Atomic Physics

1. Atomic Physics “Driving Doppler Down” Doppler-free methods Laser cooling, trapping, quantum computing

2. COOLING – FUN demos…

3. Motion of an atom affects the absorption frequency, and its emission frequency A reminder: Question: Estimate the halfwidth at room temperature (a) for Balmer-alpha, (b) for the cesium resonance line From kinetic theory, we know the fraction of atoms with velocity between v and v+dv f(v)dv = {M/(2πkT)}1/2 exp {-Mv2/(2kT)} dv = (1/u√π) exp(-v2/u2) dv Where u is the most probable velocity u = √(2kT/M) Since the fractional shift in frequency is δ/ω0 = v/c Then the line profile is GDoppler(ω) = (c/(uω0√π)exp {-(c2/u2)(ω – ω0)2/ω02}

4. Example 1: Crossed beams Doppler broadening depends on the collimation of the 2 beams – generally this is much larger for the atoms – α typically about 10-3 radians But also broadening due to the short interaction time – “transit-time broadening”. Typical widths are of the order of 1 MHz for visible spectra.

5. “Co-linear” – Saturated Absorption spectroscopy A laser beam approaching the atoms with a weak intensity, and a wavelength off-resonance by a small amount – the atoms with a certain velocity relative to the laser will be promoted to level 2, “burning a hole” in the Gaussian thermal distribution of level 1 atoms.

6. Saturated Absorption spectroscopy • Improvement in “halfwidth” is typically at least 1000, with consequent improvement in precision of measuring the resonance frequency. • Most experiments • utilize double-beam geometries – comparing the difference of the 2 beams – one crossing, the other not-crossing. • And/or chop the pump beam to remove time variations.

7. 2-photon spectroscopy Excitation can be to a virtual level, midway between the initial and final state (of opposite parity) – e.g. a 1s to 2s transition in hydrogen Lyman-α has a wavelength of 126 nm; hence each of the 2 photons needs a wavelength of 252 nm; each of these produced by frequency –doubling a green line (at 504 nm)

8. Two vital factors for precision measurements • Measuring frequency directly • Cooling the source

9. Measuring the wavelength/frequency A femtosecond laser is so short that its energy distribution can span frequencies from f to 2f – then direct comparisons of different frequency-doubled components (from the doubling crystal) allows measurement of the mode number. The unknown frequency can then be compared with these accurately known absolute frequencies. – to precisions of parts in 1012 to 1014. Important corollary: Stepping up frequency ranges from a direct lower frequency measurement (say, 109 Hz) to frequencies in the 1014 Hz range

10. Laser cooling of atoms Photon momentum can be used to apply a force to each atom: radiation of intensity I exerts a force on an area A: Frad = IA/c In the figure, the atom then radiates in all directions so that its momentum long the beam is reduced – i.e. the beam is cooled. Example: a sodium beam at T=900K (from the oven shown) v0 = 1000m/s Acceleration is a = - (ħk/M) (γ/2) = v(recoil) (γ/2) For sodium τ = 1/ γ = 16 ns, v(recoil) = 3 cm/s Hence the stopping distance x = 2v02τ/v(recoil) = 1.1 m But note - if the velocity changes the resonant wavelength changes – so how do we fix that?

11. Use Zeeman splitting to match cooling velocity!! (the first time) Matching the magnetic field can bring the atom to rest just at the end of the solenoid – such an experiment indicates a concentration of atoms which gradually move away perpendicular to the beam, where no laser cooling has occurred - W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596-599 (1982). So now let’s do it in all directions… (three are enough) => TRAP

12. Alternative longitudinal cooling: Tuning the laser slightly below the resonance absorption of a stationary sodium atom. The atom sees the head-on photon as Doppler shifted upward toward its resonant frequency and it is therefore more strongly absorbed than a photon traveling in the opposite direction which is Doppler shifted away from the resonance. For room temperature sodium atom, the incoming photon is Doppler shifted up 0.97 GHz, so to get the head on photon to match the resonant frequency would require that the laser be tuned below the resonant peak by that amount. This method of cooling sodium atoms was proposed by Theodore Hansch and Arthur Schawlow at Stanford University in 1975 and achieved by Chu at AT&T Bell Labs in 1985. Sodium atoms were cooled from a thermal beam at 500K to about 240 mK. The experimental technique involved directing laser beams from opposite directions upon the sample, linearly polarized at 90° with respect to each other. Six lasers could then provide a pair of beams along each coordinate axis. The effectively "viscous" effect of the laser beams in slowing down the atoms was dubbed "optical molasses" by Chu.

13. Three-dimensional cooling – optical molasses The effective molasses force (off resonance by kv)

14. Another way: Use laser “chirping” to match cooling velocity!! Vary the laser frequency to match the change in velocity: we must vary the frequency by GHz in milliseconds! – how do we do that? Electro-optic modulators and rf techniques can do it fast enough – the crystals produce sideband frequencies which can then be varied rapidly by applying an rf signal. Velocity distribution of “chirped” atoms The slower atoms in the distribution have been brought to zero velocity, to give a narrow low velocity peak.

15. Chirping of a laser The chirp of an optical pulse is usually understood as the time dependence of its instantaneous frequency. Specifically, an up-chirp (down-chirp) means that the instantaneous frequency rises (decreases) with time. Example: consider a pulse with a Gaussian envelope and a quadratic temporal phase: This is associated with a linear chirp, i.e., with a linear variation of the instantaneous frequency: (Fourier transform)

16. Magnetic field trap The “wrongly-connected” Helmholtz coils give zero magnetic field at the center, increasing in all directions away from the center.

17. Combination – the MOT Adding the (6) molasses cooling lasers yields an imbalance of the net forces always towards the center. The Zeeman splitting in the magnetic field. The physical geometry

18. Dipole cooling – example: the atomic fountain An intense laser beam can change the energy levels of an atom (AC Stark shift). If the laser frequency is less than the resonance frequency, this forms a potential well attracting the atoms into a volume of high laser intensity – this is called a “dipole-force” trap which can be loaded with the “molasses cooled” atoms. This can be produced neatly on a microscopic scale by producing a standing wave, and thus a string of small traps. After the cold atoms have been trapped, the lasers can be turned off, allowing the atoms to fall – and then be detected lower down by a probe laser…

19. The atomic fountain – in principle, and in reality

20. The fountain geometry increases the time between the two interrogations by gently tossing the atoms up and letting them fall back down under the influence of gravity, all under high vacuum. Atoms are collected and then launched through a single microwave cavity, which interrogates the atoms both on the way up and again on the way down. The atoms are then detected optically to determine the information about the microwave frequency. This cycle is then repeated. The longer time between interrogations improves the precision of the measurement,.

21. Who was Sisyphus? The gods had condemned Sisyphus to ceaselessly rolling a rock to the top of a mountain, whence the stone would fall back of its own weight. They had thought with some reason that there is no more dreadful punishment than futile and hopeless labor. Sisyphus, being near to death, rashly wanted to test his wife's love. He ordered her to cast his unburied body into the middle of the public square. Sisyphus woke up in the underworld. And there, annoyed by an obedience so contrary to human love, he obtained from Pluto permission to return to earth in order to chastise his wife. But when he had seen again the face of this world, enjoyed water and sun, warm stones and the sea, he no longer wanted to go back to the infernal darkness. Recalls, signs of anger, warnings were of no avail. Many years more he lived facing the curve of the gulf, the sparkling sea, and the smiles of the earth. A decree of the gods was necessary. Mercury came and seized the impudent man by the collar and, snatching him from his joys, led him forcibly back to the underworld, where his rock was ready for him.

22. A modern Sisyphus – the atom

23. Manipulating atoms, part 1 The Ioffe-Pritchard trap adds magnetic fields from coils (much further apart but with the Helmholtz phase) to pinch and “Ioffe” coils which hold the atoms in the center. By slowly reducing the fields, evaporative cooling can take place – i.e. the hotter atoms jump out of the potential lattice, leaving the cooler atoms. Such methods can lead to atom temperatures less than 10-9K. There is no theoretical limit – just the number of atoms trapped.

24. TIME August 2009: The NIST ytterbium clock is based on about 30,000 heavy metal atoms that are cooled to 15 microkelvins (close to absolute zero) and trapped in a column of several hundred pancake-shaped wells N.B. The NIST Cs-clock is stable to 1 second in 10,000 years! This photo shows about 1 million ytterbium atoms illuminated by a blue laser in an experimental atomic clock that holds the atoms in a lattice made of intersecting laser beams. The photo was taken with a digital camera through the window of a vacuum chamber. NIST is studying the possible use of ytterbium atoms in next-generation atomic clocks based on optical frequencies, which could be more stable and accurate than today's best time standards, which are based on microwave frequencies.

25. Manipulating atoms, part 2- to a Bose-Einstein condensate - As the temperature gets colder, the interatomic interactions will maintain their “collision memory”, leading to coherence in any scattering (and a “coherence time”. The atoms then behave as a single quantum entity. If they are bosons, this can lead to a Bose-Einstein condensation. The thermal DeBroglie wavelength can be defined as λ = h (2πMkT)-1/2 When the inter-atomic spacing reaches (about) this value the bosons tend to condense… when N/V ≈ (λ)-1/3 = 2.6 (λ)-1/3 see Bose-Einstein statistics for more exact formulation…

26. Bose-Einstein condensate and Quantum Computing The atoms then fit into the potential well, with all energy levels filled up to an effective Fermi level. -> well potential example on left. -> picture of cooling “atom blob” below. -> optical density cuts on right. Quantum computing comes next! Manipulation of a string or volume of optical traps…. Quantum mechanics allows optical entanglement and further manipulation of phases as well as just populations in each trap.

27. The Qubit The qubit is the quantum analogue of the bit, the classical fundamental unit of information. It is a mathematical object with specific properties that can be realized physically in many different ways as an actual physical system. Just as the classical bit has a state (either 0 or 1), a qubit also has a state. Note: any linear combination (superposition) is physically possible. In general, thus, the physical state of a qubit is the superposition ψ = α |0>+ β |1> (where α and β are complex numbers). The state of a qubit can be described as a vector in a two-dimensional Hilbert space, a complex vector space .The states |0> and |1> form an orthonormal basis of quantum states for this vector space. Fundamental theorem quantifying the improved speed of quantum computers (phase information) first formulated in Shor’s algorithm (1994).

28. Visualizing the Qubit Any general 2-component state can be written |Ψ> = cos(θ)|0> + eiφsin(θ)|1>, where the numbers θ and φ define a point on the unit three-dimensional sphere, as shown here. This sphere is often called the Bloch sphere, and it provides a useful means to visualize the state of a single qubit. Note that the act of measurement yields either the |1> state or the |0> state, but the computer can store much more (an infinite?) amount of information! This is relevant to the first definition of a “computer” by Alan Turing in 1936. The Turing machine (1936) was essentially a table of look-up values for any calculation. For a good history of the development of quantum computing see http://plato.stanford.edu/entries/qt-quantcomp/#2.1

29. Atomic quantum systems in optical micro-structures by T. M¨uther et al Abstract(Journal of Physics: Conference Series 19 (2005) 97–101) We describe an experiment on evaporative cooling in a far-detuned optical dipole trap for 87Rb. The dipole trap is created by a solid state laser at a wavelength of 1030 nm. To achieve high initial phase space densities allowing for efficient evaporative cooling, we have optimised the loading process from a magneto-optical trap into the dipole trap. These investigations aim at the creation of an ‘all-optical’ BEC based on a simple experimental scheme. As an example, we present the transport of atoms in a ring-shaped guiding structure, i.e. optical storage ring, for cold atoms which is produced by a micro-fabricated ring lens. Absorption images after 10 ms TOF (left), and measured temperatures (right, in nK) for different laser powers. Schematic of the ring lens (left) and fluorescence image of the atoms in the storage ring (right)

30. Quantum computing differs from classical computing in that a classical computer works by processing “bits” that exist in two states, either one or zero. Quantum computing uses quantum bits, or qubits, which, in addition to being one or zero can also be in a "superposition," which is both one and zero simultaneously. This is possible because qubits are quantum units like atoms, ions, or photons that operate under the rules of quantum mechanics instead of classical mechanics. The "superposition" state allows a quantum computer to process significantly more information than a classical computer and in a much shorter time. The area of quantum computing took off about 14 years ago after Peter Shor created a quantum algorithm that could factor large integers much more efficiently than a classical computer. Though researchers are still many years away from creating a quantum computer capable of running the Shor algorithm, progress has been made. Kumar’s group, which uses photons as qubits, found that they can entangle two indistinguishable photons together in an optical fiber very efficiently by using the fiber’s inherent nonlinear response. They also found that no matter how far you separate the two photons in standard transmission fibers they remain entangled and are "mysteriously" connected to each other’s quantum state.

31. 2 graduate students at Georgia Tech Science Daily (Dec. 8, 2008) — Physicists have taken a significant step toward creation of quantum networks by establishing a new record for the length of time that quantum information can be stored in and retrieved from an ensemble of very cold atoms. Though the information remains usable for just milliseconds, even that short lifetime should be enough to allow transmission of data from one quantum repeater to another on an optical network. The new record – 7 milliseconds for rubidium atoms stored in a dipole optical trap – is scheduled to reported December 7 in the online version of the journal Nature Physics by researchers at the Georgia Institute of Technology. The previous record for storage time was 32 microseconds, a difference of more than two orders of magnitude.

32. Optical entanglement Entangled photons remain interconnected even when separated by large distances. Merely observing one instantaneously affects the properties of the other. The entanglement can be used in quantum communication to pass an encryption key that is by its nature completely secure, as any attempt to eavesdrop or intercept the key would be instantly detected! Example 1 – 2 linearly polarized photons of perpendicular polarizations See http://www.davidjarvis.ca/entanglement/

33. Example of a photon entangler An ultraviolet laser sends a single photon through Beta Barium Borate. As the photon travels through the crystal, there is a chance it will split into 2 photons, each of half the energy (twice the wavelength). If it splits, the photon will exit from the Beta Barium Borate as two photons. The resulting photon pair are entangled! Result

34. a Bell-state quantum eraser The Bell-state quantum eraser has one more feature: each slit is covered by a substance that filters the (circular) polarization of a photon. Consequently, the left-hand slit will receive photons with a counter-clockwise polarization, and the right-hand slit will pass photons with a clockwise polarization. Note: Polarization does not affect interference patterns. Initially, neither detector shows an interference pattern. Since we control the polarization of photons passing through the slits and we know the polarization accepted by each slit, we can deduce which way the photons travelled (counter-clockwise through the left; clockwise through the right). Thus no interference patterns are detected. However, if we rotate the polarizing filter in front of detector A so that the polarizations of the photons that hit the detector are the same (that is, we can no longer distinguish between clockwise and counter-clockwise polarizations), then the interference pattern appears at both detectors! How do the photons arriving at detector B know that the polarizations have been "erased" at detector A?

35. Entangled source point-to-point link (information going to “Bob” and “Alice”) fiber or free-space transmission ALICE near IR or telecom IR Single-photon or number sensitive continuous or pulsed BOB phase or polarization qubits Generation Propagation Detection

36. Quantum Dots - 1 Top: Cross-section scanning tunneling microscope (STM) image shows indium arsenide quantum dot regions embedded in gallium arsenide. Each 'dot' is approximately 30 nanometers long–faint lines are individual rows of atoms. (Color added for clarity.) Bottom: Schematic of NIST-JQI experimental set up. Orienting the resonant laser at a right angle to the quantum dot light minimizes scattering (Credit: Top: J.R. Tucker; Bottom: Solomon/NIST) Quantum dots are nanoscale regions of a semiconductor material similar to the material in computer processors but with special properties due to their tiny dimensions. Though they can be composed of tens of thousands of atoms, quantum dots in many ways behave almost as if they were single atoms. Unfortunately, almost is not good enough when it comes to the fragile world of quantum cryptography and next-generation information technologies. When energized, a quantum dot emits photons, or “particles” of light, just as a solitary atom does. But imperfections in the shape of a quantum dot cause what should be overlapping energy levels to separate. This ruins the delicate balance of the ideal state required to emit entangled photons.

37. Quantum Dots - 2 Two lasers—one shining from above the quantum dot and the other illuminating it from the side—the researchers were able to manipulate energy states in a quantum dot and directly measure its emissions. Andrew Shields at Toshiba and colleagues at the University of Cambridge, produced entangled photons with an efficiency of 70% -- compared to a previous best figure of 49%. The improved performance approaches that required for useful applications, which means that devices emitting entangled light could one day be as common as lasers and light-emitting diodes New J. Physics8, 29 (2006) The team produced entangled photons from a crystal just 12 nm in diameter made from indium arsenide embedded within a gallium arsenide and aluminium arsenide cavity. When excited by a laser pulse, the quantum dot captures two electrons and two holes to form a "biexciton" state in the dot. One of the electrons recombines with a hole to create a photon, leaving behind an intermediate "exciton" state in the dot of one electron and one hole. The other electron-hole pair then combines to create a second photon.

38. BEC in a dilute gas In a Bose Einstein Condensate there is a macroscopic number of atoms in the ground state In 1995 teams in Colorado and Massachusetts achieved BEC in super-cold gas. This feat earned those scientists the 2001 Nobel Prize in physics. A. Einstein, 1925 S. Bose, 1924 W. Ketterle E. Cornell C. Wieman Atoms Using Rb and Na atoms Light

39. Optical lattices When atoms are illuminated by laser beams they feel a force which depends on the laser intensity. Two counter-propagating beams Standing wave

40. Perfect Crystals Mimic electrons in solids: understand their physics Quantum Information Why Optical lattices? Atomic Physics

41. ? 1946 2000 Atoms ~ 0.0000000001 m Microchip ~ 0.000001 m ENIAC ~ m Information Information is physical! • Any processing of information is always performed by physical means • Bits of information obey laws of classical physics. Every 18 monthsmicroprocessorsdouble in speed: Faster=Smaller

42. Quantum Computer ? Size Year Computer technology will reach a point where classical physics is no longer a suitable model for the laws of physics. We need quantum mechanics.

43. Quantum Weirdness weirdness

44. Bits and Qubits

45. Bits and Qubits n 2n 2 bits 4 states: 00, 01, 10, 11 3 bits 8 states 10 bits 1024 states 30 bits 1 073 741 824 states 500 bits More than our estimate of the number of atoms in the universe • A classical register with n bits can be in one of the 2n posible states. • A quantum register can be in a superposition of ALL 2n posible states.

46. Quantum Paralelism A quantum computer can perform 2n operations at the same time due to superposition : However we get only one answer when we measure the result: F[000] F[001] F[010] . . F[111] Only one answer F[a,b,c]

47. Quantum Measurement • Classical bit: Deterministic. We can find out if it is in state 0 or 1 and the measurement will not change the state of the bit. • Qubit: Probabilistic |Y =a |0+b |1 • We get either |0 or |1 with corresponding • probabilities |a|2 and |b|2 |a|2+|b|2=1 • The measurement changes the state of the qubit! • |Y |0 or |Y |1

48. What should we do? Strategy: Develop quantum algorithms • Use superposition to calculate 2n values of function simultaneously and do not read out the result until a useful outout is expected with reasonably high probability. • Use entanglement: measurement of states can be highly correlated

49. Entanglement • “Spooky action at a distance” - A. Einstein • “ The most fundamental issue in quantum mechanics” –E. Schrödinger Quantum entanglement:Is a quantum phenomenon in which the quantum states of two or more objects have to be described with reference to each other. Entanglement Correlation between observable physical properties e.g. |Y =( |0A 0B+ |1A 1B)/√2 Product states are not entangled |Y =|0 0

50. Public Cryptographic Systems Use mathematical hard problems: factoring a large number 870901 198043 172475846743 Shared privately with Bob