Physics:Quantum limit

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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 $\vartheta$. 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 $x(t_j)$ at the moment of reflection:

${\hat{\varphi }}_j^{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}={\hat{\varphi }}_j-2k_p\hat{x}(t_j),$

(1)

where $k_p={\omega }_p/c$, ${\omega }_p$ is the light frequency, $j=\dots ,-1,0,1,\dots$ is the pulse number and ${\hat{\varphi }}_j$ is the initial (random) phase of the $j$-th pulse. We assume that the mean value of all these phases is equal to zero, $⟨{\hat{\varphi }}_j⟩=0$, and their root mean square (RMS) uncertainty $(⟨\widehat{{\varphi }^2}⟩-⟨\hat{\varphi }{⟩}^2{)}^{1/2}$ is equal to $\mathrm{\Delta }\varphi$.

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 and references therein), or other such read-out techniques.

In this example, light pulse phase ${\hat{\varphi }}_j$ serves as the readout observable ${𝒪}$ of the Meter. Then we suppose that the phase ${\hat{\varphi }}_j^{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}$ measurement error introduced by the detector is much smaller than the initial uncertainty of the phases $\mathrm{\Delta }\varphi$. In this case, the initial uncertainty will be the only source of the position measurement error:

$\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=\frac{\mathrm{\Delta }\varphi }{2k_p}.$

(2)

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

${\tilde{x}}_j\equiv -\frac{{\hat{\varphi }}_j^{\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{l}}}{2k_p}=\hat{x}(t_j)+{\hat{x}}_{\mathrm{f}\mathrm{l}}(t_j),$

(3)

where

${\hat{x}}_{\mathrm{f}\mathrm{l}}(t_j)=-\frac{{\hat{\varphi }}_j}{2k_p}$

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

${\hat{p}}_j^{\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r}}-{\hat{p}}_j^{\mathrm{b}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}}={\hat{p}}_j^{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}=\frac{2}{c}{\widehat{{𝒲}}}_j,$

(4)

where ${\hat{p}}_j^{\mathrm{b}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}}$ and ${\hat{p}}_j^{\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r}}$ are the test-mass momentum values just before and just after the light pulse reflection, and ${{𝒲}}_j$ is the energy of the $j$-th pulse, that plays the role of back action observable $\widehat{{ℱ}}$ of the Meter. The major part of this perturbation is contributed by classical radiation pressure:

$⟨{\hat{p}}_j^{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}⟩=\frac{2}{c}{𝒲},$

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:

${\hat{p}}^{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_j)={\hat{p}}_j^{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}-⟨{\hat{p}}_j^{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}⟩=\frac{2}{c}({\widehat{{𝒲}}}_j-{𝒲}),$

and its RMS uncertainly is equal to

$\mathrm{\Delta }{p}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}=\frac{2\mathrm{\Delta }{𝒲}}{c},$

(5)

with $\mathrm{\Delta }{𝒲}$ 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, $\vartheta \ll T$), 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 $\vartheta$ later:

${\hat{x}}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_j)=\frac{{\hat{p}}^{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_j)\vartheta }{M}.$

Its uncertainty will be simply

$\mathrm{\Delta }{x}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_j)=\frac{\mathrm{\Delta }{p}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_j)\vartheta }{M}.$

If we now want to estimate how much has the mirror moved between the $j$ and $j+1$ pulses, i.e. its displacement $\delta {\tilde{x}}_{j+1,j}=\tilde{x}(t_{j+1})-\tilde{x}(t_j)$, we will have to deal with three additional uncertainties that limit precision of our estimate:

$\mathrm{\Delta }{\tilde{x}}_{j+1,j}=[(\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}(t_{j+1}){)}^2+(\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}(t_j){)}^2+(\mathrm{\Delta }{x}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_j){)}^2{]}^{1/2},$

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 $\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}(t_{j+1})=\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}(t_j)\equiv \mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}=\mathrm{\Delta }\varphi /(2k_p)$.

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:

$\mathrm{\Delta }{𝒲}\mathrm{\Delta }\varphi \ge \frac{\hslash {\omega }_p}{2}.$

Therefore, it follows from Eqs. (2 and 5) that the position measurement error $\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$ and the momentum perturbation $\mathrm{\Delta }{p}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}$ due to back action also satisfy the uncertainty relation:

$\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\mathrm{\Delta }{p}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}\ge \frac{\hslash }{2}.$

Taking this relation into account, the minimal uncertainty, $\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$, the light pulse should have in order not to perturb the mirror too much, should be equal to $\mathrm{\Delta }{x}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}$ yielding for both $\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\sqrt{\frac{\hslash \vartheta }{2M}}$. Thus the minimal displacement measurement error that is prescribed by quantum mechanics read:

$\mathrm{\Delta }{\tilde{x}}_{j+1,j}⩾[2(\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}{)}^2+(\frac{\hslash \vartheta }{2M\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}}{)}^2{]}^{1/2}⩾\sqrt{\frac{3\hslash \vartheta }{2M}}.$

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, $\mathrm{\Delta }{x}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}(t_{j+1})$, can, in theory, be reduced to 0 (it will yield, of course, $\mathrm{\Delta }{p}_{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_{j+1})\to \mathrm{\infty }$) and the limit of displacement measurement error will reduce to:

$\mathrm{\Delta }{\tilde{x}}_{SQL}=\sqrt{\frac{\hslash \vartheta }{M}},$

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 (${\hat{x}}_{\mathrm{f}\mathrm{l}}(t_j)$ and ${\hat{p}}^{\mathrm{b}\mathrm{.}\mathrm{a}\mathrm{.}}(t_j)$ 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.

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.

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

• Classical limit
• Heisenberg limit
• Ultrarelativistic limit

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

9. Physics:Quantum mechanics measurements

10. Physics:Quantum state

11. Physics:Quantum system

12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

19. Physics:Quantum Measurement problem

20. Physics:Quantum Bell's theorem

21. Physics:Quantum Hidden variable theory

22. Physics:Quantum nonlocality

23. Physics:Quantum contextuality

24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

37. Physics:Quantum Runge–Lenz vector

38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (21) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Aharonov-Casher effect

77. Physics:Quantum Scattering theory

78. Physics:Quantum Scattering matrix

79. Physics:Quantum S-matrix

80. Physics:Quantum tunnelling

81. Physics:Quantum speed limit

82. Physics:Quantum revival

83. Physics:Quantum reflection

84. Physics:Quantum oscillations

85. Physics:Quantum jump

86. Physics:Quantum boomerang effect

87. Physics:Quantum chaos

7. Measurement and information (9) Back to index

88. Physics:Quantum Measurement theory

89. Physics:Quantum Measurement operators

90. Physics:Quantum Projective measurement

91. Physics:Quantum POVM

92. Physics:Quantum Weak measurement

93. Physics:Quantum Measurement collapse

94. Physics:Quantum entanglement

95. Physics:Quantum Zeno effect

96. Physics:Quantum limit

8. Quantum information and computing (15) Back to index

97. Physics:Quantum information theory

98. Physics:Quantum Qubit

99. Physics:Quantum Entanglement

100. Physics:Quantum Bell state

101. Physics:Quantum Gates and circuits

102. Physics:Quantum BB84

103. Physics:Quantum No-cloning theorem

104. Physics:Quantum Computing Algorithms in the NISQ Era

105. Physics:Quantum Noisy Qubits

106. Physics:Quantum error correction

107. Physics:Quantum Boson sampling

108. Physics:Quantum random access code

109. Physics:Quantum pseudo-telepathy

110. Physics:Quantum network

111. Physics:Quantum money

9. Quantum optics and experiments (10) Back to index

112. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

113. Physics:Quantum Stern-Gerlach experiment

114. Physics:Quantum Experimental quantum physics

115. Physics:Quantum optics

116. Physics:Quantum optics beam splitter experiments

117. Physics:Quantum Mach-Zehnder interferometer

118. Physics:Quantum Hong-Ou-Mandel effect

119. Physics:Quantum eraser experiment

120. Physics:Quantum delayed-choice quantum eraser

121. Physics:Quantum Ultra fast lasers

122. Template:Quantum optics operators

10. Open quantum systems (15) Back to index

123. Physics:Quantum Open systems

124. Physics:Quantum channel

125. Physics:Quantum Kraus operators

126. Physics:Quantum Amplitude damping

127. Physics:Quantum Phase damping

128. Physics:Quantum Depolarizing channel

129. Physics:Quantum Master equation

130. Physics:Quantum Lindblad equation

131. Physics:Quantum Decoherence

132. Physics:Quantum Dynamical decoupling

133. Physics:Quantum dissipation

134. Physics:Quantum Markov semigroup

135. Physics:Quantum Markovian dynamics

136. Physics:Quantum Non-Markovian dynamics

137. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

138. Physics:Quantum field theory (QFT) basics

139. Physics:Quantum field theory (QFT) core

140. Physics:Quantum Fields and Particles

141. Physics:Quantum Second quantization

142. Physics:Quantum Fock space

143. Physics:Quantum Harmonic Oscillator field modes

144. Physics:Quantum Creation and annihilation operators

145. Physics:Quantum vacuum fluctuations

146. Physics:Quantum Casimir effect

147. Physics:Quantum Propagators in quantum field theory

148. Physics:Quantum Feynman diagrams

149. Physics:Quantum Path integral formulation

150. Physics:Quantum Renormalization in field theory

151. Physics:Quantum Renormalization group

152. Physics:Quantum Field Theory Gauge symmetry

153. Physics:Quantum Spontaneous symmetry breaking

154. Physics:Quantum Non-Abelian gauge theory

155. Physics:Quantum Electrodynamics (QED)

156. Physics:Quantum chromodynamics (QCD)

157. Physics:Quantum Electroweak theory

158. Physics:Quantum Standard Model

159. Physics:Quantum triviality

160. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

161. Physics:Quantum Statistical mechanics

162. Physics:Quantum Partition function

163. Physics:Quantum Distribution functions

164. Physics:Quantum Liouville equation

165. Physics:Quantum Kinetic theory

166. Physics:Quantum Boltzmann equation

167. Physics:Quantum BBGKY hierarchy

168. Physics:Quantum Relaxation and thermalization

169. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (17) Back to index

170. Physics:Quantum Band structure

171. Physics:Quantum Fermi surfaces

172. Physics:Quantum Landau levels

173. Physics:Quantum fractional Hall effect

174. Physics:Quantum Semiconductor physics

175. Physics:Quantum Phonons

176. Physics:Quantum Electron-phonon interaction

177. Physics:Quantum Exchange interaction

178. Physics:Quantum Superconductivity

179. Physics:Quantum Topological phases of matter

180. Physics:Quantum anyon

181. Physics:Quantum well

182. Physics:Quantum spin liquid

183. Physics:Quantum spin Hall effect

184. Physics:Quantum phase transition

185. Physics:Quantum critical point

186. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

187. Physics:Quantum Fusion reactions and Lawson criterion

188. Physics:Quantum Plasma (fusion context)

189. Physics:Quantum Magnetic confinement fusion

190. Physics:Quantum Inertial confinement fusion

191. Physics:Quantum Plasma instabilities and turbulence

192. Physics:Quantum Tokamak core plasma

193. Physics:Quantum Tokamak edge physics and recycling asymmetries

194. Physics:Quantum Stellarator

15. Timeline (8) Back to index

195. Physics:Quantum mechanics/Timeline

196. Physics:Quantum mechanics/Timeline/Pre-quantum era

197. Physics:Quantum mechanics/Timeline/Old quantum theory

198. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

199. Physics:Quantum mechanics/Timeline/Quantum field theory era

200. Physics:Quantum mechanics/Timeline/Quantum information era

201. Physics:Quantum mechanics/Timeline/Quantum technology era

202. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

203. Physics:Quantum topology

204. Physics:Quantum battery

205. Physics:Quantum Supersymmetry

206. Physics:Quantum Black hole thermodynamics

207. Physics:Quantum Holographic principle

208. Physics:Quantum gravity

209. Physics:Quantum De Sitter invariant special relativity

210. Physics:Quantum Doubly special relativity

211. Physics:Quantum arithmetic geometry

212. Physics:Quantum unsolved problems

213. Physics:Quantum Yang-Mills mass gap

214. Physics:Quantum gravity problem

215. Physics:Quantum black hole information paradox

216. Physics:Quantum dark matter problem

217. Physics:Quantum neutrino mass problem

218. Physics:Quantum matter-antimatter asymmetry problem

References

Author: Harold Foppele

Source attribution: Physics:Quantum limit