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{{Short description|Limit on measurements at quantum scales}}
{{Short description|Limit on measurements at quantum scales}}
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'''limit''' a quantum limit in physics is a limit on measurement accuracy at quantum scales. 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 states (e.g. the standard quantum limit in interferometry) and can be circumvented with advanced state preparation and measurement schemes. The usage of the 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 commute with itself at different times leads to such limits. In short, it is the Heisenberg uncertainty principle that is the cause. A more detailed explanation would be that any measurement in quantum mechanics involves at least two parties, an Object and a Meter. the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as indirect measurements (see pp. So any measurement is a result of interaction and that acts in both ways. This is known as back action (quantum) of the Meter on the system under measurement. This one is known as measurement imprecision or measurement noise. The equality is reached if the system is in a minimum uncertainty state. 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. Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement.
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[[File:Scheme of quantum measurement process.png|thumb|280px|Quantum limit.]]
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{{Quantum book backlink|Measurement and information}}
==Examples==
A '''quantum limit''' in physics is a limit on measurement accuracy at quantum scales.<ref name=QMeas_Br_Kh>
{{cite book
|last1=Braginsky |first1=V. B.
|last2=Khalili |first2=F. Ya.
|year=1992
|title=Quantum Measurement
|publisher=Cambridge University Press
|isbn=978-0-521-48413-8
}}</ref>  Depending on the context, the limit may be absolute (such as the [[Physics:Heisenberg limit|Heisenberg limit]]), or it may only apply when the experiment is conducted with naturally occurring [[Physics:Quantum state|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 the term '''standard quantum limit''' or SQL is, however, broader than just interferometry. In principle, any linear measurement of a quantum mechanical [[Physics:Observable|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.
===Displacement measurement===
[[File:Scheme of quantum measurement process.png|thumb|A schematic description of how physical measurement process is described in quantum mechanics]]
Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement. In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body <math>M</math>. The position <math>x</math> of <math>M</math> is probed periodically with time interval <math>\vartheta</math>. We assume mass <math>M</math> large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process.
A more detailed explanation would be that any measurement in [[Physics:Quantum mechanics|quantum mechanics]] involves at least two parties, an Object and a Meter. The former is the system whose 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, is a model of most of the measurements happening in physics, known as ''indirect'' measurements (see pp.&nbsp;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 [[Physics:Back action (quantum)|back action (quantum)]] 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 of 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 cannot be arbitrary and is linked to the back-action perturbation by the [[Uncertainty principle|uncertainty relation]]:
[[File:Position measurement using reflected light.png|thumb|Simplified scheme of optical measurement of mechanical object position]]


:<math>\Delta {\mathcal{O}} \Delta {\mathcal{F}} \geqslant \hbar/2\,,</math>
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:


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|expectation value]] of <math>a</math> in whatever [[Physics:Quantum state|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:
{{NumBlk|:|<math>
  \hat{\phi}_j^{\mathrm{refl}} = \hat{\phi}_j - 2 k_p\hat{x}(t_j) \,,
</math>|{{EquationRef|1}}}}


:<math>\hat{\mathcal{O}} = \hat x_{\mathrm{free}} + \delta \hat{\mathcal{O}} + \delta \hat x_{BA}[\hat{\mathcal{F}}]\,,</math>
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>.


where <math>\hat x_{\mathrm{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>
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 or heterodyne detection schemes (see Section 2.3 in <ref name=LRR_Da_Kh>
{{cite journal
{{cite journal
|last=Danilishin|first=S. L.
|last=Danilishin|first=S. L.
Line 39: Line 48:
| pmc=5256003  
| pmc=5256003  
|arxiv = 1203.1706 |bibcode = 2012LRR....15....5D }}
|arxiv = 1203.1706 |bibcode = 2012LRR....15....5D }}
</ref><ref>{{cite journal
</ref> and references therein), or other such read-out techniques.
|last=Chen|first=Yanbei
|title=Macroscopic quantum mechanics: theory and experimental concepts of optomechanics
|journal=J. Phys. B: At. Mol. Opt. Phys.
|year=2013
|volume=46
|issue=10
|article-number=104001
|doi=10.1088/0953-4075/46/10/104001
|arxiv = 1302.1924 |bibcode = 2013JPhB...46j4001C |s2cid=118570800
}}</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.
 
==Examples==
 
===Displacement measurement===
Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a [[General position|general position]] measurement. In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body <math>M</math>. The position <math>x</math> of <math>M</math> is probed periodically with time interval <math>\vartheta</math>. We assume mass <math>M</math> large enough to neglect the displacement inflicted by the pulses regular (classical) [[Astronomy:Radiation pressure|radiation pressure]] in the course of measurement process.
 
[[File:Position measurement using reflected light.png|thumb|Simplified scheme of optical measurement of mechanical object position]]
 
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:
 
{{NumBlk|:|<math>
  \hat{\phi}_j^{\mathrm{refl}} = \hat{\phi}_j - 2 k_p\hat{x}(t_j) \,,
</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|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>.
 
The reflected pulses are detected by a phase-sensitive device (the [[Engineering:Phase detector|phase detector]]). The implementation of an optical phase detector can be done using ''e.g.'' [[Homodyne detection|homodyne]] or [[Optical heterodyne detection|heterodyne]] detection schemes (see Section 2.3 in <ref name=LRR_Da_Kh /> and references therein), or other such read-out techniques.


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:
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:
Line 137: Line 117:
</math>
</math>


where we assumed all contributions to our [[Measurement uncertainty|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>.
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>.


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:
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:
Line 168: Line 148:


===Usage in quantum optics===
===Usage in quantum optics===
In the context of [[Astronomy:Interferometry|interferometry]] or other optical measurements, the standard quantum limit usually refers to the minimum level of [[Physics:Quantum noise|quantum noise]] which is obtainable without [[Physics:Squeezed coherent state|squeezed states]].<ref>
In the context of interferometry or other optical measurements, the standard quantum limit usually refers to the minimum level of [[Physics:Quantum noise|quantum noise]] which is obtainable without squeezed states.<ref>
{{cite journal
{{cite journal
  |last1=Jaekel |first1=M. T.
  |last1=Jaekel |first1=M. T.
Line 182: Line 162:
  }}</ref>
  }}</ref>


There is additionally a quantum limit for [[Phase noise|phase noise]], reachable only by a [[Physics:Laser|laser]] at high noise frequencies.
There is additionally a quantum limit for phase noise, reachable only by a laser at high noise frequencies.


In [[Physics:Spectroscopy|spectroscopy]], the shortest wavelength in an X-ray spectrum is called the quantum limit.<ref>
In spectroscopy, the shortest wavelength in an X-ray spectrum is called the quantum limit.<ref>
{{cite journal
{{cite journal
  |last1=Piston |first1=D. S.
  |last1=Piston |first1=D. S.
  |year=1936
  |year=1936
  |title=The Polarization of X-Rays from Thin Targets
  |title=The Polarization of X-Rays from Thin Targets
  |journal=[[Physics:Physical Review|Physical Review]]
  |journal=Physical Review
  |volume=49 |issue=4 |pages=275–279
  |volume=49 |issue=4 |pages=275–279
  |bibcode= 1936PhRv...49..275P
  |bibcode= 1936PhRv...49..275P
Line 196: Line 176:


==Misleading relation to the classical limit==
==Misleading relation to the classical limit==
Note that due to an overloading of the word "limit", the [[Physics:Classical limit|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 [[Astronomy:Armstrong limit|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 [[Physics:Classical mechanics|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.)
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 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.)


==See also==
== See also ==
*[[Physics:Classical limit|Classical limit]]
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
*[[Physics:Heisenberg limit|Heisenberg limit]]
*[[Ultrarelativistic limit]]


==References and notes==
==References and notes==
= References =
<references/>
<references/>


[[Category:Quantum mechanics]]
[[Category:Quantum mechanics]]


{{Sourceattribution|Quantum limit}}
{{Author|Harold Foppele}}
 
{{Sourceattribution|Physics:Quantum limit|1}}

Latest revision as of 00:31, 24 May 2026

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limit a quantum limit in physics is a limit on measurement accuracy at quantum scales. 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 states (e.g. the standard quantum limit in interferometry) and can be circumvented with advanced state preparation and measurement schemes. The usage of the 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 commute with itself at different times leads to such limits. In short, it is the Heisenberg uncertainty principle that is the cause. A more detailed explanation would be that any measurement in quantum mechanics involves at least two parties, an Object and a Meter. the position of the pointer on a scale of the Meter. This, in a nutshell, is a model of most of the measurements happening in physics, known as indirect measurements (see pp. So any measurement is a result of interaction and that acts in both ways. This is known as back action (quantum) of the Meter on the system under measurement. This one is known as measurement imprecision or measurement noise. The equality is reached if the system is in a minimum uncertainty state. 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. Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement.

Quantum limit.

Examples

Displacement measurement

Consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement. In the scheme shown in the Figure, a sequence of very short light pulses are used to monitor the displacement of a probe body M. The position x of M is probed periodically with time interval ϑ. We assume mass M large enough to neglect the displacement inflicted by the pulses regular (classical) radiation pressure in the course of measurement process.

Simplified scheme of optical measurement of mechanical object position

Then each j-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position x(tj) at the moment of reflection:

ϕ^jrefl=ϕ^j−2kpx^(tj),

 

 

 

 

(1)

where kp=ωp/c, ωp is the light frequency, j=…,−1,0,1,… is the pulse number and ϕ^j is the initial (random) phase of the j-th pulse. We assume that the mean value of all these phases is equal to zero, ⟨ϕ^j⟩=0, and their root mean square (RMS) uncertainty (⟨ϕ2^⟩−⟨ϕ^⟩2)1/2 is equal to Δϕ.

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 or heterodyne detection schemes (see Section 2.3 in [1] and references therein), or other such read-out techniques.

In this example, light pulse phase ϕ^j serves as the readout observable 𝒪 of the Meter. Then we suppose that the phase ϕ^jrefl measurement error introduced by the detector is much smaller than the initial uncertainty of the phases Δϕ. In this case, the initial uncertainty will be the only source of the position measurement error:

Δxmeas=Δϕ2kp.

 

 

 

 

(2)

For convenience, we renormalise Eq. (1) as the equivalent test-mass displacement:

x~j≡−ϕ^jrefl2kp=x^(tj)+x^fl(tj),

 

 

 

 

(3)

where

x^fl(tj)=−ϕ^j2kp

are the independent random values with the RMS uncertainties given by Eq. (2).

Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to

p^jafter−p^jbefore=p^jb.a.=2c𝒲^j,

 

 

 

 

(4)

where p^jbefore and p^jafter are the test-mass momentum values just before and just after the light pulse reflection, and 𝒲j is the energy of the j-th pulse, that plays the role of back action observable ℱ^ of the Meter. The major part of this perturbation is contributed by classical radiation pressure:

⟨p^jb.a.⟩=2c𝒲,

with 𝒲 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:

p^b.a.(tj)=p^jb.a.−⟨p^jb.a.⟩=2c(𝒲^j−𝒲),

and its RMS uncertainly is equal to

Δpb.a.=2Δ𝒲c,

 

 

 

 

(5)

with Δ𝒲 the RMS uncertainty of the pulse energy.

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, ϑ≪T), one can estimate an additional displacement caused by the back action of the j-th pulse that will contribute to the uncertainty of the subsequent measurement by the j+1 pulse time ϑ later:

x^b.a.(tj)=p^b.a.(tj)ϑM.

Its uncertainty will be simply

Δxb.a.(tj)=Δpb.a.(tj)ϑM.

If we now want to estimate how much has the mirror moved between the j and j+1 pulses, i.e. its displacement δx~j+1,j=x~(tj+1)−x~(tj), we will have to deal with three additional uncertainties that limit precision of our estimate:

Δx~j+1,j=[(Δxmeas(tj+1))2+(Δxmeas(tj))2+(Δxb.a.(tj))2]1/2,

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 Δxmeas(tj+1)=Δxmeas(tj)≡Δxmeas=Δϕ/(2kp).

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:

Δ𝒲Δϕ≥ℏωp2.

Therefore, it follows from Eqs. (2 and 5) that the position measurement error Δxmeas and the momentum perturbation Δpb.a. due to back action also satisfy the uncertainty relation:

ΔxmeasΔpb.a.≥ℏ2.

Taking this relation into account, the minimal uncertainty, Δxmeas, the light pulse should have in order not to perturb the mirror too much, should be equal to Δxb.a. yielding for both Δxmin=ℏϑ2M. Thus the minimal displacement measurement error that is prescribed by quantum mechanics read:

Δx~j+1,j⩾[2(Δxmeas)2+(ℏϑ2MΔxmeas)2]1/2⩾3ℏϑ2M.

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, Δxmeas(tj+1), can, in theory, be reduced to 0 (it will yield, of course, Δpb.a.(tj+1)→∞) and the limit of displacement measurement error will reduce to:

Δx~SQL=ℏϑM,

which is known as the Standard Quantum Limit for the measurement of free mass displacement.

This example represents a simple particular case of a linear measurement. This class of measurement schemes can be fully described by two linear equations of the form~(3) and (4), provided that both the measurement uncertainty and the object back-action perturbation (x^fl(tj) and p^b.a.(tj) 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

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 states.[2]

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.[3]

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 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.)

See also

Table of contents (217 articles)

Index

Full contents

References and notes

References

  1. ↑ Danilishin, S. L.; Khalili F. Ya. (2012). "Quantum Measurement Theory in Gravitational-Wave Detectors". Living Reviews in Relativity 15 (5): 60. doi:10.12942/lrr-2012-5. PMID 28179836. Bibcode: 2012LRR....15....5D. 
  2. ↑ Jaekel, M. T.; Reynaud, S. (1990). "Quantum Limits in Interferometric Measurements". Europhysics Letters 13 (4): 301–306. doi:10.1209/0295-5075/13/4/003. Bibcode: 1990EL.....13..301J. 
  3. ↑ Piston, D. S. (1936). "The Polarization of X-Rays from Thin Targets". Physical Review 49 (4): 275–279. doi:10.1103/PhysRev.49.275. Bibcode: 1936PhRv...49..275P. 


Author: Harold Foppele


Source attribution: Physics:Quantum limit

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