Skip to main content
SpringerLink
Log in
Book cover

Macroscopic Matter Wave Interferometry pp 9–83Cite as

Interaction of Polarizable Particles with Light

  • Chapter
  • First Online:

  • 539 Accesses

Part of the book series: Springer Theses ((Springer Theses))

Abstract

A proper understanding of the mechanics and susceptibility of nanoparticles under the influence of coherent light fields will be a core ingredient throughout this thesis. It will be required for the description of new matter-wave interferometry schemes and of optical methods to manipulate the motion or measure the optical properties of molecules and clusters. This chapter is dedicated to the light–matter interaction in the presence of coherent laser fields, of high-finesse cavity modes and, not least, of a (thermally occupied) radiation field.

More light!—Johann Wolfgang von Goethe’s last words

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The full mathematical description of Gaussian light fields, as generated by focused laser beams or found in curved-mirror cavities, is a little bit more involved than presented here. I give a detailed formula for symmetric Gaussian mode functions with \(w_{x}=w_{y}=w\) in Appendix A.2. Strictly speaking, the above representations (2.2) and (2.3) are zeroth order approximations of the Gaussian mode fields in the waist parameter \(1/kw\), and additional polarization components must be taken into account for higher orders.

  2. 2.

    Note that the light-atom interaction can also be modeled by a complex linear polarizability provided the light is far detuned from any internal electronic transition and the transition is not strongly driven. In the latter case the atom’s response saturates at sufficiently high field intensities, as described by the Jaynes-Cummings model [4, 5].

  3. 3.

    The vector identities [7],

    $$\begin{aligned} \nabla \left( {\varvec{a}}\cdot {\varvec{b}}\right)&={\varvec{a}}\times \left( \nabla \times {\varvec{b}}\right) +{\varvec{b}}\times \left( \nabla \times {\varvec{a}}\right) +\left( {\varvec{b}}\cdot \nabla \right) {\varvec{a}}+\left( {\varvec{a}}\cdot \nabla \right) {\varvec{b}},\\ \left( \nabla \circ {\varvec{b}}\right) {\varvec{a}}&={\varvec{a}}\times \left( \nabla \times {\varvec{b}}\right) +\left( {\varvec{a}}\cdot \nabla \right) {\varvec{b}}, \end{aligned}$$

    might occasionally be useful here and in the following. The dyadic term \(B=\nabla \circ {\varvec{b}}\) is defined as the matrix \(B_{jk}=\partial b_{k}/\partial x_{j}\).

  4. 4.

    Stochastic differential equations can serve to describe the effective time evolution of open systems in contact with an environment inducing rapid (uncontrollable) state transitions that cannot be examined with the coarse-grained time resolution of observation [9]. The transitions thus show up as random events, or ‘jumps’. Using a Poissonian model we assume single infrequent jumps that can be clearly distinguished.

  5. 5.

    Using a fixed polarization \(\varvec{\epsilon }\left( {\varvec{r}}\right) = \varvec{\epsilon }\) is a good approximation in many practical cases such as Gaussian TEM modes, where position-dependent corrections are negligibly small. A detailed description of modes with a position-dependent polarization vector is more involved and requires a specific physical model of the particle’s response during the absorption process. This is because the orientation of the induced dipole moment then contains information about the position of the particle in the field mode, which is traced out when only the center-of-mass state is monitored.

  6. 6.

    Here I have used the identity \(\left[ {\mathbf {\mathsf{{p}}}} ^{2},f\left( {\mathbf {\mathsf{{r}}}} \right) \right] =-\hbar ^{2}\varDelta f\left( {\mathbf {\mathsf{{r}}}} \right) -2i\hbar \nabla f\left( {\mathbf {\mathsf{{r}}}} \right) \cdot {\mathbf {\mathsf{{p}}}} =\hbar ^{2}\varDelta f\left( {\mathbf {\mathsf{{r}}}} \right) -2i\hbar {\mathbf {\mathsf{{p}}}} \cdot \nabla f\left( {\mathbf {\mathsf{{r}}}} \right) \), as well as the fact that the mode function by construction solves the Helmholtz equation \(\varDelta u=-k^{2}u\).

  7. 7.

    In this approximation, the intensity profile merely limits the interaction time between the field mode and the PPP traversing the cavity volume.

  8. 8.

    A cavity linewidth of \(1\,\)MHz corresponds to a so-called cavity finesse parameter \(F=\pi c / 2\kappa L \approx 5 \times 10^{-5} \). The latter is related to the reflectivity \(R\) of both mirrors via the relation \(F=\pi \sqrt{R}/\left( 1-R \right) \) in the absence of additional losses in the resonator [28]. The suggested cavity setup requires \(1-R\approx 7\times 10^{-6}\).

  9. 9.

    Since the wavelength of the laser is required to be sufficiently well defined for the present purposes, ultrashort pulses with a broad frequency spectrum are excluded here.

  10. 10.

    To be more concrete, the travelled distance \(v_{z}\tau \) during the interaction period \(\tau \) between the particle and the field must be small compared to the laser wavelength, \(\left| v_{z}\tau \right| \ll \lambda \). Given the reduced one-dimensional quantum state of motion \(\rho _{z}\), the condition should cover its entire velocity distribution \(\langle mv_{z}|\rho |mv_{z}\rangle \).

  11. 11.

    Note that the direction of the grating is commonly referred to as the \(x\)-axis in the interferometry literature, whereas the standing wave is directed along \(z\) in the present notation, which is conventionally used in the description of light scattering at spherical particles. I will resort to the \(x\)-notation in Chap. 3.

  12. 12.

    A realistic description of the molecular beam state involves a broad distribution of velocities \(v\), and the resulting \(\phi _{0}\)-dependent interferogram must be averaged accordingly.

  13. 13.

    Also referred to as OTIMA: optical time-domain ionizing matter-wave interferometer.

  14. 14.

    The ionized particles are in practice removed from the ensemble with the help of a constant electric field applied to the interferometer setup.

  15. 15.

    This is intuitively clear since the absorption of a photon reveals the information that the particle is not located at a node. On the other hand, the photon cannot distinguish two positions \(z\) and \(z'\), which differ by an integer multiple of the wavelength.

  16. 16.

    I made use of the integral representation of the Bessel function \(\int _{0}^{2\pi }\text {d}\varphi \,\exp \left( i\xi \sin \varphi \right) =2\pi J_{0}\left( \xi \right) \), as well as of the integral identity [43]

    $$ \int \limits _{0}^{\pi /2}\text {d}\theta \, J_{0}\left( \beta \sin \theta \right) \sin \theta \cos ^{2r+1}\theta =2^{r}\beta ^{-1-r}\varGamma \left( r+1\right) J_{r+1}\left( \beta \right) , $$

    with \(r=\pm 1/2\) and the Gamma function \(\varGamma \left( 3/2\right) =\sqrt{\pi }/2=2\varGamma \left( 1/2\right) \). The identity leads naturally to spherical Bessel function expressions \(j_{n}\left( \beta \right) =\sqrt{\pi /2\beta }J_{n+1/2}\left( \beta \right) \), where \(j_{0}\left( \beta \right) =\text {sinc}\beta \).

  17. 17.

    The coupling strength between fields of different polarizations through a PPP may vary, most notably if the particle is described by a tensorial polarizability. I omit this additional modulation of the coupling for simplicity.

  18. 18.

    The form is easily obtained by adding the quantized physical fields (2.54) of the modes \( {\mathsf {a}} _{0}\) and \( {\mathsf {b}} \) to an overall electric and magnetic field. When the corresponding field energy density is integrated over the volume in (2.4) the cross terms between both modes yield the above linear coupling Hamiltonian. Terms of the form \( {\mathsf {a}} _{0} {\mathsf {b}} \) and \( {\mathsf {a}} _{0}^{{\dagger }} {\mathsf {b}} \) are omitted in the rotating wave approximation [45], since they oscillate rapidly at twice an optical frequency, \(\omega _{P}+\omega _{0}\), and thus do not affect the actual mode coupling.

  19. 19.

    A rotating frame is defined through the unitary state transformation \( {\mathsf {U}} \left( t\right) =\exp \left( i\omega _{P}\sum _{n} {\mathsf {a}} _{n}^{{\dagger }} {\mathsf {a}} _{n}\right) \), with \(\omega _{P}\) the corresponding rotation frequency. Given the quantum state \(\rho \left( t\right) \) of a system of field modes \(\left\{ {\mathsf {a}} _{n}\right\} \) in the Schrödinger picture, the state in the rotating picture reads as \(\rho '= {\mathsf {U}} \left( t\right) \rho \left( t\right) {\mathsf {U}} ^{{\dagger }}\left( t\right) \). Analogously, a displaced frame is defined via the unitary displacement operator, \(\rho '= {\mathsf {D}} ^{{\dagger }}\left( \beta \right) \rho {\mathsf {D}} \left( \beta \right) \). Field observables are displaced as \( {\mathsf {D}} ^{{\dagger }}\left( \beta \right) {\mathsf {b}} {\mathsf {D}} \left( \beta \right) = {\mathsf {b}} + \beta \).

  20. 20.

    Optical frequencies in the environmental mode spectrum are practically unoccupied even at finite temperatures. I thus use the zero-temperature vacuum state here, since the cavity only couples to modes of similar frequencies.

  21. 21.

    Practical implementations of the present cavity dissipation scheme are restricted to non-absorbing particle species. The typically large pump field intensities might otherwise lead to the destruction of the particle.

  22. 22.

    Applying the interaction Hamiltonian to the vacuum state leads to nondiagonal elements of the form \(|1_{n}\rangle \langle \text {vac}|\), with \(|1_{n}\rangle = {\mathsf {a}} _{n}^{{\dagger }}|\text {vac}\rangle \) a single-excitation multimode Fock state. One can easily show that these nondiagonals evolve like \(\exp \left( \mathcal {L} _{C}t\right) |1_{n}\rangle \langle \text {vac}|=\exp \left( -\kappa _{n}t-i\varDelta _{n}t\right) |1_{n}\rangle \langle \text {vac}|\).

  23. 23.

    Running-wave modes do not exhibit a wavelength-scale oscillatory intensity pattern. Their influence on the particle through the optical potential is considerably weaker.

  24. 24.

    Condition (2.83) would be alleviated by a factor \(kw\ll 1\) in the case of a Gaussian profile with waist \(w\).

  25. 25.

    Here I make use of the procedure of integration by parts and of the fact that a well-behaved and normalizable Wigner function should vanish at the infinities.

  26. 26.

    The diffusion matrix generally should be positive semidefinite in order to ensure that the occupied phase-space area increases and that the time evolution produces physical states at all times.

  27. 27.

    This would be a bad approximation if the underlying quantum state would be a delocalized matter-wave state that could be diffracted by the standing-wave structure of the pump mode (see Sect. 2.1.4).

  28. 28.

    Rayleigh scattering can be viewed as absorption and immediate reemission of a pump photon into free space. The absorption is responsible for the radiation pressure force, whereas the reemission does not contribute on average because there is no preferred direction of scattering, \(\left\langle n_{z} \right\rangle =\int \text {d}^{2}n\, R\left( {\varvec{n}}\right) n_{z}=0\).

  29. 29.

    The expression (2.112) ceases to be valid for large velocities and at \(\varDelta =0\), where the first-order friction term (2.108) vanishes.

  30. 30.

    This corresponds to the experimental situation when a dilute beam of molecules or nanoparticles crosses the cavity, such that there is on average only one particle inside the cavity at a given time.

  31. 31.

    Laguerre-Gaussian modes can only be used if the mirror system exhibits a cylindrical symmetry. This symmetry may be violated in applications with birefringent mirrors. In this case one must use the rectangular Hermite-Gaussian modes instead [64], which yield similar results as presented here.

  32. 32.

    Second order terms of the form \(z^{2}/d^{2}\), \(1/k^{2}d^{2}\) and \(z/kd^{2}\) are dropped. The inverse tangent is linearized as \(\arctan \left( 2z/d\right) \approx 2z/d\).

  33. 33.

    The average friction rate depends on the size of the averaging area. A larger area covers more space outside the cavity where the friction effect is zero. Nevertheless, the average value is a meaningful quantity to assess the net slowing of particles that are trapped in or passing the chosen region in a given amount of time.

  34. 34.

    Nonspherical particles scatter light in different patterns, that is, the multipole composition of the scattered field deviates from the spherical case. The basic consequences of leaving the subwavelength size regime are similar to the spherical case, with the additional complication of coupling to the rotation of the object.

  35. 35.

    I use that the spherical Bessel functions can be approximated by \(j_{\ell }\left( x\right) \approx x^{\ell }/ \left[ 1\cdot 3\cdot \ldots \cdot \left( 2\ell +1\right) \right] \) and \(y_{\ell }\left( x\right) \approx -\left[ 1\cdot 3\cdot \ldots \cdot \left( 2\ell -1\right) \right] / x^{\ell +1}\) to lowest order in \(x\ll 1\), while the spherical Hankel function becomes \(h_{\ell }\left( x\right) \approx iy_{\ell }\left( x\right) \).

  36. 36.

    All the calculations can be done in the vacuum surrounding the sphere, thus avoiding the discussion [72] which form of the Poynting vector to choose inside the medium.

  37. 37.

    This argument holds as long as the sphere is not too far away from the focus of the Gaussian mode. The reason is that the representation of the Gaussian TEM\(_{00}\) mode given in Appendix A.2 ceases to be valid for far-off center coordinates \(\left| {\varvec{r}}_{0}\right| \gg w\). This limit should hardly be of relevance in any practical implementation.

  38. 38.

    Decoherence by absorption becomes relevant, too, but only if the optical depletion effect requires more than one photon to be triggered.

  39. 39.

    The following integral identities for spherical Bessel functions must be used [43]:

    $$\begin{aligned} \int \limits _{0}^{X}\text {d}x\, x^2 j_{\ell }\left( x\right) j_{\ell }\left( nx\right)&=\frac{X^{2}}{1-n^{2}}\left[ j_{\ell +1}\left( X\right) j_{\ell }\left( nX\right) -nj_{\ell }\left( X\right) j_{\ell +1}\left( nX\right) \right] \\ \int \limits _{0}^{X}\!\! \text {d}x\left\{ \ell \left( \ell +1\right) j_{\ell }\left( x\right) j_{\ell }\left( nx\right) +\left[ xj_{\ell }\left( x\right) \right] '\left[ xj_{\ell }\left( nx\right) \right] '\right\}&=\frac{n X^{2}}{1-n^{2}}\left[ n j_{\ell +1}\left( X\right) j_{\ell }\left( nX\right) -j_{\ell }\left( X\right) j_{\ell +1}\left( nX\right) \right] \\&+\left( \ell +1\right) Xj_{\ell }\left( X\right) j_{\ell }\left( nX\right) \end{aligned}$$

References

  1. S. Nimmrichter, K. Hammerer, P. Asenbaum, H. Ritsch, M. Arndt, Master equation for the motion of a polarizable particle in a multimode cavity. New J. Phys. 12, 083003 (2010)

    Article  ADS  Google Scholar 

  2. H.C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981)

    Google Scholar 

  3. C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998)

    Google Scholar 

  4. A.P. Kazantsev, G.I. Surdutovich, V.P. Yakovlev, Mechanical Action of Light on Atoms (World Scientific, Singapore, 1990)

    Book  Google Scholar 

  5. S. Haroche, J.-M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford Graduate Texts) (Oxford University Press, Oxford, 2006)

    Google Scholar 

  6. S.M. Barnett, R. Loudon, On the electromagnetic force on a dielectric medium. J. Phys. B Atomic Mol. Opt. Phys. 39, S671 (2006)

    Article  ADS  Google Scholar 

  7. B. Thidé, Electromagnetic Field Theory (Online Version) (Dover, New York, 2011)

    Google Scholar 

  8. J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999)

    Google Scholar 

  9. H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)

    Google Scholar 

  10. H.M. Wiseman, G.J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2010)

    Google Scholar 

  11. M. Gallis, G. Fleming, Environmental and spontaneous localization. Phys. Rev. A 42, 38 (1990)

    Article  ADS  Google Scholar 

  12. S. Nimmrichter, K. Hornberger, H. Ulbricht, M. Arndt, Absolute absorption spectroscopy based on molecule interferometry. Phys. Rev. A 78, 063607 (2008)

    Article  ADS  Google Scholar 

  13. H. Du, R.-C.A. Fuh, J. Li, L.A. Corkan, J.S. Lindsey, PhotochemCAD: a computer-aided design and research tool in photochemistry. Photochem. Photobiol. 68, 141 (1998)

    Google Scholar 

  14. J.M. Dixon, M. Taniguchi, J.S. Lindsey, PhotochemCAD 2: a refined program with accompanying spectral databases for photochemical calculations. Photochem. Photobiol. 81, 212 (2007)

    Article  Google Scholar 

  15. K. Hornberger, J.E. Sipe, M. Arndt, Theory of decoherence in a matter mave Talbot-Lau interferometer. Phys. Rev. A 70, 53608 (2004)

    Article  ADS  Google Scholar 

  16. K. Hansen, E. Campbell, Thermal radiation from small particles. Phys. Rev. E 58, 5477 (1998)

    Article  ADS  Google Scholar 

  17. K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, M. Arndt, Colloquium: quantum interference of clusters and molecules. Rev. Mod. Phys. 84, 157 (2012)

    Article  ADS  Google Scholar 

  18. P. Horak, G. Hechenblaikner, K. Gheri, H. Stecher, H. Ritsch, Cavity-induced atom cooling in the strong coupling regime. Phys. Rev. Lett. 79, 4974 (1997)

    Article  ADS  Google Scholar 

  19. M. Gangl, H. Ritsch, Collective dynamical cooling of neutral particles in a high-Q optical cavity. Phys. Rev. A 61, 011402 (1999)

    Article  ADS  Google Scholar 

  20. V. Vuletić, S. Chu, Laser cooling of atoms. Ions Mol. Coherent Scattering Phys. Rev. Lett. 84, 3787 (2000)

    Google Scholar 

  21. H. Chan, A. Black, V. Vuletić, Observation of collective-emission-induced cooling of atoms in an optical cavity. Phys. Rev. Lett. 90, 063003 (2003)

    Article  ADS  Google Scholar 

  22. P. Maunz, T. Puppe, I. Schuster, N. Syassen, P.W.H. Pinkse, G. Rempe, Cavity cooling of a single atom. Nature 428, 50 (2004)

    Article  ADS  Google Scholar 

  23. S. Nußmann, K. Murr, M. Hijlkema, B. Weber, A. Kuhn, G. Rempe, Vacuum-stimulated cooling of single atoms in three dimensions. Nat. Phys. 1, 122 (2005)

    Article  Google Scholar 

  24. M. Wolke, J. Klinner, H. Keßler, A. Hemmerich, Cavity cooling below the recoil limit. Science 337, 75 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  25. B. Lev, A. Vukics, E. Hudson, B. Sawyer, P. Domokos, H. Ritsch, J. Ye, Prospects for the cavity-assisted laser cooling of molecules. Phys. Rev. A 77, 023402 (2008)

    Article  ADS  Google Scholar 

  26. C. Gardiner, P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer Series in Synergetics) (Springer, Berlin, 2010)

    Google Scholar 

  27. P. Asenbaum, Private Communication (University of Vienna, Austria, 2012)

    Google Scholar 

  28. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics 2nd edn. (Wiley, New York, 2007)

    Google Scholar 

  29. E.D. Palik, G. Ghosh, Handbook of Optical Constants of Solids (Academic Press, New York, 1985)

    Google Scholar 

  30. M.L. Gorodetsky, A.D. Pryamikov, V.S. Ilchenko, Rayleigh scattering in high-Q microspheres. J. Opt. Soc. Am. B 17, 1051 (2000)

    Article  ADS  Google Scholar 

  31. U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters (Springer Series in Materials Science) (Springer, Berlin, 1995)

    Google Scholar 

  32. M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerenes and Carbon Nanotubes: Their Properties and Applications (Academic Press, New York, 1996)

    Google Scholar 

  33. G. Lach, B. Jeziorski, K. Szalewicz, Radiative corrections to the polarizability of helium. Phys. Rev. Lett. 92, 233001 (2004)

    Article  ADS  Google Scholar 

  34. A. Miffre, M. Jacquey, M. Büchner, G. Trénec, J. Vigué, Measurement of the electric polarizability of lithium by atom interferometry. Phys. Rev. A 73, 011603 (2006)

    Article  ADS  Google Scholar 

  35. R.J. Glauber, High-energy collision theory. in Lectures in Theoretical Physics: Lectures Delivered at the Summer Institute for Theoretical Physics (University of Colorado, Boulder, Wiley, 1959), p. 315

    Google Scholar 

  36. S. Nimmrichter, Matter Wave Talbot-Lau Interferometry Beyond the Eikonal Approximation (Diploma thesis, Technische Universität München, 2007)

    Google Scholar 

  37. S. Nimmrichter, K. Hornberger, Theory of near-field matter-wave interference beyond the eikonal approximation. Phys. Rev. A 78, 023612 (2008)

    Article  ADS  Google Scholar 

  38. P.R. Berman, Atom Interferometry (Academic Press, San Diego, 1996)

    Google Scholar 

  39. A.D. Cronin, D.E. Pritchard, Optics and interferometry with atoms and molecules. Rev. Mod. Phys. 81, 1051 (2009)

    Article  ADS  Google Scholar 

  40. K. Hornberger, S. Gerlich, H. Ulbricht, L. Hackermüller, S. Nimmrichter, I.V. Goldt, O. Boltalina, M. Arndt, Theory and experimental verification of Kapitza-Dirac-Talbot-Lau interferometry. New J. Phys. 11, 43032 (2009)

    Article  Google Scholar 

  41. S. Nimmrichter, P. Haslinger, K. Hornberger, M. Arndt, Concept of an ionizing time-domain matter-wave interferometer. New J. Phys. 13, 075002 (2011)

    Article  ADS  Google Scholar 

  42. A. Turlapov, A. Tonyushkin, T. Sleator, Talbot-Lau effect for atomic de Broglie waves manipulated with light. Phys. Rev. A 71, 043612 (2005)

    Article  ADS  Google Scholar 

  43. I.S. Gradshteyn, I.M. Ryzhik, A. Jeffrey, D. Zwillinger, Table of Integrals, Series, and Products (Academic Press, New York, 2007)

    Google Scholar 

  44. S.M. Dutra, Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box (Wiley, New York, 2005)

    Google Scholar 

  45. R.J. Glauber, Quantum Theory of Optical Coherence (Wiley, New York, 2006)

    Google Scholar 

  46. D. Walls, G.J. Milburn, Quantum Optics 2nd edn. (Springer, Berlin, 2007)

    Google Scholar 

  47. G.J. Milburn, Lorentz invariant intrinsic decoherence. New J. Phys. 8, 96 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  48. G. Hechenblaikner, M. Gangl, P. Horak, H. Ritsch, Cooling an atom in a weakly driven high-Q cavity. Phys. Rev. A 58, 3030 (1998)

    Article  ADS  Google Scholar 

  49. M. Poot, H.S. van der Zant, Mechanical systems in the quantum regime. Phys. Rep. 511, 273 (2012)

    Article  ADS  Google Scholar 

  50. F. Marquardt, A. Clerk, S. Girvin, Quantum theory of optomechanical cooling. J. Mod. Opt. 55, 3329 (2008)

    Article  MATH  Google Scholar 

  51. S. Gröblacher, K. Hammerer, M.R. Vanner, M. Aspelmeyer, Observation of strong coupling between a micromechanical resonator and an optical cavity field. Nature 460, 724 (2009)

    Article  ADS  Google Scholar 

  52. M. Aspelmeyer, S. Gröblacher, K. Hammerer, N. Kiesel, Quantum optomechanics-throwing a glance. J. Opt. Soc. Am. B 27, A189 (2010)

    Article  ADS  Google Scholar 

  53. P. Domokos, H. Ritsch, Mechanical effects of light in optical resonators. J. Opt. Soc. Am. B 20, 1098 (2003)

    Article  ADS  Google Scholar 

  54. S. Stenholm, The semiclassical theory of laser cooling. Rev. Mod. Phys. 58, 699 (1986)

    Article  ADS  Google Scholar 

  55. J. Cirac, R. Blatt, P. Zoller, W. Phillips, Laser cooling of trapped ions in a standing wave. Phys. Rev. A 46, 2668 (1992)

    Article  ADS  Google Scholar 

  56. K. Jaehne, K. Hammerer, M. Wallquist, Ground-state cooling of a nanomechanical resonator via a Cooper-pair box qubit. New J. Phys. 10, 095019 (2008)

    Article  ADS  Google Scholar 

  57. P. Domokos, P. Horak, H. Ritsch, Semiclassical theory of cavity-assisted atom cooling. J. Phys. B At. Mol. Opt. Phys. 34, 187 (2001)

    Article  ADS  Google Scholar 

  58. H. Risken, T. Frank, The Fokker-Planck Equation: Methods of Solution and Applications (Springer Series in Synergetics) (Springer, Berlin, 2008)

    Google Scholar 

  59. W.P. Schleich, Quantum Optics in Phase Space (Wiley, Berlin, 2001)

    Google Scholar 

  60. S. Barnett, S. Stenholm, Hazards of reservoir memory. Phys. Rev. A 64, 033808 (2001)

    Article  ADS  Google Scholar 

  61. J. Salo, S.M. Barnett, S. Stenholm, Non-Markovian thermalisation of a two-level atom. Opt. Commun. 259, 772 (2006)

    Article  ADS  Google Scholar 

  62. J. Piilo, K. Härkönen, S. Maniscalco, K.-A. Suominen, Open system dynamics with non-Markovian quantum jumps. Phys. Rev. A 79, 062112 (2009)

    Article  ADS  Google Scholar 

  63. G.D. Boyd, J.P. Gordon, Confocal multimode resonator for millimeter through optical wavelength masers. Bell Syst. Tech. J 40, 489 (1961)

    Article  Google Scholar 

  64. N. Hodgson, H. Weber, Optical Resonators: Fundamentals, Advanced Concepts, Applications (Springer Series in Optical Sciences) (Springer, New York, 2005)

    Google Scholar 

  65. P. Asenbaum, Towards Cavity Cooling of a Molecular Beam (Diploma thesis, Universität Wien, 2009)

    Google Scholar 

  66. P. Domokos, H. Ritsch, Collective cooling and self-organization of atoms in a cavity. Phys. Rev. Lett. 89, 253003 (2002)

    Article  ADS  Google Scholar 

  67. A. Black, H. Chan, V. Vuletić, Observation of collective friction forces due to spatial self-organization of atoms: from rayleigh to bragg scattering. Phys. Rev. Lett. 91, 203001 (2003)

    Article  ADS  Google Scholar 

  68. A. Pflanzer, O. Romero-Isart, J.I. Cirac, Master-equation approach to optomechanics with arbitrary dielectrics. Phys. Rev. A 86, 013802 (2012)

    Article  ADS  Google Scholar 

  69. O. Romero-isart, M.L. Juan, R. Quidant, J.I. Cirac, Toward quantum superposition of living organisms. New J. Phys. 12, 33015 (2010)

    Article  Google Scholar 

  70. D.E. Chang, C.A. Regal, S.B. Papp, D.J. Wilson, J. Ye, O. Painter, H.J. Kimble, P. Zoller, Cavity opto-mechanics using an optically levitated nanosphere. Proc. Nat. Acad. Sci. 107, 1005 (2010)

    Article  ADS  Google Scholar 

  71. B. Dalton, E. Guerra, P. Knight, Field quantization in dielectric media and the generalized multipolar Hamiltonian. Phys. Rev. A 54, 2292 (1996)

    Article  ADS  Google Scholar 

  72. S.M. Barnett, R. Loudon, The enigma of optical momentum in a medium. Philos. Trans. A 368, 927 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  73. L.D. Landau, L.P. Pitaevskii, E. Lifshitz, Electrodynamics of Continuous Media (Course of Theoretical Physics), 2nd edn, vol. 8 (Butterworth-Heinemann, 1984)

    Google Scholar 

  74. L. Tsang, J.A. Kong, K.-H. Ding, Scattering of Electromagnetic Waves: Theories and Applications (Wiley, New York, 2000)

    Google Scholar 

  75. J.A. Stratton, Electromagnetic Theory (Wiley, New York, 2007)

    Google Scholar 

  76. S. Nimmrichter, K. Hornberger, P. Haslinger, M. Arndt, Testing spontaneous localization theories with matter-wave interferometry. Phys. Rev. A 83, 43621 (2011)

    Article  ADS  Google Scholar 

  77. O. Romero-Isart, A.C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, J.I. Cirac, Large Quantum Superpositions and Interference of Massive Nanometer-Sized Objects. Phys. Rev. Lett. 107, 20405 (2011)

    Article  ADS  Google Scholar 

  78. D.M. Pozar, Microwave Engineering 4th edn. (Wiley, New York, 2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Duisburg-Essen, Duisburg, Germany

    Stefan Nimmrichter

Authors
  1. Stefan Nimmrichter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Nimmrichter .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Nimmrichter, S. (2014). Interaction of Polarizable Particles with Light. In: Macroscopic Matter Wave Interferometry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-07097-1_2

Download citation

Publish with us

Policies and ethics

Search

Navigation