Hyperfine structure

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Hyperfine_structure 

In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds.

In atoms, hyperfine structure arises from the energy of the nuclear magnetic dipole moment interacting with the magnetic field generated by the electrons and the energy of the nuclear electric quadrupole moment in the electric field gradient due to the distribution of charge within the atom. Molecular hyperfine structure is generally dominated by these two effects, but also includes the energy associated with the interaction between the magnetic moments associated with different magnetic nuclei in a molecule, as well as between the nuclear magnetic moments and the magnetic field generated by the rotation of the molecule.

Hyperfine structure contrasts with fine structure, which results from the interaction between the magnetic moments associated with electron spin and the electrons' orbital angular momentum. Hyperfine structure, with energy shifts typically orders of magnitudes smaller than those of a fine-structure shift, results from the interactions of the nucleus (or nuclei, in molecules) with internally generated electric and magnetic fields.

Schematic illustration of fine and hyperfine structure in a neutral hydrogen atom

History

The first theory of atomic hyperfine structure was given in 1930 by Enrico Fermi for an atom containing a single valence electron with an arbitrary angular momentum. The Zeeman splitting of this structure was discussed by S. A. Goudsmit and R. F. Bacher later that year. In 1935, H. Schüler and Theodor Schmidt proposed the existence of a nuclear quadrupole moment in order to explain anomalies in the hyperfine structure.

Theory

The theory of hyperfine structure comes directly from electromagnetism, consisting of the interaction of the nuclear multipole moments (excluding the electric monopole) with internally generated fields. The theory is derived first for the atomic case, but can be applied to each nucleus in a molecule. Following this there is a discussion of the additional effects unique to the molecular case.

Atomic hyperfine structure

Magnetic dipole

Main article: Dipole

The dominant term in the hyperfine Hamiltonian is typically the magnetic dipole term. Atomic nuclei with a non-zero nuclear spin ${\displaystyle \mathbf{I}}$ have a magnetic dipole moment, given by:

${\displaystyle {\mathit{\mu }}_{\text{I}}=g_{\text{I}}{\mu }_{\text{N}}\mathbf{I},}$

where ${\displaystyle g_{\text{I}}}$ is the g-factor and ${\displaystyle {\mu }_{\text{N}}}$ is the nuclear magneton.

There is an energy associated with a magnetic dipole moment in the presence of a magnetic field. For a nuclear magnetic dipole moment, μ_I, placed in a magnetic field, B, the relevant term in the Hamiltonian is given by:

${\displaystyle {\hat{H}}_{\text{D}}=-{\mathit{\mu }}_{\text{I}}\cdot \mathbf{B}.}$

In the absence of an externally applied field, the magnetic field experienced by the nucleus is that associated with the orbital (ℓ) and spin (s) angular momentum of the electrons:

${\displaystyle \mathbf{B}\equiv {\mathbf{B}}_{\text{el}}={\mathbf{B}}_{\text{el}}^{\ell }+{\mathbf{B}}_{\text{el}}^s.}$

Electron orbital angular momentum results from the motion of the electron about some fixed external point that we shall take to be the location of the nucleus. The magnetic field at the nucleus due to the motion of a single electron, with charge –e at a position r relative to the nucleus, is given by:

${\displaystyle {\mathbf{B}}_{\text{el}}^{\ell }=\frac{{\mu }_0}{4\pi }\frac{-e\mathbf{v}\times -\mathbf{r}}{r^3},}$

where −r gives the position of the nucleus relative to the electron. Written in terms of the Bohr magneton, this gives:

${\displaystyle {\mathbf{B}}_{\text{el}}^{\ell }=-2{\mu }_{\text{B}}\frac{{\mu }_0}{4\pi }\frac{1}{r^3}\frac{\mathbf{r}\times m_{\text{e}}\mathbf{v}}{\hslash }.}$

Recognizing that m_ev is the electron momentum, p, and that r×p/ħ is the orbital angular momentum in units of ħ, ℓ, we can write:

${\displaystyle {\mathbf{B}}_{\text{el}}^{\ell }=-2{\mu }_{\text{B}}\frac{{\mu }_0}{4\pi }\frac{1}{r^3}\ell .}$

For a many-electron atom this expression is generally written in terms of the total orbital angular momentum, ${\displaystyle \mathbf{L}}$, by summing over the electrons and using the projection operator, ${\displaystyle {\phi }_i^{\ell }}$, where $\sum _i{\ell }_i=\sum _i{\phi }_i^{\ell }\mathbf{L}$. For states with a well defined projection of the orbital angular momentum, L_z, we can write ${\displaystyle {\phi }_i^{\ell }={\hat{\ell }}_{z_i}/L_z}$, giving:

${\displaystyle {\mathbf{B}}_{\text{el}}^{\ell }=-2{\mu }_{\text{B}}\frac{{\mu }_0}{4\pi }\frac{1}{L_z}\sum _i\frac{{\hat{\ell }}_{zi}}{r_i^3}\mathbf{L}.}$

The electron spin angular momentum is a fundamentally different property that is intrinsic to the particle and therefore does not depend on the motion of the electron. Nonetheless it is angular momentum and any angular momentum associated with a charged particle results in a magnetic dipole moment, which is the source of a magnetic field. An electron with spin angular momentum, s, has a magnetic moment, μ_s, given by:

${\displaystyle {\mathit{\mu }}_{\text{s}}=-g_s{\mu }_{\text{B}}\mathbf{s},}$

where g_s is the electron spin g-factor and the negative sign is because the electron is negatively charged (consider that negatively and positively charged particles with identical mass, travelling on equivalent paths, would have the same angular momentum, but would result in currents in the opposite direction).

The magnetic field of a point dipole moment, μ_s, is given by:

${\displaystyle {\mathbf{B}}_{\text{el}}^s=\frac{{\mu }_0}{4\pi r^3}\left(3\left({\mathit{\mu }}_{\text{s}}\cdot \hat{\mathbf{r}}\right)\hat{\mathbf{r}}-{\mathit{\mu }}_{\text{s}}\right)+{\displaystyle \frac{2{\mu }_0}{3}}{\mathit{\mu }}_{\text{s}}{\delta }^3(\mathbf{r}).}$

The complete magnetic dipole contribution to the hyperfine Hamiltonian is thus given by:

${\displaystyle \begin{array}{rl}{\hat{H}}_D=& 2g_{\text{I}}{\mu }_{\text{N}}{\mu }_{\text{B}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{1}{L_z}}\sum _i{\displaystyle \frac{{\hat{l}}_{zi}}{r_i^3}}\mathbf{I}\cdot \mathbf{L}\\ & +g_{\text{I}}{\mu }_{\text{N}}{g}_{\text{s}}{\mu }_{\text{B}}\frac{{\mu }_0}{4\pi }\frac{1}{S_z}\sum _i\frac{{\hat{s}}_{zi}}{r_i^3}\left\{3\left(\mathbf{I}\cdot \hat{\mathbf{r}}\right)\left(\mathbf{S}\cdot \hat{\mathbf{r}}\right)-\mathbf{I}\cdot \mathbf{S}\right\}\\ & +\frac{2}{3}{g}_{\text{I}}{\mu }_{\text{N}}{g}_{\text{s}}{\mu }_{\text{B}}{\mu }_0\frac{1}{S_z}\sum _i{\hat{s}}_{zi}{\delta }^3\left({\mathbf{r}}_i\right)\mathbf{I}\cdot \mathbf{S}.\end{array}}$

The first term gives the energy of the nuclear dipole in the field due to the electronic orbital angular momentum. The second term gives the energy of the "finite distance" interaction of the nuclear dipole with the field due to the electron spin magnetic moments. The final term, often known as the Fermi contact term relates to the direct interaction of the nuclear dipole with the spin dipoles and is only non-zero for states with a finite electron spin density at the position of the nucleus (those with unpaired electrons in s-subshells). It has been argued that one may get a different expression when taking into account the detailed nuclear magnetic moment distribution.

For states with ${\displaystyle \ell \ne 0}$ this can be expressed in the form

${\displaystyle {\hat{H}}_D=2g_I{\mu }_{\text{B}}{\mu }_{\text{N}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{\mathbf{I}\cdot \mathbf{N}}{r^3}},}$

where:

${\displaystyle \mathbf{N}=\ell -\frac{g_s}{2}\left[\mathbf{s}-3(\mathbf{s}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}\right].}$

If hyperfine structure is small compared with the fine structure (sometimes called IJ-coupling by analogy with LS-coupling), I and J are good quantum numbers and matrix elements of ${\displaystyle {\hat{H}}_{\text{D}}}$ can be approximated as diagonal in I and J. In this case (generally true for light elements), we can project N onto J (where J = L + S is the total electronic angular momentum) and we have:

${\displaystyle {\hat{H}}_{\text{D}}=2g_I{\mu }_{\text{B}}{\mu }_{\text{N}}{\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{\mathbf{N}\cdot \mathbf{J}}{\mathbf{J}\cdot \mathbf{J}}}{\displaystyle \frac{\mathbf{I}\cdot \mathbf{J}}{r^3}}.}$

This is commonly written as

${\displaystyle {\hat{H}}_{\text{D}}=\hat{A}\mathbf{I}\cdot \mathbf{J},}$

with $⟨\hat{A}⟩$ being the hyperfine-structure constant which is determined by experiment. Since I·J = 1⁄2{F·F − I·I − J·J} (where F = I + J is the total angular momentum), this gives an energy of:

${\displaystyle \mathrm{\Delta }{E}_{\text{D}}=\frac{1}{2}⟨\hat{A}⟩[F(F+1)-I(I+1)-J(J+1)].}$

In this case the hyperfine interaction satisfies the Landé interval rule.

Electric quadrupole

Main article: Quadrupole

Atomic nuclei with spin ${\displaystyle I\ge 1}$ have an electric quadrupole moment. In the general case this is represented by a rank-2 tensor, ${\displaystyle \underset{\_}{\underset{\_}{Q}}}$, with components given by:

${\displaystyle Q_{ij}=\frac{1}{e}\int \left(3x_i^{\mathrm{\prime }}{x}_j^{\mathrm{\prime }}-{\left(r^{\mathrm{\prime }}\right)}^2{\delta }_{ij}\right)\rho \left({\mathbf{r}}^{\mathrm{\prime }}\right)d^3{r}^{\mathrm{\prime }},}$

where i and j are the tensor indices running from 1 to 3, x_i and x_j are the spatial variables x, y and z depending on the values of i and j respectively, δ_ij is the Kronecker delta and ρ(r) is the charge density. Being a 3-dimensional rank-2 tensor, the quadrupole moment has 3² = 9 components. From the definition of the components it is clear that the quadrupole tensor is a symmetric matrix (Q_ij = Q_ji) that is also traceless (Σ_iQ_ii = 0), giving only five components in the irreducible representation. Expressed using the notation of irreducible spherical tensors we have:

${\displaystyle T_m^2(Q)=\sqrt{\frac{4\pi }{5}}\int \rho \left({\mathbf{r}}^{\mathrm{\prime }}\right){\left(r^{\mathrm{\prime }}\right)}^2{Y}_m^2\left({\theta }^{\mathrm{\prime }},{\phi }^{\mathrm{\prime }}\right)d^3{r}^{\mathrm{\prime }}.}$

The energy associated with an electric quadrupole moment in an electric field depends not on the field strength, but on the electric field gradient, confusingly labelled $\underset{\_}{\underset{\_}{q}}$, another rank-2 tensor given by the outer product of the del operator with the electric field vector:

${\displaystyle \underset{\_}{\underset{\_}{q}}=\mathrm{\nabla }\otimes \mathbf{E},}$

with components given by:

${\displaystyle q_{ij}=\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}.}$

Again it is clear this is a symmetric matrix and, because the source of the electric field at the nucleus is a charge distribution entirely outside the nucleus, this can be expressed as a 5-component spherical tensor, ${\displaystyle T^2(q)}$, with:

${\displaystyle \begin{array}{rl}{T}_0^2(q)& =\frac{\sqrt{6}}{2}{q}_{zz}\\ T_{+1}^2(q)& =-q_{xz}-i{q}_{yz}\\ T_{+2}^2(q)& =\frac{1}{2}(q_{xx}-q_{yy})+i{q}_{xy},\end{array}}$

where:

${\displaystyle T_{-m}^2(q)=(-1{)}^m{T}_{+m}^2(q{)}^{\ast }.}$

The quadrupolar term in the Hamiltonian is thus given by:

${\displaystyle {\hat{H}}_Q=-e{T}^2(Q)\cdot T^2(q)=-e\sum _m(-1{)}^m{T}_m^2(Q)T_{-m}^2(q).}$

A typical atomic nucleus closely approximates cylindrical symmetry and therefore all off-diagonal elements are close to zero. For this reason the nuclear electric quadrupole moment is often represented by Q_zz.

Molecular hyperfine structure

The molecular hyperfine Hamiltonian includes those terms already derived for the atomic case with a magnetic dipole term for each nucleus with ${\displaystyle I>0}$ and an electric quadrupole term for each nucleus with ${\displaystyle I\ge 1}$. The magnetic dipole terms were first derived for diatomic molecules by Frosch and Foley, and the resulting hyperfine parameters are often called the Frosch and Foley parameters.

In addition to the effects described above, there are a number of effects specific to the molecular case.

Direct nuclear spin–spin

Each nucleus with ${\displaystyle I>0}$ has a non-zero magnetic moment that is both the source of a magnetic field and has an associated energy due to the presence of the combined field of all of the other nuclear magnetic moments. A summation over each magnetic moment dotted with the field due to each other magnetic moment gives the direct nuclear spin–spin term in the hyperfine Hamiltonian, ${\displaystyle {\hat{H}}_{II}}$.

${\displaystyle {\hat{H}}_{II}=-\sum _{\alpha \ne {\alpha }^{\mathrm{\prime }}}{\mathit{\mu }}_{\alpha }\cdot {\mathbf{B}}_{{\alpha }^{\mathrm{\prime }}},}$

where α and α' are indices representing the nucleus contributing to the energy and the nucleus that is the source of the field respectively. Substituting in the expressions for the dipole moment in terms of the nuclear angular momentum and the magnetic field of a dipole, both given above, we have

${\displaystyle {\hat{H}}_{II}={\displaystyle \frac{{\mu }_0{\mu }_{\text{N}}^2}{4\pi }}\sum _{\alpha \ne {\alpha }^{\mathrm{\prime }}}\frac{g_{\alpha }{g}_{{\alpha }^{\mathrm{\prime }}}}{R_{\alpha {\alpha }^{\mathrm{\prime }}}^3}\left\{{\mathbf{I}}_{\alpha }\cdot {\mathbf{I}}_{{\alpha }^{\mathrm{\prime }}}-3\left({\mathbf{I}}_{\alpha }\cdot {\hat{\mathbf{R}}}_{\alpha {\alpha }^{\mathrm{\prime }}}\right)\left({\mathbf{I}}_{{\alpha }^{\mathrm{\prime }}}\cdot {\hat{\mathbf{R}}}_{\alpha {\alpha }^{\mathrm{\prime }}}\right)\right\}.}$

Nuclear spin–rotation

The nuclear magnetic moments in a molecule exist in a magnetic field due to the angular momentum, T (R is the internuclear displacement vector), associated with the bulk rotation of the molecule, thus

${\displaystyle {\hat{H}}_{\text{IR}}=\frac{e{\mu }_0{\mu }_{\text{N}}\hslash }{4\pi }\sum _{\alpha \ne {\alpha }^{\mathrm{\prime }}}\frac{1}{R_{\alpha {\alpha }^{\mathrm{\prime }}}^3}\left\{\frac{Z_{\alpha }{g}_{{\alpha }^{\mathrm{\prime }}}}{M_{\alpha }}{\mathbf{I}}_{{\alpha }^{\mathrm{\prime }}}+\frac{Z_{{\alpha }^{\mathrm{\prime }}}{g}_{\alpha }}{M_{{\alpha }^{\mathrm{\prime }}}}{\mathbf{I}}_{\alpha }\right\}\cdot \mathbf{T}.}$

Small molecule hyperfine structure

A typical simple example of the hyperfine structure due to the interactions discussed above is in the rotational transitions of hydrogen cyanide (¹H¹²C¹⁴N) in its ground vibrational state. Here, the electric quadrupole interaction is due to the ¹⁴N-nucleus, the hyperfine nuclear spin-spin splitting is from the magnetic coupling between nitrogen, ¹⁴N (I_N = 1), and hydrogen, ¹H (I_H = 1⁄2), and a hydrogen spin-rotation interaction due to the ¹H-nucleus. These contributing interactions to the hyperfine structure in the molecule are listed here in descending order of influence. Sub-doppler techniques have been used to discern the hyperfine structure in HCN rotational transitions.

The dipole selection rules for HCN hyperfine structure transitions are ${\displaystyle \mathrm{\Delta }J=1}$, ${\displaystyle \mathrm{\Delta }F=\{0,\pm 1\}}$, where J is the rotational quantum number and F is the total rotational quantum number inclusive of nuclear spin (${\displaystyle F=J+I_{\text{N}}}$), respectively. The lowest transition (${\displaystyle J=1\to 0}$) splits into a hyperfine triplet. Using the selection rules, the hyperfine pattern of ${\displaystyle J=2\to 1}$ transition and higher dipole transitions is in the form of a hyperfine sextet. However, one of these components (${\displaystyle \mathrm{\Delta }F=-1}$) carries only 0.6% of the rotational transition intensity in the case of ${\displaystyle J=2\to 1}$. This contribution drops for increasing J. So, from ${\displaystyle J=2\to 1}$ upwards the hyperfine pattern consists of three very closely spaced stronger hyperfine components (${\displaystyle \mathrm{\Delta }J=1}$, ${\displaystyle \mathrm{\Delta }F=1}$) together with two widely spaced components; one on the low frequency side and one on the high frequency side relative to the central hyperfine triplet. Each of these outliers carry ~${\displaystyle \frac{1}{2}{J}^2}$ (J is the upper rotational quantum number of the allowed dipole transition) the intensity of the entire transition. For consecutively higher-J transitions, there are small but significant changes in the relative intensities and positions of each individual hyperfine component.

Measurements

Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra and in electron paramagnetic resonance spectra of free radicals and transition-metal ions.

Applications

Astrophysics

The hyperfine transition as depicted on the Pioneer plaque

As the hyperfine splitting is very small, the transition frequencies are usually not located in the optical, but are in the range of radio- or microwave (also called sub-millimeter) frequencies.

Hyperfine structure gives the 21 cm line observed in H I regions in interstellar medium.

Carl Sagan and Frank Drake considered the hyperfine transition of hydrogen to be a sufficiently universal phenomenon so as to be used as a base unit of time and length on the Pioneer plaque and later Voyager Golden Record.

In submillimeter astronomy, heterodyne receivers are widely used in detecting electromagnetic signals from celestial objects such as star-forming core or young stellar objects. The separations among neighboring components in a hyperfine spectrum of an observed rotational transition are usually small enough to fit within the receiver's IF band. Since the optical depth varies with frequency, strength ratios among the hyperfine components differ from that of their intrinsic (or optically thin) intensities (these are so-called hyperfine anomalies, often observed in the rotational transitions of HCN). Thus, a more accurate determination of the optical depth is possible. From this we can derive the object's physical parameters.

Nuclear spectroscopy

In nuclear spectroscopy methods, the nucleus is used to probe the local structure in materials. The methods mainly base on hyperfine interactions with the surrounding atoms and ions. Important methods are nuclear magnetic resonance, Mössbauer spectroscopy, and perturbed angular correlation.

Nuclear technology

The atomic vapor laser isotope separation (AVLIS) process uses the hyperfine splitting between optical transitions in uranium-235 and uranium-238 to selectively photo-ionize only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuned dye lasers are used as the sources of the necessary exact wavelength radiation.

Use in defining the SI second and meter

The hyperfine structure transition can be used to make a microwave notch filter with very high stability, repeatability and Q factor, which can thus be used as a basis for very precise atomic clocks. The term transition frequency denotes the frequency of radiation corresponding to the transition between the two hyperfine levels of the atom, and is equal to f = ΔE/h, where ΔE is difference in energy between the levels and h is the Planck constant. Typically, the transition frequency of a particular isotope of caesium or rubidium atoms is used as a basis for these clocks.

Due to the accuracy of hyperfine structure transition-based atomic clocks, they are now used as the basis for the definition of the second. One second is now defined to be exactly 9192631770 cycles of the hyperfine structure transition frequency of caesium-133 atoms.

On October 21, 1983, the 17th CGPM defined the meter as the length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second.

Precision tests of quantum electrodynamics

The hyperfine splitting in hydrogen and in muonium have been used to measure the value of the fine-structure constant α. Comparison with measurements of α in other physical systems provides a stringent test of QED.

Qubit in ion-trap quantum computing

The hyperfine states of a trapped ion are commonly used for storing qubits in ion-trap quantum computing. They have the advantage of having very long lifetimes, experimentally exceeding ~10 minutes (compared to ~1 s for metastable electronic levels).

The frequency associated with the states' energy separation is in the microwave region, making it possible to drive hyperfine transitions using microwave radiation. However, at present no emitter is available that can be focused to address a particular ion from a sequence. Instead, a pair of laser pulses can be used to drive the transition, by having their frequency difference (detuning) equal to the required transition's frequency. This is essentially a stimulated Raman transition. In addition, near-field gradients have been exploited to individually address two ions separated by approximately 4.3 micrometers directly with microwave radiation.

Laser cooling

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Laser_cooling 

 

Simplified principle of Doppler laser cooling:

1	A stationary atom sees the laser neither red- nor blue-shifted and does not absorb the photon.
2	An atom moving away from the laser sees it red-shifted and does not absorb the photon.
3.1	An atom moving towards the laser sees it blue-shifted and absorbs the photon, slowing the atom.
3.2	The photon excites the atom, moving an electron to a higher quantum state.
3.3	The atom re-emits a photon. As its direction is random, there is no net change in momentum over many absorption-emission cycles.

In condensed matter physics, laser cooling includes a number of techniques in which atoms, molecules, and small mechanical systems are cooled, often approaching temperatures near absolute zero. Laser cooling techniques rely on the fact that when an object (usually an atom) absorbs and re-emits a photon (a particle of light) its momentum changes. For an ensemble of particles, their thermodynamic temperature is proportional to the variance in their velocity. That is, more homogeneous velocities among particles corresponds to a lower temperature. Laser cooling techniques combine atomic spectroscopy with the aforementioned mechanical effect of light to compress the velocity distribution of an ensemble of particles, thereby cooling the particles.

The 1997 Nobel Prize in Physics was awarded to Claude Cohen-Tannoudji, Steven Chu, and William Daniel Phillips "for development of methods to cool and trap atoms with laser light".

History

Radiation pressure

Radiation pressure is the force that electromagnetic radiation exerts on matter. In 1873 Maxwell published his treatise on electromagnetism in which he predicted radiation pressure.  The force was experimentally demonstrated for the first time by Lebedev and reported at a conference in Paris in 1900,  and later published in more detail in 1901. Following Lebedev's measurements Nichols and Hull also demonstrated the force of radiation pressure in 1901, with a refined measurement reported in 1903.  

In 1933, Otto Frisch deflected an atomic beam of sodium atoms with light. This was the first realization of radiation pressure acting on a resonant transition.

Laser cooling proposals

The introduction of lasers in atomic manipulation experiments acted as the advent of laser cooling proposals in the mid 1970s. Laser cooling was proposed separately in 1975 by two different research groups: Hänsch and Schawlow, and Wineland and Dehmelt. Both proposals outlined a process of slowing heat-based velocity in atoms with "radiative forces." In the paper by Hänsch and Schawlow, the effect of radiation pressure on any object that reflects light is described. That concept was then connected to the cooling of atoms in a gas. These early proposals for laser cooling only relied on "scattering force", the name for the radiation pressure.

In the late 1970s, Ashkin described how radiation forces can be used to simultaneously cool and trap atoms. He emphasized how this process could allow for long spectroscopic measurements without the atoms escaping the trap and proposed the overlapping of optical traps in order to study interactions between different atoms.

Initial realizations

Closely following Ashkin's letter in 1978, two research groups: Wineland, Drullinger and Walls, and Neuhauser, Hohenstatt, Toscheck and Dehmelt further refined that work. In specific, Wineland, Drullinger, and Walls were concerned with the improvement of spectroscopy. The group wrote about experimentally demonstrating the cooling of atoms through a process using radiation pressure. They cite a precedence for using radiation pressure in optical traps, yet criticize the ineffectiveness of previous models due to the presence of the Doppler effect. In an effort to lessen the effect, they applied an alternative take on cooling magnesium ions below the room temperature precedent. Using the electromagnetic trap to contain the magnesium ions, they bombarded them with a laser barely out of phase from the resonant frequency of the atoms. The research from both groups served to illustrate the mechanical properties of light. Around this time, laser cooling techniques had allowed for temperatures lowered to around 40 kelvins.

William Phillips was influenced by the Wineland paper and attempted to mimic it, using neutral atoms instead of ions. In 1982, he published the first paper outlining the cooling of neutral atoms. The process he used is now known as the Zeeman slower and became one of the standard techniques for slowing an atomic beam.

Modern advances

Atoms

Now, temperatures around 240 microkelvins were reached. That threshold was the lowest researchers thought was possible. When temperatures then reached 43 microkelvins in an experiment by Steven Chu, the new low was explained by the addition of more atomic states in combination to laser polarization. Previous conceptions of laser cooling were decided to have been too simplistic. The major breakthroughs in the 70s and 80s in the use of laser light for cooling led to several improvements to preexisting technology and new discoveries with temperatures just above absolute zero. The cooling processes were utilized to make atomic clocks more accurate and to improve spectroscopic measurements, and led to the observation of a new state of matter at ultracold temperatures. The new state of matter, the Bose–Einstein condensate, was observed in 1995 by Eric Cornell, Carl Wieman, and Wolfgang Ketterle.

Laser cooling was primarily used to create ultracold atoms. For example, the experiments in quantum physics need to perform near absolute zero where unique quantum effects such as Bose–Einstein condensation can be observed. Laser cooling is also a primary tool in optical clock experiments. Laser cooling has primarily been used on atoms, but recent progress has been made toward laser cooling more complex systems. For example, a team at UT Austin demonstrated the use of laser cooling for noninvasive optical trapping and manipulation, a technique they termed opto-refrigerative tweezers.

Molecules

In 2010, a team at Yale successfully laser-cooled a diatomic molecule. In 2016, a group at MPQ successfully cooled formaldehyde to 420 μK via optoelectric Sisyphus cooling.  In 2022, a group at Harvard successfully trapped and laser cooled CaOH to a minimum temperature of 720(40) μK in a magneto-optical trap. 

Mechanical systems

In 2007, an MIT team successfully laser-cooled a macro-scale (1 gram) object to 0.8 K. In 2011, a team from the California Institute of Technology and the University of Vienna became the first to laser-cool a (10 μm x 1 μm) mechanical object to its quantum ground state.

Methods

The first example of laser cooling, and also still the most common method (so much so that it is still often referred to simply as 'laser cooling') is Doppler cooling.

Doppler cooling

Main article: Doppler cooling

 

The lasers needed for the magneto-optical trapping of rubidium-85: (a) & (b) show the absorption (red detuned to the dotted line) and spontaneous emission cycle, (c) & (d) are forbidden transitions, (e) shows that if a cooling laser excites an atom to the F=3 state, it is allowed to decay to the "dark" lower hyperfine, F=2 state, which would stop the cooling process, if it were not for the repumper laser (f).

Doppler cooling, which is usually accompanied by a magnetic trapping force to give a magneto-optical trap, is by far the most common method of laser cooling. It is used to cool low density gases down to the Doppler cooling limit, which for rubidium-85 is around 150 microkelvins.

In Doppler cooling, initially, the frequency of light is tuned slightly below an electronic transition in the atom. Because the light is detuned to the "red" (i.e., at lower frequency) of the transition, the atoms will absorb more photons if they move towards the light source, due to the Doppler effect. Thus if one applies light from two opposite directions, the atoms will always scatter more photons from the laser beam pointing opposite to their direction of motion. In each scattering event the atom loses a momentum equal to the momentum of the photon. If the atom, which is now in the excited state, then emits a photon spontaneously, it will be kicked by the same amount of momentum, but in a random direction. Since the initial momentum change is a pure loss (opposing the direction of motion), while the subsequent change is random, the probable result of the absorption and emission process is to reduce the momentum of the atom, and therefore its speed—provided its initial speed was larger than the recoil speed from scattering a single photon. If the absorption and emission are repeated many times, the average speed, and therefore the kinetic energy of the atom, will be reduced. Since the temperature of a group of atoms is a measure of the average random internal kinetic energy, this is equivalent to cooling the atoms.

Anti-Stokes cooling

The idea for anti-Stokes cooling was first advanced by Pringsheim in 1929. While Doppler cooling lowers the translational temperature of a sample, anti-Stokes cooling decreases the vibrational or phonon excitation of a medium. This is accomplished by pumping a substance with a laser beam from a low-lying energy state to a higher one with subsequent emission to an even lower-lying energy state. The principal condition for efficient cooling is that the anti-Stokes emission rate to the final state be significantly larger than that to other states as well as the nonradiative relaxation rate. Because vibrational or phonon energy can be many orders of magnitude larger than the energy associated with Doppler broadening, the efficiency of heat removal per laser photon expended for anti-Stokes cooling can be correspondingly larger than that for Doppler cooling. The anti-Stokes cooling effect was first demonstrated by Djeu and Whitney in CO₂ gas. The first anti-Stokes cooling in a solid was demonstrated by Epstein et al. in a ytterbium doped fluoride glass sample.

Potential practical applications for anti-Stokes cooling of solids include radiation balanced solid state lasers and vibration-free optical refrigeration.

Other methods

Other methods of laser cooling include:

• Sisyphus cooling
• Resolved sideband cooling
• Raman sideband cooling
• Velocity selective coherent population trapping (VSCPT)
• Gray molasses
• Optical molasses
• Cavity-mediated cooling
• Use of a Zeeman slower
• Electromagnetically induced transparency (EIT) cooling
• Anti-Stokes cooling in solids
• Polarization gradient cooling

Trapped ion quantum computer

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Trapped_ion_quantum_computer

 

Chip ion trap for quantum computing from 2011 at NIST.

A trapped ion quantum computer is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic states of each ion, and quantum information can be transferred through the collective quantized motion of the ions in a shared trap (interacting through the Coulomb force). Lasers are applied to induce coupling between the qubit states (for single qubit operations) or coupling between the internal qubit states and the external motional states (for entanglement between qubits).

The fundamental operations of a quantum computer have been demonstrated experimentally with the currently highest accuracy in trapped ion systems. Promising schemes in development to scale the system to arbitrarily large numbers of qubits include transporting ions to spatially distinct locations in an array of ion traps, building large entangled states via photonically connected networks of remotely entangled ion chains, and combinations of these two ideas. This makes the trapped ion quantum computer system one of the most promising architectures for a scalable, universal quantum computer. As of April 2018, the largest number of particles to be controllably entangled is 20 trapped ions.

History

The first implementation scheme for a controlled-NOT quantum gate was proposed by Ignacio Cirac and Peter Zoller in 1995, specifically for the trapped ion system. The same year, a key step in the controlled-NOT gate was experimentally realized at NIST Ion Storage Group, and research in quantum computing began to take off worldwide.

Simplified scale model

In 2021, researchers from the University of Innsbruck presented a quantum computing demonstrator that fits inside two 19-inch server racks, the world's first quality standards-meeting compact trapped ion quantum computer.

Paul trap

Classical linear Paul trap in Innsbruck for a string of Calcium ions.

The electrodynamic quadrupole ion trap currently used in trapped ion quantum computing research was invented in the 1950s by Wolfgang Paul (who received the Nobel Prize for his work in 1989). Charged particles cannot be trapped in 3D by just electrostatic forces because of Earnshaw's theorem. Instead, an electric field oscillating at radio frequency (RF) is applied, forming a potential with the shape of a saddle spinning at the RF frequency. If the RF field has the right parameters (oscillation frequency and field strength), the charged particle becomes effectively trapped at the saddle point by a restoring force, with the motion described by a set of Mathieu equations.

This saddle point is the point of minimized energy magnitude, ${\displaystyle |E(\mathbf{x})|}$, for the ions in the potential field. The Paul trap is often described as a harmonic potential well that traps ions in two dimensions (assume ${\displaystyle \hat{x}}$ and ${\displaystyle \hat{y}}$ without loss of generality) and does not trap ions in the ${\displaystyle \hat{z}}$ direction. When multiple ions are at the saddle point and the system is at equilibrium, the ions are only free to move in ${\displaystyle \hat{z}}$. Therefore, the ions will repel each other and create a vertical configuration in ${\displaystyle \hat{z}}$, the simplest case being a linear strand of only a few ions. Coulomb interactions of increasing complexity will create a more intricate ion configuration if many ions are initialized in the same trap. Furthermore, the additional vibrations of the added ions greatly complicate the quantum system, which makes initialization and computation more difficult.

Once trapped, the ions should be cooled such that ${\displaystyle k_{\mathrm{B}}T\ll \hslash {\omega }_z}$(see Lamb Dicke regime). This can be achieved by a combination of Doppler cooling and resolved sideband cooling. At this very low temperature, vibrational energy in the ion trap is quantized into phonons by the energy eigenstates of the ion strand, which are called the center of mass vibrational modes. A single phonon's energy is given by the relation ${\displaystyle \hslash {\omega }_z}$. These quantum states occur when the trapped ions vibrate together and are completely isolated from the external environment. If the ions are not properly isolated, noise can result from ions interacting with external electromagnetic fields, which creates random movement and destroys the quantized energy states.

Requirements for quantum computation

See also: DiVincenzo's criteria

 

Magnesium ions in a trap.

The full requirements for a functional quantum computer are not entirely known, but there are many generally accepted requirements. David DiVincenzo outlined several of these criterion for quantum computing.

Qubits

Any two-level quantum system can form a qubit, and there are two predominant ways to form a qubit using the electronic states of an ion:

1. Two ground state hyperfine levels (these are called "hyperfine qubits")
2. A ground state level and an excited level (these are called the "optical qubits")

Hyperfine qubits are extremely long-lived (decay time of the order of thousands to millions of years) and phase/frequency stable (traditionally used for atomic frequency standards). Optical qubits are also relatively long-lived (with a decay time of the order of a second), compared to the logic gate operation time (which is of the order of microseconds). The use of each type of qubit poses its own distinct challenges in the laboratory.

Initialization

Ionic qubit states can be prepared in a specific qubit state using a process called optical pumping. In this process, a laser couples the ion to some excited states which eventually decay to one state which is not coupled to the laser. Once the ion reaches that state, it has no excited levels to couple to in the presence of that laser and, therefore, remains in that state. If the ion decays to one of the other states, the laser will continue to excite the ion until it decays to the state that does not interact with the laser. This initialization process is standard in many physics experiments and can be performed with extremely high fidelity (>99.9%).

The system's initial state for quantum computation can therefore be described by the ions in their hyperfine and motional ground states, resulting in an initial center of mass phonon state of ${\displaystyle |0⟩}$ (zero phonons).

Measurement

Measuring the state of the qubit stored in an ion is quite simple. Typically, a laser is applied to the ion that couples only one of the qubit states. When the ion collapses into this state during the measurement process, the laser will excite it, resulting in a photon being released when the ion decays from the excited state. After decay, the ion is continually excited by the laser and repeatedly emits photons. These photons can be collected by a photomultiplier tube (PMT) or a charge-coupled device (CCD) camera. If the ion collapses into the other qubit state, then it does not interact with the laser and no photon is emitted. By counting the number of collected photons, the state of the ion may be determined with a very high accuracy (>99.99%).

Arbitrary single qubit rotation

One of the requirements of universal quantum computing is to coherently change the state of a single qubit. For example, this can transform a qubit starting out in 0 into any arbitrary superposition of 0 and 1 defined by the user. In a trapped ion system, this is often done using magnetic dipole transitions or stimulated Raman transitions for hyperfine qubits and electric quadrupole transitions for optical qubits. The term "rotation" alludes to the Bloch sphere representation of a qubit pure state. Gate fidelity can be greater than 99%.

The rotation operators ${\displaystyle R_x(\theta )}$ and ${\displaystyle R_y(\theta )}$ can be applied to individual ions by manipulating the frequency of an external electromagnetic field from and exposing the ions to the field for specific amounts of time. These controls create a Hamiltonian of the form ${\displaystyle H_I^i=\hslash \mathrm{\Omega }/2(S_{+}\exp (i\varphi )+S_{-}\exp (-i\varphi ))}$. Here, ${\displaystyle S_{+}}$ and ${\displaystyle S_{-}}$ are the raising and lowering operators of spin (see Ladder operator). These rotations are the universal building blocks for single-qubit gates in quantum computing.

To obtain the Hamiltonian for the ion-laser interaction, apply the Jaynes–Cummings model. Once the Hamiltonian is found, the formula for the unitary operation performed on the qubit can be derived using the principles of quantum time evolution. Although this model utilizes the rotating wave approximation, it proves to be effective for the purposes of trapped-ion quantum computing.

Two qubit entangling gates

Besides the controlled-NOT gate proposed by Cirac and Zoller in 1995, many equivalent, but more robust, schemes have been proposed and implemented experimentally since. Recent theoretical work by JJ. Garcia-Ripoll, Cirac, and Zoller have shown that there are no fundamental limitations to the speed of entangling gates, but gates in this impulsive regime (faster than 1 microsecond) have not yet been demonstrated experimentally. The fidelity of these implementations has been greater than 99%.

Scalable trap designs

Quantum computers must be capable of initializing, storing, and manipulating many qubits at once in order to solve difficult computational problems. However, as previously discussed, a finite number of qubits can be stored in each trap while still maintaining their computational abilities. It is therefore necessary to design interconnected ion traps that are capable of transferring information from one trap to another. Ions can be separated from the same interaction region to individual storage regions and brought back together without losing the quantum information stored in their internal states. Ions can also be made to turn corners at a "T" junction, allowing a two dimensional trap array design. Semiconductor fabrication techniques have also been employed to manufacture the new generation of traps, making the 'ion trap on a chip' a reality. An example is the quantum charge-coupled device (QCCD) designed by D. Kielpinski, Christopher Monroe and David J. Wineland. QCCDs resemble mazes of electrodes with designated areas for storing and manipulating qubits.

The variable electric potential created by the electrodes can both trap ions in specific regions and move them through the transport channels, which negates the necessity of containing all ions in a single trap. Ions in the QCCD's memory region are isolated from any operations and therefore the information contained in their states is kept for later use. Gates, including those that entangle two ion states, are applied to qubits in the interaction region by the method already described in this article.

Decoherence in scalable traps

When an ion is being transported between regions in an interconnected trap and is subjected to a nonuniform magnetic field, decoherence can occur in the form of the equation below (see Zeeman effect). This effectively changes the relative phase of the quantum state. The up and down arrows correspond to a general superposition qubit state, in this case the ground and excited states of the ion.

${\displaystyle |\uparrow ⟩+|\downarrow ⟩⟶\exp (i\alpha )|\uparrow ⟩+|\downarrow ⟩}$

Additional relative phases could arise from physical movements of the trap or the presence of unintended electric fields. If the user could determine the parameter α, accounting for this decoherence would be relatively simple, as known quantum information processes exist for correcting a relative phase. However, since α from the interaction with the magnetic field is path-dependent, the problem is highly complex. Considering the multiple ways that decoherence of a relative phase can be introduced in an ion trap, reimagining the ion state in a new basis that minimizes decoherence could be a way to eliminate the issue.

One way to combat decoherence is to represent the quantum state in a new basis called the decoherence-free subspaces, or DFS., with basis states ${\displaystyle |\uparrow \downarrow ⟩}$ and ${\displaystyle |\downarrow \uparrow ⟩}$. The DFS is actually the subspace of two ion states, such that if both ions acquire the same relative phase, the total quantum state in the DFS will be unaffected.

Challenges

Trapped ion quantum computers theoretically meet all of DiVincenzo's criteria for quantum computing, but implementation of the system can be quite difficult. The main challenges facing trapped ion quantum computing are the initialization of the ion's motional states, and the relatively brief lifetimes of the phonon states. Decoherence also proves to be challenging to eliminate, and is caused when the qubits interact with the external environment undesirably.

CNOT gate implementation

The controlled NOT gate is a crucial component for quantum computing, as any quantum gate can be created by a combination of CNOT gates and single-qubit rotations. It is therefore important that a trapped-ion quantum computer can perform this operation by meeting the following three requirements.

First, the trapped ion quantum computer must be able to perform arbitrary rotations on qubits, which are already discussed in the "arbitrary single-qubit rotation" section.

The next component of a CNOT gate is the controlled phase-flip gate, or the controlled-X gate (see quantum logic gate). In a trapped ion quantum computer, the state of the center of mass phonon functions as the control qubit, and the internal atomic spin state of the ion is the working qubit. The phase of the working qubit will therefore be flipped if the phonon qubit is in the state ${\displaystyle |1⟩}$.

Lastly, a SWAP gate must be implemented, acting on both the ion state and the phonon state.

Two alternate schemes to represent the CNOT gates are presented in Michael Nielsen and Isaac Chuang's Quantum Computation and Quantum Information and Cirac and Zoller's Quantum Computation with Cold Trapped Ions.