Abstract

We propose a scheme to measure the quantum state of photons in a cavity. The proposal is based on the concept of quantum weak values and applies equally well to both the solid-state circuit and atomic cavity quantum electrodynamics (QED) systems. The proposed scheme allows us to access directly the superposition components in Fock state basis, rather than the Wigner function as usual in phase space. Moreover, the separate access feature held in the direct scheme does not require a global reconstruction for the quantum state, which provides a particular advantage beyond the conventional method of quantum state tomography.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (1)

W. Wang, L. Hu, Y. Xu, K. Liu, Y. Ma Shi-Biao Zheng, R. Vijay, Y. P. Song, L.-M. Duan, and L. Sun, “Converting quasiclassical states into arbitrary Fock state superpositions in a superconducting cavity,” Phys. Rev. Lett. 118, 223604 (2017).
[Crossref]

2016 (1)

G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, “Direct Measurement of the Density Matrix of a Quantum System,” Phys. Rev. Lett. 117, 120401 (2016).
[Crossref] [PubMed]

2015 (1)

D. Tan, S. J. Weber, I. Siddiqi, K. Molmer, and K.W. Murch, “Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit,” Phys. Rev. Lett. 114, 090403 (2015).
[Crossref] [PubMed]

2014 (2)

L. Sun, A. Petrenko, Z. Leghtas, B. Vlastakis, G. Kirchmair, K. M. Sliwa, A. Narla, M. Hatridge, S. Shankar, J. Blumoff, L. Frunzio, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “ Tracking photon jumps with repeated quantum non-demolition parity measurements,” Nature 511, 444 (2014).
[Crossref] [PubMed]

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nature Communications 5, 3115 (2014).
[Crossref] [PubMed]

2013 (3)

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterisation of polarisation states of light via direct measurement,” Nature Photonics 7, 316 (2013).
[Crossref]

M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, “Quantum Back-Action of an Individual Variable-Strength Measurement,” Science 339, 178 (2013).
[Crossref] [PubMed]

K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, “Observing Single Quantum Trajectories of a Superconducting Quantum Bit,” Nature 502, 211 (2013).
[Crossref] [PubMed]

2012 (1)

J. S. Lundeen and C. Bamber, “Procedure for Direct Measurement of General Quantum States Using Weak Measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[Crossref] [PubMed]

2011 (2)

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188 (2011).
[Crossref] [PubMed]

C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J. Raimond, and S. Haroche, “Real-time quantum feedback prepares and stabilizes photon number states,” Nature 477, 73 (2011)
[Crossref] [PubMed]

2010 (1)

K Laiho, K N Cassemiro, D Gross, and C Silberhorn, “Probing the Negative Wigner Function of a Pulsed Single Photon Point by Point,” Phys. Rev. Lett. 105, 253603 (2010).
[Crossref]

2009 (4)

A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299 (2009).
[Crossref]

M. Hofheinz, H Wang, M Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and AN Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature 459, 546 (2009).
[Crossref] [PubMed]

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, P. Traina, S. Olivares, Matteo G. A. Paris, and G Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[Crossref]

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444 (2009).
[Crossref] [PubMed]

2006 (1)

A. A. Semenov, D. Yu. Vasylyev, W. Vogel, M. Khanbekyan, and D.-G. Welsch, “Leaky cavities with unwanted noise,” Phys. Rev. A 74, 033803 (2006).
[Crossref]

2005 (2)

J. F. Kanem, S. Maneshi, S. H. Myrskog, and A. M. Steinberg, “Phase Space Tomography of Classical and Nonclassical Vibrational States of Atoms in an Optical Lattice,” J. Opt. B 7, S705 (2005).
[Crossref]

B J. Smith, B Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, “Measurement of the transverse spatial quantum state of light at the single-photon level,” Opt. Lett. 30, 3365 (2005).
[Crossref]

2004 (2)

A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
[Crossref]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162 (2004).
[Crossref] [PubMed]

2003 (1)

2002 (1)

P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J.M. Raimond, and S. Haroche, “Direct Measurement of the Wigner Function of a One-Photon Fock State in a Cavity,” Phys. Rev. Lett. 89, 200402 (2002).
[Crossref] [PubMed]

2001 (1)

M. Franca Santos, L. G. Lutterbach, S. M. Dutra, N. Zagury, and L. Davidovich, “Reconstruction of the state of the radiation field in a cavity through measurements of the outgoing field,” Phys. Rev. A 63, 033813 (2001).
[Crossref]

1999 (2)

K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674 (1999).
[Crossref]

A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally Entangled States: Production, Characterization, and Utilization,” Phys. Rev. Lett. 83, 3103 (1999).
[Crossref]

1997 (2)

G. Breitenbach, S. Schiller, and J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471 (1997).
[Crossref]

L. G. Lutterbach and L. Davidovich, “Method for Direct Measurement of the Wigner Function in Cavity QED and Ion Traps,” Phys. Rev. Lett. 78, 2547 (1997).
[Crossref]

1996 (4)

J. F. Poyatos, R. Walser, J. I. Cirac, P. Zoller, and R. Blatt, “Motion tomography of a single trapped ion,” Phys. Rev. A 53, R1966 (1996).
[Crossref] [PubMed]

C. D’Helon and G. J. Milburn, “Reconstructing the vibrational state of a trapped ion,” Phys. Rev. A 54, R25 (1996).
[Crossref]

P. J. Bardroff, C. Leichtle, G. Schrade, and W. P. Schleich, “Endoscopy in the Paul Trap: Measurement of the Vibratory Quantum State of a Single Ion,” Phys. Rev. Lett. 77, 2198 (1996).
[Crossref] [PubMed]

D. Leibfried, D. M. Meekhof, B. E. King, C. Monroe, W. M. Itano, and D. J. Wineland, “Experimental Determination of the Motional Quantum State of a Trapped Atom,” Phys. Rev. Lett. 77, 4281 (1996).
[Crossref] [PubMed]

1995 (4)

S. Wallentowitz and W. Vogel, “Reconstruction of the Quantum Mechanical State of a Trapped Ion,” Phys. Rev. Lett. 75, 2932 (1995).
[Crossref] [PubMed]

T. J. Dunn, I. A. Walmsley, and S. Mukamel, “Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography,” Phys. Rev. Lett. 74, 884 (1995).
[Crossref] [PubMed]

G. Breitenbach, T. Müller, S. F. Pereira, J. Ph. Poizat, S. Schiller, and J. Mlynek, “Squeezed vacuum from a monolithic optical parametric oscillator,” J. Opt. Soc. B 12, 2304 (1995).
[Crossref]

P. J. Bardroff, E. Mayr, and W. P. Schleich, “Quantum state endoscopy: Measurement of the quantum state in a cavity,”, “Simulation of quantum-state endoscopy”, Phys. Rev. A 51, 4963 (1995).
[Crossref] [PubMed]

1994 (3)

M. Freyberger and A. M. Herkommer, “Probing a Quantum State via Atomic Deflection,” Phys. Rev. Lett. 72, 1952 (1994).
[Crossref] [PubMed]

S. M. Dutra and P. L. Knight, “Atomic probe for quantum states of the electromagnetic field,” Phys. Rev. A 49, 1506 (1994).
[Crossref] [PubMed]

J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Quantum collapse and revival in the motion of a single trapped ion,” Phys. Rev. A 49, 1202 (1994).
[Crossref] [PubMed]

1993 (1)

D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244 (1993).
[Crossref] [PubMed]

1992 (1)

C. A. Blockley, D. F. Walls, and H. Risken, “Quantum Collapses and Revivals in a Quantized Trap,” Europhys. Lett. 77, 509 (1992).
[Crossref]

1991 (1)

M. Wilkens and P. Meystre, “Nonlinear atomic homodyne detection: A technique to detect macroscopic superpositions in a micromaser,” Phys. Rev. A 43, 3832 (1991).
[Crossref] [PubMed]

1990 (1)

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11 (1990).
[Crossref] [PubMed]

1989 (2)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a “weak measurement” of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112 (1989).
[Crossref]

K. Vogel and H. Risken, “Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase,” Phys. Rev. A 40, 2847 (1989).
[Crossref]

1988 (1)

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[Crossref] [PubMed]

Abdo, B.

M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, “Quantum Back-Action of an Individual Variable-Strength Measurement,” Science 339, 178 (2013).
[Crossref] [PubMed]

Agnew, M.

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterisation of polarisation states of light via direct measurement,” Nature Photonics 7, 316 (2013).
[Crossref]

Aharonov, Y.

Y. Aharonov and L. Vaidman, “Properties of a quantum system during the time interval between two measurements,” Phys. Rev. A 41, 11 (1990).
[Crossref] [PubMed]

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[Crossref] [PubMed]

Albert, D.

Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
[Crossref] [PubMed]

Allevi, A.

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, P. Traina, S. Olivares, Matteo G. A. Paris, and G Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[Crossref]

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444 (2009).
[Crossref] [PubMed]

Amini, H.

C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J. Raimond, and S. Haroche, “Real-time quantum feedback prepares and stabilizes photon number states,” Nature 477, 73 (2011)
[Crossref] [PubMed]

Andreoni, A.

M. Bondani, A. Allevi, and A. Andreoni, “Wigner function of pulsed fields by direct detection,” Opt. Lett. 34, 1444 (2009).
[Crossref] [PubMed]

A. Allevi, A. Andreoni, M. Bondani, G. Brida, M. Genovese, M. Gramegna, P. Traina, S. Olivares, Matteo G. A. Paris, and G Zambra, “State reconstruction by on/off measurements,” Phys. Rev. A 80, 022114 (2009).
[Crossref]

Ansmann, M

M. Hofheinz, H Wang, M Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and AN Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature 459, 546 (2009).
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Auffeves, A.

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W. Wang, L. Hu, Y. Xu, K. Liu, Y. Ma Shi-Biao Zheng, R. Vijay, Y. P. Song, L.-M. Duan, and L. Sun, “Converting quasiclassical states into arbitrary Fock state superpositions in a superconducting cavity,” Phys. Rev. Lett. 118, 223604 (2017).
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D. Tan, S. J. Weber, I. Siddiqi, K. Molmer, and K.W. Murch, “Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit,” Phys. Rev. Lett. 114, 090403 (2015).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162 (2004).
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K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, “Observing Single Quantum Trajectories of a Superconducting Quantum Bit,” Nature 502, 211 (2013).
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L. Sun, A. Petrenko, Z. Leghtas, B. Vlastakis, G. Kirchmair, K. M. Sliwa, A. Narla, M. Hatridge, S. Shankar, J. Blumoff, L. Frunzio, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “ Tracking photon jumps with repeated quantum non-demolition parity measurements,” Nature 511, 444 (2014).
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Nature Communications (1)

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nature Communications 5, 3115 (2014).
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Nature Photonics (1)

J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterisation of polarisation states of light via direct measurement,” Nature Photonics 7, 316 (2013).
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M. Wilkens and P. Meystre, “Nonlinear atomic homodyne detection: A technique to detect macroscopic superpositions in a micromaser,” Phys. Rev. A 43, 3832 (1991).
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J. I. Cirac, R. Blatt, A. S. Parkins, and P. Zoller, “Quantum collapse and revival in the motion of a single trapped ion,” Phys. Rev. A 49, 1202 (1994).
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M. Franca Santos, L. G. Lutterbach, S. M. Dutra, N. Zagury, and L. Davidovich, “Reconstruction of the state of the radiation field in a cavity through measurements of the outgoing field,” Phys. Rev. A 63, 033813 (2001).
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A. A. Semenov, D. Yu. Vasylyev, W. Vogel, M. Khanbekyan, and D.-G. Welsch, “Leaky cavities with unwanted noise,” Phys. Rev. A 74, 033803 (2006).
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K. Banaszek, C. Radzewicz, K. Wódkiewicz, and J. S. Krasiński, “Direct measurement of the Wigner function by photon counting,” Phys. Rev. A 60, 674 (1999).
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Phys. Rev. D (1)

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Y. Aharonov, D. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60, 1351 (1988).
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T. J. Dunn, I. A. Walmsley, and S. Mukamel, “Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography,” Phys. Rev. Lett. 74, 884 (1995).
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J. S. Lundeen and C. Bamber, “Procedure for Direct Measurement of General Quantum States Using Weak Measurement,” Phys. Rev. Lett. 108, 070402 (2012).
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W. Wang, L. Hu, Y. Xu, K. Liu, Y. Ma Shi-Biao Zheng, R. Vijay, Y. P. Song, L.-M. Duan, and L. Sun, “Converting quasiclassical states into arbitrary Fock state superpositions in a superconducting cavity,” Phys. Rev. Lett. 118, 223604 (2017).
[Crossref]

D. Tan, S. J. Weber, I. Siddiqi, K. Molmer, and K.W. Murch, “Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit,” Phys. Rev. Lett. 114, 090403 (2015).
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Science (1)

M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, “Quantum Back-Action of an Individual Variable-Strength Measurement,” Science 339, 178 (2013).
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M. Malik and R. W. Boyd, Quantum Imaging Technologies, arXiv:1406.1685; Rivista del Nuovo Cimento 37, 5 (2014) p. 273

S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons (Oxford University, 2006).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1 Schematic plot for measuring the unknown state of photons in a cavity, say, in the central one which can be expressed in general as |Ψ〉 = Σn cn|n〉 with |n〉 the Fock state of n photons. In connection with the superconducting circuit-QED realization, the two artificial atoms (qubits) in the side cavities are employed to probe the photons state in the central cavity: the left qubit performs weak measurement selectively for Πn = |n〉〈n|; and the right qubit performs post-selection which will result in a post-selected cavity state |Ψf〉 = Σn cn(αn|n〉 + βn |n − 1〉). The coupling between the cavities, the required rotations of qubits and their measurements are also schematically indicated, while keeping more detailed explanations referred to the main text.
Fig. 2
Fig. 2 Schematic illustration for implementing the proposed scheme in atomic cavity-QED set-up. The high-Q cavity (‘C’) is prepared in an initial state described in general by |Ψ〉 = Σn cn|n〉, while the second low-Q Ramsey cavity (‘R’) is employed to rotate the atomic states of the crossing atoms by introducing external classical fields. The upper and lower panels show, respectively, a meter and post-selection atom crossing sequentially the two cavities.

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

H m e a s = γ n σ 1 x .
| Φ ( τ ) = [ | g 1 i ( γ τ ) Π n w | e 1 ] / N ,
Π n w = Ψ f | Π n | Ψ Ψ f | Ψ .
Π n w = ( 2 γ τ ) ( i σ 1 x Φ σ 1 y Φ ) .
| Ψ 2 = 1 2 n = 0 c n [ ( α n | n β n | n 1 ) | g 2 + ( α n | n + β n | n 1 ) | e 2 ] .
| Ψ f = [ c 0 α 0 | 0 + n = 1 c n ( α n | n β n | n 1 ) ] / N .
0 w = | c 0 | 2 [ α 0 ( c 1 / c 0 ) β 1 ] * , 1 w = | c 1 | 2 [ α 1 ( c 2 / c 1 ) β 2 ] * ,     n w = | c n | 2 [ α n ( c n + 1 / c n ) β n + 1 ] * .
H 0 ( n ) = ( Δ ˜ 1 ( n ) 0 0 Δ ˜ 1 ( n ) ) ,
H ( n ) = ( Δ ˜ 1 ( n ) ω 2 γ γ ( Δ ˜ 1 ( n ) ω 2 ) ) .
| g 1 , n | g 1 , n i ( γ τ / ) | e 1 , n .
U ( τ ) [ | g 1 ( n c n | n ) ] = n n c n | g 1 , n + c n [ | g 1 , n i ( γ τ / ) | e 1 , n ] = | g 1 | Ψ i ( γ τ / ) c n | e 1 , n = | g 1 | Ψ i ( γ τ / ) | e 1 ( ^ n | Ψ ) .
P e 1 ( t ) = ( γ / γ ˜ ) 2 sin 2 ( γ ˜ t ) ,
P e 1 ( t ) = sin 2 ( γ t ) ( γ t ) 2 .
P e 1 ( t ) = ( γ t ) 2 [ sin 2 ( γ ˜ t ) ( γ ˜ t ) 2 ] .

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