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We present a precise fully relativistic numerical solution of the two-center Coulomb problem. The special case of unit nuclear charges is relevant for the accurate description of the ${\rm H}_2^+$ molecular ion and its isotopologues, systems that are an active experimental topic. The computation utilizes the 2-spinor minmax approach and the finite-element method. The computed total energies have estimated fractional uncertainties of a few times $10^{-20}$ for unit charges and a bond length of 2 atomic units. The fractional uncertainty of the purely relativistic contribution is $1\times10^{-17}$. The result is relevant for future precision experiments, whereas at present the uncertainties arising from the quantum electrodynamic treatment of the rovibrational transition frequencies. are dominant.

A possible implication of an ultralight dark matter (UDM) field interacting wibeginth the Standard Model (SM) degrees of freedom is oscillations of fundamental constants. Here, we establish direct experimental bounds on the coupling of an oscillating UDM field to the up, down, and strange quarks and to the gluons, for oscillation frequencies between 10 Hz and 10^8 Hz. We employ spectroscopic experiments that take advantage of the dependence of molecular transition frequencies on the nuclear masses. Our results apply to previously unexplored frequency bands, and improve on existing bounds at frequencies > 5 MHz. We identify a sector of UDM - SM coupling space where the bounds from Equivalence Principle tests may be challenged by next-generation experiments of the present kind., Comment: 23 pages,9 figures

Optical spectroscopy in the gas phase is a key tool to elucidate the structure of atoms and molecules and of their interaction with external fields. The line resolution is usually limited by a combination of first-order Doppler broadening due to particle thermal motion and of a short transit time through the excitation beam. For trapped particles, suitable laser cooling techniques can lead to strong confinement (Lamb-Dicke regime, LDR) and thus to optical spectroscopy free of these effects. For non-laser coolable spectroscopy ions, this has so far only been achieved when trapping one or two atomic ions, together with a single laser-coolable atomic ion [1,2]. Here we show that one-photon optical spectroscopy free of Doppler and transit broadening can also be obtained with more easily prepared ensembles of ions, if performed with mid-infrared radiation. We demonstrate the method on molecular ions. We trap approximately 100 molecular hydrogen ions (HD$^{+}$) within a Coulomb cluster of a few thousand laser-cooled atomic ions and perform laser spectroscopy of the fundamental vibrational transition. Transition frequencies were determined with lowest uncertainty of 3.3$\times$10$^{-12}$ fractionally. As an application, we determine the proton-electron mass ratio by matching a precise ab initio calculation with the measured vibrational frequency., Comment: This is the final version accepted for publication in Nature Physics

Molecular hydrogen ions are of metrological relevance due to the possibility of precise theoretical evaluation of their spectrum and of external-field-induced shifts. In homonuclear molecular ions the electric dipole $E1$ transitions are strongly suppressed, and of primary laser spectroscopy interest is the electric quadrupole ($E2$) transition spectrum. In continuation of previous work on the H$_2^+$ ion, we report here the results of the calculations of the hyperfine structure of the laser-induced electric quadrupole transitions between a large set of ro-vibrational states of D$_2^+$; the inaccuracies of previous evaluations have been corrected. The effects of the laser polarization are studied in detail. We show that the electric quadrupole moment of the deuteron can in principle be determined with low fractional uncertainty $(\simeq1\times10^{-4})$ by comparing the results presented here with future data from precision spectroscopy of D$_2^+$., Comment: 19 pages, 4 figures. Comparison with the available experimental data about the hyperfine structure of D2+ has been added in Section IIC; A new Section IIID has been added with a thorough analysis of the determination of the electric quadrupole moment of the deuteron from D2+ spectroscopy; A new Section IIIE has been added discussing further applications of the composite frequency method

We study the prospects of using the electric quadrupole transitions from the ground states of Cu, Ag and Au to the metastable state $^2{\rm D}_{5/2}$ as clock transitions in optical lattice clocks. We calculate lifetimes, transition rates, systematic shifts, and demonstrate that the fractional uncertainty of the clocks can be similar to what is achieved in the best current optical clocks. The use of these proposed clocks for the search of new physics, such as time variation of the fine structure constant, search for low-mass scalar dark matter, violation of Local Position Invariance and violation of Lorenz Invariance is discussed., Comment: 10 pages, 2 figures

The proposed mission "Space Atomic Gravity Explorer" (SAGE) has the scientific objective to investigate gravitational waves, dark matter, and other fundamental aspects of gravity as well as the connection between gravitational physics and quantum physics using new quantum sensors, namely, optical atomic clocks and atom interferometers based on ultracold strontium atoms., Comment: Published in Eur. Phys. J. D 73 (2019) 228 in the Topical Issue Quantum Technologies for Gravitational Physics, Guest editors Tanja Mehlstaubler, Yanbei Chen, Guglielmo M. Tino and Hsien-Chi Yeh

We consider the application of a basic principle of quantum theory, the tracelessness of a certain class of hamiltonians, to the precision spectroscopy of the molecular hydrogen ions. We show that it is possible to obtain the spin-averaged transition frequencies between states from a simple weighted sum of experimentally accessible spin-dependent transition frequencies. We discuss the cases ${\rm H}_{2}^{+}$ and ${\rm HD}^{+}$, which are distinct in the multipole character of their rovibrational transitions. Inclusion of additional frequencies permits canceling also the electric quadrupole shift, the Zeeman shift and partially the Stark shift. In this context, we find that measuring electric quadrupole transitions in ${\rm HD}^{+}$ is advantageous. The required experimental effort appears reasonable.

We study the prospects of using transitions from the ytterbium ground state to metastable states $^3{\rm P}^{\rm o}_2$ at $E=19\,710.388~$cm$^{-1}$ and $4f^{13}5d6s^2\,(J=2)$ at $E=23\,188.518~$cm$^{-1}$ as clock transitions in an optical lattice clock. Having more than one clock transition in Yb could benefit the search for new physics beyond the Standard Model via studying the non-linearity of King's plot or the time-variation of the ratio of the frequencies of two clock transitions. We calculate the lifetime of the states, relevant transition amplitudes, systematic shifts, and the sensitivities of the clock transitions to a variation of the fine structure constant and to the gravitational potential. We find that both transitions can probably support ultra-high accuracy, similar to what is already achieved for the $^1$S$_0$ - $^3$P$^{\rm o}_0$ clock transition., Comment: Extended version, 13 pages, 7 tables, 2 figures

In order to investigate the long-term dimensional stability of matter, we have operated an optical resonator fabricated from crystalline silicon at 1.5$\,$K continuously for over one year and repeatedly compared its resonance frequency $f_{res}$ with the frequency of a GPS-monitored hydrogen maser. After allowing for an initial settling time, over a 163-day interval we found a mean fractional drift magnitude $|f_{res}^{-1}df_{res}/dt|<1.4\times10^{-20}$/s. The resonator frequency is determined by the physical length and the speed of light, and we measure it with respect to the atomic unit of time. Thus, the bound rules out, to first order, a hypothetical differential effect of the universe's expansion on rulers and atomic clocks. We also constrain a hypothetical violation of the principle of Local Position Invariance for resonator-based clocks and derive bounds for the strength of space-time fluctuations., Comment: to appear in Physical Review Letters (2016)

The ESA mission "Space Optical Clock" project aims at operating an optical lattice clock on the ISS in approximately 2023. The scientific goals of the mission are to perform tests of fundamental physics, to enable space-assisted relativistic geodesy and to intercompare optical clocks on the ground using microwave and optical links. The performance goal of the space clock is less than $1 \times 10^{-17}$ uncertainty and $1 \times 10^{-15} {\tau}^{-1/2}$ instability. Within an EU-FP7-funded project, a strontium optical lattice clock demonstrator has been developed. Goal performances are instability below $1 \times 10^{-15} {\tau}^{-1/2}$ and fractional inaccuracy $5 \times 10^{-17}$. For the design of the clock, techniques and approaches suitable for later space application are used, such as modular design, diode lasers, low power consumption subunits, and compact dimensions. The Sr clock apparatus is fully operational, and the clock transition in $^{88}$Sr was observed with linewidth as small as 9 Hz., Comment: 12 pages, 8 figures, SPIE Photonics Europe 2016

Ultra-precise optical clocks in space will allow new studies in fundamental physics and astronomy. Within an European Space Agency (ESA) program, the Space Optical Clocks (SOC) project aims to install and to operate an optical lattice clock on the International Space Station (ISS) towards the end of this decade. It would be a natural follow-on to the ACES mission, improving its performance by at least one order of magnitude. The payload is planned to include an optical lattice clock, as well as a frequency comb, a microwave link, and an optical link for comparisons of the ISS clock with ground clocks located in several countries and continents. Within the EU-FP7-SPACE-2010-1 project no. 263500, during the years 2011-2015 a compact, modular and robust strontium lattice optical clock demonstrator has been developed. Goal performance is a fractional frequency instability below 1x10^{-15}, tau^{-1/2} and a fractional inaccuracy below 5x10^{-17}. Here we describe the current status of the apparatus' development, including the laser subsystems. Robust preparation of cold {88}^Sr atoms in a second stage magneto-optical trap (MOT) is achieved., Comment: 27 Pages, 15 figures, Comptes Rendus Physique 2015

The use of ultra-precise optical clocks in space ("master clocks") will allow for a range of new applications in the fields of fundamental physics (tests of Einstein's theory of General Relativity, time and frequency metrology by means of the comparison of distant terrestrial clocks), geophysics (mapping of the gravitational potential of Earth), and astronomy (providing local oscillators for radio ranging and interferometry in space). Within the ELIPS-3 program of ESA, the "Space Optical Clocks" (SOC) project aims to install and to operate an optical lattice clock on the ISS towards the end of this decade, as a natural follow-on to the ACES mission, improving its performance by at least one order of magnitude. The payload is planned to include an optical lattice clock, as well as a frequency comb, a microwave link, and an optical link for comparisons of the ISS clock with ground clocks located in several countries and continents. Undertaking a necessary step towards optical clocks in space, the EU-FP7-SPACE-2010-1 project no. 263500 (SOC2) (2011-2015) aims at two "engineering confidence", accurate transportable lattice optical clock demonstrators having relative frequency instability below 1\times10^-15 at 1 s integration time and relative inaccuracy below 5\times10^-17. This goal performance is about 2 and 1 orders better in instability and inaccuracy, respectively, than today's best transportable clocks. The devices will be based on trapped neutral ytterbium and strontium atoms. One device will be a breadboard. The two systems will be validated in laboratory environments and their performance will be established by comparison with laboratory optical clocks and primary frequency standards. In this paper we present the project and the results achieved during the first year., Comment: Contribution to European Frequency and Time Forum 2012, Gothenburg, Sweden

Advanced techniques for manipulation of internal states, standard in atomic physics, are demonstrated for a charged molecular species for the first time. We address individual hyperfine states of ro-vibrational levels of a diatomic ion by optical excitation of individual hyperfine transitions, and achieve controlled transfer of population into a selected hyperfine state. We use molecular hydrogen ions (HD^{+}) as a model system and employ a novel frequency-comb-based, continuous-wave 5 \mum laser spectrometer. The achieved spectral resolution is the highest obtained so far in the optical domain on a molecular ion species. As a consequence, we are also able to perform the most precise test yet of the ab-initio theory of a molecule.

We demonstrate rotational excitation of molecular ions that are sympathetically cooled by laser-cooled atomic ions to a temperature as low as ca. 10 mK. The molecular hydrogen ions HD+ and the fundamental rotational transition $(v=0,\, N=0)\rightarrow(v'=0,\, N'=1)$ at 1.3 THz, the most fundamental dipole-allowed rotational transition of any molecule, are used as a test case. This transition is here observed for the first time directly. Rotational laser cooling was employed in order to increase the signal, and resonance-enhanced multiphoton dissociation was used as detection method. The black-body-radiation-induced rotational excitation is also observed. The extension of the method to other molecular species is briefly discussed.

In this work we demonstrate vibrational spectroscopy of polyatomic ions that are trapped and sympathetically cooled by laser-cooled atomic ions. We use the protonated dipeptide tryptophane-alanine (HTyrAla+) as a model system, cooled by Barium ions to less than 800mK secular temperature. The spectroscopy is performed on the fundamental vibrational transition of a local vibrational mode at 2.74 {\mu}m using a continuous-wave optical parametric oscillator (OPO). Resonant multi-photon IR dissociation spectroscopy (without the use of a UV laser) generates charged molecular fragments, which are sympathetically cooled and trapped, and subsequently released from the trap and counted. We measured the cross section for R-IRMPD under conditions of low intensity, and found it to be approximately two orders smaller than the vibrational excitation cross section. The observed rotational bandwidth of the vibrational transition is larger than the one expected from the combined effects of 300 K black-body temperature, conformer-dependent line shifts, and intermolecular vibrational relaxation broadening (J. Stearns et al., J. Chem. Phys., 127, 154322-7 (2007)). This indicates that as the internal energy of the molecule grows, an increase of the rotational temperature of the molecular ions well above room temperature (up to on the order of 1000K), and/or an appreciable shift of the vibrational transition frequency (approx. 6-8 cm$^{-1}$) occurs.

Gas-phase multiply charged proteins have been sympathetically cooled to translational temperatures below 1 K by Coulomb interaction with laser-cooled barium ions in a linear ion trap. In one case, an ensemble of 53 cytochrome c molecules (mass ~ 12390 amu, charge +17 e) was cooled by ~ 160 laser-cooled barium ions to less than 0.75 K. Storage times of more than 20 minutes have been observed and could easily be extended to more than an hour. The technique is applicable to a wide variety of complex molecules., Comment: same version as published in Phys. Rev. A

In this work, we demonstrate measurements of photodestruction rates of translationally cold, charged biomolecules. The long-term stable storage of the molecular ions in an ion trap at ultra-high vacuum conditions allows measurement of small rates and verification that rates are linear in photodestruction laser intensity. Measurements were performed on singly protonated molecules of the organic compound glycyrrhetinic acid (C30H46O4), dissociated by a continuous-wave UV laser (266 nm) using different intensities. The molecules were sympathetically cooled by simultaneously trapped laser-cooled barium ions to translational temperatures of below 150 mK. Destruction rates of less than 0.05 s^-1 and a cross section of (1.1 +/- 0.1) * 10^-17 cm^2 have been determined. An extension to tunable UV laser sources would permit high-resolution dissociation spectroscopic studies on a wide variety of cold complex molecules., Comment: same version as published in J. Phys. B

We demonstrate a diode laser system which is suitable for high-resolution spectroscopy in the 1200 nm and yellow spectral ranges. It is based on a two-facet quantum dot chip in a Littrow-type external cavity configuration. The laser is tunable in the range 1125 -1280 nm, with an output power of more than 200 mW and exhibits a free-running linewidth of 200 kHz. Amplitude and frequency noise were characterized, including the dependence of frequency noise on the cavity length. Frequency stabilization to a high-finesse reference cavity is demonstrated reducing the linewidth to about 20 - 30 kHz. Yellow light (> 3 mW) at 578 nm was generated by frequency doubling in an enhancement cavity containing a PPLN crystal. The source has potential application for precision spectroscopy of ultra-cold Yb atoms and molecular hydrogen ions.

We describe the current status of high-precision ab initio calculations of the spectra of molecular hydrogen ions (H_2^+ and HD^+) and of two experiments for vibrational spectroscopy. The perspectives for a comparison between theory and experiment at a level of 1 ppb are considered., Comment: 26 pages, 13 figures, 1 table, to appear in "Precision Physics of Simple Atomic Systems", Lecture Notes in Physics, Springer, 2007

The performance of optical clocks has strongly progressed in recent years, and accuracies and instabilities of 1 part in 10^18 are expected in the near future. The operation of optical clocks in space provides new scientific and technological opportunities. In particular, an earth-orbiting satellite containing an ensemble of optical clocks would allow a precision measurement of the gravitational redshift, navigation with improved precision, mapping of the earth's gravitational potential by relativistic geodesy, and comparisons between ground clocks., Comment: Proc. III International Conference on Particle and Fundamental Physics in Space (SpacePart06), Beijing 19 - 21 April 2006, to appear in Nucl. Phys. B

An improved analysis of the Michelson-Morley-type experiment by P. Antonini et al. (Phys. Rev. A. 71, 050101 (2005)) yields the Robertson-Mansouri-Sexl theory parameter combination $\beta-\delta-1/2=(-0.6\pm 2.1\pm 1.2)\cdot10^{-10}$ and the Standard Model Extension theory parameter $(\tilde{\kappa}_{e-})^{ZZ} = (-2.9\pm 2.2)\cdot 10^{-14}$., Comment: 2 pages

A test of Lorentz invariance for electromagnetic waves was performed by comparing the resonance frequencies of two stable optical resonators as a function of orientation in space. The crystalline resonators were operated at 3.4 K in a cryostat employing a pulse-tube refrigerator. A new analysis yields the Robertson-Mansouri-Sexl theory parameter combination $\beta-\delta-1/2=(-0.6\pm 2.1\pm1.2)\cdot10^{-10}$ and one parameter of the Standard Model Extension theory, $(\tilde{\kappa}_{e-})^{ZZ} = (-2.9\pm2.2)\cdot 10^{-14}$., Comment: 15 pages, 9 figures. Improved analysis of the paper Antonini et al, Phys. Rev. A 71, 050101 (2005). To appear in: "Special Relativity: will it hold another 100 years", Springer Lect. Notes in Physics, J. Ehlers and C. L\"ammerzahl, eds. (2006)

Silicon double paddle oscillators are well suited for the detection of weak forces because of their high Q factor (about 10^5 at room temperature). We describe an experiment aimed at the detection of gravitational forces between masses at sub-mm distance using such an oscillator. Gravitational excitation is produced by a rotating aluminium disk with platinum segments. The force sensitivity of this apparatus is about 10 fN at room temperature for 1000 s averaging time at room temperature. The current limitations to detection of the gravitational force are mentioned., Comment: 19 pages, to appear in Proceedings of the Tenth Marcel Grossmann Meeting on General Relativity, edited by M. Novello, S. Perez-Bergliaffa and R. Ruffini, World Scientific. Revision: portable format and revised figures

A test of Lorentz invariance for electromagnetic waves was performed by comparing the resonance frequencies of two optical resonators as a function of orientation in space. In terms of the Robertson-Mansouri-Sexl theory, we obtain $\beta-\delta-1/2=(+0.5\pm 3\pm 0.7) E-10$, a ten-fold improvement compared to the previous best results. We also set a first upper limit for a so far unknown parameter of the Standard Model Extension test theory, $|(\tilde{\kappa}_{e-})^{ZZ}| < 2\cdot E-14$., Comment: 4 pages, 2 figures, accepted for publication Phys. Rev. A (2005)

A static quadrupole potential breaks the cylindrical symmetry of the effective potential of a linear rf trap. For a one-component fluid plasma at low temperature, the resulting equilibrium charge distribution is predicted to be an ellipsoid. We have produced laser-cooled Be$^+$ ellipsoidal ion crystals and found good agreement between their shapes and the cold fluid prediction. In two-species mixtures, containing Be$^+$ and sympathetically cooled ions of lower mass, a sufficiently strong static quadrupole potential produces a spatial separation of the species., Comment: 4 pages, 3 figures

We have generated Coulomb crystals of ultracold $^4$He$^+$ ions in a linear radiofrequency trap, by sympathetic cooling via laser--cooled $^9$Be$^+$. Stable crystals containing up to 150 localized He$^+$ ions at $\sim$20 mK were obtained. Ensembles or single ultracold He$^+$ ions open up interesting perspectives for performing precision tests of QED and measurements of nuclear radii. The present work also indicates the feasibility of cooling and crystallizing highly charged atomic ions using $^9$Be$^+$ as coolant., Comment: 4 pages, 2 figures

It has been suggested that space-time might undergo fluctuations because of its intrinsic quantum nature. These fluctuations would pose a fundamental limit to the ability of measuring distances with arbitrary precision, beyond any limitations due to standard quantum mechanics. Laser interferometers have recently been proposed as being suited for a search for the existence of space-time fluctuations. Here we present results of a search for space-time fluctuations of very low fluctuation frequencies, in the range from 1 microHz to 0.5 Hz. Rigid optical interferometers made out of sapphire and operated at cryogenic temperature were used. We find an upper limit of 1.10^-24 Hz^-1 for the normalized distance noise spectral density at 6 microHz, and of 1.10^-28 Hz^-1 above 5 mHz, and establish an experimental limit for the parameter of a recently proposed random-walk hypothesis., Comment: 15 pages, 2 figures

It is shown that the translational degrees of freedom of a large variety of molecules, from light diatomic to heavy organic ones, can be cooled sympathetically and brought to rest (crystallized) in a linear Paul trap. The method relies on endowing the molecules with an appropriate positive charge, storage in a linear radiofrequency trap, and sympathetic cooling. Two well--known atomic coolant species, ${}^9{\hbox{Be}}^+$ and ${}^{137}{\hbox{Ba}}^+$, are sufficient for cooling the molecular mass range from 2 to 20,000 amu. The large molecular charge required for simultaneous trapping of heavy molecules and of the coolant ions can easily be produced using electrospray ionization. Crystallized molecular ions offer vast opportunities for novel studies., Comment: Accepted for publication in Phys. Rev. A

A detailed theoretical analysis of the spatiotemporal mode of a single photon prepared via conditional measurements on a photon pair generated in the process of parametric down-conversion is presented. The maximum efficiency of coupling the photon into a transform-limited classical optical mode is calculated and ways for its optimization are determined. An experimentally feasible technique of generating the optimally matching classical mode is proposed. The theory is applied to a recent experiment on pulsed homodyne tomography of the single-photon Fock state (A. I. Lvovsky et al., Phys Rev. Lett. 87, 050402 (2001) - preprint at quant-ph/0101051), Comment: submitted to the special issue of the European Physical Journal D "Quantum interference and cryptographic keys: novel physics and advancing technologies"

We present a reliable, narrow linewidth (100 kHz) continuous-wave optical parametric oscillator (OPO) suitable for high-resolution spectroscopy applications. The OPO is based on a periodically-poled lithium-niobate crystal and features a specially designed intracavity etalon which permits its continuous tuning and stable operation at any desired wavelength in a wide operation range. We demonstrate Doppler-free spectroscopy on a rovibrational transition of methane at 3.39 um., Comment: 3 pages, submitted to Optics Letters

A pulsed balanced homodyne detector has been developed for precise measurements of electric field quadratures of pulsed optical quantum states. A high level of common mode suppression (> 85 dB) and low electronic noise (730 electrons per pulse) provide a signal to noise ratio of 14 dB for the measurement of the quantum noise of individual pulses. Measurements at repetition rates up to 1 MHz are possible. As a test, quantum tomography of the coherent state is performed and the Wigner function and the density matrix are reconstructed with a 99.5% fidelity. The detection system can also be used for ultrasensitive balanced detection in cw mode, e.g. for weak absorption measurements., Comment: 3 pages, submitted to Optics Letters

We have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The single-photon Fock state |1> was prepared using conditional measurements on photon pairs born in the process of parametric down-conversion. A probability distribution of the phase-averaged electric field amplitudes with a strongly non-Gaussian shape is obtained with the total detection efficiency of (55+-1)%. The angle-averaged Wigner function reconstructed from this distribution shows a strong dip reaching classically impossible negative values around the origin of the phase space., Comment: 4 pages, 4 figures, to appear in Physical Review Letters. Avoid downloading PDF due to extremely poor figure resolution. Use Postscript

We perform a quantum theoretical calculation of the noise power spectrum for a phase measurement of the light output from a coherently driven optical cavity with a freely moving rear mirror. We examine how the noise resulting from the quantum back action appears among the various contributions from other noise sources. We do not assume an ideal (homodyne) phase measurement, but rather consider phase modulation detection, which we show has a different shot noise level. We also take into account the effects of thermal damping of the mirror, losses within the cavity, and classical laser noise. We relate our theoretical results to experimental parameters, so as to make direct comparisons with current experiments simple. We also show that in this situation, the standard Brownian motion master equation is inadequate for describing the thermal damping of the mirror, as it produces a spurious term in the steady-state phase fluctuation spectrum. The corrected Brownian motion master equation [L. Diosi, Europhys. Lett. {\bf 22}, 1 (1993)] rectifies this inadequacy., Comment: 12 pages revtex, 2 figures

We report the first observation of the quantum effects of competing $\chi^{(2)}$ nonlinearities. We also report new classical signatures of competition, namely clamping of the second harmonic power and production of nondegenerate frequencies in the visible. Theory is presented that describes the observations as resulting from competition between various $\chi^{(2)}$ upconversion and downconversion processes. We show that competition imposes hitherto unsuspected limits to both power generation and squeezing. The observed signatures are expected to be significant effects in practical systems., Comment: 5 pages, 6 figures

We directly sample the exponential moments of the canonical phase for various quantum states from the homodyne output. The method enables us to study the phase properties experimentally, without making the detour via reconstructing the density matrix or the Wigner function and calculating the phase statistics from them. In particular, combing the measurement with a measurement of the photon-number variance, we verify fundamental number-phase uncertainty., Comment: 1 Table and 4 Postscript figures, using Latex