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Figure 1. (a) Physical schematic of a polarisable particle in a unidirectional ring cavity, showing the input field Bin. (b) Equivalent ‘transfer matrix’—style (unfolded) model; the particle is drawn on both sides of this schematic to illustrate the recursive nature of the cavity. The various components are defined in Section 2.

Figure 1 (a) Physical schematic of a polarisable particle in a unidirectional ring cavity, showing the input field Bin. (b) Equivalent ‘transfer matrix’—style (unfolded) model; the particle is drawn on both sides of this schematic to illustrate the recursive nature of the cavity. The various components are defined in Section 2.

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from each configuration under realistic experimental conditions.
from each configuration under realistic experimental conditions.
since H is Hermitian.? Eq. (1.15) is valid for systems that have no dissipation. Incoherent processes—such as the decay of the electromagnetic field inside a cavity, spontaneous where we use the notation that a dot above a symbol represents its derivative with
since H is Hermitian.? Eq. (1.15) is valid for systems that have no dissipation. Incoherent processes—such as the decay of the electromagnetic field inside a cavity, spontaneous where we use the notation that a dot above a symbol represents its derivative with
with the superscript ‘st’ again denoting steady-state values. In particular, then,
with the superscript ‘st’ again denoting steady-state values. In particular, then,
physical meaning by using Eqs. (1.29) to obtain We note that, in the limit of high intensity (s > oo), the force acting on the TLA tend: since d.g has no spatial dependence on this scale. We can give this equation more towards a limit of a dissipative force whose strength is independent of the intensity. In the following sections we will restrict ourselves to the limit of small s. This is useful
physical meaning by using Eqs. (1.29) to obtain We note that, in the limit of high intensity (s > oo), the force acting on the TLA tend: since d.g has no spatial dependence on this scale. We can give this equation more towards a limit of a dissipative force whose strength is independent of the intensity. In the following sections we will restrict ourselves to the limit of small s. This is useful
In |] Al ui q. (1.67) we made the approximation that the detuning of the Pump beam is A picture [29],/? can be written out in state vector notation as from each of the excited states, and that of the Stokes beam is A+ 6. This approx- Hamiltonian for this system, after making the RWA and transforming to the interaction
In |] Al ui q. (1.67) we made the approximation that the detuning of the Pump beam is A picture [29],/? can be written out in state vector notation as from each of the excited states, and that of the Stokes beam is A+ 6. This approx- Hamiltonian for this system, after making the RWA and transforming to the interaction
FIGURE 2.1: Close to the centre of a magneto-optical trap, the magnetic field has a constant gradient, taken here to be positive along the positive x direction. An atom to the right of the origin will see the a+ beam as further detuned than the o~ beam, even though both have the same frequency wy, and therefore preferentially scatter from the latter, causing it to experience a restoring force towards x = 0. A similar principle operates when the atom is to the left of the origin, whereby it will preferentially scatter from the a+ beam. We depict a J = 0 - J’ = 1 transition here. The designation o* relates to the effective circular polarisation of the light, and not its helicity.
FIGURE 2.1: Close to the centre of a magneto-optical trap, the magnetic field has a constant gradient, taken here to be positive along the positive x direction. An atom to the right of the origin will see the a+ beam as further detuned than the o~ beam, even though both have the same frequency wy, and therefore preferentially scatter from the latter, causing it to experience a restoring force towards x = 0. A similar principle operates when the atom is to the left of the origin, whereby it will preferentially scatter from the a+ beam. We depict a J = 0 - J’ = 1 transition here. The designation o* relates to the effective circular polarisation of the light, and not its helicity.
FIGURE 2.2: Space-time diagrams showing the interaction of an atom with light when the atom is (a) inside an optical resonator, and (b) in front of a mirror. It is clear that in case (a) one gets multiple interactions between the two and a ‘memory time’ is not well-defined, whereas in case (b) a definite memory time can be assigned. The blank arrow indicates the pump beam.
FIGURE 2.2: Space-time diagrams showing the interaction of an atom with light when the atom is (a) inside an optical resonator, and (b) in front of a mirror. It is clear that in case (a) one gets multiple interactions between the two and a ‘memory time’ is not well-defined, whereas in case (b) a definite memory time can be assigned. The blank arrow indicates the pump beam.
FIGURE 2.3: Spatial dependence of the longitudinal cooling coefficient 0 (thick solid line). The shaded areas promote cooling in the longitudinal direction. Also drawn is the transverse heating coefficient (thin solid line) and the field amplitude (dotted line). Parameters are for °°Rb atoms and |A|? = 62.50 /(27), A, = —101, rT = 0.5/T, w= 1.4 um.
FIGURE 2.3: Spatial dependence of the longitudinal cooling coefficient 0 (thick solid line). The shaded areas promote cooling in the longitudinal direction. Also drawn is the transverse heating coefficient (thin solid line) and the field amplitude (dotted line). Parameters are for °°Rb atoms and |A|? = 62.50 /(27), A, = —101, rT = 0.5/T, w= 1.4 um.
FIGURE 2.4: Dependence of dp*/dt on the square of the initial momentum, pj, for w, = 0.45 x 227T and x = 3A,/16. Other parameters are as in Fig. 2.3. Cooling is achieved only for a finite range of initial momenta. where J; is the order-1 Bessel function of the first kind and Hy is the order-1 Struve
FIGURE 2.4: Dependence of dp*/dt on the square of the initial momentum, pj, for w, = 0.45 x 227T and x = 3A,/16. Other parameters are as in Fig. 2.3. Cooling is achieved only for a finite range of initial momenta. where J; is the order-1 Bessel function of the first kind and Hy is the order-1 Struve
FIGURE 2.5: Calculated steady-state temperature Ty, for an atom confined in a har- monic trap as a function of position whilst keeping the detuning and pump field con- stant. w, = 0.1 x 271; other parameters are as in Fig. 2.3.
FIGURE 2.5: Calculated steady-state temperature Ty, for an atom confined in a har- monic trap as a function of position whilst keeping the detuning and pump field con- stant. w, = 0.1 x 271; other parameters are as in Fig. 2.3.
FIGURE 2.6: Comparison of dp?/dt for the simulations without noise (data points) with the analytical approximation, Eq. (2.40), including the harmonic trap (solid line). (a) Weak harmonic trap, w, = 0.3 x 27I, showing also the linear dependence in the limit of small momenta, Eq. (2.37) (dotted line). (b) Stiff trap, w, = 0.5 x 27T. The trap position 7) = —3A/16 and other parameters are as in Fig. 2.3.
FIGURE 2.6: Comparison of dp?/dt for the simulations without noise (data points) with the analytical approximation, Eq. (2.40), including the harmonic trap (solid line). (a) Weak harmonic trap, w, = 0.3 x 27I, showing also the linear dependence in the limit of small momenta, Eq. (2.37) (dotted line). (b) Stiff trap, w, = 0.5 x 27T. The trap position 7) = —3A/16 and other parameters are as in Fig. 2.3.
FIGURE 2.7: Capture range extracted from the simulations (data points) as compared to the analytical solution (dotted line) for various values of uw. The solid line is a quadratic fit to the data for w, > 0.3 x 27T and is only intended as a guide to the eye. Other parameters are as in Fig. 2.6.
FIGURE 2.7: Capture range extracted from the simulations (data points) as compared to the analytical solution (dotted line) for various values of uw. The solid line is a quadratic fit to the data for w, > 0.3 x 27T and is only intended as a guide to the eye. Other parameters are as in Fig. 2.6.
FIGURE 2.8: Cooling rate (—dT/dt) extracted from the simulations starting at a number of initial temperatures. The solid line represents a quadratic fit to the data. wt; = 0.5 x 271; other parameters are as in Fig. 2.6.
FIGURE 2.8: Cooling rate (—dT/dt) extracted from the simulations starting at a number of initial temperatures. The solid line represents a quadratic fit to the data. wt; = 0.5 x 271; other parameters are as in Fig. 2.6.
FIGURE 2.9: Steady-state temperature for a number of simulations (circles) compared to the analytical formula Eq. (2.43) (solid line). The solid square represents the equiva- lent data from Fig. 2.8, resulting from a much larger number of simulations. Parameters are as in Fig. 2.6.
FIGURE 2.9: Steady-state temperature for a number of simulations (circles) compared to the analytical formula Eq. (2.43) (solid line). The solid square represents the equiva- lent data from Fig. 2.8, resulting from a much larger number of simulations. Parameters are as in Fig. 2.6.
FIGURE 2.10: Comparison between the calculated steady-state temperatures for mirror-mediated cooling Ty (dash-dotted line), Doppler cooling Tp (dashed), and in the presence of both effects T (solid), drawn as a function of detuning whilst keeping the saturation parameter constant. w; = 0.1 x 271; other parameters are as in Fig. 2.6.
FIGURE 2.10: Comparison between the calculated steady-state temperatures for mirror-mediated cooling Ty (dash-dotted line), Doppler cooling Tp (dashed), and in the presence of both effects T (solid), drawn as a function of detuning whilst keeping the saturation parameter constant. w; = 0.1 x 271; other parameters are as in Fig. 2.6.
2.7 Exploiting an optical memory in other geometries [is the HWHM linewidth of the upper state. For the D2 line of some common alkali Section 2.6 provided us with a sound theoretical basis, in the case of the simplest possible
2.7 Exploiting an optical memory in other geometries [is the HWHM linewidth of the upper state. For the D2 line of some common alkali Section 2.6 provided us with a sound theoretical basis, in the case of the simplest possible
FIGURE 2.11: Space-time diagram showing the distance folding argument used to explain the action of external cavity cooling. This figure should be compared to Fig. 2.2. The blank arrow indicates the pump beam.
FIGURE 2.11: Space-time diagram showing the distance folding argument used to explain the action of external cavity cooling. This figure should be compared to Fig. 2.2. The blank arrow indicates the pump beam.
FIGURE 2.12: In (a) mirror-mediated cooling, the time delay depends on the atomic position (7, 4 72), whereas in (b, c) ring cavity cooling, it is independent of the atomic position (71 = 72). The blank arrow indicates the pump beam.
FIGURE 2.12: In (a) mirror-mediated cooling, the time delay depends on the atomic position (7, 4 72), whereas in (b, c) ring cavity cooling, it is independent of the atomic position (71 = 72). The blank arrow indicates the pump beam.
FIGURE 2.13: In (a) optical binding, particle 2 is in a potential well caused by particle 1. Self-tweezing, (b), occurs when a particle is in a potential well caused by itself; if the particle is moving, (c), it will experience a force opposing its motion. The blank arrow indicates the pump beam and the thick black arrows the force.
FIGURE 2.13: In (a) optical binding, particle 2 is in a potential well caused by particle 1. Self-tweezing, (b), occurs when a particle is in a potential well caused by itself; if the particle is moving, (c), it will experience a force opposing its motion. The blank arrow indicates the pump beam and the thick black arrows the force.
In the second form of the transfer matrix M, we expressed it in terms of the reflectivity t the right of the scatterer and those on the left can be derived from the wave equation [47 where the index k has been dropped. A linear relation between the field amplitudes on
In the second form of the transfer matrix M, we expressed it in terms of the reflectivity t the right of the scatterer and those on the left can be derived from the wave equation [47 where the index k has been dropped. A linear relation between the field amplitudes on
In the approximation we assumed that the frequency bandwidth of the excited modes, due to the Doppler shift, A ~ 2wo°%, where v is the speed of the beamsplitter. Fo
In the approximation we assumed that the frequency bandwidth of the excited modes, due to the Doppler shift, A ~ 2wo°%, where v is the speed of the beamsplitter. Fo
The distance between the atom and the mirror is denoted by d. Note that the free- shift has an influence on the phase shift accumulated between two scattering events.
The distance between the atom and the mirror is denoted by d. Note that the free- shift has an influence on the phase shift accumulated between two scattering events.
FIGURE 3.3: The position dependence of the cooling coefficient o in the relation F = —ov for the velocity-dependent force acting on the mobile scatterer in Fig. 3.2, for various scattering parameters ¢, evaluated by using Eq. (3.31) and Eq. (3.36) with ko L = 2n x 100. The fixed mirror is assumed to be a perfect mirror. In order to fit all the curves into the same range, they are divided by the factors indicated in the figure. Note that x is defined differently from Fig. 2.3; the mobile scatterer here is closer to the fixed mirror for increasing «x.
FIGURE 3.3: The position dependence of the cooling coefficient o in the relation F = —ov for the velocity-dependent force acting on the mobile scatterer in Fig. 3.2, for various scattering parameters ¢, evaluated by using Eq. (3.31) and Eq. (3.36) with ko L = 2n x 100. The fixed mirror is assumed to be a perfect mirror. In order to fit all the curves into the same range, they are divided by the factors indicated in the figure. Note that x is defined differently from Fig. 2.3; the mobile scatterer here is closer to the fixed mirror for increasing «x.
FIGURE 3.4: (a) The position of the maximum friction force, kox,,, as a function of the dimensionless scattering parameter ¢ (on a semilog scale) acting on the scatterer in Fig. 3.2, with the fixed mirror being a perfect mirror. This position shifts from 77/8 to m on increasing ¢. (b) A similar plot, showing the maximum friction force as a function of ¢ (on a log-log scale) with ko L = 2a x 100. In the limit of small ¢, the force scales as ¢? [cf. Eq. (3.41); dashed line] whereas in the limit of large zeta it scales as €° [cf. Eq. (3.48); dotted line].
FIGURE 3.4: (a) The position of the maximum friction force, kox,,, as a function of the dimensionless scattering parameter ¢ (on a semilog scale) acting on the scatterer in Fig. 3.2, with the fixed mirror being a perfect mirror. This position shifts from 77/8 to m on increasing ¢. (b) A similar plot, showing the maximum friction force as a function of ¢ (on a log-log scale) with ko L = 2a x 100. In the limit of small ¢, the force scales as ¢? [cf. Eq. (3.41); dashed line] whereas in the limit of large zeta it scales as €° [cf. Eq. (3.48); dotted line].
FIGURE 3.5: Characteristic temperature for the two-scatterer system of Fig. 3.2, given by the ratio of the diffusion and cooling coefficients in the points where the friction is maximum, as a function of the dimensionless scattering parameter ¢ on a log-log scale. Constant and 1/¢? dependence can be read off in the limits of small and large ¢, respectively. The fixed mirror is a perfect mirror.
FIGURE 3.5: Characteristic temperature for the two-scatterer system of Fig. 3.2, given by the ratio of the diffusion and cooling coefficients in the points where the friction is maximum, as a function of the dimensionless scattering parameter ¢ on a log-log scale. Constant and 1/¢? dependence can be read off in the limits of small and large ¢, respectively. The fixed mirror is a perfect mirror.
Then, the transfer matrix for the scatterer moving with velocity v is given by unit vector along the ith coordinate axis, v is the velocity vector of the scatterer, and where M is the original transfer matrix for the scatterer, obtained in a manner such a in three dimensions, the beamsplitter would be oriented at 45° to all three coordinate
Then, the transfer matrix for the scatterer moving with velocity v is given by unit vector along the ith coordinate axis, v is the velocity vector of the scatterer, and where M is the original transfer matrix for the scatterer, obtained in a manner such a in three dimensions, the beamsplitter would be oriented at 45° to all three coordinate
FIGURE 3.7: The model we consider in this section, drawn schematically. A scatterer S interacts with two ‘general optical systems’ in one dimension, composed of immobile linear optical elements, one to either side. S and these two systems are each represented by a 2 x 2 matrix. express the fields it interacts with, A, B, C, and D, in terms of the input fields By anc
FIGURE 3.7: The model we consider in this section, drawn schematically. A scatterer S interacts with two ‘general optical systems’ in one dimension, composed of immobile linear optical elements, one to either side. S and these two systems are each represented by a 2 x 2 matrix. express the fields it interacts with, A, B, C, and D, in terms of the input fields By anc
matrix, obtaining: vhere Mg is included as a 2 x 2 submatrix in the new 2 x 3 matrix, which can in turn be embedded in a 3 x 3 square matrix:°
matrix, obtaining: vhere Mg is included as a 2 x 2 submatrix in the new 2 x 3 matrix, which can in turn be embedded in a 3 x 3 square matrix:°
This assumes that the noise mode is essentially unaffected by the presence of the scat- terer. In summary, then, each transfer matrix in the optical system is replaced by one of two matrices:
This assumes that the noise mode is essentially unaffected by the presence of the scat- terer. In summary, then, each transfer matrix in the optical system is replaced by one of two matrices:
FIGURE 3.9: Linear and quadratic optomechanical couplings as a function of mir- ror position for a very good cavity and for (a) ¢ = —1, and (b) ¢ = —10. In each figure we show the linear (solid curve) and quadratic (dashed curve) couplings, from Eggs. (3.114) and (3.115). Note that the peak value of w’’ is roughly proportional to ¢ whereas w’ is bounded.
FIGURE 3.9: Linear and quadratic optomechanical couplings as a function of mir- ror position for a very good cavity and for (a) ¢ = —1, and (b) ¢ = —10. In each figure we show the linear (solid curve) and quadratic (dashed curve) couplings, from Eggs. (3.114) and (3.115). Note that the peak value of w’’ is roughly proportional to ¢ whereas w’ is bounded.
In the following sections we will restrict our discussion to the case where the input field is not modified by other transfer matrices. We will apply this mechanism to investigate
In the following sections we will restrict our discussion to the case where the input field is not modified by other transfer matrices. We will apply this mechanism to investigate
Suppose, now, that we illuminate the atom with two counterpropagating beams having orthogonal linear polarisation and equal intensity. This can be represented by setting
Suppose, now, that we illuminate the atom with two counterpropagating beams having orthogonal linear polarisation and equal intensity. This can be represented by setting
FIGURE 3.13: Clebsch-Gordan coefficients for a J = 1— J’ = 2 transition where we have assumed that ||k 0¢/0k|| >> ||¢||. The velocity-dependent force terms in the above expression arise through the Doppler shifting of photons both between fiel
FIGURE 3.13: Clebsch-Gordan coefficients for a J = 1— J’ = 2 transition where we have assumed that ||k 0¢/0k|| >> ||¢||. The velocity-dependent force terms in the above expression arise through the Doppler shifting of photons both between fiel
using the Clebsch-Gordan coefficients in Fig. 3.13. Together, these give the excited state five (J’ = 2). In this case, then, we can express pe and ¢ as in the next simplest, where the ground state has three magnetic sublevels (J = 1) and
using the Clebsch-Gordan coefficients in Fig. 3.13. Together, these give the excited state five (J’ = 2). In this case, then, we can express pe and ¢ as in the next simplest, where the ground state has three magnetic sublevels (J = 1) and
FIGURE 4.1: The amplitude of the friction force acting on the scatterer, for various near-mirror transmissivities, is shown as a function of the mirror separation in the cavity. The different curves represent different near-mirror transmissivities: |t] = 0.45 (dashed—dotted curve), |t] = 0.20 (dotted), |¢} = 0.10 (dashed), |t| = 0.05 (solid). (Scatterer polarisability ¢ = —1, scatterer-cavity separation x ~ 400Ao, |T'| = 0.01, Ao = 780 nm.)
FIGURE 4.1: The amplitude of the friction force acting on the scatterer, for various near-mirror transmissivities, is shown as a function of the mirror separation in the cavity. The different curves represent different near-mirror transmissivities: |t] = 0.45 (dashed—dotted curve), |t] = 0.20 (dotted), |¢} = 0.10 (dashed), |t| = 0.05 (solid). (Scatterer polarisability ¢ = —1, scatterer-cavity separation x ~ 400Ao, |T'| = 0.01, Ao = 780 nm.)
FIGuRE 4.2: Amplitude of the friction acting on a scatterer of polarisability ¢ = —1 interacting with a cavity tuned to achieve maximum friction, for varying transmissivity of the near mirror. The friction amplitude (solid curve) approaches that for mirror- mediated cooling using the far (dotted line, t > 1) or the near (dashed-dotted line, t — 0) mirror only in the appropriate limits. The arrow indicates the point at which the two cavity mirrors have the same reflectivity. Also shown is the intracavity field (dashed). (a % 400A9, LD = 20000, |T| = 0.01, Ao = 780 nm, finesse at peak friction 5.0 x 104.)
FIGuRE 4.2: Amplitude of the friction acting on a scatterer of polarisability ¢ = —1 interacting with a cavity tuned to achieve maximum friction, for varying transmissivity of the near mirror. The friction amplitude (solid curve) approaches that for mirror- mediated cooling using the far (dotted line, t > 1) or the near (dashed-dotted line, t — 0) mirror only in the appropriate limits. The arrow indicates the point at which the two cavity mirrors have the same reflectivity. Also shown is the intracavity field (dashed). (a % 400A9, LD = 20000, |T| = 0.01, Ao = 780 nm, finesse at peak friction 5.0 x 104.)
FIGURE 4.3: In region (b) of Fig. 4.2, the cooling coefficient amplitude (solid curves) is attenuated due to the attenuation in the field reflected from the cavity (dashed). |T| = 0.01 in every plot; |¢| is, from left to right, 6.7 x 107%, 8.3 x 1073, 1.0 x 107?, 1.2 x 10-7, and 1.5 x 10~?. (Parameters as in Fig. 4.2.)
FIGURE 4.3: In region (b) of Fig. 4.2, the cooling coefficient amplitude (solid curves) is attenuated due to the attenuation in the field reflected from the cavity (dashed). |T| = 0.01 in every plot; |¢| is, from left to right, 6.7 x 107%, 8.3 x 1073, 1.0 x 107?, 1.2 x 10-7, and 1.5 x 10~?. (Parameters as in Fig. 4.2.)
FIGURE 4.4: (a) Model of a scatterer, S, inside a symmetric Fabry—Pérot cavity of length L,. The cavity mirrors have reflection and transmission coefficients, t and t, and ¢ is the polarisability of S. (b) Cooling coefficient @ per unit input power, experienced by the scatterer at different positions in the cavity for realistic parameters (see text for details).
FIGURE 4.4: (a) Model of a scatterer, S, inside a symmetric Fabry—Pérot cavity of length L,. The cavity mirrors have reflection and transmission coefficients, t and t, and ¢ is the polarisability of S. (b) Cooling coefficient @ per unit input power, experienced by the scatterer at different positions in the cavity for realistic parameters (see text for details).
FIGURE 4.6: Spatial dependence of the friction force acting on an atom inside a cav- ity with different mode waists (i.e., different polarisabilities) but equal detuning from resonance, 101 to the red. The smaller the mode waist the stronger the friction force, by several orders of magnitude, but the more significant localisation issues become. (Parameters as in Section 4.2.1.1 but with 0¢/0k = 0.)
FIGURE 4.6: Spatial dependence of the friction force acting on an atom inside a cav- ity with different mode waists (i.e., different polarisabilities) but equal detuning from resonance, 101 to the red. The smaller the mode waist the stronger the friction force, by several orders of magnitude, but the more significant localisation issues become. (Parameters as in Section 4.2.1.1 but with 0¢/0k = 0.)
FIGURE 4.7: Spatial dependence of the friction force acting on an atom outside a cavity, with different pumping field waists but equal detuning from resonance, 10I to the red. The friction force scales roughly as the inverse fourth power of the waist [cf. Eq. (3.41)], but the length scale of the cooling and heating regions is unaffected. (Parameters as in Section 4.2.1.2 but with 0¢/0k = 0.)
FIGURE 4.7: Spatial dependence of the friction force acting on an atom outside a cavity, with different pumping field waists but equal detuning from resonance, 10I to the red. The friction force scales roughly as the inverse fourth power of the waist [cf. Eq. (3.41)], but the length scale of the cooling and heating regions is unaffected. (Parameters as in Section 4.2.1.2 but with 0¢/0k = 0.)
FIGURE 4.8: Spatial dependence of the friction force acting on an atom outside a cavity, with detuning 101 to the red of resonance. Three different cavity lengths are shown; the friction force scales almost linearly with the cavity length. Note the different scaling factors and cavity lengths, given above each curve. (Parameters as in Section 4.2.1.2 but with 0¢/0k = 0.)
FIGURE 4.8: Spatial dependence of the friction force acting on an atom outside a cavity, with detuning 101 to the red of resonance. Three different cavity lengths are shown; the friction force scales almost linearly with the cavity length. Note the different scaling factors and cavity lengths, given above each curve. (Parameters as in Section 4.2.1.2 but with 0¢/0k = 0.)
FIGURE 4.9: (a) Physical schematic of a polarisable particle in a unidirectional ring cavity, showing the input field Bin. (b) Equivalent ‘transfer matrix’-style (unfolded) model; the particle is drawn on both sides of this schematic to illustrate the recursive nature of the cavity. The various components are defined in Section 4.3.1. cooling mechanism. Finally, we will conclude by mentioning some possible extensions
FIGURE 4.9: (a) Physical schematic of a polarisable particle in a unidirectional ring cavity, showing the input field Bin. (b) Equivalent ‘transfer matrix’-style (unfolded) model; the particle is drawn on both sides of this schematic to illustrate the recursive nature of the cavity. The various components are defined in Section 4.3.1. cooling mechanism. Finally, we will conclude by mentioning some possible extensions
FIGURE 4.10: Cooling rate —(du/dt)/v for °°Rb pumped —101 from Dz resonance inside a ring cavity with a round-trip length L = 300m, for two different values of the amplifier gain y. Note that the curve for y = 1, as drawn, is scaled up by a factor of 10. The cavity waist is taken to be 10pm. (|t;|? = |t3|? = 0.5, |te|? = 0.99, Bo is chosen such as to give an atomic saturation s = 0.1.)
FIGURE 4.10: Cooling rate —(du/dt)/v for °°Rb pumped —101 from Dz resonance inside a ring cavity with a round-trip length L = 300m, for two different values of the amplifier gain y. Note that the curve for y = 1, as drawn, is scaled up by a factor of 10. The cavity waist is taken to be 10pm. (|t;|? = |t3|? = 0.5, |te|? = 0.99, Bo is chosen such as to give an atomic saturation s = 0.1.)
FIGURE 4.11: Equilibrium temperature predicted by the transfer matrix model for two values of the amplifier gain y. The Doppler temperature for ®°Rb is also indicated. The horizontal axis differs from that in Fig. 4.10 mainly because the temperature is only well-defined for regions where the friction force promotes cooling. (Parameters are as in Fig. 4.10.)
FIGURE 4.11: Equilibrium temperature predicted by the transfer matrix model for two values of the amplifier gain y. The Doppler temperature for ®°Rb is also indicated. The horizontal axis differs from that in Fig. 4.10 mainly because the temperature is only well-defined for regions where the friction force promotes cooling. (Parameters are as in Fig. 4.10.)
FIGuRE 5.1: Polarised by incident radiation €°(t,1r), a particle moving with velocity v is bound to its retarded reflection in a surface, characterised by round-trip time 7 and field propagator g.
FIGuRE 5.1: Polarised by incident radiation €°(t,1r), a particle moving with velocity v is bound to its retarded reflection in a surface, characterised by round-trip time 7 and field propagator g.
FIGURE 5.2: Retarded binding of a normally-illuminated particle, moving with velocity v, to its own reflection, depicted (a) in the laboratory frame, in which the image lags behind; (b) in the rest frame of particle, whereby the ‘wake’ trails behind.
FIGURE 5.2: Retarded binding of a normally-illuminated particle, moving with velocity v, to its own reflection, depicted (a) in the laboratory frame, in which the image lags behind; (b) in the rest frame of particle, whereby the ‘wake’ trails behind.
under circularly-polarised illumination and taking into account the near-field effects in non-zero force, when averaging over the x coordinate,
under circularly-polarised illumination and taking into account the near-field effects in non-zero force, when averaging over the x coordinate,
FIGURE 6.1: Templating process used to make the structured surfaces. (a) A colloidal suspension of latex spheres in water is allowed to evaporate on a gold substrate within a masked region. (b) An ordered, close-packed monolayer of spheres is formed on the substrate. (c) Gold is electrodeposited and grows from the substrate upwards, around the latex spheres. (d) The spheres and mask are removed by using conventional solvents.
FIGURE 6.1: Templating process used to make the structured surfaces. (a) A colloidal suspension of latex spheres in water is allowed to evaporate on a gold substrate within a masked region. (b) An ordered, close-packed monolayer of spheres is formed on the substrate. (c) Gold is electrodeposited and grows from the substrate upwards, around the latex spheres. (d) The spheres and mask are removed by using conventional solvents.
FIGURE 6.2: First sample used in the vacuum chamber, pictured being secured into the holder. This sample consisted of a microscope slide onto which a layer of gold was evaporated.
FIGURE 6.2: First sample used in the vacuum chamber, pictured being secured into the holder. This sample consisted of a microscope slide onto which a layer of gold was evaporated.
FIGURE 6.3: Second sample used, mounted on the vacuum-compatible translation stage. This sample was templated with hemispherical concave mirrors; the bright dot in the mirror is the reflection of light off the hemispheres.
FIGURE 6.3: Second sample used, mounted on the vacuum-compatible translation stage. This sample was templated with hemispherical concave mirrors; the bright dot in the mirror is the reflection of light off the hemispheres.
FIGURE 6.4: Technical drawing for the Kimball Physics MCF600-SO200800 spheri- cal octagon. All dimensions are in inches (and millimetres in parentheses). Image reproduced from Ref. [160]. The sample, Fig. 6.3, is mounted on a vacuum-compatible translation stage, allowing
FIGURE 6.4: Technical drawing for the Kimball Physics MCF600-SO200800 spheri- cal octagon. All dimensions are in inches (and millimetres in parentheses). Image reproduced from Ref. [160]. The sample, Fig. 6.3, is mounted on a vacuum-compatible translation stage, allowing
FIGURE 6.5: Schematic of one of the two beam paths involved in our MOT geometry. S is the incoming beam; A, B, and C are mirrors. The component marked ‘\/4’ is a quarter-wave plate. The cold atom cloud forms in the intersection region, O. In this diagram we do not show a second, identical, beam, which provides trapping and cooling forces in the plane normal to the paper. The area of mirror A immediately adjacent to the trapped atoms is not illuminated, and can therefore be patterned or structured to explore atom-surface interactions. Inset: The lower surface of mirror A, showing the MOT beams and the sample area. which is not illuminated by anv of the beams.
FIGURE 6.5: Schematic of one of the two beam paths involved in our MOT geometry. S is the incoming beam; A, B, and C are mirrors. The component marked ‘\/4’ is a quarter-wave plate. The cold atom cloud forms in the intersection region, O. In this diagram we do not show a second, identical, beam, which provides trapping and cooling forces in the plane normal to the paper. The area of mirror A immediately adjacent to the trapped atoms is not illuminated, and can therefore be patterned or structured to explore atom-surface interactions. Inset: The lower surface of mirror A, showing the MOT beams and the sample area. which is not illuminated by anv of the beams.
FIGURE 6.6: Image of our MOT in operation, corresponding to Fig. 6.5; mirror A is indicated in the picture.
FIGURE 6.6: Image of our MOT in operation, corresponding to Fig. 6.5; mirror A is indicated in the picture.
FIGURE 6.7: The four-level system in *°Rb that we use to image our atoms. The MOT lasers (780nm) and a laser at 776nm are used to induce a ladder transition. The population decays back to the ground state, via an intermediate state, and emits a 420nm photon in the process. The hyperfine splitting of the excited states is not drawn for clarity.
FIGURE 6.7: The four-level system in *°Rb that we use to image our atoms. The MOT lasers (780nm) and a laser at 776nm are used to induce a ladder transition. The population decays back to the ground state, via an intermediate state, and emits a 420nm photon in the process. The hyperfine splitting of the excited states is not drawn for clarity.
FIGURE 6.8: 776 nm spectroscopy and locking system. (P)BS: (polarising) beam split- ter cube; A/4: quarter-wave plate; \/2: half-wave plate; VC: heated vapour cell; PD: filtered photodiode.
FIGURE 6.8: 776 nm spectroscopy and locking system. (P)BS: (polarising) beam split- ter cube; A/4: quarter-wave plate; \/2: half-wave plate; VC: heated vapour cell; PD: filtered photodiode.
FIGURE 6.9: 420nm fluorescence from the vapour cell, observed on PD (see Fig. 6.8) as a function of the detuning of the 776nm beam, with the cooling and repump beams locked and shifted by 80 MHz with respect to the frequencies required to make a MOT. The various peaks are due the hyperfine structure in °°Rb. To obtain these data, we removed the quarter-wave plates on either end of the vapour cell, thus having linearly polarised light entering the cell from both ends.
FIGURE 6.9: 420nm fluorescence from the vapour cell, observed on PD (see Fig. 6.8) as a function of the detuning of the 776nm beam, with the cooling and repump beams locked and shifted by 80 MHz with respect to the frequencies required to make a MOT. The various peaks are due the hyperfine structure in °°Rb. To obtain these data, we removed the quarter-wave plates on either end of the vapour cell, thus having linearly polarised light entering the cell from both ends.
FIGURE 6.10: 420nm fluorescence observed on PD (solid black line, see Fig. 6.8) and on a PMT imaging the MOT cloud (solid red line) as a function of the detuning of the 776nm beam. The zero on the frequency axis corresponds to the point at which the signal from the MOT cloud is highest; we lock to this point. The magnitude and sign of the shift between the two curves can be set arbitrarily by varying the magnetic field generated by the coils around the vapour cell. Inset: MOT cloud imaged at 420nm (scale in 10° counts per second). This image is naturally background-free and has a spatial resolution, limited by the optics used, of ca. 2 um.
FIGURE 6.10: 420nm fluorescence observed on PD (solid black line, see Fig. 6.8) and on a PMT imaging the MOT cloud (solid red line) as a function of the detuning of the 776nm beam. The zero on the frequency axis corresponds to the point at which the signal from the MOT cloud is highest; we lock to this point. The magnitude and sign of the shift between the two curves can be set arbitrarily by varying the magnetic field generated by the coils around the vapour cell. Inset: MOT cloud imaged at 420nm (scale in 10° counts per second). This image is naturally background-free and has a spatial resolution, limited by the optics used, of ca. 2 um.
FIGURE 6.11: A sequence of four false colour fluorescence images, taken at 8 ms inter- vals, of the cloud before and after it has been given a magnetic impulse. The first shot (leftmost picture) shows the cloud just before the magnetic field is pulsed. The second, and subsequent, shots show the cloud at later times. The transfer efficiency after 24 ms is over 40%.
FIGURE 6.11: A sequence of four false colour fluorescence images, taken at 8 ms inter- vals, of the cloud before and after it has been given a magnetic impulse. The first shot (leftmost picture) shows the cloud just before the magnetic field is pulsed. The second, and subsequent, shots show the cloud at later times. The transfer efficiency after 24 ms is over 40%.
FIGURE 7.1: Optical micrograph of a section of a templated gold surface. The bright spot in each of the dimples is the focus. The centre-to—centre distance is 100 um and the depth of the electrodeposited gold is 15 um. (Courtesy Nathan Cooper.) The derivation of the Clausius—Mossotti equation itself has some interesting subtleties
FIGURE 7.1: Optical micrograph of a section of a templated gold surface. The bright spot in each of the dimples is the focus. The centre-to—centre distance is 100 um and the depth of the electrodeposited gold is 15 um. (Courtesy Nathan Cooper.) The derivation of the Clausius—Mossotti equation itself has some interesting subtleties
FIGURE 7.3: Highly schematic illustration of a three-dimensional ring cavity cooling setup. The pump light does not couple directly into any cavity, allowing one to use gain media in the fibres similarly to the amplified optomechanics scheme (see Section 4.3).
FIGURE 7.3: Highly schematic illustration of a three-dimensional ring cavity cooling setup. The pump light does not couple directly into any cavity, allowing one to use gain media in the fibres similarly to the amplified optomechanics scheme (see Section 4.3).
FIGURE 7.4: Origin of the friction force for tangential “racetrack” motion of a particle around the centre of curvature of a hemispherical mirror. (a) In the absence of delay, the particle and its image are at diametrically opposite points in the circular locus of the motion of the particle. (b) Due to delay, the image trails behind its equilibrium position slowing the particle down due to the repulsive interaction between the two. Note that the length scales are greatly exaggerated; in reality, the particle, and the path it moves along, would be much smaller than depicted.
FIGURE 7.4: Origin of the friction force for tangential “racetrack” motion of a particle around the centre of curvature of a hemispherical mirror. (a) In the absence of delay, the particle and its image are at diametrically opposite points in the circular locus of the motion of the particle. (b) Due to delay, the image trails behind its equilibrium position slowing the particle down due to the repulsive interaction between the two. Note that the length scales are greatly exaggerated; in reality, the particle, and the path it moves along, would be much smaller than depicted.
which averages over a wavelength? to give the non-oscillatory terms, yz plane, there arises a dominant (oscillatory) component of the friction force,
which averages over a wavelength? to give the non-oscillatory terms, yz plane, there arises a dominant (oscillatory) component of the friction force,
It must be noted that these expressions are valid for kx large enough that the point-
It must be noted that these expressions are valid for kx large enough that the point-
or example, denote the force in the x direction proportional to the x-component of the
or example, denote the force in the x direction proportional to the x-component of the
and averaged over a wavelength to give The oscillatory forces have the same origin as those in the longitudinal case. Note. however, that because of the geometry of this situation the only non-oscillatory force
and averaged over a wavelength to give The oscillatory forces have the same origin as those in the longitudinal case. Note. however, that because of the geometry of this situation the only non-oscillatory force
FIGURE 7.5: Finite-difference time-domain analysis of the electric field intensity on a 2D slice of a 10 um diameter hemispherical void in an ideal metal substrate. The incident field is a linearly polarised plane wave, with a wavelength of 780 nm, propagat- ing downwards. Two sections, both intersecting the focus, are also shown: the vertical (horizontal) section runs along the vertical (horizontal) dashed line.
FIGURE 7.5: Finite-difference time-domain analysis of the electric field intensity on a 2D slice of a 10 um diameter hemispherical void in an ideal metal substrate. The incident field is a linearly polarised plane wave, with a wavelength of 780 nm, propagat- ing downwards. Two sections, both intersecting the focus, are also shown: the vertical (horizontal) section runs along the vertical (horizontal) dashed line.
FIGURE 7.6: Electric field intensity inside a 100 um diameter hemispherical void in an ideal metal substrate. The size of the hemispherical template and the depth of the void (15 um vertically from the bottom to the lip) match the surface in Fig. 7.1.
FIGURE 7.6: Electric field intensity inside a 100 um diameter hemispherical void in an ideal metal substrate. The size of the hemispherical template and the depth of the void (15 um vertically from the bottom to the lip) match the surface in Fig. 7.1.
FIG. 1: The four different modes that interact through a point-like beamsplitter in 1D. The system we will consider in some detail is com- posed of two mirrors; one of them is fixed in space, whilst the other one is mobile. This is the generic scheme for radiation-pressure cooling of moving mirrors [14, 15, 16]. At the same time, in the limit of low reflection the moving mirror can equally well represent a single atomic dipole interacting with its mirror image in front of a highly re- flecting surface [17, 18, 19]. The scattering model de- scription of this example gives a clear recipe for general- izing the method to more complex systems.
FIG. 1: The four different modes that interact through a point-like beamsplitter in 1D. The system we will consider in some detail is com- posed of two mirrors; one of them is fixed in space, whilst the other one is mobile. This is the generic scheme for radiation-pressure cooling of moving mirrors [14, 15, 16]. At the same time, in the limit of low reflection the moving mirror can equally well represent a single atomic dipole interacting with its mirror image in front of a highly re- flecting surface [17, 18, 19]. The scattering model de- scription of this example gives a clear recipe for general- izing the method to more complex systems.
FIG. 2: Physical parameters of our model. A, B, etc. repre- sent the field mode amplitudes.
FIG. 2: Physical parameters of our model. A, B, etc. repre- sent the field mode amplitudes.
matrices for the atom, propagation and mirror, respec- tively, we obtain the relation: The distance between the atom and the mirror is denoted by d. Note that the free propagation transfer matrix M, is non-uniform in the k-space, and therefore the Doppler- shift has an influence on the phase shift accumulated be- tween two scattering events.
matrices for the atom, propagation and mirror, respec- tively, we obtain the relation: The distance between the atom and the mirror is denoted by d. Note that the free propagation transfer matrix M, is non-uniform in the k-space, and therefore the Doppler- shift has an influence on the phase shift accumulated be- tween two scattering events.
This result is valid for arbitrary ¢. The main virtue of our approach is clearly seen, in that we can smoothly move from ¢ = 0, which indicates the absence of the mobile scatterer, to |¢| + co, which corresponds to a perfectly reflecting mirror, i.e., a moving boundary condition. It is worth introducing the reference point at a distance L = 2Nr/ko from the fixed mirror, where the integer N is such that the moving atom’s position x is within a wavelength of this reference point. Then the atom— mirror distance can be replaced by d = L — a, and koL drops from all the trigonometric functions. The solution, Eq. (17), has a clear physical meaning, in that the re- flected field, A, can be decomposed into an interfering sum of fields: the first term is the reflection directly from the atom, whereas the summation is over the electric field undergoing successive atom—mirror round-trips. We can also write the preceding expression in closed form:
This result is valid for arbitrary ¢. The main virtue of our approach is clearly seen, in that we can smoothly move from ¢ = 0, which indicates the absence of the mobile scatterer, to |¢| + co, which corresponds to a perfectly reflecting mirror, i.e., a moving boundary condition. It is worth introducing the reference point at a distance L = 2Nr/ko from the fixed mirror, where the integer N is such that the moving atom’s position x is within a wavelength of this reference point. Then the atom— mirror distance can be replaced by d = L — a, and koL drops from all the trigonometric functions. The solution, Eq. (17), has a clear physical meaning, in that the re- flected field, A, can be decomposed into an interfering sum of fields: the first term is the reflection directly from the atom, whereas the summation is over the electric field undergoing successive atom—mirror round-trips. We can also write the preceding expression in closed form:
FIG. 3: (Color online.) The position dependence of the lin- ear coefficient of the velocity-dependent force acting on the mobile scatterer in Fig. 2, for various scattering parameters ¢, evaluated by using Eq. (18) and Eq. (20) with koL = 100. The fixed mirror is assumed to be a perfect mirror. In order to fit all the curves into the same range, they are divided by the factors indicated in the figure.
FIG. 3: (Color online.) The position dependence of the lin- ear coefficient of the velocity-dependent force acting on the mobile scatterer in Fig. 2, for various scattering parameters ¢, evaluated by using Eq. (18) and Eq. (20) with koL = 100. The fixed mirror is assumed to be a perfect mirror. In order to fit all the curves into the same range, they are divided by the factors indicated in the figure.
FIG. 4: (a) The position of the maximum friction force, kotm, as a function of the dimensionless scattering parameter ¢ (on a semilog scale) acting on the scatterer in Fig. 2, with the fixed mirror being a perfect mirror. This position shifts from 7/8 to m on increasing ¢. (b) A similar plot, showing the maximum friction force as a function of ¢ (on a log-log scale) with koL = 100. In the limit of small ¢, the force scales as ¢? (cf. Eq. (25); dashed line) whereas in the limit of large zeta it scales as ¢° (cf. Eq. (27); dotted line).
FIG. 4: (a) The position of the maximum friction force, kotm, as a function of the dimensionless scattering parameter ¢ (on a semilog scale) acting on the scatterer in Fig. 2, with the fixed mirror being a perfect mirror. This position shifts from 7/8 to m on increasing ¢. (b) A similar plot, showing the maximum friction force as a function of ¢ (on a log-log scale) with koL = 100. In the limit of small ¢, the force scales as ¢? (cf. Eq. (25); dashed line) whereas in the limit of large zeta it scales as ¢° (cf. Eq. (27); dotted line).
FIG. 5: Characteristic temperature for the two-scatterer sys- tem of Fig. 2, given by the ratio of the diffusion and friction coefficients in the points where the friction is maximum, as @ function of the dimensionless scattering parameter ¢ on a log- log scale. Constant and 1/¢? dependence can be read off in the limits of small and large ¢, respectively. The fixed mirror is a perfect mirror.
FIG. 5: Characteristic temperature for the two-scatterer sys- tem of Fig. 2, given by the ratio of the diffusion and friction coefficients in the points where the friction is maximum, as @ function of the dimensionless scattering parameter ¢ on a log- log scale. Constant and 1/¢? dependence can be read off in the limits of small and large ¢, respectively. The fixed mirror is a perfect mirror.
FIG. 1: Schematic of the cooling scheme. (a) In the lab- oratory frame, an atom moving with velocity v parallel to a mirror, a distance Tc away, interacts with a pump beam and its time-delayed reflection. (b) In the frame of the atom, the (relativistically transformed) pump beam and its reflec- tion are tilted by an angle 6 = arcsin(v/c) and produce a net retarding force, F.
FIG. 1: Schematic of the cooling scheme. (a) In the lab- oratory frame, an atom moving with velocity v parallel to a mirror, a distance Tc away, interacts with a pump beam and its time-delayed reflection. (b) In the frame of the atom, the (relativistically transformed) pump beam and its reflec- tion are tilted by an angle 6 = arcsin(v/c) and produce a net retarding force, F.
FIG. 2: Schematic representation of the key components of the system under consideration. The atom is separated from the mirror by a distance Tc and lies in a standing wave main- tained by the pump beam.
FIG. 2: Schematic representation of the key components of the system under consideration. The atom is separated from the mirror by a distance Tc and lies in a standing wave main- tained by the pump beam.
FIG. 3: (Color online) Spatial dependence of the longitudinal heating coefficient T (thick solid line). The shaded areas pro- mote cooling in the longitudinal direction. Also drawn is the transverse heating coefficient (thin solid line) and the field amplitude (dotted line). Parameters are for Rb atoms and |A/? = 62.51 /(2r), A = —10P, 7 = 0.25/T, w = 0.7 pm.
FIG. 3: (Color online) Spatial dependence of the longitudinal heating coefficient T (thick solid line). The shaded areas pro- mote cooling in the longitudinal direction. Also drawn is the transverse heating coefficient (thin solid line) and the field amplitude (dotted line). Parameters are for Rb atoms and |A/? = 62.51 /(2r), A = —10P, 7 = 0.25/T, w = 0.7 pm.
FIG. 4: Dependence of the heating rate (dp”/dt) on the square of the initial momentum, p?, for w, = 0.45 x 2nT and x) = 3\/16. Other parameters are as in Fig. 3. Cooling is achieved only for a finite range of initial momenta.
FIG. 4: Dependence of the heating rate (dp”/dt) on the square of the initial momentum, p?, for w, = 0.45 x 2nT and x) = 3\/16. Other parameters are as in Fig. 3. Cooling is achieved only for a finite range of initial momenta.
FIG. 5: Calculated steady-state temperature Tm for an atom confined in a harmonic trap as a function of position whilst keeping the detuning and pump field constant. w, = 0.1x 27T; other parameters are as in Fig. 3.
FIG. 5: Calculated steady-state temperature Tm for an atom confined in a harmonic trap as a function of position whilst keeping the detuning and pump field constant. w, = 0.1x 27T; other parameters are as in Fig. 3.
FIG. 6: Comparison of heating rate (dp?/dt) for the simula tions without noise (data points) with the analytic approx. imation, Eq. (13), including the harmonic trap (solid line) (a) Weak harmonic trap, w, = 0.3 x 27IT, showing also the linear dependence in the limit of small momenta, Eq. (11 (dotted line). (b) Stiff trap, w, = 0.5 x 27I’. The trap posi- tion x9 = —3\/16 and other parameters are as in Fig. 3.
FIG. 6: Comparison of heating rate (dp?/dt) for the simula tions without noise (data points) with the analytic approx. imation, Eq. (13), including the harmonic trap (solid line) (a) Weak harmonic trap, w, = 0.3 x 27IT, showing also the linear dependence in the limit of small momenta, Eq. (11 (dotted line). (b) Stiff trap, w, = 0.5 x 27I’. The trap posi- tion x9 = —3\/16 and other parameters are as in Fig. 3.
FIG. 8: Heating rate (dT'/dt) extracted from the simulations starting at a number of initial temperatures. The solid line represents a quadratic fit to the data. w, = 0.5 x 27T; other parameters are as in Fig. 6. FIG. 7: Capture range extracted from the simulations (data points) as compared to the analytic solution (dotted line) for various values of u;. The solid line is a quadratic fit to the data for w; > 0.3 x 27T and is only intended as a guide to the eye. Other parameters are as in Fig. 6.
FIG. 8: Heating rate (dT'/dt) extracted from the simulations starting at a number of initial temperatures. The solid line represents a quadratic fit to the data. w, = 0.5 x 27T; other parameters are as in Fig. 6. FIG. 7: Capture range extracted from the simulations (data points) as compared to the analytic solution (dotted line) for various values of u;. The solid line is a quadratic fit to the data for w; > 0.3 x 27T and is only intended as a guide to the eye. Other parameters are as in Fig. 6.
FIG. 9: Steady-state temperature for a number of simulations (circles) compared to the analytic formula (16) (solid line). The solid square represents the equivalent data from Fig. 8, resulting from a much larger number of simulations. Param- eters are as in Fig. 6.
FIG. 9: Steady-state temperature for a number of simulations (circles) compared to the analytic formula (16) (solid line). The solid square represents the equivalent data from Fig. 8, resulting from a much larger number of simulations. Param- eters are as in Fig. 6.
FIG. 10: Comparison between the calculated steady-state temperatures for mirror-mediated cooling Ty (dash-dotted line), Doppler cooling Tp (dashed), and in the presence of both effects T (solid), drawn as a function of detuning whilst keeping the saturation parameter constant. wt = 0.1 x 27T; other parameters are as in Fig. 6.
FIG. 10: Comparison between the calculated steady-state temperatures for mirror-mediated cooling Ty (dash-dotted line), Doppler cooling Tp (dashed), and in the presence of both effects T (solid), drawn as a function of detuning whilst keeping the saturation parameter constant. wt = 0.1 x 27T; other parameters are as in Fig. 6.
Fig. 1. (Color online.) Schematic of one of the two beam paths involved in our MOT ge- ometry. S is the incoming beam; A, B, and C are mirrors. The component marked ‘A /4’ is a quarter-wave plate. The cold atom cloud forms in the intersection region, O. In this diagram we do not show a second, identical, beam, which provides trapping and cooling forces in the plane normal to the paper. The area of mirror A immediately adjacent to the trapped atoms is not illuminated, and can therefore be patterned or structured to explore atom- surface interactions. Inset: The lower surface of mirror A, showing the MOT beams and the sample area, which is not illuminated by any of the beams.
Fig. 1. (Color online.) Schematic of one of the two beam paths involved in our MOT ge- ometry. S is the incoming beam; A, B, and C are mirrors. The component marked ‘A /4’ is a quarter-wave plate. The cold atom cloud forms in the intersection region, O. In this diagram we do not show a second, identical, beam, which provides trapping and cooling forces in the plane normal to the paper. The area of mirror A immediately adjacent to the trapped atoms is not illuminated, and can therefore be patterned or structured to explore atom- surface interactions. Inset: The lower surface of mirror A, showing the MOT beams and the sample area, which is not illuminated by any of the beams.
Fig. 2. Image of our MOT in operation, corresponding to Fig. 1; mirror A is indicated ir the picture.
Fig. 2. Image of our MOT in operation, corresponding to Fig. 1; mirror A is indicated ir the picture.
Fig. 3. (Color online.) The four-level system in ®°Rb that we use to image our atoms. The MOT lasers (780 nm) and a laser at 776nm are used to induce a ladder transition. The population decays back to the ground state, via an intermediate state, and emits a 420 nm photon in the process. The hyperfine splitting of the excited states is not drawn for clarity.
Fig. 3. (Color online.) The four-level system in ®°Rb that we use to image our atoms. The MOT lasers (780 nm) and a laser at 776nm are used to induce a ladder transition. The population decays back to the ground state, via an intermediate state, and emits a 420 nm photon in the process. The hyperfine splitting of the excited states is not drawn for clarity.
Fig. 6. (Color online.) 420 nm fluorescence observed on PD (solid black line, see Fig. 4) and on a PMT imaging the MOT cloud (solid red line) as a function of the detuning of the 776 nm beam. The zero on the frequency axis corresponds to the point at which the signal from the MOT cloud is highest; we lock to this point. The magnitude and sign of the shif between the two curves can be set arbitrarily by varying the magnetic field generated by the coils around the vapor cell. Inset: MOT cloud imaged at 420 nm (scale in 10° counts per second). This image is naturally background-free. Fig. 5. 420 nm fluorescence from the vapor cell, observed on PD (see Fig. 4) as a function of the detuning of the 776 nm beam, with the cooling and repump beams locked and shifted by 80 MHz with respect to the frequencies required to make a MOT. The various peaks are due the hyperfine structure in ®°Rb. To obtain these data, we removed the quarter-wave plates on either end of the vapor cell, thus having linearly polarized light entering the cell from both ends.
Fig. 6. (Color online.) 420 nm fluorescence observed on PD (solid black line, see Fig. 4) and on a PMT imaging the MOT cloud (solid red line) as a function of the detuning of the 776 nm beam. The zero on the frequency axis corresponds to the point at which the signal from the MOT cloud is highest; we lock to this point. The magnitude and sign of the shif between the two curves can be set arbitrarily by varying the magnetic field generated by the coils around the vapor cell. Inset: MOT cloud imaged at 420 nm (scale in 10° counts per second). This image is naturally background-free. Fig. 5. 420 nm fluorescence from the vapor cell, observed on PD (see Fig. 4) as a function of the detuning of the 776 nm beam, with the cooling and repump beams locked and shifted by 80 MHz with respect to the frequencies required to make a MOT. The various peaks are due the hyperfine structure in ®°Rb. To obtain these data, we removed the quarter-wave plates on either end of the vapor cell, thus having linearly polarized light entering the cell from both ends.
Fig. 7. (Color online.) A sequence of four false color fluorescence images, taken at 8 ms intervals, of the cloud before and after it has been given a magnetic impulse. The first shot (leftmost picture) shows the cloud just before the magnetic field is pulsed. The second, and subsequent, shots show the cloud at later times. The transfer efficiency after 24 ms is over 40%.
Fig. 7. (Color online.) A sequence of four false color fluorescence images, taken at 8 ms intervals, of the cloud before and after it has been given a magnetic impulse. The first shot (leftmost picture) shows the cloud just before the magnetic field is pulsed. The second, and subsequent, shots show the cloud at later times. The transfer efficiency after 24 ms is over 40%.
FIG. 1: A simple three-level “A” system, in which a Raman transition between states |0) and |1) is driven by “pump” and “Stokes” fields via intermediate state |2). For each field, the coupling strength and frequency are shown in parentheses, and frequencies are chosen to be near two-photon resonance: wpe = (we — wo) + A; ws = (wi — wo) + (A +5). The single- photon detuning A is large compared with the couplings, |A| >> Qps, and, in this illustration, is negative: A < 0. The two-photon detuning 6 is small compared with the sep- aration between the ground states and, in this illustration, is also negative.
FIG. 1: A simple three-level “A” system, in which a Raman transition between states |0) and |1) is driven by “pump” and “Stokes” fields via intermediate state |2). For each field, the coupling strength and frequency are shown in parentheses, and frequencies are chosen to be near two-photon resonance: wpe = (we — wo) + A; ws = (wi — wo) + (A +5). The single- photon detuning A is large compared with the couplings, |A| >> Qps, and, in this illustration, is negative: A < 0. The two-photon detuning 6 is small compared with the sep- aration between the ground states and, in this illustration, is also negative.
The time dependence of the bare state coefficients Ao,1(#) is readily calculable from the procedure described above and is stated here for completeness: Hence the system behaves as a two-level system with the coupling strength Qp = Qp and detuning Ap = Ap — 6, relative to the effective detuning Ap. This justifies our previous identification of Ag with the lightshift.
The time dependence of the bare state coefficients Ao,1(#) is readily calculable from the procedure described above and is stated here for completeness: Hence the system behaves as a two-level system with the coupling strength Qp = Qp and detuning Ap = Ap — 6, relative to the effective detuning Ap. This justifies our previous identification of Ag with the lightshift.
FIG. 2: Illustration of the dependence of coupling strength described by Eqs (27) (solid) and (28) (dashed). The values have physical meaning at integer and half-integer mp only (dots); continuous lines are shown to guide the eye. A line is shown for each of several nuclear spins, beginning at J = 1/2 and increasing in steps of one half out from the center (solid) and from bottom to top (dashed). The strongest coupling is between extremal mp states by linear and circular polarisa- tions.
FIG. 2: Illustration of the dependence of coupling strength described by Eqs (27) (solid) and (28) (dashed). The values have physical meaning at integer and half-integer mp only (dots); continuous lines are shown to guide the eye. A line is shown for each of several nuclear spins, beginning at J = 1/2 and increasing in steps of one half out from the center (solid) and from bottom to top (dashed). The strongest coupling is between extremal mp states by linear and circular polarisa- tions.
with a sign change. The first term of this determinant has prefactor zero and so can be ignored. For the second term, the minor matrix (A?°)!° is upper diagonal and hence has determinant |(A?°)!°| = 22(6 — A)N~3 and, similarly, |(A”°)!°| = (—1)"2,(6 — A)N~? for n > 2. Before deconstructing A?°, we note that the next term, A®°, is of the same form if we interchange the columns headed x and 23:
with a sign change. The first term of this determinant has prefactor zero and so can be ignored. For the second term, the minor matrix (A?°)!° is upper diagonal and hence has determinant |(A?°)!°| = 22(6 — A)N~3 and, similarly, |(A”°)!°| = (—1)"2,(6 — A)N~? for n > 2. Before deconstructing A?°, we note that the next term, A®°, is of the same form if we interchange the columns headed x and 23:
FIG. 1: (a) The general system consisting of a mobile scat- terer, 5S, between two sets of generic immobile optical elements (we show a Bragg reflector, on the left, and a Fabry—Pérot- type cavity, on the right, as an example). The mobile scatterer can, @ priori, represent anything, e.g., an atom or a mirror. We discuss two specific configurations in this Letter: an atom in front of (b) a two-mirror cavity, and (c) a plane mirror [8].
FIG. 1: (a) The general system consisting of a mobile scat- terer, 5S, between two sets of generic immobile optical elements (we show a Bragg reflector, on the left, and a Fabry—Pérot- type cavity, on the right, as an example). The mobile scatterer can, @ priori, represent anything, e.g., an atom or a mirror. We discuss two specific configurations in this Letter: an atom in front of (b) a two-mirror cavity, and (c) a plane mirror [8].
FIG. 2: The amplitude of the friction force acting on the scatterer, for various near-mirror transmissivities, is shown as a function of the mirror separation in the cavity. The different curves represent different near-mirror transmissivities: |t| = 0.45 (dashed—dotted curve), |t} = 0.20 (dotted), |t] = 0.10 (dashed), |t} = 0.05 (solid). (Scatterer polarizability ¢ = 1, scatterer—cavity separation x = 400Xo, |Z| = 0.01, Ao = 780 nm.)
FIG. 2: The amplitude of the friction force acting on the scatterer, for various near-mirror transmissivities, is shown as a function of the mirror separation in the cavity. The different curves represent different near-mirror transmissivities: |t| = 0.45 (dashed—dotted curve), |t} = 0.20 (dotted), |t] = 0.10 (dashed), |t} = 0.05 (solid). (Scatterer polarizability ¢ = 1, scatterer—cavity separation x = 400Xo, |Z| = 0.01, Ao = 780 nm.)
FIG. 3: Amplitude of the friction acting on a scatterer of po- larizability ¢ = 1 interacting with a cavity tuned to achieve maximum friction, for varying transmissivity of the near mir- ror. The friction amplitude (solid curve) approaches that for mirror mediated cooling using the far (dotted line, t > 1) or the near (dashed—dotted line, t + 0) mirror only in the appropriate limits. The arrow indicates the point at which the two cavity mirrors have the same reflectivity. Also shown is the intra-cavity field (dashed). (x ~ 400A0, L & 20000, |T| = 0.01, Ao = 780 nm, finesse at peak friction 5.0 x 10*.)
FIG. 3: Amplitude of the friction acting on a scatterer of po- larizability ¢ = 1 interacting with a cavity tuned to achieve maximum friction, for varying transmissivity of the near mir- ror. The friction amplitude (solid curve) approaches that for mirror mediated cooling using the far (dotted line, t > 1) or the near (dashed—dotted line, t + 0) mirror only in the appropriate limits. The arrow indicates the point at which the two cavity mirrors have the same reflectivity. Also shown is the intra-cavity field (dashed). (x ~ 400A0, L & 20000, |T| = 0.01, Ao = 780 nm, finesse at peak friction 5.0 x 10*.)
FIG. 4: In region (b) of Fig. 3, the friction coefficient ampli- tude (solid curves) is attenuated due to the attenuation in the field reflected from the cavity (dashed). |Z'| = 0.01 in every plot; |t| is, from left to right, 6.7x 10~°, 8.3x 10-3, 1.0x 1077, 1.2 x 10~°, and 1.5 x 107°. (Parameters as in Fig. 3.)
FIG. 4: In region (b) of Fig. 3, the friction coefficient ampli- tude (solid curves) is attenuated due to the attenuation in the field reflected from the cavity (dashed). |Z'| = 0.01 in every plot; |t| is, from left to right, 6.7x 10~°, 8.3x 10-3, 1.0x 1077, 1.2 x 10~°, and 1.5 x 107°. (Parameters as in Fig. 3.)
FIG. 1: Schematics of optical cooling methods: (a) Free- space laser cooling, (b) cavity-mediated cooling of a particle between two partially reflective mirrors, (c) mirror-mediated cooling of a particle using one perfectly reflecting and one partially reflective mirror.
FIG. 1: Schematics of optical cooling methods: (a) Free- space laser cooling, (b) cavity-mediated cooling of a particle between two partially reflective mirrors, (c) mirror-mediated cooling of a particle using one perfectly reflecting and one partially reflective mirror.
FIG. 3: Averaged friction coefficient (p(x))5 for a trap cen- tered at a point of maximum friction versus trap size 6 for s =0.1, 7 = 10 ns, and w = 0.5 yum (solid line), w = 1 pm (dashes), w = 2 wm (dotted). FIG. 2: Cooling time of rubidium atoms at position of max- imum friction versus beam waist w for s = 0.1 and 7 = 1 ns (solid line), 10 ns (dashed), 100 ns (dotted), and 1 ps (dash- dotted).
FIG. 3: Averaged friction coefficient (p(x))5 for a trap cen- tered at a point of maximum friction versus trap size 6 for s =0.1, 7 = 10 ns, and w = 0.5 yum (solid line), w = 1 pm (dashes), w = 2 wm (dotted). FIG. 2: Cooling time of rubidium atoms at position of max- imum friction versus beam waist w for s = 0.1 and 7 = 1 ns (solid line), 10 ns (dashed), 100 ns (dotted), and 1 ps (dash- dotted).
FIG. 4: Stationary temperature T at position of maximum friction versus beam waist w for s = 0.1 and 7 = 1 ns (solid line), 10 ns (dashed), 100 ns (dotted), and 1 jus (dash-dotted).
FIG. 4: Stationary temperature T at position of maximum friction versus beam waist w for s = 0.1 and 7 = 1 ns (solid line), 10 ns (dashed), 100 ns (dotted), and 1 jus (dash-dotted).
FIG. 5: Momentum of a single-particle trajectory in the ab- sence of diffusion. (a) Square of momentum versus time. (b) Detail of (a) with fitted turning points (crosses) using method 1. (c) Fitting method 2: squared momentum after subtraction of a least-square fit with a sine function (solid line) and linear fit to the result (dashed). Parameters are for Rb atoms, A = —10L, 7 = 0.25/T, w = 0.7 um, Aw = 0.17, w, = 0.5 x 2x1, s = 0.076. The trap is centered at a point of maximum friction.
FIG. 5: Momentum of a single-particle trajectory in the ab- sence of diffusion. (a) Square of momentum versus time. (b) Detail of (a) with fitted turning points (crosses) using method 1. (c) Fitting method 2: squared momentum after subtraction of a least-square fit with a sine function (solid line) and linear fit to the result (dashed). Parameters are for Rb atoms, A = —10L, 7 = 0.25/T, w = 0.7 um, Aw = 0.17, w, = 0.5 x 2x1, s = 0.076. The trap is centered at a point of maximum friction.
FIG. 6: Cooling rate dT'/dt versus initial temperature. The data points (crosses) and error bars are obtained from Monte- Carlo simulations using 10* trajectories per initial tempera- ture. The solid line is a linear fit to the numerical data. The parameters are as in Fig. 5.
FIG. 6: Cooling rate dT'/dt versus initial temperature. The data points (crosses) and error bars are obtained from Monte- Carlo simulations using 10* trajectories per initial tempera- ture. The solid line is a linear fit to the numerical data. The parameters are as in Fig. 5.
whereas its off-diagonal elements, (i|p**|j), are the respective coherences. The matrix elements of p** are obtained from the appropriate optical Bloch equations (see, for example, the procedure outlined in ref. [14]). We note here that, through its dependence on p*', M depends on the fields that it helps to determine, and thus Eq. (2) will in general become a set of nonlinear equations. In cases, like the ones considered in the following sections, where only one multilevel atom is interacting with a linear optical system, this problem may be solved using a procedure similar to the one outlined below: the fields surrounding the atom are obtained from the input fields through linear operations and then used with the optical Bloch equations to obtain the populations and coherences of the atom’s various levels. Knowledge of these quantities then determines the fields, and hence the forces acting on the atom, completely.
whereas its off-diagonal elements, (i|p**|j), are the respective coherences. The matrix elements of p** are obtained from the appropriate optical Bloch equations (see, for example, the procedure outlined in ref. [14]). We note here that, through its dependence on p*', M depends on the fields that it helps to determine, and thus Eq. (2) will in general become a set of nonlinear equations. In cases, like the ones considered in the following sections, where only one multilevel atom is interacting with a linear optical system, this problem may be solved using a procedure similar to the one outlined below: the fields surrounding the atom are obtained from the input fields through linear operations and then used with the optical Bloch equations to obtain the populations and coherences of the atom’s various levels. Knowledge of these quantities then determines the fields, and hence the forces acting on the atom, completely.
Here we will discuss what is perhaps the simplest tran- sition between two levels with multiple magnetic sub- levels: the J = 3 oJ = 3 transition. In this case, we have two ground sublevels so that py is a2 x 2 matrix. Fig. 2 tabulates the Clebsch-Gordan coefficients required to evaluate ¢. We thus have: Suppose, now, that we illuminate the atom with two counterpropagating beams having orthogonal linear po- larization and equal intensity. This can be represented by setting
Here we will discuss what is perhaps the simplest tran- sition between two levels with multiple magnetic sub- levels: the J = 3 oJ = 3 transition. In this case, we have two ground sublevels so that py is a2 x 2 matrix. Fig. 2 tabulates the Clebsch-Gordan coefficients required to evaluate ¢. We thus have: Suppose, now, that we illuminate the atom with two counterpropagating beams having orthogonal linear po- larization and equal intensity. This can be represented by setting
If we illuminate an atom with two counterpropagating beams of light in a ot-o~ configuration, rich dynamic: are obtained not in the simplest (J = $ + J’ = 2 case, but in the next simplest, where the ground stat« has three magnetic sublevels (J = 1) and the excitec state five (J’ = 2). In this case, then, we can express p and x as
If we illuminate an atom with two counterpropagating beams of light in a ot-o~ configuration, rich dynamic: are obtained not in the simplest (J = $ + J’ = 2 case, but in the next simplest, where the ground stat« has three magnetic sublevels (J = 1) and the excitec state five (J’ = 2). In this case, then, we can express p and x as
Figure 2. Cooling rate —(dv/dt)/v for ®°Rb pumped —10P from Dg resonance inside a ring-cavity with a round-trip length L = 300m, for two different values of the amplifier gain y. Note that the curve for y = 1, as drawn, is scaled up by a factor of 10. The cavity waist is taken to be 10 wm. (|t1|? = |t3|? = 0.5, |t2|? = 0.99, Bo is chosen such as to give an atomic saturation s = 0.1.)
Figure 2. Cooling rate —(dv/dt)/v for ®°Rb pumped —10P from Dg resonance inside a ring-cavity with a round-trip length L = 300m, for two different values of the amplifier gain y. Note that the curve for y = 1, as drawn, is scaled up by a factor of 10. The cavity waist is taken to be 10 wm. (|t1|? = |t3|? = 0.5, |t2|? = 0.99, Bo is chosen such as to give an atomic saturation s = 0.1.)
Figure 3. Equilibrium temperature predicted by the transfer matrix model for two values of the amplifier gain y. The Doppler temperature for ®°Rb is also indicated. The horizontal axis differs from that in Fig. 2 mainly because the temperature is only well-defined for regions where the friction force promotes cooling. (Parameters are as in Fig. 2.)
Figure 3. Equilibrium temperature predicted by the transfer matrix model for two values of the amplifier gain y. The Doppler temperature for ®°Rb is also indicated. The horizontal axis differs from that in Fig. 2 mainly because the temperature is only well-defined for regions where the friction force promotes cooling. (Parameters are as in Fig. 2.)
Fig. 2. (a) Model of a scatterer, S, inside a symmetrical Fabry—Pérot cavity of length L,. The cavity mirrors have reflection and transmission coefficients, t and t, and ¢ is the polarisability of S. (b) Friction coefficient, per unit input power, experienced by the scatterer at different po- sitions in the cavity, for realistic parameters (see text for details). André Xuereb et al.: Cavity cooling of atoms
Fig. 2. (a) Model of a scatterer, S, inside a symmetrical Fabry—Pérot cavity of length L,. The cavity mirrors have reflection and transmission coefficients, t and t, and ¢ is the polarisability of S. (b) Friction coefficient, per unit input power, experienced by the scatterer at different po- sitions in the cavity, for realistic parameters (see text for details). André Xuereb et al.: Cavity cooling of atoms
Fig. 3. External cavity cooling. (a) Model, similar to Fig. 2, but with the atom at a distance L — x outside the cavity. (b) Friction coefficient per unit input power experienced by the scatterer as x is varied, for realistic parameters (see text for details). Note the change of scale, on the vertical axis, from Fig. 2(b).
Fig. 3. External cavity cooling. (a) Model, similar to Fig. 2, but with the atom at a distance L — x outside the cavity. (b) Friction coefficient per unit input power experienced by the scatterer as x is varied, for realistic parameters (see text for details). Note the change of scale, on the vertical axis, from Fig. 2(b).
Fig. 4. Spatial dependence of the friction force acting on an atom inside a cavity with different mode waists but equal detuning from resonance, 10I° to the red. The smaller the mode waist the stronger the friction force, by several orders of magnitude, but the more significant lo calisation becomes. (Parameters as in Section 2.2 but witl aC /ak = 0.) Fig. 5. Spatial dependence of the friction force acting on an atom outside a cavity, with different pumping field waists but equal detuning from resonance, 10I° to the red. The friction force scales roughly as the inverse fourth power of the waist [26], but the length scale of the cool- ing and heating regions is unaffected. (Parameters as in Section 2.3 but with 0¢/0k = 0.)
Fig. 4. Spatial dependence of the friction force acting on an atom inside a cavity with different mode waists but equal detuning from resonance, 10I° to the red. The smaller the mode waist the stronger the friction force, by several orders of magnitude, but the more significant lo calisation becomes. (Parameters as in Section 2.2 but witl aC /ak = 0.) Fig. 5. Spatial dependence of the friction force acting on an atom outside a cavity, with different pumping field waists but equal detuning from resonance, 10I° to the red. The friction force scales roughly as the inverse fourth power of the waist [26], but the length scale of the cool- ing and heating regions is unaffected. (Parameters as in Section 2.3 but with 0¢/0k = 0.)
Fig. 6. Spatial dependence of the friction force acting on an atom outside a cavity, with detuning 10I to the red of resonance. Three different cavity lengths are shown; the friction force scales almost linearly with the cavity length. Note the different scaling factors and cavity lengths, given above each curve. (Parameters as in Section 2.3 but with aC¢/k = 0.)
Fig. 6. Spatial dependence of the friction force acting on an atom outside a cavity, with detuning 10I to the red of resonance. Three different cavity lengths are shown; the friction force scales almost linearly with the cavity length. Note the different scaling factors and cavity lengths, given above each curve. (Parameters as in Section 2.3 but with aC¢/k = 0.)
FIGURE B.2: A more in-depth treatment of mirror-mediated cooling and some specu- lation about possible experimental realisations.
FIGURE B.2: A more in-depth treatment of mirror-mediated cooling and some specu- lation about possible experimental realisations.
FicurE B.3: An explanation of laser cooling intended for a lay audience.
FicurE B.3: An explanation of laser cooling intended for a lay audience.
FIGURE B.4: Introduction to our transfer matrix theory and its use in explaining both mirror-mediated cooling and external cavity cooling.
FIGURE B.4: Introduction to our transfer matrix theory and its use in explaining both mirror-mediated cooling and external cavity cooling.
Figure B.5: An exploration of mirror-mediated cooling from the semi-classical and transfer matrix approaches, as well as external cavity cooling. Also some comments on the experimental apparatus used to conduct these investigations.
Figure B.5: An exploration of mirror-mediated cooling from the semi-classical and transfer matrix approaches, as well as external cavity cooling. Also some comments on the experimental apparatus used to conduct these investigations.
FIGURE B.6: Introduction of a classical, fully 3D and vectorial scattering theory to explain the interaction of a polarisable particle with its own reflection.
FIGURE B.6: Introduction of a classical, fully 3D and vectorial scattering theory to explain the interaction of a polarisable particle with its own reflection.
Related topics:
Quantum PhysicsScattering TheoryCold Atoms PhysicsPhysical OpticsLaser Cooling
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