Annuaire du Collège de France 2011-2012

# Single Atoms: Controlling the Quantum World

**Dieter Meschede**

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# Single Atoms: Controlling the Quantum World

- 1 W. Neuhauser
*et al*.,*Phys.**Rev*., A 22, 1980, 1137.

1Atoms and photons are the principal cases of microscopic particles, the hallmarks of quantum objects. For more than a century they have served as conceptual models to understand and investigate the microscopic structure of matter. More than 30 years ago it became possible to isolate and observe single ions in a trapping device. This experiment1 may be taken as a turning point in the history of approaching the quantum world: experimenters left the role of pure observers and turned into quantum engineers striving to control the properties of matter down to the level of a single atom, molecule, or photon.

- 2 H. Dehmelt, “Stored Ion Spectroscopy”,
*in*Arecchi F., Strumia F., Walther H.,*Advances in Laser S*(...) - 3 D. Meschede, A. Rauschenbeutel, “Manipulating Single Atoms”,
*Adv. At. Mol. Opt.Phys*. 53, 2006, 75 (...)

2Ion storage relies on the interaction of its Coulomb charge with electromagnetic fields and has long since realized the “spectroscopist’s dream of a single stored ion in free space” (H. Dehmelt2). Neutral atoms can only be trapped by means of their feeble electric or magnetic dipole moments which furthermore mix motional and internal degrees of freedom. However, in contrast to the long-range ionic Coulomb force, they offer short-range interactions which can be controlled by the experimenter, making them ideally suited for constructing many-body quantum systems atom by atom in the so-called bottom-up approach3.

3A light field induces an electric dipole moment in an atom, and a laser, red detuned from the atomic resonance frequency, exerts a force towards maximal intensity. Thus, in a standing wave with a transverse bell shape, atoms— after laser cooling — become trapped at the maxima of the standing wave which provide micro potentials with a trap depth energy *U *= k_{B}· 1 mK, 100 times or more above the typical kinetic energy of the trapped atoms. Illumination of the atoms with near resonant laser light allows to record, by means of a photon counting CCD camera real time movies of atoms travelling with a moving lattice (optical conveyor belt).

- 4 Karski M.
*et al*.,*Phys. Rev. Lett*., 102, 2009, 053001.

4Adjacent micro-potentials are spaced by about 0.5µm only, and hence below the Rayleigh resolution limit of about 2µm of the microscope objective. It is thus an interesting question whether atoms (which are point-like radiation sources just like stars in space) in adjacent potentials can be detected. Neighboring atoms define one of the most interesting situations since they can be made to interact. It turns out that it is not necessary to obtain full Rayleigh resolution4: the point-spread function of an individual atom can be measured with very high precision, and hence the position quantum number can be inferred by deconvolution methods with high fidelity.

5The atoms have two long-lived quantum states (hyperfine states) forming a pseudo spin ½ system. Pure spin quantum states ǀ↑⟩ and ǀ↓⟩ are prepared by optical hyperfine pumping. Microwave transitions at 9.2 GHz induce arbitrary spin (Rabi) rotations, *e.g*. an even superposition (ǀ↑⟩+ǀ↓⟩)/√2 is created in less than 10µs by a so called π/2-pulse. The internal quantum states can be measured by *e.g*. the so called “push-out” technique with excellent fidelity, and we can observe Rabi oscillations with excellent contrast.

6For certain “magic wavelengths’” the conveyor belt can be made sensitive to the internal spin quantum number of the trapped atom, thus there is a right hand polarized (RHP) conveyor belt for the ǀ↑⟩ and its left hand companion for the ǀ↓⟩-state. RHP- and LHP-standing waves are created from a superposition of two counterpropagating linearly polarized waves. If one of the linear polarizations is rotated with respect to the other, the RHP- and LHP-waves walk in opposite directions. At present, the magic conveyor belt can only be shuttled back and forth for one lattice site, but changing the spin state after each transport step makes the atom continue its journey unidirectionally with a fidelity exceeding 99%/step and making transport distances of about 50 sites available.

- 5 L. Förster
*et al*.,*Phys. Rev. Lett*., 103, 2009, 233001.

7Shifting spin-dependent potentials with respect to each other opens a highly efficient way to cool and manipulate the vibrational degree of freedom of trapped atoms: microwave transitions with Δn_{vib} ≠ 0 are now allowed with high and variable transition strengths enabling spectroscopy of all micro-potential levels, and moreover opening the route to cooling and coherent vibrational state engineering.5

- 6 N. Spethmann
*et al,**Phys. Rev. Lett.,*109, 2012, 235301.

8An approach alternative to micro-wave cooling has been realized with a single Cs atom interacting with an ultracold gas of Rb atoms. In future experiments, single atoms can be used to probe quantum many-body systems like a dopant in conventional condensed matter.6

- 7 A. Steffen
*et al.,**Proc. Nat.**Acad. Science,*109, 2012, 9770.

9We are thus able to control all degrees of freedom of the trapped atoms. Based on quantum states ǀn⟩_{site}ǀn⟩_{vib}ǀ↑,↓⟩ we have engineered a single atom interferometer. The single atom interferometer has achieved a sensitivity for measuring accelerations of 10^{-4} g, it is of interest for nanometric applications, and extended versions may even rival the sensitivity promised by ballistic interferometers.7

# Single Atoms and Single Photons: Retrieving Information

10Both simple atoms and photons can be described by two quantum states only, *e.g*. two selected internal states for atoms, two polarization states for photons. In a new language, introduced over the past two decades, such objects are now considered carriers of quantum information where in addition to the pure states ǀ↑⟩ and ǀ↓⟩, available for classical bits, too, superposition states play a central role. Trapped atoms store qubits, photons transport qubits from one node to another. A single atom interacting with a single photon is thus not only an elementary process of light matter interaction but also of quantum information science.

- 8 H.J. Kimble
*in*P. Berman,*Cavity quantum electrodynamics*, Academic Press, Boston, 1994.

11The interaction of an atom with the light field of a high finesse resonator can be understood in terms of two coupled oscillators: at resonance, the modes are split and the transmission of the empty resonator is strongly suppressed by the factor 1/(1+*C*)^{2}, where *C* = *g*^{2}/κγ is the so called cooperativity quantifying the coupling of the atom-resonator field (rate *g*) in terms of the loss rates of atom (γ) and field (κ). At *C* = 25, very strong resonant suppression occurs.8

- 9 M. Khudaverdyan
*et al*.,*Phys. Rev. Lett*. 103, 2009, 123006; J. Volz*et al*., 475, 2011, 210. - 10 S. Reick
*et al., J. Opt. Soc. Am., B 27*, A152, 2010.

12In atoms, two different long lived ground states make good qubit states, *e.g*. the hyperfine states of the Cs atomic clock. Transitions between those states are induced by micro-waves or by Raman two photon transitions. Resonant interaction with the cavity field occurs for only one of the two qubits states: For one state the cavity looks empty and hence transmits the full probe laser light; for the other one strong atom cavity coupling suppresses transmission. We observe a random telegraph signal exhibiting quantum jumps between the two qubit states caused by incidental excitation from the probe and a weak repumping laser. This measurement approaches a QND (quantum non-demolition) measurement since it continuously monitors the system quantum state, ideally without scattering photons.9 We detect single photons (“clicks”) (lower trace), and by suitable binning we can straightforwardly assign quantum states. The information content carried by a single detected photon can be used in an optimal way using Bayes’ rule of conditional probabilities: every photon click “updates” our knowledge about the state of the system, provided a suitable model is available.10

- 11 M. Mücke,
*Nature*(London), 465, 2010, 755; T. Kampschulte*et al*.,*Phys. Rev. Lett*., 105, 2010, 153 (...)

13An atom with two ground states has more to offer: its optical properties (the refractive index) can be controlled by an additional, strong light field connecting the two ground states via Raman transitions. The most interesting region occurs around the two photon resonance:11 in a narrow window, the atomic refractive index varies from dispersive to transparent to strongly absorptive, showing *e.g*. the celebrated electromagnetically induced transparency. We therefore expect correspondingly strong variations of the atom cavity system transmission.

14The surprising observation of strong cooling action near the two photon resonance can be explained by sideband cooling in the vicinity of the narrow two photon EIT resonance. It offers an efficient cooling scheme to address one of the big challenges of experiments in optical cavity QED: the quest for tightly controlled atom cavity coupling which is impaired by atomic motion.

- 12 A. Rauschenbeutel
*et al*.,*Phys. Rev. Lett*., 83, 1999, 5166. - 13 T. Wilk
*et al*.,*Phys. Rev. Lett*., 104, 2010, 010502; L. Isenhower*et al*.,*Phys. Rev. Lett*., 104, 2 (...) - 14 A. Sorensen
*et al*.,*Phys. Rev. Lett*., 91, 2003, 097905. - 15 S. Brakhane
*et al,**Phys.**Rev. Lett.,*109, 2012, 173601.

15The next interesting question is whether we can coherently control the interaction of exactly two atoms. Direct unitary interactions have so far only been shown with next-nearest neighbors of a quantum degenerate gas in an optical lattice. At larger separations, other enhancement mechanisms must be employed, such as enhanced collision cross sections12, or Rydberg blockade13. For atom-optical cavity systems, variations of the projective state detection scheme presented on a figure showing the quantum jumps of a strongly coupled atom-cavity system seem natural14: away from resonance 3 different two atom spin states (α=0 {ǀ↑↑⟩}, α=1 {ǀ↑↓⟩,ǀ↓↑⟩} α=2 {ǀ↓↓⟩}) can be discriminated. In order to overcome the fluctuations of the freely evolving two atom state one can implement a feedback loop (pump lasers, micro-wave transitions).15 The stabilized α=1 state is considered. Starting with this state, simultaneous π/2 rotations of the atoms and subsequent measurements will progressively project to the maximally entangled singlet state, the only state invariant under 2atom rotations.

# Quantum Walks and Digital Quantum Simulation

- 16 S. Lloyd,
*Science,*273, 1996, 1073.

16Quantum systems which are discrete both in space and time can be realized with pseudo-spin-½ atoms stored in the spin-dependent optical lattice presented in lecture I: the quantum state evolves in a digital way through finite time steps, with spatial coordinates taking only discrete values on the lattice. At individual sites *q*, the quantum state is defined by the internal spin (s={↑,↓}) and vibrational degrees of freedom, ǀψ⟩ = ǀq⟩_{site}ǀn⟩_{vib}ǀs⟩_{spin}. Simultaneously, temporal discreteness is imposed through the application of periodic actions, *e.g*., transport steps. It was shown by S.Lloyd16 that systems whose quantum dynamics is ruled by finite-time local operations can be regarded as universal quantum simulators, capable of simulating the evolution of any continuous-time quantum system. Note that this system makes a very large Hilbert space available, which is accessible with well-controlled properties at every position.

17System dynamics is implemented with discrete operations. The shift operator is

Ŝ = ǀq+1⟩⟨qǀ_{ site} ǀ↑⟩⟨↑ǀ+ǀq-1⟩⟨qǀ_{ site} ǀ↓⟩⟨↓ǀ

- 17 J. Kempe,
*Cont. Phys*., 44, 2003, 307. - 18 M. Karski
*et al*.,*Science,*325, 2009, 174.

18which connects each site with its adjacent sites conditioned on the spin state. So-called coin operations cause spin rotations R (φ,θ) which may or may not be site dependent, Ĉ = R (φ,θ)ǀs⟩⟨sǀ. A quantum walk17 consists of n-fold applications (Ŵ^{ (n)}) of a fixed combination of coin and shift operation Ŵ= Ŝ⊗Ĉ. After n steps, an atom initially positioned at site 0 is coherently delocalized over 2*n*+1 sites. The final distribution appears as the result of a multi-path interference of the delocalized atom; its details depend on the initial spin state of the atom. Examples of an experiment averaging about 1000 atoms are considered and compared to their theoretical prediction18. As a consequence of its unitary character, the *rms* width of the quantum walk scales as *n *(“ballistically”)*,* in contrast to the random walk, which shows the well-known √*n*, or diffusive behavior.

19In the idealized case, reversing the quantum walk after n steps for another n steps leads to a complete refocusing of the delocalized atom to a single site. After 6+6 steps we observe a peak of almost 30% at the original site accompanied by a Gaussian distribution which is caused by decoherence processes and shows the potential of the system to investigate the interplay of unitary and dissipative evolution.

20The strict periodicity of the lattice calls for interpretations in momentum (*k*) space, too. For the lattice with periodicity λ/2, the Brillouin zone extends from -2π/*λ* to 2π/*λ*. With periodic system time τ, the energies Eτ/*ħ*=ωτ are bounded as well (-π ≤ ωτ ≤ π, see Floquet theory). Let us consider the energy-momentum dispersion relation: without any spin rotation the initial spin-up/spin-down components of the atom remain unaltered by the walk operation and move “ballistically” away from each other, reflected by a linear dispersion relation (blue trace). The velocity is independent of *k*, *v*=dω/d*k*= λ/(2τ), *i.e*. there is no dispersion. With π/2 spin rotation (so-called Hadamard coins), the two spin states are maximally mixed. The splitting at k=0 can be interpreted as an avoided-crossing due to spin-orbit coupling. An atom initially prepared in a single site (delta-like) homogeneously fills the Brillouin zone. Therefore, the asymptotic distribution will be dominated by the nearly-dispersionfree zones of the energy bands, where the group velocity is times the unity coin velocity. This explains the emergence of two ballistic peaks near the maximum range of the walk.

21The *k*-states are eigenstates of the walk operators and hence the initial distribution in *k*-space is preserved unless forces are acting onto the walkers. Forces, for instance acceleration of the lattice or a field gradient, will lead to *e.g*. Bloch oscillations like free electrons in a crystal lattice.

- 19 A. Ahlbrecht
*et al.*,*New J. Phys.*, 2012, accepted.

22The next level of complexity is obtained by inserting a second walker which can interact with its companion. Interactions are characterized by a phase shift which occurs if and only if the walkers meet at the same site (on-site collisions). The problem can be simulated on a computer but is also tractable in analytic terms.19 In the absence of interactions, the distribution resembles the single particle quantum walk. With interactions, a bound two-particle state with maximum probability near x_{1}=x_{2} is found (“molecule formation”). This situation resembles the textbook problem of molecular binding by a *delta*-like potential.

23Not only increasing the number of quantum particles but also the number of spatial dimensions for the spin dependent transport is a very attractive topic offering novel properties for studying *e.g*. artificial gauge fields.

# Did We Forget to Properly Celebrate the 50^{th} Anniversary of the Laser?

- 20 A. Schawlow, C. Townes,
*Phys. Rev*., 112, 1958, 1940. - 21 T. Maiman,
*Nature,*187, 1960, 493

24In 2010, the 50^{th} anniversary of the laser celebrated the completion of a scientific race, initiated by A. Schawlow and C. Townes with a famous publication in 195820 predicting the laser along with many of its properties in detail. The race was won —mostly unexpected by the large mainstream technical laboratories— by T. Maiman who collected suitable components in his laboratory and succeeded in igniting the world’s first (ruby) laser in 196021.

25Almost immediately after its invention, the laser (Light Amplification by Stimulated Emission of Radiation) was exploited for entertainment. For instance, James Bond was among its earliest clients. To this day, society loves these futuristic aspects, but does it pay sufficient tribute to the laser’s deeper relevance?

- 22 Laserfest.

26The laser was clearly initiated and invented in the United States —and 50 years later well recognized for its role in science, technology, and society by an address of President B. Obama to the congress and by numerous other activities22. In Europe, the anniversary of the laser seems to have attracted much less attention, in spite of the contribution by European scientists and in spite of the driving force that the laser keeps exerting on science and technology.

27The laser is a beautiful result of fundamental research in physics, and it may well keep transforming technology with a potential sisterly rivaling that of semiconductor electronics. Addressing the origin and future of the laser offers interesting insight into the interplay of basic and applied research and asks for the attention of society at an appropriate level.

28Let us look back at the history of the laser beginning 125 years ago, with the foundation of the Physikalisch-Technische Reichsanstalt in 1887 in Berlin by farsighted scientists and entrepreneurs: W. von Siemens and H. von Helmholtz. Already in 1888, one of the first laboratories began investigating the properties of black bodies. This work —in plain words— performed technology analyses for the rising industry producing electric lanterns for street lighting. It led directly into the discovery of quantum physics in 1900 by M. Planck, laying no less than the foundations of modern physics. At about the same time, the electron was discovered by J.J. Thomson, triggering the birth of the electronics industry. Moreover, in 1886 H. Hertz proved the transmission of electromagnetic waves predicted by Maxwell’s equations, perhaps one of the most significant milestones of modern day technology.

29One way to understand the creation of the laser is a continuously evolving process of electromagnetic oscillators with ever increasing frequency – a continued process to this date. Probably at all times during the past 150 years, the evolution of electromagnetic oscillators at higher and higher frequencies had important consequences in history, *e.g*. it strongly influenced the outcome of the second world war, when compact magnetron cm-wave sources made the use of radar equipment aboard planes efficient.

- 23 C. Townes,
*How the laser happened*, Oxford University Press, New York, 1999.

30An extension of the magnetron to ever shorter wavelengths would have called for continued miniaturization, a principle we might straightforwardly follow today but which deemed unrealistic in the early 1950s. This hurdle provoked C. Townes23 (in parallel with N. Basov and N. Prokhorov in Russia) to invent the maser (micro wave amplification by stimulated emission of radiation) based on the principle of stimulated emission: he left the machined electron tube type devices and resorted to oscillating molecules instead. From there it was a small step only to extend the principles to laser radiation at visible wavelengths.

- 24 R. Ladenburg, H. Kopfermann
*Z. Physik*, 65, 1930, 167.

31In fact there was no fundamental obstacle in the 1930s to invent the laser, it might have been conceived when Kopfermann and Ladenburg24 measured dispersion by neon atoms in a discharge. Extending their observation of what they called “negative dispersion” to *e.g*. higher discharge currents would have brought inversion! However, the concept of achieving inversion, laid out by Einstein already but corresponding *e.g*. to negative temperatures, was not yet ready in the minds of physicists.

- 25 J. von Neumann (1953), reprinted
*in**IEEE J. Quant. Electron*, QE-23, 1987, 659.

32Following the initial demonstration in 1960, a rapid series of further inventions was observed in less than 4 years: the pulsed ruby laser was followed by the iconic helium-neon laser, the workhorses solid state and CO_{2} gas laser; mode locking for generating the shortest pulses ever was demonstrated. Perhaps the most relevant beacon was set in 1962, when the operation of the first semiconductor laser —which had been predicted by J. von Neumann in 1953 already25— was shown.

- 26 H. Welker and H. Mataré had also realized a European counterpart of the invention of the transist (...)

33It is interesting to note that, although the concepts for probably all laser sources used today were laid out in an extremely short time, many of them had to wait for 20 to 30 more years until further technological breakthroughs paved their way for widespread applications. Semiconductor lasers are a case in point where samples for room temperature laboratory use became available in the mid-to-late 1980s only. Here, the final breakthrough was owed to the invention of heterostructures in GaAs, a material first used by H. Welker at Siemens laboratories.26

34The semiconductor laser made numerous advances possible: diode lasers replaced exciter lamps bringing the “wall plug efficiency” of high power lasers to 50%; they activated the worldwide optical communication network by offering large bandwidth; tunable diode laser sources opened the way to the simultaneous application of many affordable and narrowband lasers, enabling the complex experiments in today’s fundamental quantum optical and other laboratories. It is no surprise that the present world market of lasers is dominated by diode lasers.

35In conclusion, we may call the laser world we are witnessing today a sister of the semiconductor electronic revolution that will continue to modify the life style of generations to come. The arrival of laser based technologies has been slower but they offer the potential for a similar impact —the 21^{st} century may indeed become a century of the photon.

## Notes

1 W. Neuhauser *et al*., *Phys. **Rev*., A 22, 1980, 1137.

2 H. Dehmelt, “Stored Ion Spectroscopy”, *in* Arecchi F., Strumia F., Walther H., *Advances in Laser Spectroscopy*, Plenum Press, New York, 1981.

3 D. Meschede, A. Rauschenbeutel, “Manipulating Single Atoms”, *Adv. At. Mol. Opt.Phys*. 53, 2006, 75-104.

4 Karski M. *et al*., *Phys. Rev. Lett*., 102, 2009, 053001.

5 L. Förster *et al*., *Phys. Rev. Lett*., 103, 2009, 233001.

6 N. Spethmann*et al, **Phys. Rev. Lett.,* 109, 2012, 235301.

7 A. Steffen*et al., **Proc. Nat. **Acad. Science,* 109, 2012, 9770.

8 H.J. Kimble *in* P. Berman, *Cavity quantum electrodynamics*, Academic Press, Boston, 1994.

9 M. Khudaverdyan *et al*., *Phys. Rev. Lett*. 103, 2009, 123006; J. Volz *et al*., 475, 2011, 210.

10 S. Reick *et al., J. Opt. Soc. Am., B 27*, A152, 2010.

11 M. Mücke, *Nature* (London), 465, 2010, 755; T. Kampschulte *et al*., *Phys. Rev. Lett*., 105, 2010, 153603.

12 A. Rauschenbeutel *et al*., *Phys. Rev. Lett*., 83, 1999, 5166.

13 T. Wilk *et al*., *Phys. Rev. Lett*., 104, 2010, 010502; L. Isenhower *et al*., *Phys. Rev. Lett*., 104, 2010, 010503.

14 A. Sorensen *et al*., *Phys. Rev. Lett*., 91, 2003, 097905.

15 S. Brakhane*et al, **Phys. **Rev. Lett.,* 109, 2012, 173601.

16 S. Lloyd, *Science,* 273, 1996, 1073.

17 J. Kempe, *Cont. Phys*., 44, 2003, 307.

18 M. Karski *et al*., *Science,* 325, 2009, 174.

19 A. Ahlbrecht *et al.*, *New J. Phys.*, 2012, accepted.

20 A. Schawlow, C. Townes, *Phys. Rev*., 112, 1958, 1940.

21 T. Maiman, *Nature,* 187, 1960, 493

22 Laserfest.

23 C. Townes, *How the laser happened*, Oxford University Press, New York, 1999.

24 R. Ladenburg, H. Kopfermann *Z. Physik*, 65, 1930, 167.

25 J. von Neumann (1953), reprinted *in**IEEE J. Quant. Electron*, QE-23, 1987, 659.

26 H. Welker and H. Mataré had also realized a European counterpart of the invention of the transistor by Bardeen *et al*. at the Paris Westinghouse laboratory in 1948.

## Pour citer cet article

### Référence papier

*Cours et travaux du Collège de France. Annuaire 112*^{e}* année*, Collège de France, Paris, avril 2013, p. 854-861. ISBN 978-2-7226-0198-7

### Référence électronique

**Dieter Meschede**, « Single Atoms: Controlling the Quantum World », *L’annuaire du Collège de France* [En ligne], 112 | 2013, mis en ligne le 28 août 2013, consulté le 24 février 2018. URL : http://journals.openedition.org/annuaire-cdf/1130