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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Sudarshan-Glauber representation ： ウィキペディア英語版 Glauber–Sudarshan P representation The Glauber-Sudarshan P representation is a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation is the quasiprobability distribution in which observables are expressed in normal order. In quantum optics, this representation, formally equivalent to several other representations,〔L. Cohen, "Generalized phase-space distribution functions," ''Jou. Math. Phys.'', vol.7, pp. 781–786, 1966.〕〔L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''Jou. Math. Phys.'', vol.7, pp. 1863–1866, 1976.〕 is sometimes championed over alternative representations to describe light in optical phase space, because typical optical observables, such as the particle number operator, are naturally expressed in normal order. It is named after George Sudarshan〔E. C. G. Sudarshan "Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams", ''Phys. Rev. Lett.'',10 (1963) pp. 277–279. 〕 and Roy J. Glauber, who were working on the topic in 1963. It was the subject of a controversy when Glauber was awarded a share of the 2005 Nobel Prize in Physics for his work in this field and George Sudarshan's contribution was not recognized. Despite many useful applications in laser theory and coherence theory, the Glauber-Sudarshan P representation has the drawback that it is not always positive, and therefore is not a true probability function. ==Definition== (詳細はdensity matrix $\backslash hat$ is diagonal in the basis of coherent states $\backslash$, i.e. :$\backslash hat\; =\; \backslash int\; P(\backslash alpha)\; |\backslash rangle\; \backslash langle\; |\backslash ,\; d^\backslash alpha,\; \backslash qquad\; d^2\backslash alpha\; \backslash equiv\; d\backslash ,\; (\backslash alpha)\; \backslash ,\; d\backslash ,\; (\backslash alpha).$ We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the optical equivalence theorem. This implies that the density matrix should be in ''anti''-normal order so that we can express the density matrix as a power series :$\backslash hat\_A=\backslash sum\_\; c\_\backslash cdot\backslash hat^j\backslash hat^.$ Inserting the identity operator :$\backslash hat=\backslash frac\; \backslash int\; |\backslash rangle\; \backslash langle\; |\backslash ,\; d^\backslash alpha\; ,$ we see that :$\backslash begin$ \rho_A(\hat,\hat^)&=\frac\sum_ \int c_\cdot\hat^j|\rangle \langle |\hat^ \, d^\alpha \\ &= \frac \sum_ \int c_ \cdot \alpha^j|\rangle \langle |\alpha^ \, d^\alpha \\ &= \frac \int \sum_ c_ \cdot \alpha^j\alpha^|\rangle \langle | \, d^\alpha \\ &= \frac \int \rho_A(\alpha,\alpha^ *)|\rangle \langle | \, d^\alpha,\end and thus we formally assign :$P(\backslash alpha)=\backslash frac\backslash rho\_A(\backslash alpha,\backslash alpha^$ *). More useful integral formulas for ''P'' are necessary for any practical calculation. One method〔C. L. Mehta and E. C. G. Sudarshan "Relation between Quantum and Semiclassical Description of Optical Coherence", ''Phys. Rev.'',138 (1965) pp. B274–B280. 〕 is to define the characteristic function :$\backslash chi\_N(\backslash beta)=\backslash operatorname(\backslash hat\; \backslash cdot\; e^\}e^\backslash int\; \backslash chi\_N(\backslash beta)\; e^\; \backslash ,\; d^2\backslash beta.$ Another useful integral formula for ''P'' is〔C. L. Mehta "Diagonal Coherent-State Representation of Quantum Operators", ''Phys. Rev. Lett.'',18 (1967) pp. 752–754. 〕 :$P(\backslash alpha)=\backslash frac\backslash int\; \backslash langle\; -\backslash beta|\backslash hat|\backslash beta\backslash rangle\; e^\; \backslash ,\; d^2\backslash beta.$ Note that both of these integral formulas do ''not'' converge in any usual sense for "typical" systems . We may also use the matrix elements of $\backslash hat$ in the Fock basis $\backslash$. The following formula shows that it is ''always'' possible〔 to write the density matrix in this diagonal form without appealing to operator orderings using the inversion (given here for a single mode): :$P(\backslash alpha)=\backslash sum\_\; \backslash sum\_\; \backslash langle\; n|\backslash hat|k\backslash rangle\; \backslash frac\; e^\; \backslash left(-\; \backslash frac\; \backslash right)^\; \backslash delta\; (r)\; \backslash right),$ where ''r'' and ''θ'' are the amplitude and phase of ''α''. Though this is a full formal solution of this possibility, it requires infinitely many derivatives of Dirac delta functions, far beyond the reach of any ordinary tempered distribution theory. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Glauber–Sudarshan P representation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.