4 Representations of the Electromagnetic Field 4.1 Basics A quantum mechanical state is completely described by its density matrix ½. The density matrix ½ can be expanded in di®erent basises fjÃig: X ­ ¯ ¯ ½ = Dij jÃii Ãj for a discrete set of basis states (158) i;j ¯ ® ¯ The c-number matrix Dij = hÃij ½ Ãj is a representation of the density operator. One example is the number state or Fock state representation of the density ma- trix. X ½nm = Pnm jni hmj (159) n;m The diagonal elements in the Fock representation give the probability to ¯nd exactly n photons in the ¯eld! A trivial example is the density matrix of a Fock State jki: ½ = jki hkj and ½nm = ±nk±km (160) Another example is the density matrix of a thermal state of a free single mode ¯eld: + exp[¡~!a a=kbT ] ½ = + (161) T rfexp[¡~!a a=kbT ]g In the Fock representation the matrix is: X ½ = exp[¡~!n=kbT ][1 ¡ exp(¡~!=kbT )] jni hnj (162) n X hnin = jni hnj (163) (1 + hni)n+1 n with + ¡1 hni = T r(a a½) = [exp(~!=kbT ) ¡ 1] (164) 37 This leads to the well-known Bose-Einstein distribution: hnin ½ = hnj ½ jni = P = (165) nn n (1 + hni)n+1 4.2 Glauber-Sudarshan or P-representation The P-representation is an expansion in a coherent state basis jf®gi: Z ½ = P (®; ®¤) j®i h®j d2® (166) A trivial example is again the P-representation of a coherent state j®i, which is simply the delta function ±(® ¡ a0). The P-representation of a thermal state is a Gaussian: 1 2 P (®) = e¡j®j =n (167) ¼n Again it follows: Z ¤ 2 2 Pn = hnj ½ jni = P (®; ® ) jhnj®ij d ® (168) Z 2n 1 2 j®j 2 = e¡j®j =n e¡j®j d2® (169) ¼n n! The last integral can be evaluated and gives again the thermal distribution: hnin P = (170) n (1 + hni)n+1 4.3 Optical Equivalence Theorem Why is the P-representation useful? ² j®i corresponds to a classical state. ² The expectation values of normally ordered operator functions are similiar to their classical counterpart. 38 Let g(N) be a normally ordered operator function (i.e., all creation operators left, all annihilation operators right): X (N) + n m g = cnm(a ) a (171) n;m The expectation value of g(N) is: ­ ® g(N) = T r[½g(N)(a+; a)] (172) ( ) Z X + n m 2 = T r P (®) j®i h®j cnm(a ) a d ® (173) n;m Z X + n m 2 = P (®) cnm h®j (a ) a j®i d ® (174) n;m Z X ¤ n m 2 = P (®) cnm(® ) ® d ® (175) Z n;m = P (®)g(N)(®; ®¤) d2® (176) The last row gives the classical average value of the c-number function g(N)(®; ®¤) wheighted be the c-number function P (®). Formally one can write ­ (N) + ® ­ (N) ¤ ® g (a; a ) = g (®; ® ) P (177) This is the optical equivalence theorem. Example: The g(2) function: Classically: ha+a+aai g(2)(0) = ¸ 1 (178) ha+ai2 Using the optical equivalence theorem the numerator of the expression above is Z µZ ¶ ¡ ¢ 2 ­ ® P (®) j®j2 2 d2® ¸ P (®)j®j2 d2® = j®j2 2 (179) , (180) Z Z ³¡ ¢ ­ ® ´ P (®) j®j2 2 ¡ j®j2 2 d2® = P (®)V ar(®) d2® ¸ 0 (181) But, we have seen that g(2)(0) can be smaller than 1 in some cases! Thus: Z P (®)V ar(®) d2® < 0 for some cases (182) 39 Since V ar(®) is a positive function, P (®) has to be negative for some cases. Thus, P (®) cannot be interpreted as a probability function! For this reason P (®) and other functions are referred to as quasi-probability func- tions. For non-classical ¯eld states the quasi-probability functions can have negative values! 4.4 Wigner Function The ¯rst quasi-probability function was introduced by Wigner in 1932. The so-called Wigner function or Wigner distribution can be derived from the P- distribution via convolution with a Gaussian: Z 2 1 £ ¤ W (®) = P (¯) exp ¡2j¯ ¡ ®j2 d2¯ (183) ¼ ¡1 It is easy to show that W (®) can be written in terms of the conjugate variables q = ® + ®¤ and p = ¡i (® ¡ ®¤) Z µ ¶ 1 1 iq0p W (q; p) = hq + q0j ½ jq ¡ q0i exp dq0 (184) 4¼ ¡1 2 W (q; p) is the quasi-probability distribution of the conjugate variables q; p. From the Wigner function the probability density Pr(q) and Pr(p) of both q and p can be derived via integration: Z 1 Pr(q) = W (q; p) dp (185) ¡1Z Z µ ¶ 1 1 1 iq0p = hq + q0j ½ jq ¡ q0i dq0 exp dp (186) 4¼ 2 Z¡1 ¡1 1 1 = hq + q0j ½ jq ¡ q0i 4¼±(q0) dq0 (187) 4¼ ¡1 = hqj ½ jqi (188) = ä(q)Ã(q) = jÃ(q)j2 for a pure state (189) 40 Similar: Z 1 Pr(p) = W (q; p) dq = hpj ½ jpi (190) ¡1 Examples for Wigner functions are: Wigner function of a coherent state: · ¸ 2 1 ¡ ¢ W = exp ¡ q2 + p2 (191) coherent ¼ 2 Wigner function of a Fock state: 2 £ ¤ 2 2 W = (¡1)nL 4(q2 + p2) e¡2(q +p ) (192) n ¼ n The following pictures show the Wigner functions of a coherent state and a single photon Fock-state. The non-classical character of the Fock-state shows up as a negativity of the Wigner function at the origin. Figure 19: Wigner function of a harmonic oscillator eigenstate. 41 Figure 20: Wigner function of a single photon Fock state [from Lvovsky et al., Phys. Rev. Lett. 87, 050402 (2001)] 4.5 Quantum Tomography The Wigner function can be projected on any axis (not only on the p- and q-axis). The probability distribution along an arbitrary axis is: Z 1 Pr(q; #) = W (q cos # ¡ p sin #; q sin # + p cos #) dp (193) ¡1 The Wigner function and thus the complete information of the state ½ can be re- constructed by measuring Pr(q; #) for all possible #. For the special case of a rotationally symmetric Wigner function one ¯nds: Z 1 d 1 dq Pr(q) W (r) = ¡ 2 2 dq (194) ¼ 0 q ¡ r A more complex transformation holds for the general case. Such a reconstruction of the Wigner function is called quantum state tomography. How can one measure the probability distribution P r(q)? One way to do this is using homodyne detection. A typical homodyne experiment is shown in the following picture: 42 Figure 21: Principle of homodyne detection. We assume single mode ¯elds in both arms of the input. One arm is called the local oscillator. It contains a strong ¯eld which we assume to be classical: (0) ELO / ELO ® + c:c where ® is a c-number (195) The other input arms contains a weak quantum ¯eld, the signal ¯eld: (0) ES / ES aS + h:c: (196) The two photodetectors detect (positive frequency part only!) in arm 1 or 2: ­ ® 1 ­ ® I = E+ E = (E § E )+ (E § E ) (197) 1;2 1;2 1;2 2 LO S LO S 1 n­ ® ­ ® D Eo = E+E+ + E+ E+ § E(0)aE(0) ® + E(0)a+E(0) ®¤ (198) 2 S S LO LO S LO S LO 1 n ­ ®o = hI i + hI i § E(0)E(0) j®j aei# + a+e¡i# (199) 2 S LO S LO Thus, the di®erence of the detected intensity (measured, e.g., as a di®erence in the photo current) is: (0) (0) ­ i# + ¡i#® I1 ¡ I2 = ES ELOj®j ae + a e (200) (0) (0) = ES ELOj®j hq(#)i (201) The homodyne detection measures the ampli¯ed (by the large amplitude of the clas- sical ¯eld j®j) expectation value of the quadrature component q(#). The phase # can be changed very easily by changing the ¯eld of the classical ¯eld, e.g., by a delay line. 43 Thus, plotting the di®erence of hq(#)iT versus the time gives P r(q(#)) and from the measurement of P r(q(#)) for many di®erent # the Wigner function can be de- rived. P r(q(#)) is also called the marginal distribution. 4.6 Tomography of a Single Photon Fock State Fock-states are non-classical states. However, it is very di±cult to create them. It is important to divide single photon Fock states j1i from the more general class of single photon states jÃ1i. We may de¯ne a single photon state as a state of the form X X + jÃ1i = cfng jfngi = ckak jf0gi (202) fng k where for all the n-tupel fng there is only one ni di®erent from 0, e.g., jnij = j000:::1:::i. The single photon Fock-state is a special case (only one speci¯c k is relevant). Such as state can be created by: ² Spontaneous decay of a single two level system ² Detection of one of the photons from an entangled photon pair (e.g., from parametric down-conversion) If such a single photon (state) is detected with a photodetector, the photocurrent is proportional to (+) ª1 = h0j E (r; t) jÃ1i (203) We will see in a later chapter that for a detector at a distance r from a sponta- neously decaying atom: A r ª = 0 £(t ¡ )e¡i(t¡r=c)(!¡i¡=2) (204) 1 r c 44 where £ is the step function, which accounts for the ¯nite speed of light. The rate ¡ is the rate of spontaneous exponential decay of the upper state of the atom. The frequency spectrum (which is proportional to G(1)(¿) ) has a Lorentzian shape and thus a non-zero width.