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| The '''Sudarshan-Glauber 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 [[observable]]s are expressed in [[normal order]]. In [[quantum optics]], this representation, formally equivalent to several other representations,<ref>L. Cohen, "Generalized phase-space distribution functions," ''Jou. Math. Phys.'', vol.7, pp. 781–786, 1966.</ref><ref>L. Cohen, "Quantization Problem and Variational Principle in the Phase Space Formulation of Quantum Mechanics," ''Jou. Math. Phys.'', vol.7, pp. 1863–1866, 1976.</ref> 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]]<ref name="Sudarshan">E. C. G. Sudarshan "Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams", ''Phys. Rev. Lett.'','''10''' (1963) pp. 277–279. {{doi|10.1103/PhysRevLett.10.277}}</ref> and [[Roy J. Glauber]],<ref>R. J. Glauber "Coherent and Incoherent States of the Radiation Field", ''Phys. Rev.'','''131''' (1963) pp. 2766–2788. {{doi|10.1103/PhysRev.131.2766}}</ref> both working on the subject around 1963. (It was the subject of a [[Nobel Prize controversies#Physics|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.) | | The title of the writer is Figures. She is a librarian but she's always wanted her personal business. Puerto Rico is where he's been residing for many years and he will never move. What I love doing is to gather badges but I've been taking on new things lately.<br><br>my weblog - [http://chatbook.biz/blogs/post/922 http://chatbook.biz] |
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| ==Definition==
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| {{main|Quasiprobability distribution}}
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| We wish to construct a function <math>P(\alpha)</math> with the property that the [[density matrix]] <math>\hat{\rho}</math> is [[diagonal matrix|diagonal]] in the basis of [[coherent states]] <math>\{|\alpha\rangle\}</math>, i.e.
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| :<math>\hat{\rho} = \int P(\alpha) |{\alpha}\rangle \langle {\alpha}|\, d^{2}\alpha, \qquad d^2\alpha \equiv d\, {\rm Re}(\alpha) \, d\, {\rm Im}(\alpha).</math>
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| 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
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| :<math>\hat{\rho}_A=\sum_{j,k} c_{j,k}\cdot\hat{a}^j\hat{a}^{\dagger k}.</math>
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| Inserting the [[identity operator]]
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| :<math>\hat{I}=\frac{1}{\pi} \int |{\alpha}\rangle \langle {\alpha}|\, d^{2}\alpha ,</math>
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| we see that
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| :<math>\begin{align}
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| \rho_A(\hat{a},\hat{a}^{\dagger})&=\frac{1}{\pi}\sum_{j,k} \int c_{j,k}\cdot\hat{a}^j|{\alpha}\rangle \langle {\alpha}|\hat{a}^{\dagger k} \, d^{2}\alpha \\
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| &= \frac{1}{\pi} \sum_{j,k} \int c_{j,k} \cdot \alpha^j|{\alpha}\rangle \langle {\alpha}|\alpha^{*k} \, d^{2}\alpha \\
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| &= \frac{1}{\pi} \int \sum_{j,k} c_{j,k} \cdot \alpha^j\alpha^{*k}|{\alpha}\rangle \langle {\alpha}| \, d^{2}\alpha \\
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| &= \frac{1}{\pi} \int \rho_A(\alpha,\alpha^*)|{\alpha}\rangle \langle {\alpha}| \, d^{2}\alpha,\end{align}</math>
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| and thus we formally assign
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| :<math>P(\alpha)=\frac{1}{\pi}\rho_A(\alpha,\alpha^*).</math>
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| More useful integral formulas for ''P'' are necessary for any practical calculation. One method<ref>C. L. Mehta and E. C. G. Sudarshan "Relation between Quantum and Semiclassical Description of Optical Coherence", ''Phys. Rev.'','''138''' (1965) pp. B274–B280. {{doi|10.1103/PhysRev.138.B274}}</ref> is to define the [[characteristic function (probability theory)|characteristic function]]
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| :<math>\chi_N(\beta)=\operatorname{tr}(\hat{\rho} \cdot e^{i\beta\cdot\hat{a}^{\dagger}}e^{i\beta^*\cdot\hat{a}})</math>
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| and then take the [[Fourier transform]]
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| :<math>P(\alpha)=\frac{1}{\pi^2}\int \chi_N(\beta) e^{-\beta\alpha^*+\beta^*\alpha} \, d^2\beta.</math>
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| Another useful integral formula for ''P'' is<ref>C. L. Mehta "Diagonal Coherent-State Representation of Quantum Operators", ''Phys. Rev. Lett.'','''18''' (1967) pp. 752–754. {{doi|10.1103/PhysRevLett.18.752}}</ref>
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| :<math>P(\alpha)=\frac{e^{|\alpha|^2}}{\pi}\int \langle -\beta|\hat{\rho}|\beta\rangle e^{|\beta|^2-\beta\alpha^*+\beta^*\alpha} \, d^2\beta.</math>
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| Note that both of these integral formulas do ''not'' converge in any usual sense for "typical" systems <!--(see example below)-->. We may also use the matrix elements of <math>\hat{\rho}</math> in the [[Fock state|Fock basis]] <math>\{|n\rangle\}</math>. The following formula shows that it is ''always'' possible<ref name="Sudarshan" /> to write the density matrix in this diagonal form without appealing to operator orderings using the inversion (given here for a single mode):
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| :<math>P(\alpha)=\sum_{n} \sum_{k} \langle n|\hat{\rho}|k\rangle \frac{\sqrt{n! k!}}{2 \pi r (n+k)!} e^{r^2-i(n-k)\theta} \left[\left( - \frac{\partial}{\partial r} \right)^{n+k} \delta (r) \right],</math>
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| 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 function]]s, far beyond the reach of any ordinary [[distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution theory]].
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| ==Discussion==
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| If the quantum system has a classical analog, e.g. a coherent state or [[thermal radiation]], then ''P'' is non-negative everywhere like an ordinary probability distribution. If, however, the quantum system has no classical analog, e.g. an incoherent [[Fock state]] or [[quantum entanglement|entangled system]], then ''P'' is negative somewhere or more singular than a Dirac delta function. Such "[[negative probability]]" or high degree of singularity is a feature inherent to the representation and does not diminish the meaningfulness of expectation values taken with respect to ''P''. Even if ''P'' does behave like an ordinary probability distribution, however, the matter is not quite so simple. According to Mandel and Wolf: "The different coherent states are not [mutually] orthogonal, so that even if <math>\scriptstyle P(\alpha) \,</math> behaved like a true probability density [function], it would not describe probabilities of mutually exclusive states."<ref>{{harvnb|Mandel|Wolf|1995|page=541}}</ref>
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| ==Examples==
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| ===Thermal radiation===
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| From [[statistical mechanics]] arguments in the Fock basis, the mean photon number of a mode with [[wavevector]] '''''k''''' and polarization state ''s'' for a [[black body]] at temperature ''T'' is known to be
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| :<math>\langle\hat{n}_{\mathbf{k},s}\rangle=\frac{1}{e^{\hbar \omega / k_B T}-1}.</math>
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| The ''P'' representation of the black body is
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| :<math>P(\{\alpha_{\mathbf{k},s}\})=\prod_{\mathbf{k},s} \frac{1}{\pi \langle\hat{n}_{\mathbf{k},s}\rangle} e^{-|\alpha|^2 / \langle\hat{n}_{\mathbf{k},s}\rangle}.</math>
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| In other words, every mode of the black body is [[normal distribution|normally distributed]] in the basis of coherent states. Since ''P'' is positive and bounded, this system is essentially classical. This is actually quite a remarkable result because for thermal equilibrium the density matrix is also diagonal in the Fock basis, but Fock states are non-classical.
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| ===Highly singular example===
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| Even very simple-looking states may exhibit highly non-classical behavior. Consider a superposition of two coherent states
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| :<math>|\psi\rangle=c_0|\alpha_0\rangle+c_1|\alpha_1\rangle</math>
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| where ''c<sub>0</sub>'' ''c<sub>1</sub>'' are constants subject to the normalizing constraint
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| :<math>1=|c_0|^2+|c_1|^2+2e^{-(|\alpha_0|^2+|\alpha_1|^2)/2}\operatorname{Re}\left( c_0^*c_1 e^{\alpha_0^*\alpha_1} \right).</math> | |
| Note that this is quite different from a [[qubit]] because <math>|\alpha_0\rangle</math> and <math>|\alpha_1\rangle</math> are not orthogonal. As it is straightforward to calculate <math>\langle -\alpha|\hat{\rho}|\alpha\rangle=\langle -\alpha|\psi\rangle\langle\psi|\alpha\rangle</math>, we can use the Mehta formula above to compute ''P'':
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| :<math>\begin{align}P(\alpha)&=|c_0|^2\delta^2(\alpha-\alpha_0)+|c_1|^2\delta^2(\alpha-\alpha_1) \\
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| &\, \, \, \, \, +2c_0^*c_1
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| e^{|\alpha|^2-\frac{1}{2}|\alpha_0|^2-\frac{1}{2}|\alpha_1|^2}
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| e^{(\alpha_1^*-\alpha_0^*)\cdot\partial/\partial(2\alpha^*-\alpha_0^*-\alpha_1^*)}
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| e^{(\alpha_0-\alpha_1)\cdot\partial/\partial(2\alpha-\alpha_0-\alpha_1)}
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| \cdot \delta^2(2\alpha-\alpha_0-\alpha_1) \\
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| &\, \, \, \, \, +2c_0c_1^*
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| e^{|\alpha|^2-\frac{1}{2}|\alpha_0|^2-\frac{1}{2}|\alpha_1|^2}
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| e^{(\alpha_0^*-\alpha_1^*)\cdot\partial/\partial(2\alpha^*-\alpha_0^*-\alpha_1^*)}
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| e^{(\alpha_1-\alpha_0)\cdot\partial/\partial(2\alpha-\alpha_0-\alpha_1)}
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| \cdot \delta^2(2\alpha-\alpha_0-\alpha_1).
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| \end{align}</math>
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| Despite having infinitely many derivatives of delta functions, ''P'' still obeys the optical equivalence theorem. If the expectation value of the number operator, for example, is taken with respect to the state vector or as a phase space average with respect to ''P'', the two expectation values match:
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| :<math>\begin{align}\langle\psi|\hat{n}|\psi\rangle&=\int P(\alpha) |\alpha|^2 \, d^2\alpha \\
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| &=|c_0\alpha_0|^2+|c_1\alpha_1|^2+2e^{-(|\alpha_0|^2+|\alpha_1|^2)/2}\operatorname{Re}\left( c_0^*c_1 \alpha_0^*\alpha_1 e^{\alpha_0^*\alpha_1} \right).\end{align}</math>
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| == References ==
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| ===Citations===
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| {{reflist}}
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| ===Citation bibliography===
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| {{refbegin}}
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| {{Citation
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| | last = Mandel
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| | first = L.
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| | author-link = Leonard Mandel
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| | last2 = Wolf
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| | first2 = E.
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| | author2-link = Emil Wolf
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| | title = Optical Coherence and Quantum Optics
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| | place = Cambridge UK
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| | publisher = Cambridge University Press
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| | series =
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| | volume =
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| | origyear =
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| | year = 1995
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| | month=
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| | edition =
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| | chapter =
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| | chapterurl =
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| | page =
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| | pages =
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| | language =
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| | url =
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| | archiveurl =
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| | archivedate =
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| | doi =
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| | id =
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| | isbn = 0-521-41711-2
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| | mr =
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| | jfm = }}
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| {{refend}}
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| ==See also==
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| *[[Nonclassical light]]
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| *[[Wigner quasiprobability distribution]]
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| *[[Husimi Q representation]]
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| *[[Nobel Prize controversies]]
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| {{DEFAULTSORT:Glauber-Sudarshan P representation}}
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| [[Category:Quantum optics]]
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