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Dispersion forces in Micromechanics: Casimir and Casimir-Polder forces affected by geometry and non-zero temperature
Norwegian University of Science and Technology, Faculty of Engineering Science and Technology, Department of Energy and Process Engineering.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The present thesis focuses on several topics within three separate but related branches of the overall field of dispersion forces. The three branches are: temperature corrections to the Casimir force between real materials (Part 1), explicit calculation of Casimir energy in wedge geometries (Part 2), and Casimir-Polder forces on particles out of thermal equilibrium (Part 3).

Part 1 deals primarily with analysis of a previously purported thermodynamic inconsistency in the Casimir-Lifshitz free energy of the interaction of two plane mirrors – violation of the third law of thermodynamics – when the latter’s dielectric response is described with dissipative models. It is shown analytically and numerically that the Casimir entropy of the interaction between two metallic mirrors described by the Drude model does tend to zero at zero temperature, provided electronic relaxation does not vanish. The leading order terms at low temperature are found. A similar calculation is carried out for the interaction of semiconductors with small but non-zero DC conductivity. In a generalisation, it is shown that a violation of the third law can only occur for permittivities whose low-frequency behaviour is temperature dependent near zero temperature. A calculation using path integral methods shows that the low temperature behaviour of the interaction of fluctuating Foucault currents in two mirrors of Drude metal is identical to that of the full Casimir-Lifshitz free energy, reasserting a previous finding by Intravaia and Henkel that such fluctuating bulk currents are the physical reason for the anomalous entropy behaviour.

In a related effort, an analysis of the frequency dependence of the Casimir force by Ford is generalised to imperfectly reflecting mirrors. A paradox is pointed out, in that the effects of a perturbation of the reflecting properties of the mirrors in a finite frequency window can be calculated in two ways giving different results. It is concluded that optimistic conclusions reached by Ford based on one of these methods, which seems to allow radically changing and tailoring the Casimir force with engineered materials, can not be realised.

Part 2 presents several explicit calculations of the Casimir energy of different wedge and cylinder geometries. The Casimir energy of a perfectly conducting wedge intercut by a circularly cylindrical arc, either perfectly conducting or (magneto)dielectric, is calculated. The energy is found to include a singular and non-regularisable term due to the corners where the arc meets the wedge, whereas the finite part is an immediate generalisation of the previously known results for a circular cylinder. The energy of a magnetodielectric wedge obeying a criterion of isorefractivity (spatially uniform speed of light) superimposed coaxially on a perfectly conducting cylindrical shell is calculated. This is the first expression for the energy of a wedge which is not perfectly reflecting. Finally, the energy of the perfectly conducting wedge and arc (and, as a special case, cylinder) is extended to the case of non-zero temperatures. After a regularisation procedure making use of the Chowla-Selberg formula an analytical expression for the temperature-dependent energy at all temperatures is derived, and showed to coincide with previously calculated high-temperature asymptotics by Bordag, Nesterenko and Pirozhenko.

Part 3 considers numerical and analytical studies of the Casimir-Polder forces acting on particles prepared in a given eigenstate (or superposition of such) in an environment which is otherwise at thermal equilibrium. We first consider cold polar molecules outside a metallic halfspace. It is found that the force in the near-zone (non-retarded regime) is much weaker than what would result from a naïve perturbative calculation, and that in the far-zone (retarded regime) the force becomes spatially oscillatory. It is demonstrated how these spatial oscillations may be enhanced in a resonating planar cavity, although for polar molecules the resulting amplitude is still insufficient for observation. A cylindrical cavity, however, can achieve a better enhancement factor. The Casimir-Polder forces on Rydberg atoms near a surface are calculated; because of the very large transition dipole moments of Rydberg transitions, the force is enormous on an atomic scale. We show that the oscillating force on Rydberg atoms can be enhanced into the observable regime by use of a fine-tuned cylindrical cavity. A particle in an eigenstate which is in the non-retarded regime with respect to all its dominant transitions is shown to feel a Casimir-Polder force which is virtually independent of temperature from zero to room temperature and beyond. Both for cold polar molecules and Rydberg atoms, the temperature-independent regime extends to a few and hundreds of micrometers, respectively, and includes the separations generally accessed in experiments

Place, publisher, year, edition, pages
NTNU, 2011.
Series
Doctoral theses at NTNU, ISSN 1503-8181 ; 2011:29
Identifiers
URN: urn:nbn:no:ntnu:diva-12365ISBN: 978-82-471-2576-2 (printed ver.)ISBN: 978-82-471-2577-9 (electronic ver.)OAI: oai:DiVA.org:ntnu-12365DiVA: diva2:413131
Public defence
2011-03-04, 00:00
Available from: 2011-04-27 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
List of papers
1. Analytical and numerical verification of the Nernst theorem for metals
Open this publication in new window or tab >>Analytical and numerical verification of the Nernst theorem for metals
2007 (English)In: PHYS REV E, ISSN 1539-3755, Vol. 75, no 5, 051127- p.Article in journal (Refereed) Published
Abstract [en]

In view of the current discussion on the subject, an effort is made to show very accurately both analytically and numerically how the Drude dispersion model gives consistent results for the Casimir free energy at low temperatures. Specifically, for the free energy near T=0 we find the leading term proportional to T-2 and the next-to-leading term proportional to T-5/2. These terms give rise to zero Casimir entropy as T -> 0 and are thus in accordance with Nernst's theorem.

Identifiers
urn:nbn:no:ntnu:diva-12346 (URN)10.1103/PhysRevE.75.051127 (DOI)000246890100038 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
2. Frequency spectrum of the Casimir force: Interpretation and a paradox
Open this publication in new window or tab >>Frequency spectrum of the Casimir force: Interpretation and a paradox
2008 (English)In: Europhysics letters, ISSN 0295-5075, E-ISSN 1286-4854, Vol. 82, no 5, 53001- p.Article in journal (Refereed) Published
Abstract [en]

The frequency spectrum of the Casimir force between two plates separated by vacuum as it appears in the Lifshitz formalism is reexamined and generalised as compared to previous works to allow for imperfectly reflecting plates. As previously reported by Ford (Phys. Rev. A, 48 (1993) 2962), the highly oscillatory nature of the frequency dependence of the Casimir force points to possibilities for very large and indeed negative Casimir forces if the frequency-dependent dielectric response, epsilon(omega), of the materials could be tuned. A paradox occurs, however, because an alternative calculation of the effect of a perturbation of epsilon(omega) involving a Wick rotation to imaginary frequencies indicates only very modest effects. A recent experiment appears to convincingly rule out the reality of Ford's optimistic predictions, although given the enormous technological promise of such frequency effects, further theoretical and experimental study is called for. Copyright (c) EPLA, 2008.

Identifiers
urn:nbn:no:ntnu:diva-12347 (URN)10.1209/0295-5075/82/53001 (DOI)000256301100009 ()
Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
3. Temperature correction to Casimir-Lifshitz free energy at low temperatures: Semiconductors
Open this publication in new window or tab >>Temperature correction to Casimir-Lifshitz free energy at low temperatures: Semiconductors
2008 (English)In: PHYS REV E, ISSN 1539-3755, Vol. 78, no 2, 021117- p.Article in journal (Refereed) Published
Abstract [en]

The Casimir force and free energy at low temperatures have been the subject of focus for some time. We calculate the temperature correction to the Casimir-Lifshitz free energy between two parallel plates made of dielectric material possessing a constant conductivity at low temperatures, described through a Drude-type dielectric function. For the transverse magnetic (TM) mode such a calculation is made. A further calculation for the case of the TE mode is thereafter presented which extends and generalizes previous work for metals. A numerical study is undertaken to verify the correctness of the analytic results.

Identifiers
urn:nbn:no:ntnu:diva-12348 (URN)10.1103/PhysRevE.78.021117 (DOI)000259263600026 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
4. Nernst's heat theorem for Casimir-Lifshitz free energy
Open this publication in new window or tab >>Nernst's heat theorem for Casimir-Lifshitz free energy
2008 (English)In: PHYS REV E, ISSN 1539-3755, Vol. 78, no 2, 021120- p.Article in journal (Refereed) Published
Abstract [en]

Consideration of the Lifshitz expression for the Casimir free energy on the real frequency axis rather than the imaginary Matsubara frequencies as is customary sheds light on the ongoing debate regarding the thermodynamical consistency of this theory in combination with common permittivity models. It is argued that when permittivity is temperature independent over a temperature interval including zero temperature, a cavity made of causal material with continuous dispersion properties separated by vacuum cannot violate Nernst's theorem (the third law of thermodynamics). The purported violation of this theorem pertains to divergencies in the double limit in which frequency and temperature vanish simultaneously. While any model should abide by the laws of thermodynamics within its range of applicability, we emphasize that the Nernst heat theorem is a relevant criterion for choosing among candidate theories only when these theories are fully applicable at zero temperature and frequency.

Identifiers
urn:nbn:no:ntnu:diva-12349 (URN)10.1103/PhysRevE.78.021120 (DOI)000259263600029 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
5. Casimir Lifshitz pressure and free energy: exploring a simple model
Open this publication in new window or tab >>Casimir Lifshitz pressure and free energy: exploring a simple model
2008 (English)In: The Casimir Effect and Cosmology: A volume in honour of Professor Iver H. Brevik on the occasion of his 70th birthday / [ed] Odintsov, Sergei D.; Elizalde, Emilio; Gorbunova, Olesya G., (Tomsk State Pedagogical University Press , 2008, 45-60 p.Chapter in book (Refereed)
Place, publisher, year, edition, pages
(Tomsk State Pedagogical University Press, 2008
Identifiers
urn:nbn:no:ntnu:diva-12362 (URN)978-5-89428-311-1 (ISBN)
Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
6. Low temperature Casimir-Lifshitz free energy and entropy:the case of poor conductors
Open this publication in new window or tab >>Low temperature Casimir-Lifshitz free energy and entropy:the case of poor conductors
2009 (English)In: Journal of Physics, Conference Series, ISSN 1742-6588, E-ISSN 1742-6596, Vol. 161Article in journal (Refereed) Published
Identifiers
urn:nbn:no:ntnu:diva-12363 (URN)10.1088/1742-6596/161/1/012010 (DOI)
Note
© IOP Publishing 2011 Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
7. The Casimir frequency spectrum: can it be observed?
Open this publication in new window or tab >>The Casimir frequency spectrum: can it be observed?
2009 (English)In: Journal of Physics, Conference Series, ISSN 1742-6588, E-ISSN 1742-6596, Vol. 161Article in journal (Refereed) Published
Identifiers
urn:nbn:no:ntnu:diva-12364 (URN)10.1088/1742-6596/161/1/012011 (DOI)
Note
© IOP Publishing 2011 Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
8. Casimir-Foucault interaction: Free energy and entropy at low temperature
Open this publication in new window or tab >>Casimir-Foucault interaction: Free energy and entropy at low temperature
2010 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 82, no 3, 032504- p.Article in journal (Refereed) Published
Abstract [en]

It was recently found that thermodynamic anomalies which arise in the Casimir effect between metals described by the Drude model can be attributed to the interaction of fluctuating Foucault (or eddy) currents [F. Intravaia and C. Henkel, Phys. Rev. Lett. 103, 130405 (2009).] We focus on the transverse electric (TE) polarization, where the anomalies occur, and show explicitly that the two leading terms of the low-temperature correction to the Casimir free energy of interaction between two plates are identical to those pertaining to the Foucault current interaction alone, up to a correction which is very small for good metals. Moreover, a mode density along real frequencies is introduced, showing that the TE contribution to the Casimir free energy, as given by the Lifshitz theory, separates in a natural manner into contributions from eddy currents and propagating cavity modes, respectively. The latter have long been known to be of little importance to the low-temperature Casimir anomalies. This convincingly demonstrates that eddy current modes are responsible for the large temperature correction to the Casimir effect between Drude metals, predicted by the Lifshitz theory, but not observed in experiments.

Identifiers
urn:nbn:no:ntnu:diva-12350 (URN)10.1103/PhysRevA.82.032504 (DOI)000281779000005 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
9. Electrodynamic Casimir effect in a medium-filled wedge
Open this publication in new window or tab >>Electrodynamic Casimir effect in a medium-filled wedge
2009 (English)In: PHYS REV E, ISSN 1539-3755, Vol. 79, no 4, 041120- p.Article in journal (Refereed) Published
Abstract [en]

We re-examine the electrodynamic Casimir effect in a wedge defined by two perfect conductors making dihedral angle alpha=pi/p. This system is analogous to the system defined by a cosmic string. We consider the wedge region as filled with an azimuthally symmetric material, with permittivity and permeability epsilon(1), mu(1) for distance from the axis r < a, and epsilon(2), mu(2) for r>a. The results are closely related to those for a circular-cylindrical geometry, but with noninteger azimuthal quantum number mp. Apart from a zero-mode divergence, which may be removed by choosing periodic boundary conditions on the wedge, and may be made finite if dispersion is included, we obtain finite results for the free energy corresponding to changes in a for the case when the speed of light is the same inside and outside the radius a, and for weak coupling, parallel to epsilon(1)-epsilon(2)parallel to < 1, for purely dielectric media. We also consider the radiation produced by the sudden appearance of an infinite cosmic string, situated along the cusp line of the pre-existing wedge.

Keyword
Casimir effect, cosmology, free energy, geometry, quantum electrodynamics, quantum theory
Identifiers
urn:nbn:no:ntnu:diva-12352 (URN)10.1103/PhysRevE.79.041120 (DOI)000265941300031 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
10. Electrodynamic Casimir effect in a medium-filled wedge. II
Open this publication in new window or tab >>Electrodynamic Casimir effect in a medium-filled wedge. II
2009 (English)In: PHYS REV E, ISSN 1539-3755, Vol. 80, no 2, 021125- p.Article in journal (Refereed) Published
Abstract [en]

We consider the Casimir energy in a geometry of an infinite magnetodielectric wedge closed by a circularly cylindrical, perfectly reflecting arc embedded in another magnetodielectric medium, under the condition that the speed of light be the same in both media. An expression for the Casimir energy corresponding to the arc is obtained and it is found that in the limit where the reflectivity of the wedge boundaries tends to unity the finite part of the Casimir energy of a perfectly conducting wedge-shaped sheet closed by a circular cylinder is regained. The energy of the latter geometry possesses divergences due to the presence of sharp corners. We argue how this is a pathology of the assumption of ideal conductor boundaries and that no analogous term enters in the present geometry.

Keyword
Casimir effect, dielectric bodies, quantum electrodynamics, reflectivity, shapes (structures)
Identifiers
urn:nbn:no:ntnu:diva-12351 (URN)10.1103/PhysRevE.80.021125 (DOI)000269637800035 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
11. Casimir effect at nonzero temperature for wedges and cylinders
Open this publication in new window or tab >>Casimir effect at nonzero temperature for wedges and cylinders
2010 (English)In: PHYS REV D, ISSN 1550-7998, Vol. 81, no 6, 065031- p.Article in journal (Refereed) Published
Abstract [en]

We consider the Casimir-Helmholtz free energy at nonzero temperature T for a circular cylinder and perfectly conducting wedge closed by a cylindrical arc, either perfectly conducting or isorefractive. The energy expression at nonzero temperature may be regularized to obtain a finite value, except for a singular corner term in the case of the wedge which is present also at zero temperature. Assuming the medium in the interior of the cylinder or wedge be nondispersive with refractive index n, the temperature dependence enters only through the nondimensional parameter 2 pi naT, a being the radius of the cylinder or cylindrical arc. We show explicitly that the known zero-temperature result is regained in the limit aT -> 0 and that previously derived high-temperature asymptotics for the cylindrical shell are reproduced exactly.

Identifiers
urn:nbn:no:ntnu:diva-12353 (URN)10.1103/PhysRevD.81.065031 (DOI)000276195700114 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
12. Dynamics of thermal Casimir-Polder forces on polar molecules
Open this publication in new window or tab >>Dynamics of thermal Casimir-Polder forces on polar molecules
2009 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 79, no 5, 052903- p.Article in journal (Refereed) Published
Abstract [en]

We study the influence of thermal Casimir-Polder forces on the near-surface trapping of cold polar molecules, with emphasis on LiH and YbF near a Au surface at room temperature. We show that even for a molecule initially prepared in its electronic and rovibrational ground state, the Casimir-Polder force oscillates with the molecule-wall separation. The nonresonant force and the evanescent part of the resonant force almost exactly cancel at high temperature which results in a saturation of the (attractive) force in this limit. This implies that the Casimir-Polder force on a fully thermalized molecule can differ dramatically from that obtained using a naive perturbative expansion of the Lifshitz formula based on the molecular ground-state polarizability. A dynamical calculation reveals how the spatial oscillations die out on a typical time scale of several seconds as thermalization of the molecule with its environment sets in.

Keyword
Casimir effect, ground states, laser cooling, molecule-photon collisions, radiation pressure, rotational-vibrational states
Identifiers
urn:nbn:no:ntnu:diva-12354 (URN)10.1103/PhysRevA.79.052903 (DOI)000266500900108 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
13. Enhancement of thermal Casimir-Polder potentials of ground-state polar molecules in a planar cavity
Open this publication in new window or tab >>Enhancement of thermal Casimir-Polder potentials of ground-state polar molecules in a planar cavity
2009 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 80, no 2, 022901- p.Article in journal (Refereed) Published
Abstract [en]

We analyze the thermal Casimir-Polder potential experienced by a ground-state molecule in a planar cavity and investigate the prospects for using such a setup for molecular guiding. The resonant atom-field interaction associated with this nonequilibrium situation manifests itself in oscillating standing-wave components of the potential. While the respective potential wells are normally too shallow to be useful, they may be amplified by a highly reflecting cavity whose width equals a half-integer multiple of a particular molecular transition frequency. We find that with an ideal choice of molecule and the use of a cavity bounded by Bragg mirrors of ultrahigh reflectivity, it may be possible to boost the potential by up to two orders of magnitude. We analytically derive the scaling of the potential depth as a function of reflectivity and analyze how it varies with temperature and molecular properties. It is also shown how the potential depth decreases for standing waves with a larger number of nodes. Finally, we investigate the lifetime of the molecular ground state in a thermal environment and find that it is not greatly influenced by the cavity and remains in the order of several seconds.

Identifiers
urn:nbn:no:ntnu:diva-12356 (URN)10.1103/PhysRevA.80.022901 (DOI)000269638200110 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
14. Temperature-Independent Casimir-Polder Forces Despite Large Thermal Photon Numbers
Open this publication in new window or tab >>Temperature-Independent Casimir-Polder Forces Despite Large Thermal Photon Numbers
2010 (English)In: Physical Review Letters, ISSN 0031-9007, Vol. 104, no 22, 223003- p.Article in journal (Refereed) Published
Abstract [en]

We demonstrate that Casimir-Polder potentials can be entirely independent of temperature even when allowing for the relevant thermal photon numbers to become large. This statement holds for potentials that are due to low-energy transitions of a molecule placed near a plane metal surface whose plasma frequency is much larger than any atomic resonance frequencies. For a molecule in an energy eigenstate, the temperature independence is a consequence of strong cancellations between nonresonant potential components and those due to evanescent waves. For a molecule with a single dominant transition in a thermal state, upward and downward transitions combine to form a temperature-independent potential. The results are contrasted with the case of an atom whose potential exhibits a regime of linear temperature dependence. Contact with the Casimir force between a weakly dielectric and a metallic plate is made.

Identifiers
urn:nbn:no:ntnu:diva-12357 (URN)10.1103/PhysRevLett.104.223003 (DOI)000278303000004 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
15. Thermal Casimir-Polder shifts in Rydberg atoms near metallic surfaces (vol 82, 010901, 2010)
Open this publication in new window or tab >>Thermal Casimir-Polder shifts in Rydberg atoms near metallic surfaces (vol 82, 010901, 2010)
Show others...
2010 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 82, no 2, 029902- p.Article in journal (Refereed) Published
Identifiers
urn:nbn:no:ntnu:diva-12358 (URN)10.1103/PhysRevA.82.029902 (DOI)000280812300017 ()
Note
© 2011 American Physical Society Available from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved
16. Casimir-Polder potential and transition rate in resonating cylindrical cavities
Open this publication in new window or tab >>Casimir-Polder potential and transition rate in resonating cylindrical cavities
2010 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, E-ISSN 1094-1622, Vol. 82, no 3, 032516- p.Article in journal (Refereed) Published
Abstract [en]

We consider the Casimir-Polder potential of particles placed inside a metallic cylindrical cavity at finite temperatures, taking account of thermal nonequilibrium effects. In particular, we study how the resonant ( thermal nonequilibrium) potential and transition rates can be enhanced by fine tuning the radius of the cavity to match the transition wavelength of the dominant transitions of the particle. Numerical calculations show that the cavity-induced energy-level shift of atoms prepared in low-lying Rydberg states can be enhanced beyond 30 kHz, which is within the range of observability of modern experiments. Because the magnitude of the resonance peaks depends sensitively on the low-frequency dissipation of the cavity metal, experiments in this setup could be a critical test of the disputed thermal correction to the Casimir force between metal plates.

Identifiers
urn:nbn:no:ntnu:diva-12359 (URN)10.1103/PhysRevA.82.032516 (DOI)000282269100010 ()
Note
© 2011 American Physical SocietyAvailable from: 2011-03-29 Created: 2011-03-29 Last updated: 2011-04-27Bibliographically approved

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