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| [[File:Optical phase space.jpg|thumb|400px|right|Optical phase diagram of a coherent state's distribution across phase space.]]
| | If you present photography effectively, it helps you look much more properly at the globe around you. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can customize the appearance with PSD to Word - Press conversion ''. <br><br> |
| In [[quantum optics]], an '''optical phase space''' is a [[phase space]] in which all [[quantum state]]s of an [[optical system]] are described. Each point in the optical phase space corresponds to a unique state of an ''optical system''. For any such system, a plot of the ''quadratures'' against each other, possibly as functions of time, is called a [[phase diagram]]. If the quadratures are functions of time then the optical phase diagram can show the evolution of a quantum optical system with time. | |
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| An optical phase diagram can give insight into the properties and behaviors of the system that might otherwise not be obvious. This can allude to qualities of the system that can be of interest to an individual studying an optical system that would be very hard to deduce otherwise. Another use for an optical phase diagram is that it shows the evolution of the state of an optical system. This can be used to determine the state of the optical system at any point in time.
| | Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Some of the Wordpress development services offered by us are:. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. To turn the Word - Press Plugin on, click Activate on the far right side of the list. There are plenty of tables that are attached to this particular database. <br><br>It is the convenient service through which professionals either improve the position or keep the ranking intact. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Thus it is difficult to outrank any one of these because of their different usages. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Thus, Word - Press is a good alternative if you are looking for free blogging software. If you have any kind of inquiries relating to where and how to use [http://snipitfor.me/wordpress_dropbox_backup_8159670 wordpress backup], you could contact us at our internet site. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
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| ==Background information==
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| When discussing the quantum theory of light, it is very common to use an electromagnetic [[oscillator]] as a model.<ref name="Measuring the Quantum State of Light">{{cite book|first=Ulf |last=Leonhardt |title=Measuring the Quantum State of Light |pages=18–29| publisher=[[Cambridge University Press]] |location=Cambridge |year=2005 |isbn=0-521-02352-1}}</ref> An electromagnetic oscillator describes an oscillation of the electric field. Since the magnetic field is proportional to the rate of change of the electric field, this too oscillates. Such oscillations describe light. Systems composed of such oscillators can be described by an optical phase space.
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| Let '''u'''('''x''',t) be a [[vector function]] describing a [[single mode]] of an [[simple harmonic oscillator|electromagnetic oscillator]]. For simplicitity, it is assumed that this electromagnetic oscillator is in vacuum. An example is the [[plane wave]] given by
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| :<math> \mathbf{u}(\mathbf{x},t) = \mathbf{u_{0}}e^{i(\mathbf{k} \cdot \mathbf{x} - wt)} </math>
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| where '''u<sub>0</sub>''' is the [[polarization vector]], '''k''' is the [[wave vector]], '''w''' the frequency, and '''A'''<math>\cdot </math>'''B''' denotes the [[dot product]] between the [[Euclidean vector|vectors]] '''A''' and '''B'''. This is the equation for a [[plane wave]] and is a simple example of such an electromagnetic oscillator. The oscillators being examined could either be free waves in space or some normal mode contained in some [[cavity]].
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| A single mode of the electromagnetic oscillator is isolated from the rest of the system and examined. Such an oscillator, when quantized, is described by the mathematics of a [[quantum harmonic oscillator]].<ref name="Measuring the Quantum State of Light"/> Quantum oscillators are described using [[creation and annihilation operators]] <math>\widehat a^\dagger</math> and <math>\widehat a</math>. Physical quantities, such as the [[electric field strength]], then become [[operator (physics)|quantum operators]].
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| In order to distinguish a physical quantity from the quantum mechanical operator used to describe it, a "hat" is used over the operator symbols. Thus, for example, where <math>E_i</math> might represent (one component of) the [[electric field]], the symbol <math>\widehat E_i</math> denotes the quantum-mechanical operator that describes <math>E_i</math>. This convention is used throughout this article, but is not in common use in more advanced texts, which avoid the hat, as it simply clutters the text.
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| In the quantum oscillator mode, most operators representing physical quantities are typically expressed in terms of the creation and annihilation operators. In this example, the electric field strength is given by:
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| :<math>\widehat{E}_{i}=u_{i}^{*}(\mathbf{x},t)\widehat{a}^{\dagger} + u_{i}(\mathbf{x},t)\widehat{a}</math><ref name="Quantum Optics">{{cite book |author=Scully, Marlan; Zubairy, M. Suhail |title=Quantum Optics |pages=5| publisher=[[Cambridge University Press]] |location=Cambridge |year=1997 |isbn=0-521-43595-1}}</ref>
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| (where ''x<sub>i</sub>'' is a single component of '''x''', position). The [[Hamiltonian (quantum mechanics)|Hamiltonian]] for an electromagnetic oscillator is found by [[Quantization (physics)|quantizing]] the [[electromagnetic field]] for this oscillator and the formula is given by:
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| :<math>\widehat{H} = \hbar\omega (\widehat{a}^{\dagger}\widehat{a} + 1/2)</math><ref name="Quantum Optics"/>
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| where <math>\omega</math> is the frequency of the (spatio-temportal) mode. The annihilation operator is the bosonic annihilation operator and so it obeys the [[canonical commutation relation]] given by:
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| :<math>[\widehat{a},\widehat{a}^{\dagger}] = 1</math>
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| The eigenstates of the annihilation operator are called [[coherent states]]:
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| :<math>\widehat{a}|\alpha\rangle = \alpha|\alpha\rangle</math>
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| It is important to note that the annihilation operator is not [[Hermitian]]; therefore its eigenvalues <math>\alpha</math> can be complex. This has important consequences. | |
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| Finally, the [[photon number]] is given by the operator <math> \widehat{N} = \widehat{a}^{\dagger} \widehat{a},</math> which gives the number of photons in the given (spatial-temporal) mode '''u'''.
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| ==Quadratures==
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| [[Operator (mathematics)|Operators]] given by
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| :<math> \widehat q = \tfrac 1 {\sqrt 2}(\widehat a^\dagger + \widehat a)</math>
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| and
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| :<math> \widehat p = \tfrac i {\sqrt 2}(\widehat a^\dagger - \widehat a)</math>
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| are called the quadratures and they represent the [[real number|real]] and [[imaginary number|imaginary]] parts of the complex amplitude represented by <math> \widehat a</math>.<ref name="Measuring the Quantum State of Light"/> The commutation relation between the two quadratures can easily be calculated:
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| :<math>
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| \begin{align}
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| \left[ \widehat q, \widehat p \right]
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| &= \tfrac i 2 [\widehat a^\dagger + \widehat a, \widehat a^\dagger - \widehat a] \\
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| &= \tfrac i 2 ([\widehat a^\dagger, \widehat a^\dagger] - [\widehat a^\dagger, \widehat a] +
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| [\widehat a, \widehat a^\dagger] - [\widehat a, \widehat a]) \\
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| &= \tfrac i 2 (-(-1) + 1) \\
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| &= i
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| \end{align}
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| </math>
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| This looks very similar to the commutation relation of the position and momentum operator. Thus, it can be useful to think of and treat the quadratures as the position and momentum of the oscillator although in fact they are the "in-phase and out-of-phase components of the electric field amplitude of the spatial-temporal mode", or '''u''', and have nothing really to do with the position or momentum of the electromagnetic oscillator (as it is hard to define what is meant by position and momentum for an electromagnetic oscillator).<ref name="Measuring the Quantum State of Light"/>
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| ===Properties of quadratures===
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| The [[eigenstates]] of the quadrature operators <math>\widehat{q}</math> and <math>\widehat{p}</math> are called the quadrature states. They satisfy the relations:
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| :*<math> \widehat{q}|q\rangle = q |q\rangle</math> and <math>\widehat{p}|p\rangle = p |p\rangle</math>
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| :*<math> \langle q | q'\rangle = \delta(q-q')</math> and <math>\langle p | p'\rangle = \delta(p-p')</math>
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| :*<math> \int_{-\infty}^{\infty} |q\rangle \langle q| dq = 1 </math> and <math>\int_{-\infty}^{\infty} |p\rangle \langle p| dp = 1 </math>
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| as these form [[Orthonormal basis|complete basis]] sets.
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| ===Important result===
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| The following is an important relation that can be derived from the above which justifies our interpretation that the quadratures are the real and imaginary parts of a complex <math>\alpha</math> (i.e. the in-phase and out-of-phase components of the electromagnetic oscillator)
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| :<math> \langle\alpha|\widehat{q}|\alpha\rangle = 2^{-1/2}(\langle\alpha|\widehat{a}^{\dagger}|\alpha\rangle + \langle\alpha|\widehat{a}|\alpha\rangle) = 2^{-1/2}(\alpha^{*}\langle\alpha|\alpha\rangle + \alpha\langle\alpha|\alpha\rangle) </math>
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| The following is a relationship that can be used to help evaluate the above and is given by:
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| :<math>\langle\alpha'|\alpha\rangle = e^{(-1/2)(|\alpha'|^{2}+|\alpha|^{2}) + \alpha'^{*}\alpha}</math><ref name="Measuring the Quantum State of Light"/>
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| This gives us that:
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| :<math> \langle\alpha|\widehat{q}|\alpha\rangle = 2^{-1/2}(\alpha^{*} + \alpha) = q_{\alpha}</math>
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| :<math> \langle\alpha|\widehat{p}|\alpha\rangle = i2^{-1/2}(\alpha^{*} - \alpha) = p_{\alpha} </math> by a similar method as above.
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| :<math> \alpha = 2^{-1/2}(\langle\alpha|\widehat{q}|\alpha\rangle + i\langle\alpha|\widehat{p}|\alpha\rangle) = 2^{-1/2}(q_{\alpha} + ip_{\alpha}) </math>
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| Thus, <math>\alpha</math> is just a composition of the quadratures.
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| Another very important property of the coherent states becomes very apparent in this formalism. A coherent state is not a point in the optical phase space but rather a distribution on it. This can be seen via
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| :<math>q_{\alpha} = \langle\alpha|\widehat{q}|\alpha\rangle</math>
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| and
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| :<math>p_{\alpha} = \langle\alpha|\widehat{p}|\alpha\rangle</math>.
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| These are only the expectation values of <math>\widehat{q}</math> and <math>\widehat{p}</math> for the state <math>|\alpha\rangle</math>.
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| It can be shown that the quadratures obey [[Heisenberg's Uncertainty Principle]] given by: | |
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| :<math>\Delta q\Delta p \ge 1/2</math><ref name="Measuring the Quantum State of Light"/> (where <math>\Delta q</math> and <math>\Delta p</math> are the [[variance]]s of the distributions of q and p, respectively)
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| This inequality does not necessarily have to be saturated and a common example of such states are [[squeezed coherent states]]. The coherent states are [[Gaussian probability distribution]]s over the phase space localized around <math>\alpha</math>.
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| ==Operators on phase space==
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| It is possible to define operators to move the coherent states around the phase space. These can produce new coherent states and allow us to move around phase space.
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| ===Phase-shifting operator===
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| [[File:Rotation operator.jpg|thumb|350px|right|Phase shifting operator acting on a coherent state rotating it by an angle <math>\theta</math> in phase space.]]
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| The phase-shifting operator rotates the coherent state by an angle <math>\theta</math> in the optical phase space. This operator is given by:
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| :<math> \widehat{U}(\theta) = e^{-i\theta\widehat{N}} </math> <ref name="Measuring the Quantum State of Light"/>
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| The important relationship
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| :<math> \widehat{U}(\theta)^{\dagger}\widehat{a}\widehat{U}(\theta) = \widehat{a}e^{-i\theta} </math>
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| is derived as follows:
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| :<math> d/d\theta (\widehat{U}^{\dagger}\widehat{a}\widehat{U}) = i\widehat{n}\widehat{U}^{\dagger}\widehat{a}\widehat{U} - i\widehat{U}^{\dagger}\widehat{a}\widehat{U}\widehat{N} = \widehat{U}^{\dagger}i[\widehat{N},\widehat{a}]\widehat{U}</math>
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| :<math>= \widehat{U}^{\dagger}i(\widehat{a}^{\dagger}\widehat{a}\widehat{a} - \widehat{a}\widehat{a}^{\dagger}\widehat{a})\widehat{U} = \widehat{U}^{\dagger}i[\widehat{a}^{\dagger},\widehat{a}]\widehat{a}\widehat{U} = -i\widehat{U}^{\dagger}\widehat{a}\widehat{U}</math>
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| and solving this [[differential equation]] yields the desired result.
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| Thus, using the above it becomes clear that
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| :<math>\widehat{U}(\theta)|\alpha\rangle = |\alpha e^{-i\theta}\rangle</math>,
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| or a rotation by an angle theta on the coherent state in phase space. The following illustrates this more clearly:
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| :<math>\widehat{a}(\widehat{U}|\alpha\rangle) = \widehat{U}\widehat{a}e^{-i\theta}|\alpha\rangle </math>
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| (which is obtained using the fact that the phase-shifting operator is [[unitary]]{{Disambiguation needed|date=January 2012}})
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| :<math> \widehat{a}(\widehat{U}|\alpha\rangle) = \widehat{U} \alpha e^{-i\theta}|\alpha\rangle = \alpha e^{-i\theta}(\widehat{U}|\alpha\rangle) </math>
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| Thus, | |
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| :<math>(\alpha e^{-i\theta}, \widehat{U}|\alpha\rangle) </math>
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| is the [[Eigenvalue, eigenvector and eigenspace|eigenpair]] of | |
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| :<math> \widehat{a}\widehat{U}|\alpha\rangle</math>.
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| From this it is possible to see that
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| :<math> (\alpha e^{-i\theta} = 2^{-1/2}[q_{\alpha} cos(\theta) + p_{\alpha} sin(\theta)] + i2^{-1/2}[-q_{\alpha} csin(\theta) + p_{\alpha} cos(\theta)], \widehat{U}|\alpha\rangle = |\alpha e^{-i\theta}\rangle)</math>
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| which is another way of expressing the eigenpair which more clearly illustrates the effects of the phase-shifting operator on coherent states.
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| ===Displacement operator===
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| [[File:Displacement operator.jpg|thumb|350px|right|Displacement operator acting on a coherent state displacing it by some value <math>\alpha</math> in phase space.]]
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| The displacement operator takes a coherent state and moves it (by some value) to another coherent state somewhere in the phase space. The displacement operator is given by:
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| :<math>\widehat{D}(\alpha) = e^{\alpha\widehat{a}^{\dagger} - \alpha^{*}\widehat{a}}</math>
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| The relationship
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| :<math> \widehat{D}^{\dagger}\widehat{a}\widehat{D} = \widehat{a} + \alpha</math>.<ref name="Measuring the Quantum State of Light"/>
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| can be derived quite easily.
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| To do this, take an infinitesimal displacement <math>\delta\alpha</math>.
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| The operators <math>\widehat{D}</math> and <math>\widehat{D}^{\dagger}</math> can be expanded in terms of the identity
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| :<math>e^{x} = \sum_{0}^{\infty}x^{i}/i!</math>
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| and look at the first order terms and ignore all the higher order terms (all higher order terms are very close to zero as <math>\delta\alpha</math> is very small).
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| Thus:
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| :<math> \widehat{D}^{\dagger}(\delta\alpha)\widehat{a}\widehat{D}(\delta\alpha) = \sum_{i,j}(\delta\alpha^{*}\widehat{a} - \delta\alpha\widehat{a}^{\dagger})^{i}\widehat{a}(\delta\alpha\widehat{a}^{\dagger}-\delta\alpha^{*}\widehat{a})^{j}/i!j!</math>
| |
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| :<math>=\widehat{a} + (\delta\alpha^{*}\widehat{a} - \delta\alpha\widehat{a}^{\dagger})\widehat{a} + \widehat{a}(\delta\alpha\widehat{a}^{\dagger} - \delta\alpha^{*}\widehat{a}) + O(\delta\alpha^{2},(\delta\alpha^{*})^{2})</math> (but as given above, the higher order terms are very close to zero and therefore neglected)
| |
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| :<math>=\widehat{a} + \widehat{a}(\delta\alpha\widehat{a}^{\dagger} - \delta\alpha^{*}\widehat{a}) - (\delta\alpha\widehat{a}^{\dagger} - \delta\alpha^{*}\widehat{a})\widehat{a}</math>
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| :<math>=\widehat{a} + [\widehat{a},\delta\alpha\widehat{a}^{\dagger} - \delta\alpha^{*}\widehat{a}] = \widehat{a} + \delta\alpha[\widehat{a},\widehat{a}^{\dagger}] - \delta\alpha^{*}[\widehat{a},\widehat{a}]</math> (use the identity from above)
| |
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| :<math> \widehat{D}^{\dagger}(\delta\alpha)\widehat{a}\widehat{D}(\delta\alpha)=\widehat{a} + \delta\alpha</math>
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| The above identity can be applied repeatedly in the following fashion to derive the following:
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| :<math> (\widehat{D}^{\dagger}(\delta\alpha))^{k}\widehat{a}(\widehat{D}(\delta\alpha))^{k} = \widehat{a} + k\delta\alpha</math>
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| Thus, the above identity suggests that repeated use of the displacement operator generates translations in phase space.
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| ====Important consequence====
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| The following is an important consequence of the displacement vector.
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| First note that the displacement operator is a [[unitary operator]]. Use
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| :<math> \widehat{D}^{\dagger}(\delta\alpha)\widehat{a}\widehat{D}(\delta\alpha)=\widehat{a} + \delta\alpha</math>
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| to get:
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| :<math>\widehat{a}\widehat{D}(-\alpha)|\alpha\rangle = \widehat{D}(-\alpha)(\widehat{a} - \alpha)|\alpha\rangle</math>
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| :<math>=\widehat{D}(-\alpha)(\widehat{a}|\alpha\rangle - \alpha|\alpha\rangle)</math>
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| :<math>=\widehat{D}(-\alpha)(\alpha|\alpha\rangle - \alpha|\alpha\rangle)</math>
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| Thus,
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| :<math>\widehat{a}(\widehat{D}(-\alpha)|\alpha\rangle)=0</math>
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| or it follows that
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| :<math>\widehat{D}(-\alpha)|\alpha\rangle = |0\rangle</math>
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| which leads to
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| :<math>|\alpha\rangle=\widehat{D}(\alpha)|0\rangle</math>.
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| This is important as it suggests that all coherent states are just displacements of the [[ground state]], which in optics is also the [[vacuum state]]. That is, any coherent state can be generated via the displacement of the ground state of the electromagnetic oscillator from above.
| |
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| ==See also==
| |
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| * [[Nonclassical light]]
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| * [[Rotation operator (quantum mechanics)]]
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| * [[Quantum harmonic oscillator]]
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| * [[Quasiprobability distribution]]
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| * [[Husimi Q representation]]
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| * [[Squeezed coherent state]]
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| * [[Wigner function]]
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| | |
| ==References==
| |
| | |
| {{reflist}}
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| [[Category:Quantum optics| ]]
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| [[Category:Optics]]
| |
If you present photography effectively, it helps you look much more properly at the globe around you. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. You can customize the appearance with PSD to Word - Press conversion .
Word - Press is known as the most popular blogging platform all over the web and is used by millions of blog enthusiasts worldwide. Some of the Wordpress development services offered by us are:. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience.
Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. I hope this short Plugin Dynamo Review will assist you to differentiate whether Plugin Dynamo is Scam or a Genuine. To turn the Word - Press Plugin on, click Activate on the far right side of the list. There are plenty of tables that are attached to this particular database.
It is the convenient service through which professionals either improve the position or keep the ranking intact. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Thus it is difficult to outrank any one of these because of their different usages. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level.
There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Thus, Word - Press is a good alternative if you are looking for free blogging software. If you have any kind of inquiries relating to where and how to use wordpress backup, you could contact us at our internet site. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.