Abstract

There have recently been several studies of the performance of laser frequency stabilization using spectral holes in solids, instead of an external cavity, as a frequency reference. Here an analytical theory for Pound-Drever-Hall laser frequency stabilization using spectral hole-burning is developed. The interaction between the atomic medium and the phase modulated light is described using a linearized model of the Maxwell-Bloch equations. The interplay between the carrier and modulation sidebands reveals significant differences from the case of locking to a cavity. These include a different optimum modulation index, an optimum sample absorption, and the possibility to lock the laser in an inherent linear frequency drift mode. Spectral holes in solids can be permanent or transient. For the materials normally used, the dynamics and time scales of transient holes often depend on population relaxation processes between ground state hyperfine levels. These relaxation rates can be very different for different solid state materials. We demonstrate, using radio-frequency pumping, that the hyperfine population dynamics may be controlled and tailored to give optimum frequency stabilization performance. In this way also materials with initially non-optimum performance can be used for stabilization. The theoretical predictions regarding the inherent linear frequency drift is compared to experimental data from a dye laser stabilized to a spectral hole in a Pr3+:Y2SiO5 crystal.

© 2007 Optical Society of America

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References

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  1. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
    [CrossRef]
  2. E. D. Black, "An introduction to Pound-Drever-Hall laser frequency stabilization," Am. J. Phys. 69, 79-87 (2001).
    [CrossRef]
  3. P. B. Sellin, N.M. Strickland, J. L. Carlsten, and R. L. Cone, "Programmable frequency reference for subkilohertz laser stabilization by use of persistent spectral hole burning," Opt. Lett. 24, 1038-1040 (1999).
    [CrossRef]
  4. N. M. Strickland, P. B. Sellin, Y. Sun, J. L. Carlsten, and R. L. Cone, "Laser frequency stabilization using regenerative spectral hole burning," Phys. Rev. B 62, 1473-1476 (2000).
    [CrossRef]
  5. G. J. Pryde, T. Böttger, and R. L. Cone, "Numerical modeling of laser stabilization by regenerative spectral hole burning," J. Lumin. 94-95, 587-591 (2001).
    [CrossRef]
  6. G. J. Pryde, T. Böttger, R. L. Cone, and R. C. C . Ward, "Semiconductor lasers stabilized to spectral holes in rare earth crystals to a part in 10(13) and their application to devices and spectroscopy," J. Lumin. 98, 309-315 (2002).
    [CrossRef]
  7. P. B. Sellin, N. M. Strickland, T. Böttger, J. L. Carlsten, and R. L. Cone, "Laser stabilization at 1536 nm using regenerative spectral hole burning," Phys. Rev. B 63, 155111 (2001).
    [CrossRef]
  8. T . Böttger, Y. Sun, G. J. Pryde, G. Reinemer, and R. L. Cone, "Diode laser frequency stabilization to transient spectral holes and spectral diffusion in Er3+:Y2SiO5 at 1536 nm," J. Lumin. 94, 565-568 (2001).
    [CrossRef]
  9. T. Böttger, G. J . Pryde, and R. L . Cone, "Programmable laser frequency stabilization at 1523 nm by use of persistent spectral hole burning," Opt. Lett. 28, 200-202 (2003).
    [CrossRef] [PubMed]
  10. K. D. Merkel, R. D. Peters, P. B. Sellin, K. S. Repasky, and W. R. Babbitt, "Accumulated programming of a complex spectral grating," Opt. Lett. 25, 1627-1629 (2000).
    [CrossRef]
  11. R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, "Ultraslow optical dephasing in Eu3+:Y2SiO5, " Phys. Rev. Lett. 72, 2179-2181 (1994).
    [CrossRef] [PubMed]
  12. N. Ohlsson, R. K. Mohan, and S. Kroll, "Quantum computer hardware based on rare-earth-ion-doped inorganic crystals," Opt. Commun. 201, 71-77 (2002).
    [CrossRef]
  13. I. Roos and K. Mølmer, "Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping," Phys. Rev. A 69, 022321 (2004).
    [CrossRef]
  14. J. J. Longdell and M. J. Sellars, "Experimental demonstration of quantum-state tomography and qubit-qubit interactions for rare-earth-metal-ion-based solid-state qubits," Phys. Rev. A 69, 032307 (2004).
    [CrossRef]
  15. J. J. Longdell, M. J. Sellars, and N. B. Manson, "Demonstration of conditional quantum phase shift between ions in a solid," Phys. Rev. Lett. 93, 130503 (2004).
    [CrossRef] [PubMed]
  16. J. H. Wesenberg, K. Mølmer, L. Rippe, and S. Kr¨oll, "Scalable designs for quantum computing with rare-earthion- doped crystals," Phys. Rev. A 75, 012304 (2007).
    [CrossRef]
  17. L. Rippe, B. Julsgaard, A. Walther, and S. Kroll, "Laser stabilization using spectral hole burning," http://arxiv.org/abs/quant-ph/0611056.
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  19. P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons, New York, 1988).
  20. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).
  21. A. M. Stoneham, "Shapes of Inhomogeneously Broadened Resonance Lines in Solids," Rev. Mod. Phys. 41, 82 (1969).
    [CrossRef]
  22. J. H. Wesenberg and K. Mølmer, "Field Inside a Random Distribution of Parallel Dipoles," Phys. Rev. Lett. 93, 143903 (2004).
    [CrossRef] [PubMed]
  23. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
    [CrossRef]
  24. T. Böttger, G. J.  Pryde, C. W. Thiel, and R. L. Cone, "Laser frequency stabilization at 1.5 microns using ultranarrow inhomogeneous absorption profiles in Er3+:LiYF4," J. Lumin. 127, 83-88 (2007).
    [CrossRef]
  25. L. Rippe, "Quantum computing with naturally trapped sub-nanometre-spaced ions," Ph.D. thesis, Division of Atomic Physics, LTH, P.O. Box 118, SE 221 00 Lund (2006).
  26. "Laser stabilization system documentation," available at http://www.atom.fysik.lth.se/QI/.
  27. K. J. °Astrom and R. J. Murray, "Feedback Systems: An Introduction for Scientists and Engineers," preprint at http://www.cds.caltech.edu/ murray/amwiki/.
  28. F. Wolf, "Fast sweep experiments in microwave spectroscopy," J. Phys. D 27, 1774-1780 (1994).
    [CrossRef]
  29. T. Chang, M. Z. Tian, R. K. Mohan, C. Renner, K. D. Merkel, and W. R. Babbitt, "Recovery of spectral features readout with frequency-chirped laser fields," Opt. Lett. 30, 1129-1131 (2005).
    [CrossRef] [PubMed]
  30. M. Nilsson, L. Rippe, R. Klieber, D. Suter, and S. Kroll, "Holeburning techniques for isolation and study of individual hyperfine transitions in inhomogeneously broadened solids, demonstrated in Pr3+:Y2SiO5, " Phys. Rev. B 70, 214116 (2004).
    [CrossRef]
  31. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1988).

2007 (2)

J. H. Wesenberg, K. Mølmer, L. Rippe, and S. Kr¨oll, "Scalable designs for quantum computing with rare-earthion- doped crystals," Phys. Rev. A 75, 012304 (2007).
[CrossRef]

T. Böttger, G. J.  Pryde, C. W. Thiel, and R. L. Cone, "Laser frequency stabilization at 1.5 microns using ultranarrow inhomogeneous absorption profiles in Er3+:LiYF4," J. Lumin. 127, 83-88 (2007).
[CrossRef]

2005 (1)

2004 (5)

M. Nilsson, L. Rippe, R. Klieber, D. Suter, and S. Kroll, "Holeburning techniques for isolation and study of individual hyperfine transitions in inhomogeneously broadened solids, demonstrated in Pr3+:Y2SiO5, " Phys. Rev. B 70, 214116 (2004).
[CrossRef]

J. H. Wesenberg and K. Mølmer, "Field Inside a Random Distribution of Parallel Dipoles," Phys. Rev. Lett. 93, 143903 (2004).
[CrossRef] [PubMed]

I. Roos and K. Mølmer, "Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping," Phys. Rev. A 69, 022321 (2004).
[CrossRef]

J. J. Longdell and M. J. Sellars, "Experimental demonstration of quantum-state tomography and qubit-qubit interactions for rare-earth-metal-ion-based solid-state qubits," Phys. Rev. A 69, 032307 (2004).
[CrossRef]

J. J. Longdell, M. J. Sellars, and N. B. Manson, "Demonstration of conditional quantum phase shift between ions in a solid," Phys. Rev. Lett. 93, 130503 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (2)

N. Ohlsson, R. K. Mohan, and S. Kroll, "Quantum computer hardware based on rare-earth-ion-doped inorganic crystals," Opt. Commun. 201, 71-77 (2002).
[CrossRef]

G. J. Pryde, T. Böttger, R. L. Cone, and R. C. C . Ward, "Semiconductor lasers stabilized to spectral holes in rare earth crystals to a part in 10(13) and their application to devices and spectroscopy," J. Lumin. 98, 309-315 (2002).
[CrossRef]

2001 (4)

P. B. Sellin, N. M. Strickland, T. Böttger, J. L. Carlsten, and R. L. Cone, "Laser stabilization at 1536 nm using regenerative spectral hole burning," Phys. Rev. B 63, 155111 (2001).
[CrossRef]

T . Böttger, Y. Sun, G. J. Pryde, G. Reinemer, and R. L. Cone, "Diode laser frequency stabilization to transient spectral holes and spectral diffusion in Er3+:Y2SiO5 at 1536 nm," J. Lumin. 94, 565-568 (2001).
[CrossRef]

G. J. Pryde, T. Böttger, and R. L. Cone, "Numerical modeling of laser stabilization by regenerative spectral hole burning," J. Lumin. 94-95, 587-591 (2001).
[CrossRef]

E. D. Black, "An introduction to Pound-Drever-Hall laser frequency stabilization," Am. J. Phys. 69, 79-87 (2001).
[CrossRef]

2000 (2)

K. D. Merkel, R. D. Peters, P. B. Sellin, K. S. Repasky, and W. R. Babbitt, "Accumulated programming of a complex spectral grating," Opt. Lett. 25, 1627-1629 (2000).
[CrossRef]

N. M. Strickland, P. B. Sellin, Y. Sun, J. L. Carlsten, and R. L. Cone, "Laser frequency stabilization using regenerative spectral hole burning," Phys. Rev. B 62, 1473-1476 (2000).
[CrossRef]

1999 (1)

1994 (2)

F. Wolf, "Fast sweep experiments in microwave spectroscopy," J. Phys. D 27, 1774-1780 (1994).
[CrossRef]

R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, "Ultraslow optical dephasing in Eu3+:Y2SiO5, " Phys. Rev. Lett. 72, 2179-2181 (1994).
[CrossRef] [PubMed]

1983 (2)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
[CrossRef]

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

1969 (1)

A. M. Stoneham, "Shapes of Inhomogeneously Broadened Resonance Lines in Solids," Rev. Mod. Phys. 41, 82 (1969).
[CrossRef]

Am. J. Phys. (1)

E. D. Black, "An introduction to Pound-Drever-Hall laser frequency stabilization," Am. J. Phys. 69, 79-87 (2001).
[CrossRef]

Appl. Phys. B (2)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97-105 (1983).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, "Frequency modulation (FM) spectroscopy," Appl. Phys. B 32, 145-152 (1983).
[CrossRef]

J. Lumin. (4)

T. Böttger, G. J.  Pryde, C. W. Thiel, and R. L. Cone, "Laser frequency stabilization at 1.5 microns using ultranarrow inhomogeneous absorption profiles in Er3+:LiYF4," J. Lumin. 127, 83-88 (2007).
[CrossRef]

G. J. Pryde, T. Böttger, and R. L. Cone, "Numerical modeling of laser stabilization by regenerative spectral hole burning," J. Lumin. 94-95, 587-591 (2001).
[CrossRef]

G. J. Pryde, T. Böttger, R. L. Cone, and R. C. C . Ward, "Semiconductor lasers stabilized to spectral holes in rare earth crystals to a part in 10(13) and their application to devices and spectroscopy," J. Lumin. 98, 309-315 (2002).
[CrossRef]

T . Böttger, Y. Sun, G. J. Pryde, G. Reinemer, and R. L. Cone, "Diode laser frequency stabilization to transient spectral holes and spectral diffusion in Er3+:Y2SiO5 at 1536 nm," J. Lumin. 94, 565-568 (2001).
[CrossRef]

J. Phys. D (1)

F. Wolf, "Fast sweep experiments in microwave spectroscopy," J. Phys. D 27, 1774-1780 (1994).
[CrossRef]

Opt. Commun. (1)

N. Ohlsson, R. K. Mohan, and S. Kroll, "Quantum computer hardware based on rare-earth-ion-doped inorganic crystals," Opt. Commun. 201, 71-77 (2002).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (3)

I. Roos and K. Mølmer, "Quantum computing with an inhomogeneously broadened ensemble of ions: Suppression of errors from detuning variations by specially adapted pulses and coherent population trapping," Phys. Rev. A 69, 022321 (2004).
[CrossRef]

J. J. Longdell and M. J. Sellars, "Experimental demonstration of quantum-state tomography and qubit-qubit interactions for rare-earth-metal-ion-based solid-state qubits," Phys. Rev. A 69, 032307 (2004).
[CrossRef]

J. H. Wesenberg, K. Mølmer, L. Rippe, and S. Kr¨oll, "Scalable designs for quantum computing with rare-earthion- doped crystals," Phys. Rev. A 75, 012304 (2007).
[CrossRef]

Phys. Rev. B (3)

N. M. Strickland, P. B. Sellin, Y. Sun, J. L. Carlsten, and R. L. Cone, "Laser frequency stabilization using regenerative spectral hole burning," Phys. Rev. B 62, 1473-1476 (2000).
[CrossRef]

P. B. Sellin, N. M. Strickland, T. Böttger, J. L. Carlsten, and R. L. Cone, "Laser stabilization at 1536 nm using regenerative spectral hole burning," Phys. Rev. B 63, 155111 (2001).
[CrossRef]

M. Nilsson, L. Rippe, R. Klieber, D. Suter, and S. Kroll, "Holeburning techniques for isolation and study of individual hyperfine transitions in inhomogeneously broadened solids, demonstrated in Pr3+:Y2SiO5, " Phys. Rev. B 70, 214116 (2004).
[CrossRef]

Phys. Rev. Lett. (3)

J. H. Wesenberg and K. Mølmer, "Field Inside a Random Distribution of Parallel Dipoles," Phys. Rev. Lett. 93, 143903 (2004).
[CrossRef] [PubMed]

R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, "Ultraslow optical dephasing in Eu3+:Y2SiO5, " Phys. Rev. Lett. 72, 2179-2181 (1994).
[CrossRef] [PubMed]

J. J. Longdell, M. J. Sellars, and N. B. Manson, "Demonstration of conditional quantum phase shift between ions in a solid," Phys. Rev. Lett. 93, 130503 (2004).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

A. M. Stoneham, "Shapes of Inhomogeneously Broadened Resonance Lines in Solids," Rev. Mod. Phys. 41, 82 (1969).
[CrossRef]

Other (8)

L. Rippe, "Quantum computing with naturally trapped sub-nanometre-spaced ions," Ph.D. thesis, Division of Atomic Physics, LTH, P.O. Box 118, SE 221 00 Lund (2006).

"Laser stabilization system documentation," available at http://www.atom.fysik.lth.se/QI/.

K. J. °Astrom and R. J. Murray, "Feedback Systems: An Introduction for Scientists and Engineers," preprint at http://www.cds.caltech.edu/ murray/amwiki/.

L. Rippe, B. Julsgaard, A. Walther, and S. Kroll, "Laser stabilization using spectral hole burning," http://arxiv.org/abs/quant-ph/0611056.

D. Allen and J. H. Eberly, Optical resonance and two-level atoms (Wiley, New York, 1975).

P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons, New York, 1988).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1988).

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Figures (10)

Fig. 1.
Fig. 1.

The function f (x) defined in Eq. (25). At x=2.41 it attains its maximum value of 0.172. Physically, f(x) is proportional to the slope of ϕi (ω) at ω=0 in Eq. (22).

Fig. 2.
Fig. 2.

The magnitude and phase of the transfer function T(ω) calculated from Eq. (23) (heavy lines). The light lines show the asymptotic cases discussed around Eqs. (24) and (26). Parameters used are T 1=150 µs, T 2=18 µs, T rg=T gr=4 ms, b er=0.5, Ω0=2π·1 kHz, and m=0.40, giving Γh=2π·17.5 kHz, Γhole,c≈2π·21 kHz and Γhole,s≈2π·18 kHz. These parameters are close to our experimental working values, as we shall see in Sec. 4.

Fig. 3.
Fig. 3.

Thresholds for stable laser operation. If d (meas) hole,c is below the solid line the square bracket in Eq. (28) is positive and the zero-drift solution is stable. In order for the low-frequency part of the laser locking to have the correct sign (f (x c)-f(x s)>0 in Eq. (24)) we require the less stringent condition that d (meas) hole,c is below the dotted line. Between the two lines the laser can be locked in a linearly drifting mode.

Fig. 4.
Fig. 4.

Different level schemes used in this paper. We define timescales for relaxation between ground state levels and branching ratios from the excited state. The excited state lifetime is always denoted T 1. (a) The most naive scenario with two levels, considered in Sec. 2.1 and in the first row of Tab. 1. (b) Our basic model for all calculations, described in Secs. 2.2 and 2.4. Rows two to four in Tab. 1 refer to this case. Note, we may have different relaxation timescales T rgT gr. (c) and (d) Different schemes with three ground states coupled as shown with RF-magnetic fields. Hence the timescale is the same in two opposite directions. These cases are reflected by rows five and six in Tab. 1, respectively. (e) The real Pr3+:Y2SiO5 level scheme.

Fig. 5.
Fig. 5.

Optical and electronic design schematics. Abbreviations: N, notch filter; LP, low-pass diplexer; PS, phase shifter; LO, local oscillator; EOM, electro-optical modulator. See the text for more details.

Fig. 6.
Fig. 6.

Experimental setup for characterizing the spectral hole dynamics. Apart from the locking beam needed for the laser stabilization system, we place an additional probing beam for characterizing the dynamics of the locking itself. AOM 1 is in double pass configuration and allows us to scan the laser beam frequency without any beam motion. AOM 2 allows us to shift back to the original stabilized laser frequency to characterize the holes when the laser is locked. An extra crystal in another cryostat is used for measuring the laser frequency drift on long timescales.

Fig. 7.
Fig. 7.

(Color online) Example of driftmeasurements. a) Total observation time is 3 seconds and we see a drift rate of roughly 160 kHz/s, where the direction changes occasionally. b) Total observation time is 20 s and the drift is 0.3 kHz/s over this time.

Fig. 8.
Fig. 8.

(Color online) In the two upper graphs the measured absorption, αR(ω)L, is plotted for comparison under different conditions for a number of center holes (a) and side holes (b). With increasing hole depth, the asymmetry of both increases. The imaginary part, αI0), of the absorption coefficient at the hole center is a quantitative measure of this asymmetry, and can be calculated from αR(ω) using the Kramers-Krönig relations (30). In (c) we see, for several measurements under different conditions, a clear linear relationship between this asymmetry for the center and side holes. The straight line is a fit through the origin with a slope of 0.72, theoretically we expect a slope of unity.

Fig. 9.
Fig. 9.

(Color online)Measured drift rates versus measured hole depth. The hole shapes are changed by varying T 10MHz and T 17MHz. Red circles, m=0.56; green crosses, m=0.40; blue squares, m=0.28; cyan triangles up, m=0.20, purple triangles down, m=0.14. The vertical dashed lines indicate the values of the corresponding thresholds shown in Fig. 3.

Fig. 10.
Fig. 10.

(Color online) Illustration of the phase stability measurements. Inset a) shows the pulse sequence for the experiment: (1) is a 10 ms burn pulse scanning between 40MHz and 50 MHz (relative to the AOM double-pass center). After waiting 100 µs, pulse (2) with a constant frequency of zero and duration 40 µs sets up a coherence in the atoms which leads to the FID at (3). Finally, pulse (4) at a frequency of 45 MHz beats with the FID, leading to the detector signal shown (after filtering) in the main panel. The traces in insets b) and c) show this signal in a 200 ns window around time 0 and 10 µs, respectively, and the phase of these oscillations can be calculated with a good signal-to-noise ratio. Inset d) shows the calculated phase as a function of time for the stabilized (black) and unstabilized (green) laser.

Tables (1)

Tables Icon

Table 1. The value of G and R for the different setups shown in Fig. 4. The first row gives the relations for the two-level atom (Fig. 4(a)) and the second row describes the two-level plus reservoir state system considered in Sec. 2.2 (Fig. 4(b)). The third and fourth rows are special cases of the second row. In the third row we assume T gr=∞, which describes a one-way natural decay from states |r〉 to |g〉. In the fourth row we assume Trg=Tgr which describes the case when an RF magnetic field couples the otherwise uncoupled states |r〉 and |g〉. The fifth and sixth rows correspond to the cases shown in Fig. 4(c) and Fig. 4(d), respectively, where there are two reservoir states. These four-level cases are presented since they resemble our experimental case using Pr3+:Y2SiO5 as the atomic medium

Equations (34)

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t ( u iv ) = ( Γ h 2 + i Δ ) ( u iv ) i Ω w ,
t w = i 2 [ Ω ( u + iv ) Ω * ( u iv ) ] 1 T 1 ( 1 + w ) ,
( z + n b c t ) Ω = i α 0 2 π g ( Δ ) ( u iv ) d Δ .
z Ω ( z , ω ) = α 0 2 π g ( Δ ) d Δ Γ h 2 + i ( Δ ω ) Ω ( z , ω )
α R ( ω ) + i α I ( ω ) 2 Ω ( z , ω ) .
A ( z , ω ) z = α R ( ω ) 2 A ( z , ω ) ,
ϕ ( z , ω ) z = + α I ( ω ) 2 .
t ( u iv ) = ( Γ h 2 + i Δ ) ( u iv ) i Ω ( ρ e ρ g ) ,
ρ e t = i 4 [ Ω ( u + iv ) Ω * ( u iv ) ] 1 T 1 ρ e ,
ρ g t = i 4 [ Ω ( u + iv ) Ω * ( u iv ) ]
+ b eg T 1 ρ e 1 T gr ρ g + 1 T rg ρ r ,
ρ r t = b er T 1 ρ e + 1 T gr ρ g 1 T rg ρ r .
ρ g ρ e = G ( 1 d hole Γ hole 2 4 Γ hole 2 4 + ( Δ Δ 0 ) 2 ) ,
d hole = ( 1 + R ) s 0 2 1 + ( 1 + R ) s 0 2 , Γ hole = Γ h 1 + ( 1 + R ) s 0 2 ,
s 0 = Ω 0 2 T 1 T 2 ,
R = 1 + b er T rg T 1 1 + T rg T gr , G = 1 1 + T rg T gr .
g ( Δ ) = ρ g ρ e G = 1 d hole Γ hole 2 4 Γ hole 2 4 + ( Δ Δ 0 ) 2 .
1 d hole = ( Γ h Γ hole ) 2 .
g ( Δ ) = Γ inh 2 4 Γ inh 2 4 + Δ 2 ( 1 d hole Γ hole 2 4 Γ hole 2 4 + ( Δ Δ 0 ) 2 ) ,
α R ( ω ) α 0 = Γ inh 2 4 Γ inh 2 4 + ω 2 Γ inh 2 4 Γ inh 2 4 + Δ 0 2 · Γ hole ( Γ hole + Γ h ) d d hole ( Γ hole + Γ h ) 2 4 + ( Δ 0 ω ) 2 1 Γ hole ( Γ hole + Γ h ) 4 d hole ( Γ hole + Γ h ) 2 4 + ( Δ 0 ω ) 2 ,
α I ( ω ) α 0 = Γ inh 2 Γ inh 2 4 + ω 2 + Γ inh 2 4 Γ inh 2 4 + Δ 0 2 · Γ hole 2 ( Δ 0 ω ) d hole ( Γ hole + Γ h ) 2 4 + ( Δ 0 ω ) 2 Γ hole 2 ( Δ 0 ω ) d hole ( Γ hole + Γ h ) 2 4 + ( Δ 0 ω ) 2 ,
Γ hole ( meas ) = Γ hole + Γ h , d hole ( meas ) = Γ hole d hole Γ hole + Γ h .
η i ( ω ) = exp ( α 0 L 2 [ 1 Γ hole , i ( Γ hole , i + Γ h ) 4 d hole , i ( Γ hole , i + Γ h ) 2 4 + ω 2 ] ) ,
ϕ i ( ω ) = α 0 L 2 Γ hole , i 2 d hole , i ω ( Γ hole , i + Γ h ) 2 4 + ω 2 ,
P ω m ( out ) ( t ) = 4 P ( in ) J 0 J 1 Re { T ( ω ) · ε ω e i ω t } · sin ( ω m t ) ,
T ( ω ) = η c ( ω ) η s ( 0 ) e i ϕ c ( ω ) η c ( 0 ) η s ( ω ) e i ϕ s ( ω ) i ω + 1 T rg .
T ( ω ) α 0 L Γ h i ω T rg 1 + i ω T rg e α 0 L 2 ( 1 x c + 1 x s ) [ f ( x c ) f ( x s ) ] .
f ( x ) = x 1 x ( x + 1 ) , x c = Γ hole , c Γ h , x s = Γ hole , s Γ h .
T ( ω ) 1 i ω e α 0 L 2 [ e α 0 L 2 x c e α 0 L 2 x s ] .
α I ( Δ 0 ) = α 0 ( β T rg Γ hole ) d hole × T 1 T rg [ 1 + 3 2 T rg T gr + 1 2 T rg 2 T gr 2 ] + 1 2 [ b er ( 1 + T rg T 1 ) b eg T rg T gr ] ( 1 + T rg T gr ) ( 2 + T rg T gr + b er T rg T 1 ) .
P ω m ( out ) ( t ) = P ( in ) J 0 J 1 α 0 L e α 0 L 2 ( Γ h Γ hole , c + Γ h Γ hole , s ) β T rg 1 + T rg T gr [ d hole , c Γ hole , c d hole , s Γ hole , s ] sin ( ω m t ) .
g ( ω ) = R 2 R 1 . i ω ( R 3 + R 4 ) C + 1 i ω R 3 C .
α R ( ω 0 ) = + lim δ 0 1 π α I ( ω ) ( ω ω 0 ) d ω ( ω ω 0 ) 2 + δ 2 ,
α I ( ω 0 ) = lim δ 0 1 π α R ( ω ) ( ω ω 0 ) d ω ( ω ω 0 ) 2 + δ 2 .

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