Introduction to Super-radiance and Laser

Transcription

1 Introducton to Super-radance and Laser Jong Hu Department of Physcs and Astronomy, Oho Unversty Abstract Brefly dscuss the absorpton and emsson processes wth the energy levels of an atom. Introduce and dscuss the Superradance phenomenon n an atom-atom entanglement model. Dscuss the basc physcs theory of LASER and three basc condtons of an operatonal LASER. Introduce dfferent knds of operatonal LASERs Absorpton and Emsson Processes Fg Fg demonstrates dfference between stmulated absorpton, spontaneous emsson, and stmulated emsson. In stmulated absorpton, a photon of h v was absorbed by the atom and stmulated a transton from the ground state E to the excted state E. In the spontaneous emsson process, the atom ump to E, also emt a photon of h v. If the atoms n an ensemble

2 have good coherence between each other, a process called Superradance would lkely to occur, whch we wll dscuss later. In the stmulated emsson process, an atom s stmulated by a propagatng photon of h v, the emtted photon would be dentcal to be propagatng photon, the basc atom process durng the LASER operaton to be dscussed later. Superradance Frst, let's dscuss a smple system of two two-level atoms (two qubts) and ts relaton to quantum nterference: an atom-atom entanglement model. The term entanglement, one of the most ntrgung propertes of multpartcle systems, was ntroduced by Schrodnger n hs dscussons of the foundatons of quantum mechancs. It descrbes a multpartcle system whch has the astonshng property that the results of a measurement on one partcle cannot be specfed ndependently of the results of measurements on the other partcles. Although entangled systems can be physcally separated, they can no longer be consdered as ndependent, even when they are very far from one another. In the absence of the nteratomc nteractons and the drvng feld, the space of the two-atom system s spanned by for product states g g, g e, g e, e e () Wth correspondng energes E gg = hϖ, E eg = h, E ge = h, E ee = hϖ () Where ϖ = ( ϖ + ϖ ), = ( ϖ ϖ ), and g, e denotes the ground and

3 excted states of the th atom. The product states e g, g e form a par of nearly degenerated states. Now, suppose that the atoms can nteract wth each other through the vacuum feld. The nteracton s represented by the dpole-dpole nteracton, whch for small nteratomc separatons s represented by the dpole-dpole potental 3Γ Ω µ, (3) [ 3( ˆ r )] = ˆ 3 4( kr ) Where ˆ µ, rˆ are unt vectors n the drecton of the atomc dpole moments and the nteratomc axs, respectvely, and Γ s the spontaneous emsson dampng rate of the atoms. When we nclude the dpole-dpole nteracton nto the Hamltonan, the product states combne nto tow lnear superpostons (entangled states), wth ther energes shfted from ± h by the dpole-dpole nteracton energy. To see ths, we begn wth the Hamltonan of two atoms ncludng the dpole-dpole nteracton = z Hˆ = hϖ S + h Ω S S. (4) aa + In the bass of the product states (), the Hamltonan (4) can be wrtten n a matrx form as ϖ Ω H ˆ aa = h Ω (5) ϖ Consder a system of two dentcal atoms ( = ). In order to fnd energes and correspondng egenstates of the system, we have to dagonalze the matrx (5). The resultng energes and correspondng egenstates of the system are

4 E g = hϖ, g = g g E, s = ( e g + g e ) s = hω E a = hω, a = ( e g g e ) (6) E e = hϖ, e = e e The egenstates (6), frst ntroduced by R.H.Dcke, who was also the frst to defne the term super-radant, are known as the collectve states of two nteractng atoms. The energes of the ground state g and the upper state e are not affected by the dpole-dpole nteracton, whereas the states s and a are shfted from ther unperturbed energes by the amount ± Ω, the dpole-dpole energy. The most mportant property of the collectve states s and a s that they are an example of maxmally entangled sates of the two-atom system. The states are lnear superpostons of the product states whch cannot be separated nto product states of the ndvdual atoms. Fg We show the collectve states of two dentcal atoms n Fg. It s seen that n the collectve states representaton, the two-atom system behaves as a sngle four-level system, wth the ground state g, the upper state e, and two ntermedate states: the symmetrc state s and the

5 antsymmmetrc state a. The energes of the ntermedate states depend on the dpole-dpole nteracton and these states suffer a large shft when the nteratomc separaton s small. There are two transton channels e s g and e a g, each wth two cascade nondegenerate transtons. For two dentcal atoms, these two channels are uncorrelated, but the transtons n these channels are damped wth sgnfcantly dfferent rates. The transtons through the symmetrc state are damped wth an enhanced rate (superradant), whle the transtons through the antsymmmetrc state are damped wth a reduced (subradant) rate. A system of two dentcal two-level atoms may be prepared n the symmetrc states s by a short laser pulse. The condtons for a selectve exctaton of the collectve atomc states can be analyzed from the nteracton Hamltonan of the laser feld wth the two-atom system. Hˆ z + S + h Ω S S + = = hϖ Hˆ L (7) Where Ĥ L s the nteracton Hamltonan of the atoms wth the laser feld of the Rab frequency Ω ( r ). From the Schrödnger equaton, the tme evoluton of the populaton () t s as P s () t = sn Ωt (8), P s of the state Where Ω = Ω = Ω.The populaton oscllates wth the Rab frequency of the s g transton and at certan tmes () t = P s ndcatng that all the populaton s n the symmetrc state. Ths happens at tmes T n π = n Ω ( n + ), =,... (9) Hence, the system then got prepared n the symmetrc state s, from where the spontaneous

6 radance would occur to the g state n the system. When the radance reaches ts maxm, the system enters the Super-radant State. Because of the coherent addton between the atoms, the output of the Superradance system wll be proportonal to N, where N s the number of atoms n the system. That s a sgnfcant dfference compared to other spontaneous emssons, n whch the output power s proportonal to N. LASER (Lght Amplfcaton by Stmulated Emsson of Radaton) In order for most lasers to operate, three basc condtons must be satsfed. Frst, there must be an actve medum, that s, a collecton of atoms, molecules, or ons that emt radaton n the optcal part of the electromagnetc spectrum. Second, a condton known as a populaton nverson must exst. Ths condton s hghly abnormal n nature. It s created n a laser by an exctaton process known as pumpng. Fnally, for true laser oscllaton to take place there must be some form of optcal feedback present n the laser system. If ths were not present, the laser mght serve as an amplfer of narrowband lght, but t could never produce the hghly collmated, monochromatc beam that makes the laser so useful. THE ACTIVE MEDIUM Let s frst brefly recall the selecton rules and metastable states n quantum theory. Consder the helum atom as an example. In the ground state, the two electrons n a helum atom have spns of opposte sgn. The total spn S = (+/) + (-/) =. In an excted energy state, the two electrons

7 have dfferentl, and the electron spns can be ether alke or opposte n sgn wthout volatng the excluson prncple. The selecton rule then mples that transtons from the excted state to the ground state are allowed only f the spns of the electrons n the excted state are pared,.e. the spns of the electrons n the excted state are of opposte sgn. Thus tells us that transtons between snglet states (S = / - / = ) or between trplet states (S = / + / = ) are allowed. Transtons from trplet to snglet or from snglet to trplet, however, are forbdden. Suppose there exst such energy state that once the electrons have been smulated to the state, forbdden by the selecton rule, the transton lfetme,τ (whch could be consdered to be ether the perod of tme or the probablty of a transton) would be long enough. Consequently, the spontaneous transton to the lower level would be less lkely to occur. Such a state s called metastable state. However, f stmulated by external source, transton to the lower state would occur and photons would be emtted. Energy levels of the ruby, used n the frst operatonal LASER. Fg 3 So, the actve medum are sometmes called gan medum or workng medum. They are the medum n whch the process populaton nverson (To be dscussed later) could occur after the atoms n the medum have been stmulated. Another crtcal property of the actve medum s

8 where the metastable states could be found n the ensemble of atoms. LASER PUMPING: CREATING A POPULATION INVERSION The populaton nverson requred for lght amplfcaton consttutes an abnormal dstrbuton of atoms among the varous avalable energy levels. To understand how lght amplfcaton can be acheved n a medum, let's recall the Boltzmann's prncple. Boltzmann's prncple specfes what fracton of atoms are found, on the average, n any partcular energy state for any gven equlbrum temperature. Stated mathematcally, N E / kt N e = () where state; N s the number of atoms n the excted state; N s the number of atoms n the ground E s the energy of the excted state measured relatve to the ground state energy; T s the absolute temperature; and k s the Boltzmann's constant. Ths relatonshp, known as the Boltzmann rato, s llustrated n Fg 4, where the populatons of several energy levels are shown measured relatve to the populaton of the ground state. From ths basc relatonshp, t s possble to obtan several alternatve expressons. The rato of the atomc populatons n the gas for two arbtrary energy levels, E > E for example, s easly shown to be N N ( E E ) E = exp = exp () kt kt Thus, for a specfed number of energy levels, a specfc temperature, and a known number of atoms n the collecton, t s possble to specfy the equlbrum, number of atoms exstng n any partcular energy state: E.

9 Fg 4 Boltzmann s dstrbuton for several energy level. Dashed lne ndcates the populaton of levels f the dstrbuton of energy levels were contnuous rather than dscrete, as shown here. To obtan some feel for ths dstrbuton, t s useful to consder Eq. () for two extreme condtons: E << kt and E >> kt. When E << kt, the hgh-temperature case, the Boltzmann rato s near unty and the populaton at energy level E s nearly equal to the populaton at energy level E. If E should be the ground-level energy, then Eq. () predcts that atomc states wth energes (measured relatve to the ground-state energy) small compared to kt are populated to about the same extent as the ground state, f, on the other hand kt s much smaller than E E the low-temperature case, the Boltzmann rato s qute small and vrtually no atoms are found n the hgher energy state. A Boltzmann rato of /e s often taken as a convenent dvdng lne between them. Such a rato occurs when E = kt, or, equvalently, when the frequency of the transton satsfes the equaton hv = kt () As T n Eq. () s ncreased, N approaches N but cannot exceed t. Under thermal equlbrum

10 condtons, the populaton of a hgher energy state s never larger than that of a lower energy state. How then s a populaton nverson obtaned n a laser? The answer s that the atoms n the laser medum must be excted or PUMPED to a non-thermal-equlbrum dstrbuton through the applcaton of some external source of energy. Once a populaton nverson s acheved, lght amplfcaton by stmulated emsson can take place. For example, optcal pumpng scheme, where a populaton nverson s brought about by stmulated absorpton, s one typcal pumpng method. In the three-and four-level schemes typcal of optcally-pumped doped nsulator lasers overcome the restrctons mposed on a two-level scheme by pumpng atoms n the actve medum ndrectly to the upper state of the transton. Fg 5 The three-level scheme, llustrated n Fg 5, was frst proposed by Bloembergen at Harvard Unversty n 956. Intally, the dstrbuton of atomc-state populatons obeys Boltzmann's law, as shown n Fg 5 (a), and the optcal transton between energy state E and the ground state s absorptve. If the collecton of atoms s ntensely llumnate for example, wth a xenon flash lamp--a large number of atoms can be excted' through stmulated absorpton to the hghest energy level, E. From there they decay to level E as shown n Fg 5 (b). Wth suffcently ntense pumpng, a sgnfcant number of ground-state atoms can be pumped to level E. A populaton nverson occurs when the populaton of E exceeds that of the ground states, as shown n Fg 5 (c). In order for nverson to be acheved easly, t s necessary for the transton from E to E

11 to be rapd (.e., for the ( ) transton to have a very short lfetme) and for energy state E to be metastable. If these two condtons are satsfed, ground state atoms are quckly pumped to level E, where they tend to accumulate. OPTICAL FEEDBACK: THE LASER RESONATOR The most drect applcaton of the amplfcaton-by-stmulated-emsson prncple s ncorporated n a sngle-pass lght amplfer, a devce that takes as ts nput a collmated beam of narrowband lght, typcally from a laser, and amplfes t. Lght amplfers are regularly employed to ncrease the output of ruby lasers and neodymum-doped glass lasers. A resonator Fg 6 Drect amplfcaton of lght n the manner descrbed above s practcal only for a few select materals. In most lasers, ths lmtaton s crcumvented by the use of mrrors that drect the lght beam back and forth through the actve medum many tmes. Wth such an arrangement, the effectve length of the amplfyng medum becomes many tmes the actual length of the laser. A laser, for example, that has a -percent reflectve mrror at one end and a 98-percent reflectve mrror at the other end has an effectve length roughly 5 tmes the actual separaton dstance between mrrors. Although sngle-pass amplfcaton may be qute small for the nverted medum,

12 the total amplfcaton provded by multple passes can be substantal. OPERATIONAL LASERs The frst operatonal LASER was ntroduced by T.H.MAIMAN n 96. After that, dfferent types of LASER usng dfferent actve medum and dfferent pumpng method have been developed. The structure of the frst operatonal LASER Fg 7 In the dscusson above, we speak of the actve medum as though t were a collecton of atoms n a gas. Just as easly, however, the actve medum of a laser mght be atoms, molecules, or ons n lqud or sold form. Typcal actve medum could be ruby, used n Sold-state Laser wth optcal pumpng; helum and neon, used n Gas Laser wth dscharge pumpng; gallum, used n n Sem-conductor Laser pumped by electrcal current, dyes n Dye Laser pumped by other laser or a beam of electrons n Free Electron Lasers pumped by relatvstc electron beam. Reference Coherence n Spontaneous Radaton Processes Dcke, R.H. 954, Physcal Revew, vol. 93, Issue, pp. 99- Stmulated Optcal Emsson n Fluorescent Solds. II. Spectroscopy and Stmulated Emsson n Ruby,Maman, T. H. 96, Physcal Revew, vol. 3, Issue 4, pp. 5-57

13 Stmulated Optcal Radaton n Ruby, T. H. Maman,Nature, August 6, 96, Vol. 87, No. 4736, pp Collectve spontaneous emsson (Dcke superradance), A V Andreev, V I Emel'yanov, Yu A Il'nskĭ, SOV PHYS USPEKHI, 98, 3 (8), An Introducton to Lasers and Ther Applcatons, by Donald C. O Shea, W. Russell Callen and Wllam T. Rhodes Quantum Optcs (Fundamentals Applcatons), by Zbgnew Fcek and Mohamed R. Wahddn. Coherence and Quantum Optcs, by L. Mandel and E. Wolf