [MATH].', '1804.01505-1-26-3': 'The resulting probability of measuring the [MATH] ancilla qubits in the binary representation of the integer [MATH] is (see eg. [CITATION] for more details) [EQUATION] where [MATH] is the well-known Fejer kernel from Fourier analysis (see eg. [CITATION]).', '1804.01505-1-26-4': 'The probability distribution [MATH] is a good approximation of [MATH] since this kernel can be seen as a representation of the delta function with width [MATH].', '1804.01505-1-26-5': 'Therefore if we require a frequency resolution [MATH] we will need [MATH] auxiliary qubits and a polynomial number of applications of the time evolution operator to obtain a sample from [MATH].', '1804.01505-1-27-0': 'As mentioned above, for most Hamiltonians of interest the energy gap [MATH] scales only polynomially with the size of the system.', '1804.01505-1-28-0': 'We now need to estimate [MATH] from [MATH] samples drawn from it.', '1804.01505-1-28-1': 'Since [MATH] is a discrete variable an efficient way of reconstructing the probability distribution is by producing an histogram [MATH] from the samples.', '1804.01505-1-28-2': "Using Hoeffding's inequality [CITATION] we find that [EQUATION] which implies in order to obtain a precision [MATH] with probability [MATH] we need approximately [EQUATION] independent samples.", '1804.01505-1-29-0': 'In Fig. [REF] we plot the approximate response [MATH] for the model Hamiltonian Eq. [REF] at three different values of [MATH] (6,8,12).', '1804.01505-1-29-1': 'By comparing with the exact result shown as black dots, we see that for [MATH] the effect of energy resolution is negligible but already with [MATH] we obtain a rather accurate estimate for [MATH].', '1804.01505-1-29-2': 'Even [MATH] reproduces important features of the response, which in experiments is convoluted with the detector resolution.', '1804.01505-1-29-3': 'The inset shows the convergence of the maximum error [EQUATION] as a function of the sample size [MATH].', '1804.01505-1-29-4': 'Response functions relevant for [MATH] and [MATH] scattering are typically smooth at high energy and hence require small [MATH] and short propagation times.', '1804.01505-1-30-0': 'Finally, in order to obtain a negligible bias from the state preparation we need the parameter [MATH] to scale as [EQUATION] for some constant [MATH].', '1804.01505-1-30-1': 'Note that the Hamiltonian evolution implemented in [MATH] has to have an error [MATH] to be negligible (luckily algorithms with only logarithmic dependence on [MATH] are known [CITATION]).', '1804.01505-1-31-0': 'This concludes the proof of the scalings [REF] and [REF].', '1804.01505-1-32-0': '# Final state measurements', '1804.01505-1-33-0': 'In electron- or neutrino-nuclear scattering experiments [CITATION] one would like to infer the probability [MATH] that the probe transferred energy-momentum [MATH] to the nucleus and simultaneously that the final state includes a nucleon (or neutron or proton) of momentum [MATH].', '1804.01505-1-33-1': 'More concretely this amounts to an inference procedure of the form [EQUATION] where [MATH] results from the experimental measure, [MATH] is the momentum distribution of the final states for a process with given [MATH] and [MATH].', '1804.01505-1-33-2': 'The prior probability [MATH] depends on the static response of the nucleus and the characteristic of the probe beam and can be updated given the other ones by a Bayesian procedure.', '1804.01505-1-33-3': 'The above section explains how to obtain [MATH] with a given accuracy and in the following we will show how to evaluate few-body momentum distributions given by the final state of the algorithm above.', '1804.01505-1-33-4': 'Note that after measuring the [MATH] ancilla qubits of Sec. [REF] the main register will be left in a state [MATH] composed by a linear superposition of final states corresponding to energy transfer [MATH].', '1804.01505-1-33-5': 'Imagine we want now to compute exclusive 1 and 2-body momentum distributions [EQUATION] where [MATH] is the number operator for a state with momentum [MATH], spin [MATH] and isospin [MATH].', '1804.01505-1-33-6': 'We can define a unitary operator [MATH] (which is efficiently implementable) and run the following circuit with an ancilla qubit [EQUATION]', '1804.01505-1-33-7': 'By using the idempotence of [MATH] we find [EQUATION] and we can then extract the expectation value by estimating these probabilities.', '1804.01505-1-33-8': 'Note that we may use the same procedure with [MATH] to estimate [MATH] (and possibly higher body momentum distributions).', '1804.01505-1-33-9': 'We can get a better strategy by reusing the final state of circuit [REF] upon measuring the ancilla in [MATH] and running it again with [MATH] since the probabilities now will be [EQUATION]', '1804.01505-1-33-10': 'Note that [MATH] will in general be contaminated by final state interactions but we can access a better approximation to an asymptotic state by evolving it in time using [MATH].', '1804.01505-1-34-0': 'This measurement procedure will need to then be repeated a polynomial number of times for all the observables of interest.', '1804.01505-1-34-1': 'Given the expensive procedure needed to generate the final states a better strategy to estimate multiple observables per iteration may be needed for greater efficiency.', '1804.01505-1-34-2': "One option is using state reconstruction techniques developed in quantum tomography [CITATION] or devising strategies tailored to the particular system studied and it's encoding on the quantum computer.", '1804.01505-1-35-0': '# Conclusions', '1804.01505-1-36-0': 'We presented a complete quantum algorithm for calculating the linear response of a quantum system to external perturbations with controllable accuracy.', '1804.01505-1-36-1': 'This is achieved by probabilistically preparing the perturbed state (even though a deterministic preparation with polynomial cost is in general available) and then analyzing it by using the standard Phase Estimation Algorithm [CITATION].', '1804.01505-1-36-2': 'Our approach is efficient (scaling is polynomial in system size and required accuracy) and provides direct access to the final states resulting from the perturbation, a property that potentially makes it extremely valuable to the interpretation of ongoing and planned scattering experiments.'} | {'1804.01505-2-0-0': 'The dynamic linear response of a quantum system is critical for understanding both the structure and dynamics of strongly-interacting quantum systems, including neutron scattering from materials, photon and electron scattering from atomic systems, and electron and neutrino scattering by nuclei.', '1804.01505-2-0-1': 'We present a general algorithm for quantum computers to calculate the dynamic linear response function with controlled errors and to obtain information about specific final states that can be directly compared to experimental observations.', '1804.01505-2-1-0': 'Quantum computers should enable dramatic new capabilities in simulating quantum many-body systems, particularly their dynamic properties [CITATION].', '1804.01505-2-1-1': 'Quantum dynamics is in general extremely difficult to treat on a classical computer except for a few special cases such as very low-energy scattering where spectral decomposition in finite volumes enable direct connections between spectra and phase shifts in the scattering of 2- or 3-clusters [CITATION] or very high-energy scattering that can be treated as nearly non-interacting final states, including y-scaling in neutron or electron scattering [CITATION] or inclusive deep inelastic scattering in QCD [CITATION].', '1804.01505-2-1-2': 'The general problem is essentially intractable because of quantum interference, the rapidly oscillating phases that arise in the relevant path integrals.', '1804.01505-2-2-0': 'Perhaps the simplest quantum dynamics problem is the dynamic linear response, framed as the response of a quantum system to a small perturbation.', '1804.01505-2-2-1': 'Examples are ubiquitous, including for example neutron scattering on materials, photon scattering in atomic systems, and electron and neutrino scattering from atomic nuclei.', '1804.01505-2-2-2': 'The response of the system can in principle tell us much about the structure of the system being probed as well as important properties of the dynamics.', '1804.01505-2-2-3': 'In the case of neutrinos scattered by nuclei it is also used to infer properties of the neutrino itself including masses, mixing angles, the mass hierarchy and CP violation in the neutrino sector (eg. [CITATION]).', '1804.01505-2-3-0': 'The ability to accurately calculate the dynamic response over a wide range of energy and momentum transfers, augmented by the possibility of determining specific features of the final states, would revolutionize our ability to extract information from many kinds of scattering experiments.', '1804.01505-2-3-1': 'Some information on quantum dynamics can be obtained using classical computers even for relatively large systems, typically by computing imaginary-time correlation functions [CITATION].', '1804.01505-2-3-2': 'Even for systems where the ground state or thermal ensembles can be simulated free of any sign problem, it is extremely difficult to invert these correlation functions to obtain the exact dynamic response.', '1804.01505-2-3-3': 'In this paper we discuss methods to determine the dynamic response on a quantum computer, as well as to detect important features of explicit final states that can be directly compared to experimental data.', '1804.01505-2-3-4': 'Our approach is particularly well suited for problems defined on a lattice, but these lattice methods can of course also be used to simulate systems in the continuum over a wide but finite range of energy and momenta, for example in lattice studies of cold atoms [CITATION] and nuclear systems [CITATION].', '1804.01505-2-3-5': 'Also we restrict ourselves to the response from the quantum ground state (T=0), generalizations to finite temperature are possible by preparing states in thermal equilibrium rather than the ground state [CITATION].', '1804.01505-2-4-0': 'We note that the similar problem of evaluating chemical reaction rates [CITATION] and time-dependent correlation functions [CITATION] have been already investigated in the quantum-chemistry literature.', '1804.01505-2-4-1': 'Our proposed algorithm improves on these earlier techniques in that our strategy is completely general, does not depend on simplifying assumptions on the excitation operator (for example, being able to diagonalize it as in [CITATION]) and requires only a polynomial number of measurements (instead of exponential like in [CITATION]).', '1804.01505-2-4-2': 'Also, working directly in frequency space allows us a direct access to the final states of a reaction which can be further analyzed.', '1804.01505-2-4-3': 'This is particularly important for neutrinos where the momentum and energy transfer are a priori unknown.', '1804.01505-2-5-0': 'Furthermore we are able to provide rigorous cost and error estimates of the computed dynamical properties.', '1804.01505-2-5-1': 'Available algorithms for evaluating energy spectra [CITATION] can in principle be adapted to compute response functions but they require resolution of individual excited states which grows exponentially in number for large systems.', '1804.01505-2-6-0': 'The paper is organized as follows, in Sec. [REF] we provide detailed definition of the Dynamical Response Function and describe the implementation of our method.', '1804.01505-2-6-1': 'In Sec. [REF] we provide an example of final state characterization by discussing the estimation of the one- and two-body momentum distribution and conclude in Sec. [REF].', '1804.01505-2-7-0': '# Method', '1804.01505-2-8-0': 'In the linear regime the response of a system of interacting particles due to a perturbative probe characterized by the excitation operator [MATH] can be fully characterized using the Dynamical Response Function, which can be expressed as [EQUATION] where [MATH] is the ground-state of the system with energy [MATH], [MATH] are the final states of the reaction with energies [MATH] and [MATH] is the energy transfer.', '1804.01505-2-8-1': "It is convenient to rescale the response function so that it's zero moment (the integral over frequencies) is [MATH]; this can be achieved by defining [EQUATION]", '1804.01505-2-8-2': 'The final normalization can be restored by either using the knowledge of one of the sum rules or by direct evaluation of the ground state expectation value [MATH].', '1804.01505-2-8-3': 'Understanding this, in the following we will drop the superscript [MATH].', '1804.01505-2-9-0': 'Our goal is to estimate the dynamical response function [MATH] with energy resolution [MATH] and a precision [MATH] with probability [MATH].', '1804.01505-2-9-1': 'We will indicate the difference between the largest eigenvalue of [MATH] and the ground state energy by: [MATH].', '1804.01505-2-9-2': 'Note that this quantity grows only polynomially with system size for most Hamiltonians of interest (see discussion below).', '1804.01505-2-10-0': 'In the following we will assume to have access to three black-box quantum procedures (oracles):', '1804.01505-2-11-0': 'Even though the oracle [MATH] may be impractical to implement for a general Hamiltonian, for most systems of interest many different algorithms are available in the literature ([CITATION]) and some have already be tested on simple nuclear systems [CITATION].', '1804.01505-2-11-1': 'Also, close to optimal strategies to implement the time-evolution operator for sparse Hamiltonians are known [CITATION] and for Hubbard-type Hamiltonians (like those derived within lattice-EFT [CITATION]) efficient implementations of Trotter steps with sub-linear circuit depth are available [CITATION].', '1804.01505-2-11-2': 'For the common case where [MATH] is a one-body operator the latter strategies can be used to implement [MATH] efficiently.', '1804.01505-2-12-0': 'Our scheme is composed of two quantum circuits', '1804.01505-2-13-0': 'For typical situations where the implementation of [MATH] requires considerable effort the success probability of the first routine can be amplified to [MATH] with additional [MATH] calls to the oracle [MATH].', '1804.01505-2-13-1': 'An alternative algorithm which removes the dependence of [MATH] on [MATH] but is more difficult to make deterministic is also presented in Sec. [REF].', '1804.01505-2-14-0': 'This whole circuit needs to be run a number of times given approximately by [EQUATION] independent of the target resolution [MATH].', '1804.01505-2-15-0': 'In summary, for a given choice of the excitation operator [MATH] our algorithm can be described by the following steps:', '1804.01505-2-16-0': 'iteration number less than [MATH] prepare the ground state using [MATH] run the first quantum algorithm (Sec. [REF]) algorithm succeeds we have prepared [MATH] run the second quantum algorithm (Sec. [REF]) store result for classical post-processing final state information needed measure final state (eg.', '1804.01505-2-16-1': 'Sec [REF])', '1804.01505-2-17-0': 'In the next sections we describe in detail the implementation of the two quantum routines introduced above.', '1804.01505-2-17-1': 'We also present examples obtained by classical simulation of a simple 2D fermionic system described by the Hubbard hamiltonian [EQUATION] where [MATH] indicates the nearest-neighbor lattice sites and [MATH] denotes the number operator.', '1804.01505-2-17-2': 'The results shown here were obtained for [MATH] "nucleons", [MATH] lattice sites and [MATH].', '1804.01505-2-17-3': 'These parameters are chosen to give a bound state considerably smaller than the lattice.', '1804.01505-2-18-0': '## State preparation algorithm', '1804.01505-2-19-0': 'The first problem we have to solve is the preparation of the state [MATH] given a quantum register initialized in the ground-state [MATH].', '1804.01505-2-19-1': "Let's start by adding an ancilla qubit and defining the unitary operator [EQUATION] where the Pauli [MATH] operator acts on the ancilla and the final matrix representation is on the basis spanned by the states [MATH] of the ancilla.", '1804.01505-2-19-2': 'Note that this unitary can be implemented efficiently with just [MATH] calls to a controlled version of the oracle [MATH] and additional [MATH] one-qubit gates.', '1804.01505-2-20-0': 'By initializing the ancilla register to [MATH], applying [MATH] and measuring the state [MATH] we have effectively produced [EQUATION] which differs from the wanted state by corrections of order [MATH].', '1804.01505-2-20-1': 'The error in the implementation of the unitary [MATH] needs to be at least of the same order, which means a simple single Trotter step will suffice.', '1804.01505-2-20-2': 'The state preparation has a success probability of [EQUATION]', '1804.01505-2-20-3': 'This approach for the application of a non-unitary transformation is similar in spirit to earlier work (see eg. [CITATION]) and it suffers from a possibly very low efficiency since we may need [MATH] trials to succeed.', '1804.01505-2-20-4': 'One option is to perform the algorithm at a few relatively large values of [MATH] and fit a quadratic function to extrapolate out the error from the final response function.', '1804.01505-2-20-5': 'This approach is however complicated if one is interested also in properties of the final states.', '1804.01505-2-20-6': 'A second approach, already proposed in [CITATION], is to repeat the application of the unitary [MATH] until success.', '1804.01505-2-20-7': 'This works because [MATH] is approximately the identity.', '1804.01505-2-20-8': 'In order to obtain a success probability [MATH] we will need [MATH] repetitions.', '1804.01505-2-20-9': "In addition, if the inverse [MATH] of the ground-state preparation circuit is available then it's possible to use Amplitude Amplification [CITATION] to gain a quadratic speedup over this .", '1804.01505-2-21-0': 'Note that by using the normalized state [MATH] we will compute the normalized response function Eq. [REF].', '1804.01505-2-21-1': 'If no sum-rules are known one can estimate the normalization constant by estimating the success probability Eq. [REF] at different values of [MATH] and extrapolating.', '1804.01505-2-22-0': 'Since the state preparation through the unitary [MATH] is only approximate, the parameter [MATH] would need to depend on the final target accuracy.', '1804.01505-2-22-1': 'As mentioned in the introduction an alternative scheme that avoids this problem by removing the error in Eq. [REF] can be obtained by representing the excitation operator [MATH] as a linear combination of [MATH] unitary matrices [EQUATION] which can be efficiently implemented employing additional [MATH] ancilla qubits using known techniques [CITATION].', '1804.01505-2-22-2': 'The success probability in this case is given by [EQUATION] which depending on the particular case may be larger than Eq. [REF].', '1804.01505-2-22-3': 'The main drawback of this approach is that Amplitude Amplification is the only process that can make the algorithm deterministic since upon failure the output state can in general be very different from the starting point.', '1804.01505-2-23-0': '## Response Function estimation', '1804.01505-2-24-0': 'We now present our strategy to obtain the response function trough the standard Phase Estimation Algorithm (PEA) [CITATION].', '1804.01505-2-24-1': 'It is convenient to shift and scale the original Hamiltonian: [EQUATION] so that we map the energy spectrum to [MATH].', '1804.01505-2-25-0': 'By direct calculation we see that the response function [MATH] obtained from [MATH] is related to the original one by [EQUATION] for a scaled frequency [MATH].', '1804.01505-2-26-0': 'Our goal is to estimate [MATH] efficiently.', '1804.01505-2-26-1': 'We do this by using PEA on an auxiliary register of [MATH] qubits with the evolution operators [EQUATION] for [MATH].', '1804.01505-2-26-2': 'The resulting circuit will have depth [MATH], where the first term comes from the inverse Quantum Fourier Transform [CITATION] and [MATH] is the gate count needed for a time evolution of [MATH] using the oracle [MATH].', '1804.01505-2-26-3': 'The resulting probability of measuring the [MATH] ancilla qubits in the binary representation of the integer [MATH] is (see eg. [CITATION] for more details) [EQUATION] where [MATH] is the well-known Fejer kernel from Fourier analysis (see eg. [CITATION]).', '1804.01505-2-26-4': 'The probability distribution [MATH] is a good approximation of [MATH] since this kernel can be seen as a representation of the delta function with width [MATH].', '1804.01505-2-26-5': 'Therefore if we require a frequency resolution [MATH] we will need [MATH] auxiliary qubits and a polynomial number of applications of the time evolution operator to obtain a sample from [MATH].', '1804.01505-2-27-0': 'As mentioned above, for most Hamiltonians of interest the energy gap [MATH] scales only polynomially with the size of the system.', '1804.01505-2-28-0': 'We now need to estimate [MATH] from [MATH] samples drawn from it.', '1804.01505-2-28-1': 'Since [MATH] is a discrete variable an efficient way of reconstructing the probability distribution is by producing an histogram [MATH] from the samples.', '1804.01505-2-28-2': "Using Hoeffding's inequality [CITATION] we find that [EQUATION] which implies in order to obtain a precision [MATH] with probability [MATH] we need approximately [EQUATION] independent samples.", '1804.01505-2-29-0': 'In Fig. [REF] we plot the approximate response [MATH] for the model Hamiltonian Eq. [REF] at three different values of [MATH] (6,8,12).', '1804.01505-2-29-1': 'By comparing with the exact result shown as black dots, we see that for [MATH] the effect of energy resolution is negligible but already with [MATH] we obtain a rather accurate estimate for [MATH].', '1804.01505-2-29-2': 'Even [MATH] reproduces important features of the response, which in experiments is convoluted with the detector resolution.', '1804.01505-2-29-3': 'The inset shows the convergence of the maximum error [EQUATION] as a function of the sample size [MATH].', '1804.01505-2-29-4': 'Response functions relevant for [MATH] and [MATH] scattering are typically smooth at high energy and hence require small [MATH] and short propagation times.', '1804.01505-2-30-0': 'Finally, in order to obtain a negligible bias from the state preparation we need the parameter [MATH] to scale as [EQUATION] for some constant [MATH].', '1804.01505-2-30-1': 'Note that the Hamiltonian evolution implemented in [MATH] has to have an error [MATH] to be negligible (luckily algorithms with only logarithmic dependence on [MATH] are known [CITATION]).', '1804.01505-2-31-0': 'This concludes the proof of the scalings [REF] and [REF].', '1804.01505-2-32-0': '# Final state measurements', '1804.01505-2-33-0': 'In electron- or neutrino-nuclear scattering experiments [CITATION] one would like to infer the probability [MATH] that the probe transferred energy-momentum [MATH] to the nucleus and simultaneously that the final state includes a nucleon (or neutron or proton) of momentum [MATH].', '1804.01505-2-33-1': 'More concretely this amounts to an inference procedure of the form [EQUATION] where [MATH] results from the experimental measure, [MATH] is the momentum distribution of the final states for a process with given [MATH] and [MATH].', '1804.01505-2-33-2': 'The prior probability [MATH] depends on the static response of the nucleus and the characteristic of the probe beam and can be updated given the other ones by a Bayesian procedure.', '1804.01505-2-33-3': 'The above section explains how to obtain [MATH] with a given accuracy and in the following we will show how to evaluate few-body momentum distributions given by the final state of the algorithm above.', '1804.01505-2-33-4': 'Note that after measuring the [MATH] ancilla qubits of Sec. [REF] the main register will be left in a state [MATH] composed by a linear superposition of final states corresponding to energy transfer [MATH].', '1804.01505-2-33-5': 'Imagine we want now to compute exclusive 1 and 2-body momentum distributions [EQUATION] where [MATH] is the number operator for a state with momentum [MATH], spin [MATH] and isospin [MATH].', '1804.01505-2-33-6': 'We can define a unitary operator [MATH] (which is efficiently implementable) and run the following circuit with an ancilla qubit [EQUATION]', '1804.01505-2-33-7': 'By using the idempotence of [MATH] we find [EQUATION] and we can then extract the expectation value by estimating these probabilities.', '1804.01505-2-33-8': 'Note that we may use the same procedure with [MATH] to estimate [MATH] (and possibly higher body momentum distributions).', '1804.01505-2-33-9': 'We can get a better strategy by reusing the final state of circuit [REF] upon measuring the ancilla in [MATH] and running it again with [MATH] since the probabilities now will be [EQUATION]', '1804.01505-2-33-10': 'Note that [MATH] will in general be contaminated by final state interactions but we can access a better approximation to an asymptotic state by evolving it in time using [MATH].', '1804.01505-2-34-0': 'This measurement procedure will need to then be repeated a polynomial number of times for all the observables of interest.', '1804.01505-2-34-1': 'Given the expensive procedure needed to generate the final states a better strategy to estimate multiple observables per iteration may be needed for greater efficiency.', '1804.01505-2-34-2': "One option is using state reconstruction techniques developed in quantum tomography [CITATION] or devising strategies tailored to the particular system studied and it's encoding on the quantum computer.", '1804.01505-2-35-0': '# Conclusions', '1804.01505-2-36-0': 'We presented a complete quantum algorithm for calculating the linear response of a quantum system to external perturbations with controllable accuracy.', '1804.01505-2-36-1': 'This is achieved by probabilistically preparing the perturbed state (even though a deterministic preparation with polynomial cost is in general available) and then analyzing it by using the standard Phase Estimation Algorithm [CITATION].', '1804.01505-2-36-2': 'Our approach is efficient (scaling is polynomial in system size and required accuracy) and provides direct access to the final states resulting from the perturbation, a property that potentially makes it extremely valuable to the interpretation of ongoing and planned scattering experiments.'} | [['1804.01505-1-34-0', '1804.01505-2-34-0'], ['1804.01505-1-34-1', '1804.01505-2-34-1'], ['1804.01505-1-34-2', '1804.01505-2-34-2'], ['1804.01505-1-0-0', '1804.01505-2-0-0'], ['1804.01505-1-0-1', '1804.01505-2-0-1'], ['1804.01505-1-4-0', '1804.01505-2-4-0'], ['1804.01505-1-4-1', '1804.01505-2-4-1'], ['1804.01505-1-4-2', '1804.01505-2-4-2'], ['1804.01505-1-4-3', 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1605.00633 | "{'1605.00633-1-0-0': 'The interaction of charged particles and photons with intense electromagnetic(...TRUNCATED) | "{'1605.00633-2-0-0': 'The interaction of charged particles and photons with intense electromagnetic(...TRUNCATED) | "[['1605.00633-1-0-0', '1605.00633-2-0-0'], ['1605.00633-1-0-1', '1605.00633-2-0-1'], ['1605.00633-1(...TRUNCATED) | "[['1605.00633-1-0-0', '1605.00633-2-0-0'], ['1605.00633-1-0-1', '1605.00633-2-0-1'], ['1605.00633-1(...TRUNCATED) | "[['1605.00633-1-0-4', '1605.00633-2-0-4'], ['1605.00633-1-8-1', '1605.00633-2-9-0'], ['1605.00633-1(...TRUNCATED) | [] | "[['1605.00633-1-6-3', '1605.00633-2-7-2'], ['1605.00633-1-6-4', '1605.00633-2-7-2'], ['1605.00633-1(...TRUNCATED) | [['1605.00633-1-13-3', '1605.00633-2-14-10']] | "['1605.00633-1-1-0', '1605.00633-1-5-0', '1605.00633-1-6-0', '1605.00633-1-8-0', '1605.00633-1-14-0(...TRUNCATED) | "{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexc(...TRUNCATED) | https://arxiv.org/abs/1605.00633 | null | null | null | null | null |
cond-mat-0411186 | "{'cond-mat-0411186-1-0-0': 'The Mott-Hubbard metal-insulator transition is investigated in a two-ba(...TRUNCATED) | "{'cond-mat-0411186-2-0-0': 'The Mott-Hubbard metal-insulator transition is investigated in a two-ba(...TRUNCATED) | "[['cond-mat-0411186-1-43-0', 'cond-mat-0411186-2-50-0'], ['cond-mat-0411186-1-43-1', 'cond-mat-0411(...TRUNCATED) | "[['cond-mat-0411186-1-43-0', 'cond-mat-0411186-2-50-0'], ['cond-mat-0411186-1-43-1', 'cond-mat-0411(...TRUNCATED) | "[['cond-mat-0411186-1-46-1', 'cond-mat-0411186-2-53-1'], ['cond-mat-0411186-1-46-2', 'cond-mat-0411(...TRUNCATED) | [] | "[['cond-mat-0411186-1-42-0', 'cond-mat-0411186-2-49-0'], ['cond-mat-0411186-1-31-3', 'cond-mat-0411(...TRUNCATED) | [] | ['cond-mat-0411186-1-33-0', 'cond-mat-0411186-1-42-3', 'cond-mat-0411186-2-40-0'] | "{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-(...TRUNCATED) | https://arxiv.org/abs/cond-mat/0411186 | null | null | null | null | null |
1807.05692 | "{'1807.05692-1-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price p(...TRUNCATED) | "{'1807.05692-2-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price p(...TRUNCATED) | "[['1807.05692-1-14-0', '1807.05692-2-14-0'], ['1807.05692-1-22-0', '1807.05692-2-22-0'], ['1807.056(...TRUNCATED) | "[['1807.05692-1-14-0', '1807.05692-2-14-0'], ['1807.05692-1-22-0', '1807.05692-2-22-0'], ['1807.056(...TRUNCATED) | "[['1807.05692-1-14-2', '1807.05692-2-14-2'], ['1807.05692-1-6-0', '1807.05692-2-6-0'], ['1807.05692(...TRUNCATED) | [] | "[['1807.05692-1-20-18', '1807.05692-2-20-16'], ['1807.05692-1-20-18', '1807.05692-2-20-17'], ['1807(...TRUNCATED) | [] | "['1807.05692-1-2-0', '1807.05692-1-14-1', '1807.05692-1-18-0', '1807.05692-1-18-1', '1807.05692-1-1(...TRUNCATED) | "{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexc(...TRUNCATED) | https://arxiv.org/abs/1807.05692 | "{'1807.05692-3-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price p(...TRUNCATED) | "{'1807.05692-4-0-0': 'On SDEs with Lipschitz coefficients, driven by continuous, model-free price p(...TRUNCATED) | "{'1807.05692-5-0-0': \"Using similar assumptions as in Revuz and Yor's book [CITATION] we prove the(...TRUNCATED) | null | null |
1904.11783 | "{'1904.11783-1-0-0': 'Word embeddings have recently been shown to reflect many of the pronounced so(...TRUNCATED) | "{'1904.11783-2-0-0': 'Word embeddings have recently been shown to reflect many of the pronounced so(...TRUNCATED) | "[['1904.11783-1-0-0', '1904.11783-2-0-0'], ['1904.11783-1-0-1', '1904.11783-2-0-1'], ['1904.11783-1(...TRUNCATED) | "[['1904.11783-1-0-0', '1904.11783-2-0-0'], ['1904.11783-1-0-1', '1904.11783-2-0-1'], ['1904.11783-1(...TRUNCATED) | [] | [] | [] | [] | "['1904.11783-1-11-0', '1904.11783-1-12-0', '1904.11783-1-13-0', '1904.11783-1-14-0', '1904.11783-1-(...TRUNCATED) | "{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexc(...TRUNCATED) | https://arxiv.org/abs/1904.11783 | null | null | null | null | null |
1512.07837 | "{'1512.07837-1-0-0': 'In models with extended Higgs sector and additional matter fields, the decay (...TRUNCATED) | "{'1512.07837-2-0-0': '# Introduction', '1512.07837-2-1-0': 'In models with extended Higgs sector an(...TRUNCATED) | "[['1512.07837-1-33-0', '1512.07837-2-32-0'], ['1512.07837-1-33-1', '1512.07837-2-32-1'], ['1512.078(...TRUNCATED) | "[['1512.07837-1-33-0', '1512.07837-2-32-0'], ['1512.07837-1-33-1', '1512.07837-2-32-1'], ['1512.078(...TRUNCATED) | "[['1512.07837-1-51-1', '1512.07837-2-50-1'], ['1512.07837-1-16-0', '1512.07837-2-15-0'], ['1512.078(...TRUNCATED) | [] | "[['1512.07837-1-54-0', '1512.07837-2-53-0'], ['1512.07837-1-54-0', '1512.07837-2-53-1'], ['1512.078(...TRUNCATED) | [] | ['1512.07837-1-19-5', '1512.07837-2-18-5'] | "{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexc(...TRUNCATED) | https://arxiv.org/abs/1512.07837 | null | null | null | null | null |
hep-th-0203094 | "{'hep-th-0203094-1-0-0': 'We show that a formalism for analyzing the near-horizon conformal symmetr(...TRUNCATED) | "{'hep-th-0203094-2-0-0': 'We show that a formalism for analyzing the near-horizon conformal symmetr(...TRUNCATED) | "[['hep-th-0203094-1-12-1', 'hep-th-0203094-2-13-1'], ['hep-th-0203094-1-12-3', 'hep-th-0203094-2-13(...TRUNCATED) | "[['hep-th-0203094-1-12-1', 'hep-th-0203094-2-13-1'], ['hep-th-0203094-1-12-3', 'hep-th-0203094-2-13(...TRUNCATED) | "[['hep-th-0203094-1-12-0', 'hep-th-0203094-2-13-0'], ['hep-th-0203094-1-12-2', 'hep-th-0203094-2-13(...TRUNCATED) | [] | "[['hep-th-0203094-1-6-3', 'hep-th-0203094-2-6-3'], ['hep-th-0203094-1-4-4', 'hep-th-0203094-2-4-3'](...TRUNCATED) | [] | "['hep-th-0203094-1-1-0', 'hep-th-0203094-1-2-0', 'hep-th-0203094-1-2-1', 'hep-th-0203094-2-1-0', 'h(...TRUNCATED) | "{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-(...TRUNCATED) | https://arxiv.org/abs/hep-th/0203094 | null | null | null | null | null |
1808.04807 | "{'1808.04807-1-0-0': 'We consider the problem of testing graph cluster structure: given access to a(...TRUNCATED) | "{'1808.04807-2-0-0': 'We consider the problem of testing graph cluster structure: given access to a(...TRUNCATED) | "[['1808.04807-1-10-0', '1808.04807-2-10-0'], ['1808.04807-1-11-0', '1808.04807-2-11-0'], ['1808.048(...TRUNCATED) | "[['1808.04807-1-10-0', '1808.04807-2-10-0'], ['1808.04807-1-11-0', '1808.04807-2-11-0'], ['1808.048(...TRUNCATED) | [] | [] | [] | [] | "['1808.04807-1-9-0', '1808.04807-1-16-1', '1808.04807-1-18-2', '1808.04807-1-23-0', '1808.04807-1-2(...TRUNCATED) | "{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexc(...TRUNCATED) | https://arxiv.org/abs/1808.04807 | null | null | null | null | null |
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Dataset Card for ArXivEdits (Sentence-aligned documents)
ArXivEdits is a dataset comprising 751 English scientific papers from arXiv, each with sentence alignments across multiple revisions. It also includes fine-grained, span-level edits which are annotated with the revision type and the underlying intention for 1000 sentences. This dataset consists of only the sentence-aligned documents subset of the whole dataset. The annotated edits can be found in this dataset.
Dataset Sources
Check the original paper and repository for a more detailed description:
Repository: https://github.com/chaojiang06/arXivEdits/tree/main
Licenses:
- http://arxiv.org/licenses/assumed-1991-2003/
- http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- http://creativecommons.org/licenses/by-nc-sa/4.0/
- http://creativecommons.org/licenses/by-sa/4.0/
- http://creativecommons.org/licenses/by/3.0/
- http://creativecommons.org/licenses/by/4.0/
- http://creativecommons.org/licenses/publicdomain/
- http://creativecommons.org/publicdomain/zero/1.0/
Citation
@article{jiang-etal-2022-arXivEdits,
title={arXivEdits: Understanding the Human Revision Process in Scientific Writing},
author={Jiang, Chao and Xu, Wei and Stevens, Samuel},
journal={In Proceedings of EMNLP 2022},
year={2022}
}
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