QUANTUM SQUEEZING TO QUANTUM ENTANGLEMENT
by
RAY-KUANG LEE
A DISSERTATION
Presented to the Department of Photonics
and the Institute of Electro-Optical Engineering, National Chiao-Tung University, Hsinchu, Taiwan
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in Electrical Engineering and Computer Science Hsinchu, Taiwan, December 2004
QUANTUM OPTICAL PHENOMENA IN PHOTONIC CRYSTALS -FROM QUANTUM SQUEEZING TO QUANTUM ENTANGLEMENT
Student: Ray-Kuang Lee Advisor: Dr. Yinchieh Lai
Department of Photonics and the Institute of Electro-Optical Engineering National Chiao-Tung University
Abstract
In this dissertation we study the quantum optical phenomena in photonic crystals. In the part of atom-light interaction, the steady state fluorescence spectra of a two-level atom embedded in a three-dimensional photonic bandgap crystal are predicted to get squeezed in the in-phase quadrature spectra. In the part of nonlinear photonic crystals, we use the back-propagation method to study the quantum fluctuations of optical Bragg solitons propagating in nonlinear fiber Bragg gratings and matter-wave gap solitons in optical lattices. Finally, new schemes for generating continuous-variable entangled states through continuous interaction of two solitons are proposed to produce entangled optical sources for quantum communication and computation.
CURRICULUM VITA
NAME OF AUTHOR : Ray-Kuang Lee PLACE OF BIRTH : Keelung, Taiwan DATE OF BIRTH : October 24, 1974 CONTACT : [email protected] DEGREES OF EDUCATION :
Doctor of Philosophy in EECS,
National Chiao-Tung University, Sep. 2000 - Dec. 2004. Master of Science in EECS,
National Chiao-Tung University, Sep. 1997 - Jun. 1999. Bachelor of Engineering in EECS,
National Taiwan University, Sep. 1993 - Jun. 1997. AREAS OF SPECIAL INTEREST :
Quantum optics, Nonlinear Physics, and Fiber Lasers. PROFESSIONAL EXPERIENCE :
Visiting Student at Nonlinear Physics Centre (Head: Yuri S. Kivshar), The Australian National University, Apr. 2004 - Dec. 2004. Assistant Research Scientist,
National Laboratory, Oct. 1999 - Apr. 2004. AWARDS AND HONORS :
• Award for Graduate Students Study Abroad Program, supported by National Science Council, Taiwan, 2004.
• Student Paper Award, ”Classical-interference analog of quantum fluc-tuations for bound-soliton pairs in fiber lasers,” in Optics and Pho-tonics Taiwan 2004.
PUBLICATION LIST:
Refereed Journal Papers
1. Ray-Kuang Lee and Yinchieh Lai, ”Amplitude-squeezed fiber-Bragg-grating solitons,” Phys. Rev. A 69, 021801(R) (2004); also as part of Virtual Journal of Nanoscale Science & Technology 9, Issue 8 (2004).
2. Ray-Kuang Lee and Yinchieh Lai, and Boris A. Malomed, ”Quan-tum Fluctuations around Bistable Solitons in the Cubic-Quintic nonlinear Schr¨odinger equation,” J. Opt. B 6, 367 (2004). 3. Ray-Kuang Lee and Yinchieh Lai, ”Quantum theory of fiber
Bragg grating solitons,” J. Opt. B 6, S638 (2004).
4. Ray-Kuang Lee and Yinchieh Lai, ”Fluorescence squeezing spec-tra near a photonic bandgap,” J. Opt. B 6, S715 (2004).
5. Ray-Kuang Lee, Yinchieh Lai, and Boris Malomed, ”Quantum correlations in bound-soliton pairs and trains in fiber lasers,” Phys. Rev. A 70, xxxxxx (2004); quant-ph/0408175.
6. Ray-Kuang Lee, Yinchieh Lai, and Boris Malomed, ”Generation of photon-number entangled soliton pairs through interactions,” Phys. Rev. A 71, xxxxxx (2005); quant-ph/0405138.
7. Ray-Kuang Lee, Elena A. Ostrovskaya, Yuri S. Kivshar, and Yinchieh Lai, ”Squeezing and entanglement of matter-wave gap solitons,” submitted for publication; quant-ph/0412036.
8. Ray-Kuang Lee, Yinchieh Lai, and Boris A. Malomed, ”Classical-interference analog of quantum fluctuations for bound-state soli-ton pairs,” submitted for publication.
9. Ray-Kuang Lee, and Yinchieh Lai, Yuri S. Kivshar, ”Quantum correlations in the soliton collisions,” submitted for publication.
Proceedings of Conference
1. Ray-Kuang Lee and Yinchieh Lai, ”Resonance Fluorescence Spectrum near Photonic Bandgap,” SPIE Photonics West 2003, 5000-20, San Jose, California, USA, 25-31 January (2003). 2. Ray-Kuang Lee and Yinchieh Lai, ”Resonance Fluorescence
Spectrum in a Two-Band Photonic Bandgap Crystal,” SPIE First International Symposium on Fluctuation and Noise, 5111-27, Santa Fe, New Mexico, USA, 1-4 June (2003).
3. Ray-Kuang Lee and Yinchieh Lai, ”Steady-state fluorescence spectrum of a two-level atom in three-dimensional photonic crys-tals,” CLEO/Pacific-Rrm 2003, P0405, Taipei, Taiwan 15-19 De-cember (2003).
4. Ray-Kuang Lee and Yinchieh Lai, ”Quantum theory of Bragg solitons: Linearized approach,” CLEO/Pacific-Rim 2003, P0401, Taipei, Taiwan, 15-19 December (2003).
5. Ray-Kuang Lee and Yinchieh Lai, ”Quantum Theory of Fiber Bragg Grating Solitons,” Optics and Photonics Taiwan 2003, PD1-4, National Taipei University of Technology, Taipei, Taiwan, 25-26 December (2003).
6. Ray-Kuang Lee and Yinchieh Lai, ”Quantum fluctuations of Bragg grating solitons in apodized fiber gratings,” Annual Meet-ings of The Physical Society of Taiwan, PE-57, National Tsing Hua University, Hsinchu, Taiwan, 9-11 February (2004).
7. Ray-Kuang Lee, Yinchieh Lai, and Boris Malomed, ”Squeezing of Bi-solitons in Cubic-Quintic Schr¨odinger Equations,” OSA Non-linear Guided Waves and Their Applications, MC-5, Toronto, Canada, 28-31 March (2004).
8. Ray-Kuang Lee and Yinchieh Lai, ”Interaction induced photon-number entangled temporal soliton pairs,” OSA Nonlinear Optics, FA3, Waikoloa, Hawaii, USA, 2-6 August (2004).
9. Ray-Kuang Lee and Yinchieh Lai, ”Classical-interference analog of quantum fluctuations for bound-soliton pairs in fiber lasers,” Optics and Photonics Taiwan 2004, C-SA-IV2-8, National Cen-tral University, Taoyuan, Taiwan, 18-19 December (2004).
Talks in Seminar/Workshop and Others
1. Ray-Kuang Lee, ”Quantum Information with Optical Systems,” Quantum Information Workshop, The Industrial Technology Re-search Institute (ITRI), Hsinchu, Taiwan, 1 October (2003). 2. Yinchieh Lai, and Ray-Kuang Lee, ”Quantum Optical Effects in
Photonic Crystals,” Workshop on Advanced Photonic-Crystal De-vices, National Taiwan Normal University, Taipei, Taiwan, 18-19 October (2003).
3. Ray-Kuang Lee, ”Quantum Theory of Bragg Solitons,” Depart-ment Seminar, Institute of Electro-Optical Engineering, National Chiao-Tung University, Hsinchu, Taiwan, 12 December (2003). 4. Ray-Kuang Lee and Yinchieh Lai, ”Quantum Optical Effects in
Photonic Crystals,” Optical Engineering Journal (in Traditional Chinese), 83, 48 (2003).
5. Ray-Kuang Lee, ”Quantum Noise Squeezing and Quantum Cor-relations of Optical Solitons,” Department Seminar, Nonlinear Physics Centre, Research School of Physical Sciences and Engi-neering, The Australian National University, Canberra, Australia, 19 May (2004).
6. Ray-Kuang Lee and Yinchieh Lai, ”Fluorescence Spectra of a Two-Level Atom Embedded in a Three-Dimensional Photonic Crystal,” quant-ph/0308106 (un-published paper).
TABLE OF CONTENTS
Chapter Page
1. Introduction . . . 1
1.1. Motivation . . . 1
1.2. Dissertation Organization . . . 7
2. Fluorescence Squeezing Spectra in 3D Photonic Crystals . . . 8
2.1. Introduction . . . 8
2.2. Fluorescence Spectra . . . 10
2.3. Fluorescence Squeezing Spectra . . . 18
2.4. Summary . . . 21
3. Quantum theory of Bragg solitons and gap solitons . . . 22
3.1. Introduction . . . 22
3.2. Quantum theory of FBG solitons . . . 24
3.3. Quantum theory of gap solitons . . . 35
3.4. Summary . . . 42
4. Soliton Entanglement . . . 44
4.1. Introduction . . . 44
4.2. Interaction induced soliton entanglement, NLSE model . . . 47
4.3. Entangled bound-state of solitons in CGLE model . . . 53
4.4. Entangled bound gap soliton pairs . . . 58
4.5. Summary . . . 59
5. Conclusions . . . 62
List of Figures
Figure (Color Online) Page
1.1 Australian Opal. . . 2
1.2 Illustration of a fiber Bragg grating soliton. . . 4
2.1 Spectra of the memory function. . . 13
2.2 Comparison of fluorescence spectra far from and near the bandedge. . . . 16
2.3 Evolution of resonance fluorescence spectrum near the bandedge. . . 17
2.4 Resonance fluorescence quadrature spectra near the bandedge. . . 19
2.5 Evolution of in-phase quadrature spectra near the bandedge. . . 20
3.1 Evolution of the fiber Bragg grating soliton. . . 26
3.2 Transmittance and squeezing ratio for fiber Bragg grating solitons. . . . 29
3.3 Optimal squeezing ratio for Bragg solitons. . . 30
3.4 Squeezing ratio versus FBG lengths and local oscillator phases. . . 31
3.5 Measurement scheme of direct FBG soliton squeezing detection. . . 33
3.6 Photon number squeezing ratio versus the FBG length. . . 34
3.7 Band-gap diagram for Bloch waves and the family of the gap solitons. . . 36
3.8 Time evolution of the optimal squeezing ratio for the gap soliton. . . 38
3.9 Quantum correlation spectra in the (x)-domain for gap solitons. . . 40
3.10 Squeezing ratios and the homodyne detection phases for gap solitons. . . 41
4.1 Time-domain photon-number correlations of two interacting solitons. . . 48
4.2 The photon-number correlation parameter for the soliton pair. . . 50 4.3 The photon-number correlation coefficient of interacting vectorial solitons. 51 4.4 Photon-number correlations for a bound soliton pair in the CGLE model. 55 4.5 Evolution of the photon-number correlation parameter for bound solitons. 56 4.6 Evolution of the photon-number correlation parameters for soliton trains. 57 4.7 Squeezing ratios and atom number correlation parameter gap-soliton pairs. 58
CHAPTER 1
Introduction
1.1. Motivation
The study of Photonic Crystals (PhCs), which is the electromagnetic analog of semiconductor crystals, can be tracked back to Lord Rayleigh in 1887 when he identi-fied the fact that the crystalline mineral with periodic ”twinning” planes has a narrow band gap prohibiting light propagation. This angle-dependent band gap effect is due to the multilayer thin film structure in one-dimension, similar to many other irides-cent colors in nature, such as butterfly wings, abalone shells, and Australian opals in Fig. 1.1 1.
Extending the idea to two and three dimensions, Yablonovitch [1] and John [2] in 1987 introduced the new kind of man-made crystals, called Electromagnetic Crystals or Photonic Bandgap Crystals, which also stirred the imagination toward compact photonic integrated circuits. In recent years, with the advance of new fabrication technologies, it has become more feasible to actually utilize higher dimensional peri-odic dielectric structures (or especially the photonic bandgap crystals) for modifying the properties of the photon states as well as the properties of the spontaneous
FIGURE 1.1: Due to the periodic structures in the surface, Australian Opal is the only gemstone that holds all colours of the spectrum. Every opal has a unique play of colour and pattern.
sion. Generally, photonic crystals possess photonic band gaps where light cannot propagate through the structure within a certain range of wavelengths (the bandgap) due to the lack of available photon states. The introduction of defects in the photonic crystals (analogous to electronic dopants) can give rise to localized electromagnetic states, which can act as waveguides and point-like cavities. The photonic crystals thus can provide novel possibilities for the control of electromagnetic phenomena.
With the existence of a forbidden electromagnetic bandgap, photonic crystals can be used to not only control the flow of lights but also their quantum optical properties. It has been well known that the spontaneous emission from an excited atom can be modified by the electromagnetic reservoir that surrounds the atom (the Purcell effect [3]). In contrast to the free space case where the distribution of the photon density states is more uniform, the impacts of such a non-uniform photon state distribution upon the characteristics of the resonance fluorescence from a two-level atom with its
emission wavelength near the bandgap are investigated in Chapter 2. Such a non-uniform distribution of the photon density of states has been investigated by many authors and has provided a new and experimentally feasible platform for investigating photon-atom interaction. Many new quantum optical phenomena [4, 5, 6, 7, 8, 9] have been theoretically discovered in the presence of the bandgap. However, all of the above studies only focused on the transient behavior of the atom-photon interactions and to the best of our knowledge there is still no theoretical treatment on calculating the steady-state fluorescence spectra in PhCs. To investigate the steady-state problem of photon-atom interaction in PhCs, we use a new approach to treat the photon states of the photonic crystal as the background reservoir and introduce non-Markovian noise operators caused by the non-uniform DOS distribution near the band edge. Based on the theory, we show that the spectral shape of the fluorescence intensity spectra vary with the wavelength offset between the atomic transition wavelength and the bandedge. More interestingly, squeezing phenomena are found to be present in the in-phase quadrature spectra instead of the out-of-phase quadrature spectra.
In optical fibers, which are three-dimensional structures but with translational invariance along the propagation axis, one can induce one-dimensional Bragg gratings inside the fiber core by the side-illumination of the UV interference lights. The fiber Bragg gratings (FBGs) formed this way can also be viewed as a one-dimensional photonic bandgap crystal for the guiding mode of the single-mode fiber. It has been shown that a fiber Bragg grating with Kerr nonlinearity can exhibit optical soliton-like
FIGURE 1.2: Illustration of a fiber Bragg grating soliton proprogating inside a optical fiber.
phenomena known as the fiber Bragg grating solitons [10]. Intuitively the fiber Bragg grating soliton can be formed if the input pulse has the suitable pulse-width and peak intensity such that the nonlinear Kerr effect is large enough to compensate the high anomalous dispersion near one of the bandedge of the FBG, see the illustration in Fig. 1.2.
From the theoretical point of view, fiber Bragg grating solitons belong to the class of bi-directional pulse propagation problems, where the quantum theory is still lack of enough consideration. In Chapter 3, we extend the quantum theory of nonlinear traveling wave pulses into the case of nonlinear bi-directional propagation problems and give special consideration to the quantum propagation effects of fiber Bragg grating solitons. It will be shown that the fiber Bragg grating solitons will quantum-mechanically get amplitude-squeezed after passing through the fiber grating and the squeezing ratio can be calculated theoretically. With the use of apodized FBGs, we also find that one can tailor the squeezing ratio of the FBG solitons as long as the soliton pulses evolve adiabatically.
wave-packets in periodic structures, can also be found in the two- or three-dimensional nonlinear photonic bandgap crystals [11, 12]. The Bose-Einstein condensate (BEC) matter wave loaded into the optical lattices [13, 14, 15] is another important example, which have provided a new platform to understand the interactions between the nonlinearity and the periodicity. In Chapter 3 we also study quantum fluctuations of matter-wave gap solitons in an optical lattice and investigate the band-gap effect on the quantum noise squeezing. We employ the soliton perturbation approach [16] to analyze the quantum fluctuations around the soliton solutions of the Gross-Pitaevskii equation with a periodic potential. Using this approach, we demonstrate enhancement of quantum noise squeezing effects induced by the evolution of gap solitons in an atomic band-gap structure.
After examining the generation of squeezed states from Bragg and gap solitons, in Chapter 4 we propose new schemes for generating continuous-variable entangled states through continuous interaction of two fiber-optic solitons. In contrast to the known method for achieving an EPR pulse source by combining the two output pulse squeezed states through a beam splitter, almost perfect correlation between the photon-number fluctuations of the soliton pair can be achieved after their propagating a certain distance, if with a suitable initial separation between the two solitons. Nearly maximum photon-number entanglement in the soliton pairs can be produced in the systems including the single-polarization fiber soliton system with two time-separated solitons, the bimodal fiber soliton system with two polarizations, and the bound-state
system of gap solitons. These results offer novel possibilities to provide a favorable environment for the generation of entangled soliton pairs, which maybe the first step for developing new light sources for quantum communication and computation.
1.2. Dissertation Organization
The thesis is organized as follows. Following the introduction part in Chapter 1, the fluorescence intensity and quadrature spectra from a two-level atom embedded in a photonic bandgap crystal and resonantly driven by a classical pump light are calculated in Chapter 2. Then we use the nonlinear coupled mode equations to de-scribe the waves propagating in one-dimensional PhCs and study the quantum effects of optical solitons in FBGs by developing a general quantum theory of bi-directional nonlinear optical pulse propagation in Chapter 3. In the same chapter, we also extend the work to the matter-wave gap solitons described by the Gross-Pitaevskii equation with a periodic potential. In Chapter 4, new schemes for generating macro-scopic (many-photon) continuous-variable entangled states by means of continuous interactions between solitons are proposed and investigated. Finally, in Chapter 5 we briefly conclude the thesis and discuss the possible research directions that can be pursued in the future.
CHAPTER 2
Fluorescence Squeezing Spectra in 3D Photonic Crystals
2.1. Introduction
The study of fluorescence spectra from two-level atoms have been a central topic in quantum optics since the beginning era of quantum mechanics in 1930’s. From the view point of light scattering, both elastic Rayleigh scattering and inelastic Raman scattering processes are involved [17] and thus the fluorescence spectra will have a triplet shape as first calculated by Mollow [18]. Theoretical calculations of the fluo-rescence spectra has been explored [19, 20] and also verified in experiments [21]. The squeezing phenomena in the phase-dependent fluorescence spectra of the quadrature field components were first predicted by Walls and Zoller [22] and Mandel [23]. It has been theoretically shown that the squeezing can be found in the out-of-phase quadrature component spectra under the condition that Ω2 < 4Γ2 [24, 25], where Ω is the Rabi frequency and Γ is the atomic decay rate. Squeezed fluorescence spectra have also been experimentally observed in an experiment using 174Yb atoms [26].
In recent years the atom-photon interaction in photonic crystals (PhCs) [1, 2] has been found to exhibit many interesting new phenomena such as photon-atom bound states [4], spectral splitting [5], quantum interference dark line effect [6], phase
control of spontaneous emission [7], transparency near band edge [8], and single-atom switching [9]. From the Aulter-Townes spectra for single-atoms coupled to a photonic bandgap structure [5] (or equivalently a frequency-depended photon density of states [27]), the modification of the spontaneous emission caused by the environment (the Purcell effect [3]) actually can be verified. However, all of the above studies only focused on the transient behavior of the atom-photon interactions and to the best of our knowledge there is still no theoretical treatment on calculating the steady-state fluorescence spectra in photonic bandgap crystals.
In the theoretical studies of fluorescence spectra for the free space case, the Marko-vian approximation is usually used to describe the statistical properties of the optical noises. This is a good assumption for the free space case but is not applicable for the case of photonic bandgap crystals. This is because in a photonic bandgap crystal, the distributions of the photonic density of states (DOS) are typically highly non-uniform near the bandedge. Such a property has prohibited the direct applicability of the Markovian approximation to simplify the derivation for the problem we are going to consider.
The aim of this Chapter is to investigate the properties of the steady state res-onance fluorescence emitted by a two-level atom embedded in a photonic bandgap crystal and driven by a classical pumping light. We treat the photon states in the pho-tonic crystal as the background reservoir and obtain a set of generalized Bloch equa-tions for the atomic operators by eliminating the reservoir field operators. The
non-uniform DOS distributions near the bandedge are modeled by the three-dimensional anisotropic dispersion relation [28] and the Liouville operator expansion is used to reduce the two-time atomic operator products into equal-time atomic operator prod-ucts. In this way the nonlinear generalized Bloch equations are reduced into a set of linear equations with memory function terms caused by the atom-reservoir interac-tion. This set of linear equations can then be easily solved in the frequency domain and the resonance fluorescence spectra can be directly obtained from the correla-tion funccorrela-tions of the atomic operators in the frequency domain without applying the quantum regression theorem. After performing numerical calculation, we find that the spectral shape of the fluorescence intensity spectra will vary with the wavelength offset between the atomic transition wavelength and the bandedge. More interest-ingly, squeezing phenomena are found to be present in the in-phase quadrature spectra instead of the out-of-phase quadrature spectra.
2.2. Fluorescence Spectra
To begin the derivation, the Hamiltonian for the system to be considered can be written as: H = ~ωa 2 σz+ ~ X k ωka†kak+ Ω~ 2 (σ−e iωLt+ σ +e−iωLt) + ~ X k (gkσ+ak+ gk∗a † kσ−). (2.1)
Here the transition frequency of the atom and the frequency of the pumping light are denoted by ωa and ωL respectively, a†k and ak are the creation and annihilation
operators of the photon states in the photonic bandgap crystals, Ω is the Rabi-flopping frequency of the atom under the external pumping light and it also represents the relative magnitude of the pumping light, σz ≡ (|2ih2| − |1ih1|), σ+ ≡ |2ih1| = σ−†
are the usual Pauli matrices for the two-level atom, and gk is the atom-field coupling
constant.
Starting from the Hamiltonian, we treat the photon field operators as the back-ground reservoir and eliminate their corresponding equations to obtain the following set of generalized Bloch equations.
˙σ−(t) = i Ω 2σz(t)e −i∆t+ Z t −∞ d t′G(t − t′)σz(t)σ−(t′) + n−(t), (2.2) ˙σ+(t) = −i Ω 2σz(t)e i∆t+ Z t −∞ d t′G c(t − t′)σ+(t′)σz(t) + n+(t), (2.3)
˙σz(t) = iΩ [σ−(t)ei∆t− σ+(t)e−i∆t] + nz(t) (2.4)
− 2 Z t
−∞
dt′[G(t − t′)σ+(t)σ−(t′) + Gc(t − t′)σ+(t′)σ−(t)].
Here ∆ ≡ ωL− ωa and ∆k ≡ ωa− ωk. The two functions G(τ ) and Gc(τ ) are the
memory functions of the system caused by the atom-reservoir interaction and they are defined as G(τ ) ≡ X k |gk|2ei∆kτΘ(τ ), and Gc(τ ) ≡ X k |gk|2e−i∆kτΘ(τ ). Here n−(t),
n+(t), and nz(t) are three noise operators originated from the original photon filed
operator before interaction.
Supposing that the reservoir is in thermal equilibrium, it can be easily shown that the three noise operators n−(t), n+(t), and nz(t) are zero mean with their correlation
functions given below:
hn−(t)iR= hn+(t)iR= hnz(t)iR= hn−(t)n−(t′)iR= hn+(t)n+(t′)iR = 0, (2.5)
hn−(t)n+(t′)iR = X k |gk|2(¯nk+ 1)ei∆k(t−t ′) hσz(t)σz(t′)i, (2.6) hn+(t)n−(t′)iR = X k |gk|2n¯ke−i∆k(t−t ′) hσz(t)σz(t′)i, (2.7) hnz(t)nz(t′)iR = 4 X k |gk|2[(¯nk+ 1)ei∆k(t−t ′) hσ+(t)σ−(t′)i (2.8) + ¯nke−i∆k(t−t ′) hσ−(t)σ+(t′)i].
Since in general the correlation functions of these noise operators are not delta correlated at time (non-Markovian), we cannot directly use the Born-Markovian ap-proximation to solve the problem. One can see that the correlation functions depend not only on the photon density of states, but also on the correlations of the atomic operators.
To actually evaluate the memory functions as well as the correlation functions of the noise operators, one needs to know the spectral distribution of the photonic density of states. Although in general the DOS of photonic bandgap crystals is very complicated and also varies with the geometrical structure and the dielectric constants of the material, it is still possible to approximately model the DOS near the bandedge with a simple formula. According to the results from the full vectorial numerical calcu-lation, the DOS near the bandgap for three-dimensional photonic crystals increases from zero and behaves more like the anisotropic model proposed in the literature [28]. To be more specific, for a three dimensional photonic bandgap crystal, if the
ω/β a b s [G ( ω )] a rg [G ( ω )] -100 0 100 200 300 400 0.05 0.06 0.07 0.08 0.09 0.1 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 ωc/β
FIGURE 2.1: Amplitude and phase spectra of the memory function G(ω) with band-edge frequency ωc = 100β. The memory function is non-uniform around the bandedge
and becomes pure imaginary inside the bandgap.
wavevector that corresponds to the bandedge is ki0, then the dispersion relation in the anisotropic model is described by the following form: ωk = ωc+ A|k − ki0|2, where A
is a model dependent constant and ωc is the bandedge frequency. Based on this
dis-persion relation, the corresponding DOS is given by: D(ω) = 1 A3/2
√
ω − ωcΘ(ω −ωc).
The memory functions under this anisotropic model also can be derived as:
˜ G(ω) = β3/2√ −i ωc+√ωc− ωa− ω , (2.9) ˜ Gc(ω) = β3/2 i √ω c+√ωc− ωa+ ω . (2.10) where β3/2 = ω2ad2 6~ǫ0πA3/2
η, and we have used the space average coupling strength η ≡ 8π3
Z
dΩ|ˆd · E|2 in the derivation.
From Fig. 2.1 the spectrum of the memory function G(ω) is non-uniform and asymmetric as we expect. When the frequency is far from the bandedge frequency ωc,
the memory function is a pure real function which corresponding to the decay rate of the atom. When the frequency is below the bandedge frequency, the memory function becomes pure imaginary, indicating the inhibition of the spontaneous emission inside the bandgap. And in between the memory function is a complex function, of which the real part is related to the decay process and the imaginary part is related to the oscillation process. The spectrum for another memory function Gc(ω) is also similar.
The generalized Bloch equations are a set of nonlinear operator equations and can-not be solved easily. To overcome this difficulty, we introduce the following Liouville operator expansion: σij(t) = e−iL(t−t ′) σij(t′) = ∞ X n=0 [−i(t − t′)]n n! L nσ ij(t′). (2.11)
Here the Liouville super operator L is defined as Lnσij(t′) =
1
~n[· · · [σij(t ′
), H], H], · · · , H]. If the atom we consider is with a longer lifetime and under weak pumping (small Rabi frequency), which is the usual case in the optical domain, then the time scale of the atomic evolution will be longer than the time scale of the memory functions. There-fore under such assumptions it should be legitimate to simply apply the zero-th order perturbation term (zeroth-order Born approximation) [9]. This is equivalent to use the equal time operator products to replace the two-time operator products. With these approximations and by using the identities of Pauli matrices, the generalized
optical Bloch equations can be reduced into the following form: ˙σ−(t) = i Ω 2σz(t)e −i∆t− Z t −∞ dt′ G(t − t′)σ −(t′) + n−(t), (2.12) ˙σ+(t) = −i Ω 2σz(t)e i∆t − Z t −∞ dt′Gc(t − t′)σ+(t′) + n+(t), (2.13)
˙σz(t) = iΩ(σ−(t)ei∆t− σ+(t)e−i∆t) (2.14)
− Z t
−∞
dt′[G(t − t′) + Gc(t − t′)](1 + σz(t′)) + nz(t).
The approximation we have used should be valid as long as the time scale of the memory function is still much shorter than the time scale of the atomic response (i.e., the inverse of the Rabi frequency and the decay rate). It can be easily checked that the full-width-half-maximum (FWHM) bandwidth of the memory functions in Eq. (2.9) and Eq. (2.10) are 4ωc. For a bandgap in the optical domain, the order of ωc is
about 1014−15 Hz, and the typical lifetime of the atom is from 10−3 sec to 10−9 sec, which is much longer than the response time of the memory functions.
Theoretically the fluorescence spectrum can be calculated by taking the Fourier transform of the first order correlation function of the atomic dipole moment opera-tor. By using Fourier transform, we can directly solve these modified optical Bloch equations in Eq. (2.12). Because the two-time correlation function of the atomic dipole is proportional to the first order coherence function g(1)(τ ) [29] of the radiated photon field and the fluorescence spectrum can be obtained by taking the Fourier transform of the first order coherence function, one has:
S(ω) = Z ∞
−∞
(ω-ωa)/β S ( ω ) [A .U .] -0.5 -0.25 0 0.25 0 20 40 60 80 100 Ω= 0.25β ωc= 100β
FIGURE 2.2: Comparison of resonance fluorescence spectra far from the bandedge (dashed-line, ωa − ωc = 1000β) and near the bandedge (solid-line ωa − ωc = Ω);
Ω = 0.25β, ωc = 100β.
In this way the fluorescence spectrum can be easily determined after determining the noise correlation functions. By using the anisotropic model from Eqs. (2.9, 2.10), it can be easily to show that the statistics of the quantum noises of the photonic bandgap reservoir are not only color noises but also exhibit bandgap behavior.
Based on the above formula, in Fig. 2.2 we plot the resonance fluorescence spectra at a constant Rabi frequency when the atomic transition frequencies ωa are far from
(dashed-line) and near (solid-line) the bandedge frequency ωc respectively. The
evolu-tion of the resonance fluorescence spectra with different offsets between the transievolu-tion frequency and the bandedge frequency are also plotted in Fig. 2.3. It can be noted that the separation of each adjacent peaks is determined by the Rabi frequency as in the case of free space. When the atomic transition frequency is far away from the
0 50 100 150 200 S ( ω ) [A .U .] 0 0.5 1 (ωa-ω c)/β -0.4 -0.2 0 0.2 0.4 (ω-ωa)/β Ω= 0.25β ωc= 100β
FIGURE 2.3: Evolution of resonance fluorescence spectrum near the bandedge at constant Rabi frequency: Ω = 0.25β, ωc = 100β.
bandedge (ωa≫ ωc), the normal resonance fluorescence spectrum of Mollow’s triplets
with three Lorentzian profiles is obtained just as expected [18]. The contribution from the elastic Rayleigh scattering in the center part (which is a delta function with zero detuning frequency) has been ignored and only the contribution from the inelastic Raman scattering (the three peaked profiles) are considered here. The linewidth of each peak is proportional to the decay rate of atom and the separation of adjacent peaks is proportional to the Rabi frequency.
When the atomic transition frequency moving toward the bandedge, the profiles due to incoherent scattering processes become sharper and sharper because there are fewer and fewer DOS available. The narrowing of the fluorescence spectra also indicates a smaller decay rate due to the forbidden effect of the bandgap. The profile in the lower frequency is not only suppressed but also becomes asymmetrical due to
the existence of the bandgap. It should also be noted that the peak in the higher frequency is enhanced a lot as can be clearly seen in the figure. Eventually the peak in the lower frequency will be totally suppressed when the atomic transition frequency is moving more toward the bandedge. At this time the resonance fluorescence spectrum now only has two peaks.
2.3. Fluorescence Squeezing Spectra
The noise spectra of a two-level atom driven by a classical pumping light can also exhibit non-classical phenomena (squeezing spectra) if the phase-dependent fluores-cence spectra are measured. To observe squeezing in the phase-dependent fluoresfluores-cence spectra, one needs to calculate the fluorescence spectra for the quadrature field com-ponents. Theoretically the quadrature field operator in the θ phase angle is defined as,
ˆ
Eθ(t) = eiθEˆ(+)(t) + e−iθEˆ(−)(t). The two cases corresponding to θ = 0 and θ = π/2
represent the in-phase and out-of-phase components of the electric field respectively. The spectra for the quadrature fields can be obtained by calculating the following normally order variance:
Sθ(ω) ≡ < ˜Eθ(ω), ˜Eθ(−ω) >, (2.16) = Γ1 4[< ˜σ−(ω)˜σ−(−ω) > e −2iθ+ < ˜σ +(ω)˜σ−(−ω) > + < ˜σ+(−ω)˜σ−(ω) > + < ˜σ+(−ω)˜σ+(ω) > e2iθ].
(ω-ωa)/β Q u a d ra tu re S p e c tr u m -1 -0.5 0 0.5 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Ω= 0.25β ωc= 100β
FIGURE 2.4: Resonance fluorescence quadrature spectra near the bandedge; Solid line: in-phase, S1(ω), Dashed line: out-of-phase, S2(ω) with ωc = 100β and Ω =
0.25β.
and Γ is the decay rate of the two-level atom [24, 25].
In the free space case, the in-phase quadrature S1(ω) produces the central peak of
the Mollow’s triplet while the out-of-phase quadrature S2(ω) produces the two
side-peaks when the separation of the two sidebands is large. When the Rabi frequency is small (small driving field), Ω2 < 4Γ2, squeezing can be observed in the out-of-phase quadrature spectra.
The situations are different for the case of photonic bandgap crystals. When the transition frequency is near the bandedge of the photonic crystals, the bandgap effects modify the fluorescence intensity spectrum and cause asymmetric spectral profiles as we have seen in Fig. (2.2). As shown in Fig. 2.4, for the case of photonic bandgap crystals, the in-phase quadrature not only contributes to the central component but
-0.1 0 0.1 0.2 0.3 S 1(ω ) [A .U .] 0.2 0.4 0.6 0.8 1 1.2 1.4 (ωa-ω c)/ β -1 -0.5 0 0.5 1 (ω-ω a)/β Ω= 0.25β ωc=100β
FIGURE 2.5: Evolution of in-phase quadrature spectra near the bandedge with ωc =
100β and Ω = 0.25β.
also the two sidebands. The out-of-phase quadrature still only contributes to the two sidebands as in the case of free space. Moreover, we find both sidebands of the in-phase quadrature now exhibit squeezing even when Ω2 > 4Γ2. This squeezing effect in the in-phase quadrature near the bandedge is similar to the case of a atom dressed by the finite-bandwidth cavity-field excitation [30].
In the above calculation all the quadrature noise spectra are normalized with respect to the decay rate of the atom, Γ, as shown in Eq. (2.16). For the case of free space, the decay rate Γ is frequency independent due to the white noise reservoir. However, for photonic crystals, the decay rate will be modified according to the offset between the atomic transition frequency and the bandedge frequency. According to the works in references [5, 9], the spontaneous emission rate of a two-level atom near the bandedge contains non-exponential terms. For simplicity we have used the
following formula to estimate the decay rate of the atom: Γ ≈ Re[ ˜G(ω = 0)] = β3/2 √ ωa− ωc ωa (2.17)
In Fig. (2.5), we plot the evolution of the in-phase quadrature spectra for different frequency (wavelength) offset. One can see that the higher frequency peak exhibits larger squeezing as well as larger fluorescence intensity when the offset frequency is approaching the bandedge frequency.
2.4. Summary
In summary, by introducing the Liouville operator expansion, we have successfully overcome some of the difficulties associated with the non-Markovian nature of the problem caused by the non-uniform distribution of the photon states in the photonic bandgap crystal. Our calculated results have indicated that the resonance fluores-cence spectra near a photonic bandgap can exhibit interesting behavior including the suppression and enhancement of the Mollow’s triplet peaks, and the squeezing phe-nomena in the in-phase quadrature spectra. Besides the atom-light interaction in the linear photonic crystals, in next Chapter we will switch the gear to discuss quantum properties of waves in nonlinear PhCs, which can support unique wave solutions - the gap solitons.
CHAPTER 3
Quantum theory of Bragg solitons and gap solitons
3.1. Introduction
In the literature, various types of optical soliton phenomena have been studied extensively in the area of nonlinear optical physics. These include the nonlinear Schr¨odinger solitons in dispersive optical fibers, spatial and vortex solitons in pho-torefractive materials/waveguides, and cavity solitons in resonators [31]. Gap solitons are also found to exist in nonlinear periodic systems due to combination of unique properties of nonlinear periodic systems.
In fiber optics, it has been well known that fiber Bragg gratings (FBGs) with Kerr nonlinearity can exhibit optical soliton-like phenomena known as the fiber Bragg grating solitons [10, 32]. The FBGs are one-dimensional photonic bandgap crystals with weak index modulation. By utilizing the high dispersion of the FBGs near the bandedges, one can produce optical solitons in the anomalous dispersion spectral side if the input pulse have suitable pulse-width and peak intensity.
From the theoretical point of view, fiber Bragg grating solitons belong to the class of bi-directional pulse propagation problems, where the quantum theory is still lack of enough consideration. Most of the previous studies on fiber Bragg grating
solitons have been on the classical effects and there is almost no reported result on their quantum properties. The quantum theory of traveling-wave optical solitons has been intensively developed during the past 15 years and several approaches have been successfully carried out to calculate the quantum properties of different traveling-wave optical solitons including the family of nonlinear Schr¨odinger solitons [33, 34, 35, 36, 37] as well as the self-induced-transparency solitons [38]. In the first part of this Chapter, we develop a general quantum theory for bi-directional pulse propagation problems and particularly by applying the theory to the case of fiber Bragg grating solitons. It will be shown that the fiber Bragg grating soliton pulses will quantum-mechanically get amplitude-squeezed after passing through the fiber grating and the squeezing ratio will be calculated theoretically. The squeezing ratio of FBG solitons is found to exhibit interesting relations with the fiber grating length as well as with the intensity of the input pulse. With the use of apodized FBGs, we also find that one can tailor the squeezing ratio of FBG solitons al long as the soliton pulses evolve adiabatically.
More generally, FBG solitary waves are only a subset of gap solitons in shallow periodic media. For examples, supported by the nonlinear photonic bandgap crystals made of Kerr materials [11, 12], or by loading Bose-Einstein condensates (BEC) into an optical lattices [13, 39, 40, 41], optical and matter-wave gap solitons have been predicted to display lots of novel nonlinear optical phenomena. Recently, demon-strations of optical gap solitons in the dynamically reconfigurable PhCs [42, 43, 44]
and bright gap solitons of an attractive or a repulsive BEC in an optical potential [15, 45, 46] show that the dynamics of these system are very rich and complex.
In the second part of this Chapter, we will study the quantum fluctuations of matter-wave gap solitons in an optical lattice and investigate the band-gap effects on quantum noise squeezing. We employ the soliton perturbation approach [16] to ana-lyze quantum fluctuations around the soliton solutions of Gross-Pitaevskii equation (GPE) with a periodic potential. Using this approach, we demonstrate the enhance-ment of quantum noise squeezing induced by gap soliton evolution in an atomic band-gap structure.
3.2. Quantum theory of FBG solitons
In our modeling, we use the nonlinear coupled mode equations (NCMEs) to de-scribe the bi-directional waves propagating in a uniform FBG. To be more explicit, let us consider the wave propagation problem in a one dimension fiber grating struc-ture with the nonlinearity coming from the third order nonlinearity of the optical fiber. With the self-phase modulation and cross-phase modulation effects, we model Bragg solitons by using the following NCMEs that describe the coupling between the forward and the backward propagating waves in a uniform FBG.
1 vg
∂
∂tUa(z, t) + ∂
∂zUa = iδUa+ iκUb+ iΓ|Ua|
2U a+ 2iΓ|Ub|2Ua, (3.1) 1 vg ∂ ∂tUb(z, t) − ∂
∂zUb = iδUb + iκUa+ iΓ|Ub|
2U
Here Ua(z, t) and Ub(z, t) represent the forward and backward propagation pulses
respectively. They are in the units of GW1/2/cm. Moreover, vg is the group velocity of
the pulse, κ is the coupling coefficient, λBis the Bragg wavelength, δ is the wavelength
detuning parameter, and Γ represents the self-phase modulation coefficient.
This set of NCMEs has analytical soliton solutions for the case of infinite grating length, as is shown by Aceves and Wabnitz with the introduction of the massive Thirring model [10]. However, for gratings of finite length, no analytic solution can be found. So in our studies we directly use the finite difference numerical simulation method with the parameters based on the first experimental reported in the literature [32]. We consider a 60 ps FWHM sech-shaped pulse incidents into a uniform grating with 15.0 cm−1 wavenumber detuning from the center of the bandgap. The coupling strength of the fiber grating is 10 cm−1, the nonlinear coefficient Γ is 0.018 cm/GW , and the group velocity vg is chosen to be c/n with n = 1.5 and c being the speed
of light in free space. When the input peak intensity is below the required value for forming a solitary pulse in the FBGs (about 4.5 GW/cm2 in this case), the peak intensity of the pulse will decrease along the propagation. On the other hand, as shown in Fig. 3.1, when the input peak intensity is above 4.5 GW/cm2, the peak intensity of the pulse oscillates during the propagation within the grating. Only when the nonlinearity can exactly compensate the dispersion induced by the FBGs, one can have a stable solitary pulse inside the grating.
0 50 z (cm) 1000 2000 3000 Tim e(p s)
FIGURE 3.1: Evolution of the fiber Bragg grating soliton with the input peak inten-sity I = 9.0GW/cm2.
quantum properties. In quantum theory the NCMEs become the quantum nonlinear coupled mode equations (QNCMEs):
1 vg
∂
∂tUˆa(z, t) + ∂
∂zUˆa = iδ ˆUa+ iκ ˆUb+ iΓ ˆU
† aUˆaUˆa+ 2iΓ ˆUb†UˆbUˆa, (3.3) 1 vg ∂ ∂tUˆb(z, t) − ∂
∂zUˆb = iδ ˆUb + iκ ˆUa+ iΓ ˆU
†
bUˆbUˆb + 2iΓ ˆU †
aUˆaUˆb, (3.4)
where ˆUa and ˆUb represent the forward and backward normalized fields which satisfy
the usual equal time Bosonic commutation relations.
[ ˆUa(z1, t), ˆUa†(z2, t)] = δ(z1− z2),
[ ˆUb(z1, t), ˆUb†(z2, t)] = δ(z1− z2),
[ ˆUa(z1, t), ˆUa(z2, t)] = [ ˆUa†(z1, t), ˆUa†(z2, t)] = 0,
[ ˆUb(z1, t), ˆUb(z2, t)] = [ ˆUb†(z1, t), ˆUb†(z2, t)] = 0,
This is a set of coupled operator equations in the Heisenberg picture and can be derived from the following Hamiltonian under the effective-mass approximation [47]
H = −vg{δ Z dz ( ˆUa†Uˆa+ ˆUb†Uˆb) + i Z dz ( ˆUa† ∂ ∂zUˆa− ˆU † b ∂ ∂zUˆb) + κ Z dz ( ˆU† aUˆb+ ˆUb†Uˆa) + Γ 2 Z dz ( ˆU† aUˆa†UˆaUˆa+ ˆUb†Uˆ † bUˆbUˆb) + Γ Z dz ( ˆU† aUˆ † bUˆbUˆa+ ˆU † bUˆ † aUˆaUˆb)}. (3.5)
This derivation automatically proves that the QNCMEs preserve the commutation brackets.
Since for optical solitons the average photon number is usually very large, we can safely use the linearization approximation to study their quantum effects. By setting
Ua(z, t) = Ua0(z, t) + ˆua(z, t),
Ub(z, t) = Ub0(z, t) + ˆub(z, t),
and substituting them into Eq. (3.3-3.4) for linearization, we obtain the linear quan-tum operator equations in Eq. (3.6) that describe the evolution of the quanquan-tum fluctuations associated with the fiber Bragg grating solitons. The quantum pertur-bation fields ˆua(z, t) and ˆub(z, t) in Eq. (3.6) also have to satisfy the same equal time
commutation relations as the original field operators ˆUa(z, t) and ˆUb(z, t). 1 vg ∂ ∂t ˆ ua ˆ ub = iΓUa02 2iΓUa0Ub0 2iΓUa0Ub0 iΓUb02 ˆ u† a ˆ u†b + (3.6)
−∂z∂ + iδ + 2iΓ|Ua0|2+ 2iΓ|Ub0|2 iκ + 2iΓUa0Ub0∗
iκ + 2iΓU∗ a0Ub0 ∂ ∂z + iδ + 2iΓ|Ua0| 2 + 2iΓ|Ub0|2 ˆ ua ˆ ub , 1 vg ∂ ∂t uAa uAb = −iΓUa02 −2iΓUa0Ub0 −2iΓUa0Ub0 −iΓUb02 uA∗a uA∗b + (3.7) − ∂ ∂z + iδ + 2iΓ|Ua0| 2
+ 2iΓ|Ub0|2 iκ + 2iΓUa0Ub0∗
iκ + 2iΓUa0∗ Ub0 ∂ ∂z + iδ + 2iΓ|Ua0| 2 + 2iΓ|Ub0|2 uAa uAb .
If we define the inner product operation according to
h ~f|~ˆgi = 12 Z
dz [f∗
agˆa+ fagˆ†a+ fb∗gˆb+ fbgˆ†b], (3.8)
then Eq. (3.7) is the corresponding set of adjoint equations for the perturbed QNCMEs, which have the following desired property:
d d th~u
A|~ˆui = 0, (3.9)
where ~uA= (uAa, uAb )T is the solution of the adjoint equation defined in Eq. (3.7). The important thing is that the inner product between the solutions of the two equation sets is preserved along the time axis.
By taking advantage of the preservation of the inner product, we can express the inner product of the output quantum perturbation operator with a projection func-tion in terms of the input quantum field operators by the back-propagafunc-tion method.
Input pulse intensity (GW/cm2) T ra n s m it ta n c e 5 10 15 20 0.722 0.724 0.726 0.728 0.73 0.732 0.734 0.736 0.738 0.74
Peak Intensity of Input Pulse(GW/cm2
) O p ti m a l S q u e e z in g R a ti o (d B ) 0 5 10 15 20 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
FIGURE 3.2: Transmittance (left) and photon number squeezing ratio (right) for fiber Bragg grating solitons with different input intensities.
This will allow us to calculate the quantum uncertainty for the inner product of the output quantum operator with any given projection function. Under the lineariza-tion approximalineariza-tion, any measurement of a physical quantity can be expressed as an inner product between a measurement characteristic function and the perturbed quantum field operator [16]. The squeezing ratio of the measured quantity thus can be calculated according to:
R(T ) = var[h ~f|~ˆu(t = T )i] var[h ~f|~ˆu(t = 0)i] =
var[h ~FT|~ˆu(t = 0)i]
var[h ~f |~ˆu(t = 0)i] . (3.10) Here var[·] means the variance, ~f is the original projection function and ~FT is the
back-propagated projection function. The choice of the characteristic function ~f will depend on the measurement to be performed. For the photon number measurement,
~
f is simply the normalized output classical pulse from the grating [48]. For the homodyne detection, it will be the local oscillator pulse. In the following we consider
Grating Length (cm) O p ti m a l S q u e e z in g R a ti o (d B ) 0 20 40 60 80 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
FIGURE 3.3: Optimal squeezing ratio for Bragg solitons propagating through differ-ent length of FBGs.
a solitary pulse incident into a uniform FBG, and calculate the quantum fluctuation of its first transmitted pulse based on the formulation given above.
The transmittance of the FBGs with different input intensities of solitons for a constant FBG length (50 cm) is shown in the top curve of Fig. 3.2. The calculated photon number squeezing ratio is shown in the bottom for the same parameters. When the input peak intensity is smaller than that of the fundamental soliton, the output squeezing ratio monotonically decreases when the input peak intensity is in-creased. The output squeezing ratio will begin to oscillate strongly with respect to the changing input intensity when the input intensity is much larger than that of the fundamental soliton. The oscillation behaviors of the FBG transmittance and the squeezing ratio match very well. That is, the squeezing ratio has a local minimum when the transmission has a local maximum. Intuitively the periodic grating structure
-10 0 10 20 S q u e e z in gR a ti o (d B ) -1.5 -1 -0.5 0 0.5 1 1.5 θ(rad) 0 20 40 60 Le ng th (cm )
FIGURE 3.4: Squeezing ratio for different FBG lengths and different local oscillator phases.
acts like a spectral filter which can filter out the noisier high frequency components in the soliton spectrum and produce a net amplitude squeezing effect just as in the previous soliton amplitude squeezing experiments where a spectral filter is cascaded after a nonlinear fiber [49, 50]. And the minimum squeezing ratio occurs when the pulse energy of soliton is slightly large than that of the fundamental soliton. It is also intuitively clear that larger amplitude squeezing should occur when the trans-mittance curve is saturated. Fig. 3.3 shows the dependence of the optimal squeezing ratios for different FBG lengths with a constant input intensity (I = 4.5GW/cm2). If we only consider the gratings with the length longer than 1 cm, we find that the squeezing ratio monotonically decreases with the FBG length and saturates at the length around 60 cm. Intuitively this is because the filtering effect of the grating will unavoidably introduce additional noises on the light fields and eventually cause the
squeezing ratio to become saturated.
So far we have shown that the FBG solitons will get amplitude squeezed during propagation. Under the linearization approximation, the amplitude squeezing cor-responds to the squeezing of the in-phase quadrature field component. To further determine the maximum squeezing phase angle of the quadrature field components of the FBG solitons, we perform another calculation to simulate the squeezing ratio when the homodyne detection scheme is used and when the local oscillator pulse is exactly the classical output pulses. With the homodyne detection scheme, one has the additional degree of freedom to adjust the relative phase between the local os-cillator and the signal for detecting different quadrature field components. Fig. 3.4 plots the squeezing ratio for different FBG lengths and for different local oscillator phases with a constant input intensity (I = 4.5GW/cm2). One can see that for short FBG lengths the quadrature squeezing direction is close to but not exactly in the in-phase (or amplitude) quadrature, θ = 0. However, when the FBG length is long enough, the squeezing direction will approach the in-phase quadrature. This proves that the FBG solitons will indeed be squeezed in the amplitude direction when the FBG length is long enough.
In Fig. 3.5 we illustrate the possible direct detection scheme for measuring the photon number squeezing of the FBG solitons. To avoid the complication due to the multiple transmitted pulses of the FBG, it may be necessary to use a time-gating device to make sure that only the first transmitted pulse is detected
FBG
G
D
FIGURE 3.5: Measurement scheme of direct detection for observing FBG soliton squeezing. Here G is a gating device which will block out all the transmitting pulses but the first one; D is an optical detector.
It is well known that one can engineer the dispersion along the FBG by using an apodized FBG. Such apodized nonlinear FBGs have been used for adiabatic soliton pulse compression within a very short length of several centimeters [51]. Intuitively the solitons with higher peak intensities will exhibit large squeezing due to higher nonlinear effects. It is thus expected to be able to compress the pulsewidth of the FBG soliton and enhance its squeezing ratio simultaneously. To verify this idea, here we consider an apodized FBG which has a position dependent coupling coefficient described by
κ(z) = κ0 + αz, (3.11)
where κ0 = ω0˜ǫ/2¯nc is the initial coupling coefficient and α is the slope of the coupling
coefficient.
In the following calculation we consider the same 60 ps FWHM sech-shaped input pulse with the peak intensity of I = 4.5GW/cm2 for the apodized grating without changing any parameter. In Fig. 3.6 (Left) we plot the squeezing ratio versus the FBG length with a constant input intensity (I = 4.5GW/cm2) and different apodization
Grating Length (cm) S q u e e z in g R a ti o (d B ) 0 20 40 60 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 α= +0.04 α= 0.0 α= -0.04 Grating Length (cm) S q u e e z in g R a ti o (d B ) 0 20 40 60 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 α= +0.08 α= 0.0 α= -0.08
FIGURE 3.6: Photon number squeezing ratio versus the FBG length with a constant input intensity (I = 4.5GW/cm2). (Left): The three curves correspond to different
slopes of κ(z); Solid-line: α = 0; Dashed-line: α = −0.04; Dash-dotted line: α = +0.04. (Right): Solid-line: α = 0; Dashed-line: α = −0.08; Dash-dotted line: α = +0.08.
slopes. The three curves in Fig. 3.6 (Left) correspond to different slops of κ(z): α = +0.04, 0, −0.04 respectively. When α = 0.04(1/cm−2), the FWHM of the original FBG soliton (60 ps) will be adiabatically compressed into 30 ps after propagating through a 70 cm FBG. Because of this, the achievable optimal squeezing ratio thus increases from 8 dB to 9.7 dB for the propagation distance of 70 cm. On the other hand, if the slope of the coupling coefficient is positive, we have a broaden FBG soliton and the optimal squeezing ratio will be degraded. If the slope of the coupling coefficient is too large, then the soliton cannot be compressed adiabatically and thus the optimal squeezing ratio will be degraded eventually. This can be seen in Fig. 3.6 (Right), where we plot the squeezing ratio versus the grating length with larger slopes (α = −0.08, +0.08). The result for the uniform FBG case (α = 0) is also plotted for
comparison.
3.3. Quantum theory of gap solitons
Due to similarities between the physics of atomic solitons in attractive BEC and optical solitons in a Kerr medium, the two systems are expected to have similar quan-tum noise properties. It has been recognized [52] that general methods of quanquan-tum noise squeezing and entanglement developed in quantum optics could apply to other nonlinear bosonic fields, such as weakly interacting ultracold atoms in a Bose-Einstein condensate (BEC). Consequently, a number of theoretical proposals were put forward [53, 54] and some experiments were carried out [55, 56, 57] for generating macroscopic entangled number-squeezed states in BEC. Production of the quantum correlations among the macroscopic quantum states relies on the nonlinear interaction of the atomic waves. The mechanism is analogous to the production of squeezing through optical Kerr nonlinearity.
For instance, particle number squeezing of an atomic soliton during its evolution may result from coupling between the soliton amplitude and phase, akin to those induced in optical solitons formed through the Kerr nonlinearity [58, 59]. Recently, however, a new kind of gap solitons in a repulsive condensate, supported in periodic potentials of optical lattices, have attracted a great deal of attention due to their potential for controllable interaction and robust evolution uninhibited by collapse [13, 15, 39, 40, 41]. The current techniques for gap solitons generation suffer greatly
X -1 0 1 2 (a) -1 0 1 (b) -50 -25 0 25 50 -1 0 1 (c) µ P 1 2 3 4 0 2 4 6 8 10 12 c a b
FIGURE 3.7: Left: Band-gap diagram for Bloch waves (bands are shaded), and the family of the gap solitons in the first finite gap for V0 = 4.0. Right: Profiles of gap
solitons with different chemical potentials; µ = 1.91, 3.0, and 3.85 corresponding to the points a, b, and c respectively.
from “technical” noise [15]. Provided this problem can eventually be overcome, e.g. by employing low-noise stabilized atom laser sources [60], gap solitons may represent an attractive high-density source for atomic interferometry, quantum measurements and quantum information processing with ultracold atoms. To understand and control the quantum noise associated with atomic gap solitons are therefore fundamental issues which so far have not been explored.
As a model, we consider an elongated cigar-shape BEC loaded into a one-dimensional optical lattice and described by the mean-field Gross-Pitaevskii equation for the macroscopic wave function (see, e.g. Ref. [40]):
i~∂Ψ ∂t = − 1 2 ∂2Ψ ∂x2 + V (x)Ψ + g1D|Ψ| 2Ψ, (3.12)
nonlinear interaction coefficient, and V (x) = V0sin2(x) is the one-dimensional
pe-riodic potential. Stationary states of the condensate can be presented in the form, Ψ(t, x) = ψ(x) exp(−iµt), where µ is the chemical potential. In the linear, non-interacting limit, g1D → 0, the spectrum of matter waves has the characteristic
band-gap structure [40]. The nonlinear localization of matter waves in the form of band-gap solitons occurs in the gaps of the linear spectrum, and the family of the lowest-order gap solitons in the first (finite) spectral gap is shown in Fig. 3.7 as the dependence of the soliton norm, P =
Z
ψ2dx, vs. chemical potential.
It is well known that the degree of localization of gap solitons varies across the gap [see Fig. 3.7(a-c)]. Near the bottom edge of the gap, µ ≈ µ0 the weakly localized
soliton profile is well described by the “envelope” approximation [61],
ψ(x; µ) = AF (x)Φ(x; µ0), (3.13)
where Φ(x; µ0) is the periodic Bloch state at the corresponding band edge and F (x)
is a slowly varying function of the spatial coordinate. The inset of Fig. 3.8 shows the oscillating wavefunction ψ(x) of a gap soliton near the band edge, at µ = 1.91, together with the corresponding Bloch-wave envelope F (x). The envelope function F (x) is the solution of the lattice-free GPE or nonlinear Schr¨odinger (NLS) equation with the effective anomalous diffraction and interaction energy modified by the lattice [61].
To study the quantum fluctuations of the gap solitons, we replace the ‘classical’ mean field described by Eq. (3.12) by the bosonic field operator ˆΨ. Then, we use
X A m p . -100 -50 0 50 100 -0.2 -0.1 0 0.1 0.2 t R [d B ] 0 2 4 6 8 10 -2 -1.5 -1 -0.5 0 Gap soliton NLS soliton
FIGURE 3.8: Time evolution of the optimal squeezing ratio, R, for the gap soliton (solid), its near-band-edge envelope approximation (dot-dashed), and the envelope (NLS) soliton (dashed). Inset: profiles of the gap soliton found from Eq. (3.12) at µ = 1.91 and its envelope used in approximation of Eq. (3.13).
the linearization approach with the perturbed quantum field operator ˆψ around the mean-field solution Ψ0. By using the back-propagation method [16], we calculate the
optimal squeezing ratio for gap solitons as a function of the evolution time. The optimal squeezing ratio of the quadrature field for gap solitons, as a result of the homodyne detection scheme, can be defined as the ratio of variances [16]:
R(t) = min[varhΨL(t)| ˆψ(t)i/varhΨL(t)| ˆψ(0)i],
where the inner product between the operator perturbation field, ˆψ, and the local oscillator profile, ΨL, is defined as:
hΨL| ˆψi = 1 2 Z Ψ∗Lψ + Ψˆ Lψˆ† dx
oscil-lator profile, ΨL = Ψ0eiθ/P . By varying the phase of the homodyne detection, θ, we
can determine the minimum value of the squeezing ratio of the quadrature component of the quantum field. The result of the calculations of R(t) for the gap soliton near the bottom edge of the spectral gap is presented in Fig. 3.8 by a solid curve.
The optimal squeezing ratio for gap solitons can be compared with that for a conventional NLS soliton coinciding with the soliton envelope F (x) and evolving in a lattice-free anomalous diffraction regime (dashed line in Fig. 3.8). The NLS soliton of the same envelope as the gap state is only weakly quadrature squeezed during propagation due to the nonlinearity of the matter-wave, whereas the gap soliton shows enhanced quantum noise squeezing. In contrast, the near-band-edge envelope approximation of Eq. (3.13) (dot-dashed line in Fig. 3.8) provides a good estimate for soliton squeezing in the initial stage of evolution.
To understand the band-gap effect on the quantum fluctuation of matter-wave gap solitons, we analyze the number correlations between different spectral components of the gap soliton induced by its nonlinear evolution. The intra-soliton correlation coefficients, Cij, are found by calculating the normally-ordered covariance,
Cij ≡ h: ∆ˆni ∆ˆnj :i q ∆ˆn2 i∆ˆn2j , (3.14)
where ∆ˆnj is the atom-number fluctuation in the j-th slot ∆sj in the spatial (s = x)
or momentum (s = k) domain:
∆ˆnj =
Z
∆sj
ki kj -50 0 50 -50 0 50 0.04 0.03 0.02 0.02 0.01 0.00 -0.01 -0.02 -0.02 -0.03 -0.04 (a)µ= 1.91 ki kj -50 0 50 -50 0 50 0.13 0.10 0.08 0.05 0.03 0.00 -0.03 -0.05 -0.08 -0.10 -0.13 (b)µ= 2.4 V1 -50 0 50 0 1 2 V1 -50 0 50 0 1 2 ki kj -50 0 50 -50 0 50 0.08 0.06 0.05 0.03 0.02 0.00 -0.02 -0.03 -0.05 -0.06 -0.08 (c)µ= 3.0 V1 -50 0 50 0 1 2 ki kj -50 0 50 -50 0 50 0.14 0.11 0.08 0.06 0.03 0.00 -0.03 -0.06 -0.08 -0.11 -0.14 (d)µ= 3.85 V1 -50 0 50 0 1 2
FIGURE 3.9: Quantum correlation spectra in the spatial (x)-domain for gap solitons at different points within the gap. Insets show the corresponding components of solitons in the x-domain. For distinctness, all figures are plotted with the different set of correlation densities.
The corresponding number correlation spectra in the spatial (x)-domain for gap solitons at different points within the gap are shown in Figs. 3.9 (a-d), respectively. Compared to the well-known correlation spectrum of NLS soliton, the near-band-edge gap soliton has enlarged and discrete correlation pattern “shaped” by the Bragg scat-tering of the matter-wave on the periodic potential. And in the momentum domain NLS soliton displays the well-known symmetric correlation pattern with the noisy, strongly correlated outer regions of the soliton spectrum. For this reason, an efficient number squeezing of NLS solitons can be produced by spectral filtering that removes the noisy spectral components [49]. On the contrary, the correlation spectrum of a near-band-edge gap soliton displays a periodic pattern with regions of strong anti-correlations at the Bragg condition, k = ±(2m + 1) (m is an integer). This ensures
t θ [r a d ] 0 5 10 0 0.2 0.4 0.6 0.8 1 µ= 3.85 µ= 1.91 µ= 2.4 µ= 3.0 µ= 3.4 (b) µ R [d B ] 2 2.5 3 3.5 -25 -20 -15 -10 -5 0 (a) t = 9.0 t = 1.0 t = 3.0 t = 5.0 t = 7.0
FIGURE 3.10: (a)Optimal squeezing ratios, R, for gap solitons with different chemical potentials, µ, at different time. (b) Time evolution of the phase in the homodyne detection for the optimal squeezing ratios at different values of chemical potentials within the gap.
enhanced number squeezing of the BEC soliton without additional spectral filtering. When the chemical potential of the gap soliton moves away from the band edge, the envelope approximation (3.13) becomes invalid. In Fig. 3.10(a), we show the de-pendence of the optimal squeezing ratios of gap solitons at different chemical poten-tials inside the gap for different times. Towards the middle of the gap, the squeezing ratio improves with the chemical potential as the peak density of the gap soliton and the nonlinear interaction is increasing. It must be emphasized that near the band edge the quantum noises of gap solitons is squeezed in the quadrature field compo-nent, Fig. 3.10 (b). But like amplitude-squeezed fiber Bragg grating solitons [64], gap solitons become atom-number squeezed, θ ≈ 0, when their chemical potentials move toward the band gap (θ is not exactly zero for we don’t use an optimal profile
for the local oscillator.) For a fixed time (say at t = 5) and varying chemical po-tential, the best squeezing occurs in the middle of the gap, at µ = 3.0. Analysis of the number correlation structure reveals that the gap soliton in the middle of gap is entirely composed of strongly localized, weakly anti-correlated components [Fig. 3.9 (b)], which assists the atom number squeezing. However, near the top edge of the gap, the localization of the gap soliton degrades due to resonance with the Bloch state at the corresponding edge, and strongly correlated, noisy components associated with the periodic Bloch wave structure start to dominate in the soliton spectrum [Fig. 3.9 (d)]. Subsequently, the squeezing is reduced on that edge of the spectral gap. In between, the balance of spectral filtering induced by the periodic potential and the delocalization of Bloch wave states makes the squeezing ratio of gap soliton to a minimum value [Fig. 3.9 (c)].
3.4. Summary
In summary, we have developed a general quantum theory for bi-directional non-linear optical pulse propagation problems and have especially used it to study the squeezing phenomena of fiber Bragg grating solitons. It has been shown for the first time that the output FBG soliton pulses will get amplitude squeezed automatically. The squeezing ratio of the FBG solitons exhibits interesting relation with the fiber grating length as well as with the intensity of the input pulse. The squeezing ratio saturates after a certain grating length and the optimal squeezing ratio occurs when
the intensity of the FBG soliton is slightly large than that of the fundamental soliton. With the use of apodized FBGs, we also find that one can compress the FBG solitons and enhance its squeezing ratio simultaneously, as long as the soliton pulses evolve adiabatically. To actually measure the quantum fluctuations of the fiber Bragg grat-ing solitons experimentally, we propose to use a time-gatgrat-ing device to block out other smaller multiple transmitted pulses and only directly detect the first transmitted pulse from the grating.
We also have investigated the effect of the periodic potential on quantum fluc-tuations of gap solitons in repulsive BEC confined by an optical lattices. We have revealed that the quantum correlation spectra of gap solitons show discrete and de-localized patterns in the spatial domain which are introduced by the periodic nature of Bloch states. This property of intra-soliton quantum correlations causes the en-hanced squeezing of gap states compared the envelope solitons described by the NLS equation. And we find that gap soliton gets quadrature squeezed near the band edge and atom number squeezed inside the band gap. We would like to emphasize that the basic results of our analysis can be useful in the study of the bandgap effects on the quantum squeezed states in other fields, such as quantum optics and gap solitons in photonic crystals.
CHAPTER 4
Soliton Entanglement
4.1. Introduction
Quantum-noise squeezing and correlations are two key quantum properties that can exhibit completely different characteristics when compared to the predictions of the classical theory. Almost all the proposed applications to quantum measurements and quantum information treatment utilize either one or both of these properties. In particular, solitons in optical fibers have been known to serve as a platform for demonstrating macroscopic quantum properties in optical fields, such as quadrature squeezing [33, 34, 36, 37, 62, 63], amplitude squeezing [49, 64], and both intra-pulse and inter-pulse correlations [58, 65].
Recently, experimental progress in demonstrating various quantum information processes by using two-mode squeezed states in optical solitons has been reported, see Refs. [66, 67, 68] and references therein. In previous works, continuous-variable entangled beams have been generated by letting two squeezed fields (squeezed vacuum states [69], or amplitude-squeezed fields [70]) interfere through a beam splitter, which mathematically acts as the Hadamard transformation. By utilizing the continuous EPR-like correlations of optical beams, one can also realize quantum-key
