Editing Resolved sideband cooling

'''Resolved sideband cooling''' is a [[laser cooling]] technique allowing cooling of tightly bound atoms and ions beyond the [[Doppler cooling limit]], potentially to their motional [[ground state]]. Aside from the curiosity of having a particle at zero point energy, such preparation of a particle in a definite state with high probability (initialization) is an essential part of state manipulation experiments in [[quantum optics]] and [[quantum computing]].

== Historical notes ==

As of the writing of this article, the scheme behind what we refer to as ''resolved sideband cooling'' today is attributed<ref name = monroe /><ref name = eschner /> to [[David J. Wineland|D.&nbsp;J.&nbsp;Wineland]] and [[Hans Dehmelt|H.&nbsp;Dehmelt]], in their article "Proposed <math>10^{14}\delta\nu/\nu</math> laser [[fluorescence spectroscopy]] on {{chem|Tl|+}} mono-ion oscillator III (sideband cooling)".<ref name = wineland_75 /> The clarification is important, as at the time of the latter article, the term also designated what we call today [[Doppler cooling]],<ref name = eschner /> which was experimentally realized with atomic ion clouds in 1978 by W.&nbsp;Neuhauser<ref name  = neuhauser /> and independently by D.&nbsp;J.&nbsp;Wineland.<ref name = wineland_78 /> An experiment that demonstrates resolved sideband cooling unequivocally in its contemporary meaning is that of Diedrich ''et al.''<ref name = diedrich /> Similarly unequivocal realization with non-Rydberg neutral atoms was demonstrated in 1998 by S.&nbsp;E.&nbsp;Hamann ''et al.''<ref name = hamann /> via [[Raman cooling#Two photon Raman process|Raman cooling]].

== Conceptual description ==

[[File: Sideband cooling level diagram.png|thumb|right|An atom undergoing resolved sideband cooling. Driven transitions are shown with straight arrows and spontaneous transitions with wiggly arrows. After each driven transition, the atom reaches an excited state with one less motional quanta than the state it came from. For example, the atom starts in the ground ''n''&nbsp;=&nbsp;3 state and is driven to the excited ''n''&nbsp;=&nbsp;2 state. The motional quantum number n does not change in spontaneous transitions.]]

'''Resolved sideband cooling''' is a laser-cooling technique that can be used to cool strongly trapped atoms to the quantum [[ground state]] of their motion. The atoms are usually precooled using the [[Doppler cooling|Doppler laser cooling]]. Subsequently, the resolved [[sideband]] cooling is used to cool the atoms beyond the [[Doppler cooling limit]].

A cold trapped atom can be treated to a good approximation as a [[quantum-mechanical]] [[harmonic oscillator]]. If the spontaneous decay rate is much smaller than the vibrational frequency of the atom in the trap, the [[energy level]]s of the system will be an evenly spaced frequency ladder, with adjacent levels spaced by an energy <math>\hbar \nu</math>. Each level is denoted by a motional quantum number ''n'', which describes the amount of motional energy present at that level. These motional quanta can be understood in the same way as for the [[quantum harmonic oscillator]]. A ladder of levels will be available for each internal state of the atom. For example, in the figure at right both the ground (''g'') and excited (''e'') states have their own ladder of vibrational levels.

Suppose a two-level atom whose ground state is denoted by ''g'' and [[excited state]] by ''e''. Efficient laser cooling occurs when the frequency of the laser beam is tuned to the red sideband i.e.
<math display="block">
 \omega = \omega_0 - \nu,
</math>
where <math>\omega_0</math> is the internal atomic transition frequency corresponding to at transition between ''g'' and ''e'', and <math>\nu</math> is the harmonic-oscillation frequency of the atom. In this case the atom undergoes the transition
<math display="block">
 |g, n\rangle \to |e, n - 1\rangle,
</math>
where <math>|a, m\rangle</math> represents the state of an ion whose internal atomic state is ''a'', and the motional state is ''m''.

If the recoil energy of the atom is negligible compared with the vibrational quantum energy, subsequent [[spontaneous emission]] occurs predominantly at the [[carrier frequency]]. This means that the vibrational quantum number remains constant. This transition is
<math display="block">
 |e, n - 1\rangle \to |g, n - 1\rangle.
</math>

The overall effect of one of these cycles is to reduce the vibrational quantum number of the atom by one. To cool to the ground state, this cycle is repeated many times until <math>|g, n = 0\rangle</math> is reached with a high probability.<ref name = mech />

== Theoretical basis ==

The core process that provides the cooling assumes a two level system that is well localized compared to the wavelength (<math>2\pi c/\omega_0</math>) of the transition (Lamb–Dicke regime), such as a trapped and sufficiently cooled ion or atom. Modeling the system as a harmonic oscillator interacting with a classical monochromatic electromagnetic field<ref name = eschner /> yields (in the rotating wave approximation) the Hamiltonian
<math display="block">
 H = H_\text{HO} + H_\text{AL}
</math>
with
<math display="block">
 H_\text{HO} = \hbar\nu\left(n + \frac 1 2\right),
</math>
<math display="block">
 H_\text{AL} = -\hbar\Delta |e\rangle \langle e| + \hbar \frac \Omega 2 \big(|e\rangle \langle g| e^{i\mathbf k\cdot\mathbf r} + |g\rangle \langle e|e^{-i\mathbf k\cdot\mathbf r}\big),
</math>
and where
: <math>n</math> is the number operator,
: <math>\nu</math> is the frequency spacing of the oscillator,
: <math>\Omega</math> is the Rabi frequency due to the atom-light interaction,
: <math>\Delta</math> is the laser detuning from <math>\omega_0</math>,
: <math>\mathbf k</math> is the laser [[wave vector]].

That is, incidentally, the Jaynes–Cummings Hamiltonian used to describe the phenomenon of an atom coupled to a cavity in cavity QED.<ref name = wineland_98 /> The absorption(emission) of photons by the atom is then governed by the off-diagonal elements, with probability of a transition between vibrational states <math>m, n</math> proportional to <math>\big|\langle m| e^{i\mathbf k\cdot\mathbf r} |n\rangle \big|^2</math>, and for each <math>n</math> there is a manifold <math>\big\{|g, n\rangle, |e, n\rangle\big\}</math> coupled to its neighbors with strength proportional to <math>\big|\langle m| e^{i\mathbf k\cdot\mathbf r} |n\rangle\big|</math>. Three such manifolds are shown in the picture.

If the <math>\omega_0</math> transition linewidth <math>\Gamma</math> satisfies <math>\Gamma \ll \nu</math>, a sufficiently narrow laser can be tuned to a red sideband, <math>\omega_0 - q\nu, q \in \{1, 2, 3, \dots\}</math>. For an atom starting at <math>|g, n\rangle</math>, the predominantly probable transition will be to <math>|e, n - q\rangle</math>. This process is depicted by arrow "1" in the picture. In the Lamb–Dicke regime, the spontaneously emitted photon (depicted by arrow "2") will be, on average, at frequency <math>\omega_0</math>,<ref name = diedrich /> and the net effect of such a cycle, on average, will be the removing of <math>q</math> motional quanta. After some cycles, the average phonon number is <math>\bar n = R_q^{1/q}/(1 - R_q^{1/q})</math>, where <math>R_q</math> is the ratio of the intensities of the red to blue <math>q</math>-th sidebands.<ref name = turchette /> In practice, this process is normally done on the first motional sideband <math>q = 1</math> for optimal efficiency. Repeating the processes many times while ensuring that spontaneous emission occurs provides cooling to <math>\bar n \approx (\Gamma/\nu)^2 \ll 1</math>.<ref name = eschner /><ref name = wineland_98 /> More rigorous mathematical treatment is given in Turchette et al.<ref name = turchette /> and Wineland et al.<ref name = wineland_98 /> Specific treatment of cooling multiple ions can be found in [[Giovanna Morigi|Morigi]] et al.<ref name=morigi />

== Experimental implementations ==

For resolved sideband cooling to be effective, the process needs to start at sufficiently low <math>\bar n</math>. To that end, the particle is usually first cooled to the Doppler limit, then some sideband cooling cycles are applied, and finally, a measurement is taken or state manipulation is carried out. A more or less direct application of this scheme was demonstrated by Diedrich et al.<ref name=diedrich /> with the caveat that the narrow quadrupole transition used for cooling connects the ground state to a long-lived state, and the latter had to be pumped out to achieve optimal cooling efficiency. It is not uncommon, however, that additional steps are needed in the process, due to the atomic structure of the cooled species. Examples of that are the cooling of {{chem|Ca|+}} ions and the Raman sideband cooling of {{chem|Cs}} atoms.

=== Example: cooling of {{chem|Ca|+}} ions ===

[[File:Internal structure of Ca 40 ion with zeeman splitting.png|thumb|right|Relevant {{chem|Ca|+}} structure and light: blue - Doppler cooling; red - sideband cooling path; yellow - spontaneous decay; green - spin polarization <math>\sigma^-</math> pulses]]
The energy levels relevant to the cooling scheme for {{chem|Ca|+}} ions are the S<sub>1/2</sub>, P<sub>1/2</sub>, P<sub>3/2</sub>, D<sub>3/2</sub>, and D<sub>5/2</sub>, which are additionally split by a static magnetic field to their Zeeman manifolds. Doppler cooling is applied on the dipole S<sub>1/2</sub> - P<sub>1/2</sub> transition (397&nbsp;nm), however, there is about 6% probability of spontaneous decay to the long-lived D<sub>3/2</sub> state, so that state is simultaneously pumped out (at 866&nbsp;nm) to improve Doppler cooling. Sideband cooling is performed on the narrow quadrupole transition S<sub>1/2</sub> - D<sub>5/2</sub> (729&nbsp;nm), however, the long-lived D<sub>5/2</sub> state needs to be pumped out to the short lived P<sub>3/2</sub> state (at 854&nbsp;nm) to recycle the ion to the ground S<sub>1/2</sub> state and maintain cooling performance. One possible implementation was carried out by Leibfried et al.<ref name = leibfried /> and a similar one is detailed by Roos.<ref name = roos /> For each data point in the 729&nbsp;nm absorption spectrum, a few hundred iterations of the following are executed:
* the ion is Doppler cooled with 397&nbsp;nm and 866&nbsp;nm light, with 854&nbsp;nm light on as well
* the ion is spin polarized to the S<sub>1/2</sub>(m=-1/2) state by applying a <math>\sigma^-</math> 397&nbsp;nm light for the last few moments of the Doppler cooling process
* sideband cooling loops are applied at the first red sideband of the D<sub>5/2</sub>(m=-5/2) 729&nbsp;nm transition
* to ensure the population ends up in the S<sub>1/2</sub>(m=-1/2) state, another <math>\sigma^-</math> 397&nbsp;nm pulse is applied
* manipulation is carried out and analysis is carried out by applying 729&nbsp;nm light at the frequency of interest
* detection is carried out with 397&nbsp;nm and 866&nbsp;nm light: discrimination between dark (D) and bright (S) state is based on a pre-determined threshold value of fluorescence counts
Variations of this scheme relaxing the requirements or improving the results are being investigated/used by several ion-trapping groups.

=== Example: Raman sideband cooling of {{chem|Cs}} atoms ===

A [[Raman cooling#Two photon Raman process|Raman transition]] replaces the one-photon transition used in the sideband above by a two-photon process via a virtual level. In the {{chem|Cs}} cooling experiment carried out by Hamann et al.,<ref name=hamann /> trapping is provided by an isotropic [[optical lattice]] in a magnetic field, which also provides Raman coupling to the red sideband of the Zeeman manifolds. The process followed in <ref name=hamann /> is:
* preparation of cold sample of <math>10^6</math> {{chem|Cs}} atoms is carried out in [[optical molasses]], in a [[magneto-optic trap]]
* atoms are allowed to occupy a 2D, near resonance lattice
* the lattice is changed adiabatically to a far off resonance lattice, which leaves the sample sufficiently well cooled for sideband cooling to be effective ([[Lamb Dicke regime|Lamb-Dicke regime]])
* a magnetic field is turned on to tune the Raman coupling to the red motional sideband
* relaxation between the hyperfine states is provided by a pump/repump laser pair 
* after some time, pumping is intensified to transfer the population to a specific hyperfine state
* lattice is turned off and [[time of flight]] techniques are employed to perform Stern-Gerlach analysis

==See also==
* [[Laser cooling]]
* [[Amplitude modulation]]

== References ==

<!--- See [[Wikipedia:Footnotes]] on how to create references using<ref></ref> tags which will then appear here automatically -->
{{Reflist|
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J. | last7=Gould | first7=P. | title=Resolved-Sideband Raman Cooling of a Bound Atom to the 3D Zero-Point Energy | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=75 | issue=22 | date=27 November 1995 | issn=0031-9007 | doi=10.1103/physrevlett.75.4011 | pmid=10059792 | pages=4011–4014| bibcode=1995PhRvL..75.4011M }}</ref><ref name=wineland_75>D. Wineland and H. Dehmelt, ‘‘Proposed <math>10^{14}\delta\nu/\nu</math> laser fluorescence spectroscopy on {{chem|Tl|+}} mono-ion oscillator III (sideband cooling),’’ Bull. Am. Phys. Soc. 20, 637 (1975).</ref><ref name=neuhauser>{{cite journal | last1=Neuhauser | first1=W. | last2=Hohenstatt | first2=M. | last3=Toschek | first3=P. | last4=Dehmelt | first4=H. | title=Optical-Sideband Cooling of Visible Atom Cloud Confined in Parabolic Well | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=41 | issue=4 | date=24 July 1978 | issn=0031-9007 | doi=10.1103/physrevlett.41.233 | pages=233–236| bibcode=1978PhRvL..41..233N }}</ref><ref name=wineland_78>{{cite journal | last1=Wineland | first1=D. J. | last2=Drullinger | first2=R. E. | last3=Walls | first3=F. L. | title=Radiation-Pressure Cooling of Bound Resonant Absorbers | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=40 | issue=25 | date=19 June 1978 | issn=0031-9007 | doi=10.1103/physrevlett.40.1639 | pages=1639–1642| bibcode=1978PhRvL..40.1639W |doi-access=free}}</ref><ref name=hamann>{{cite journal | last1=Hamann | first1=S. 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Roos|title=Controlling the quantum state of trapped ions|type=Ph.D.|url=http://heart-c704.uibk.ac.at/publications/dissertation/roos_diss.pdf|access-date=2014-03-17|archive-url=https://web.archive.org/web/20070111202411/http://heart-c704.uibk.ac.at/publications/dissertation/roos_diss.pdf|archive-date=2007-01-11|url-status=dead}}</ref><ref name = mech>{{cite journal | last1=Schliesser | first1=A. | last2=Rivière | first2=R. | last3=Anetsberger | first3=G. | last4=Arcizet | first4=O. | last5=Kippenberg | first5=T. J. | title=Resolved-sideband cooling of a micromechanical oscillator | journal=Nature Physics | publisher=Springer Science and Business Media LLC | volume=4 | issue=5 | date=13 April 2008 | issn=1745-2473 | doi=10.1038/nphys939 | pages=415–419| arxiv=0709.4036 | bibcode=2008NatPh...4..415S | s2cid=119203324 }}</ref>
}}

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