Topics to be covered

1. Boundary effects of electromagnetic vacuum fluctuations on charged particlesDepartment of Physics National Dong Hwa University Da-Shin LeeTalk given atNational Tsing-Hua Univeristy4 December 2008

2. Topics to be covered • Influence on electron coherence from quantum electromagnetic fields in the presence of conducting plates Jen-Tsung Hsiang and Da-Shin Lee: Phys. Rev. D 73, 065022 (2006) • Stochastic Lorentz forces on a point charge moving near the conducting plate Jen-Tsung Hsiang, Tai-Hung Wu and Da-Shin Lee: Phys. Rev. D 77, 105021 (2008) • Effects of smeared quantum noise on the stochastic motion of the charged particle near a conducting plate Jen-Tsung Hsiang, Tai-Hung Wu and Da-Shin Lee: submitted to Phys. Rev. A

3. Coherence reduction of the electron due to electromagnetic vacuum fluctuations The interest in the decoherence phenomenon is motivated by the study of the experimental realization of quantum computers in which the central obstacle is to prevent the degradation of quantum coherence arising from a unavoidable coupling to the environment.

4. Coherence reduction of the electron due to electromagnetic vacuum fluctuations Influence of electron coherence from the coupling to quantum electromagnetic fields can be studied with an interference experiment through the effects of phase shift and contrast change of the interference pattern. The Lagrangian for a nonrelativistic electron coupled to electromagnetic fields is given by such a particle-field interaction ( the Coulomb gauge): Imposition of the boundary condition on quantum fields will result in modification of vacuum fluctuations that may further influence electron interference.

6. The closed time path formalism The initial density matrix for the electron and gauge fields is assumed to be factorizable as: The fields are assumed to be in thermal equilibrium with the density matrix given by: where is the free field Hamiltonian. Then, in the Schroedinger picture, the density matrix evolves in time as: We will take the limits:

7. The reduced density matrix of the electron by tracing out the fields becomes: Here we have introduced an identity in terms of a complete set of eigenstates Then, the matrix element of the time evolution operator can be expressed by the path integral.

8. Reduced density matrix

10. The closed-time-path formalism Suggested review article: D. Boyanovsky, M. D'Attanasio, H.J. de Vega, R. Holman, D.-S. Lee, and A. Singh : Proceedings of International School of Astrophysics, D. Chalonge: 4th Course: String Gravity and Physics at the Planck Energy Scale, Erice, Italy (1995) , hep-ph/9511361.

11. Decoherence functional & Phase shift Consider the electron initially being in a coherent superposition of two localized states with the distinct mean trajectories. Phase shift Decoherence functional Leading order effect comes from the contribution of the mean trajectory given by the external potential where the width of the wavefunction is ignored ( discussed later).

12. Gauge invariant decoherence functional where the closed worldline is for a moving electron along its path in the forward time direction and then along the path in the backward time direction. By means of the 4-dimensional Stokes' theorem, where the area element of the integral is bounded by a closed worldline of the electron in Minkowski spacetime. Decoherence is found sensitive to the field strength in the region in Minkowski spacetime where the electron is excluded. The decoherence effect is essentially driven by the non-static features of quantum fields.

13. Dipole approximation by considering small k modes consistent with nonrelativistic motion has been applied to account for E fields only. Evaluation of decoherence functional Unbounded case: worldlines of the electrons are given by Lorentz invariance of the W functional allows us to chose the observe moving with the velocity , in which the electrons are seen to have transverse motion in the z direction only.

14. Single plate: The tangential component of E fields and the normal component of B fields on the perfectly conducting plate surface located at the z=0 plane vanish. The image charge method:

15. Decoherence for a single plate (parallel) Single plate: worldlines of the electrons are given by Under the dipole approximation (small k), Electron coherence is restored for small z as in the case with no influence from electromagnetic fields due to the fact that E fields parallel to the plate surface vanish on the boundary. The boundary effect becomes irrelevant for large z.

16. Decoherence for a single plate (perpendicular) Single plate: worldlines of the electrons are given by Under the dipole approximation (small k), Boundary induced effects of vacuum fluctuations suppress electron coherence for small z. In particular, near the plate, since large E fields normal to the plate surface are induced. Decoherence reduces to the result without the boundary for large z .

17. Decoherence for double plates (parallel) Double plates: an additional plate is located at z=a plane The double prime in the summation assigns an extra normalization factor to the n=0 mode. Worldlines of the electrons: The presence of the second parallel plate further suppresses vacuum fluctuations of E fields in the direction parallel to the plate surface, thus again restores electron coherence.

18. Double plates: worldlines of the electrons are given by Decoherence for double plates (perpendicular) In this case, an additional parallel plate seems to boost vacuum fluctuations of E fields in the direction normal to the plate surface so as to further reduce electron coherence significantly. Thus, the presence of the conducting plate anisotropically modifies the electromagnetic vacuum fluctuations that in turn influence electron coherence.

19. Discussion on involved approximations • The finite conductivity effect: Now consider the boundary plate with finite • conductivity . path length Anglin & Zurek, quant-ph/9611049 The Joule energy loss rate for bulk currents inside the conductor induced by the motion of the surface charge with the same velocity of the electron can be given by: However, mean energy fluctuations of the electron owing to electromagnetic vacuum fluctuations along the plate surface are given by: Yu &Ford, PRD 70, 065009 (2004) Thus, the finite conductivity effect can be ignored as long as the Joule energy loss during the electron’s flight time is much smaller than its mean energy fluctuations driven by vacuum fluctuations:

20. Discussion on involved approximations • The electrostatic attraction arising from the image charge on the • electron: • It can be neglected as the time scale for the electron with a trajectory • at a height z above the plate, which might fall into the boundary due • to this attraction force, is much larger than the electron’s flight time. • Thus, • The spreading of the quantum state: • The increase in the size of the localized quantum state during the • electron’s flight time can be estimated as: • The spreading effect can be ignored when • leading to • The backreaction from the fields on the mean trajectory of the • electron ( for example: radiation reaction ) will contribute to the decoherence function of order , and thus, is ignored.

21. Summary • Coherence reduction of the electron due to electromagnetic vacuum fluctuations in the presence of the conducting plates is studied with an interference experiment within the context of the closed time path formalism where corrections beyond involved approximations can be systematically incorporated. • Decoherence of the electron driven by non-static quantum electromagnetic fields is found sensitive to the field strength in the region in Minkowski spacetime bounded by a closed worldline of the electron. • The plate boundary anisotropically modifies vacuum fluctuations that in turn affect the electron coherence, and it is found that electron coherence is restored as in the case with no influence from electromagnetic fields when the path plane is parallel to the plate surface, but reduced in the normal case. • Decoherence effect for localized states turns out too weak to be detected.

23. Q & A

24. Stochastic Lorentz forces on a point charge moving near the conducting plate When a charged particle interacts with quantized electromagnetic fields, a nonuniform motion of the charge will result in radiation that backreacts on itself through electromagnetic self-forces as well as the stochastic noise manifested from quantum field fluctuations will drive the charge into a zig-zag motion. We wish to explore further the anisotropic nature of vacuum fluctuations under the boundary by the motion of the charged particle near the conducting plate.

25. The Lagrangian for a nonrelativistic charged particle coupled to electromagnetic fields is given by such a particle-field interaction ( the Coulomb gauge): The initial density matrix for the particle and fields is assumed to be factorizable by ignoring the initial correlations: The fields are assumed to be in thermal equilibrium with the density matrix given by: where is the free field Hamiltonian. Then, in the Schroedinger picture, the density matrix evolves in time as:

26. The reduced density matrix of the particle by tracing out the fields becomes: Here we have introduced an identity in terms of a complete set of eigenstates Then, the matrix element of the time evolution operator can be expressed by the path integral.

27. Reduced density matrix

28. We also assume that the particle is initially in a localized quantum state, which can be approximated by the position eigenstate: The nonequilibrium partition function can be defined by taking the trace of the reduced density matrix over the particle variable. The limits have be taken at this moment.

30. The stochastic Langevin equation is then obtained by extremizing the stochastic effective action. We ignore intrinsic quantum fluctuations of the particle by assuming that the resolution of the length scale measurement is greater than its position uncertainty.

31. Remarks: The influence of electromagnetic fields appears as the nonMarkovian backreaction in terms of electromagnetic self forces , and stochastic noise, driving the charge into a fluctuating motion. This is the nonlinearLangevin equation on the charge's trajectory since the dissipation kernel as well as noise correlation are the functional of the trajectory. The noise-averaged result arises from classical effects. Fluctuations on the particle’s trajectory driven by the noise entirely are of the quantum origin as seen from an explicit dependence on the noise term.

32. Fluctuation-Dissipation theorem Fluctuation-Dissipation theorem plays a vital role in balancing between these two effects to dynamically stabilize the nonequilibrium Brownian motion in the presence of external fluctuation forces. The tangential component of E fields and the normal component of B fields on the perfectly conducting plate surface located at the z=0 plane vanish.

33. The corresponding fluctuation-dissipation theorem can be derived from the first principles calculation: The F-D theorem at finite-T The F-D theorem in vacuum

34. Gauge invariant expression Retarded E and B fields are obtained by introducing the Lienard-Wiechert potentials together with the Coulomb potential. Stochastic E and B fields involve only the transverse components of the gauge potentials because in the Coulomb gauge, the Coulomb potential is not a dynamical variable, and hence it has no corresponding stochastic component.

35. Langevin equation under the dipole approximation Dipole approximation will be applied for this nonrelativistic motion to account for the backreaction solely from E fields. The charged particle undergoes the harmonic motion with the small amplitude at . An additional component of the external potential is applied to counteract the Coulomb attraction from its image charge. The initial conditions are specified as which can be achieved by applying an appropriate external potential to hold the particle at the starting position with zero velocity. Then the applied potential is suddenly switched off to the harmonic motion potential.

36. The noise-averaged equation ( classical effect ) Backreaction from the free-space contribution entails the retarded Green's function nonvanishing for the lightlike spacetime intervals. The charge follows a timelike trajectory where radiation due to the charge’s nonuniform motion can backreact on itself at the moment just when radiation is emitted. It is given by , electromagnetic self force + UV-divergence absorbed by mass renormalization =the ADL equation. Backreaction owing to the boundary has a memory effect where emitted radiation backscatters off the boundary, and in turn alters the charge's motion at a later time.

37. ． The kernel can be found from inverse Laplace transform: where the Browish contour is to enclose all singularities counterclockwisely on the complex s plane. The branch-cut arises from discontinuity of the kernel. Since the cut lies within the region of where imaginary part of the self-energy nonvanishing. The pole equation: The poles originally in the first Rienmann sheet move to the second sheet due to the interaction with environment fields as long as the poles are in the cuts. The pole on the first sheet located in the positive real s axis corresponds to the runaway solution to be discarded.

38. Breit-Wigner shape The resonance mode with the peak around the oscillation frequency is found to have dominant contributions to the late time behavior: High frequency modes relevant to very early evolution are ignored.

40. Velocity fluctuations ( quantum effect ) It is of interest to study velocity fluctuations of this charged oscillator under fluctuating electromagnetic fields to see how they are affected by the boundary and asymptotically saturated as a result of the fluctuation-dissipation relation. Velocity fluctuations grow linearly in time at early stages, and then saturate to a constant at late times. Although they for two different orientations of the motion start off at different rates, the same saturated value is reached asymptotically.

41. The spectral density reveals the oscillatory behavior on k space over the change in k by . The function has a Breit-Wigner feature on k space peaked at about and its width being approximately of order at early times or at late times. The integrand has the linear k dependence for large k, leading to quadratic UV-divergence with the weak time dependence in velocity fluctuations.

42. Growing regime: Backreaction dissipation can be ignored. Velocity fluctuations thus mainly result from the stochastic noise. Velocity fluctuations are found to grow linearly with time. The growing rate is related to the relaxation constant out of the dissipation kernel due to the F-D relation. Quadratic UV-divergence is found to vary slowly in time. The effect of the stochastic noise on the oscillator is much weaker, leading to a smaller growing rate on the parallel motion than the normal one since E field fluctuations parallel to the plate vanish, but its normal components become doubled, compared with that without the boundary. The relaxation constant shares the similar feature as a result of the F-D relation. The presence of the boundary apparently modifies the behavior of the charged oscillator in an anisotropic way.

43. Saturation regime: We investigate the behavior of velocity fluctuations at late times by incorporating backreaction dissipation. Backreaction from the contribution of the resonance is isotropic due to delicate balancing effects between fluctuations and dissipation, and thus is solely determined by the motion of the charge. The high-k modes probe UV-divergence as well as the strong boundary dependence for small z on backreaction. As expected, the enhancement in velocity fluctuations arises in the normal motion for small z resulting from large E fields induced in that direction.

44. Fluctuations induced by the motion of the charge Discussion on the saturated value of velocity fluctuations The change in velocity fluctuations, as compared with a static charge interacting with electromagnetic fields in its Minkowski vacuum state, arises from the imposition of the conducting plate as well as the motion of the charge . The relative importance between two effects will be estimated by taking an electron as an example. Fluctuations induced by the boundary : Velocity fluctuations owing to the electron's motion are overwhelmingly dominant constrained by the electron’s plasma frequency as well as the width of the wave function

45. Summary • The influence of electromagnetic fields on a nonrelativistic point charge moving near the conducting plate is studied by deriving the nonlinear, nonMarkovian stochastic Langevin equation from Feynman-Vernon influence functional within the context of the closed time path formalism. • This stochastic approach incorporates not only backreaction dissipation on a charge in the form of retarded Lorentz forces, but also the stochastic noise manifested from electromagnetic vacuum fluctuations. • Under the dipole approximation, noise-averaged result reduces to the known ADL equation plus the corrections from the boundary, resulting from classical effects. Fluctuations on the trajectory driven by the noise are of quantum origins where the dynamics obeys the F-D relation. • Velocity fluctuations of the charged oscillator are to grow linearly with time in the early stage of the evolution at the rate, smaller in the parallel motion than that of the normal case. • Same saturated value is obtained asymptotically for both orientations of the motions due to delicate balancing effects between F & D by taking the electron as an example.

46. Q & A