Contents 1 Acknowledgments
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2 Introduction 2.1 Experimental laser cooling and quantum optics at the University of Toronto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Overview and definitions . . . . . . . . . . . . . . . . . . . . . . . .
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3 Theoretical background 3.1 Light-Matter interaction forces . . 3.1.1 The light pressure . . . . . 3.1.2 The dipole force . . . . . . 3.2 Laser cooling and magneto-optical 3.2.1 Optical Molasses . . . . . 3.2.2 Magneto-optical Trap . . . 3.3 Optical Lattice . . . . . . . . . .
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4 Experimental Setup 4.1 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Doppler-Free Saturation Absorption Spectroscopy and Repumping beam . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Doppler-Free Laser Polarization Rotation Spectroscopy and Trapping beam . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Lattice beam Specifications . . . . . . . . . . . . . . . . . . . . . 5 Building and testing the 3-D lattice 5.1 Motivations . . . . . . . . . . . . . 5.2 Implementation . . . . . . . . . . . 5.3 Building the apparatus . . . . . . . 5.4 Alignment procedure . . . . . . . . 5.5 Results . . . . . . . . . . . . . . . .
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6 Conclusion
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A Optical lattice band structure
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B The Optical Bloch Equations
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C The optical isolator
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D Atomic cloud profiles
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E Usefull numbers and definitions
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1
List of Figures 2.1
Hyperfine energy levels of 85 Rb . . . . . . . . . . . . . . . . . . . .
3.1
Working scheme of magneto-optical trap . . . . . . . . . . . . . . . 12
4.1 4.2 4.3 4.4
Repumper beam and absorption Trapping beam and polarization Vertical Optical lattice setup . . Cold cloud of Rb 85 falling . . .
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14 16 17 18
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10
Free oscillations combined with pulse oscillations . Pulse Echo oscillation . . . . . . . . . . . . . . . . Extra lattice beams . . . . . . . . . . . . . . . . . Laser diode holder . . . . . . . . . . . . . . . . . Temperature Controller . . . . . . . . . . . . . . . Anamorphic prism and collimator setup . . . . . . Integrated intensity . . . . . . . . . . . . . . . . . Profile of the beam . . . . . . . . . . . . . . . . . The atomic cloud before and after the push . . . The atomic cloud in the assumed-to-be 3D lattice
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20 20 22 23 24 24 25 25 26 27
spectroscopy spectroscopy . . . . . . . . . . . . . . . .
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A.1 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 C.1 Schematic picture of an optical isolator . . . . . . . . . . . . . . . . 33 D.1 Profile of the atomic cloud in the vertical lattice . . . . . . . . . . . 34 D.2 Profile of the atomic cloud in the 3D lattice . . . . . . . . . . . . . 35
2
1
Acknowledgments
This work would not have been possible without the support of many people. I have been in a laboratory for six months and I have to say thank you to the whole lab team for their advice and their help. Especially I want to thank professor Aephraim Steinberg for having me in his quantum optics crew and for giving me such precious insights on quantum effects. He demonstrated a remarkable breadth of knowledge which he was more than happy to share with everyone. I could not have done anything without the very presence of Samansa Maneshi, who spent so many hours explaining to me the experimental setup. She has also been of great help in the installation of the diode lasers and she shown a pronounced patience while doing optical alignments, which really was an extra when in the darkness and with thirty degrees and a bright sun outside. Jalani Fox had his own way of helping me out but he did. Jeff Lundeen and Matt Partlow also deserve my respect for their smart tricks. I should mention Jan Henneberger for being supportive as a graduate fellow. Thanks as well to Krister Shalm for introducing me to swing dance and to Robert Adamson for the soccer game and the delicious fondue. The time spent with Mirco Siercke and Fan Wang has been as enjoyable and relaxing as can be. They have managed to get the stress out of me when needed and I owe them many thanks for that. A french fellow Alpha Ga¨etan took part of the relaxing process as well and it goes without saying that he was of a good help when I could not find my words in english. My last thought goes to Christopher Ellenor. He has been a very good deskmate and a very friendly coffee partner, in other words, a wise person whom it is worth discussing with.
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2
Introduction
2.1
Experimental laser cooling and quantum optics at the University of Toronto
The main research interests of Aephraim Steinberg group are fundamental quantummechanical phenomena, and particularly quantum information processing and the control and characterization of the quantum states of systems ranging from lasercooled atoms to individual photons. The group is a member of the Quantum Optics Group at University of Toronto, of the Center for Quantum Information and Quantum Control, and of IOS, the new Institute for Optical Sciences. A.M. Steinberg is a Fellow in the Quantum Information Processing programme of the Canadian Institute for Advanced Research, and an affiliate member of the Perimeter Institute for Theoretical Physics. Their projects are supported by NSERC, by the DARPA QuIST Program, and by Photonics Research Ontario. They have also enjoyed support from CFI, PREA, CIPI, and ORDCF at various times. Their experimental program is two-pronged, using both nonclassical two-photon interference and laser-cooled atoms to study issues such as quantum information and computation, decoherence and the quantum-classical boundary, tunneling times, weak measurement and retrodiction in quantum mechanics, and the control and characterisation of novel quantum states. Specific projects of interest at the present time include: Characterisation and control of coherence of ultracold atoms Cold atoms trapped in optical potentials can be used to store quantum information in the discrete energy levels of their centre-of-mass oscillations. Groups around the world are studying using such optical lattices for quantum information, and one of the biggest stumbling blocks is likely to be the poorly-understood nature of the loss of quantum coherence. That leads to the following issues: investigating decoherence mechanisms, developing pulse-echo and other error-correcting techniques, implementing quantum control of quantum chaos, studying the possibility of spontaneous coherence induced by tunneling, quantum measurement and quantum information with entangled photon pairs. We have several systems for the generation of pairs of photons with strong quantum correlations, and work on the development of newer, more powerful sources as well. Such pairs have been the workhorse of most tests of quantum nonlocality, and at the heart of many proposals for quantum cryptography and computation. How much does quantum mechanics allow one to conclude from measurements about past (unmeasured) events? Can one particle be in two places at the same time? How can our recently developed ”two-photon switch” be used in quantum-information applications? How can one generate entangled states of 3 or 4 photons, and to what end? How can one most fully characterize the evolution of a two-photon quantum system, and use this characterisation to develop error-correction methods?
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Linear-optics for quantum information Many quantum-information tasks can be carried out using single photons in linear optical interferometers. While not on their own scalable, in the computer-science sense of the term, these systems could be crucial, at the few-qubit level, for nearterm implementations of secure quantum communications links. How well can generalized quantum measurements (”POVMs”) be used to enhance our ability to distinguish between imperfectly distinguishable states? How can this be applied to quantum communications and other interesting tasks? Can linear-optical quantum computing be practically extended to 3 or 4 or 5 qubits, and applied to interesting communications tasks? Bose-Einstein condensation (BEC) Since the first demonstration in 1995 of the Bose-Einstein condensation of trapped alkali atoms below one one-millionth of a degree above absolute zero, dozens of groups worldwide have set up similar systems. They believe that such a revolutionary source of atoms will play a role in future atomic-physics research akin to that played over the past 40 years in optics by the invention of the laser. They are presently completing development of a system for generating BECs of 87Rb in a TOP trap. What truly quantum (beyond-mean-field) effects have observable signatures in the finite-temperature quantum hydrodynamics of this system? Application of this coherent atom source to novel atom-optics experiments, notably to study the question of what happens when you try to observe a tunneling particle in the ”forbidden” region. How long does tunneling take? Where is the particle while it tunnels? What is the impact of measurement or decoherence on these issues?
2.2
The project
Results in quantum optics have made many significant contributions to all scientific fields but particularly in atomic and molecular physics where the laser is the basis of any experiment nowadays. The latest technological advances now make possible experiments that no one could have even thought of ten years ago. One can easily cool and trap atoms and even move them along a determined path using the dipole force. For few years now, the laboratory that has hosted me has been focusing on purely quantum phenoma such as Bose-Einstein condensation. The experiment I have worked on is concerned with the dynamics of quantum systems, see [12] for reference. The principle is simple: Rubidium atoms are pumped into the 52 S1/2 F = 3 state and trapped using combination of laser beams and strong magnetic fields. Then we apply a strong confining potential that can be modulated spatially. The purpose is to be able to control and manipulate the atoms and observe their dynamics. This experiment probes fundamental quantum mechanics. For example, the decoherence which occurs in these sytems is not well understood. The laboratory is using a technique similar to nuclear magnetic resonance to preserve as much coherence as possible. This has been a very hot topic in recent years because preserving coherence is one of the most important issues in the exploding field of quantum computation. The atoms are trapped along the vertical direction. As a result, they are free to move in the two other directions. It has been proposed to build a 3D lattice to verify if this free motion is responsible in part for these
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decoherence processes. This is what will be discussed in the present report after some explanations on the setup.
2.3
Overview and definitions
Experiment timeline Here we give an overall picture of the whole process as it operates in the lab. A vapor of Rb 85 is loaded into a high vacuum chamber by heating up a piece of solid rubidium. The pressure is maintained at 4.10−11 bars. The natural proportion of Rb 85 is about 75% for 25% of Rb 87. It is then easier to trap more atoms if we use Rb 85. In order to clarify some ideas, we introduce some useful expressions. For instance, we will call trapping beams the beams tuned to the F = 3 → F ′ = 4 transition, which is to say the MOT beams, and Repumper beams the beams to the F = 2 → F ′ = 3 transition and the current vertical dipole trap beams are called lattice beams. What is called the lattice itself is the periodical light-induced shift in energy, see [3]. The MOT may refer to the Magneto-Optical Trap as well as to the cold atomic sample itself. The experiment consists in running an automated loop of a unique process. The MOT is on for 2 seconds, then off for few milliseconds during which only the molasses beams are on. The lattice is turned on during the molasses cooling because it increases its loading. Each process is run 3 times with the same parameters. We collect and then average the corresponding data. We make use of Quantum State Tomography to reconstruct the initial and final density matrixes, see [12, 13, 14] for details. Our statistical technique has a fundamental impact on the data as it determines in part their quality, beyond the experimental errors. A sample of such plotted data is shown and explained in the last part. It is related to the Echo experiment which will also be reviewed.
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Measurements overview The temperature of the atoms in the z direction is roughly defined as follows: 1 kB T = hEzKin i 2 where kB is the Boltzmann constant. To measure the temperature of the atomic cloud, we resort to a time-of-flight technique. Once the lattice beams are turned off, the atoms fall down because of gravity. We record the increase of the cloud size after a given time of free expansion. Images are obtained by flashing the atomic cloud with near resonant pulse beams (molasses and repumper beams in our case). The comparison of the fluorescence images obtained with and without the turningoff allows us to reconstruct the evolution of the momentum distribution related to the temperature as specified above. The relative number of atoms in vibrational states of the optical lattice is determined using fluorescence images profile.
Figure 2.1: Hyperfine energy levels of 85 Rb
The states we are concerned with are shown in the diagram (2.1). The hyperfine structure of Rubidium atoms has been studied extensively and a satisfying treatment can be found on numerous websites 1 .
1
http://mxp.physics.umn.edu/s05/Projects/S05Rb/theory.htm
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3
Theoretical background
Cooling atoms by using their interaction with an electromagnetic field flows directly out of fundamental conservation principles. Momentum conservation during emission or absorption of a photon plays an essential role in every laser-cooling scheme. The quantum mechanical picture of an atom usually separates the internal electronic states from the external degrees of freedom. Laser coolig is obtained when an atom, in a gaz, loses its translational energy (associated with the motion of center of mass) because of repeated transitions between two internal states. The next section is devoted to a brief review of light-matter interaction processes involved in laser cooling and trapping of neutral atoms.
3.1
Light-Matter interaction forces
It is well known since the early begining of quantum physics that atoms have discret energy levels and that they can absorb and emit photons to jump, thanks to Einstein’s intuition, from one state to another. Through out the developement of quantum physics and its promising implementations as the laser appeared to be, the scientists started to ask themselves if it would be possible to cool and even trap atoms in order to study them. They went through different ideas about cooling but they finally ended up with the current laser cooling theory. We will consider the simplest interacting system for exerting optical forces on atoms, which is to say a single frequency light field interacting with a twolevel atom at rest within the low-saturation scheme. It will be treated in a semiclassical way, the fully detailed description can be found in [1, 2, 3, 4]. From the correspondence principle and following Ehrenfest theorem, we find that a force F on an atom is the expectation value of the quantum mechanical force operator F. F = hFi =
d hpi . dt
(3.1)
The time evolution of the expectation value of the operator p is commonly given by i d hpi = h[H, p]i (3.2) dt ~ The operator p can been replaced by −i~ (∂/∂z), which yields to the final expression of a force on an atom: ∂H F =− (3.3) ∂z If we now introduce the interaction hamiltonian H(t) = −e~ε(~r, t) · ~r , where ~ε is the electric field, into (3.3) and we make use of the dipole approximation which allows us the interchange of the gradient with the expectation value, the force simply becomes: ∂ ∂ (~ε(~r, t) · ~r) = e h~ε(~r, t) · ~ri . (3.4) F = hFi = e ∂z ∂z 8
The dipole approximation is justified by the fact that the spatial variation of the electric field occurs over a wavelength scale, let say 500nm and that the ”size” of an atom is of the order of the angstr¨om which leads to a negligible ratio of 10−3 . It will prove useful to define the Rabi frequency as: Ω ≡ − eε~0 he|r|gi. If one goes through the whole mathematics, one would find the following expression for the force: ∂Ω ∗ ∂Ω∗ F =~ ρ + ρef (3.5) ∂z ef ∂z where ρef is the coherence between the ground and excited states. It is instructive at that point to write the derivative of Ω as follows: ∂Ω = (qr + iqi ) Ω ∂z Which, introduced in equation (3.5), leads to the general equation: F = ~qr (Ωρ∗eg + Ω∗ ρeg ) + i~qi (Ωρ∗eg − Ω∗ ρeg )
(3.6)
It should be mentioned that the force remains real and therefore the first term is proportional to the real part of Ωρ∗eg whereas the second is proportional to the imaginary part. From this equation, we should now focus on the two relevant cases for the electric field which is to say the case of a traveling wave and the case of a standing wave. The difference lies in amplitude and phase gradient of the electric field. This will give rise to the two fundamental forces which are briefly reviewed below.
3.1.1
The light pressure
Subtituing the solution for ρeg of the Optical Bloch Equations (See Appendix B), we find the following expression for the force: 1 ~s −δqr + γqi (3.7) F = 1+s 2 where s =
s0 1+(2δ/γ)2
is the saturation parameter with:
Ω the Rabbi frequency. s0 =
2|Ω|2 γ2
=
I Is
the on-resonance saturation parameter.
Is ≡ πhc/3λ3 τ is the saturation intensity. γ the decay rate from the excited state to the ground state. δ = ωl − ω0 the detuning from the atomic resonance frequency ω0 (ωl being the frequency of the laser). Let us now consider the case of a wave traveling along z. The electric field can be written as E0 i(kz−wt) E(z) = e + c.c . 2 The calculations then show(using the Rotating Wave Approximation) qr = 0 and qi = k 9
which leads to the final and significant expression for the force acting on the atom at rest: ~ks0 γ/2 Fsp = (3.8) 1 + s0 + (2δ/γ)2 It is relevant to consider the case of zero detuning ωl = ω0 to understand the nature of the force. Equation (3.8) thus becomes Fsp =
~ks0 γ/2 1 + s0
This corresponds to the absorption of a photon of momentum ~k times the scattering rate γp = γρee . The force saturates at large intensity as a result of the factor s0 in the denominator, its maximum value being Fmax = ~kγ/2 as M ax(ρee ) = 1/2. This force is called the light pressure, the scattering force or the dissipative force. This is the force used in the optical molasses to cool down atomic clouds.
3.1.2
The dipole force
Now let us consider the case of a standing wave, which is to say the superposition of two counterpropagating traveling waves. The electric field can be written as: E(z, t) = E0 cos(kz) e−iωt) + c.c. (3.9)
From resolving the OBE it appears that the energy levels of an atom interacting with an electromagnetic field are shifted. This shift is called the light shift and is usually proportional to the intensity gradient of the field. It gives rise to a new kind of force that is used to trap neutral atoms. The derivative of Ω turns out to be qr Ω with qr = −k tan(kz). We can then derive the expression for the force: Fdip =
2~δks0 sin(2kz) 1 + 4s0 cos2 (kz) + (2δ/γ)2
(3.10)
For δ < 0 the force drives the atoms to positions where the intensity has a maximum whereas the atoms are attracted to the minima of the intensity for δ > 0. This force is called the dipole force, it is a reactive force as it arises from the redistibution of the photons from one laser to another. It is the basis of optical trapping of neutral atoms.
3.2 3.2.1
Laser cooling and magneto-optical trap Optical Molasses
The generalization to an atom in motion in a traveling wave yields to the following expression: Fdissip = F0 − βv (3.11) with: β = −~k 2
4s0 (δ/γ) (1 + s0 + (2δ/γ)2 )2
It is important to note that such a dissipative force would reduce the mean velocity of an atomic sample for red detuned light, e.g. for δ < 0. In order to 10
cool, in fact, we have to be capable of narrowing the velocity distribution instead of just changing its mean value. The first scheme proposed to obtain this effect is called ”Doppler Cooling”. It has its origin in the Doppler effect: when an atom is moving towards a laser beam with a velocity v, it sees an effective frequency νef f = νL (1 ± v/c). When its velocity is such that it matches with the detuning of the laser, the atom absorbs a photon and therefore gains a recoil momentum ~k in the opposite direction. It spontaneously (28ns) reemits a photon in a random direction. We need to accumulate 105 basic processes to achieve an observationally significant change in the atomic velocity as the recoil momentum is small compared to the momentum of the atom. Note that the friction coefficient β depends on the detuning and the intensity of the beam through s0 . It takes its maximum value for δ = −γ/2 and s0 = 2. If one decides to narrow the z velocity distribution, one can focus two counterpropagating laser beams along te z direction with red detuned frequency. In this situation, the atoms experience a net radiation pressure against their motion. The scattering force is : Fsc = −~k 2 v
8s0 (δ/γ) (1 + (2δ/γ)2 )2
The atoms with zero velocity along the z direction are subject to a force that averages to zero because the radiation pressures of the two beams exactly compensate. In our case, we send three pairs of such beams oriented along the Cartesian axes, which make possible a three-dimensional (3D) cooling of the atomic cloud. However, the radiation force can not compress without limit the velocity distribution and hence it can only cool the sample to a minimum mean value (Doppler limit, see [7]). This has to do with the discretness of the momentum change induced by photon absorption. Such a system is usually called ”Opical Molasses” because of the presence of a viscous force that slows down the atoms without the presence of a restoring force that could gather them. We use a slightly different scheme called Corkscrew Molasses which allows us to cool below the Doppler limit. It is important to note that this kind of cooling process produces an atomic sample that is cold but not necessarily dense.
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3.2.2
Magneto-optical Trap
Figure 3.1: Working scheme of magneto-optical trap To combine the process of cooling and trapping, we simultaneously make use of magnetic fields and laser beams. When the atoms experience a magnetic field, their energy levels are split. This Zeeman splitting is the basis of the MOT. The diagram of the hyperfine levels of rubidium 85 is given in the overview. To understand the principle, it is instructive to consider a fictitious atome of zero spin on a J = 0 → J ′ = 1 transition placed in a region where a static magnetic field B = bz is present(Anti-Helmholtz configuration). Two red detuned beams counterpropagate along the z axis with a σ + and σ − polarization. An atom at rest at z = 0 will not move as the radiation forces compensate at this point. Consider now an atom in the ground state at z > 0, the detuning for the transition |g 0i → |e − 1i is then decreased while the detuning for the transition |g 0i → |e + 1i is raised, which means the atom is more sensitive to the σ − polarized beam and less sensitive to the σ + beam. This arises from the selection rules, see [1]. In other words, the atom is more sensitive to the beam that pushes it towards the origin. Needless to say that the same atom in the negative z region will be pushed towards the origin. We hence do have a restoring force which allows us to confine the atoms in the region where the field is near zero. This is called a Magneto-Optical Trap (MOT). In our case, we cool around 109 atoms to 10µK.
3.3
Optical Lattice
Recall equation (3.10), as the force is conservative it derives from a potential F = −∇Udip which is given by: 1 1 + 4s0 cos2 (kz) + (2δ/γ)2 Udip = ~δ ln (3.12) 2 1 + (2δ/γ)2 With our actual experimental parameters I ≈ 103 Is and δ ≈ 25GHz for a natural linewidth γ of 5.98 · 2π Mhz -the ratio ξ = (2δ/γ)2 is more than 106 - we are in the situation of a low saturation parameter s 0, the optical potential minima correspond to the nodes of the standing wave and the atoms can be trapped in sites with negligible photon-scattering rate.
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4
Experimental Setup
4.1
Spectroscopy
4.1.1
Doppler-Free Saturation Absorption Spectroscopy and Repumping beam
One has to recall that our first goal is to maximize the cooling of an atomic sample of Rubidium and hence maximize the number of fluorescence cyle F = 3 → F ′ = 4 → F = 3. We use this transition because the coupling coefficient is the stongest for a σ+ polarized light. It appears that the coupling for the transition F ′ = 3 → F = 2 is also strong. That is why we make use of a repumping beam. To find the resonant frequency we proceed through Doppler-Free Saturation Absorption Spectroscopy. We will briefly explain the principle and the setup, more details may be found in [8]. Our pimary light source is a commercial gratingstabilized laser diode (or ECL, which stands for External Cavity Laser), which frequency can be tuned by varying the applied voltage and the grating orientation. It emits photons at 780.03 nm according to the manufacturer and the bandwidth is very narrow with about 50 kHz.
Figure 4.1: Repumper beam and absorption spectroscopy 14
The beam passes through two optical isolators1 , it hits a thin piece of glass that sends few per cent to the spectroscopy apparatus while the rest is directed towards a slave laser diode in order to injection lock it to the desired frequency. The deflected beam is split into three others, two of them will play the role of probe beams while the third (and strongest) will be the ”pump” beam. The latter and one of the probe beams counterpropagate through the vapor cell. While the ”pump” beam has a high intensity and serves to bleach the atomic gas, i.e. to make the gas transparent, the transmittance of the ”probe” beam through the atomic vapour is recorded with a photodiode. We record the signal of the other probe the same way and subtract it from the other. In that case, we get rid of the Doppler Broadening. Once the frequency is tuned to resonance, we lock it electronically so that we can work a whole afternoon with no real need to fix it manually every time we run an experiment. The power provided by the ECL being too weak, we do injection lock a common laser diode which delivers almost 50 mW, enough to pump a large number of atoms into the desired F = 3 state.
4.1.2
Doppler-Free Laser Polarization Rotation Spectroscopy and Trapping beam
The apparatus The setup is shown in figure (4.2). The principle is based on a polarization change of one probe beam, see [9] for more details. As in the Saturation-Absorption scheme, we use a second probe beam to weaken the Doppler broadening. This time, the pumping beam is σ + polarized using a quarter wave plate. The two probe beams are vertically polarized and their amplitude is weak enough to break down their linear polarization into a superposition of σ + and σ − polarization components. It is known from the selection rules that the coupling between an electromagnetic wave and an alkali depends on the polarization of the electric field. Without any pumping beam, the absorption for the σ + component would approximately be the same as for the σ − component as a result of a random population distribution. The saturating or polarizing beam will therefore induce a change in absorption ∆α+ and ∆α− , as a consequence it will also change the refractive index ∆n+ and ∆n− . As the speed of light changes for different refractive indeces, the σ + and σ − components of the concerned probe beam will acquire different phases. This will make the axis of polarization to rotate. We do not have to worry about the hyperfine sublevels as we do only focus on the F = 3 → F ′ = 4 transition. The probe beams are collected after a polarizing beam splitter that plays the role of a polarizer. Since probe beams are vertically polarized both will be reflected by the PBS if there is no rotation in polarization. When on resonance the σ+ component of the probe beam couterpropagating the pump sees almost free space (n+ = 1) and the σ− component sees n− which makes it to propagate at a different speed, therefore lagging behing the σ+ component and subsequently leading to polarization rotation. Because of the polarizer, the intensity recorded by the photodiode depends on the axis of polarization. It even drops to zero when on resonance. The calculations show that the resolution is higher than usual saturation-absorption spectroscopy.
1
See Appendix C for details
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Figure 4.2: Trapping beam and polarization spectroscopy
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The trapping beam This beam is first detuned 140 MHz by double passing it through an AOM and then locked to the F = 3 → F ′ = 4 transition. The mechanism will be briefly reviewed in the Lattice beam section. Then the beam passes through a MOPA (Master Oscillator Power Amplifier) which amplifies it to a maximum of 400 mW. The spot size is then reduced using a telescope in order to maximize the transmission through an optical isolator. Afterwards, it is returned close to resonance with a red detuning of 2 linewidths ∆ = 12 ∗ 2π MHz. Another telescope that is not drawn expands the beam which is then combined with the repumper beam using a polarizing beam splitter. The combined beams finally go to an upper table where they are split as equally as possible in the 3 dimensions. A set of irises and quarter wave plates ensures the right polarization and alignment of the beams.
4.2
Lattice beam Specifications
Figure 4.3: Vertical Optical lattice setup We formed the vertical lattice by intersecting two vertically polarized beams at an angle of θ = 49.6˚ overlapping the MOT. The resulting lattice vector is kL = λ2πL sin( 2θ ) = πa = 3.38 · 106 m−1 . λL = 780nm is the wavelength of the lattice light and a = 0.93µm is the spatial period of the lattice. As for the trapping beam (MOT beam), the primary source of light is a commercial ECL which is amplified by a MOPA. The output beam is linearly polarized at 45˚. The horizontal and vertical components are then split using a PBS and each of them passes through an AOM driven at 70MHz. They are then recombined, and spatially filtered to obtain a smooth gaussian profile. That is to say we have flat wavefronts, which ensures a good contrast of the fringes. They are then directed to the upper table where they are split again. A half wave plate rotates the polarization of the previously horizontaly polarized beam by 90˚. By controlling the frequency and phase of the signal with which we drive each AOM independently we are able to control 17
the position, velocity and acceleration of the lattice. We can shift the relative phase of the lattice beams using a home-made electronic circuit controlling the RF signal through the AOMs. This results in a spatial shift of the lattice. As it has been said in section (3.3), our lattice has been designed to carry from 1 to 3 motional states. The theory shows that displacing the lattice sharply excites the atoms into a higher motional state. We can therefore prepare the atoms in a certain state within experimental limits (imperfections of the excitation pulse). The lattice is created along the z axis, such that gravity also plays a role. It actually induces an undesirable tilt in the optical potential. However, gravity helps us in the preparation of ground state. Indeed, we proceed as follows: we softly lower the intensity of the lattice beams until one bound state remains, the higher states then become unbound and fall out of the interaction region. Afterwards we raise the intensity back to its original level but are left with the ground state in each well, with a typical contamination of the first excited state of about 5 − 15%. We can see on the figure (4.4) the atomic cloud trapped in the lattice(small spot) and the MOT (the big bright spot) falling down.
Figure 4.4: Cold cloud of Rb 85 falling
18
5 Building and testing the 3-D lattice 5.1
Motivations
As it has been seen above, the atoms are cooled in all three dimensions. However, they are only trapped by the current lattice in the vertical direction. They are consequently free to move in the other directions. This causes significant loss while running the experiments because of the abscence of a restoring force along the x and y axis. Even if it does not prevent relevant results, it is always a problem one has to take into account while doing his analysis. Hence the idea of building a three dimensional lattice that could limit loss and decoherence. To get a better understanding, we present the Echo experiment. Our system can be mapped onto a two-level system of eigenstates |0i (ground motional state) and |1i (first excited motional state). Let us first consider the case of free oscillations. We prepare atoms in the ground state |0i. We then shift the lattice by 60˚ ( equivalent to 1/6 of the lattice spacing a = 0.93µm ) which has the effect of coupling the two levels. This is mathematically represented by the displacement ˆ operator D(α) |0i = a(α) |0i + b(α) |1i + ǫ where ǫ represents any loss. We let the hE −E i system evolve for a time t ( U (t) = exp(−i · ω · t) where ω = |1i ~ |0i is the mean frequency difference between ground and first excited state). We then shift back ˆ −1 (α = 60˚)) 1 . the lattice to its original position (D We finally project onto the ground state. The formula (5.1) thus describes the evolution of the ground state population as a function of time. E 2 D ˆ −1 ˆ (5.1) P0 = 0 D (α)U (t)D(α) 0 We repeat the operation for successive times and we built the consequent blue graph in figure(5.1), extracted from a presentation of Samansa Maneshi. The first 800 ms are the free oscillations. The latter can be fitted to a sine with a nearly gaussian envelop. The second part, from 800 ms to the end represents the echo. We focus on the first part. If the system was free of loss and decoherence processes, one should observe infinite oscillations. However, we do see a sharp gaussian decay. The transverse Gaussian profile of the lattice beams are mostly responsible for the dephasing. This profile induces spatial inhomogeneities in the depth of the wells E −E which will cause the frequency ω = |1i ~ |0i to change from well to well. As we image over a stripe of width ∆x, we measure average oscillation frequency of all atoms within this stripe and observe decay of oscillation as a result of Gaussian distribution of frequencies. However, this does not take part in the decoherence processes. As we discuss below, there is still some coherence but we just can not observe it because of the predominant dephasing. 1
α in general is a complex number; to a good approximation we can consider it to be real and the real part is related to the spatial amount we shift the lattice which in itself is related to what we refer to in the lab as for example 60-degree.
19
0,8
0,6
P
0
0,4
0,2
0,0 0
500
1000
1500
2000
2500
t (ms)
Figure 5.1: Free oscillations combined with pulse oscillations Now let’s consider the pulse-echo sequence (5.2) (from 800 ms to the end). When the oscillations have died out, we perform another ”pulse” (shift, wait and shift back). At time τ we apply a pulse we see a drop in the average number of atoms in the ground state, which is due to the pulse imperfections that causes loss of atoms and creates new coherence. The new coherence created by this pulse decays on the same time scale as the original coherence, but we observe an echo signal which is a revival of the original oscillations at time 2τ . These results and
Figure 5.2: Pulse Echo oscillation theirs explaination are similar to the ones of the Spin-Echo experiments suggested in the 50’s by E.L. Hahn 2 .The reviving signal is centered at t = 1600µs = 2τ and has a smaller amplitude -which witnesses a loss in coherence- because atoms have had the time to escape the wells. If we run the experiment with the same parameters but apply the echo pulse at a time τ + dτ , we will observe the revival oscillations after a time t + dt but with a smaller amplitude and so on until it would totally disappear. There are hence two processes that induce decreasing of the peaks: the imperfections of the pulse and the decoherence. We can not avoid the loss because our pulses are imperfect but we could preserve coherence. If we assume the transverse motion of the atoms to be one source of decoherence in 2
Spin Echoes, E.L. Hahn, Physical Review Letter Vol 80, #4, november 15, 1950
20
our system (Inter-Well tunneling could be another), submitting the atoms to a 3D lattice would have the effect of preserving coherence over a longer timescale. This means that no matter the time we apply the pulse(with reasonable assumptions concerning the lifetime of the lattice), even if the reviving signal is weak, we should be able to observe it. Nonetheless we do not have a clear idea of how this free motion of the atoms affects the decoherence. The imlementation of a 3 D lattice will help understanding the phenomena and hopefully it will provide quantitative clues. It would in any case ensure a better confining in the x and y direction and hence prevent the atoms to move in those directions. Much theoretical and experimental studies have been performed in the 90’s and consequent descriptions can be found in the scientific litterature, see [3, 5, 6]. Without going into details, we would just say that among the different configurations we have been through, we have selected the simplest. The two additional lattices will be formed by counterpropagating beams of the same polarization in the x0y plane at nearly right angle. The right angle could not be achieved because of geometrical constraints. They will be blue detuned δ ≈ 23 GHz with respect to the F = 3 → F ′ = 4 transition. In this scheme, the atoms will experience minima and maxima of potential in all three directions, redistributing them in space. We will use slightly different frequencies for the laser fields in order to time average any residual interferences between different standing waves.
5.2
Implementation
We use laser diodes as our light sources because one can tune their frequency by changing their temperature [10] and the current going through by ±10 nm and because they are affordable. We rotate an AR coated collimating lens so as to make the beam as focused as can be. The beam shape is asymmetrical, showing an elliptical pattern, z being the great axis. We therefore use a set of anamorphic prisms to squeeze the vertical expansion of the beam and make it as rounder as possible. The whole diagram is shown in figure (5.3). An optical isolator (based on Faraday rotation) is required in both cases as the reflected beams will match the ingoing ones. [10] tell us how sensitive a laser diode can be and how annoying it can be. The telescope is only here to reduce the size of the beam while it goes through the Faraday rotator. It ensures maximum transmission. The room available on the upper table imposes some restrictions on us. This includes setting up a retro-reflected beam. Several drawbacks appear for this kind of implementation. First of all, it makes the soverlap of the incoming and reflected beams a bit trickier than usual. Then, the intensity of the reflected beam is lower than the one of the incoming beam because it has to cross six extra glass walls, which, even if they are Anti-Reflection coated, induce loss. They may also cause an undesirable change in polarization. We have mounted several irises along both paths to make alignments easier. We have placed a 50/50 PBS after the iris to get rid of any vertical component. The other horizontal lattice is formed by sending two counterpropagating beams coming out of the same laser diode. The same precautions have been taken to maximize the intensity and roundness of the beam. One has to recall that after passing the optical isolator the light is linearly polarized at an angle of 45˚. That is the reason why we have mounted a half wave plate on the retro-reflected beam path. In the other case, the components are split according to their axis of polarization. We do then rotate the vertical one to the horizontal position. This configuration present a lot of advantages. Both beams 21
Figure 5.3: Extra lattice beams are controlled independently, making the alignment more accurate. We can also fix their intensity properly.
5.3
Building the apparatus
Laser diodes are very sensitive to various kinds of parameters [10, 11]. Among them, temperature and current have decisive impacts. They may contribute to change in frequency and intensity. Indeed, the frequency of the emitted light is determined by the manufactured semiconductor materials and the dimensions of the cavity. Layers of different material are stamped together. This yields an effective periodic potential. An electron in such a potential has a resolved energy band structure. When one applies a current in the direction perpendicular to the sheets, it forces the electrons to cross the different layers. When the voltage is high enough, the electrons may jump from one energy band to another, releasing one or more photons. The bandgap is what determines the frequency of the emitted light. It is then easily conceivable that a small change in the layer spacing will result in changing the frequency. Beyond the current threshold, one would think that the 22
Figure 5.4: Laser diode holder frequency does not depend on the current. However, the dimensions of the resonant cavity also play a role. Indeed, the cavity length is in part responsible for the gain of the emitted light. Changing its value will modify the gain for a particular frequency. The flow of electrons also modifies the refractive index and effective lenght of the cavity. At this scale (µm), a small variation of the temperature also distorts the dimension significantly. Those two considerations lead us to build a special enclosure for the laser diode and an appropriate temperature controller. The laser diode is first installed in a commercial cylindrical tube which includes the collimating lens. This same tube is inserted into an aluminum cube which presents good thermal conductivity, see figure (5.4). We then glue this cube to a peltier plate using thermal epoxy to ensure a good thermal contact. A thermister is inserted into the cube through a small drilled hole. The whole assembly is glued to a thin aluminum sheet, which is screwed to an optical support. The post is taken to be 2 inches to limit the height and hence the amplitude of stray vibrations. All electrical wires are soldered to serial ports making it easy to plug and unplug. A final plastic cap covers the whole assembly, preventing residual air streams to interfere. The peltier plate is combined with a commercial temperature controller (5.5) to maintain a constant temperature. The purpose is to minimize fluctuations in frequency. We monitor the actual temperature Tact of the apparatus by monitoring the voltage across the thermistor. A set screw enables us to set this voltage to the desired value. This set point is monitored through the control channel Tset . A feedback loop ensures that the peltier plate is supplied in a way which tends to match the set and the actual temperatures. By this means the temperature of the aluminum cube can be modulated from 17 to 24˚C with a typical error of 10−3 . This is a quite satisfying result knowing that a change of 1˚C shifts the frequency by almost 1 nm. The stability of the temperature is fixed by the gain of the integrator circuit. If the gain is too high, the device will respond faster than the material and one will observe undesirable oscillations. Only running the apparatus 23
Figure 5.5: Temperature Controller will prove the best tuning. In our case, the temperature remains stable over a very long time if the outer environment is not submitted to dramatic variations.
5.4
Alignment procedure
We first make sure that the beam coming out of the laser diodes is well collimated. That means that we look at the spot size after a long distance and adjust the position of a lens. We then set up the anamorphic prisms. We verify the roundness by eye putting a lens with a small focal length afterwards and looking at the spot on a white paper sheet. If the prisms are well aligned, the big spot should be round. This is just a rough method but precise enough for what is required.
Figure 5.6: Anamorphic prism and collimator setup Aligning the optical isolator is difficult. Before doing so, we make sure the beam goes as parallel as possible to the optical table. Afterwards, we install the two lenses forming a telescope with a ratio of 1/2. One of the lenses is mounted on a translational stage to have more control on the output beam collimation. We then insert the optical isolator. It is roughly done by changing its height and orientation and by looking at the outgoing light. In order to be accurate, we monitor the power coming out of the optical isolator. We remove the two polarizers and move the apparatus so as to maximize the power. We then replace the front polarizer and turn it until the power is maximized. We do the same thing with 24
the back polarizer. This procedure ensures a maximum transmission but does not guarantee good extinction. For this, we turn back the optical isolator, scale the power probe to µW and very softly turn the back polarizer to minimize the transmited light power. In both cases, we can reach an extinction ratio of 40dB, which corresponds to the maximum value given by the manufacturer.
50
Power (mW)
40
30
20
10
0
0
1
2
3
4
5
x (mm)
Figure 5.7: Integrated intensity
Signal proportionnal to the intensity
50
40
30
20
10
0
0
1
2
3
4
5
x (mm)
Figure 5.8: Profile of the beam At this point, we measure the profile of the beam so as to have an idea of the intensity and hence the depth of the optical potential. We perform this measurement by monitoring the power of the incident light while slowly moving a razor blade across the beam. By doing so, we integrate the power across the profile(if we assume the beam to be radially symmetric). We obtain the intensity profile by deriving the latter integral. The width is compared to that of a fitted Gaussian profile. The result for the retro reflected beam is shown below. The error bars are too small to appear on the plot. 25
Pmax = 6.7 · 10−3 mW ⇒ w = x2 (6.7 · 10−3 ) − x1 (6.7 · 10−3 ) ≈ 1.8mm e2 which gives the average intensity of the beam I≈
Ptot ≈ 19.6 mW/mm2 π(w/2)2
where we assume all the power to be within the width w. This yields an optical potential of depth U0 ≈ 103 ER where we define the recoil 2 2 2 energy ER = ~2Mk = 2Mh λ2 with λ = 780nm. It is a strong confining potential. A set of mirrors leads the beams to the upper table. Much precaution is taken to ensure a 45˚ angle between the incoming beam and the mirrors. This makes the setup look clear and reduce the polarizing effects induced by the reflection on the mirrors. The vertical alignment requires a lot of attention. Geometrical considerations on the actual setup give us the heights at which we have to fix our mirrors. The additional beams are then focused to the assumed-to-be region of interference of the lattice beams. We roughly align to overlap them but we have to perform a fine alignment to make sure we form a standing wave. For this purpose, we tune the frequency to near resonance in the red detuned regime. If the beams hit the atoms, they will be pushed out of the optical trap. The procedure is different for the two cases but the measurement is realized the same way. We continually run the trapping process so that we can observe the push. It has to be emphazised that the two extra lattice beams are not turned on at the same time. For the case of the retro-reflected beam, we proceed as follows: we cover the beam after the glass cell and we adjust the mirror until the bright spot representing the trapped atoms has disappeared. We then uncover it and tune the last mirror to overlap the incoming and reflected beams. For the counterpropagating beams, one of the beams is covered while we adjust the other. Same thing, we tune the mirror until the bright spot disappears. We repeat the same process for the other beam.
Figure 5.9: The atomic cloud before and after the push We want to stress the difficulty of this procedure. For the retro-reflected beam, the superimposition is easy but maintaining the beam in the horizontal position is not because of the spatial constraints. For the other lattice beams, the height is always the same but it is harder to superimpose them. One has to recall that this setting happens on a raised table and so we have to stand on the optical table and try to avoid any contact with other optical components. The pictures (5.9) represent the atomic sample in the lattice before and after alignment of the 26
retro-reflected beam. Those pictures have been taken on a ”bad” day for the MOT and we have had to enhance the contrast to make them interpretable. The second figure clearly shows a dark region in the center, which disappears when we cover the incoming beam. We turn the knobs until the dark spot is centered and ”maximized”. We are then certain that the beam hits the cloud in the middle, the region from which we extract the relevant datas.
5.5
Results
Few results will be presented here as a lot of problems has disturbed the apparatus. We will define along the following lines what would be a good 3D lattice. To begin with, let us describe the following picture. It represents the atomic cloud trapped in the 3D potential with a detuning of (∆ = 30 GHz) and an intensity of I = 19.6 mW/mm2 for both extra lattice beams. The first number corresponds to the height of the cloud, the second describes the brightness of the brighter pixel picked in the shiniest stripe on a range of [0, 254] and the third number corresponds to the brighter pixel from the top of the picture.
Figure 5.10: The atomic cloud in the assumed-to-be 3D lattice We clearly see a bright stripe in the middle of the cloud and darker sides. The fluorescence profile given in figure (D.2) confirms that oservation. If we compare with a profile of an atomic cloud in the 1D lattice (5.9 on the left and D.1), one would notice that the distribution along the y axis is narrower in the first case, tending to prove that the atoms are more localized in the 3D lattice. This narrowing is surely a result of the dipole trap induced by the laser beams counterpropagating in the y direction. 27
Let us describe what is happening physically in both cases. In the case of the 1D vertical lattice, the atoms are trapped along the z axis but they are free to move in the transverse direction. The intensity of the beams being small on the sides (gaussian decay of the waist), the potential the atoms experience is not as deep as in the center, where the intensity is high, and their kinetic energy is big enough for them to escape. In the 3D case, the atoms experience a restoring force along the 3 directions and they can not escape as easily as in the first case. That is why the shape of the atomic cloud seems rounder in the 3D lattice. Actually, if our configuration was perfect, meaning all the beams of same intensity, perfectly aligned, etc...the atomic cloud shoud take the form of a dense and bright sphere surrounded with a sort of faint halo corresponding to hotter atoms escaping the trap. But in our case, the lattice vector (ky ≈ 8.106 m−1 ) along the y axis corresponds to a smaller lattice spacing λ = 0.78µm and the intensity is different for each lattice beam. The spot sizes are not even the same. This is probably what causes the warm halo surrounding the brighter part of the cloud. At this point, we realized that it was necessary to use a second camera watching the horizontal plane to show confinement along the x axis. What first came to mind was to set up this additional camera directly along the z axis but this appeared to be inappropriate as this camera recorded only the fluorescence of the MOT falling down and covering the one of interest, the one of the trapped atoms. Unfortunately, the spatial constraints limited the room where the camera could have been placed and we were left with one unique camera watching the z0y plane. We can also notice that the fluorescence intensity recorded for the 3D lattice is weaker than the one of a 1D lattice, showing a smaller number of trapped atoms. We could not find any good explanation for this effect. But this has a significant effect on our data. Indeed, as we collect data in a thin stripe cut in the middle of the cloud, we expect the ratio signal to noise to be as high as possible and it goes without saying that the smaller number of atoms the bigger the errors on the average. In other words, we would like the atomic cloud to be as dense and cold as possible. It is needless to say that everytime the experiment was run, we made sure the MOT was dense and cold as can be acheived with our setup. This ensured we were focusing on the trapping process itself. However, the ratio signal to noise was too small in our case to infer a true trapping in all 3 directions from this observation. We also observe speckle light resulting from the reflections of the beams on the cuvette. Those stray beams do not interfer with the trapping process but they introduce noise in the imaging process. We overcame this issue by implementing electronical shutters. This recquired a lot of work because we were not sure of the vibrations and we had to build appropriate electronic circuits to drive them fast enough to close for the time of imaging. It has been impossible to perform more measurement at this point because the optical feedback to the diode for the retro-reflected beam induced instability in frequency. Many tries have been given to ensure a better extinction rate but nothing conclusive came out and no time was left to consider another setup. Apart from all these testing methods, it remains to run the Echo experiment to complete the characterization of the 3D lattice.
28
6
Conclusion
Our objectives were to implement two extra lattice beams in order to produce an optical trap in all three dimensions. By considering the geometrical constraints on our actual setup, we decided to make the x-oriented beam to reflect on itself with a horizontal polarization and make counterpropagate two other horizontally polarized beams along the y axis. For this purpose, we chose the use of low-cost laser diodes. We have set up two temperature controlers that ensure thermal stability of the two devices. We have managed to keep these two diodes stable in frequency with a detuning of δ ≈ 25 − 30GHz to the F = 3 → F ′ = 4 transition and in intensity with almost 20 mW/mm2 before doing any alignment. Special attention has been paid on the quality of the beams. We have tried to make them as round and bright as possible so as to form clean standing waves. We have established their profile using a razor blade technique, from which we have deduced the potential depth of the trap (around 1000 recoil energies). Unfortunately, it was hard to align correctly the beams and when doing so, the feedback made our diodes jump from the desired frequency to another. We had a hard time maximizing the extinction and transmission rates at the same time. We barely had a chance to take pictures of the supposed 3D lattice. When we finally managed to do so, there was no time left to improve alignments and achieve a correct caracterization of the lattice in three dimensions.
29
Bibliography [1] Claude Cohen-Tannoudji, Bernard Diu & Franck Lalo¨e, Quantum Mechanics [2] Claude Cohen-Tannoudji, Cours au Coll`ege de France, ann´ees 1982-1983, 1983-1984 [3] L. Guidoni & P. Verkerk, Optical lattices: cold atoms ordered by light, J. Optics B: Quantum Semiclass. Opt. 1, 1999 [4] Harold J. Metcalf & Peter van der Straten, Laser Cooling and Trapping, Springer Publishing [5] P.S. Jessen & I.H. Deutsch, Optical Lattices, Advances in Atomic, Molecular and Optical Physics 37 [6] Markus Greiner, Ultracold quantum gases in three-dimensional optical lattice potentials, Thesis Ludwig-Maximilians-Universitt Mnchen [7] C. Cohen-Tannoudji, Laser cooling and trapping of neutral atoms: theory, Phys. Rep., 219, 153-164, 1992 [8] Dr. Kevin Wagner, Doppler-free satured absorption spectroscopy, Advanced Optics Laboratory KAOS Advanced Optoelectronic System Research Group, University of Colorado [9] Carl E. Wieman & T. W. H¨ansch, Doppler-free Laser Polarization Spectroscopy, Physical Review Letters V.36 No. 20, 17 May 1976 [10] Carl E. Wieman & Leo Hollberg, Using diode lasers for atomic physics, Rev. Sci. V 62, January 1991 [11] Diode laser DL 714 201 S specifications,http://www.sanyo.com/ [12] Daniel F.V. James & Paul G. Kwiat & William J.Munro & Andrew G. White, Measurement of Qubits, Phys. Rev. A 64, 052312 (November 2001) [13] Isaac L. Chuang & M. A. Nielsen, Quantum Process Tomography in ”Quantum Computation and Quantum Information”, Cambridge university Press [14] Myrskog, SH, Fox, JK, Mitchell, MW, et al, Quantum Process Tomography on vibrational states of atoms in an Optical Lattice, Physical Review A 72 (1), July 2005
30
A
Optical lattice band structure 30 Ground band First excited band 2nd excited band 3rd excited band 25
20
E(Er)
Potential depth
15
10
5
0 −2
−1.5
−1
−0.5
0
0.5
k (First Brillouin zone)
Figure A.1: Band structure
31
1
1.5
2
B
The Optical Bloch Equations
The Optical Bloch Equations arise from the resolution of the time-dependent Schr¨odinger equation for an two-level atom interacting with an electromagnetic field. One can write down this equation using the density matrix formalism. It leads to: h i dρ ~ i~ = [H, ρ] = −eE(~r, t) · ~r, ρ (B.1) dt where the density matrix is given by: ∗ ce ce ce c∗g ρee ρeg ρ= = (B.2) ρge ρgg cg c∗e cg c∗g The cg (t) and ce (t) respectively describe the evolution in time for the ground and excited eigeinstates. Combining equation (B.1) with (B.2) and introducing the effect of spontaneous emission caracterized by dρeg γ = − ρeg dt spon 2 one would find the following equations which are called Optical Bloch Equations: dρgg dt dρee dt d˜ ρge dt d˜ ρeg dt
i ∗ (Ω ρ˜eg − Ω˜ ρge ) 2 i = −γρee + (Ω˜ ρge − Ω∗ ρ˜eg ) 2 γ i =− + iδ ρ˜ge + Ω∗ (ρee − ρgg ) 2 2 γ i =− − iδ ρ˜eg + Ω (ρgg − ρee ) 2 2 = +γρee +
(B.3) (B.4) (B.5) (B.6)
with δ the detuning of the laser from the resonant frequency, Ω the Rabi frequency as defined in Appendix E and ρ˜ge ≡ ρge e−iδt . The solutions with the Rotating Wave Approximation (RWA) yields to the following results: s0 /2 ρee = 1 + s0 + (2δ/γ)2 ρeg = s0 with s = 1+(2δ/γ) 2 and s0 = section (3.1.1).
2|Ω|2 γ2
iΩ 2(γ/2 − iδ)(1 + s) =
I Is
the saturation parameters mentioned in
32
C
The optical isolator
Figure C.1: Schematic picture of an optical isolator The optical isolator is an unavoidable device when using diode laser, which uses the Faraday effect. The Faraday effect is a rotation of the plane of polarization of a light beam in the presence of a strong magnetic field along the propagation axis. There is a simple classical picture to get a feel of it. An incoming liht beam imposes an oscillating electric field on the electrons in the solid, which causes the electrons to oscillate. Normally the oscillating electrons re-radiate the light in the same direction as the original beam, which doesn’t change the polarization of the light(it does change the phase, however, which is the cause of the material index of refraction). With the application of a strong magnetic field, you can see that the → → Lorentz force e v × B will shift the motion of the electrons, and rotate their plane of oscillation. As the electrons re-radiate this tends to rotate the polarization of the light beam. In our configuration, the optical isolator uses the Faraday effect to rotate the polarization angle of the input beam by 45 degrees, and the output beam exits through a 45-degree polarizer. Note that the diode laser’s beam is polarized, in our case along the horizontal axis. If one reflects the beam back into the optical isolator, the polarization experiences another 45-degree rotation, in the same direction as the first, and the beam is then extinguished by the input polarizer. You can see that the rotation has the correct sens using the classical picture. Thus the overall effect is that of an ”optical diode”-light can go through in one direction, but not in the reverse direction. The Faraday effect is typically very weak, so the optical isolator uses a special crystal, which exhibits an anomalously large Faraday effect, and a very strong longitudinal magnetic field produced by state-of-the-art permanents magnets. Optical isolators have gotten much smaller over the last couple of decades as magnet technology has improved. The magnetic field is strong only near the axis of the device, which therefore has a small aperture. Also too much light intensity will burn a spot in the Faraday cristal, so one must be careful not to focus the diode laser to a tight spot inside the optical isolator.
33
D
Atomic cloud profiles
Figure D.1: Profile of the atomic cloud in the vertical lattice
34
Figure D.2: Profile of the atomic cloud in the 3D lattice
35
E Usefull numbers and definitions ~=
h 2π
= 1, 054 · 10−34 J s.
For F = 3 → F ′ = 4, λ = 780, 24nm and γ = 2π · 5, 98 MHz The saturation intensity Is = 1, 64 mW/cm2 corresponds to the intensity at which no more photons can be absorbed by the atoms because they do not have the time to spontaneously emit a previously absorbed photon. The atomic mass of Rb 85 is M = 85.4678 amu. The commonly called Doppler temperature TDop = 143 µK is the lowest temperature achievable by a Doppler cooling scheme. 2 2
2
The recoil energy ER = ~2Mk = 2Mh λ2 , often associated with the recoil temperature Tr = 0, 370 µK for Rb 85, correspond to the kinetic energy gained by an atom after the absorption of a photon of momentum p = ~k. The Rabi frequency Ω ≡ − eE~0 he|r|gi arise from the Rabi two-level problem. |α| =
mω 1/2 · d, 2~
θ where d = a · 2π and a = vertical lattice spacing = 0.93µm.
36
