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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Two-photon correlation of photon pairs: near field and polarization effects (Plenary Paper)

C. H. Raymond Ooi, Anil K. Patnaik, Marlan O. Scully

Published May 23, 2005• 8 citations
Publisher

Abstract

The original quantum eraser scheme [Scully and Druhl, Phys. Rev. A 25, 2208 (1982)] is based on the entanglement of the two sequentially and spontaneously emitted photons. We consider the scheme in which the first emitted photon is largely detuned from resonance. We use the Schrodinger equation approach and the Laplace transform method to obtain the Raman Emission Doublet(RED) state vector. An exact analytical expression for the two-photon correlation function is derived in a general form that applies to any polarization of the two detectors located at any position, including the near field regime where the distances are comparable to the wavelengths of the photons. Frequency dependent refractive indexes of the RED photons are also included.

DOI
10.1117/12.613119

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IMPROVING QUANTUM MICROSCOPY VIA RAMAN PHOTON PAIRS 1

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In ordinary (classical) microscopy and lithography, the limit of resolution is governed by the wavelength of light used to il-luminate the sample. However, by using the correlated photon pairs produced in Raman quantum erasure, the resolution of two-photon quantummicroscopy is substantially improved. To that end, we here derive the two-photon quantum state gener-ated by sequential Raman emissions and investigate potential applications. The photon-photon correlation function for Ra-man pairs is of interest in itself and has features in common with photon antibunching. 1Asim Barut enjoyed and brought joy to all of Physics. He was interested in Concepts of Physics, Vol. II (2005) 261

•Jan 1, 2005•1 citation
The original quantum eraser scheme [Scully and Druhl, Phys. Rev. A 25, 2208 (1982)] is based on the entanglement of the two sequentially and spontaneously emitted photons. We consider the scheme in which the first emitted photon is largely detuned from resonance. We use the Schrodinger equation approach and the Laplace transform method to obtain the Raman Emission Doublet(RED) state vector. An exact analytical expression for the two-photon correlation function is derived in a general form that applies to any polarization of the two detectors located at any position, including the near field regime where the distances are comparable to the wavelengths of the photons. Frequency dependent refractive indexes of the RED photons are also included.
The photon correlations of resonance fluorescence emitted by a degenerate two-level atom near a metal surface are studied. An intense non-monochromatic optical laser drives a low-lying atomic transition, and the emitted fluorescence is detected by a photon counter. An expression for the spontaneous decay operator is derived, and it appears that the various substates of the upper level have a different lifetime due to the presence of the surface, or equivalently, due to the loss of spherical symmetry. The temporal photon correlation function is then evaluated in terms of a photon-emission operator and the time evolution operator for the atomic density matrix. The general theory is exemplified with a detailed elaboration of the j/sub g/ = 0 yields j/sub e/ = 1 transition, for which the two-photon correlation is obtained explicitly as a function of the lifetime A(h), depending on the atom surface distance h, the laser linewidth, the detuning from resonance, and the dipole-coupling strength with the external field.
We show that by using the strongly correlated photon pairs generated in a Raman quantum erasure scheme (Scully M and Drühl K 1982 Phys. Rev. A 25 2208), it is possible to exceed the Rayleigh resolution limit of classical microscopy. The complete analysis of the underlying physics is given here. Further discussion of the physics and potential applications are presented in a companion paper (Scully M O 2004 Improving quantum microscopy and lithography via Raman photon pairs: I. Biological applications, submitted).
The probability of an atom undergoing a transition to a given final state by means of two-photon absorption is expressed for arbitrary field states in terms of the second-order normally-ordered field correlation function, evaluated at the position of the atom. In the case of stationary fields, an expression for the absorption rate is found which reduces to a particularly simple form for narrow-bandwidth fields near resonance. The effect of field statistics is illustrated by comparing the absorption rate w2(ch)w_{2}^{}{}_{}{}^{(\mathrm{ch})}w2​​(ch) for chaotic or Gaussian light to the rate w2(l)w_{2}^{}{}_{}{}^{(l)}w2​​(l) for laser light. The rate for laser light is calculated within the context of a particular model based on the assumption of fixed field amplitude and random frequency modulation, and the chaotic light to which it is compared is assumed to have the same (Lorentzian) power spectrum. When the width ${\ensuremath{\kappa}}_{f}$ of the final atomic level is much greater than the bandwidth bbb of the field, we obtain the previously derived result w2(ch)w2(l)=2\frac{w_{2}^{}{}_{}{}^{(\mathrm{ch})}}{w_{2}^{}{}_{}{}^{(l)}}=2w2​​(l)w2​​(ch)​=2. When ${\ensuremath{\kappa}}_{f}\ensuremath{\ll}b$, on the other hand, the ratio between the two rates depends on the (mean) frequency of the field, and assumes the (maximum) value 4 when the field is exactly on resonance.
We present a largely analytical theory for two-photon correlations G(2){G}^{(2)}G(2) between Stokes (s)(s)(s) and anti-Stokes (a)(a)(a) photon pairs from an extended medium (amplifier) composed of double-$\ensuremath{\Lambda}$ atoms in counterpropagating geometry. We generalize the parametric coupled equations with quantum Langevin noise given in a beautiful experimental paper of Balic et al. [Phys. Rev. Lett. 94, 183601 (2005)] beyond adiabatic approximation and valid for arbitrary strength and detuning of laser fields. We derive an analytical formula for cross correlation ${G}_{as}^{(2)}=⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}^{\ifmmode\dagger\else\textdagger\fi{}}(L){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}(L)⟩$ and use it to obtain results that are in good quantitative agreement with the experimental data. Results for Gas(2){G}_{as}^{(2)}Gas(2)​ obtained using our coupled equations are in good quantitative agreement with the results using the equations of Balic et al., while perfect agreement is obtained for sufficiently large detuning. We also compute the reverse correlation Gsa(2){G}_{sa}^{(2)}Gsa(2)​ which turns out to be negligibly small and remains classical while the cross correlation violates the Cauchy-Schwartz inequality by a factor of more than a hundred.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
In ordinary (classical) microscopy and lithography, the limit of resolution is governed by the wavelength of light used to il-luminate the sample. However, by using the correlated photon pairs produced in Raman quantum erasure, the resolution of two-photon quantummicroscopy is substantially improved. To that end, we here derive the two-photon quantum state gener-ated by sequential Raman emissions and investigate potential applications. The photon-photon correlation function for Ra-man pairs is of interest in itself and has features in common with photon antibunching. 1Asim Barut enjoyed and brought joy to all of Physics. He was interested in Concepts of Physics, Vol. II (2005) 261