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Laser Cooling and Trapping – An Overview Eyal Fleminger, 38500575 December 2004 Foreword ........................................................................................................................ 3 Forces on an Atom in a Light Field ............................................................................... 3 Scattering (Radiative) Force ...................................................................................... 3 Dipole Forces ............................................................................................................. 6 Trapping of Atoms ......................................................................................................... 6 Optical Traps .............................................................................................................. 6 Magnetic Trapping ..................................................................................................... 9 Magneto-Optical Trap .............................................................................................. 12 Sub-Doppler cooling .................................................................................................... 15 Evaporative Cooling ............................................................................................ 17 Trapping on a Microchip ............................................................................................. 21 Mirror MOT ............................................................................................................. 22 Magnetic Trapping ................................................................................................... 22 Implementation of a MOT ........................................................................................... 25 Conclusion ................................................................................................................... 26 References .................................................................................................................... 27 2 Foreword The subject of the 1997 Nobel Prize for physics, the field of laser trapping and cooling has produced a toolbox of techniques for creating confined ensembles of ultracold atoms. These techniques have various applications, such as the study of ultracold matter, atom optics, Bose-Einstein Condensation (BEC), and so forth. In this paper, I will explore the basic theory of cooling and confining atoms with laser light. I will begin with a description of the forces induced on atoms by a laser beam, describing how they can serve to both cool and confine atoms; following that, I will describe several optical trapping schemes (in addition, I will describe purely magnetic trapping schemes, as a prelude to the MOT) as well as a hybrid trap (MOT). I will then describe the theory of sub-Doppler cooling (including evaporative cooling). Following that will be a brief overview of a few of the variations on trapping schemes (both hybrid and magnetic) used in creating a MOT on a microchip (part of the subject of my group and my current work). I will conclude with a description of one of the earlier MOT experiments. Forces on an Atom in a Light Field Scattering (Radiative) Force When an atom of mass m absorbs a photon whose frequency ν matches a resonance frequency of the atom, the photon’s energy ħν causes transition to an excited state, while the photon’s momentum h is absorbed as an addition to the c atom’s momentum in the direction of the photon’s movement (the converse is true for photon emission from an atom). The momentum exchange induces a force (1) F dp h p ˆ A dt c γp is the atom’s excitation (or scattering) rate (see below). In the case of absorption, the arbitrary direction vector  is the same as the laser’s direction of propagation; in the case of photon emission, the vector’s direction is opposite that of the emitted photon. The change in the atom’s velocity is of the magnitude 3 v (2) h cm If we have a gas of atoms of mass m and temperature T in a volume (assuming the gas is dilute or otherwise approximates an ideal gas), their velocity is governed by the Maxwell-Boltzmann distribution 3/ 2 (3) m f (v) 4v 2k BT 2 e mv2 2k BT The characteristic velocity for a given temperature is vrms, given by (4) 3k BT m vrms Let us consider an atom moving in a direction opposing the laser beam (for the moment, we will consider only the velocity component in the same axis as the beam). If it absorbs a photon, its velocity is reduced, since the direction of Δv is directly opposed to that of the atom’s velocity v. As the atom shifts back to its ground state, it gives off a photon, further changing its velocity; but as the probability of the direction of the emission is spatially symmetrical, the velocity change due to emission has a mean value of zero over multiple instances of photon absorptions, and the total deceleration of the atom is in the direction of the laser beam. In order to allow absorption by the atom, ν must be equal to the atom’s resonant frequency. There is, however, a problem; atoms moving in the same direction as the laser beam will also absorb photons, accelerating them. To prevent this, the laser frequency selected is slightly below the resonant frequency of the atom. From the view point of an atom moving in the gas, the frequency is Doppler shifted by (note that v in equation (5) is positive if its direction is the same as that of the beam’s propagation) (5) v D c Thus, for atoms heading “into” the laser beam, ν is blue-shifted toward the resonant frequency, while for atoms moving “with” the beam, ν is red-shifted and therefore those atoms will not absorb photons. The excitation (or scattering) rate γp depends on the laser’s detuning (designated by δ) from resonance, defined as the difference between the atomic resonance frequency ωa and the laser frequency ωL, and is given (for a two-level atom) by the Lorentzian (Metcalf, van der Straten, 2003) 4 (6) p s0 d 2 2 D 2 1s0 d γd is an angular frequency corresponding to the excited state’s rate of decay (1/τ). s0 is defined as the ratio between the light intensity I and the saturation intensity; the latter is the intensity at which the rates of spontaneous emission and stimulated emission are equal, and is given as Is (7) h 3 3c 2 Is is dependent on the atoms in question, and is typically of the order of several mW/cm2. As intensity increases, so does the deceleration. However, at high intensities, the rate of stimulated emission increases. Since, in the case of stimulated emission, the photon is emitted in the same direction as the laser beam’s direction of propagation, the momentum “kick” is in the opposite direction, nullifying the deceleration caused by absorption. At these intensities, the atom has an equal chance of being in the excited or ground states, and the maximum deceleration is (8) a max h d 2mc According to equation (5), v , the Doppler shift becomes smaller as the atoms slow down, and eventually the shifted frequency will be too far from the resonance frequency to allow excitation. There are various methods to compensate for this effect (Metcalf, van der Straten, 1999). The two most common ones are changing the laser’s frequency as cooling progresses (known as “chirping”), and spatially varying the resonance frequency by means of an inhomogeneous magnetic field (such as in a MOT, described later). With a pair of counterpropagating lasers, it is possible to use this effect to slow atoms. Atoms moving toward a laser have their momentum reduced, while atoms moving away from it are not affected by it (due to the Doppler shift away from resonance). This type of cooling is known as Doppler cooling, and the resulting process is known as optical molasses. 5 Dipole Forces Atoms in an inhomogeneous field, such as a standing wave (such as created by two counterpropagating laser beams), experience a force derived from the spatial gradient of their light shifts (see below).When δ>>γd, spontaneous emission may occur less frequently than stimulated emission. The force arising from stimulated emission is known as the dipole force and derives from the gradient of the light shift (the Stark shift due to the electric field of the wave). For a laser beam traveling in the  direction, the Hamiltonian is 2 H * 0 2 The Rabi frequency Ω is given by (9) p (10) s0 2 The light shift is provided by the eigenvalues of equation (9), and is given by (Metcalf, van der Straten, 2003) (11) ls 2 2 2 For large detuning, equation (11) is approximately ls (12) 2 2 ps0 4 8 In a standing wave, the light shift, as well as the probability of an atom undergoing spontaneous rather than stimulated decay, varies sinusoidally from node to anti-node. Atoms are excited by one beam and are then stimulated into emission by the other, slowing them. The force generated is (13) Fx d2 I x 8Is where I is the total intensity distribution of the light field. Trapping of Atoms Optical Traps It is possible to use the forces outlined above to spatially confine a cloud atoms. One type of tap utilizes the dipole force. 6 Consider a Gaussian laser beam with a waist width of w0 whose intensity at the focus is (14) I r I0e r w0 2 When δ<0, the ground-state light shift is negative, and has its largest value at the beam center. Therefore, atoms in the beam experience a force toward the beam’s center arising from the gradient of the light shift given by equation (12). For δ>>Ω and δ>>γ this transverse force is given by equation (13), and for a Gaussian beam is (15) F I0 r e 4 Is w 02 2 d r w0 2 In order to trap atoms with this force, it is necessary to overcome the radiation pressure on the atoms in the laser beam’s direction. This can be done by selecting the appropriate laser parameters. As per equation (6), the radiation pressure force decreases as δ-2, while the dipole force decreases as δ-1 (equation(11)) for δ>>Ω, choosing sufficiently large δ means that atoms spend little time in the untrapped (repelled) excited state, since its population is proportional to δ-2. Thus, a sufficiently large absolute magnitude of δ will produce both longitudinal and transverse trapping, maintaining the atomic population (mostly) in the trapped ground state. The maximum intensity, and therefore light shift and trap depth, attainable by a given laser is proportional to the area of the beam spot πw02, requiring a large numerical aperture for focusing (ibid). This form of dipole trap is perhaps the simplest imaginable. One important drawback is that negative detuning causes attraction to the regions of highest intensity, but once their, a higher incidence of spontaneous emission is caused (defeating the trap) unless the detuning is a large fraction of the optical frequency. In addition, the reliance of the trap on Stark shifts (equal to the trap depth) can compromise the use of the trap for applications such as spectroscopy. A solution to this problem is the use of positive detuning (blue-shifted traps). In such a case, atoms are attracted to the areas of lowest intensity. In that case, however, the simple Gaussian beam described above could not be used, since the atoms would be drawn to its fringes and escape. One solution (Davidson et al, 1995) is the use of two adjacent beams, forming a trough between them where the atoms accumulate. Another possible solution under research is the use of a hollow laser beam. 7 It is also possible to use the scattering force to confine atoms, by using a system of six lasers, with each pair orthogonal to the other two pairs. However, the optical Earnshaw theorem (Ashkin, Gordon, 1983) precludes such a trap from being stable, so long as the trapping force is proportional to the light intensity. Poynting’s theorem states u P 0 t (16) with u designating the density of the electromagnetic field and P being the Poynting vector. For static beams (in the absence of sources or sinks of radiation), the time derivative of u is zero, and therefore P 0 (17) Since the force is proportional to the Poynting vector Fr (18) p P r c (σp being the cross section of the particle), it follows that F 0 (19) In accordance with the divergence theorem, (20) F nˆ dS F dV 0 S V In order for equation (20) to be valid, the direction of the field on the surface of a sphere enclosing the trap must change direction (inward/outward) on at least one point, or else be zero at all points (an alternative way of viewing this is that a divergenceless force is represented by continuous lines, which must leave any volume they enter). In either case, the trap is not stable; in the former case, there is a t least one point at which the particles can escape the trap, while in the latter case, the particles will not be kept in the trap. In the absence of sources or sinks of radiation, the divergence of the Poynting vector of a static laser beam is zero, and therefore so is the divergence of the force (since it is proportional to the intensity). Therefore, there cannot be a closed surface on which the force is inward at all points. It is possible, however, to overcome this limitation for atoms with multiple ground states with different absorption probabilities (Pritchard et al, 1986). 8 Magnetic Trapping It is also possible to trap atoms with a “purely” magnetic trap. This is necessary because the emission and absorption of photons set a lower limit (see equation (39) below) on the achievable cooling by lasers, and cooling below that limit requires the removal of the lasers from the system, so a trap utilizing lasers cannot be used at that stage. This section describes the principles of magnetic traps, as well as several trap configurations. The magnetic moment of an atom interacts with an inhomogeneous magnetic field, producing a force F B (21) For a positive moment, the force drives the atom towards higher potential (such states are referred to as high-field seekers) and for a negative moment the force drives the atom toward lower potentials (low field seekers). A magnetic trap uses this force to confine the atoms. There are various methods of doing so; several will be described below. The simplest type of trap is the quadrupole trap. Two identical coils with identical but counterpropagating currents are used (Figure 1). Figure 1 – Schematic of a magnetic quadrupole trap (Bergeman et al, 1987) In cylindrical coordinates, the magnitude of the field is (Bergeman et al, 1987) proportional to the coordinates as (22) B 2 4z 2 The field at the center of the trap is zero, and therefore will trap low field seekers. The problem with this type of trap is that a moving atom experiences a time-varying magnetic field, which induces state transitions. Atoms which make a transition from a low-seeking state to a high seeking state will be ejected from the trap. This effect becomes serious when the frequency of the time-varying magnetic field exceeds that 9 of the transitions between magnetic sublevels, which is of the order of μBB (where μB is the Bohr magneton). Therefore, since the field magnitude is zero at the origin (equation (22)), losses become significant as the center of the trap is approached, imposing a limit on the time an atom can be trapped. An atom with velocity v and mass m, passing within a minimum distance r of the center of the trap, will undergo a nonadiabatic spin flip if the rate of change v/r is greater than the Larmor frequency (the frequency at which the spin precesses around the magnetic field), which is approximately (Petrich et al, 1995) (23) L r Bg where (24) Bg B is the radial field gradient. Therefore, losses occur within an ellipsoid of radius (25) v Bg r0 The loss rate is given by the flux of atoms through the ellipsoid and is (26) 1 N r02 v 0 l3 where N is the number of atoms in the cloud and l is its radius. The viral theorem relates mean velocity to cloud size by (ibid) (27) mv2 l Bg By isolating Bg in equation (27) and placing it in equation (25), and placing the resultant r0 in equation (26), we get (28) 1 N 0 m l 2 One approach to overcome this is by use of a modified quadrupole trap known as a time-averaged orbiting potential (TOP) trap. A rotating spatially-uniform bias field is superimposed on the quadrupole field, changing the location of the field’s zero point faster than the atoms can respond. For a quadrupole trap with the axis of symmetry in the z direction, the bias field rotates in the xy planes at a frequency ωb. The zero point of the total field orbits around the trapped cloud (Figure 2). The radius 10 of the orbit is designated R0, and equals the ratio of the magnitude of the bias field (Bb) to the horizontal gradient of the quadrupole field R0 (29) Bb Bg Figure 2 – The zero point (the meeting point of the field gradient arrows) of a TOP trap orbits around (the dotted line) the trapped atom cloud (the grey area) at a radius of R0 (Petrich et al, 1995) The instantaneous potential is therefore (30) U(x, y, z, t) xBg Bb cos b t xˆ yBg Bb sin b t yˆ 2zBG zˆ The frequency ωb is chosen to be less then the Larmor frequency, thus ensuring that the atoms will remain in the same quantum state relative to the instantaneous field. Averaged over one field rotation, the potential is (31) U b 2 2 / b U(t)dt Bb 0 Bg2 4Bb 2 8z 2 .... Therefore, the time-averaged field never vanishes, and there is no “hole” in the trap. Another way to remove the “hole” is to use a magnetic trap with a non-zero minimum. Consider a trap similar to the quadrupole trap, but here the current in both coils flows in the same direction. At the center of the system, the potential is given by (Pethick, Smith, 2002) z U A l r l Pl (32) where Pl are Legendre polynomials and Al are coefficients. In the vicinity of the origin, the magnetic field is (33) 2 B(, , z) 3A 3z, 0, A1 3A 3 z 2 2 If A1 and A3 have the same sign, the magnetic field increases with the distance from the origin in the z direction. This configuration is known as a magnetic bottle, 11 and is used in plasma physics to confine charged particles. However, the field does not have a local minimum at the origin, and neutral atoms moving perpendicular to the field are not constrained. One way of creating a local minimum is known as the Ioffe trap. Four conducting bars (with identical current flowing in each of them) are placed parallel to the axis of symmetry (Figure 3). Figure 3 - Ioffe trap, side (left) and frontal (right) view (Bergeman et al, 1987) The potential near the origin must be of the form (34) U 2 Ce2i C*e2i 2 The constant C is determined by the current in and geometry of the bar. If the azimuthal angle φ is zero between two adjacent bars, C is real. Symmetry requires the potential function be proportional to x2-y2. The field generated by the bars is therefore (35) BI x, y, z C,C,0 And the magnitude of the total field (expanded to the second order) is (36) 2 C 2 2 B A1 3A3 z 2 2 2A1 Therefore, the magnitude of the field will have a minimum for C 2 3A1A 3 It is possible to generate fields with different degrees of curvature in the axial and radial directions by varying the current in the bars. Magneto-Optical Trap The most common method of trapping atoms is the magneto-optical trap (MOT). As its name suggests, this traps utilizes a combination of lasers and a magnetic field to trap and cool the atoms. 12 Figure 4 – Schematic of a MOT The field used for a MOT is a linearly inhomogeneous field, such as that generated by a quadrupole trap (equation (22)). For each axis, two counterpropagating lasers with opposite circular polarizations are used. Each laser is detuned by δ below the atomic resonance for zero magnetic field (Figure 5). Figure 5 – (a) Schematic view of the field and beams in a MOT. (b) The transitions induced by each beam. (c) Influence of the magnetic field on the atomic sublevels (Pethick, Smith, 2002) At z=0, both laser beams have an equal probability of being absorbed by an atom. However, at z>0 the frequency of transition to the m-1 substate is reduced, and is closer to the laser frequency, while the frequency of the transition to the m=1 substate is tuned farther away from the lasers’ frequency. Therefore, atoms at z>0 have a higher probability of absorbing photons from the σ- beam than photons from the σ+ beam, and therefore the net force is toward z=0. At z<0, the situation reverses itself. Therefore, atoms throughout the trap tend to move toward its center. 13 Figure 6 – Force operating on an atom in a MOT. The dotted lines represent the force from a single laser, while the dark line is the net force. As can be seen in Figure 6, the net force is almost linear, and takes the form of the restoring force (37) F kz This description is simplified. In practice, MOTs tend to be more complicated than described above. One problem is that many atoms have multiple hyperfine states. Since the laser does not match the resonances between all the levels, the population of excited atoms at certain levels may grow to the point the MOT ceases to function. For example, the 3S1/2 ground state of sodium has hyperfine levels F=1,2, while the excited state 3P3/2 has hyperfine levels F’=0,1,2,3. If the laser frequency is resonant with the F=2→F’=3 transition, some atoms will be excited to F’=2, and will then decay either to F=2 or F=1. Since the laser does not match the resonance of the F=1→F’=2 transition, there will be a buildup in the population of atoms in the F=1 substate compared to the F=2 substate. This process is known as optical pumping. Eventually, the number of atoms in the F=2 level (referred to as the bright state) will be sufficiently low that the MOT will cease functioning. To avoid this, radiation resonant with the F=1→F’=2 transition is applied, thus depopulating the F=1 level in a process known as repumping (Ketterle et al, 1993). In evaporative cooling (discussed later) it is necessary to achieve a high density of atoms. However, the emission of photons from excited atoms creates a force which drives the atoms apart at high densities. In addition, at high densities, the atom cloud becomes opaque to the laser. One way to reduce the impact of these effects is to reduce the level of repumping radiation so that only a small proportion of the atoms are in a level where the laser beams can induce a transition. Though this reduces the effective force of the trap, the attainable density increases. One method (ibid) of doing so is repumping light preferentially to the outer portions of the cloud, causing atoms 14 on the fringes to be driven inward, while atoms in the center will experience a high degree of optical pumping, reducing the radiation forces in the interior. This configuration is known as a dark-spot MOT, and can achieve densities two orders of magnitude greater than a conventional MOT (Pethick, Smith, 2002). Sub-Doppler cooling Doppler cooling can only reduce the temperature of an atom gas so far. As the velocity of the atoms decreases, so does the differential cooling rate, and at some point it becomes small enough to be counteracted by the random momentum kicks imparted by the atoms’ photon emissions. The temperature at which this occurs (known as the cooling limit or the Doppler temperature) is given by (38) TD 2k B where γ is the natural width associated with the resonance frequency. For example, for sodium, TD is approximately 240 μK (Philips et al, 1988) In 1988, it was discovered that the temperature of Doppler-cooled atoms was well below the Doppler limit (ibid). This is caused by the inhomogeneity of the light field as a result of the opposing lasers’ polarization, and the fact that while the discussion above referred to a simple two-level atomic model, in reality atoms are more complex, containing sublevels as well (such as those cause by Zeeman shifts). Figure 7 – Polarization of the superposed field of two counterpropagating and perpendicularly polarized lasers. The field has a minimum at 0, a maximum at λ/8, and both maxima and minima have a sinusoidal periodicity of λ/8 (Metcalf, van der Straten, 2003) If the lasers are linearly polarized perpendicular to one another, the total electric field potential’s magnitude varies sinusoidally along the lasers’ axis of propagation. Due to the light shifts, each of the ground state substates (e.g. m=±1/2) has a maximum at the other substate’s minimum, and vice versa. The electric field’s polarization also varies, from linear at the minima to circular at the maxima, with alternating directions (Figure 7). 15 This mechanism is dependent on the fact that optical pumping between two sublevels takes a finite and nonzero amount of time. If an atom begins in a potential “valley” (say at the substate m=1/2), it can move a certain distance, climbing the potential “hill”. When it reaches the maximum, the light is now polarized in the σdirection, optically pumping the atom to the m=-1/2 substate. The potential difference between the levels is emitted as a photon in the transition. The atom is now at the minimum for the m=-1/2 substate, and climbs the potential to the next maximum, where the polarization is now σ+, inducing a transition to m=1/2 (Figure 8). In such a fashion, the atom continues to climb potential “hills” without ever descending them, translating its kinetic energy into potential energy (Dalibard, Cohen-Tannoudji, 1989). The process repeats until the kinetic energy is too small to climb the next “hill”. This cooling mechanism is known as Sisyphus cooling, after the mythological Greek figure who was condemned to eternally roll a boulder up a hill. Through this mechanism, very low temperatures can be reached (e.g. 35 μK for sodium). In the case that the lasers are circularly polarized, the resulting electric field is linearly polarized everywhere and of constant magnitude, but the polarization’s orientation rotates through an angle of 2π over one wavelength. In this case, an effect similar to the force in a MOT (described above) occurs. The m=1 sublevel (with m being the magnetic quantum number) will have a higher population for atoms moving in the positive direction (Metcalf, van der Straten, 1999), while in atoms moving in the negative direction the m=-1 sublevel will have a greater population. Because of the different Clebsch-Gordan coefficients involved in the various transitions (ibid, Appendix D), the m=1 sublevel scatters σ+ at an efficiency six times greater than σphotons. Therefore, atoms moving against the σ+ beam scatter more of its photons and experience a greater momentum shift in the negative direction, while atoms moving in 16 the negative direction are preferentially pumped to the m=-1 sublevel and recoil in the positive direction. Though difficult to quantify, the final cooling derived from this mechanism is on the same order as for Sisyphus cooling (Dalibard, Cohen-Tannoudji, 1989). In all the methods involved, there is constant absorption and emission of photons by the atoms. That sets a lower temperature limit due to the fact that each time an atom emits a photon, it receives a “kick” in some direction. At low temperatures, the velocity changed caused by emission is of the same order as the atom’s total velocity, and thus the atom cannot be cooled further. This limit, known as the recoil limit, is given by (39) Tr h 2 m While optical methods have been developed to cool atoms beneath this limit, description of those methods is beyond the scope of this paper. Further material can be found in the advanced background notes for the 1997 Nobel Prize in physics. Evaporative Cooling Laser cooling has a limit to temperatures of several μK. At higher densities, photons emitted by a given atom are absorbed by other atoms, causing a repulsion effect. In addition, as density increases, so does the incidence of interatomic collisions. As these are inelastic in nature, increasing the density increases the heat. Evaporative cooling is a method based on the preferential removal (from an ensemble of trapped atoms) those atoms possessing a higher energy. The principle used is the same as that which exists when a hot liquid (e.g. a cup of coffee) cools. Particles whose energy is higher than average are removed from the ensemble, thus lowering its average energy (and hence temperature). Eventually, the particles tend to occupy the lowest energy states of the trap at high densities. I will describe here a simple model of evaporative cooling (Metcalf, van der Straten, 1999). It makes the following assumptions: 1. The gas possesses sufficient ergodicity – the phase space distribution of atoms is dependent on their atoms and the trap alone. 2. The gas is governed by classical statistics 3. Evaporation preserves the thermal nature of the distribution. 17 4. Atoms escaping from the trap do not exchange energy with the remaining atoms. Consider an ensemble of N atoms with a temperature of T held in an infinitely deep trap. The model (Davis et al, 1995) assumes the evaporation is composed of discrete cycles. The trap depth is lowered to a depth ηkBT (η is known as the truncation parameter), allowing the escape of atoms with a higher temperature; the atoms still inside the trap then achieve a new mean temperature as a result of mutual collisions. We define N’ as the number of atoms in the trap at the end of this cycle, and T’ as the temperature; for convenience, we define υ=N’/N. The parameter β is a measure of the temperature decrease, and is defined (40) ln T '/ T ln N '/ N ln T '/ T ln Therefore, the decrease in temperature is the power-law function T ' T (41) However, for atoms in a trap of potential U, 2m g(E) 3/ 2 (42) 4 2 3 E Ud 3r The fraction of atoms remaining in the trap (for chemical potential μ) equals 1 N (43) k BT g(E)e (E ) k BT dE 0 Let us assume a potential which can be described as a power law function (Bagnato et al, 1987) so that (44) x U x, y, z 1 a1 s1 y 2 a2 s2 z 3 a3 s3 We further define (45) 1 1 1 s1 s 2 s3 For this potential, the volume is proportional to the temperature so that (46) V T 18 Property Number of Sy mbol q N 1 Temperature T β Volume V βξ Density N 1- βξ atoms Phase space density Collision rate ρ K 2 2 1 3 1 1 Table 1 – scaling of thermodynamic properties, where for property X X’=Xυ q Table 1 shows the exponential scaling for various thermodynamic properties. At infinite η, υ=1, from which the chemical potential is determined (Bagnato, et al, 1987). Equation (43) becomes (47) e d 0 where the reduced energy is defined as (48) E k BT and the reduced density of states is (49) 1 2 3 2 Using the incomplete gamma function, equation (47) can be written as (50) inc 3 , 2 3 2 Therefore, the fraction of remaining atoms is dependent solely on the truncation parameter and on the trap parameter ξ. 19 The average reduced energy before truncation is (Metcalf, van der Straten, 1999) (51) e 0 e d d 3 2 0 while the average reduced energy after truncation is e (52) d inc 5 , 2 ' 3 , 2 e d inc 0 0 Due to equation (48), ' T' T (53) The energy carried away by each evaporated atom is (54) l ' 1 1 3 2 1 1 For large values of η, the denominator in equation (54) is small and therefore (55) l 1 3 2 In such a case, β represents the energy above the mean energy, which is carried away. As ξ grows (stronger potential), the decrease in volume (and therefore the increase in density) as the temperature decreases becomes larger. Elastic collision rates also increase for a larger potential, speeding up the rethermalization. At weak potentials, the collision rate decreases and the cooling eventually stops. The speed of evaporation can be estimated using the principle that elastic collisions produce atoms whose energy is greater than ηkBT at a rate determined by the number of atoms with energy above this divided by their collision rate (Ketterle, van Druten, 1996). The velocity of atoms with energy ηkBT is (56) v 2k BT 3 v m 2 For large values of η, the fraction of atoms with energy greater than ηkBT is (57) f e 20 3 2 The elastic collision rate is defined as (58) k el nv where the collisions’ cross section for scattering length a is (59) 8a 2 Therefore, the rate of evaporation is (60) dN N Nf k el nve N dt ev Since the average value of kel depends on the relative rather than the mean velocity of the atom, (61) k el 4nv 3 Thus, the ratio of the evaporation time τev and the collision time τel increases exponentially with η (62) ev 2e el So far, it has been assumed all collisions are elastic. This is not necessarily true, however, which may cause problems, since the release of internal energy can heat the atoms, and the state of the atoms may be changed to a new state which is no longer trapped. Thus, the final limit cooling is dependent on the ration of elastic to inelastic collisions. The magnitude of inelastic collisions varies with atomic species; for alkali atoms, the final temperature is on the order of picokelvins (ibid). Trapping on a Microchip There are advantages to small trap sizes. Large field gradients and curvatures can be achieved with small currents. This means high gradients can be achieved for less energy dissipation, which leads (Table 1) to faster rethermalization and therefore faster evaporative cooling rates. In addition, by their nature small traps allow the entire device to be small, which is important for various practical applications. There is considerable research being carried out in the field of creating traps on a microchip; this section will discuss a few MOT and magnetic trap configurations specific to such chips. 21 Mirror MOT The small volume of a chip trap presents certain drawback. Its small size presents difficulties in loading the atoms into the trap; in addition, the proximity of the trap to the substrate on which it resides presents difficulties when implementing a six-laser MOT, since the laser must be positioned between the substrate and the trapping area (otherwise, the substrate blocks the beam). One solution (Reichel et al, 1999) for this problem is a mirror MOT (Figure 9). Figure 10 – Schematic view of a mirror MOT (Reichel et al, 1999) This type of MOT uses four lasers instead of six. By reflecting two of the lasers from a mirror, their polarizations are reversed, and for a particle above the mirror, there are effectively four lasers Figure 11 – Path of the lasers in a mirror MOT (there are two additional lasers, not indicated here, perpendicular to the page) Magnetic Trapping As discussed above, once the atom cloud reaches the limit for laser cooling, it must be confined by a purely magnetic trap for evaporative cooling. For obvious reasons of size, normal traps such as the quadrupole or Ioffe traps described above cannot be used on a chip (or at least, fabrication is extremely difficult). It is possible, however, to achieve similar potentials by means of the use of wires on the chip, in 22 conjunction with an external homogenous bias field. Here, I will describe two configurations to do so. According to the law of Biot-Savart, the field created by a wire1 of length 2a placed on the y axis (in such a way that it is bisected by the origin), through which a current I is flowing, is given by B x, y, z (63) 0 I 4 ay x 2 a y 2 z 2 ay x 2 a y 2 z 2 x 2 z2 z, 0, x One method of trapping is to bend the wire into a U shape. An example of a field generated by such a wire configuration can be seen in Figure 112 (using arbitrary values for current I and crosspiece length 2a). Y Z a Z b 2 4 4 1 3 3 0 2 2 -1 1 1 -2 X -2 -1 0 1 2 0 -2 Y -1 0 1 2 0 -2 c X -1 0 1 2 Figure 12 – U trap field (without bias field from (a) above (b) in front (c) the side The field generated by this configuration is (64) ay ay x x x x 1 z a y 1 a y 1 1 0 I yˆ B x, y, z xˆ z 2 2 2 a y 2 z 2 4 x 2 z2 a y z2 a y z2 a y z2 ay ay x zˆ x 2 z2 where (65) x 2 a y z 2 2 By inserting the (x,y,z) coordinates for the desired trap location into equation (64), an appropriate magnitude for the bias field can be calculated, such that the total 1 Because of the scale, unless stated otherwise, I will assume here the wires are one-dimensional The plots in this section were generated with Mathematica. Magnitude ascends from red through orange hues, then greens, blues, purples, and red again for maximal intensity. All the traps lie in the xy plane, with orientations as shown in Figure 11(a), Figure 12(a) Figure 13(a), and Figure 14(a). For other orientations, the values x,y,z in equations (64) and (66) are transposed as needed. 2 23 field will be zero at some point. Doing so results in a quadrupole trap, as seen in Figure 12 Y Z a Z b 2 4 4 1 3 3 0 2 2 -1 1 1 X -2 -2 -1 0 1 2 0 -2 Y -1 0 1 2 0 -2 c X -1 0 1 2 Figure 13 – Quadrupole trap generated by a U-wire trap in a bias field, from (a) above, (b) the front, (c) the side A second configuration for a wire trap is the Z-trap, so named because the wires are twisted into a Z shape. The field caused by that configuration is shown in Figure 13. Y Z a Z b 2 4 4 1 3 3 0 2 2 -1 1 1 -2 X -2 -1 0 1 2 Y 0 -2 -1 0 1 2 0 -2 c X -1 0 1 2 Figure 14 – (a) top (b) front and (c) side views of the field magnitude of a Z-trap The field for this trap is (66) ay ay x x x x 1 z a y 1 a y 1 1 0 I yˆ B x, y, z xˆ z 2 2 2 a y 2 z 2 4 x 2 z2 a y z2 a y z2 a y z2 Introducing a suitable bias field, we get an Ioffe trap (Figure 14) 24 ay ay x zˆ x 2 z2 Y Z a Z b 2 4 4 1 3 3 0 2 2 -1 1 1 -2 X -2 -1 0 1 2 Y 0 -2 -1 0 1 2 0 -2 c X -1 0 1 2 Figure 15 – (a) Top (b) front and (c) side views of the resulting trapping potential of the Z-trap + bias field Implementation of a MOT This section will describe the construction of a “basic” MOT. The experiment in question is described by Monroe et al (1990). Figure 16 – The vapor cell used in the experiment (Monroe et al, 1990) The experiment’s configuration is shown in Figure 14. The vapor cell (containing a vapor of cesium atoms) is a vertical cylinder with two crosswise cylinders placed at the top, with a window at each of the six openings (note that henceforth, the “center” of the cell will refer to the intersection of this cross, rather than the cell’s physical center halfway down the vertical cylinder). Two coils are placed above and below the center (the larger coil shown in Figure 15 is used to create a purely magnetic trap later in the experiment, and will not be discussed here). Two smaller tubes are attached to the vertical tube. One contains a reservoir of cesium, whose temperature (which can 25 be reduced to -23ºC) controls the vapor pressure in the cell. The second tube leads to an ion pump which removes any residual gas which may diffuse through the walls of the cell. A diode laser, tuned to the 6S1/2,F=4→6P3/2,F=5 transition is split into three beams, which are circularly polarized and directed to intersect perpendicularly in the center of the cell. After leaving the cell, each beam passes through a quarter-wave plate, and is then reflected back into the cell, thus reversing its polarization (since it passes the cell twice). A second laser, tuned to the 6S1/2,F=3→P3/2,F=4 transition illuminates the intersection to prevent atoms from accumulating in the F=3 ground state. The coils produce a magnetic field gradient of about 10 Gauss/cm. When the apparatus was activated, with the laser red-detuned by 1-10 MHz below resonance (optimal results were attained for about 6 MHz below resonance), a cloud of atoms with a volume of less than 1 cubic millimeter appeared in the center of the cell. By measuring the resulting fluorescence, it was determined that the cloud contained about 1.8x107 atoms, at a temperature of about 30 μK. Varying the pressure from about 10-7 Torr to about 1.5x10-9 Torr reduced the number of trapped atoms (though by less than 30%), presumably because of the loss due to collisions with noncesium atoms. At high pressures, the number of atoms in the trap also decreases, as the vapor attenuated the laser beams. Conclusion This overview is, of course, by no means exhaustive; just the variations on a few of the traps shown here would each merit a substantial article of their own.. Further variations and techniques are created constantly. Likewise, various aspects of the theory can be expanded at great length. However, this paper goes over the principles involved, giving an overview of the basics of this field. Further information can be found in the references given below, as well as the web sites and publications of various groups and researchers involved in this field. 26 References 1) Metcalf, H.J, van der Straten, P. (2003) Laser Cooling and Trapping of Atoms, Journal of the Optical Society of America B, volume 20, 887 2) Ashkin, A., Gordon, J.P. (1983) Stability of Radiation-Pressure Traps: An Optical Earnshaw Theorem, Optics Letters, volume 8, 511 3) Pritchard, D.E., Raab, E.L., Bagnato, V. (1986) Light Traps Using Spontaneous Forces, Physical Review Letters, volume 57, 310 4) Davidson, N., Lee, H.J., Adams, C.S., Kasevich, M., Chu, S. (1995) Long Atomic Coherence Times in an Optical Dipole Trap, Physical Review Letters, volume 74, 1311 5) Smith, H., Pethick, C.J. (2002) Bose-Einstein Condensation in Dilute Gases, Cambridge University Press 6) Metcalf, H.J., van der Straten, P. (1999) Laser Cooling and Trapping, Springer 7) Lett, P.D, Watts, R.N., Westbrook, C.I., Philips, W.D, Gould, P.L., Metcalf, H.J. (1988) Observation of Atoms Laser Cooled Below the Doppler Limit, Physical Review Letters, volume 61, 169 8) Dalibard, J., Cohen-Tannoudji, C. (1989) Laser Cooling Below the Doppler Limit by Polarization Gradients: Simple Theoretical Models, Journal of the Optical Society of America B, volume 6, 2023 9) Nobel Prize in physics website (http://www.nobel.se/physics/index.html) 10) Bergeman, T., Erez, G., Metcalf H. (1987) Magnetostatic Trapping Fields for Neutral Atoms, Physical Review A, volume 35, 1535 11) Petrich, W., Anderson, M.H., Ensher, J.R., Cornell, E.A (1995) Stable, Tightly Confining Magnetic Trap for Evaporative Cooling of Neutral Atoms, Physical Review Letters, volume 74, 3352 12) Ketterle, W., Davis, K.B., Joffe, M.A., Martin, A., Pritchard, D.E. (1993) High Densities of Cold Atoms in a Dark Spontaneous-Force Optical Trap, Physical Review Letters, volume 70, 2253 13) Bagnato, V., Pritchard, E.E., Kleppner D. (1987) Bose-Einstein Condensation in an External Potential, Physical Review A, volume 35, 4354 14) Davis, K.B, Mewes, M.O., Ketterle, W. (1995) An Analytical Model for Evaporative Cooling of Atoms, Applied Physics B, volume 60, 155 27 15) Ketterle, E., van Druten, N.J. (1996) Evaporative Cooling of Trapped Atoms, Advances in Atomic, Molecular and Optical Physics, volume 37, 181 16) Reichel, J., Hänsel, W., Hänsch, T.W. (1999) Atomic Micromanipulation with Magnetic Surface Traps, Physical Review Letters, volume 83, 3398 17) Monroe, C., Swann, W., Robinson, H., Wieman, C. (1990) Very Cold Trapped Atoms in a Vapor Cells, Physical Review Letters, volume 65, 1571 28