Abstract

The standard expressions for the noise that is due to photon fluctuations in thermal background radiation typically apply only for a single detector and are often strictly valid only for single-mode illumination. I describe a technique for rigorously calculating thermal photon noise, which allows for arbitrary numbers of optical inputs and detectors, multiple-mode illumination, and both internal and external noise sources. Several simple examples are given, and a general result is obtained for multimode detectors. The formalism uses scattering matrices, noise correlation matrices, and some fundamentals of quantum optics. The covariance matrix of the photon noise at the detector outputs is calculated and includes the Hanbury Brown and Twiss photon-bunching correlations. These correlations can be of crucial importance, and they explain why instruments such as autocorrelation spectrometers and pairwise-combined interferometers are competitive (and indeed common) at radio wavelengths but have a sensitivity disadvantage at optical wavelengths. The case of autocorrelation spectrometers is studied in detail.

© 2003 Optical Society of America

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2003 (2)

2002 (1)

S. Savasta, O. Di Stefano, R. Girlanda, “Light quantization for arbitrary scattering systems,” Phys. Rev. A 65, 043801 (2002); see Eq. (4.28).
[CrossRef]

2001 (3)

A. I. Harris, J. Zmuidzinas, “A wideband lag correlator for heterodyne spectroscopy of broad astronomical and atmospheric spectral lines,” Rev. Sci. Instrum. 72, 1531–1538 (2001).
[CrossRef]

A. Shumovsky, “Quantum multipole radiation,” Adv. Chem. Phys. 119, 395–490 (2001).
[CrossRef]

M. Kamionkowski, A. H. Jaffe, “Detection of gravitational waves from inflation,” Int. J. Mod. Phys. A (Suppl. 1A) 16, 116–128 (2001).
[CrossRef]

1999 (1)

L. Knöll, S. Scheel, E. Schmidt, D.-G. Welsch, A. V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

1998 (4)

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

C. W. J. Beenakker, “Thermal radiation and amplified spontaneous emission from a random medium,” Phys. Rev. Lett. 81, 1829–1832 (1998).
[CrossRef]

S. Withington, J. Murphy, “Modal analysis of partially coherent submillimeter-wave quasi-optical systems,” IEEE Trans. Antennas Propag. 46, 1651–1659 (1998).
[CrossRef]

D. Benford, T. Hunter, T. Phillips, “Noise equivalent power of background limited thermal detectors at submillimeter wavelengths,” Int. J. Infrared Millim. Waves 19, 931–938 (1998).
[CrossRef]

1996 (2)

J. Murphy, S. Withington, “Perturbation analysis of Gaussian-beam-mode scattering at off-axis ellipsoidal mirrors,” Infrared Phys. Technol. 37, 205–219 (1996).
[CrossRef]

T. Gruner, D.-G. Welsch, “Quantum-optical input-output relations for dispersive and lossy multilayer dielectric plates,” Phys. Rev. A 54, 1661–1677 (1996).
[CrossRef] [PubMed]

1995 (1)

R. Matloob, R. Loudon, S. M. Barnett, J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823–4838 (1995).
[CrossRef] [PubMed]

1994 (4)

E. Berglind, L. Gillner, “Optical quantum noise treated with classical electrical network theory,” IEEE J. Quantum Electron. 30, 846–853 (1994).
[CrossRef]

P. L. Richards, “Bolometers for infrared and millimeter waves,” J. Appl. Phys. 76, 1–24 (1994).
[CrossRef]

M. Torres, “A frequency-agile hybrid spectral correlator for mm-wave radio interferometry,” Rev. Sci. Instrum. 65, 1537–1540 (1994).
[CrossRef]

S. Prasad, “Implications of light amplification for astronomical imaging,” J. Opt. Soc. Am. A 11, 2799–2803 (1994).
[CrossRef]

1993 (4)

S. Padin, T. Clark, M. Ewing, R. Finch, R. Lawrence, J. Navarro, S. Scott, N. Scoville, C. Seelinger, T. Seling, “A high-speed digital correlator for radio astronomy,” IEEE Trans. Instrum. Meas. 42, 793–798 (1993).
[CrossRef]

S. W. Wedge, D. B. Rutledge, “Wave computations for microwave education,” IEEE Trans. Educ. 36, 127–131 (1993).
[CrossRef]

J. R. Jeffers, N. Imoto, R. Loudon, “Quantum optics of traveling-wave attenuators and amplifiers,” Phys. Rev. A 47, 3346–3359 (1993).
[CrossRef] [PubMed]

J. Murphy, S. Withington, A. Egan, “Mode conversion at diffracting apertures in millimeter and submillimeter-wave optical-systems,” IEEE Trans. Microwave Theory Tech. 41, 1700–1702 (1993).
[CrossRef]

1992 (1)

S. W. Wedge, D. B. Rutledge, “Wave techniques for noise modeling and measurement,” IEEE Trans. Microwave Theory Tech. 40, 2004–2012 (1992).
[CrossRef]

1991 (2)

J. A. Tauber, N. R. Erickson, “A low-cost filterbank spectrometer for submm observations in radio astronomy,” Rev. Sci. Instrum. 62, 1288–1292 (1991).
[CrossRef]

S. W. Wedge, D. B. Rutledge, “Noise waves and passive linear multiports,” IEEE Microwave Guid. Wave Lett. 1, 117–119 (1991).
[CrossRef]

1990 (1)

K. J. Blow, R. Loudon, S. J. D. Phoenix, T. J. Shepherd, “Continuum fields in quantum optics,” Phys. Rev. A 42, 4102–4114 (1990).
[CrossRef] [PubMed]

1987 (1)

H. A. Haus, Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” IEEE J. Quantum Electron. QE-23, 212–221 (1987).
[CrossRef]

1986 (1)

1985 (2)

R. S. Bondurant, “Response of ideal photodetectors to photon flux and/or energy flux,” Phys. Rev. A 32, 2797–2802 (1985).
[CrossRef] [PubMed]

B. Yurke, “Wideband photon counting and homodyne detection,” Phys. Rev. A 32, 311–323 (1985).
[CrossRef] [PubMed]

1984 (2)

H. J. Kimble, L. Mandel, “Photoelectric detection of polychromatic light,” Phys. Rev. A 30, 844–850 (1984).
[CrossRef]

B. Yurke, J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984).
[CrossRef]

1982 (3)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

J. C. Mather, “Bolometer noise: nonequilibrium theory,” Appl. Opt. 21, 1125–1129 (1982).
[CrossRef] [PubMed]

R. W. Boyd, “Photon bunching and the photon-noise-limited performance of infrared detectors,” Infrared Phys. 22, 157–162 (1982).
[CrossRef]

1970 (1)

H. Haus, “Steady-state quantum analysis of linear systems,” Proc. IEEE 58, 1599–1611 (1970).
[CrossRef]

1968 (1)

B. R. Mollow, “Quantum theory of field attenuation,” Phys. Rev. 168, 1896–1919 (1968).
[CrossRef]

1967 (2)

K. M. vanVliet, “Noise limitations in solid-state photodetectors,” Appl. Opt. 6, 1145–1169 (1967).
[CrossRef]

H. Bosma, “On the theory of linear noisy systems,” Philips Res. Rep. Suppl. 10, 1–190 (1967).

1966 (2)

V. V. Karavaev, “Output fluctuations of thermal radiation detectors,” Sov. Phys. JETP 22, 570–577 (1966).

B. L. Morgan, L. Mandel, “Measurement of photon bunching in a thermal light beam,” Phys. Rev. Lett. 16, 1012–1015 (1966).
[CrossRef]

1965 (1)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

1964 (1)

P. L. Kelley, W. H. Kleiner, “Theory of electromagnetic field measurement and photoelectron counting,” Phys. Rev. 136, A316–A334 (1964).
[CrossRef]

1963 (2)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[CrossRef]

1960 (1)

M. Harwit, “Measurement of thermal fluctuations in radiation,” Phys. Rev. 120, 1551–1556 (1960).
[CrossRef]

1959 (2)

P. Felgett, R. C. Jones, R. Q. Twiss, “Fluctuations in photon streams,” Nature 184, 967–969 (1959).
[CrossRef]

C. W. McCombie, “Fluctuations in photon streams,” Nature 184, 969–970 (1959).
[CrossRef]

1957 (5)

G. A. Rebka, R. V. Pound, “Time-correlated photons,” Nature 180, 1035–1036 (1957).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light: I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. London Ser. A 242, 300–324 (1957).
[CrossRef]

P. Felgett, “The question of correlation between photons in coherent beams of light,” Nature 179, 956–957 (1957).
[CrossRef]

R. Q. Twiss, R. Hanbury Brown, “The question of correlation between photons in coherent beams of light,” Nature 179, 1128–1129 (1957).
[CrossRef]

R. M. Sillitto, “Correlation between events in photon detectors,” Nature 179, 1127–1128 (1957).
[CrossRef]

1956 (3)

R. Hanbury Brown, R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[CrossRef]

E. M. Purcell, “The question of correlation between photons in coherent light rays,” Nature 178, 1449–1450 (1956).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, “Correlation between photons in 2 coherent beams of light,” Nature 177, 27–29 (1956).
[CrossRef]

1951 (1)

H. B. Callen, T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

1949 (1)

1947 (3)

W. B. Lewis, “Fluctuations in streams of thermal radiation,” Proc. Phys. Soc. 59, 34–40 (1947).
[CrossRef]

M. J. E. Golay, “Theoretical consideration in heat and infra-red detection, with particular reference to the pneumatic detector,” Rev. Sci. Instrum. 18, 347–356 (1947).
[CrossRef] [PubMed]

R. C. Jones, “The ultimate sensitivity of radiation detectors,” J. Opt. Soc. Am. 37, 879–890 (1947).
[PubMed]

1946 (1)

R. H. Dicke, “The measurement of thermal radiation at microwave frequencies,” Rev. Sci. Instrum. 17, 268–275 (1946).
[CrossRef] [PubMed]

1943 (1)

J. M. W. Milatz, H. A. van der Velden, “Natural limit of measuring radiation with a bolometer,” Physica 10, 369–380 (1943).
[CrossRef]

1928 (2)

J. B. Johnson, “Thermal agitation of electricity in conductors,” Phys. Rev. 32, 97–109 (1928).
[CrossRef]

H. Nyquist, “Thermal agitation of electric charge in conductors,” Phys. Rev. 32, 110–113 (1928).
[CrossRef]

Barnett, S. M.

S. M. Barnett, J. Jeffers, A. Gatti, R. Loudon, “Quantum optics of lossy beam splitters,” Phys. Rev. A 57, 2134–2145 (1998).
[CrossRef]

R. Matloob, R. Loudon, S. M. Barnett, J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823–4838 (1995).
[CrossRef] [PubMed]

Beenakker, C. W. J.

C. W. J. Beenakker, “Thermal radiation and amplified spontaneous emission from a random medium,” Phys. Rev. Lett. 81, 1829–1832 (1998).
[CrossRef]

Benford, D.

D. Benford, T. Hunter, T. Phillips, “Noise equivalent power of background limited thermal detectors at submillimeter wavelengths,” Int. J. Infrared Millim. Waves 19, 931–938 (1998).
[CrossRef]

Berglind, E.

E. Berglind, L. Gillner, “Optical quantum noise treated with classical electrical network theory,” IEEE J. Quantum Electron. 30, 846–853 (1994).
[CrossRef]

Blow, K. J.

K. J. Blow, R. Loudon, S. J. D. Phoenix, T. J. Shepherd, “Continuum fields in quantum optics,” Phys. Rev. A 42, 4102–4114 (1990).
[CrossRef] [PubMed]

Bondurant, R. S.

R. S. Bondurant, “Response of ideal photodetectors to photon flux and/or energy flux,” Phys. Rev. A 32, 2797–2802 (1985).
[CrossRef] [PubMed]

Bosma, H.

H. Bosma, “On the theory of linear noisy systems,” Philips Res. Rep. Suppl. 10, 1–190 (1967).

Boyd, R. W.

R. W. Boyd, “Photon bunching and the photon-noise-limited performance of infrared detectors,” Infrared Phys. 22, 157–162 (1982).
[CrossRef]

Brown, R. Hanbury

R. Q. Twiss, R. Hanbury Brown, “The question of correlation between photons in coherent beams of light,” Nature 179, 1128–1129 (1957).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, “Interferometry of the intensity fluctuations in light: I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. London Ser. A 242, 300–324 (1957).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, “The question of correlation between photons in coherent light rays,” Nature 178, 1447–1448 (1956).
[CrossRef]

Callen, H. B.

H. B. Callen, T. A. Welton, “Irreversibility and generalized noise,” Phys. Rev. 83, 34–40 (1951).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Chattopadhyay, G.

J. Ward, F. Rice, G. Chattopadhyay, J. Zmuidzinas, “SuperMix: a flexible software library for high-frequency circuit simulation, including SIS mixers and superconducting elements,” in Tenth International Symposium on Space Terahertz Technology: Symposium Proceedings (University of Virginia, Charlottesville, Va., 1999), pp. 268–281.

Chizhov, A. V.

L. Knöll, S. Scheel, E. Schmidt, D.-G. Welsch, A. V. Chizhov, “Quantum-state transformation by dispersive and absorbing four-port devices,” Phys. Rev. A 59, 4716–4726 (1999).
[CrossRef]

Clark, T.

S. Padin, T. Clark, M. Ewing, R. Finch, R. Lawrence, J. Navarro, S. Scott, N. Scoville, C. Seelinger, T. Seling, “A high-speed digital correlator for radio astronomy,” IEEE Trans. Instrum. Meas. 42, 793–798 (1993).
[CrossRef]

Denker, J. S.

B. Yurke, J. S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419–1437 (1984).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “The measurement of thermal radiation at microwave frequencies,” Rev. Sci. Instrum. 17, 268–275 (1946).
[CrossRef] [PubMed]

Egan, A.

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Figures (10)

Fig. 1
Fig. 1

General linear N-port network represented by its scattering matrix S. The incoming waves are represented by complex power amplitudes a i , whereas the outgoing waves are represented by b i . The scattering matrix relates the outgoing wave amplitudes to the incoming amplitudes. The matrix element S ij can be considered to be the quantum-mechanical probability amplitude for a photon that enters port j to emerge at port i.

Fig. 2
Fig. 2

Any lossy linear N-port network can be represented by a lossless (N + M)-port network, in which the extra internal ports, labeled by greek indices α = 1, …, M, are terminated by resistors R α. In turn, these resistors can be replaced by semi-infinite transmission lines with characteristic impedances R α. In this diagram, the semi-infinite transmission line is represented by the dashed box with one ragged edge.

Fig. 3
Fig. 3

Top: a discrete ladder approximation to a transmission line, including resistors R that account for ohmic loss. Bottom: the resistors have been replaced by ports that are attached to semi-infinite transmission lines (not shown) that have characteristic impedance Z 0 = R.

Fig. 4
Fig. 4

Schematic diagram of a direct-detection correlation spectrometer. The input signal is first split N lags ways; the power transmission is assumed to be 1/N lags. Each of the splitter outputs is then fed to a Mach-Zehnder interferometer (see Fig. 5), represented by the boxes labeled M l , which incorporate the necessary delays and detectors to produce the lag outputs. Interferometer M l is set with a fixed time delay (or path-length difference) equal to δt l .

Fig. 5
Fig. 5

Mach-Zehnder interferometer made with 90° hybrids that are equivalent to 50% beam splitters. The path-length difference between the two arms is δt l . The drawing shows detectors for both outputs of the interferometer (D 1 and D 2). It is possible to use just one detector per lag; to do this, D 2 should be replaced by a termination (a perfect absorber). The power transmission from the input to D 1 is cos2νδt l ), and the transmission from the input to D 2 is sin2νδt l ). With both detectors present, the total quantum efficiency is unity; with only one detector, the average quantum efficiency is 50%.

Fig. 6
Fig. 6

Comparison of the noise in the spectral channels in the low-n 0 limit for 64-lag direct-detection correlators that utilize one (upper line) and two (lower line) detectors per lag.

Fig. 7
Fig. 7

Variation of the spectral noise in the low-n 0 limit as a function of the number of lags. The number of spectral channels is equal to the number of lags, and two detectors are used per lag. The nearly constant spacing between traces indicates that the noise varies as Nlags.

Fig. 8
Fig. 8

Variation of the spectral noise in the low-n 0 limit for a 64-lag, two-detector correlator as a function of the number of channels in the output spectrum.

Fig. 9
Fig. 9

Variation of the spectral noise in the high-n 0 (classical) limit for a 64-lag, two-detector correlator, as a function of the number of channels in the output spectrum.

Fig. 10
Fig. 10

Solid curve: the variation of the mid-band spectral noise [Eq. (82)] relative to an ideal instrument [Eq. (85)] for a 64-lag, two-detector correlator, producing a 64-channel spectrum, as a function of the photon occupation number at the input. Dashed curve: relative noise for the same correlator, now preceded by a high-gain quantum-limited amplifier [Eq. (86)]. Dotted curve: relative noise for a two-detector FTS [Eq. (88)].

Equations (125)

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σP=hνΔνηΔντηn01+ηn01/2,
biν=j Sijνajν.
bi=j Sijaj+ci.
cicj*=I-SSijhν2cothhν2kTΔν,
ciνcj*ν=I-SSijhν2cothhν2kTδν-ν=Cijclassνδν-ν,
δViνδVj*ν=2kTZ+Zijδν-ν,
biν=j Sijνajν+ciν,
aiν, ajν=δijδν-ν,
biν, bjν=δijδν-ν.
ciν, cjν=I-SνSνjiδν-ν.
bi=j Sijaj+α Siβaβ,
bα=j Sαjaj+β Sαβaβ.
ci=β Siβaβ.
I-SSij=δij-k SikSjk*=β SiβSjβ*.
ciνcjν=α,β SiανSjβ*νaανaβν.
aανaβν=nthν, Tαδαβδν-ν,
ciνcjν=Cijνδν-ν,
Cijν=β SiβνSjβ*νnthν, Tβ.
Cijν=I-SSijnthν, T,
hν2 ciνcjν+cjνciν=hνI-SSijnthν, T+1/2δν-ν,
di=1τ0τ dtbitbit,
bit=0+ dν expi2πνtbiν,
bit=0+ dν exp-i2πνtbiν.
diB=1τ0τ dtbiBtbiBt,
biBt=0+ dν expi2πνtbiνhν,
biBt=0+ dν exp-i2πνtbiνhν.
biνbjν=k,l SikνSjl*νakνalν+ciνcjν.
akνalν=nkνδklδν-ν.
biνbjν=δν-νBijν,
Bijν=k SikνSjk*νnkν+Cijν.
di=1τ0τ dtbitbit=1τ  dνdν 0τ dt×expi2πν-νtbiνbiν= dνBiiν.
σij2=δdiδdj=didj-didj.
didj=1τ20τ dt10τ dt2Nit1Njt2,
ijδt=Nit+δtNjt+NitNjt+δt.
didj=1τ20τ dδtτ-δtijδt.
ijδt= dν1dν2dν3dν4biν1biν2bjν3bjν4×exp-i2πν1-ν2t+δt×exp-i2πν3-ν4t+exp-i2πν1-ν2t×exp-i2πν3-ν4t+δt= dν1dν2dν3dν4biν1bjν3biν2bjν4+δijδν2-ν3biν1bjν4×exp-i2πν1-ν2t+δt×exp-i2πν3-ν4t+exp-i2πν1-ν2t×exp-i2πν3-ν4t+δt= dν1dν2dν3dν4Biiν1Bjjν3δν1-ν2×δν3-ν4+Bijν1Bjiν3+δijδν1-ν4δν3-ν2×exp-i2πν1-ν2t+δt×exp-i2πν3-ν4t+exp-i2πν1-ν2t×exp-i2πν3-ν4t+δt= dν1dν32Biiν1Bjjν3+2Bijν1Bjiν3+δijcos2πν1-ν3δt.
2 0τ dδtτ-δtcos[2πν1-ν3δt]τδν1-ν3,
σij2=1τ  dνBijνBjiν+δij.
Cijquantν=1hν Cijclassν-12I-SSij,
Bijν=Si0νSj0*νn0ν+Cijν,
di= dν|Si0ν|2n0ν+Ciiν.
σij2=1τ  dν|Si0ν|2n0ν|Sj0ν|2n0ν+δij+CijνCjiν+δij+2 ReSi0*νSj0νCijνn0ν.
σ2=1τ  dνηνn0νηνn0ν+1.
d= dνηνn0ν.
d=ηn0Δν
σ2=1τ ηn01+ηn0Δν.
σP=hνση=hνΔντn01+ηn0η Δν,
σPkT0Δντ Δν,
σPhνητηn0Δντ.
Cijν=SS-Iij.
σ222=1τ  dνB22νB22ν+1=1τ  dνG2νnν+1-G-1νnν+1.
σP=Δντ hνnν+1.
σP=Δντhνηηnν+1,
σ2=1τ0 dνhνηνPν+0 dνη2νP2νNν,
σ2=1τ0 dνhν2ηνnν1+ηνnν.
d=o ηodo.
d=o do.
σ2=o,p δdoδdp, where o, pMO=o,p σop2,
σ2=o,pMO1τ  dνBopνBpoν+δop.
Bopν=ηνnνδop,
σ2=N 1τ  dνηνnν1+ηνnν,
Bopν=SoiνSpi*νnν.
σ2=1τ  dνo |Soiν|2nν1+p |Spiν|2nν=1τ  dνηνnν1+ηνnν,
Aδt=VtVt-δt=A-δt,
Aδt=-+ dνexpi2πνδtSν=2 0+ dνcos2πνδtSν,
Sν=2 0+ dδtcos2πνδtAδt.
μi=τ  dν|Si0ν|2n0ν.
CijD=δDiδDj=τ  dν|Si0ν|2n0ν|Sj0ν|2n0ν+δij.
n0ν=c=1N chann¯cUcν,
pic=1Δνcνcνc+1 dν|Si0ν|2.
λc=n¯cΔνcτ.
μi=c picλc,
CijD=μiδij+c picpjcρijcλcn¯c,
ρijc=1picpjcΔνcνcνc+1 dν|Si0ν|2|Sj0ν|2.
Cijλˆ=diagλ1, λ2,,
λˆ=P-1D.
λˆ=PTP-1PTD,
Cijλˆ=δλˆδλˆT=ACDAT,
A=PTP-1PT.
CijD=c picpjcλc2Δνcτ.
Di=c picNc.
CccN=δNcδNc=λc2Δνcτ δcc.
CD=PCNPT,
Cijλˆ=CN=diagλ12Δν1τ, λ22Δν2τ,,
|Si0ν|2=1Nlagscos2πδtlνfor i oddsin2πδtlνfor i even,
Cλˆ=ACDAT=λ0CPoiss+n0Cclass,
CccPoiss=i Acic picAci,
Cccclass=ij Acic picpjcρijcAcj.
Cccideal=λ01+n0.
Cλˆamp=Cclassλ01+n01+n0-1,
plν=1Nlagscos2πνδtl.
CλˆFTS=λ0CPoiss+n0CclassFTS,
CccclassFTS=Nlagsliljl Acic picpjcρijcAcj,
Vix, tx=- Iix, tt,
Iix, tx=-C Vix, tt,
Vix, t=Z020 dν exp+j2πνtaiνexp-jkx+biνexp+jkx+c.c.
Iix, t=12Z00 dν exp+j2πνtaiν×exp-jkx-biνexp+jkx+c.c.
Uit=-+ dx12 CVi2x, t+12 Ii2x, t
Ui=0 dν|aiν|2+|biν|2,
aiν=-+ dt exp-j2πνt×Vi0, t+Z0Ii0, t2Z0,
biν=-+ dt exp-j2πνt×Vi0, t-Z0Ii0, t2Z0,
aiν=aiδν-ν0.
Vi0, t=2Z0Reai exp+j2πνt,
Ii0, t=2Z0Reai exp+j2πνt,
Pi=Ii0, tVi0, tt=|ai|2.
aiνai*ν=Aiνδν-ν,
ai=ν0-Δν/2ν0+Δν/2 dνaiν=-+ aitexp-j2πν0tsincπΔνtΔν,
biν=j Sijνajν,
Vclx, t=Z020 dνaclνexp+j2πνtexp-jkx+acl*νexp-j2πνtexp+jkx.
Vopx, t=Z020 dνhνaν×exp-i2πνtexp+ikx+aνexp+i2πνtexp-ikx,
i att=at, H,
aν=ψ|aν|ψ=αν, aν=ψ|aν|ψ=α*ν.
acl*ν=ανhν.
βi*ν=j Sijναj*ν,
A=TrρA.
ρi=Ci exp- dνhνNiνkTi,
ρ=i ρi,
ρiaiν=exphν/kTiaiνρi.
aiνajν=Tr[ρaiνajν]=Tr[ρajνaiν]-Tr{ρ[ajν, aiν]}=exphν/kTjTr[ajνρaiν]-Trρδijδν-ν=exphν/kTjaiνajν-δijδν-ν,
aiνajν=δijδν-ν[exphν/kTi-1]-1=nthν, Tiδijδν-ν.
aiνajν=0=aiνajν.
aiν1ajν2akν3alν4=nthν1, Tinthν3, Tk+1δilδjkδν1-ν4δν2-ν3+nthν1, Tinthν3, Tkδijδklδν1-ν2δν3-ν4.
ciνcjν=Cijνδν-ν.
ciν1cjν2ckν3clν4=α,β,γ,δ Siαν1Sjβ*ν2Skγν3Slδ*ν4×aαν1, aβν2aγν3, aδν4=α,γ Siαν1Slα*ν1nthν1, TαSkγν3Sjγ*ν3×nthν3, Tγ+1δν1-ν4δν3-ν2+Siαν1Sjα*ν1nthν1, TαSlγ*ν3Skγν3×nthν3, Tγδν1-ν2δν3-ν4.
ciν1cjν2ckν3clν4=δν1-ν4×δν3-ν2Cilν1Ckjν3+1-Sν3Sν3kj+δν1-ν2δν3-ν4Cijν1Cklν3.

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