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Quantizing the Electromagnetic Field II Quantizing the electromagnetic field II Hyeran Kong Table of Contents • Two flavors of quantum variables • Physical Consequence of field quantization - Lamb shift, Quantum beat, Casimir(-Polder) effect • Coherent state representation • Optical equivalence Theorem Quantization with different quantum variables 1. Discrete quantum variables Energy ; photon counting e.g. Single photon, Fock states .. 2. Continuous quantum variables Phase and amplitude quadratures e.g. Squeezed state, Schrodinger cat state QuantizationPlane wave expansions in cube of volume with different quantum variables 1. DiscreteHomogeneous wave equaon from Maxwell eqn. quantum variables complex analytical signal Energy ; photon coun/ng e.g. Single photon, Fock states .. 2. Continuous quantum variables Phase and amplitude e.g. Squeezed state, Schrodinger cat state Quantizationeigenstate of operator with different quantum Using the commutaon relaon variables 1. Discrete quantum variables eigenstate of Energy ; photon coun/ng e.g. Single photon, Fock states .. 2. Continuous quantum variables creaon operator Phase and amplitude e.g. Squeezed state, Schrodinger cat state annihilaon operator eigenvalue is quantized Quantization with different quantum variables 1. Discrete quantum variables Fock state representation Energy ; photon counting e.g. Single photon, Fock states .. 2. Continuous quantum variables PERSPECTIVES Phase and amplitude Coherent state e.g. Squeezed state, Schrodinger catNegative state values and positive W(Q,P) results. The Wigner function is a way W(Q,P) to describe how “quantum” a light A pulse is. Progressing from most clas- B sical to most quantum, the Wigner functions are shown for (A) the coherent state, (B) a squeezed state, (C) the single-photon state, and (D) a Schrödinger’s-cat state. The pro- Q P Q jections or “shadows” of the Wigner P function (shown on the sides) are the measured probability distributions of the quantum continuous variables Q or P. The Wigner function is a Gauss- W(Q,P) ian function for (A) and (B), but it W(Q,P) takes negative values for the strongly C quantum states (C) and (D). These D negative features vanish very quickly in the presence of decoherence. In the experiment by Lee et al., a quan- tum state similar to (C) and (D) was teleported and kept the negative Q P Q values of the Wigner function. This P result demonstrates an extraordinary degree of experimental control over such fragile quantum objects. on December 27, 2011 (see the figure, panel A). For a squeezed tions ( 10). Of particular interest here are can remain at the end (16 , 17): The cat state state, the peak is still Gaussian but now has superpositions of coherent states with oppo- must really be “erased” somewhere in order two-fold symmetry. One variance, say along site phases, which are called Schrödinger’s to be able to “reappear” elsewhere. P, is smaller than the vacuum noise, while kittens ( 11– 13) or Schrödinger’s cats ( 14, Overall, such an achievement is certainly the other one is larger (see the fi gure, panel 15), depending on their size (see the fi gure, very impressive, and it goes beyond pure B). For all optics experiments done during panel D). experimental virtuosity. It shows that the the last century, W(Q, P) was some kind of Such highly quantum states are desirable controlled manipulation of quantum objects Gaussian and therefore looked like a “real” for effi cient quantum information process- has progressed steadily and achieved objec- www.sciencemag.org (positive) probability distribution. ing tasks, such as entanglement distillation tives that seemed impossible just a few years More surprisingly, the number state, in quantum communication, or as logical ago, and that tools are now available to where the number of photons in a light pulse gates for quantum computing. These tasks tackle more ambitious goals. is well defi ned, has a W(Q, P) function that involve many operations and are quite vul- is negative at the origin (see the fi gure, panel nerable to decoherence—the degradation of References and Notes C). Negative values are allowed because entanglement by unwanted coupling to the 1. N. Lee et al., Science 332, 330 (2011). 2. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, J. F. true probabilities are obtained by integrat- environment. A natural question is how well Valley, Phys. Rev. Lett. 55, 2409 (1985). ing these distributions—the integral over P highly quantum states can be controlled in a 3. A. Furusawa et al., Science 282, 706 (1998). Downloaded from gives the true probability distribution of Q. basic processing operation. 4. C. H. Bennett et al., Phys. Rev. Lett. 70, 1895 (1993). 5. D. Bouwmeester et al., Nature 390, 575 (1997). The integral of W(Q, P) over any component The quantum teleportation of a 6. D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu, of the electric fi eld is the probability distri- Schrödinger’s-cat state by Lee et al. suc- Phys. Rev. Lett. 80, 1121 (1998). bution of its orthogonal component (these cessfully combined many operations. They 7. A. I. Lvovsky et al., Phys. Rev. Lett. 87, 050402 (2001). 8. A. Ourjoumtsev, R. Tualle-Brouri, P. Grangier, Phys. Rev. appear as “shadows” in the fi gure panels). generated a highly entangled state—the Lett. 96, 213601 (2006). This shadow idea is behind the method resource for teleportation—and indepen- 9. V. Parigi, A. Zavatta, M. Kim, M. Bellini, Science 317, used to obtain W(Q, P) experimentally. dently a Schrödinger’s-kitten state, the one 1890 (2007). Many projections are measured so that to be teleported. Then, after the appropri- 10. A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, P. Grangier, Nat. Phys. 5, 189 (2009). W(Q, P) can be reconstructed by a pro- ate steps including the destruction of the 11. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, P. Grangier, cess called quantum tomography. All of initial kitten, they “recreated” this kitten in Science 312, 83 (2006). the states with negative Wigner functions another place, still with negative features 12. J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, E. S. Polzik, Phys. Rev. Lett. 97, 083604 (2006). are called “nonclassical,” because their dis- in its Wigner function. These fi nal negative 13. H. Takahashi et al., Phys. Rev. Lett. 101, 233605 (2008). crete or continuous properties (or both) are features can only be observed if the qual- 14. A. Ourjoumtsev et al., Nature 448, 784 (2007). now mixed and are purely quantum features. ity of the teleportation is high enough. This 15. T. Gerrits et al., Phys. Rev. A 82, 031802(R) (2010). Many states with negative Wigner functions quality is measured by a number, called the 16. F. Grosshans, P. Grangier, Phys. Rev. A 64, 010301(R) (2001). have recently been realized experimentally, fi delity, which must be greater than 2/3 for 17. M. Ban, Phys. Rev. A 69, 054304 (2004). including number states with one ( 7) or two the operation to be successful. This 2/3 value 18. We thank A. Ourjoumtsev for help with the fi gures. photons ( 8), photon-added states ( 9), and is the so-called no-cloning limit, which also entangled states with negative Wigner func- ensures that no other copy of the initial state 10.1126/science.1204814 OURJOUMTSEV ALEXEI CREDIT: 314 15 APRIL 2011 VOL 332 SCIENCE www.sciencemag.org Published by AAAS Quantization with different quantum variables • Phase ? does not satisfy uncertainty principle is not Hermitian do not commute commute ! commute, orthogonal Quantization with different quantum variables D6 vacuum Input2 D5 BS Input1 D3 vacuum D4 J.W. Noh, A. Fougeres, L. Mandel, Phys. Rev. Lett. 67, 1467 (1991). Quantization with different quantum variables D6 vacuum Input2 In the regime, D5 BS Phase difference ill defined Input1 D3 vacuum D4 J.W. Noh, A. Fougeres, L. Mandel, Phys. Rev. Lett. 67, 1467 (1991). Lamb Shift • “Experimental Result” Energy difference between by level shi • Vacuum Fluctuation Theory (by Welton, 1957) W E. Lamb, JR., and R C. Rutherford, Phys. Rev. Lett. 72,241 (1947) W E. Lamb, JR. Nobel Lecture, (1955) Lamb Shift • “ExperimentalElectron Result” orbit perturbed by vacuum fluctuation perturbation Energy difference between by level shi Non-relativistic eqn. of motion for electron • Vacuum Fluctuation Theory (by Welton, 1957) ~1GHz in good agreement with Lamb shift measurement W E. Lamb, JR., and R C. Rutherford, Phys. Rev. Lett. 72,241 (1947) W E. Lamb, JR. Nobel Lecture, (1955) Quantum beat • Semiclassical Theory (SCT) configuration configuration Interference term Quantum beat • Quantum mechanics configuration configuration S. Haroche, J.A. Paisner, A.L. Schawlow Phys. Rev. Lett. 30, 948 (1973). Casimir effect Casimir effect manifested over different scales. H.B.G. Casimir, and D. Polder, Phys. Rev. Lett. 73, 360 (1947). V. Sandoghdar, C. I. Sukenik, E. A. Hinds, S. Haroche, Phys. Rev. Lett. 68, 3432 (1992). 4.2. FOCK STATES 31 Problem 4.4 The annihilation operator is defined as follows: 1 aˆ = Xˆ + iPˆ ; (4.11) √2 ! " 4 Representations of the Electromagnetic Field The operatora ˆ† is called the creation operator. Show that: 4.1 Basics a) the creation operator is A quantum mechanical state is completely described1 by its density matrix Ω. The density matrix Ω can be expanded in diaˆÆerent† = basisesXˆ √ i:Pˆ ; (4.12) √2 {|−i} Ω = D √ √ for a discrete! set of basis" states (158) ij | ii j i,j b) the creationTwo and annihilationX flavors≠ operatorsØ of arequantum not Hermitian; Ø The c-number matrix Dij = √i Ω √j is a representation of the density operator. c) their commutator is variablesh | One example is the number stateØ orÆ Fock state representation of the density ma- 1. Energy Discrete quantum variablesØ photon counting trix. [ˆa, aˆ†] = 1; (4.13) Ω = P n m (159) d) positionIn andFock-state momentum representation can benm expressednm | asih | n,m X The diagonal elements in the Fock representation give the probability to find exactly 1 34 1 A.
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