User:AnupamMitra/sandbox/Optical Lattice Clock

A few thousand {}^{87} Sr atoms were cooled to a few micro Kelvin. They were trapped in an optical lattice near the magic wavelength 813 nm with the trap depth between 40 E_r to 300 E_r, where E_r is the photon recoill energy. The clock transition {}^1S_0 \leftrightarrow {}^3P_0 clock transition was interrogated with Rabi spectroscopy over a 160 ms interval with a thermal noise limited laser. The state determination was destructive. Therefore the experiment was repeated every 1.3 s.

An optical lattice clock is an atomic clock which uses an electromagnetic transition frequency of atoms trapped in an optical lattice as a frequency standard for its time keeping element. Atomic clocks in general are the most precise and reproducible standard of frequency. They are used to define second, the SI unit of time.

Strontium based optical lattice atomic clock

In quantum mechanics, all particles of the same type are identical. In particular, every atoms of a particular isotope of an element (like Cesium 133) is identical to every other atom of the same isotope of the element. Therefore, these atoms have identical energy spectra. The energy levels of atoms are discrete and the differences between pairs of energy levels is identical for every atom of the same type. A transition between a pair of energy levels can be excited using electromagnetic radiation whose frequency is proportional to the difference of the two energy levels, with the constant of proportionality being the Planck's constant $h$.

The identical nature of all atoms makes the choice of a pair of energy levels of atom ideal for a frequency standard. This is the basis of an atomic clock. An atomic clock using two hyperfine levels in the ground state manifold of Cesium 133, with a energy difference in units of the Planck's constant is 9192631770 Hz is used to define a second.

In order to reduce the uncertainty, clocks which use higher frequencies corresponding to energy levels with larger gaps have been made where the transition frequency is an optical frequencies. Development of atomic clocks, often called "optical atomic clocks" or "optical clocks" poses several challenges. The primary obstacle is the Doppler shift in the transition frequency between the appropriate energy levels of the atom. This Doppler shift is proportional to the product of the velocity of the atoms and the base transition frequency. Therefore at higher transition frequencies, the Doppler shift becomes important. In order to control the motion of the atoms, two approaches to trapping the have been used. The first involves ionizing the atoms and trapping them in a static electric field. The second involves trapping neutral atom in optical lattices formed using a standing wave of electromagnetic fields.

Mechanism

Atoms and their energy levels

For using a quantum resonance as frequency reference, a quantum transition that provides a very good quality factor and whose frequency is insensitive to external fields is desirable. The state of the art frequency standards require long coherence times. Therefore, it is important to have precise control over the motion of the atom. This has been done with laser cooling and trapping. Historically, Alkali atoms like Cesium and Rubidium have been used for atomic clocks and have been mostly explored for laser cooling and quantum control. These elements have easy to control and manipulate, which led to the development of extremely precise Cesium and Rubidium atomic clock standards, which use laser cooled atomic samples in an atomic fountain. While the precision of the most advanced Cesium clock is about 1 part in $10^{16}$, improvements in fractional stability and ultimately precision is limited by the relatively small (microwave) hyperfine transition frequency of about 9.192 GHz.

Alkaline earth and similar atoms with two valence electrons  (Mg, Ca, Sr, Yb, Hg) have narrow inter combination transitions and simple level structures. They have become popular choices for frequency standards. These elements have two valence electrons in the s shells. These valence electrons combine to form two electron triplet and singlet states. There are strong transitions among various singlet or triplet state and weak spin forbidden transitions.

Energy levels for alkaline earth and similar atoms. The states used as clock states are shown in red.

 

The ${}^{1}S_{0}\leftrightarrow {}^{3}P_{0}$ transition in isotopes with non zero nuclear spin have been most used as states for the clock transitions. The low-lying metastable ${}^{3}P_{0}$ excited state has only very weak coupling to ${}^{1}S_{0}$ with an energy difference corresponding to an optical frequency. The ${}^{1}S_{0}\leftrightarrow {}^{3}P_{0}$ transition has a very small linewidth (a few Hz to a a few mHz). This provides a quality factor in the range $10^{18}$ making it ideal for an optical frequency standard. Moreover, these states have zero electronic angular momentum, $J=0$, making them less prone to several potential systematic uncertainties in the system. This also reduces the dependence on the polarization of light. For atomic confinement in an optical potential, alkaline earth elements are ideal as there exist Stark cancellation frequencies of light.

Other transitions are suitable for different kinds of cooling. For example, ${}^{1}S_{0}\leftrightarrow {}^{1}P_{1}$ transition is well suited for laser cooling and trapping from a thermal source. This achieves temperatures of a few milli Kelvin. The ${}^{1}S_{0}\leftrightarrow {}^{3}P_{1}$ transition allows further cooling to a few micro Kelvin.

Alkaline earth atoms have several isotopes which are both bosonic and fermionic isotopes. In general, the addition of spins of nucleons in the nucleus leads to the bosonic isotopes having numbered atomic masses zero nuclear spin and the fermionic isotopes having odd numbered atomic mass and non zero nuclear spin. The non zero nuclear spin of fermionic isotopes introduces hyperfine splittings into the level structure. As a result, bosonic isotopes are somewhat simpler for manipulations like laser cooling. Nevertheless, the hyperfine structure can sometimes bring unexpected benefits like sub Doppler cooling for fermionic isotopes.

Confinement in optical lattices

A common feature of optical lattice atomic clocks and trapped ion atomic clocks is that the atoms are confined by a trap. The confinement decouples the external degree freedom corresponding to motion of the atom from the internal degree of freedom corresponding to the state of the atom. This facilitates a precise measurement of the atomic state without being affected by atomic motion.

In optical lattice clocks, the motional state of the atom is controlled by confining the atom in an optical lattice. A neutral atom is electrically polarized by the electric field of a laser. This polarization leads to an induced electric dipole in the atom, which experiences a force due to the electric field of the laser. As a result, the atom gets a potential energy which depends on the square of the electric field, that is the intensity of the laser. 

Atoms trapped in an optical lattice

For confining atoms, lasers with frequencies far off resonance from atomic transition frequencies are used in a stand wave configuration. For a pair of states whose transition frequency is larger than the laser frequency (the case of red detuning), the lower energy state is attracted to lower intensities of the standing wave and the higher energy state is attracted to higher intensities regions of the standing wave. For a pair of states whose transition frequency is smaller than the laser frequency (the case of blue detuning), the opposite happens.

Well resolved sideband and Lamb Dicke regimes

A two level atom in the presence of electromagnetic radiation that is near resonant to the transtion frequency between the levels undergoes Rabi oscillations at Rabi frequency $\Omega$. As a function of the frequency of the electromagnetic radiation, the transition probability is square of sinc function. Decoherences changes this square of sinc into a Lorentzian whose width is the decoherence rate $\Gamma /2\pi$ In the presence of atomic motion, the line shape becomes inhomogeneously broadened from the Doppler shift across the atomic velocity distribution, making the lineshape Gaussian or Voigt. 

If an atom is confined to a bound state of a potential, the atomic motion is not a continuous variable, but is restricted to the quantized motional states of the system. The excitation of the two-level atom with initial motional state $|n\rangle$ and final motional state $|m\rangle$ is given by a modified Rabi rate

$\Omega _{mn}=\Omega \exp \left(-{\frac {\eta ^{2}}{2}}\right){\sqrt {\frac {n_{<}!}{n_{<}!}}}\eta ^{|m-n|}L_{n_{<}}^{|m-n|}(\eta ^{2})$

where $\Omega$ is the corresponding Rabi frequency of the atom at rest, $n_{<}$ is the smaller of $n,m$, $n_{>}$ is the larger of $n,m$, $L_{n}^{\alpha }$ is the generalized Laguerre polynomial.

Here $\eta$ is the Lamb Dicke parameter,

which is proportional to ratio of the wavelength of spatial extent of the ground state of the trapping potential, $x_{0}$ and the wavelength of the trapping light, $\lambda _{T}$.

$\eta ={\frac {{\sqrt {2}}\pi x_{0}}{\lambda _{T}}}$

Including the motional state of the atom leads to a modified Rabi rate $\Omega _{mn}$ Moreover the transition frequency is determined by the energy difference between initial and final states that include both electronic and motional degrees of freedom.

Spectrum in the resolved side band and Lamb Dicke regimes. In these regimes, the central peak is much taller and is more separated from the sidebands.

The line shape of the excitation spectrum has a tall central peak which corresponds to only electronic excitation. On either side of this central peak there are side bands associated with both electronic excitation and motional excitation/deexcitation. The shape of each side band peak and the distance from the central peaks is determined by the ratio of the decoherence rate $\Gamma$ and the trap frequency $\omega _{T}$. If $\Gamma >\omega _{T}$, the side band structure is not resolvable from the central peak. This makes discrimination of the purely electronic excitation (the central peak or carrier transition) from a mixed electronic and motional excitation (side band transitions) difficult. In this limit, various spectral peaks blend into each other. In the other limit of $\Gamma \ll \omega _{T}$, the motional effects occur at high modulation frequencies far from the carrier transition. The regime of $\Gamma \ll \omega _{T}$ facilitates discrimination of carrier and sidebands and is called resolved sideband or strong binding regime.

Atomic recoil due absorption and emission of a laser's electromagnetic wave also needs to be considered for the line shapes of different peaks in the excitation spectrum. The effect of recoil is determined by the parameters of the atomic confinement. The confinement of the atom is parameterized by the Lamb-Dicke parameter, $\eta$.

In the strong confinement regime $\eta \gg 1$ or the Lamb-Dicke regime the recoil effects on the line are reduced. The carrier transitions at zero detuning is the dominant peak in the spectrum. The side bands have suppressed amplitudes compared to the carrier and are displaced further away from the carrier. The suppression of sideband excitation can be understood as the recoil momentum being absorbed into the optical potential energy and not the kinetic energy of the atom. In this case, the modified Rabi rate simplifies to the $\Omega \eta {\sqrt {n}}$ for an excitation and $\Omega \eta {\sqrt {n+1}}$ for a deexcitation invoving a change in motional state.

The resolved side band regime pushes the motional effects away from the central peak and the Lamb Dicke regime supresses the sideband peak amplitudes. Therefore, the resolved side band ($\Gamma \ll \omega _{T}$) and Lamb Dicke ($\eta \ll 1$) regimes enable spectroscopy of the clock transition almost free of Doppler and recoil effects.

To realize full separation between excitation of internal and external atomic degrees of freedom, another condition is required - the effect of the confinement on the atom should be independent of the internal state of the atom in the clock states. For an ensemble of ultra cold neutral atoms confined in an optical lattice, this is accomplished using magic wavelength optical lattices.

Magic wavelengths

Lights shifts for the clock states of alkali atoms as a function of the trapping wavelength. At the magic wavelength, the light shifts are exactly equal.

An optical lattice confines atoms by inducing a dipole moment in the atom and exerting a force on this dipole through the gradient of the laser's electric field. In general the induced polarizabilities of each atomic state is different. Specifically, the light shift experienced by the two clock states are in general different, thereby changing the clock transition frequency. Moreover, the inhomogeneous intensity of the light makes the mechanical force on the atom dependent on the internal state of the atom. Unfortunately, this couples the external and internal degrees of freedom.

The ac Stark shift experienced by the clock states of the atom depend on the wavelength and polarization of the trapping laser. In some cases, these can be tailored such that the ac Stark shift of the clock states are equal. These configurations are called magic wavelengths and lead to the same perturbation on the energies of the clock state. This enables measuring the transition frequency between the clock states in pseudo Stark shift free environment.

Therefore, trapping the atoms in an optical lattice operating the resolved sideband and Lamb Dicke regimes with the magic wavelength and polarizations facilitates the pristine clock transition frequency of the atom to be measured without affecting or being influenced by the motion of the atom.

Spectroscopy of lattice confined atoms

Even in the resolved sideband and Lamb Dicke regimes, the excitation spectrum is altered by the details of the confinement, especially in one dimensional lattices. In practice, red and blue sidebands are broadened over a range of frequencies away from the central peak. These are measured and characterized by driving the carrier transition into saturation. The narrow central peak is used as the frequency standard. This carrier transition which has no change in motional state provides a narrow atomic resonance minimally affected by atomic motion in the Lamb-Dicke and resolved sideband regimes. In this case, the excitation spectrum of interest is a single tall peak at the clock transition frequency, with its width determined by the Fourier limit of the probe laser pulse, when other broadening mechanism are negligibly small and the laser is sufficiently coherent.

The narrower a resonance, the higher the frequency resolution using that resonance. Therefore, optical lattice atomic clocks need to operator with best possible spectral linewidths for both stability and accuracy. At present, the ability to observe the narrowest spectra is limited by the stability of lasers used to probe the transition and not the width of the atomic resonances. The probe time is limited by the coherence time of the lasers, which determines the minimum Fourier resolvable linewidth. The laser frequency needs to be stable not only during the spectroscopic probing but during several probings to scan the laser frequency across the spectral lineshape. Both Rabi and Ramsey spectroscopy is used to probe the clock transition frequency.

Experiment

One of the most recent optical lattice clock experiment performed with Sr that achieved an fractional frequency uncertainty of $10^{-18}$ was as follows. A few thousand ${}^{87}$Sr atoms were cooled to a few micro Kelvin. They were trapped in an optical lattice near the magic wavelength 813 nm with the trap depth between $40E_{r}$ to $300E_{r}$, where $E_{r}$ is the photon recoil energy. The clock transition ${}^{1}S_{0}\leftrightarrow {}^{3}P_{0}$ clock transition was interrogated with Rabi spectroscopy over a 160 ms interval with a thermal noise limited laser. The state determination was destructive. Therefore the experiment was repeated every 1.3 s.

Systematic shifts in frequency

Several systematic sources of shifts in frequency are mitigated in order to reach a precision in fractional frequency of $10^{-18}$. These shifts in frequency are due to shifts in energy levels due to perturbation and shifts because of relativistic effects. The frequency of clock transition is affected by shifting of the clock energy levels. The shifts occur due to several factors like the relative motion of the atom and the interrogating laser, stray electric and magnetic fields, unaccounted for interactions of atoms with electromagnetic waves.

Stark shifts

Static electric fields can be present at the site of the atoms, arising from various sources. In optical lattice atomic clocks the atom sample is usually trapped in optical potentials which are far from physical surfaces which may accumulate stray charges. Typically the clock states are chosen to be eigenstates of the parity operator so that the first order shifts are zero and the second order shifts can be calculate with sufficient precision using second order perturbation theory. For optical lattice clocks, the dc Stark shifts have been measured and been found to be corresponding to a fractional frequency shift of about $10^{-17}$ to $10^{-18}$.

Zeeman shifts

Static magnetic fields are often used to define a quantization axis for atoms. These fields are assumed to be uniform. Nevertheless, there are inhomogeneties across the atomic samples. The shifts from these fields are often the largest that must be corrected, but these can often be implemented with high accuracy. In general the energy difference between the clock states is affected the magnetic field. This can be written using a power series in powers of the magnitude of magnetic field, $B$

$\Delta \nu _{B}=\nu -\nu _{A}=C_{B1}+C_{B2}B^{2}+\cdots$

For small magnetic fields, the first two terms are usually sufficient. For atoms with zero nuclear spin, the first term is zero and the second term is very small for small magnetic field. Atoms which have non zero nuclear and electron spin have a hyperfine structure. This makes both $C_{B1}$ and $C_{B2}$ significant. In these cases, the clock transitions are chosen to be between lower states $|F=0,m_{F}=0\rangle$ and upper states states $|F'=0,m_{F'}=0\rangle$. The shift due to the first term can be found by measuring states that are symmetrically shifted about the clock state. The shift due to the second term can be found by modulating the bias fields between extreme values. They can be accounted for and for the Sr clock have a combined frequency shift below $10^{-18}$.

Light shifts

The energy levels of the clock transition can be affected by the presence of oscillating electromagnetic fields. Lasers are used to create the optical lattice and for interrogation of the clock transition. The field of these laser may contribute to light shifts of the clock states. 

Optical lattice light shifts

The confinement of neutral atoms in optical lattice uses the fact that the neutral atom is polarized by the electric field of the laser. This polarization leads to energy shifts in the energy levels of the atoms.

The polarizability of atoms is a second rank Euclidean tensor and can be written using spherical tensors of ranks $0,1,2$, which give the scalar $\alpha ^{S}$, vector $\alpha ^{V}$ and tensor $\alpha ^{T}$ polarizabilities.

$\Delta E_{LS}=-{\frac {1}{2}}\sum _{a,b}E_{a}\alpha _{ab}E_{b}=-{\frac {1}{2}}\alpha ^{S}E^{2}-{\frac {1}{2}}\sum _{a,b}\alpha _{a}^{V}E_{a}^{2}-{\frac {1}{2}}\sum _{a,b}E_{a}\alpha _{ab}^{T}E_{b}$

Here $a,b$ are Cartesian indices and $E$ is the electric field.

Optical lattice atomic clocks use clocks states that have electronic orbital angular momentum $J=0$. Therefore the light shift from the scalar polarizability is the dominant term. Using magic wavelength traps, this can be made equal for the clock states. Fortunately, the light shifts are equal for a reasonably wide range (about 500 kHz) around the magic frequency. This enables cancelling of the differential light shifts of the clock states at the fractional frequency level of about $10^{-18}$ for Sr.

For atoms that have non zero nuclear spin, causing  hyperfine levels in the clock states, the vector and tensor lights shifts need to be addressed. These have been calculated and measured and corrected to an fractional frequency shift of about $10^{-17}$.

Light shift from interrogation laser

The clock states have identical polarizabilities at the magic wavelength used for the optical lattice. However, their polarizabilities are different for the laser used to interrogate the clock transition frequency. The interrogation laser leads to some off resonant couplings to other states outside the clock subspace, causing light shifts on the clocks states. The shift depends on the difference in polarizabilities of the clock states and the intensity of the laser used to drive the transitions. At present the fractional frequency shift because of this has been found to be about $10^{-17}$ to $10^{-18}$.

Light shift from black body radiation

Blackbody radiation bathing the atoms also causes light shifts. Room temperature black body radiation has dominant electromagnetic frequencies that are smaller than the detunings of the transitions that contribute to the electric dipole polarizability of the clock state. Computing the black body radiation shifts involves calculating the polarizatibilites of all participating states and the black body radiation electric field at the temperature $T$ which bathes the atoms. These shifts have been found to be at the level of $10^{-15}$ level at room temperature for Sr and Yb. Temperature inhomogeneities makes the black body shifts more complicated than these calculations predict. These are corrected by cooling the background to a few Kelvin or by monitoring the thermal environment and accounting for the energy shifts. This leads to fractional frequency uncertainties in the range of $10^{-17}$ to $10^{-18}$.

Cold collision shift

The large number of atoms used in optical lattice atomic clocks gives very high signal to noise ratio for the frequency measurements. Nevertheless, the high atomic density can cause atoms to interact with each other, through collisions, for example. This leads to shift in the energy levels of the clock states. At present by tuning the atomic density per lattice side, the fractional frequency uncertainty has been reduced to $10^{-18}$ due to this effect.

Motion of the atoms

An atomic clock tries to tune the frequency of an electromagnetic wave in the laboratory frame to the transition frequency between a pair of levels of the atom in the rest frame of the atom. This frequency of the interrogating laser in the laboratory frame, $\nu _{L}$ is related to the transition frequency of the atom $\nu _{A}$ in the atom's rest frame as follows.

$\nu _{L}=\nu _{A}{\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\left(1-{\frac {v_{\parallel }}{c}}\right)$

Here $v$ is the velocity of the atom in the  laboratory frame, $v_{\parallel }$ is the component of the velocity of the atom in the direction of the laser beam and $c$ is the speed of light. The experiment is setup so that over the interrogation duration, the average value of the laser frequency, $\langle \nu _{L}\rangle$ is equal to the atomic transition resonance frequency in the atoms rest frame, $\nu _{A}$.

${\frac {\nu -\nu _{A}}{\nu _{A}}}=\left\langle {\frac {\sqrt {1-v^{2}/c^{2}}}{1-v_{\parallel }/c}}\right\rangle -1={\frac {\langle v_{\parallel }\rangle }{c}}+{\frac {\langle v_{\parallel }\rangle ^{2}}{c^{2}}}-{\frac {\langle v^{2}\rangle }{2c^{2}}}+O\left({\frac {v^{3}}{c^{3}}}\right)$

The first term is the first order Doppler shift. The motivation for confining atoms has been to mitigate this term. Although an appropriate optical lattice can address the Doppler effect quite well, there are other effects that can increase the sensitivity to atom motion like the quantum tunneling of atoms between sites on the optical lattice. This can be mitigated by making deep optical lattice potentials or using gravity to suppress the tunneling. These effects at present lead to a negligible effect on the transition frequency.

The higher order terms are relativistic. The second order term account for relativistic time dilation. In present experiments, at temperature of a few micro Kelvin, corresponding to an atomic velocity of a few centimeters per second, the fractional shift in frequency is below $10^{-20}$.

Applications and future directions

The high precision of optical lattice atomic clocks and trapped ion atomic clocks enables several applications involving high precision time and frequency measurements. Optical frequency atomic clocks (both optical lattice atomic clocks and trapped ion atomic clocks) enable the measurement of very small changes of time and frequency. This has facilitated the measurement of the slower passage of time in higher gravitational potentials by measuring time at two different heights above the surface of the earth. The extremely precise measurement of frequency in optical frequency atomic clocks enables probing for variation of fundamental physical constants as small time scales. Optical frequency atomic clocks have been used to put bounds on the variation of physical constants.

Moreover, the precise control of the quantum states of the atoms also suggests applications in quantum state control which has applications like analog and digital quantum information processing.