One of the main characteristics of an optical radiation receiver is its sensitivity, i.e., the minimum value of the detectable (detectable) power of the optical signal at which the specified values of the signal-to-noise ratio or error probability are ensured.
Let's define minimum detectable power(MDM) of an optical signal corresponding to the minimum sensitivity threshold of the photodetector in the absence of noise and distortion, i.e. under conditions of ideal reception.
The symbol “1” corresponds to the transmission of an optical study pulse with a duration τ , whose energy at the receiver input is equal to E in,symbol “0” - zero value of optical energy. When a photodetector is irradiated with a flux of optical energy E in electron-hole pairs - charge carriers - are generated. This is an independent random process for which the average number of emerging pairs of charge carriers is determined by the formula
The number of emerging charge carrier pairs is determined by the Poisson probability distribution, i.e.
. (20.7)
Let us assume that even with the generation of only one pair of carriers, registration of an optical radiation pulse is possible, i.e., reception of “1”. Under this assumption, the probability of an error occurring means the probability of the occurrence of a single pair of charge carriers. The probability of such an event can be determined using formula (20.7), putting P=0. Then
……………………..(20.8)
If we put that R osh = R(0)=10 -9 , then we get N=21. This means that the energy received in the optical pulse must be equal to the energy of 21 photons, i.e., to ensure an error probability of no worse than 10 -9 from (20.6) - (20.8) it follows that .
This is the minimum permissible sensitivity of the receiver for ideal reception, and the requirement to generate 21 photons for each received pulse at R osh =10 -9 is a fundamental limit that is inherent in any physically realizable photodetector and is called quantum detection limit.
It corresponds to the minimum average power of an optical signal with a duration τ =1/IN, Where IN- information transfer speed,
which is called minimum detectable power.
From (20.3) taking into account (20.9) it follows that the MDM of the optical signal
(20.10)
Inequality (20.10) determines, all other things being equal, the minimum sensitivity threshold or MDM of the photodetector.
In addition to the quantum detection limit, there are other factors: thermal, dark and shot noise, intersymbol interference that limit the MDM. The fundamental difference between these factors is that by increasing the complexity of the equipment and using appropriate transmission and reception methods, their influence can be minimized.
Control questions
1. Interference affecting the optical signal.
2. OLT and factors influencing its structure.
3. Digital repeater (circuit and principle of operation).
4. Digital regenerator (circuit and principle of operation).
5. Functions of digital repeaters and their classification.
6. Types of analog OLT repeaters.
7. AOLT repeaters of the first type.
8. AOLT repeaters of the second and third types.
9. Main noise sources of POM with LD
10. Main noise sources of POMs with LEDs
11. Methods for reducing noise in POM with LD
12. OLT noise sources
13. Calculation of the probability of regenerator error, security
14. Minimum detectable power, quantum detection limit of the photodetector
For today I will describe, as I said earlier, one of the very complex nodes of the Probable. Unfortunately, part of the lecture is understandable only to a few. But this will not prevent others from understanding different things and raising their own level of development. Actually knowledge is knowledge. I like to look beyond the threshold. We are talking about a complex conglomerate in a significant area of the globe. Although, of course, I would prefer to write the last of the Blades... But I have to be content with what I can voice. I would immediately like to warn me deeply about all kinds of poisonous statements from those who have sawdust in their skulls. Therefore, do not labor.
P.S.
If the West thought with its brains, and not with the selfish interests of the wallet, then perhaps everything would have gone much easier. However, I have strong doubts that the West has any brains. Having been hit at least 4 times in my memory over the past two years, the West has learned nothing. Well, 5 times may be the last. The fact is that some awakened forces have found a point of application, trying to restore the disturbed balance. This was inevitable and natural. If we take an analogy. The West begs the Saint for a slap in the face, then this is exactly the Case. And this point of application is far from Iraq. Observing that implicit Knot, I can only sadly state that the invasion of neo-barbarians from the Dark Ages is perhaps worse than the army of hungry Huns. As for other things... The products of such experiments showed themselves not only in Paris.
Researchers were able to increase the sensitivity of a gravitational antenna, bypassing one of the limitations imposed by quantum mechanics. The fundamental laws of physics were not violated; scientists used light in the so-called compressed state. Details are given in the article Nature Photonics.
Physicists were able to overcome a limitation known as the standard quantum limit when determining the positions of mirrors inside the LIGO gravitational wave detector. This installation, built in the USA, consists of two perpendicular tunnels about four kilometers long. Each of them has a pipe from which the air is evacuated and through which a laser beam passes. Laser beams are reflected from mirrors located at the ends of the tunnels, and then converge again. Due to the phenomenon of interference, the rays either strengthen or weaken each other, and the magnitude of the effect depends on the path traveled by the rays. Theoretically, such a device (interferometer) should record changes in the distances between the mirrors when a gravitational wave passes through the installation, but in practice the accuracy of the interferometer is still too low.
LIGO's operation from 2002 to 2010 allowed physicists and engineers to figure out how the facility could be significantly improved. Now it is being rebuilt taking into account new proposals, so an international group of scientists (including employees of the Physics Department of Moscow State University and the Institute of Applied Physics in Nizhny Novgorod) conducted an experiment to increase the sensitivity of one of the LIGO detectors above one of the quantum barriers and presented its results.
Scientists have overcome a limitation known as the standard quantum limit. It was a consequence of another prohibition (which was not violated) associated with the Heisenberg uncertainty principle. The uncertainty principle states that when two quantities are measured simultaneously, the product of their measurement errors cannot be less than a certain constant. An example of such simultaneous measurements is the determination of the coordinate and momentum of a mirror using a reflected photon.
The Heisenberg uncertainty principle indicates that as the accuracy of determining the coordinate increases, the accuracy of determining the velocity decreases sharply. When a mirror is irradiated with many photons, errors in measuring the speed lead to the fact that it becomes more difficult to determine its displacement and, as a result, its position in space (there is little sense in many precise measurements that contradict each other). To circumvent this limitation, about a quarter of a century ago it was proposed to use so-called compressed states of light (they, in turn, were obtained in 1985), but the idea was only recently implemented in practice.
The compressed state of light is characterized by the fact that the spread (dispersion) of one of the parameters between photons is minimized. Most light sources, including lasers, are not capable of creating such radiation, but with the help of special crystals, physicists have learned to produce light in a compressed state. A laser beam passing through a crystal with nonlinear optical properties undergoes spontaneous parametric scattering: some photons turn from a single quantum into a pair of entangled (quantum correlated) particles. This process plays an important role in quantum computing and data transmission along quantum lines, but physicists have been able to adapt it to produce “squeezed light” that can improve the accuracy of measurements.
Scientists have demonstrated that using quantum correlated photons can reduce measurement error to a value that is above the level predicted by the Heisenberg uncertainty relation (since it is a fundamental barrier), but less than the standard quantum limit due to the interaction of many individual photons. To simplify the essence of the work, we can say that entangled particles, due to their connections with each other, behave more consistently than independent photons and therefore make it possible to more accurately determine the position of the mirror.
The researchers emphasize that the changes they made significantly increased the sensitivity of the gravitational wave detector in the frequency range from 50 to 300 hertz, which is especially interesting to astrophysicists. It is in this range that, according to theory, waves should be emitted when massive objects merge: neutron stars or black holes. The search for gravitational waves is one of the most important tasks of modern physics, but so far it has not been possible to register them due to the too low sensitivity of existing equipment.
| The basis |
|---|
| Classical mechanics · Planck's constant · Interference · Bra and ket · Hamiltonian · Old quantum theory |
| Fundamental Concepts |
|---|
| Quantum state · Quantum observable · Wave function · Quantum superposition · Quantum entanglement · Mixed state · Measurement · Uncertainty · Pauli principle · Dualism · Decoherence · Ehrenfest's theorem · Tunnel effect |
| Experiments |
|---|
| Davisson-Germer experiment · Popper experiment · Stern-Gerlach experiment · Young experiment · Bell's inequalities test · Photoelectric effect · Compton effect |
| Formulations |
|---|
| Schrödinger representation · Heisenberg representation · Interaction representation · Matrix quantum mechanics · Path integrals · Feynman diagrams |
| Equations |
|---|
| Schrödinger equation · Pauli equation · Klein-Gordon equation · Dirac equation · Schwinger-Tomonaga equation · Von Neumann equation · Bloch equation · Lindblad equation · Heisenberg equation |
| Interpretations |
|---|
| Copenhagen · Hidden parameter theory · Many-worlds · De Broglie-Bohm theory |
| Development of the theory |
|---|
| Quantum field theory · Quantum electrodynamics · Glashow-Weinberg-Salam theory · Quantum chromodynamics · Standard Model · Quantum gravity |
| Famous scientists |
|---|
| Planck · Einstein · Schrödinger · Heisenberg · Jordan · Bohr · Pauli · Dirac · Fock · Born · de Broglie · Landau · Feynman · Bohm · Everett |
Standard quantum limit(SKP) in quantum mechanics - a limitation imposed on the accuracy of a continuous or repeatedly repeated measurement of any quantity described by an operator that does not commute with itself at different times. Predicted in 1967 by V. B. Braginsky, and the term itself standard quantum limit(English) standard quantum limit, SQL) was proposed later by Thorne. The SKP is closely related to the Heisenberg uncertainty relation.
An example of a standard quantum limit is the quantum limit of measuring the coordinate of a free mass or mechanical oscillator. The coordinate operator at different times does not commute with itself due to the fact that there is a dependence of the added coordinate fluctuations on measurements at previous times.
If instead of the coordinate of the free mass we measure its momentum, this will not lead to a change in momentum at subsequent moments of time. Therefore, momentum, which is a conserved quantity for free mass (but not for the oscillator), can be measured as accurately as desired. Such measurements are called quantum non-perturbative. Another way to get around the standard quantum limit is to use non-classical squeezed field states and variational measurements in optical measurements.
The SKP limits the resolution of LIGO's laser gravity antennas. Currently, in a number of physical experiments with mechanical micro- and nanooscillators, coordinate measurement accuracy corresponding to the standard quantum limit has been achieved.
UCS of free mass coordinates
Let us measure the coordinate of the object at some initial moment of time with some accuracy texvc not found; See math/README - help with setup.): \Delta x_0. In this case, during the measurement process, a random impulse will be transferred to the body (reverse fluctuation influence) Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \Delta p_0. And the more accurately the coordinate is measured, the greater the disturbance of the pulse. In particular, if the coordinate is measured by optical methods based on the phase shift of the wave reflected from the body, then the disturbance of the pulse will be caused by quantum shot fluctuations of the light pressure on the body. The more accurately a coordinate is required to be measured, the greater the required optical power, and the greater the quantum fluctuations in the number of photons in the incident wave.
According to the uncertainty relation, the disturbance of the body's momentum is:
Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \Delta p_0=\frac(\hbar)(2\Delta x_0),
Where Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \hbar- reduced Planck constant. This change in momentum and the associated change in the speed of the free mass will lead to the fact that when the coordinate is re-measured after time Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \tau it will additionally change by an amount.
Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \Delta x_\text(add)=\frac(\Delta p_0\tau)(m)=\frac(\hbar \tau)(2\Delta x_0 m).
The resulting root mean square error is determined by the relation:
Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): (\Delta X_\Sigma)^2= (\Delta x_0)^2+(\Delta x_\text(add))^2=(\Delta x_0)^2 +\left(\frac(\hbar \tau)(2m\Delta x_0)\right)^2.
This expression has a minimum value if
Unable to parse expression (Executable file texvc not found; See math/README for setup help.): (\Delta x_0)^2 = \frac(\hbar \tau)(2m).
In this case, the root-mean-square measurement accuracy is achieved, which is called the standard quantum limit for the coordinate:
Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \Delta X_\Sigma=\Delta X_\text(SQL) = \sqrt(\frac(\hbar \tau)(m)).
UPC mechanical oscillator
The standard quantum limit for the coordinate of a mechanical oscillator is given by
Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \Delta X_\text(SQL) = \sqrt(\frac(\hbar)(2m\omega_m)),
Where Unable to parse expression (Executable file texvc not found; See math/README - help with setup.): \omega_m- frequency of mechanical vibrations.
The standard quantum limit for oscillator energy is:
Unable to parse expression (Executable file texvc not found; See math/README for setup help.): \Delta E_\text(SQL) = \sqrt(\hbar\omega_m E),
Excerpt characterizing the Standard Quantum Limit
That evening, the entire park literally shone and shimmered with thousands of colored lights, which, merging with the flickering night sky, formed a magnificent continuous sparkling fireworks display. Judging by the splendor of the preparations, it was probably some kind of grandiose party, during which all the guests, at the whimsical request of the queen, were dressed exclusively in white clothes and, somewhat reminiscent of ancient priests, “organized” walked through the wonderfully illuminated, sparkling park, heading towards the beautiful stone gazebo, called by everyone - the Temple of Love.Temple of Love, antique engraving
And then suddenly, behind the same temple, a fire broke out... Blinding sparks soared to the very tops of the trees, staining the dark night clouds with bloody light. The delighted guests gasped in unison, approving the beauty of what was happening... But none of them knew that, according to the queen’s plan, this raging fire expressed the full power of her love... And the real meaning of this symbol was understood only by one person who was present that evening at holiday...
Excited, Axel leaned against a tree and closed his eyes. He still couldn't believe that all this stunning beauty was meant for him.
-Are you satisfied, my friend? – a gentle voice whispered quietly behind him.
“I’m delighted...” Axel answered and turned around: it was, of course, her.
They looked at each other with rapture for only a moment, then the queen gently squeezed Axel’s hand and disappeared into the night...
- Why was he always so unhappy in all his “lives”? – Stella was still sad for our “poor boy”.
To tell the truth, I haven’t seen any “misfortune” yet and therefore I looked in surprise at her sad face. But for some reason the little girl stubbornly refused to explain anything further...
The picture changed dramatically.
A luxurious, very large green carriage was speeding along the dark night road. Axel sat in the coachman's place and, quite skillfully driving this huge carriage, looked around and looked around with obvious anxiety from time to time. It seemed like he was in a wild hurry somewhere or was running away from someone...
Inside the carriage sat the king and queen we already knew, and also a pretty girl of about eight years old, as well as two ladies still unknown to us. Everyone looked gloomy and worried, and even the little girl was quiet, as if she sensed the general mood of the adults. The king was dressed surprisingly modestly - in a simple gray frock coat, with the same gray round hat on his head, and the queen hid her face under a veil, and it was clear that she was clearly afraid of something. Again, this whole scene was very reminiscent of an escape...
Just in case, I looked again in Stella’s direction, hoping for an explanation, but no explanation came - the little girl was very intently watching what was happening, and in her huge doll eyes there was a deep, not at all childish, sadness lurking.
“Well, why?.. Why didn’t they listen to him?!.. It was so simple!..” she suddenly became indignant.
The carriage was rushing all this time at almost crazy speed. The passengers looked tired and somehow lost... Finally, they drove into some large, unlit courtyard, with the black shadow of a stone building in the middle, and the carriage stopped abruptly. The place resembled an inn or a large farm.
Axel jumped to the ground and, approaching the window, was about to say something, when suddenly an authoritative male voice was heard from inside the carriage:
– Here we will say goodbye, Count. It is not worthy for me to expose you to further danger.
Axel, of course, who did not dare to object to the king, only managed to fleetingly touch the queen’s hand as a farewell... The carriage rushed off... and literally a second later disappeared into the darkness. And he was left standing alone in the middle of the dark road, wanting with all his heart to rush after them... Axel felt “in his gut” that he could not, had no right to leave everything to the mercy of fate! He just knew that without him, something would definitely go awry, and everything that he had organized for so long and carefully would completely fail due to some ridiculous accident...
The carriage was no longer visible for a long time, and poor Axel still stood and looked after them, clenching his fists with all his might in despair. Angry male tears flowed sparingly down his deathly pale face...
Beware, quantum mechanics below!
SKP (or SQL, Standard Quantum Limit) is a concept from quantum mechanics. This is the name for the limitation in the accuracy of measurements that are carried out repeatedly or over a long period of time. A good example, which also applies to our case, is measuring the distance to a certain mass with the highest possible accuracy. A laser beam is used for measurement. By knowing the wavelength of the laser, the initial phase of the wave, and measuring the phase of the returned beam, we can calculate the exact distance it has traveled. Unfortunately, the pressure of the beam on the body will cause disturbances in it at the quantum level (quantum shot fluctuations). The more accurately you need to measure the coordinate, the more powerful the laser beam is needed, and the greater these fluctuations will be. This quantum noise creates measurement error.
In fact, SKP is a consequence of the fundamental prohibition of quantum physics - the Heisenberg uncertainty principle. The uncertainty principle states that when two quantities are measured simultaneously, the product of errors cannot be less than a certain constant. Roughly speaking, the more accurately we measure the speed of a quantum particle, the less accurately we can determine its position. And vice versa. It is important to note that the limitations on measurement accuracy imposed by the SKP are more severe than the limitations of the Heisenberg uncertainty principle. It is in principle impossible to bypass the latter without destroying the foundations of all quantum mechanics.
A way to circumvent the limitation of the standard quantum limit was proposed in the American gravitational wave detector LIGO. The search for gravitational waves is one of the most important tasks of modern physics, but so far it has not been possible to register them due to the too low sensitivity of existing equipment. The LIGO setup is very simple. It consists of two vacuum tunnels converging at right angles. Laser beams pass through the pipes, and mirrors are installed at their far ends (see figure). It is the distance to these mirrors that is measured by the laser, as described above. Of particular importance is the intersection of laser beams returning from the mirrors. Interference occurs between them. Due to this phenomenon, the rays either strengthen or weaken each other. The amount of interference depends on the phase of the rays, and therefore on the path traveled by the rays. Theoretically, such a device should record changes in the distances between the mirrors when a gravitational wave passes through the installation, but in practice the accuracy of the interferometer is still too low.
To bypass the SKP, about a quarter of a century ago it was proposed to use the so-called squeezed states of light. They were received in 1985, but the idea was put into practice only recently. Most light sources, including lasers, are not capable of creating such radiation, but with the help of special crystals, physicists have learned to produce light in a compressed state. A laser beam passing through such a crystal undergoes spontaneous parametric scattering. In other words, some photons turn from a single quantum into a pair of entangled particles.
Scientists have demonstrated that using quantum correlated photons can reduce measurement error to below the standard quantum limit. Unfortunately, without special knowledge it is very difficult to understand (and, even more so, explain) exactly how this happens, but the behavior of entangled photons reduces the same quantum shot noise that was mentioned at the beginning.
The researchers emphasize that the changes they made significantly increased the sensitivity of the gravitational wave detector in the frequency range from 50 to 300 hertz, which is especially interesting to astrophysicists. It is in this range that, according to theory, waves should be emitted when massive objects merge: neutron stars or black holes.
