Photon Detection as a Process of Information Gain
Making use of the equivalence between information and entropy, we have shown in a recent paper that particles moving with a kinetic energy <inline-formula> <math display="inline"> <semantics> <mi>ε</mi> </semantics> </math> </inline-formula>...
| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-03-01
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| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/22/4/392 |
| _version_ | 1797571985388077056 |
|---|---|
| author | J Gerhard Müller |
| author_facet | J Gerhard Müller |
| author_sort | J Gerhard Müller |
| collection | DOAJ |
| description | Making use of the equivalence between information and entropy, we have shown in a recent paper that particles moving with a kinetic energy <inline-formula> <math display="inline"> <semantics> <mi>ε</mi> </semantics> </math> </inline-formula> carry potential information <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>ε</mi> <mo>,</mo> <mi>T</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mi>ε</mi> <mrow> <msub> <mi>k</mi> <mi>B</mi> </msub> <mo> </mo> <mi>T</mi> </mrow> </mfrac> </mrow> </semantics> </math> </inline-formula> relative to a heat reservoir of temperature <inline-formula> <math display="inline"> <semantics> <mi>T</mi> </semantics> </math> </inline-formula>. In this paper we build on this result and consider in more detail the process of information gain in photon detection. Considering photons of energy <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula> and a photo-ionization detector operated at a temperature <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> </semantics> </math> </inline-formula>, we evaluate the signal-to-noise ratio <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for different detector designs and detector operation conditions and show that the information gain realized upon detection, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, always remains smaller than the potential information <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> carried with the photons themselves, i.e.,: <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mi>ln</mi> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mi>B</mi> </msub> <msub> <mi>T</mi> <mi>D</mi> </msub> <mo> </mo> </mrow> </mfrac> </mrow> </semantics> </math> </inline-formula>. This result is shown to be generally valid for all kinds of technical photon detectors, which shows that <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> can indeed be regarded as an intrinsic information content that is carried with the photons themselves. Overall, our results suggest that photon detectors perform as thermodynamic engines that incompletely convert potential information into realized information with an efficiency that is limited by the second law of thermodynamics and the Landauer energy bounds on information gain and information erasure. |
| first_indexed | 2024-03-10T20:49:15Z |
| format | Article |
| id | doaj.art-dbe7f25db7344d3eb90ad05585500db6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T20:49:15Z |
| publishDate | 2020-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-dbe7f25db7344d3eb90ad05585500db62023-11-19T20:07:59ZengMDPI AGEntropy1099-43002020-03-0122439210.3390/e22040392Photon Detection as a Process of Information GainJ Gerhard Müller0Department of Applied Sciences and Mechatronics, Munich University of Applied Sciences, D-80335 Munich, GermanyMaking use of the equivalence between information and entropy, we have shown in a recent paper that particles moving with a kinetic energy <inline-formula> <math display="inline"> <semantics> <mi>ε</mi> </semantics> </math> </inline-formula> carry potential information <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>ε</mi> <mo>,</mo> <mi>T</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mi>ε</mi> <mrow> <msub> <mi>k</mi> <mi>B</mi> </msub> <mo> </mo> <mi>T</mi> </mrow> </mfrac> </mrow> </semantics> </math> </inline-formula> relative to a heat reservoir of temperature <inline-formula> <math display="inline"> <semantics> <mi>T</mi> </semantics> </math> </inline-formula>. In this paper we build on this result and consider in more detail the process of information gain in photon detection. Considering photons of energy <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mrow> </semantics> </math> </inline-formula> and a photo-ionization detector operated at a temperature <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> </semantics> </math> </inline-formula>, we evaluate the signal-to-noise ratio <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for different detector designs and detector operation conditions and show that the information gain realized upon detection, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, always remains smaller than the potential information <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> carried with the photons themselves, i.e.,: <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mi>ln</mi> <mrow> <mo>(</mo> <mrow> <mi>S</mi> <mi>N</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mfrac> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mi>B</mi> </msub> <msub> <mi>T</mi> <mi>D</mi> </msub> <mo> </mo> </mrow> </mfrac> </mrow> </semantics> </math> </inline-formula>. This result is shown to be generally valid for all kinds of technical photon detectors, which shows that <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>i</mi> <mrow> <mi>p</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>E</mi> <mrow> <mi>p</mi> <mi>h</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>D</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> can indeed be regarded as an intrinsic information content that is carried with the photons themselves. Overall, our results suggest that photon detectors perform as thermodynamic engines that incompletely convert potential information into realized information with an efficiency that is limited by the second law of thermodynamics and the Landauer energy bounds on information gain and information erasure.https://www.mdpi.com/1099-4300/22/4/392photonphoton detectioninformation gaindetection efficiencyfigure of merit (FOM)Landauer principle |
| spellingShingle | J Gerhard Müller Photon Detection as a Process of Information Gain Entropy photon photon detection information gain detection efficiency figure of merit (FOM) Landauer principle |
| title | Photon Detection as a Process of Information Gain |
| title_full | Photon Detection as a Process of Information Gain |
| title_fullStr | Photon Detection as a Process of Information Gain |
| title_full_unstemmed | Photon Detection as a Process of Information Gain |
| title_short | Photon Detection as a Process of Information Gain |
| title_sort | photon detection as a process of information gain |
| topic | photon photon detection information gain detection efficiency figure of merit (FOM) Landauer principle |
| url | https://www.mdpi.com/1099-4300/22/4/392 |
| work_keys_str_mv | AT jgerhardmuller photondetectionasaprocessofinformationgain |