Abstract Given the extensive applications of random walks to classical algorithms in virtually every science related discipline, we may be at the threshold of yet another problem solving paradigm with the advent of quantum walks. Over the past decade, quantum walks have been extensively explored for their non-intuitive dynamics, which may hold the key to radically new quantum algorithms. This growing interest in the theoretical applications of quantum walks has been paralleled by a flurry of research into a more practical problem: how does one physically implement a quantum walk in the laboratory? This book provides a comprehensive survey of numerous proposals, as well as actual experiments, for a physical realisation of quantum walks, underpinned by a wide range of quantum, classical and hybrid technologies. J. Wang and K. Manouchehri, Physical Implementation of Quantum Walks, 151 Quantum Science and Technology, DOI 10.1007/978-3-642-36014-5, © Springer-Verlag Berlin Heidelberg 2014 Appendix A Electromagnetic Radiation A.1 Classical Radiation Field Main References (Townsend 2000b; Shore 1990a; Griffiths 1999) Classically, the electromagnetic field represents the general solution to Maxwell’s equations which can be expressed in terms of a scalar potential ' and a vector potential A, whereby for some charge density and volume current density vector j 8 ˆ @A ˆ ˆE Dr' ; ˆ @t <ˆB DrA; @ 1 ˆ 2 (A.1) ˆr ' C .rA/ D ; ˆÂ @t à Â0 à ˆ @2A @' :ˆ r2A r rA C D j; 0 0 @t2 0 0 @t 0 with 0 and 0 representing the electric and magnetic vacuum permittivity respec- tively. Using the Coulomb gauge condition rA D 0; (A.2) and in the absence of charges and currents (i.e. D j D 0), the scalar potential ' D 0 and the vector potential A satisfies the wave equation 1 @2A r2A D 0; (A.3) c2 @t2 for which the generalized traveling plane wave solution is given by Z X i.kr!k; t/ A.r;t/D dk d Ak;; u.k;;/e C c.c. ; (A.4) D1;2 J. Wang and K. Manouchehri, Physical Implementation of Quantum Walks, 153 Quantum Science and Technology, DOI 10.1007/978-3-642-36014-5, © Springer-Verlag Berlin Heidelberg 2014 154 A Electromagnetic Radiation where c.c. denotes “complex conjugate”, k labels the spatial modes of radiation propagating along vector k, labels the frequency modes, labels the two linearly independent (real valued) polarization unit vectors u such that k u.k;;/ D 0 (satisfying Eq. A.2)andAk;; is a complex scalar. In addition, for a valid solution, the wave vector and the angular frequency are not independent and must adhere to the dispersion relation ! jkj D k; : (A.5) c The electric component of the electromagnetic field is then given by @A E.r;t/D @t Z X 1 D dk d E u.k;;/ei.k r !k;t/ C c.c. 2 k;; D1;2 X Z D dk d Ek;; cos.k r !k; t k;;/; (A.6) D1;2 where ik;; Ek;; D 2i!k; Ak;; D jEk;;j e ; (A.7) and the real valued vector Ek;; D jEk;;j u.k;;/: (A.8) It is instructive to note that when interacting with atoms and molecules, the magnetic component B.r;t/ of the radiation field is often neglected, due to much weaker interaction energy compared with the electric field (see Appendix B.1.3). A commonly used specialization of the generalized electric field in Eq. A.4 is the single mode radiation traveling in direction z,forwhich 1 1 E.z;t/D E.C/.z;t/C E./.z;t/ (A.9) 2 2 1 1 D Jei.kz!t/ C Jei.kz!t/ (A.10) 2 2 1 1 Á eE ei.kz!t/ C eE ei.kz!t/ (A.11) 2 0 2 0 1 1 Á E.C/.z/ei!t C E./.z/ei!t (A.12) 2 2 1 1 Á E.C/.!/eikz C E./.!/eikz; (A.13) 2 2 A.2 Quantized Radiation Field 155 where the complex vector  à E eix J D x (A.14) iy Ey e is known as the Jones vector, oftenq normalized to a unit vector e, Ex and Ey are 2 2 real-valued amplitudes, E0 D Ex C Ey and the last two lines can be used to explicate either the temporal or the spatial parameters of the field. This electric field is said to be linearly polarized when x D y D which yields the familiar case E.z;t/D E0 cos.!t kz /; (A.15) where E0 D .Ex;Ey /, corresponding to  à 1 1 e D p ; (A.16) 2 1 for Ex ˇD Ey . Toˇ obtain a circularly polarized radiation on the other hand, we ˇ ˇ require x y D˙=2 and Ex D Ey , corresponding to right- and left-handed orthonormal vectors  à 1 1 e˙ D p ; (A.17) 2 ˙i assuming x D 0. A.2 Quantized Radiation Field Main Reference (Townsend 2000b) A quantum description of light is obtained by quantizing the radiation field. A standard way to do this is to consider the radiation in a cubic cavity with an arbitrary volume V D L3 and the periodic boundary condition C eikj j D eikj .j L/; (A.18) for j D x;y;z. This leads to the quantization of the wave vector according to 2n k D j ; (A.19) j L where nj D 0; ˙1; ˙2;::: which in turn requires frequency modes to also be discrete due to the dispersion relation. Note that the introduction of volume V is 156 A Electromagnetic Radiation purely for mathematical convenience and does not appear in any of the observables. Now considering the classical electromagnetic field energy of a system confined to volume V Z  à 1 3 2 1 2 HEM D d r 0E0 C B0 2 V 0 (  à ) Z 2 1 3 @A 1 2 D d r 0 C .rA/ ; (A.20) 2 V @t 0 vector A can be represented using a discretized form of Eq. A.4 given by  à X ei.k r !k; t/ A.r;t/D AM u.M/ p C c.c. ; (A.21) M V where the shorthand notation M D fk;;g identifies a given mode with a unique set of spatial, frequency and polarization parameters. Taking advantage of the orthonormality relation Z ikr ik0r 3 e e d r p p D ıkk0 ; (A.22) V V V it is possible to show that X 2 HEM D 20 !M AMAM M  à X 2 pM 1 2 2 D C !MqM ; (A.23) M 2 2 where p pM Di!M 0.AM AM/; (A.24) p qM D 0.AM C AM/: (A.25) The above equation closely resembles an ensemble of harmonic oscillators for which the local energy at any one infinitesimal point in space can be determined quantum mechanically via the usual operator X pO2 1 HO D s C m!2sO2; (A.26) HO 2m 2 sDx;y;z A.2 Quantized Radiation Field 157 where for m D 1 we have r !„ pO Di .aO Oa/; (A.27) s 2 s s r „ sO D .aO COa/; (A.28) 2! s s and aOs and aOs are the annihilation and creation operators for mode s. This enables us to write equivalent quantum mechanical operators for the classical field components. Making the substitutions s „ AM ! aOM; (A.29) 20!M s „ AM ! aOM; (A.30) 20!M the vector potential in Eq. A.4 can then be transformed to an operator s X „ i.kr!Mt/ AO .r;t/D u.M/ aOM e C c.c. : (A.31) M 20!MV Using this expression to evaluate the Hamiltonian in Eq. A.20, which is now also an operator, we obtain X Á 1 HOEM D „!M aOMaOM COaMaOM 2 M X  à 1 D „!M aOMaOM C ; (A.32) M 2 where the last step takes advantage of the communication relations ŒaOM; aOM0 D ıM;M0 : (A.33) Equation A.32 represents an ensemble of the familiar harmonic oscillator Hamiltonians. The total field energy in the box is determined by the observable h jHOEMj i,wherej i represents the quantum state of the field over the entire 158 A Electromagnetic Radiation cavity volume V that can be made arbitrarily large to encapsulate the entire quantum system under investigation. We can also derive the electric field operator @OA EO .r;t/D @t X i.kr!k; t/ D ik;; u aOk;;.k;/e C c.c.; (A.34) k;; where s „!k; k;; D (A.35) 20V is the quantum mechanical unit of electric field. Subsequently the operator for the single mode field in Eq. A.9 is given by Á 1 EO .z;t/D EO .C/.z/ei!t C EO ./.z/ei!t ; (A.36) 2 where EO .C/.z/ D iu aeO ikz. As evident from the above discussion, particularly Eq. A.32, each unique radi- ation mode in the quantum field is modeled by a simple harmonic oscillator. To complete the quantization, we now represent each oscillator by a Fock state jni, with n D 0; 1; 2; : : : being the number of photons in that state. The overall quantized radiation field for a specific value of n is hence given by Y jnM1 ijnM2 ijnM3 i :::D jnMijfnMgi: (A.37) M Any arbitrary pure state can now be constructed as a superposition of the basis states, i.e.