We have developed a relativistic coupled-cluster theory to incorporate nuclear spin-dependent interaction Hamiltonians perturbatively. This theory is ideal to calculate parity violating nuclear spin-dependent electric dipole transition amplitudes, $E1_{\rm PNC}^{\rm NSD}$, of heavy atoms. Experimental observation of which is a clear signature of nuclear anapole moment, the dominant source of nuclear spin-dependent parity violation in atoms and ions. We apply the theory to calculate $E1_{\rm PNC}^{\rm NSD}$ of Cs, which to date has provided the best atomic parity violation measurements. We also calculate $E1_{\rm PNC}^{\rm NSD}$ of Ba$^+ $ and Ra$^+$, candidates of ongoing and proposed experiments.
We use configuration interaction technique to calculate parity non-conservation (PNC) in metastable Xe and Hg [proposal of the experiment in L. Bougas et al, Phys. Rev. Lett. 108, 210801 (2012)]. Both, nuclear spin-independent and nuclear spin-dependent (dominated by the nuclear anapole moment) parts of the amplitude are considered. The amplitudes are strongly enhanced by proximity of the states of opposite parity.
We calculate magnetic dipole transition amplitudes in s-s and s-d transitions of Rb, Cs, Ba+, Fr, Ra+, Yb+, Ac2+ and Th3+. These transitions were used or considered to be used for the measurements of parity non-conservation (PNC). We also calculate the magnetic g-factor anomalies for a selection of states, along with electric quadrupole transition amplitudes for s-d transitions in these systems.
We present nuclear spin-independent and dependent parity non-conservation amplitudes for the $[4f^{14}]6s\;^2S_{1/2}-[4f^{14}]5d\;^2D_{3/2}$ transition of $^{171}$Yb$^+$, calculated using perturbed relativistic coupled-cluster theory. As a proxy to estimate theoretical uncertainty of these results we calculate the excitation energies, hyperfine structure constants and E1 transition amplitudes for the important low lying states. The PNC results presented in paper shall be useful in the propose PNC experiments.
We calculate the binding energy, charge radii in $^{126}$Ba--$^{140}$Ba and $^{214}$Ra--$^{228}$Ra using the relativistic mean field theory which includes scalar and vector mesons. We then evaluate the nuclear structure corrections to the weak charges for a series of isotopes of these atoms using different parameters and estimate their uncertainty in the framework of this model. Our results will have important implication for the ongoing and planned parity non-conservation experiments and atomic structure calculations on Ba$^+$ and Ra$^+$.
Currently the theoretical uncertainty limits the interpretation of the atomic parity non-conservation (PNC) measurements. We calculate the PNC $5s$ - $6s$ electric dipole transition amplitude in rubidium and demonstrate that rubidium is a good candidate to search for new physics beyond the standard model since accuracy of the atomic calculations in rubidium can be higher than in cesium. PNC in cesium is currently the best low-energy test of the standard model, therefore, similar measurements for rubidium present a good option for further progress in the field. We also calculate nuclear spin-de…
We discuss the prospects for parity non-conservation experiments with highly charged heavy ions. Energy levels and parity mixing for heavy ions with two to five electrons are calculated. We investigate two-photon-transitions and the possibility to observe interference effects between weak-matrix elements and Stark matrix elements for periodic electric field configurations.
The contribution of the axial triangle anomalous graph to the parity non-conservation effect in atoms is evaluated. The final answer looks like the emission of the electric photon by the magnetic dipole. The relative contribution to the parity non-conservation effect in neutral atoms appears to be negligible but is essentially larger in case of multicharged ions.
We have developed a relativistic coupled-cluster theory to incorporate nuclear spin-dependent interaction Hamiltonians perturbatively. In this theory, the coupled-cluster operators in the electronic sector are defined as tensor operators of rank one and we introduce suitable diagrammatic representations. For properties calculations, the electronic part is first calculated and later coupled to the nuclear spin part. The method is ideal to calculate parity violating nuclear spin-dependent electric dipole transition amplitudes, $E1_{\rm PNC}^{\rm NSD}$, of heavy atoms. To validate the proposed me…
We report a new calculation of the parity non-conserving E1 amplitude for the 6p_{1/2} -> 6p_{3/2} transition in Tl. Our result for the reduced matrix element is $E1_{PNC}=-(66.7 \pm 1.7) i 10^{-11} (-Q_W/N)$ a.u. Comparison with the experiment of Vetter et al [PRL,{\bf 74},2658 (1995)] gives the following result for the weak charge of $^{205}$Tl: $Q_W/Q_W^{SM} = 0.97 (\pm 0.01)_{expt} (\pm 0.03)_{theor}$, where $Q_W^{SM}=-116.7$ is the standard model prediction. This result confirms an earlier conclusion based on the analysis of a Cs experiment that atomic PNC experiments are in agreement wit…
Using recent high-precision measurements of electric dipole matrix elements of atomic cesium, we make an improved determination of the scalar ($α$) and vector ($β$) polarizabilities of the cesium $6s \ ^2S_{1/2} \rightarrow 7s \ ^2S_{1/2} $ transition calculated through a sum-over-states method. We report values of $α= -268.82 \ (30) \ a_0^3$ and $β= 27.139 \ (42) \ a_0^3$ with the highest precision to date. We find a discrepancy between our value of $β$ and the past preferred value, resulting in a significant shift in the value of the weak charge $Q_w$ of the cesium nucleus. Future work to re…
We present accurate {\em ab initio} non-perturbative calculations of the Breit correction to the parity non-conserving (PNC) amplitudes of the $6s-7s$ and $6s-5d_{3/2}$ transitions in Cs, $7s-8s$ and $7s-6d_{3/2}$ transitions in Fr, $6s-5d_{3/2}$ transition in Ba$^+$, $7s-6d_{3/2}$ transition in Ra$^+$, and $6p_{1/2} - 6p_{3/2}$ transition in Tl. The results for the $6s-7s$ transition in Cs and $7s-8s$ transition in Fr are in good agreement with other calculations while calculations for other atoms/transitions are presented for the first time. We demonstrate that higher-orders many-body correc…
We present calculations of nuclear-spin-dependent and nuclear-spin-independent parity violating amplitudes in Ba, Ra, Ac+, Th and Pa. Parity nonconservation in these systems is greatly enhanced due to the presence of very close electronic energy levels of opposite parity, large nuclear charge, and strong nuclear enhancement of parity-violating effects. The presented amplitudes constitute several of the largest atomic parity-violating signals predicted so far. Experiments using these systems may be performed to determine values for the nuclear anapole moment, a P-odd T-even nuclear moment given…
We perform calculations of s-d_5/2 nuclear-spin-dependent parity nonconservation amplitudes for Rb, Cs, Ba+, Yb+, Fr, Ra+ and Ac++. These systems prove to be good candidates for the use in atomic experiments to extract the so-called anapole moment, a P-odd T-even nuclear moment important for the study of parity violating nuclear forces. We also extend our previous works by calculating the missed spin-dependent amplitudes for the s-d_3/2 transitions in the above systems.
There are several factors which lead to a huge enhancement of parity and time invariance violating effects in the Ra atom: very close electronic levels of opposite parity, the large nuclear charge Z and the collective nature of T,P-odd nuclear moments. Experiments with Radium may be used to measure it's nuclear anapole, magnetic quadrupole and Schiff moments. Such measurements provide information about parity and time invariance violating nuclear forces and electron-nucleon interactions.
Nuclear-spin-dependent (NSD) parity violating (PV) effects can be strongly enhanced in diatomic molecules containing heavy atoms. Future measurements are anticipated to provide nuclear anapole moments and strength constants for PV nuclear forces. In light molecules, the NSD electroweak electron-nucleus interaction may also be detected. Here we calculate NSD PV effects for molecular ions. Our calculations are motivated by rapid developments in trapping techniques for such systems at low temperatures.
We use correlation potential and many-body perturbation theory techniques to calculate spin-independent and nuclear spin-dependent parts of the parity nonconserving amplitudes of the transitions between the $6s_{1/2}$ ground state and the $5d_{3/2}$ excited state of Ba$^+$ and Yb$^+$ and between the $7s_{1/2}$ ground state and the $6d_{3/2}$ excited state of Ra$^+$. The results are presented in a form convenient for extracting of the constants of nuclear-spin-dependent interaction (such as, e.g., anapole moment) from the measurements.
A dynamic blue-detuned optical dipole trap with stable $^{87}Rb$ atoms produces a differential ac Stark shift of 18 Hz in the ground state hyperfine transition, and it preserves the ground state hyperfine superpositions for a long coherence time of 180 ms. The trapped atoms undergoing microwave Rabi oscillations are sensitive to a small signal, artificially generated with a second microwave source, phase locked to the first allow- ing a simple and effective method for determining signal-to-noise ratio limits through interference techniques. This provides an excellent means of calibrating sensi…
We identify a pair of near-degenerate states of opposite parity in atomic Xe, the $5p^5 10s \,\, ^2[3/2]_2^o$ at $\rm{E}=94759.927$ cm$^{-1}$ and $5p^5 6f \,\, ^2[5/2]_2$ at $\rm{E}= 94759.935$ cm$^{-1}$, for which parity- and time-odd effects are expected to be enhanced by the small energy separation. We present theoretical calculations which indicate narrow widths for both states and we report a calculated value for the weak matrix element, arising from configuration mixing, of $|W|=2.1$ Hz for $^{132}$Xe. In addition, we measured the Stark effect of the $5p^5\,6f$ $^2[5/2]_{2}$ and $5p^5 \,…
Many-body calculation of nuclear spin-dependent parity-nonconserving amplitude for (7s,F=4) --> (7s,F=5) transition between hyperfine sublevels of the ground state of $^{211}$Fr is carried out. The final result is <7s,F=5 ||d_PNC|| 7s,F=4> = -0.49 10^{-10} i kappa a.u., where kappa is the dimensionless coupling constant. This is approximately an order of magnitude larger than similar amplitude in Cs. The dominant contribution to kappa is associated with the anapole moment of the nucleus.
Due to electron-nucleus weak interaction, atomic bound states with different parities turn out to be mixed. We discuss a prospect for measuring the mixing parameter between the nearly degenerate metastable states 1s_{1/2} 2s_{1/2} : J = 0 and 1s_{1/2} 2p_{1/2} : J = 0 in heliumlike Uranium. Our analysis is based on the polarization properties of the photons emitted in the two-photon decays of such states.
We apply the sum-over-states approach to calculate partial contributions to the parity non-conservation (PNC) in cesium [Porsev {\em et al}, Phys. Rev. D {\bf 82}, 036008 (2010)]. We have found significant corrections to two non-dominating terms coming from the contribution of the core and highly excited states ($n>9$, the so called {\em tail}). When these differences are taken into account the result of Porsev {\em et al}, $E_{\rm PNC} = 0.8906\,(24) \times 10^{-11}i(-Q_W/N)$ changes to $0.8977\,(40)$, coming into good agreement with our previous calculations, $0.8980\,(45)$. The interpretati…
Parity (P) and time (T) invariance violating effects in the Ra atom are strongly enhanced due to close states of opposite parity, the large nuclear charge Z and the collective nature of P,T-odd nuclear moments. We have performed calculations of the atomic electric dipole moments (EDM) produced by the electron EDM and the nuclear magnetic quadrupole and Schiff moments. We have also calculated the effects of parity non-conservation produced by the nuclear anapole moment and the weak charge. Our results show that as a rule the values of these effects are much larger than those considered so far i…
Recent progress in calculations of QED radiative corrections to parity nonconservation in atoms is reviewed. The QED vacuum polarization, the self-energy corrections and the vertex corrections are shown to be described very reliably by different methods used by different groups. All new calculations have recently converged to very close final values. Each separate radiative correction is very large, above 1 % for heavy atoms, but having different signs they partly compensate each other. Our results for the radiative corrections for all atoms are presented. The corrections are -0.54 % for 133 C…
The weak interaction does not conserve parity, which is apparent in many nuclear and atomic phenomena. However, thus far, parity nonconservation has not been observed in molecules. Here we consider nuclear-spin-dependent parity nonconserving contributions to the molecular Hamiltonian. These contributions give rise to a parity nonconserving indirect nuclear spin-spin coupling which can be distinguished from parity conserving interactions in molecules of appropriate symmetry, including diatomic molecules. We estimate the magnitude of the coupling, taking into account relativistic corrections. Fi…
The nuclear spin of a He$^3$ quasiparticle dissolved in superfluid He$^4$ sees an apparent magnetic field proportional to the Fermi coupling constant, the superfluid condensate density, and the electron current at the He$^3$ nucleus. Whereas the direction of the current must be parallel to the quasiparticle momentum, calculating its magnitude presents an interesting theoretical challenge because it vanishes in the Born-Oppenheimer approximation. We find the effect is too small to be observed and present our results in the hope others will be inspired to look for similar effects in other system…
We have performed relativistic many-body calculations of the hyperfine interaction in the $6s$ and $7s$ states of Cs, including the off-diagonal matrix element. The calculations were used to determine the accuracy of the semi-empirical formula for the electromagnetic transition amplitude $<6s|M1|7s>$ induced by the hyperfine interaction. We have found that even though the contribution of the many-body effects into the matrix elements is very large, the square root formula $<6s|H_{hfs}|7s > = \sqrt{< 6s|H_{hfs}|6s > < 7s|H_{hfs}|7s >}$ remains valid to the accuracy of a fraction of $10^{-3}$. T…
A semi-empirical calculation of the 6s - 7s Stark amplitude $α$ in Cs has been performed using the most accurate measurements and calculations of the electromagnetic amplitudes available. This is then used to extract the parameters of the electroweak theory from experimental data. The results are: $α= 269.0 (1.3) a_0^3$, weak charge of Cs $Q_W = -72.41(25)_{exp} (80)_{theor}$, deviation from the Standard model $S = -1.0(.3)_{exp} (1.0)_{theor}$ and limit on the mass of the extra Z-boson in SO(10) model $M_{Z_x} > 550 GeV$.
We present a method for measuring nuclear-spin-dependent atomic parity violation without nuclear-spin-independent background. Such measurements can be achieved by observing interference of parity-conserving and parity-violating two-photon J=0-to-1 transitions driven by collinear photons of the same frequency in the presence of an external static magnetic field.
We use relativistic Hartree-Fock and correlation potential methods to calculate nuclear spin-dependent parity non-conserving amplitudes (dominated by the nuclear anapole moment) between hyperfine structure components of the ground state of odd isotopes of K, Rb, Cs, Ba+, Yb+, Tl, Fr, and Ra+. The results are to be used for interpretation of current and future measurements.
Fock-space perturbed relativistic coupled-cluster calculations of electric dipole polarizability and nuclear spin-dependent parity non-conservation in Cs
We discuss experiments aimed at measurements of atomic parity nonconservation (PNC) effects in the 1S0-3D1 transition (408 nm) in atomic Ytterbium (Z=70). According to theoretical predictions, the PNC-induced E1 amplitude of this transition is ~100 times larger than the analogous amplitude in Cs. Such an experiment will determine differences in PNC effects between different hyperfine components for odd-neutron-number Yb isotopes and, thereby, will allow measurements of the nuclear anapole moments in nuclei with unpaired neutrons. In addition, measurements of PNC in different isotopes would giv…
Nuclear spin-dependent parity violation arises from weak interactions between electrons and nucleons, and from nuclear anapole moments. We outline a method to measure such effects, using a Stark-interference technique to determine the mixing between opposite-parity rotational/hyperfine levels of ground-state molecules. The technique is applicable to nuclei over a wide range of atomic number, in diatomic species that are theoretically tractable for interpretation. This should provide data on anapole moments of many nuclei, and on previously unmeasured neutral weak couplings.
We propose a nuclear mechanism which can induce an atomic electric dipole moment (EDM). Parity and time reversal violating (P,T-odd) nuclear forces generate a mixed P,T-odd nuclear polarizability beta_{ik} (defined by an energy shift U = -beta_{ik} E_{i}H_{k}, E is electric field and H magnetic field). The interaction of atomic electrons with beta_{ik} produces an atomic EDM. We performed an analytical calculation of the P,T-odd nuclear polarizability and estimated the value for the induced atomic EDM. The measurements of atomic EDMs can provide information about P,T-odd nuclear forces and tes…
We propose the effective simulation of light-matter ultrastrong-coupling phenomena with strong-coupling systems. Recent theory and experiments have shown that the quantum Rabi Hamiltonian can be simulated by a Jaynes--Cummings system with the addition of two classical drives. This allows to implement nonlinear processes that do not conserve the total number of excitations. However, parity is still a conserved quantity in the quantum Rabi Hamiltonian, which forbids a wide family of processes involving virtual transitions that break this conservation. Here, we show that these parity-non-conservi…
In light of new data on neutron distributions from experiments with antiprotonic atoms [ Trzcinska {\it et al.}, Phys. Rev. Lett. 87, 082501 (2001)], we reexamine the role of nuclear-structure uncertainties in the interpretation of measurements of parity violation in atoms using chains of isotopes of the same element. With these new nuclear data, we find an improvement in the sensitivity of isotopic chain measurements to ``new physics'' beyond the standard model. We compare possible constraints on ``new physics'' with the most accurate to date single-isotope probe of parity violation in the Cs…
We review the current status of the study of parity and time invariance violation in atoms, nuclei and molecules. We focus on parity non-conservation in cesium and three of the most promising areas of research: (i) parity non-conservation in a chain of isotopes, (ii) search for nuclear anapole moments, and (iii) search for permanent electric dipole moments (EDMs) of atoms and molecules, which in turn are caused by either an electron EDM or nuclear $T,P$-odd moments such as a nuclear EDM or nuclear Schiff moment.
Nuclei with spin $I \ge 1$ have a weak quadrupole moment which leads to tensor contribution to the parity non-conserving interaction between nuclei and electrons. We calculate this contribution for Yb$^+$, Fr and Ra$^+$ and found it to be small. In contrast, in many lanthanides (e.g., Nd, Gd, Dy, Ho, Er, Pr, Sm) and Ra close levels of opposite parity lead to strong enhancement of the effect making it sufficiently large to be measured. Another possibility is to measure the PNC transitions between the hyperfine components of the ground state of Bi. Since nuclear weak charge is dominated by neutr…
Accurately calibrated effective field theories are used to compute atomic parity non-conserving (APNC) observables. Although accurately calibrated, these effective field theories predict a large spread in the neutron skin of heavy nuclei. While the neutron skin is strongly correlated to a large number of physical observables, in this contribution we focus on its impact on new physics through APNC observables. The addition of an isoscalar-isovector coupling constant to the effective Lagrangian generates a wide range of values for the neutron skin of heavy nuclei without compromising the success…
Employing the relativistic coupled-cluster method, comparative studies of the parity non-conserving electric dipole amplitudes for the $7s \ ^2S_{1/2} \rightarrow 6d \ ^2D_{5/2}$ transitions in $^{210}$Fr and $^{211}$Fr isotopes have been carried out. It is found that these transition amplitudes, sensitive only to the nuclear spin dependent effects, are enhanced by more than 3 orders compared to the low-lying $S-D_{5/2}$ transitions in Ba$^+$ and Ra$^+$ owing to the very large contributions from the electron core-polarization effects in Fr. This translates to a relatively large and, in princip…
The scaling of nuclear spin-dependent parity violating effects with increasing nuclear charge $Z$ is discussed in two series of isovalent open-shell diatomic molecules. The parameter $W_\mathrm{a}$ characterising the strength of parity violation in diatomic molecules is calculated in the framework of the zeroth-order regular approximation (ZORA) and found to be in good agreement with the $R(Z) Z^k$ scaling law derived for atoms in which $R(Z)$ represents a relativistic enhancement factor. The influence of electron correlation is studied on the molecular level, with spin-polarisation effects be…
Anapole moments are parity-odd, time-reversal-even moments of the E1 projection of the electromagnetic current. Although it was recognized, soon after the discovery of parity violation in the weak interaction, that elementary particles and composite systems like nuclei must have anapole moments, it proved difficult to isolate this weak radiative correction. The first successful measurement, an extraction of the nuclear anapole moment of 133Cs from the hyperfine dependence of the atomic parity violation, was obtained only recently. An important anapole moment bound in Tl also exists. We discuss…
Nuclear spin-dependent parity violation (NSD-PV) effects in atoms and molecules arise from $Z^0$ boson exchange between electrons and the nucleus, and from the magnetic interaction between electrons and the parity-violating nuclear anapole moment. We demonstrate measurements of NSD-PV that use an enhancement of the effect in diatomic molecules, here using the test system $^{138}$Ba$^{19}$F. Our sensitivity surpasses that of any previous atomic PV measurement. We show that systematic errors can be suppressed to at least the level of the present statistical sensitivity. We measure the matrix ele…
Electron interactions with the nuclear-spin-dependent (NSD) parity non-conserving (PNC) anapole moment are strongly enhanced within heteronuclear diatomic molecules. A novel, low-energy optical rotation experiment is being proposed with the aim of observing NSD PNC interactions in HgH. Based on the relativistic coupled cluster method we present a complete calculation of the circular polarization parameter $P = 2\,\textrm{Im}(E1_{PNC})/M1 \approx 3 \times 10^{-6}\; κ$ for the $^2Σ_{1/2} \to ^2Π_{1/2}$ optical transition of HgH, where $κ$ is a dimensionless constant determined by the nuclear ana…
Parity violation in atoms: neutrino-mediated long range forces and finite nuclear size
PRISMA ++ Cluster of Excellence, Institut für Kernphysik, Johannes Gutenberg-Universität（PRISMA++卓越集群，核物理研究所，约翰内斯·古腾堡大学）；PRISMA ++ Cluster of Excellence, Institut für Physik, Johannes Gutenberg-Universität（PRISMA++卓越集群，物理研究所，约翰内斯·古腾堡大学）
We review current status of the study of parity and time invariance phenomena in atoms, nuclei and molecules. We focus on three most promising areas of research: (i) parity non-conservation in a chain of isotopes, (ii) search for nuclear anapole moments, and (iii) search for permanent electric dipole moments (EDM) of atoms and molecules which are caused by either, electron EDM or nuclear $T,P$-odd moments such as nuclear EDM and nuclear Schiff moment.
We show that existing calculations of the interaction between nuclear Schiff moments and electrons in molecules use an inaccurate operator which gives rise to significant errors. By comparing the matrix elements of the accurate and imprecise Schiff moment operators, we calculated the correction factor as a function of the nuclear charge Z and presented corrected results for the T,P-violating interaction of the nuclear spin with the molecular axis in the TlF, RaO, PbO, TlCN, ThO, AcF molecules and in the ferroelectric solid PbTiO$_3$.
Nuclear spin-dependent parity violation (NSD-PV) effects in atoms and molecules arise from $Z^0$ boson exchange between electrons and the nucleus, and from the magnetic interaction between electrons and the parity-violating nuclear anapole moment. It has been proposed to study NSD-PV effects using an enhancement of the observable effect in diatomic molecules [D. DeMille $\textit{et al.}$, Phys. Rev. Lett. $\textbf{100}$, 023003 (2008)]. Here, we demonstrate measurements of this type with sensitivity surpassing that of any previous atomic PV measurement, using the test system ${^{138}\mathrm{Ba…
Measurements of nuclear spin-dependent parity-violating (NSD-PV) effects provide an excellent opportunity to test nuclear models and to search for physics beyond the Standard Model. Molecules possess closely-spaced states with opposite parity which may be easily tuned to degeneracy to greatly enhance the observed parity-violating effects. A high-sensitivity measurement of NSD-PV effects using light triatomic molecules is in preparation [E. B. Norrgard, et al., Commun. Phys. 2, 77 (2019)]. Importantly, by comparing these measurements in light nuclei with prior and ongoing measurements in heavie…
In the presence of P-violating interactions, the exchange of vector bosons between electrons and nucleons induces parity-nonconserving (PNC) effects in atoms and molecules, while the exchange of vector bosons between nucleons induces anapole moments of nuclei. We perform calculations of such vector-mediated PNC effects in Cs, Ba$^+$, Yb, Tl, Fr and Ra$^+$ using the same relativistic many-body approaches as in earlier calculations of standard-model PNC effects, but with the long-range operator of the weak interaction. We calculate nuclear anapole moments due to vector boson exchange using a sim…
The self-energy and the vertex radiative corrections to the effect of parity nonconservation in heavy atoms are calculated analytically in orders Z alpha^2 and Z^2 alpha^3 ln(lambda_C/r_0), where lambda_C and r_0 being the Compton wavelength and the nuclear radius, respectively. The value of the radiative correction is -0.85% for Cs and -1.41% for Tl. Using these results we have performed analysis of the experimental data on atomic parity nonconservation. The obtained values of the nuclear weak charge, Q_W=-72.90(28)_{exp}(35)_{theor} for Cs, and Q_W=-116.7(1.2)_{exp}(3.4)_{theor} for Tl, agre…
