Lecture C Light and Two-Level Atoms (continued)

Transcription

1 Lecture C Light and Two-Level Atoms (continued Peter van der Straten March 1, Introduction In presence of off-diagonal Hamiltonian matrix elements of operator H (t, energies E e,g that are eigenvalues of H 0 are no longer eigenvalues of full Hamiltonian. The energy levels are shifted by an amount that depends on Ω and vanishes for Ω = 0. The solution for eigenenergies of linear, homogeneous Eqs. (B.6 can be found by diagonalizing matrix H = 2 ( 2δ Ω Ω 0, (1 where order of states in matrix is e, g. The eigenvalues of Eq. (1 are E 1,2 = 2 ( δ Ω, (2 The light mixes states by an amount expressed in terms of a mixing angle θ given by tan(2θ Ω /δ with 0 θ π, so that each state is mixed with a component of ground and excited state. The eigenstates corresponding to E 1,2 are called dressed states of atom [1] and are given by 1 = ( cos θ sin θ e iϕ and 2 = ( sin θ cos θ e iϕ, (3 where ϕ = arg(ω takes into account that Ω can be complex. Thus identification of eigenstates with e and g is ambiguous because y are linear combinations of both ground and excited states as shown by Eqs. (3. In limit where Ω δ and δ > 0, mixing angle becomes θ π/2 and state 1 is predominantly ground state, whereas state 2 is predominantly excited state. In same limit but δ < 0, angle θ 0 and now state 2 is predominantly ground state and state 1 excited state. In both cases resulting energies are shifted by E g Ω 2 /(4δ and E e Ω 2 /(4δ. Since light intensity is proportional to Ω 2, E e,g is appropriately called light shift. In opposite limit Ω δ, solutions give E 1 = Ω /2 and E 2 = Ω /2, where both states are in an equal superposition of ground and excited state. In a standing wave, light shifts of se atomic dressed states vary from zero at nodes to a maximum at antinodes. The spatially oscillating energies found from Eq. (2 are 1

2 C2 The Dressed Atom Picture 2 ω +Ω ω ω Ω δ Ω Figure 1: Energy levels of dressed states for case of δ 0. Note that splitting of each state in two leads to four different possibilities for transitions, for which two coincide at a frequency of ω (Figure from Ref. [2]. Figure 2: The classical counterpart of Mollow triplet. Since atom can only radiate if it is in excited state, emitted light is amplitude modulated (part a leading to three frequencies (part b: one carrier and two side peaks. not sinusoidal, except in limit of δ Ω. This is apparent because se oscillatory terms will always be dominated by δ 2 in vicinity of a node. Thus, for any value of Ω δ, expansion of Eq. (2 in a standing wave as E Ω cos kz/2 will eventually fail near a node. Because eigenstates in presence of field are mixtures of ground and excited states re are several possible transitions as shown in Fig. 1. This results in a spectrum that has three peaks corresponding to se transitions. There is a clear semiclassical view of this phenomenon that arises because probability of atom being in excited state oscillates. Since it can only radiate when it is in excited state, field emitted looks like an amplitude modulated sine wave as shown in Fig. 2a whose spectrum is exactly that shown in Fig. 2b, namely a carrier and two sidebands. This spectrum is commonly called Mollow triplet [3]. 2 The Dressed Atom Picture Absorption of light by an atom can only conserve energy if energy of electromagnetic field is correspondingly reduced. But Hamiltonian of Eq. (B.2 does not have energy of

3 C3 The Bloch Vector and Bloch Sphere 3 Atomic Quantum Number n-1 n n+1... Laser Field Quantum Number Figure 3: The energy level diagram for atom plus field Hamiltonian. In each vertical column re is familiar level scheme of a typical atom, but columns are vertically displaced by addition of ω per column. The nearly degenerate pairs are indicated. In presence of coupling interaction, each of se pairs is a mixture of ground and excited states, so each can decay by spontaneous emission (Figure from Ref. [2]. field itself included, so subsequent description is lacking such information. To include interaction with field in Hamiltonian, consider operators that raise (a or lower (a energy of light field by ω since atomic energy levels are separated by approximately this amount. Then interaction operator that describes se transitions is ( H int = Ω a g e + a e g. (4 Clearly it preserves energy conservation because it changes g to e while lowering field energy with operator a and conversely. The operator H int results from making RWA on fully quantized Hamiltonian, and is called Jaynes-Cummings model [4]. Its eigenstates are often used interchangeably with dressed states of Eqs. (3 [1]. Of course, H int needs to operate on something, so optical field must also be included in wavefunction. In quantized field description, it is characterized by a total energy (n + 1 / 2 ω where n is an integer not to be confused with principle quantum number of atomic states. The energy level diagram of combined atom plus field system consists of ordinary atomic energies repeated for each value of n and vertically displaced by ω each time, as shown schematically in Fig. 3. Attention is focused on two atomic states coupled by laser light that form closely spaced pairs of one excited state and one ground state separated by δ, as shown in Fig. 3. They are each mixtures of ground and excited states, found by diagonalizing Hamiltonian of combined atom plus field system. The interaction between atom and field embodied in H int, couples ground and excited states that form each of se pairs through off-diagonal matrix elements H ge(t. This splits energy levels furr apart to Ω as given in Eq. (2. Note that Ω is independent of sign of δ, and shift (Ω δ /2 is light shift of each dressed state. 3 The Bloch Vector and Bloch Sphere Because overall phase of wavefunction has no physical meaning, re are really only three free parameters in solutions given in Eqs. (B.8 for complex coefficients c j (t. In

4 C3 The Bloch Vector and Bloch Sphere 4 z R y x Figure 4: Graphical representation of Bloch vector R on Bloch sphere (Figure from Ref. [2]. a classic paper, Feynman, Vernon, and Hellwarth [5] combined real and imaginary parts of coefficients c j (t to form three real parameters, denoted commonly as u 2Re(c g c e v 2Im(c g c e, and w c e 2 c g 2. (5 The equations of motion (Eq. (B.6 can be used to calculate time dependence of parameters u, v, and w, and one finds du dt = +δv Ω iw dv dt = δu Ω rw and dw dt = +Ω iu + Ω r v, (6 with Ω Ω r + iω i. The result bears a striking resemblance to a vector cross product, and so notation can be made more compact by defining two artificial vector quantities Ω = (Ω r, Ω i, δ and R = (u, v, w. Then evolution equation for R becomes d R dt = Ω R, (7 An algebraically different but physically equivalent way to arrive at this result uses wellknown Pauli matrices to represent Hamiltonian. The vector R is called Bloch vector after Felix Bloch. Notice that time derivative of R is always perpendicular to R. This means that magnitude R is a constant, which is unity as seen from its components in Eq. (5. The notion of Ω causing precession of R according to Eq. (7 is clearest in a reference frame where Ω is stationary, so most textbooks suggest viewing dynamics in rotating frame. Eir way, path taken after rotating frame transformation led to readily-solved time-independent Hamiltonian matrix of Eq. (1 and dressed atom picture of Sec. 2. It will become clear that dynamic solutions can lead to much more than Rabi oscillations of Fig. (B.1. The artificial vector R refore moves on surface of a unit sphere called Bloch sphere, as shown in Fig. 4. The south (north poles of this sphere correspond to ground (excited states of atom, and equatorial plane corresponds to equal superpositions with various phases.

5 C3 The Bloch Vector and Bloch Sphere 5 If R is in equatorial plane, w = 0 so c 1 c 1 = c 2 c 2 and eigenstates Ψ ± are equal mixtures of excited and ground states. This mixture can be written as Ψ + = 1 2 2( g + e iχ e and Ψ = 1 2 2( g e iχ e where χ is a phase that depends on details of state preparation. For case of δ = 0, Ω is in equatorial plane and an atom starting in state g executes polar orbits. The probability to find it in state e oscillates, and is exactly solid curve of Fig. (B.1. If δ 0, such orbits do not reach north pole because Ω is off equator, as plotted in or two curves of Fig. (B.1. In discussion of dynamical solutions of Schrödinger equation just above Eqs. (B.8, specific initial conditions were chosen. In general, response of an atom initially in any superposition of ground and excited states depends strongly on initial components of R and thus on process that produced superposition. For example, if initial state is e interaction with radiation tuned to resonance (δ = 0 causes stimulated emission. If initial state is an equal superposition of ground and excited states ( R on equator, response to on-resonance light depends on initial preparation. Then system could be driven to eir ground or excited state depending on relative phase of superposition. Section B.3 shows that a light field tuned near atomic resonance causes populations of ground and excited states to oscillate at Rabi frequency, and by contrast, Eqs. (3 give time-independent expressions for eigenstates (θ is constant. This apparent contradiction can be reconciled by examination of Bloch sphere. The eigenstates on Bloch sphere given by Eqs. (3 are parallel and antiparallel to torque vector Ω so that Ω R = 0. Thus se particular states do not evolve, and are stationary states of Hamiltonian. An interesting special case is R on equator after a π/ 2 -pulse, followed by exposure to light having δ = 0 but with a phase shift from π/ 2 -pulse such that Ω is parallel or antiparallel to R so that Ω R = 0.

6 Exercises 6 Exercises C.1 Ramsey method: In this exercise we consider Ramsey method for precision measurement of a frequency. Here we will use Bloch picture to describe action of radiation field. The evolution of Bloch vector R = (u, v, w is given by d R dt = Ω R, with vector Ω = (+Ω r, Ω i, δ describing action of field. Note that time derivative of R is perpendicular to R and thus that its magnitude is constant. The next figure describes 3 different evolutions of R. t =3π/2Ω B u v A B u v A v A t = π/2ω C C (a (b (c a Argue that evolution of R of Bloch sphere is a rotation of R in plane perpendicular to vector Ω. b Show that on resonance (δ = 0 a π/2-pulse leads to a rotation of R to point A, as shown in part a of figure. And a 3π/2-pulse to point B. c Although re is an equal population in ground and excited state in both points, describe what kind of pulse can be used to detect difference between point A and B. d In Ramsey method a π/2-pulse is followed by a time T, where atom is in dark. Finally, state of atom is measured by using anor π/2-pulse. Describe in Bloch picture, how R evolves in dark, if detuning δ is non-zero. e Describe in your own words, how detuning can be defined, even if atom is in dark. What does this imply for radiation fields in two zones? f Explain, why on resonance signal has a maximum. g Explain, why signals decreases, if radiation field is slightly detuned from resonance. h What happens to R after first π/2-pulse, if field is not on resonance? i Explain, why amplitude of oscillation decreases if detuning increases? j Why does a fountain clock increase resolution of atomic clock considerably?

7 Exercises 7 insight overview (Fig. 3 remain one of most important applications of coherent interaction of atoms with electromagnetic radiation. Improvements in our ability to parse time into increasingly selfconsistent and reproducible intervals (that is, measurement of time have been associated with our ability to count passage of ever-shorter time intervals. In essence, faster oscillators make better clocks. The caesium atomic clock standard, oscillating 9,192,631,770 times a second, has an absolute uncertainty / ~ To improve this accuracy by several orders of magnitude, Ramsey method must be extended to optical domain. In microwave domain, it is straightforward to ensure that all atoms in an experiment experience same electromagnetic phase, as dimensions /2 of standing wave in cavities are larger than transverse spread of atomic beam. However, if two optical standing waves are used, atoms with different transverse velocities can easily experience trajectories through apparatus where overall phase difference between two Ramsey zones differs by half a cycle, as shown in Fig. 4. Solutions to this problem were found by moving to a geometry with three, separated standing-wave regions 6,7, use of two-photon Doppler-free transitions 8, or interaction with four successive running waves of light 9. Initially, atomic recoil due to light was not considered, but in 1989, Christian Bordé 10 realized that four-zone geometry 11,12 created an atom interferometer with spatially separated atomic paths. Thus, direct extension of Ramsey s separated oscillatoryfield method in microwave regime to optical domain led to spatially separated equal atom interferometers. lengths. Atom interferometers based on optical transitions (and also methods adapted from work using nuclear magnetic resonance have led to precise and accurate measurements of gravity 13, gravity gradients 14, rotations 15 and 1.0 photon recoil of an atom 16. There is one very important distinction between atom interferometers designed for atomic clocks and atom interferometers 0.8 developed to measure inertial effects or fundamental constants. 0.6 Chebotayev, Bordé and ir co-workers were motivated to look for a method that would allow m to stabilize a laser to a high-q optical resonance, whereas atom interferometers developed by Kasevich 0.4 and Chu bypassed requirement of an ultra-stable laser by using Raman transitions between two ground states of atom The achieved resolution of se atom interferometers (~10 3 Hz out of a Atom Ground state Superposition state Excited state nance methods and closely related nuclear magnetic resonance methods into optical domain. But physical processes exist that do not have counterparts in radio or microwave region of electromagnetic spectra, one example being laser cooling of atoms. Laser cooling was first demonstrated with ions confined in traps in 1978 (ref. 18. Subsequently, ion-cooling methods were improved to 0.8 point where a single ion could be cooled to lowest vibrational energy level of trap This capability n led to creation of 0.4 quantum states where electronic and motional degrees of 0.2freedom of ion became intimately connected and could no longer be factored into separate Hilbert spaces Much of early progress in quantum computing (see review in this issue by Monroe, pages came from ability to coherently link and entangle se degrees of freedom. In contrast to ions, laser cooling of neutral atoms required two more decades to reach zero-point limit. frequency shift of ~10 7 Hz is determined by frequency difference 0.0 Laser cooling requires transfer of entropy from ensemble of atoms being cooled to microwave radiation detuning field. (Hz All demonstrated of two laser beams used to excite Raman transition. Because two laser frequencies can be phase-locked to each or with cooling methods use spontaneous scattering of radiation as radio-frequency methods, this configuration allowed FIG. 2: complete Measured Ramsey means fringe of transferring pattern for PTB s this CSF1 entropy; fountain unfortunately, clock (dots. however, Inset: sponta- eye. emission also is a heating mechanism. At sufficiently high central part enlarged. The solid phase control of two-photon optical transitions. lines are re to guideneous densities, cooling efficiency degrades owing to inelastic collisions Control of external degrees of freedom of atoms be reached for such a that rmal allow fountain transfer 3 m in height. of energy from In internal following to we will external mostly concentrate on The development of central atomic clocks part is one example enlarged. of considerable The solid degrees lines of freedom. are For that re reason, to final guide stages of cooling eye. The main problem for fountains using rmal beams seven primary caesium fountain used inclocks in operation today because cooling y 26,27 have. been most thoroughly charac- work done in past 50 years to extend molecular-beam is that re reso- are justbose Einstein not enough slow condensation atoms in a r- use evaporative terized ones and more information is available on m. v L ~ Figure 5: A schematic diagram of molecular beam setup for a clock based on Ramsey Figure 2 Ramsey s separated oscillatory-field method. The quantum measurement separated oscillatory field method. time t L/v, where Atoms v is velocity of travelling atom. If microwave through oscillator is tuned this to system are exposed to a precise frequency of atomic resonance, excitation is completed. As microwave field in zone 1, n frequency remain of microwave oscillator in is varied slightly, dark atomic as population y will fly toward zone 2, and n oscillate between ground and excited states, depending on phase of are exposed to a microwave field with its phase preserved from zone 1. Phase preservation is microwave radiation relative to phase difference of two atomic states. implemented by splitting microwave input as shown, and carefully adjusting both arms to trans ition prob ab Figure 6: Measured Ramsey fringe pattern for PTBs CSF1 fountain clock (dots. Inset: mal beam to give a sufficiently strong signal. The situation changed completely when laser-cooling techniques allowed one to prepare cold atom samples containing millions of atoms, and all of m with basically same A magnet velocity. The C magnet potential of such laser-cooled samples B magnet in C.2 Bloch vector: The state of a two-level atom in a radiation field is described by III. OPERATION OF A FOUNTAIN CLOCK Bloch vector. a fountain geometry was considered by Hall et al. [9] Figure 3 shows a simplified setup of vacuum subsystem of a fountain clock. Six laser beams cross in both for microwave and optical transitions and simultaneously realized by Kasevich et al. [10]. In latter center of S D preparation zone, where cold atomic a Show that experiment three acomponents laser-cooled sodium cloud ofwas launched Bloch by cloud vector is produced R (Sect. ofiii Eq. A. Above (5 that are follows real. a laser pulse and entered a radio-frequency waveguide detection zone which is traversed by laser beams for where it reached its apogee. While in waveguide fluorescence detection of falling cloud (Sect. III C. b Show that R j = atoms σwere illuminated with two π/2 pulses and n fell down and j, where σ is Pauli matrix The microwave defined interactionsas take place inside a magnetic through an ionization detection zone. A width shield in presence of a well-defined internal longitudinal magnetic field (Sect. III B. ( of central Ramsey fringe of 2 Hz was obtained in this way. ( ( Figure 1 Traditional atomic-beam resonance apparatus. Atoms emerge from a source frequency coils, and n through a B region. In absence of B magnet, The first fountain for metrological use was developed at chamber S into a vacuum chamber and are sent through three regions of Observatoire magnetic 0 1 de Paris/France atoms that pass [11]. through Its design slit will [12] not became is used a standard to for almost choice of all field subsequently gradient, B constructed magnet will focus atoms A. in a particular Preparation atomic state of ontocold atomic cloud reach 0 i detector. However, with proper 1 0 σ field. The first region consists of an inhomogeneous magnetic field x = σ that 1 0 y = σ i 0 z = 0 1 deflect atoms with a particular magnetic moment through a slit fountain C region. clocks, Theexcept for detector, variations D. in relative spatial field arrangement and radio of cavities and detection zone. The key scientific achievement which enabled op- atoms drift through a C region consisting of a uniform-bias magnetic In late 1990s fountains NIST-F1 at National eration of an atomic fountain was that of laser cooling. Institute of Standards and Technology (NIST in Boulder/USA equation [13] and CSF1 of motion at Physikalisch-Technische for Bun- Bloch is givenvector in 1997isNobel dr/dt lectures A historical review of development of this technique NATURE VOL MARCH c 2002 Show that 207 [28 30]. = Ω Here R Macmillan Magazines Ltd desanstalt (PTB in Braunschweig/Germany [14] became operational as primary standards. Recently, y were joined by caesium fountain clocks CsF1 at Isti (8 principles are recalled only briefly insofar as ir understanding is needed for explanation of operation of an atomic fountain clock. First of all, a source of caesium atoms is needed. Tra- C.3 Contradiction: tuto Elettrotecnico Since Nazionale (IEN equation in Torino/Italy of [15] motion for Bloch vector is dr/dt = Ω R, and CsF1 at National Physical Laboratory (NPL in ditionally it consists of a temperature-controlled caesium it means that R isteddington/gb time dependent [16]. With FO2 and in FOM [17] presence Observatoire de Paris (Systèmes de Référence Temps-Espace The reservoir is held at a suitable temperature near room reservoir of separated an optical from cooling field. chamber The by a valve. same problem yields SYRTE is operating two more primary fountain clocks. temperature order to obtain a caesium partial pressure solutions in dressed state picture that are time independent eigenfunctions as given above A number of or laboratories are currently operating on order of 10 6 Pa in cooling chamber. or developing fountain clocks (see Refs for an incomplete list To obtain required low temperatures of atom in terms of mixing angle of examples. θ. Most Explain of se are employing how samples same in an atomic problem fountain can atoms are yield cooled intwo solutions, one caesium as active element. a magneto-optical trap ( MOT [31] and/or an optical time dependent and or stationary.

8 Exercises 8 to pump and window Magnetic shields C-field coil Vacuum tank Ramsey cavity State-selection cavity Detection zone Preparation of cold atoms Caesium reservoir FIG. 3: Simplified setup of atomic fountain clock molasses ( OM [32, 33]. Common to both configurations is a setup consisting of three mutually orthogonal pairs of counterpropagating laser beams, which are well balanced with respect to ir intensities and usually have diameters of about two centimeters. So far two different laser beam geometries have been used. The first one uses two vertical, upward and downward directed beams (z-axis and four horizontal beams, counterpropagating along x-axis and y-axis of a Cartesian coordinate system, respectively. This setup offers advantage of easy alignment but has disadvantage that one pair of laser beams overlaps atomic trajectories. These two laser beams are limited in diameter by apertures (of typically 1 cm diameter of microwave cavity and are particularly critical in connection with laser light shift, i.e., an uncontrolled frequency shifting interaction with atoms during ir ballistic flight. These disadvantages are circumvented by so-called (1,1,1 laser beam configuration, consisting again of three orthogonal pairs of counterpropagating laser beams, but with a different spatial arrangement: when in previously described setup laser beams are imagined to run along six face normals of a cube repumping cooling = 5 = 4 = 3 = 2 = 4 = FIG. 4: Simplified 133 Cs energy level diagra showing hyperfine splitting of 6 2 S and 6 2 P 3/2 excited state. The excitation and repumping, respectively, are indicated b lying on one of its faces, in (1,1,1 co cube is balanced on one of its corners. Th (arranged symmetrically around ve refore pointing downwards at an angl accordingly three laser beams are pointin In Fig. 4 a simplified 133 Cs energy is shown. The frequency ν c of si beamsistuned2γ-3γ(γ=5.3mhz, na linewidth to red (low-frequency sid F =4 F =5 caesium transition in ter a large number of photons before tually pumped into or caesium hy state F =3. Although forbidden by sele hyperfine pumping process will happen to small polarization imperfections in c off-resonant excitation to excited sta repumping laser beam tuned to cae F =3 F =4 is refore superimp one of six cooling laser beams. It depl level so that all atoms can continue to pa cooling process. There exists a large variety of optical se ing all necessary laser beams for foun ation and here is clearly not place to detail. Generally speaking, such a setup enough power for six cooling laser to obtain for each beam an intensity of mw/cm 2. Additionally a few mw/cm 2 pumping light is needed for cooling detection region (see below a few mw/cm sity for F =4 F =5 caesium a very low intensity repumping laser be Laser linewidths of a few MHz are suffic and repumping but detection laser li be at most a few 100 khz because o be too much noise on detected numb achieve a good short-term stability of Figure 7: A schematic diagram of an atomic fountain clock. Atoms are laser cooled in a magneto-optical trap at base, and m launched upward by a walking wave that results from counterpropagating laser beams of slightly different frequencies. They pass through a probe laser beam and a microwave cavity on way up, and also on way down. The two sequential passes through microwave field constitute separated oscillatory fields for clock transition, and phase coherence is assured because it is same field: arm length differences do not exist. The first passage through laser probe beam prepares atomic ground state and second passage measures it after two separated exposures to microwaves.

9 References 9 References [1] C. Cohen-Tannoudji, B. Diu, and F. Laloë. Quantum Mechanics. Wiley, New York (1977. [2] H. Metcalf and P. van der Straten. Laser Cooling and Trapping. Springer, New York (1999. [3] B. R. Mollow. Power Spectrum of Light Scattered by Two-Level Systems. Phys. Rev. 188, (1969. [4] E.T. Jaynes and F.W. Cummings. Comparison of Quantum and Semiclassical Radiation Theories with Application To Beam Maser. Proc. IEEE 51, (1963. [5] R. Feynman, F. Vernon, and R. Hellwarth. Geometrical Representation of Schrödinger Equation for Solving Maser Problems. J. Appl. Phys. 28, 49 (1957.