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| A '''quantum limit''' in physics is a limit on measurement accuracy at quantum scales.<ref name=QMeas_Br_Kh>
| | If you are looking for a specific plugin, then you can just search for the name of the plugin. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. This CMS has great flexibility to adapt various extensions and add-ons. If you need a special plugin for your website , there are thousands of plugins that can be used to meet those needs. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br>You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. If a newbie missed a certain part of the video then they could always rewind. A Wordpress plugin is a software that you can install into your Wordpress site. They provide many such popular products which you can buy for your baby. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Setting Up Your Business Online Using Free Wordpress Websites. To turn the Word - Press Plugin on, click Activate on the far right side of the list. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites. <br><br>If you adored this write-up and you would certainly such as to obtain even more details concerning [http://urlon.com.br/wordpress_backup_plugin_97038 wordpress dropbox backup] kindly browse through the web-page. Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <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. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. If this is not possible you still have the choice of the default theme that is Word - Press 3. For your information, it is an open source web content management system. |
| {{cite book
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| |last1=Braginsky |first1=V .B.
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| |last2=Khalili |first2=F. Ya.
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| |year=1992
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| |title=Quantum Measurement
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| |publisher=[[Cambridge University Press]]
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| |isbn=978-0521484138
| |
| }}</ref> Depending on the context, the limit may be absolute (such as the [[Heisenberg limit]]), or it may only apply when the experiment is conducted with naturally occurring [[quantum state]]s (e.g. the '''standard quantum limit''' in interferometry) and can be circumvented with advanced state preparation and measurement schemes.
| |
| | |
| The usage of term '''standard quantum limit''' or SQL is, however, broader than just interferometry. In principle, any linear measurement of a quantum mechanical [[observable]] of a system under study that does not [[Commutator|commute]] with itself at different times leads to such limits. In short, it is the [[Uncertainty principle|Heisenberg uncertainty principle]] that is the cause .
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| [[File:Scheme of quantum measurement process.png|thumb|A schematic description of how physical measurement process is described in quantum mechanics]]
| |
| A more detailed explanation would be that any measurement in [[quantum mechanics]] involves at least two parties, an Object and a Meter. The former is the system which observable, say <math>\hat x</math>, we want to measure. The latter is the system we couple to the Object in order to infer the value of <math>\hat x</math> of the Object by recording some chosen observable, <math>\hat{\mathcal{O}}</math>, of this system, ''e.g.'' the position of the pointer on a scale of the Meter. This, in a nutshell, a model of most of the measurement happening in physics, known as ''indirect'' measurement (see pp. 38–42 of <ref name=QMeas_Br_Kh />)
| |
| So any measurement is a result of interaction and that acts in both ways. Therefore, the Meter acts on the Object during each measurement, usually via the quantity, <math>\hat{\mathcal{F}}</math>, conjugate to the readout observable <math>\hat\mathcal{O}</math>, thus perturbing the value of measured observable <math>\hat x</math> and modifying the results of subsequent measurements. This is known as back action of the Meter on the system under measurement.
| |
| | |
| At the same time, quantum mechanics prescribes that readout observable of the Meter should have an inherent uncertainty, <math>\delta\hat\mathcal{O}</math>, additive to and independent on the value of the measured quantity <math>\hat x</math>. This one is known as ''measurement imprecision'' or ''measurement noise''. Because of the [[Uncertainty principle|Heisenberg uncertainty principle]] this imprecision can not be arbitrary and is linked to the back-action perturbation by the [[Uncertainty principle|uncertainty relation]]:
| |
| | |
| :<math>\Delta \mathcal{O}\Delta \mathcal{F}\geqslant \hbar/2\,,</math> | |
| | |
| where <math>\Delta a = \sqrt{\langle\hat a^2\rangle-\langle\hat a\rangle^2}</math> is a standard deviation of observable <math>a</math> and <math>\langle\hat a\rangle</math> stands for [[expectation value]] of <math>a</math> in whatever [[quantum state]] the system is. The equality is reached if the system is in a ''minimum uncertainty state''. The consequence for our case is that the more precise is our measurement, ''i.e'' the smaller is <math>\Delta \mathcal{\delta O}</math>, the larger will be perturbation the Meter exerts on the measured observable <math>\hat x</math>. Therefore the readout of the meter will, in general, consist of three terms:
| |
| | |
| :<math>\hat\mathcal{O} = \hat x_{free} + \delta\hat\mathcal{O} + \delta\hat x_{BA}[\hat\mathcal{F}]\,,</math>
| |
| | |
| where <math>\hat x_{free} </math> is a value of <math>\hat x</math> that the Object would have, were it not coupled to the Meter, and <math>\delta\hat x_{BA}[\hat\mathcal{F}]</math> is the perturbation to the value of <math>\hat x</math> caused by back action force, <math>\hat\mathcal{F}</math>. The uncertainty of the latter is proportional to <math>\Delta \mathcal{F}\propto\Delta \mathcal{O}^{-1}</math>. Thus, there is a minimal value, or the limit to the precision one can get in such a measurement, provided that <math>\delta\hat\mathcal{O} </math> and <math>\hat\mathcal{F}</math> are uncorrelated <ref name=LRR_Da_Kh>
| |
| {{cite journal
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| |last=Danilishin|first=S. L.
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| |coauthors=Khalili F. Ya.
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| |title=Quantum Measurement Theory in Gravitational-Wave Detectors
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| |journal=Living Reviews in Relativity
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| |year=2012
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| |issue=5
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| |page=60
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| |doi=10.12942/lrr-2012-5
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| |url=http://www.livingreviews.org/lrr-2012-5}}</ref>
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| .<ref>{{cite journal
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| |last=Chen|first=Yanbei
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| |title=Macroscopic quantum mechanics: theory and experimental concepts of optomechanics
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| |journal=J. Phys. B: At. Mol. Opt. Phys.
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| |year=2013
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| |volume=46
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| |pages=104001
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| |doi=10.1088/0953-4075/46/10/104001
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| |url=http://arxiv.org/abs/1302.1924
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| |accessdate=6 November 2013}}</ref>
| |
| | |
| The terms "quantum limit" and "standard quantum limit" are sometimes used interchangeably. Usually, "quantum limit" is a general term which refers to ''any'' restriction on measurement due to quantum effects, while the "standard quantum limit" in any given context refers to a quantum limit which is ubiquitous in that context.
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| | |
| ==Examples==
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| | |
| ===Displacement measurement===
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| | |
| Let us consider a very simple measurement scheme, which, nevertheless,
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| embodies all key features of a general position measurement. In the
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| scheme shown in Figure, a sequence of very short light
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| pulses are used to monitor the displacement of a probe body <math>M</math>. The
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| position <math>x</math> of <math>M</math> is probed periodically with time interval
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| <math>\vartheta</math>. We assume mass <math>M</math> large enough to neglect the displacement inflicted by
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| the pulses regular (classical) [[radiation pressure]] in the course of measurement
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| process.
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| [[File:Position measurement using reflected light.png|thumb|Simplified scheme of optical measurement of mechanical object position]]
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| Then each <math>j</math>-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position <math>x(t_j)</math> at the moment of reflection:
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| | |
| {{NumBlk|:|<math>
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| \hat{\phi}_j^{\mathrm{refl}} = \hat{\phi}_j - 2 k_p\hat{x}(t_j) \,,
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| </math>|{{EquationRef|1}}}}
| |
| | |
| where <math>k_p=\omega_p/c</math>, <math>\omega_p</math> is the light frequency, <math>j=\dots,-1,0,1,\dots</math> is the pulse number and <math>\hat{\phi}_j</math> is the initial (random) phase of the <math>j</math>-th pulse. We assume that the mean value of all these phases is equal to zero, <math>\langle\hat{\phi}_j\rangle=0</math>, and their root mean square (RMS) uncertainty <math>\langle(\hat{\phi^2}\rangle-\langle\hat{\phi}\rangle^2)^{1/2}</math> is equal to <math>\Delta\phi</math>.
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| The reflected pulses are detected by a phase-sensitive device (the phase detector). The implementation of an optical phase detector can be done using, ''e.g.'' [[Homodyne detection|homodyne]] or [[Optical heterodyne detection|heterodyne]] detection scheme (see Section 2.3 in
| |
| <ref name=LRR_Da_Kh /> and references therein).
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| | |
| In this example, light pulse phase <math>\hat\phi_j</math> serves as the readout observable <math>\mathcal{O}</math> of the Meter. Then we suppose that the phase <math>\hat{\phi}_j^{\mathrm{refl}}</math> measurement error introduced by the detector is much smaller than the initial uncertainty of the phases <math>\Delta\phi</math>. In this case, the initial uncertainty will be the only source of the position measurement error:
| |
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| {{NumBlk|:|<math>
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| \Delta x_{\mathrm{meas}} = \frac{\Delta\phi}{2 k_p} \,.
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| </math>|{{EquationRef|2}}}}
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| | |
| For convenience, we renormalise Eq. ({{EquationNote|1}}) as the equivalent test-mass displacement: | |
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| {{NumBlk|:|<math>
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| \tilde{x}_j \equiv -\frac{\hat{\phi}_j^{\mathrm{refl}}}{2 k_p}
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| = \hat{x}(t_j) + \hat{x}_{\mathrm{fl}}(t_j) \,,
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| </math>|{{EquationRef|3}}}}
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| | |
| where
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| | |
| :<math>
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| \hat{x}_{\mathrm{fl}}(t_j) = -\frac{\hat{\phi}_j}{2 k_p}
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| </math>
| |
| | |
| are the independent random values with the RMS uncertainties given by Eq. ({{EquationNote|2}}).
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| | |
| Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to
| |
| | |
| {{NumBlk|:|<math>
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| \hat{p}_j^{\mathrm{after}} - \hat{p}_j^{\mathrm{before}} = \hat{p}_j^{\mathrm{b.a.}}
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| = \frac{2}{c}\hat{\mathcal{W}}_j \,,
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| </math>|{{EquationRef|4}}}}
| |
| | |
| where <math>\hat{p}_j^{\mathrm{before}}</math> and <math>\hat{p}_j^{\mathrm{after}}</math> are the test-mass momentum values just before and just after the light pulse reflection, and <math>\mathcal{W}_j</math> is the energy of the <math>j</math>-th pulse, that plays the role of back action observable <math>\hat\mathcal{F}</math> of the Meter. The major part of this perturbation is contributed by classical radiation pressure:
| |
| | |
| :<math>
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| \langle\hat{p}_j^{\mathrm{b.a.}}\rangle = \frac{2}{c}\mathcal{W} \,,
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| </math>
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| | |
| with <math>\mathcal{W}</math> the mean energy of the pulses. Therefore, one could neglect its effect, for it could be either subtracted from the measurement result or compensated by an actuator. The random part, which cannot be compensated, is proportional to the deviation of the pulse energy:
| |
| | |
| :<math>
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| \hat{p}^{\mathrm{b.a.}}(t_j) = \hat{p}_j^{\mathrm{b.a.}} - \langle\hat{p}_j^{\mathrm{b.a.}}\rangle
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| = \frac{2}{c}\bigl(\hat{\mathcal{W}}_j - \mathcal{W}\bigr) \,,
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| </math> | |
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| and its RMS uncertainly is equal to | |
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| {{NumBlk|:|<math>
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| \Delta p_{\mathrm{b.a.}} = \frac{2\Delta\mathcal{W}}{c} \,,
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| </math>|{{EquationRef|5}}}}
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| | |
| with <math>\Delta\mathcal{W}</math> the RMS uncertainty of the pulse energy.
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| Assuming the mirror is free (which is a fair approximation if time interval between pulses is much shorter than the period of suspended mirror oscillations, <math>\vartheta\ll T</math>), one can estimate an additional displacement caused by the back action of the <math>j</math>-th pulse that will contribute to the uncertainty of the subsequent measurement by the <math>j+1</math> pulse time <math>\vartheta</math> later:
| |
| | |
| :<math>
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| \hat x_{\mathrm{b.a.}}(t_j) = \frac{\hat{p}^{\mathrm{b.a.}}(t_j)\vartheta}{M} \,.
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| </math>
| |
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| Its uncertainty will be simply
| |
| | |
| :<math>
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| \Delta x_{\mathrm{b.a.}}(t_j) = \frac{\Delta {p}_{\mathrm{b.a.}}(t_j)\vartheta}{M} \,.
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| </math>
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| If we now want to estimate how much has the mirror moved between the <math>j</math> and <math>j+1</math> pulses, ''i.e.'' its ''displacement'' <math>\delta\tilde x_{j+1,j} = \tilde x(t_{j+1}) - \tilde x(t_{j+1})</math>, we will have to deal with three additional uncertainties that limit precision of our estimate:
| |
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| :<math>
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| \Delta \tilde{x}_{j+1,j} = \Bigl[(\Delta x_{\rm meas}(t_{j+1}))^2+(\Delta x_{\rm meas}(t_{j}))^2+(\Delta x_{\rm b.a.}(t_{j}))^2\Bigr]^{1/2} \,,
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| </math>
| |
| | |
| where we assumed all contributions to our measurement uncertainty statistically independent and thus got sum uncertainty by summation of standard deviations. If we further assume that all light pulses are similar and have the same phase uncertainty, thence <math>\Delta x_{\rm meas}(t_{j+1}) = \Delta x_{\rm meas}(t_{j}) \equiv \Delta x_{\rm meas} = \Delta\phi/(2k_p)</math> .
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| | |
| Now, what is the minimum this sum and what is the minimum error one can get in this simple estimate? The answer ensues from quantum mechanics, if we recall that energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:
| |
| | |
| :<math>
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| \Delta\mathcal{W}\Delta\phi \ge \frac{\hbar\omega_p}{2} \,.
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| </math>
| |
| | |
| Therefore, it follows from Eqs. ({{EquationNote|2}} and {{EquationNote|5}}) that the position measurement error <math>\Delta x_{\mathrm{meas}}</math> and the momentum perturbation <math>\Delta p_{\mathrm{b.a.}}</math> due to back action also satisfy the uncertainty relation:
| |
| | |
| :<math>
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| \Delta x_{\mathrm{meas}}\Delta p_{\mathrm{b.a.}} \ge \frac{\hbar}{2} \,.
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| </math>
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| | |
| Taking this relation into account, the minimal uncertainty, <math>\Delta x_{\mathrm{meas}}</math>, the light pulse should have in order not to perturb the mirror too much, should be equal to <math>\Delta x_{\mathrm{b.a.}}</math> yielding for both <math>\Delta x_{\mathrm{min}} = \sqrt{\frac{\hbar\vartheta}{2M}}</math>. Thus the minimal displacement measurement error that is prescribed by quantum mechanics read:
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| | |
| :<math>
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| \Delta \tilde{x}_{j+1,j} \geqslant \Bigl[2(\Delta x_{\rm meas})^2+\Bigl(\frac{\hbar\vartheta}{2M\Delta x_{\rm meas}}\Bigr)^2\Bigr]^{1/2} \geqslant \sqrt{\frac{3\hbar\vartheta}{2M}}\,,
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| </math>
| |
| | |
| This is the Standard Quantum Limit for such a 2-pulse procedure. In principle, if we limit our measurement to two pulses only and do not care about perturbing mirror position afterwards, the second pulse measurement uncertainty, <math> \Delta x_{\rm meas}(t_{j+1})</math>, can, in theory, be reduced to 0 (it will yield, of course, <math> \Delta p_{\rm b.a.}(t_{j+1})\to\infty</math>) and the limit of displacement measurement error will reduce to:
| |
| | |
| :<math>
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| \Delta \tilde{x}_{SQL} = \sqrt{\frac{\hbar\vartheta}{M}}\,,
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| </math>
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| which is known as the Standard Quantum Limit for the measurement of free mass displacement.
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| This example represents a simple particular case of a \emph{linear measurement}. This class of measurement schemes can be fully described by two linear equations of the form~({{EquationNote|3}}) and ({{EquationNote|4}}), provided that both the measurement uncertainty and the object back-action perturbation (<math>\hat{x}_{\mathrm{fl}}(t_j)</math> and <math>\hat{p}^{\mathrm{b.a.}}(t_j)</math> in this case) are statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as the measured observable and its canonically conjugate counterpart (the object position and momentum in this case).
| |
| | |
| ===Usage in quantum optics===
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| In the context of [[interferometry]] or other optical measurements, the standard quantum limit usually refers to the minimum level of [[quantum noise]] which is obtainable without [[Squeezed coherent state|squeezed states]].<ref>
| |
| {{cite journal
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| |last1=Jaekel |first1=M. T.
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| |last2=Reynaud |first2=S.
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| |year=1990
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| |title=Quantum Limits in Interferometric Measurements
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| |journal=[[Europhysics Letters]]
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| |volume=13 |issue=4 |pages=301
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| |arxiv=quant-ph/0101104
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| |bibcode=1990EL.....13..301J
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| |doi=10.1209/0295-5075/13/4/003
| |
| }}</ref>
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| | |
| There is additionally a quantum limit for [[phase noise]], reachable only by a [[laser]] at high noise frequencies.
| |
| | |
| In [[spectroscopy]], the shortest wavelength in an X-ray spectrum is called the quantum limit.<ref>
| |
| {{cite journal
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| |last1=Piston |first1=D. S.
| |
| |year=1936
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| |title=The Polarization of X-Rays from Thin Targets
| |
| |journal=[[Physical Review]]
| |
| |volume=49 |issue=4 |pages=275
| |
| |bibcode= 1936PhRv...49..275P
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| |doi=10.1103/PhysRev.49.275
| |
| }}</ref>
| |
| | |
| ==Misleading relation to the classical limit==
| |
| | |
| Note that due to an overloading of the word "limit", the [[classical limit]] is ''not'' the opposite of the quantum limit. In "quantum limit", "limit" is being used in the sense of a physical limitation (e.g. the [[Armstrong limit]]). In "classical limit", "limit" is used in the sense of a [[Limit (mathematics)|limiting process]]. (Note that there is no ''simple rigorous'' mathematical limit which fully recovers classical mechanics from quantum mechanics, the [[Ehrenfest theorem]] notwithstanding. Nevertheless, in the [[phase space formulation]] of quantum mechanics, such limits are more systematic and practical.)
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| ==See also==
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| *[[Classical limit]]
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| *[[Heisenberg limit]]
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| *[[Ultrarelativistic limit]]
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| ==References and Notes==
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| <references/>
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| [[Category:Quantum mechanics]]
| |