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Laser Sisyphus cooling in a magnetic trap
Optics Communications 106 (1994) 202-206 North-Holland
OPTICS COMMUNICATIONS
Laser Sisyphus cooling in a magnetic trap B. Hoeling
and R.J. Knize
Department of Physics, University of Southern CuL$ornia, Los Angeles, CA 90089-0484, USA
Received 2 1 September 1993; revised manuscript received 13 December 1993
A method of laser Sisyphus cooling of neutral atoms in a magnetic trap is presented. A one-dimensional model predicts a minimum temperature that approaches the recoil limit. Three-dimensional Monte Carlo simulations show that sub-mK temperatures can be reached if the atoms can be kept from ending up in orbits where no further interaction with the laser is possible.
1. Introduction During the past decade, considerable work has been devoted to the trapping and cooling of neutral atoms [ 1,2]. Part of the motivation for this research is the possibility of doing high resolution spectroscopy on the slowly moving or trapped atoms [ 31, the study of slow atomic collisions [ 41 and an attempt to produce conditions where quantum collective phenomena such as Bose-Einstein condensation can occur [ 51. For confining and further cooling of atoms, magnetic traps are of particular interest since the restoring force does not involve excitation and long confinement times are possible. However, only weakfield-seeking atomic states can be trapped by static magnetic fields, and the inhomogeneous confining magnetic fields complicate laser cooling, so that usual techniques used for laser cooling of atoms in free space may not work as well for atoms in a magnetic trap. Recently [ 6 1, one-dimensional optical molasses combined with mixing of the transverse and longitudinal motions was used to cool Na atoms in a magnetic trap from an initial temperature of 50 mK down to 2 mK. Doppler cooling of magnetically trapped hydrogen atoms to a temperature of 8 mK has also been demonstrated [ 7 1. Another method that also has been used to cool atoms in a magnetic trap is evaporative cooling [ 7,8] : Hot atoms are allowed to leave the trap, while the remaining atoms thermalize due to elastic collisions. The expression “Sisyphus cooling” [ 91 was invented to describe a cooling mechanism in optical 202
molasses by which atoms can loose energy climbing up AC Stark potential wells and sub-Doppler temperatures [ 10 ] can be achieved. For atoms in a trap, a form of magnetic Sisyphus cooling using two different ground state sublevels and a combination of rf and laser optical pumping has been proposed by Pritchard [ 111. In this scheme, the atom, originally in bublevel2 with higher magnetic energy E,, makes a rf transition to sublevel 1 with lower energy El when it is near the most outward reachable position in the trap, i.e. in a region of high magnetic field. When the atom is near the center of the trap, i.e. in low B-field, optical pumping by an appropriately tuned and polarized laser and subsequent spontaneous decay brings it back to the starting sublevel 2. However, a problem with this approach is that it requires two ground state sublevels, and it may not be possible to achieve a sufficiently long trap lifetime for atoms with 1m,l CF. These substates might undergo spin exchange transitions into untrapped states due to collisions whereas states with Im,j = F are stable against spin exchange losses [ 12 1. It also has not been explicitly stated what atom could be used in this scheme. In the present paper, we examine another method for laser cooling of neutral atoms in a trap which also takes advantage of the different magnetic energies of two states. However, there are important differences to Pritchard’s scheme: Our cooling method relies directly upon laser excitation of atoms from a state with a smaller g factor to a state with a larger g factor without the use of an additional microwave field and
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will work for atoms that possess a relatively long lifetime excited state. It can be used with both ground and excited states with 1mFl = F, thus avoiding the problem of atoms escaping the trap due to spin exchange collisions. In the strong trapping magnetic fields, losses due to off-resonant transitions to other magnetic substates that lead to untrapped states are negligible [ 61, and the confinement time will be much longer than the time necessary for cooling. Because of the long lifetime of the upper state, the excited atom will move around in the trap before decaying. If, on the average, the initial excitation takes place in a region of lower magnetic field and the deexcitation happens in a higher field region, cooling will occur due to the difference in magnetic energies. An analytic expression is derived that estimates the limits of cooling by this method. Monte Carlo calculations were performed to simulate the atoms’ three-dimensional motion in the trap and their interaction with a laser. The results show that it should be possible to achieve sub-mK temperatures that approach the recoil limit using this magnetic Sisyphus cooling.
0 0
a2
0.4
a8
I
12
Pohtio*~[8.r] Fig. 1. Magnetic energies of ground and excited states in a onedimensional harmonic B-field. E, is the energy of the absorbed and E2 the energy of the emitted photon.
dimensional harmonic B-field. For a laser of frequency fL, the resonance condition is given by
(2) 2. Cooling in one dimension The energy of a ground state atom in a magnetic field magnitude B at position (x,y,z) can be described by E,=imv2+g&,B+FGz,
(1)
where m is the mass, v is the velocity, g, is the product of the LandC splitting factor for the ground state and the magnetic quantum number m, of the total electronic angular momentum, pLgis the Bohr magneton, and FG = mg is the gravitational force in the z direction. For the energy E, of an excited atom, gg has to be replaced by g,, the product of the LandC splitting factor of the excited state and mJ, and the energy of the exciting photon has to be added. In this expression, we have neglected nuclear spin and assumed that the electron spin and orbital angular momentum are aligned with the local magnetic field. Also, we assume that the atoms remain in a pure spin state which has the highest m, value. Figure 1 shows a diagram of the position dependent magnetic energies ofground and excited state for an atom in a one-
where f0 is the frequency of a stationary atom at the trap center, kL is wave vector of the laser, Vi is the initial velocity, Ag=ge-gg, Bi is the magnitude of the magnetic field at the point of absorption, h is Plan&% constant, and U, is the magnitude of the recoil velocity. We assume that the lifetime of the excited state is large enough so that the atom can move around in the trap before decaying, and that the atomic linewidth is narrow. Silver is an example of an atom that allows initial cooling using the 5Si12 -5P3,2 transition and also has a metastable 5D,,, state (T= l/4 s) that could be used for this magnetic Sisyphus cooling. The average change in energy after one absorption-emission cycle is given by (~)=m(Vi.V,+Vf)+AgiUB(Bi-(Bf)),
(3)
where ( Bf) is the average magnetic field at the point of emission. If ( Bf) is sufficiently larger than Bi, then ( AE) will be negative, and cooling will occur. Since the atomic lifetime t is long enough for the excited atom to move around in the trap, (AZ?) is independent of both the magnetic field gradient and T. Since 203
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the magnetic field strength where absorption can take place is determined by the laser frequency f,, we can use the resonance condition eq. (2) to obtain
In order to estimate the limits of Sisyphus cooling in a trap, we used the simple one-dimensional model of an atom in a harmonic magnetic field B= kz2 for which the equation of motion can be solved analytically. At temperatures below a few tens of a mK, gravity must be included. The equations of motion for the ground and excited state then yield different frequencies and amplitudes of motion as well as different points of equilibrium: The atom oscillates around z= -FG/2gspBk, where g,=g, if it is in the excited and g,=g, if it is in the ground state. Using the equation of motion for the excited atom, we calculated the average magnetic field (Bf) at which emission will occur and obtained the average energy loss per interaction cycle, i.e. after one absorption followed by an emission: (AE)=hV;-fa)(l-$)+%
e
’
(5)
where Eg is the initial total energy of the atom in the ground state. By setting ( AE) = 0 and solving for EB, the lowest achievable energy Ef is obtained:
Er=
~[hUA)(l-2) (6)
In particular, we considered the electric quadrupole transition between the Sij2 (F= 1, rnp 1) and the metastable D s,2 (F= 3, mF= 3 ) state in silver atoms. In this case, because g, = 3 and gs= 1, the gravitational term in eq. (5) cancels, and the final energy Ef becomes E,=
2hti-j&+3&,
(7)
where E, = mvf/2 is the recoil energy. Thus, the minimum of Ef is reached if the laser is detuned as far to the red as possible while still fulfilling the reso204
nance condition then given by
15 March 1994
of eq. (2). The lowest energy Ef is
Ef=13E,+X-4E,J+xIE,,
(8)
where X= Ag F&/2g,2p,,k. With strong trapping field gradients, i.e. small X, the lowest obtainable energy Ef approaches the recoil energy E,.
3. Three-dimensional
Monte Carlo simulations
In order to examine magnetic Sisyphus cooling in three dimensions, we performed Monte Carlo calculations simulating the atoms’ motion in a magnetic trap and their interaction with a cooling laser. Using the 5S,,, to 5D,,, transition in silver and neglecting the effects of the nuclear spin as well as interactions between the atoms, we solved numerically the equations of motion for the atoms in a magnetic field with the specifications as given in ref. [ 111: the superposition of a quadrupole and a bottle field with its axis in the z-direction [ 111. The surfaces of equal potential in the vicinity of the trap center are shaped like a pear with its axis along the z-direction [ 13 1. The atom will only be in resonance with the laser light and can therefore only interact with the laser when its position lies within a certain shell, the interaction or resonance region, the thickness of which is determined by the laser and atomic linewidths and by the magnetic field gradient. In our case, the absorption and stimulated emission rates were chosen to be 100 s- ‘. Spontaneous emission was allowed to take place everywhere in the trap at a rate of 4 s-i given by the lifetime of the silver D,,, state. However, to cool the atoms more efficiently, it proved favorable to introduce in addition a much higher decay rate of 1000 s-’ for atoms outside the interaction shell. This way, when an atom has been excited and then moves out of the interaction zone towards higher magnetic field regions, its chances of undergoing spontaneous emission were greatly enhanced. This increased decay rate allows the cooling process to be considerably faster without significantly altering the final temperatures. In an actual experiment, the same effect can be achieved by using an additional field to drive the Ag transition from the D5,2 to the P3,2 state which then decays to the S,,2 state with a lifetime of 7 ns.
OPTICS COMMUNICATIONS
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The simulations were started with 100 atoms with random positions and randomly directed velocities of a magnitude of 6 m/s corresponding to a temperature of about 170 mK which can be reasonably obtained by initial cooling of a thermal beam using the S1,2-P3,2 transition. Figure 2 shows the cooling as a function of time in a typical case: The inner radius of the interaction shell was about 5 mm, the laser linewidth constant at 300 MHz. The final temperature was about 30 mK with an approximately gaussian distribution. The cooling times were of the order of a few seconds. In fig. 3, for a constant laser
\
0
I
2
3
4
5
6
7
8
Time [s] Fig. 2. Simulation showing the typical decrease in temperature of the atoms as a function of time.
120 j
0
) 3
2
I Laser
3
4
5
Dehming [GHz]
Fig. 3. Final temperature (left abscissa) and exponential time constant (right abscissa) as a function of laser detuning.
15 March 1994
linewidth of 300 MHz, the dependence of the final temperature on the laser detuning (and thus on the location of the interaction zone) is displayed. For zero detuning, the laser is resonant with the trap center. The coldest temperature was obtained by tuning the laser to be in resonance at the minimum magnetic field. This might at first seem surprising, as atoms nearly at rest at the center of the trap were still interacting with the laser and thus might be heated. However, it turned out that the atoms were not slowed down sufficiently for this re-heating to occur. When the laser detuning was gradually reduced to zero while the laser linewidth was kept constant at 300 MHz, the resultant final temperature was about 15 mK, which is much higher than the recoil temperature (0.65 uK) expected from the above one-dimensional model. The explanation for this discrepancy can be found in the observation that in three dimensions, the cooled atoms exhibit a tendency to orbit around the center of the trap. Particularly with the laser tuned to low B-field regions, the final temperature is usually not reached until all atoms have ended up in stable orbits outside of the resonance region and thus no further cooling is possible. Figure 3 also shows the exponential time constants for cooling. For constant laser linewidth, the closer the laser frequency is tuned to resonance at the trap center, the longer the cooling process takes due to the smaller interaction volume. In order to obtain lower temperatures, it is necessary to prevent the orbiting of the atoms that do not interact with the laser. To achieve this, one has to destroy the conservation of angular momentum fulfilled in the original trapping field near the center by breaking the field symmetry. Experimentally, this is probably done most easily by allowing a static defect of the magnetic field. Our solution which is easier to handle mathematically was to introduce an additional time varying B-field, a quadrupole field in the y-z-plane oscillating sinusoidally at about the orbiting frequency which should kick the atoms out of their stable orbits back into the interaction zone. The amplitude of this disturbing field has to be chosen carefully in order to achieve the desired effect: If the field is too strong, the atoms exhibit heating, whereas too low of an amplitude will not alter the orbiting motion noticeably. It turns out that with a disturbing field amplitude of 1 percent of the trapping field 205
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strength, a further reduction of temperature from 15 mK to about 3 mK was possible. In order to improve the cooling process, it was necessary to adjust the laser detuning as the atoms were loosing energy. By reducing the laser detuning (shrinking the interaction zone in three steps from an inner radius of 10 mm to 0, then 5 mm to 0, then 2.5 mm to 0), and using a linewidth of 300 MHz, 50 MHz and 5 MHz for the respective steps, it was possible to cool the atoms from 500 mK down to about 600 uK when again most of them were found in stable orbits. In the last two steps, a disturbing field as described in the previous paragraph has also been added. Although obtained for a particular field configuration, these results are not dependent on the actual shape of the magnetic field and should hold similarly for other magnetic traps of comparable strength. In particular, the problem of orbiting will arise in any confining field with approximate rotational symmetry in the vicinity of the trap center. These Monte Carlo simulations show that the described magnetic Sisyphus cooling can achieve sub-mK temperatures, although attention has to be paid to orbiting atoms that will no longer interact with the laser.
4. Conclusion We have shown that in a one-dimensional model for Sisyphus cooling in a magnetic trap, the predicted final temperature of the atoms can approach the recoil limit. Simulations demonstrate that this cooling will work in three dimensions at least down to temperatures in the sub-mK range. However, the limitations of this calculation in determining the linal temperature are given by the problem of the atoms’ orbiting that keeps them from further interacting with the laser. One idea to achieve sub-recoil temperatures is a variation of velocity space optical pumping [ 141 involving both the Doppler shift and the frequency shift due to the trapping field which are approximately of the same size at a few PK. If the laser is appropriately tuned to the red, it will only interact with atoms with velocities directed towards the beam and positions close to the trap center where the magnetic field is small. These atoms will be cooled further, whereas those that already are in the desired
206
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1994
velocity interval will remain unaffected. Some atoms, however, will be heated in this process and could then be cooled by a second laser. Another approach for achieving low temperatures in a moderately dense vapor is to cool the faster atoms using the described magnetic Sisyphus cooling and to rely on collisions to transfer energy between atoms. These methods may allow the achievement of ultra-low temperatures for spectroscopy, the study of collisions and the observation of quantum-collective phenomena.
Acknowledgements We would like to acknowledge the support of the National Science Foundation, the U.S. Army Research Office and the Deutsche Forschungsgemeinschaft.
References [l A.L. Migdall, J.V. Prodan, W.D. Phillips, T.H. Bergeman and H.J. Metcalf, Phys. Rev. Lett. 54 (1985)
2596.
12 For a review see the special issues of J. Opt. Sot. Am. B on laser cooling and trapping: 2 (1985) and 6 (1989). ]3 1J.L. Hall, M. Zhu and P. Buch, J. Opt. Sot. Am. B 6 (1989) 2194; C. Monroe, H. Robinson, and C. Wieman, Optics Lett. 16 (1991) 50. P.S. Julienne and F.H. Mies, J. Opt. Sot. Am. B 6 (1989) 2257. V. Bagnato, D.E. Pritchard and D. Kleppner, Phys. Rev. A 35 (1987) 4354. K. Helmerson, A. Martin and D.E. Pritchard, J. Opt. Sot. Am.B9 (1992) 1988. I.D. Setija, H.G.C. Werij,O.J. Luiten, M.W. Reynolds, T.W. Hijmans and J.T.M. Walraven, Phys. Rev. Lett. 70 ( 1993) 2257. ]8 1N. Masuhara, J.M. Doyle, J.C. Sandberg, D. Kleppner, T.J. Greytag, H.F. Hess and G.P. Kochanski, Phys. Rev. Lett. 61 (1988) 935. J. Opt. Sot. Am. B 2 19 J. Dalibard and C. Cohen-Tannoudji, (1985) 1707. [lo] P.D. Lett, R.N. Watts, C.J. Westbrook, W.D. Phillips, P.L. Gould and H.J. Metcalf, Phys. Rev. Lett. 61 (1988) 169. [ 1 l] D.E. Pritchard, Phys. Rev. Lett. 51 (1983) 1336. [ 12 ] E. Tiesinga, B.J. Verhaar and H.T.C. Stoof, Phys. Rev. A 47 (1993) 4114. [ 131 T. Bergeman, G. Erez and H.J. Metcalf, Phys. Rev. A 35 (1987) 1535. [ 141 D.E. Pritchard, K. Helmerson, V.S. Bagnato, G.F. Lafyatis, and A.G. Martin, in: Laser Spectroscopy VIII, Persson and Svanberg (1987) p. 68.
OPTICS COMMUNICATIONS
Laser Sisyphus cooling in a magnetic trap B. Hoeling
and R.J. Knize
Department of Physics, University of Southern CuL$ornia, Los Angeles, CA 90089-0484, USA
Received 2 1 September 1993; revised manuscript received 13 December 1993
A method of laser Sisyphus cooling of neutral atoms in a magnetic trap is presented. A one-dimensional model predicts a minimum temperature that approaches the recoil limit. Three-dimensional Monte Carlo simulations show that sub-mK temperatures can be reached if the atoms can be kept from ending up in orbits where no further interaction with the laser is possible.
1. Introduction During the past decade, considerable work has been devoted to the trapping and cooling of neutral atoms [ 1,2]. Part of the motivation for this research is the possibility of doing high resolution spectroscopy on the slowly moving or trapped atoms [ 31, the study of slow atomic collisions [ 41 and an attempt to produce conditions where quantum collective phenomena such as Bose-Einstein condensation can occur [ 51. For confining and further cooling of atoms, magnetic traps are of particular interest since the restoring force does not involve excitation and long confinement times are possible. However, only weakfield-seeking atomic states can be trapped by static magnetic fields, and the inhomogeneous confining magnetic fields complicate laser cooling, so that usual techniques used for laser cooling of atoms in free space may not work as well for atoms in a magnetic trap. Recently [ 6 1, one-dimensional optical molasses combined with mixing of the transverse and longitudinal motions was used to cool Na atoms in a magnetic trap from an initial temperature of 50 mK down to 2 mK. Doppler cooling of magnetically trapped hydrogen atoms to a temperature of 8 mK has also been demonstrated [ 7 1. Another method that also has been used to cool atoms in a magnetic trap is evaporative cooling [ 7,8] : Hot atoms are allowed to leave the trap, while the remaining atoms thermalize due to elastic collisions. The expression “Sisyphus cooling” [ 91 was invented to describe a cooling mechanism in optical 202
molasses by which atoms can loose energy climbing up AC Stark potential wells and sub-Doppler temperatures [ 10 ] can be achieved. For atoms in a trap, a form of magnetic Sisyphus cooling using two different ground state sublevels and a combination of rf and laser optical pumping has been proposed by Pritchard [ 111. In this scheme, the atom, originally in bublevel2 with higher magnetic energy E,, makes a rf transition to sublevel 1 with lower energy El when it is near the most outward reachable position in the trap, i.e. in a region of high magnetic field. When the atom is near the center of the trap, i.e. in low B-field, optical pumping by an appropriately tuned and polarized laser and subsequent spontaneous decay brings it back to the starting sublevel 2. However, a problem with this approach is that it requires two ground state sublevels, and it may not be possible to achieve a sufficiently long trap lifetime for atoms with 1m,l CF. These substates might undergo spin exchange transitions into untrapped states due to collisions whereas states with Im,j = F are stable against spin exchange losses [ 12 1. It also has not been explicitly stated what atom could be used in this scheme. In the present paper, we examine another method for laser cooling of neutral atoms in a trap which also takes advantage of the different magnetic energies of two states. However, there are important differences to Pritchard’s scheme: Our cooling method relies directly upon laser excitation of atoms from a state with a smaller g factor to a state with a larger g factor without the use of an additional microwave field and
0030-4018/94/$07.00
0 1994 Elsevier Science B.V. All tights reserved. SSDI 0030-4018(93)E0611-I
Volume 106, number 4,5,6
15 March 1994
OPTICS COMMUNICATIONS
will work for atoms that possess a relatively long lifetime excited state. It can be used with both ground and excited states with 1mFl = F, thus avoiding the problem of atoms escaping the trap due to spin exchange collisions. In the strong trapping magnetic fields, losses due to off-resonant transitions to other magnetic substates that lead to untrapped states are negligible [ 61, and the confinement time will be much longer than the time necessary for cooling. Because of the long lifetime of the upper state, the excited atom will move around in the trap before decaying. If, on the average, the initial excitation takes place in a region of lower magnetic field and the deexcitation happens in a higher field region, cooling will occur due to the difference in magnetic energies. An analytic expression is derived that estimates the limits of cooling by this method. Monte Carlo calculations were performed to simulate the atoms’ three-dimensional motion in the trap and their interaction with a laser. The results show that it should be possible to achieve sub-mK temperatures that approach the recoil limit using this magnetic Sisyphus cooling.
0 0
a2
0.4
a8
I
12
Pohtio*~[8.r] Fig. 1. Magnetic energies of ground and excited states in a onedimensional harmonic B-field. E, is the energy of the absorbed and E2 the energy of the emitted photon.
dimensional harmonic B-field. For a laser of frequency fL, the resonance condition is given by
(2) 2. Cooling in one dimension The energy of a ground state atom in a magnetic field magnitude B at position (x,y,z) can be described by E,=imv2+g&,B+FGz,
(1)
where m is the mass, v is the velocity, g, is the product of the LandC splitting factor for the ground state and the magnetic quantum number m, of the total electronic angular momentum, pLgis the Bohr magneton, and FG = mg is the gravitational force in the z direction. For the energy E, of an excited atom, gg has to be replaced by g,, the product of the LandC splitting factor of the excited state and mJ, and the energy of the exciting photon has to be added. In this expression, we have neglected nuclear spin and assumed that the electron spin and orbital angular momentum are aligned with the local magnetic field. Also, we assume that the atoms remain in a pure spin state which has the highest m, value. Figure 1 shows a diagram of the position dependent magnetic energies ofground and excited state for an atom in a one-
where f0 is the frequency of a stationary atom at the trap center, kL is wave vector of the laser, Vi is the initial velocity, Ag=ge-gg, Bi is the magnitude of the magnetic field at the point of absorption, h is Plan&% constant, and U, is the magnitude of the recoil velocity. We assume that the lifetime of the excited state is large enough so that the atom can move around in the trap before decaying, and that the atomic linewidth is narrow. Silver is an example of an atom that allows initial cooling using the 5Si12 -5P3,2 transition and also has a metastable 5D,,, state (T= l/4 s) that could be used for this magnetic Sisyphus cooling. The average change in energy after one absorption-emission cycle is given by (~)=m(Vi.V,+Vf)+AgiUB(Bi-(Bf)),
(3)
where ( Bf) is the average magnetic field at the point of emission. If ( Bf) is sufficiently larger than Bi, then ( AE) will be negative, and cooling will occur. Since the atomic lifetime t is long enough for the excited atom to move around in the trap, (AZ?) is independent of both the magnetic field gradient and T. Since 203
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OPTICS COMMUNICATIONS
the magnetic field strength where absorption can take place is determined by the laser frequency f,, we can use the resonance condition eq. (2) to obtain
In order to estimate the limits of Sisyphus cooling in a trap, we used the simple one-dimensional model of an atom in a harmonic magnetic field B= kz2 for which the equation of motion can be solved analytically. At temperatures below a few tens of a mK, gravity must be included. The equations of motion for the ground and excited state then yield different frequencies and amplitudes of motion as well as different points of equilibrium: The atom oscillates around z= -FG/2gspBk, where g,=g, if it is in the excited and g,=g, if it is in the ground state. Using the equation of motion for the excited atom, we calculated the average magnetic field (Bf) at which emission will occur and obtained the average energy loss per interaction cycle, i.e. after one absorption followed by an emission: (AE)=hV;-fa)(l-$)+%
e
’
(5)
where Eg is the initial total energy of the atom in the ground state. By setting ( AE) = 0 and solving for EB, the lowest achievable energy Ef is obtained:
Er=
~[hUA)(l-2) (6)
In particular, we considered the electric quadrupole transition between the Sij2 (F= 1, rnp 1) and the metastable D s,2 (F= 3, mF= 3 ) state in silver atoms. In this case, because g, = 3 and gs= 1, the gravitational term in eq. (5) cancels, and the final energy Ef becomes E,=
2hti-j&+3&,
(7)
where E, = mvf/2 is the recoil energy. Thus, the minimum of Ef is reached if the laser is detuned as far to the red as possible while still fulfilling the reso204
nance condition then given by
15 March 1994
of eq. (2). The lowest energy Ef is
Ef=13E,+X-4E,J+xIE,,
(8)
where X= Ag F&/2g,2p,,k. With strong trapping field gradients, i.e. small X, the lowest obtainable energy Ef approaches the recoil energy E,.
3. Three-dimensional
Monte Carlo simulations
In order to examine magnetic Sisyphus cooling in three dimensions, we performed Monte Carlo calculations simulating the atoms’ motion in a magnetic trap and their interaction with a cooling laser. Using the 5S,,, to 5D,,, transition in silver and neglecting the effects of the nuclear spin as well as interactions between the atoms, we solved numerically the equations of motion for the atoms in a magnetic field with the specifications as given in ref. [ 111: the superposition of a quadrupole and a bottle field with its axis in the z-direction [ 111. The surfaces of equal potential in the vicinity of the trap center are shaped like a pear with its axis along the z-direction [ 13 1. The atom will only be in resonance with the laser light and can therefore only interact with the laser when its position lies within a certain shell, the interaction or resonance region, the thickness of which is determined by the laser and atomic linewidths and by the magnetic field gradient. In our case, the absorption and stimulated emission rates were chosen to be 100 s- ‘. Spontaneous emission was allowed to take place everywhere in the trap at a rate of 4 s-i given by the lifetime of the silver D,,, state. However, to cool the atoms more efficiently, it proved favorable to introduce in addition a much higher decay rate of 1000 s-’ for atoms outside the interaction shell. This way, when an atom has been excited and then moves out of the interaction zone towards higher magnetic field regions, its chances of undergoing spontaneous emission were greatly enhanced. This increased decay rate allows the cooling process to be considerably faster without significantly altering the final temperatures. In an actual experiment, the same effect can be achieved by using an additional field to drive the Ag transition from the D5,2 to the P3,2 state which then decays to the S,,2 state with a lifetime of 7 ns.
OPTICS COMMUNICATIONS
Volume 106, number 4,5,6
The simulations were started with 100 atoms with random positions and randomly directed velocities of a magnitude of 6 m/s corresponding to a temperature of about 170 mK which can be reasonably obtained by initial cooling of a thermal beam using the S1,2-P3,2 transition. Figure 2 shows the cooling as a function of time in a typical case: The inner radius of the interaction shell was about 5 mm, the laser linewidth constant at 300 MHz. The final temperature was about 30 mK with an approximately gaussian distribution. The cooling times were of the order of a few seconds. In fig. 3, for a constant laser
\
0
I
2
3
4
5
6
7
8
Time [s] Fig. 2. Simulation showing the typical decrease in temperature of the atoms as a function of time.
120 j
0
) 3
2
I Laser
3
4
5
Dehming [GHz]
Fig. 3. Final temperature (left abscissa) and exponential time constant (right abscissa) as a function of laser detuning.
15 March 1994
linewidth of 300 MHz, the dependence of the final temperature on the laser detuning (and thus on the location of the interaction zone) is displayed. For zero detuning, the laser is resonant with the trap center. The coldest temperature was obtained by tuning the laser to be in resonance at the minimum magnetic field. This might at first seem surprising, as atoms nearly at rest at the center of the trap were still interacting with the laser and thus might be heated. However, it turned out that the atoms were not slowed down sufficiently for this re-heating to occur. When the laser detuning was gradually reduced to zero while the laser linewidth was kept constant at 300 MHz, the resultant final temperature was about 15 mK, which is much higher than the recoil temperature (0.65 uK) expected from the above one-dimensional model. The explanation for this discrepancy can be found in the observation that in three dimensions, the cooled atoms exhibit a tendency to orbit around the center of the trap. Particularly with the laser tuned to low B-field regions, the final temperature is usually not reached until all atoms have ended up in stable orbits outside of the resonance region and thus no further cooling is possible. Figure 3 also shows the exponential time constants for cooling. For constant laser linewidth, the closer the laser frequency is tuned to resonance at the trap center, the longer the cooling process takes due to the smaller interaction volume. In order to obtain lower temperatures, it is necessary to prevent the orbiting of the atoms that do not interact with the laser. To achieve this, one has to destroy the conservation of angular momentum fulfilled in the original trapping field near the center by breaking the field symmetry. Experimentally, this is probably done most easily by allowing a static defect of the magnetic field. Our solution which is easier to handle mathematically was to introduce an additional time varying B-field, a quadrupole field in the y-z-plane oscillating sinusoidally at about the orbiting frequency which should kick the atoms out of their stable orbits back into the interaction zone. The amplitude of this disturbing field has to be chosen carefully in order to achieve the desired effect: If the field is too strong, the atoms exhibit heating, whereas too low of an amplitude will not alter the orbiting motion noticeably. It turns out that with a disturbing field amplitude of 1 percent of the trapping field 205
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106, number
4,5,6
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COMMUNICATIONS
strength, a further reduction of temperature from 15 mK to about 3 mK was possible. In order to improve the cooling process, it was necessary to adjust the laser detuning as the atoms were loosing energy. By reducing the laser detuning (shrinking the interaction zone in three steps from an inner radius of 10 mm to 0, then 5 mm to 0, then 2.5 mm to 0), and using a linewidth of 300 MHz, 50 MHz and 5 MHz for the respective steps, it was possible to cool the atoms from 500 mK down to about 600 uK when again most of them were found in stable orbits. In the last two steps, a disturbing field as described in the previous paragraph has also been added. Although obtained for a particular field configuration, these results are not dependent on the actual shape of the magnetic field and should hold similarly for other magnetic traps of comparable strength. In particular, the problem of orbiting will arise in any confining field with approximate rotational symmetry in the vicinity of the trap center. These Monte Carlo simulations show that the described magnetic Sisyphus cooling can achieve sub-mK temperatures, although attention has to be paid to orbiting atoms that will no longer interact with the laser.
4. Conclusion We have shown that in a one-dimensional model for Sisyphus cooling in a magnetic trap, the predicted final temperature of the atoms can approach the recoil limit. Simulations demonstrate that this cooling will work in three dimensions at least down to temperatures in the sub-mK range. However, the limitations of this calculation in determining the linal temperature are given by the problem of the atoms’ orbiting that keeps them from further interacting with the laser. One idea to achieve sub-recoil temperatures is a variation of velocity space optical pumping [ 141 involving both the Doppler shift and the frequency shift due to the trapping field which are approximately of the same size at a few PK. If the laser is appropriately tuned to the red, it will only interact with atoms with velocities directed towards the beam and positions close to the trap center where the magnetic field is small. These atoms will be cooled further, whereas those that already are in the desired
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velocity interval will remain unaffected. Some atoms, however, will be heated in this process and could then be cooled by a second laser. Another approach for achieving low temperatures in a moderately dense vapor is to cool the faster atoms using the described magnetic Sisyphus cooling and to rely on collisions to transfer energy between atoms. These methods may allow the achievement of ultra-low temperatures for spectroscopy, the study of collisions and the observation of quantum-collective phenomena.
Acknowledgements We would like to acknowledge the support of the National Science Foundation, the U.S. Army Research Office and the Deutsche Forschungsgemeinschaft.
References [l A.L. Migdall, J.V. Prodan, W.D. Phillips, T.H. Bergeman and H.J. Metcalf, Phys. Rev. Lett. 54 (1985)
2596.
12 For a review see the special issues of J. Opt. Sot. Am. B on laser cooling and trapping: 2 (1985) and 6 (1989). ]3 1J.L. Hall, M. Zhu and P. Buch, J. Opt. Sot. Am. B 6 (1989) 2194; C. Monroe, H. Robinson, and C. Wieman, Optics Lett. 16 (1991) 50. P.S. Julienne and F.H. Mies, J. Opt. Sot. Am. B 6 (1989) 2257. V. Bagnato, D.E. Pritchard and D. Kleppner, Phys. Rev. A 35 (1987) 4354. K. Helmerson, A. Martin and D.E. Pritchard, J. Opt. Sot. Am.B9 (1992) 1988. I.D. Setija, H.G.C. Werij,O.J. Luiten, M.W. Reynolds, T.W. Hijmans and J.T.M. Walraven, Phys. Rev. Lett. 70 ( 1993) 2257. ]8 1N. Masuhara, J.M. Doyle, J.C. Sandberg, D. Kleppner, T.J. Greytag, H.F. Hess and G.P. Kochanski, Phys. Rev. Lett. 61 (1988) 935. J. Opt. Sot. Am. B 2 19 J. Dalibard and C. Cohen-Tannoudji, (1985) 1707. [lo] P.D. Lett, R.N. Watts, C.J. Westbrook, W.D. Phillips, P.L. Gould and H.J. Metcalf, Phys. Rev. Lett. 61 (1988) 169. [ 1 l] D.E. Pritchard, Phys. Rev. Lett. 51 (1983) 1336. [ 12 ] E. Tiesinga, B.J. Verhaar and H.T.C. Stoof, Phys. Rev. A 47 (1993) 4114. [ 131 T. Bergeman, G. Erez and H.J. Metcalf, Phys. Rev. A 35 (1987) 1535. [ 141 D.E. Pritchard, K. Helmerson, V.S. Bagnato, G.F. Lafyatis, and A.G. Martin, in: Laser Spectroscopy VIII, Persson and Svanberg (1987) p. 68.