arXiv subjects

L. P. Teo

Publications and source records attributed to L. P. Teo.

At least 19 recordsLinked to original sources

arXiv2015

Massive scalar Casimir interaction beyond proximity force approximation

Since massive scalar field plays an important role in theoretical physics, we consider the interaction between a sphere and a plate due to the vacuum fluctuation of a massive scalar field. We consider combinations of Dirichlet and Neumann boundary conditions. There is a simple prescription to obtain the functional formulas for the Casimir interaction energies, known as TGTG formula, for the massive interactions from the massless interactions. From the TGTG formulas, we discuss how to compute the small separation asymptotic expansions of the Casimir interaction energies up to the next-to-leading order terms. Unlike the massless case, the results could not be expressed as simple algebraic expressions, but instead could only be expressed as infinite sums over some integrals. Nonetheless, it is easy to show that one can obtain the massless limits which agree with previously established results. We also show that the leading terms agree with that derive using proximity force approximation. The dependence of the leading order terms and the next-to-leading order terms on the mass of the scalar field is studied both numerically and analytically. In particular, we derive the small mass asymptotic expansions of these terms. Surprisingly, the small mass asymptotic expansions are quite complicated as they contain terms that are of odd powers in mass as well as logarithms of mass terms.

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arXiv2015

Scalar cylinder-plate and cylinder-cylinder Casimir interaction in higher dimensional spacetime

We study the cylinder-plate and the cylinder-cylinder Casimir interaction in the $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations of massless scalar fields. Different combinations of Dirichlet (D) and Neumann (N) boundary conditions are imposed on the two interacting objects. For the cylinder-cylinder interaction, we consider the case where one cylinder is inside the other, and the case where the two cylinders are outside each other. By computing the transition matrices of the objects and the translation matrices that relate different coordinate systems, the explicit formulas for the Casimir interaction energies are derived. Using perturbation technique, we compute the small separation asymptotic expansions of the Casimir interaction energies up to the next-to-leading order terms. The leading terms coincide with the respective results obtained using proximity force approximation, which is of order $d^{-D+1/2}$, where $d$ is the distance between the two objects. The results on the next-to-leading order terms are more interesting and important. We find some universal behaviors. It is also noticed that for the case of Dirichlet-Dirichlet cylinder-plate interaction, the next-to-leading order term agrees with that obtained using derivative expansion. Hence, based on our results on other boundary conditions and on the cylinder-cylinder interaction, we postulate a formula for the derivative expansion to expand the Casimir interaction energy up to the next-to-leading order terms for DD, DN, ND and NN boundary conditions, for the interaction between two curved surfaces in $(D+1)$-dimensional Minkowski spacetime. It is found that the postulate agrees with our previous results on the sphere-sphere interactions except when $D=4$.

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arXiv2015

Fermionic Casimir interaction in cylinder-plate and cylinder-cylinder geometries

In this work, we consider the Casimir effect due to massless fermionic fields in the presence of long cylinders. More precisely, we consider the interaction between a cylinder parallel to a plate, between two parallel cylinders outside each other, and between a cylinder lying parallelly inside another cylinder. We derive the explicit formulas for the Casimir interaction energies and compute the leading and the next-to-leading order terms of the small separation asymptotic expansions. As expected, the leading order terms coincide with the proximity force approximations. We compare the results of the next-to-leading order terms of different quantum fields, and show that our results support the ansatz of derivative expansions.

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arXiv2015

Ferminoic Casimir effect between spheres

We consider the Casimir interaction between two spheres corresponding to massless Dirac fields with MIT-bag boundary conditions. Using operator approach, we derive the TGTG-formula for the Casimir interaction energy between the two spheres. A byproduct is the explicit formula for the translation matrix that relates the fermionic spherical waves in different coordinate systems. In the large separation limit, it is found that the order of the Casimir interaction energy is $L^{-5}$, where $L$ is the separation between the centers of the spheres. This order is intermediate between that of two Dirichlet spheres (of order $L^{-3}$) and two Neumann spheres (of order $L^{-7}$). In the small separation limit, we derive analytically the asymptotic expansion of the Casimir interaction energy up to the next-to-leading order term. The leading term agrees with the proximity force approximation. The result for the next-to-leading order term is compared to the corresponding results for scalar fields and electromagnetic fields.

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arXiv2014

Finite temperature Casimir interaction between spheres in $(D+1)$-dimensional spacetime: Exact computations and asymptotic expansions

We consider the finite temperature Casimir interaction between two Dirichlet spheres in $(D+1)$-dimensional Minkowski spacetime. The Casimir interaction free energy is derived from the zero temperature Casimir interaction energy via the Matsubara formalism. In the high temperature region, the Casimir interaction is dominated by the term with zero Matsubara frequency, and it is known as the classical term since this term is independent of the Planck constant $\hbar$. Explicit expression of the classical term is derived and it is computed exactly using appropriate similarity transforms of matrices. We then compute the small separation asymptotic expansion of this classical term up to the next-to-leading order term. For the remaining part of the finite temperature Casimir interaction with nonzero Matsubara frequencies, we obtain its small separation asymptotic behavior by applying certain prescriptions to the corresponding asymptotic expansion at zero temperature. This gives us a leading term that is shown to agree precisely with the proximity force approximation at any temperature. The next-to-leading order term at any temperature is also derived and it is expressed as an infinite sum over integrals. To obtain the asymptotic expansion at the low and medium temperature regions, we apply the inverse Mellin transform techniques. In the low temperature region, we obtain results that agree with our previous work on the zero temperature Casimir interaction.

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arXiv2014

Exact classical sphere-plate Casimir interaction in (D+1)-dimensional spacetime

We consider the high temperature limit of the Casimir interaction between a Dirichlet sphere and a Dirichlet plate due to the vacuum fluctuations of a scalar field in $(D+1)$-dimensional Minkowski spacetime. The high temperature leading term of the Casimir free interaction energy is known as the classical term since it does not depend on the Planck constant $\hbar$. From the functional representation of the zero temperature Casimir interaction energy, we use Matsubara formalism to derive the classical term. It can be expressed as a weighted sum over logarithms of determinants. Using similarity transforms of matrices, we re-express this classical term as an infinite series. This series is then computed exactly using generalized Abel-Plana summation formula. From this, we deduce the short distance asymptotic expansions of the classical Casimir interaction force. As expected, the leading term agrees with the proximity force approximation. The next two terms in the asymptotic expansion are also computed. It is observed that the ratio of the next-to-leading order term to the leading order term is proportional to the dimension of spacetime. Hence, a larger correction to the proximity force approximation is expected in spacetime with higher dimensions. This is similar to a previous result deduced for the zero temperature case.

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arXiv2014

Casimir interaction between spheres in $\boldsymbol{(D+1)}$-dimensional Minkowski spacetime

We consider the Casimir interaction between two spheres in $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar fields. We consider combinations of Dirichlet and Neumann boundary conditions. The TGTG formula of the Casimir interaction energy is derived. The computations of the T matrices of the two spheres are straightforward. To compute the two G matrices, known as translation matrices, which relate the hyper-spherical waves in two spherical coordinate frames differ by a translation, we generalize the operator approach employed in [IEEE Trans. Antennas Propag. \textbf{36}, 1078 (1988)]. The result is expressed in terms of an integral over Gegenbauer polynomials. Using our expression for the Casimir interaction energy, we derive the large separation and small separation asymptotic expansions of the Casimir interaction energy. In the large separation regime, we find that the Casimir interaction energy is of order $L^{-2D+3}$, $L^{-2D+1}$ and $L^{-2D-1}$ respectively for Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions, where $L$ is the center-to-center distance of the two spheres. In the small separation regime, we confirm that the leading term of the Casimir interaction agrees with the proximity force approximation, which is of order $d^{-\frac{D+1}{2}}$, where $d$ is the distance between the two spheres. Another main result of this work is the analytic computations of the next-to-leading order term in the small separation asymptotic expansion. This term is computed using careful order analysis as well as perturbation method. We find that when $D$ is large, the ratio of the next-to-leading order term to the leading order term is linear in $D$, indicating a larger correction at higher dimensions.

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arXiv2014

Casimir interaction between spherical and planar plasma sheets

We consider the interaction between a spherical plasma sheet and a planar plasma sheet due to the vacuum fluctuations of electromagnetic fields. We use the mode summation approach to derive the Casimir interaction energy and study its asymptotic behaviors. In the small separation regime, we confirm the proximity force approximation and calculate the first correction beyond the proximity force approximation. This study has potential application to model Casimir interaction between objects made of materials that can be modeled by plasma sheets such as graphene sheets.

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arXiv2013

Sphere-plate Casimir interaction in (D+1)-dimensional spacetime

In this paper, we derive the formula for the Casimir interaction energy between a sphere and a plate in $(D+1)$-dimensional Minkowski spacetime. It is assumed that the scalar field satisfies the Dirichlet or Neumann boundary conditions on the sphere and the plate. As in the $D=3$ case, the formula is of TGTG type. One of our main contributions is deriving the translation matrices which express the change of bases between plane waves and spherical waves for general $D$. Using orthogonality of Gegenbauer polynomials, it turns out that the final TGTG formula for the Casimir interaction energy can be simplified to one that is similar to the $D=3$ case. To illustrate the application of the formula, both large separation and small separation asymptotic behaviors of the Casimir interaction energy are computed. The large separation leading term is proportional to $L^{-D+1}$ if the sphere is imposed with Dirichlet boundary condition, and to $L^{-D-1}$ if the sphere is imposed with Neumann boundary condition, where $L$ is distance from the center of the sphere to the plane. For the small separation asymptotic behavior, it is shown that the leading term is equal to the one obtained using proximity force approximation. The next-to-leading order term is also computed using perturbation method. It is shown that when the space dimension $D$ is larger than 5, the next-to-leading order has sign opposite to the leading order term. Moreover, the ratio of the next-to-leading order term to the leading order term is linear in $D$, indicating a larger correction at higher dimensions.

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arXiv2013

Finite temperature Fermionic Casimir interaction in Anti-de Sitter spacetime

We study the finite temperature Casimir interactions on two parallel boundaries in the anti-de Sitter spacetime AdS$_{D+1}$ induced by the vacuum fluctuations of a massive fermionic field with MIT bag boundary conditions. As in the Minkowski spacetime, the Casimir interaction always tends to attract the two boundaries to each other at any temperature and for any mass. For bosonic fields, it is well known that the high temperature leading term of the Casimir interaction is linear in temperature. However, for fermionic fields, the Casimir interaction decays exponentially at high temperature due to the absence of zero Matsubara frequency.

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arXiv2013

Finite temperature Casimir effect on spherical shells in (D+1)-dimensional spacetime and its high temperature limit

We consider the finite temperature Casimir free energy acting on a spherical shell in (D+1)-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar and electromagnetic fields. Dirichlet, Neumann, perfectly conducting and infinitely permeable boundary conditions are considered. The Casimir free energy is regularized using zeta functional regularization technique. To renormalize the Casimir free energy, we compute the heat kernel coefficients $c_n$, $0\leq n\leq D+1$, from the zeta function $ζ(s)$. After renormalization, the high temperature leading term of the Casimir free energy is $-c_DT\ln T-T ζ'(0)/2$. Explicit expressions for the renormalized Casimir free energy and $ζ'(0)$ are derived. The dependence of the renormalized Casimir free energy on temperature is shown graphically.

math-ph↗

arXiv2013

Electromagnetic Casimir effect on the boundary of a D-dimensional cavity and the high temperature asymptotics

We consider the finite temperature Casimir stress acting on the boundary of a D>=3 dimensional cavity due to the vacuum fluctuations of electromagnetic fields. Both perfectly conducting and infinitely permeable boundary conditions are considered, and it is proved that they correspond mathematically to the relative and absolute boundary conditions. The divergence terms of the Casimir free energy are related to the heat kernel coefficients of the Laplace operator. It is shown that the Casimir stress is free of divergence if and only if D is exactly three. The high temperature asymptotics of the regularized Casimir free energy are also found to depend on the heat kernel coefficients. When D>3, renormalization is required to remove terms of order higher than or equal to T^2.

math-ph↗

arXiv2013

Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation

We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The leading order and next-to-leading order terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the leading order term coincides with that derived using the proximity force approximation. The result on the next-to-leading order term complements that found by Bimonte, Emig and Kardar [Appl. Phys. Lett. \textbf{100}, 074110 (2012)] using derivative expansion. Numerical results are presented when the dielectric functions are given by the plasma model or the Drude model, with the plasma frequency (for plasma and Drude models) and relaxation frequency (for Drude model) given respectively by 9eV and 0.035eV, the conventional values used for gold metal. It is found that if plasma model is used instead of Drude model, the error in the sum of the first two leading terms is at most 2%, while the error in $θ_1$, the ratio of the next-to-leading order term divided by $d/R$ to the leading order term, can go up to 4.5%.

quant-ph↗

arXiv2013

Casimir interaction between a sphere and a cylinder

We study the Casimir interaction between a sphere and a cylinder both subjected to Dirichlet, Neumann or perfectly conducting boundary conditions. Generalizing the operator approach developed by Wittman [IEEE Trans. Antennas Propag. 36, 1078 (1988)], we compute the scalar and vector translation matrices between a sphere and a cylinder, and thus write down explicitly the exact TGTG formula for the Casimir interaction energy. In the scalar case, the formula shows manifestly that the Casimir interaction force is attractive at all separations. Large separation leading term of the Casimir interaction energy is computed directly from the exact formula. It is of order $\sim \hbar c R_1/[L^2\ln(L/R_2)]$, $\sim \hbar c R_1^3R_2^2/L^6$ and $\sim \hbar c R_1^3/[L^4\ln(L/R_2)]$ respectively for Dirichlet, Neumann and perfectly conducting boundary conditions, where $R_1$ and $R_2$ are respectively the radii of the sphere and the cylinder, and $L$ is the distance between their centers.

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arXiv2012

Mode summation approach to Casimir effect between two objects

In this paper, we explore the TGTG formula from the perspective of mode summation approach. Both scalar fields and electromagnetic fields are considered. In this approach, one has to first solve the equation of motion to find a wave basis for each object. The two T's in the TGTG formula are T-matrices representing the Lippmann-Schwinger T-operators, one for each of the objects. The two G's in the TGTG formula are the translation matrices, relating the wave basis of an object to the wave basis of the other object. After discussing the general theory, we apply the prescription to derive the explicit formulas for the Casimir energies for the sphere-sphere, sphere-plane, cylinder-cylinder and cylinder-plane interactions. First the T-matrices for a plane, a sphere and a cylinder are derived for the following cases: the object is imposed with general Robin boundary conditions; the object is semitransparent; and the object is magnetodielectric. Then the operator approach is used to derive the translation matrices. From these, the explicit TGTG formula for each of the scenarios can be written down. Besides summarizing all the TGTG formulas that have been derived so far, we also provide the TGTG formulas for some scenarios that have not been considered before.

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arXiv2012

Zero and finite temperature Casimir effect of massive vector field between real metals

We consider the Casimir effect of a massive vector field between two semi-infinite dielectric slabs. We first derive the generalization of the Lifshitz formula that gives the Casimir interaction energy of two magnetodielectric slabs separated by a magnetodielectric medium due to the vacuum fluctuations of a massive vector field. We then discuss the asymptotic behaviors of the Casimir energy and the Casimir force in various limits, such as low temperature, high temperature, small mass, large mass, up to the first order in the finite conductivity correction, for two real metal semispaces whose dielectric property is described by the plasma model. Application to the Casimir effect in Randall-Sundrum spacetime is briefly discussed.

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arXiv2011

Casimir effect between two spheres at small separations

We consider the Casimir interaction between two spheres at zero and finite temperature, for both scalar fields and electromagnetic fields. Of particular interest is the asymptotic expansions of the Casimir free energy when the distance between the spheres is small. The scenario where one sphere is inside the other is discussed in detail. At zero temperature, we compute analytically the leading and the next-to-leading order terms from the functional determinant representation of the Casimir energy. As expected, the leading order term agrees with the proximity force approximation. The results for the next-to-leading order terms are new. In the limit where the radius of the outer sphere goes to infinity, the results for the sphere-plane geometry are reproduced. At finite temperature, the leading order term is computed and it is found to agree completely with the proximity force approximation in the medium and high temperature regions. For the scenario where two spheres are outside each other, analogous results are obtained. In the case of Dirichlet boundary conditions on both spheres, the next-to-leading order term of the zero temperature Casimir energy is found to agree with that computed recently using derivative expansion.

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arXiv2011

On the corrections beyond proximity force approximation (PFA)

We recalculate the first analytic correction beyond PFA for a sphere in front of a plane for a scalar field and for the electromagnetic field. We use the method of Bordag and Nikolaev [J.Phys.A, {\bf 41} (2008) p.164002]. We confirm their result for Dirichlet boundary conditions whereas we find a different one for Robin, Neumann and conductor boundary conditions. The difference can be traced back to a sign error. As a result, the corrections depend on the Robin parameter. Agreement is found with a very recent method of derivative expansion.

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