Abstract

We experimentally demonstrate the strong suppression of dephasing of a qubit stored in a single 87Rb atom in an optical dipole trap by using Carr-Purcell-Meiboom-Gill (CPMG) pulse sequences. Regarded as a repetition of spin echo, CPMG sequence is an excellent choice for suppressing both inhomogeneous and homogeneous phase decoherence mechanisms. In comparison with atomic ensembles, the dephasing due to atomic collisions disappears for individual atoms. Thus, CPMG suppression effect is efficient with a few π-pulses. In our trap with 830 nm wavelength and 0.7 mK potential depth, the reversible inhomogeneous dephasing time is T2*=1.4ms. We obtain the homogeneous dephasing time of T′2 = 103 ms in the spin echo process. By employing CPMG sequence with pulse number n = 6, the homogeneous dephasing is further suppressed by a factor of 3, and its corresponding coherence time is extended to T′2 = 304 ms.

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2012

A. Steffen, A. Alberti, W. Alt, N. Belmechri, S. Hild, M. Karski, A. Widera, and D. Meschede, “Digital atom interferometer with single particle control on a discretized space-time geometry,” Proc. Natl. Acad. Sci. U. S. A.109, 9770–9774 (2012).
[CrossRef] [PubMed]

L. P. Parazzoli, A. M. Hankin, and G. W. Biedermann, “Observation of free-space single-atom matter wave interference,” Phys. Rev. Lett.109, 230401 (2012).
[CrossRef]

X. D. He, S. Yu, P. Xu, J. Wang, and M. S. Zhan, “Combining red and blue-detuned optical potentials to form a Lamb-Dicke trap for a single neutral atom,” Opt. Express20, 3711–3724 (2012).
[CrossRef] [PubMed]

G. Li, S. Zhang, L. Isenhower, K. Maller, and M. Saffman, “Crossed vortex bottle beam trap for single-atom qubits,” Opt. Lett.37, 851–853 (2012).
[CrossRef] [PubMed]

2011

H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, “Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs,” Nat. Phys.7, 109–113 (2011).
[CrossRef]

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys.7, 565–570 (2011).
[CrossRef]

D. J. Szwer, S. C. Webster, A. M. Steane, and D. M. Lucas, “Keeping a single qubit alive by experimental dynamic decoupling,” J. Phys. B At. Mol. Opt. Phys.44, 025501 (2011).
[CrossRef]

2010

T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” Nature464, 45–53 (2010).
[CrossRef] [PubMed]

C. A. Ryan, J. S. Hodges, and D. G. Cory, “Robust decoupling techniques to extend quantum coherence in diamond,” Phys. Rev. Lett.105, 200402 (2010).
[CrossRef]

Y. Sagi, I. Almog, and N. Davidson, “Process tomography of dynamical decoupling in a dense cold atomic ensemble,” Phys. Rev. Lett.105, 053201 (2010).
[CrossRef] [PubMed]

S. Pasini and G. S. Uhrig, “Optimized dynamical decoupling for power-law noise spectra,” Phys. Rev. A81, 012309 (2010).
[CrossRef]

P. Xu, X. D. He, J. Wang, and M. S. Zhan, “Trapping a single atom in a blue detuned optical bottle beam trap,” Opt. Lett.35, 2164–2166 (2010).
[CrossRef] [PubMed]

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett.104, 010503 (2010).
[CrossRef] [PubMed]

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett.104, 010502 (2010).
[CrossRef] [PubMed]

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys.82, 2313–2363 (2010).
[CrossRef]

2009

M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, “Optimized dynamical decoupling in a model quantum memory,” Nature458, 996–1000 (2009).
[CrossRef] [PubMed]

S. Damodarakurup, M. Lucamarini, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Experimental inhibition of decoherence on flying qubits via ‘Bang-Bang’ control,” Phys. Rev. Lett.103, 040502 (2009).
[CrossRef]

J. F. Du, X. Rong, N. Zhao, Y. Wang, J. H. Yang, and R. B. Liu, “Preserving electron spin coherence in solids by optimal dynamical decoupling,” Nature461, 1265–1268 (2009).
[CrossRef] [PubMed]

C.-S. Chuu and C. Zhang, “Suppression of phase decoherence in a single atomic qubit,” Phys. Rev. A80, 032307 (2009).
[CrossRef]

2007

M. P. A. Jones, J. Beugnon, A. Gaëtan, J. Zhang, G. Messin, A. Browaeys, and P. Grangier, “Fast quantum state control of a single trapped neutral atom,” Phys. Rev. A75, 040301 (2007).
[CrossRef]

J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P. A. Jones, G. Messin, A. Browaeys, and P. Grangier, “Two-dimensional transport and transfer of a single atomic qubit in optical tweezers,” Nat. Phys.3, 696–699 (2007).
[CrossRef]

J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich, “Atom interferometer measurement of the Newtonian constant of gravity,” Science315, 74–77 (2007).
[CrossRef] [PubMed]

2006

D. D. Yavuz, P. B. Kulatunga, E. Urban, T. A. Johnson, N. Proite, T. Henage, T. G. Walker, and M. Saffman, “Fast ground state manipulation of neutral atoms in microscopic optical traps,” Phys. Rev. Lett.96, 063001 (2006).
[CrossRef] [PubMed]

2005

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Analysis of dephasing mechanisms in a standing-wave dipole trap,” Phys. Rev. A72, 023406 (2005).
[CrossRef]

2004

L.-A. Wu and D. A. Lidar, “Overcoming quantum noise in optical fibers,” Phys. Rev. A70, 062310 (2004).
[CrossRef]

M. F. Andersen, A. Kaplan, T. Grünzweig, and N. Davidson, “Suppression of dephasing of optically trapped atoms,” Phys. Rev. A70, 013405 (2004).
[CrossRef]

2003

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, and D. Meschede, “Coherence properties and quantum state transportation in an optical conveyor belt,” Phys. Rev. Lett.91, 213002 (2003).
[CrossRef] [PubMed]

L. You, X. X. Yi, and X. H. Su, “Quantum logic between atoms inside a high-Q optical cavity,” Phys. Rev. A67, 032308 (2003).
[CrossRef]

2002

N. Schlosser, G. Reymond, and P. Grangier, “Collisional blockade in microscopic optical dipole traps,” Phys. Rev. Lett.89, 023005 (2002).
[CrossRef] [PubMed]

1999

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via cold controlled collisions,” Phys. Rev. Lett.82, 1975–1978 (1999).
[CrossRef]

A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature400, 849–852 (1999).
[CrossRef]

1998

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett.81, 971–974 (1998).
[CrossRef]

1994

Alberti, A.

A. Steffen, A. Alberti, W. Alt, N. Belmechri, S. Hild, M. Karski, A. Widera, and D. Meschede, “Digital atom interferometer with single particle control on a discretized space-time geometry,” Proc. Natl. Acad. Sci. U. S. A.109, 9770–9774 (2012).
[CrossRef] [PubMed]

Almog, I.

Y. Sagi, I. Almog, and N. Davidson, “Process tomography of dynamical decoupling in a dense cold atomic ensemble,” Phys. Rev. Lett.105, 053201 (2010).
[CrossRef] [PubMed]

Alt, W.

A. Steffen, A. Alberti, W. Alt, N. Belmechri, S. Hild, M. Karski, A. Widera, and D. Meschede, “Digital atom interferometer with single particle control on a discretized space-time geometry,” Proc. Natl. Acad. Sci. U. S. A.109, 9770–9774 (2012).
[CrossRef] [PubMed]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Analysis of dephasing mechanisms in a standing-wave dipole trap,” Phys. Rev. A72, 023406 (2005).
[CrossRef]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, and D. Meschede, “Coherence properties and quantum state transportation in an optical conveyor belt,” Phys. Rev. Lett.91, 213002 (2003).
[CrossRef] [PubMed]

Andersen, M. F.

M. F. Andersen, A. Kaplan, T. Grünzweig, and N. Davidson, “Suppression of dephasing of optically trapped atoms,” Phys. Rev. A70, 013405 (2004).
[CrossRef]

Belmechri, N.

A. Steffen, A. Alberti, W. Alt, N. Belmechri, S. Hild, M. Karski, A. Widera, and D. Meschede, “Digital atom interferometer with single particle control on a discretized space-time geometry,” Proc. Natl. Acad. Sci. U. S. A.109, 9770–9774 (2012).
[CrossRef] [PubMed]

Beugnon, J.

J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P. A. Jones, G. Messin, A. Browaeys, and P. Grangier, “Two-dimensional transport and transfer of a single atomic qubit in optical tweezers,” Nat. Phys.3, 696–699 (2007).
[CrossRef]

M. P. A. Jones, J. Beugnon, A. Gaëtan, J. Zhang, G. Messin, A. Browaeys, and P. Grangier, “Fast quantum state control of a single trapped neutral atom,” Phys. Rev. A75, 040301 (2007).
[CrossRef]

Biedermann, G. W.

L. P. Parazzoli, A. M. Hankin, and G. W. Biedermann, “Observation of free-space single-atom matter wave interference,” Phys. Rev. Lett.109, 230401 (2012).
[CrossRef]

Biercuk, M. J.

M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, “Optimized dynamical decoupling in a model quantum memory,” Nature458, 996–1000 (2009).
[CrossRef] [PubMed]

Bluhm, H.

H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, “Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs,” Nat. Phys.7, 109–113 (2011).
[CrossRef]

Bollinger, J. J.

M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga, W. M. Itano, and J. J. Bollinger, “Optimized dynamical decoupling in a model quantum memory,” Nature458, 996–1000 (2009).
[CrossRef] [PubMed]

Bouyer, P.

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett.81, 971–974 (1998).
[CrossRef]

Briegel, H.-J.

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via cold controlled collisions,” Phys. Rev. Lett.82, 1975–1978 (1999).
[CrossRef]

Browaeys, A.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett.104, 010502 (2010).
[CrossRef] [PubMed]

M. P. A. Jones, J. Beugnon, A. Gaëtan, J. Zhang, G. Messin, A. Browaeys, and P. Grangier, “Fast quantum state control of a single trapped neutral atom,” Phys. Rev. A75, 040301 (2007).
[CrossRef]

J. Beugnon, C. Tuchendler, H. Marion, A. Gaëtan, Y. Miroshnychenko, Y. R. P. Sortais, A. M. Lance, M. P. A. Jones, G. Messin, A. Browaeys, and P. Grangier, “Two-dimensional transport and transfer of a single atomic qubit in optical tweezers,” Nat. Phys.3, 696–699 (2007).
[CrossRef]

Bylander, J.

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys.7, 565–570 (2011).
[CrossRef]

Chu, S.

A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature400, 849–852 (1999).
[CrossRef]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Chung, K. Y.

A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature400, 849–852 (1999).
[CrossRef]

Chuu, C.-S.

C.-S. Chuu and C. Zhang, “Suppression of phase decoherence in a single atomic qubit,” Phys. Rev. A80, 032307 (2009).
[CrossRef]

Cirac, J. I.

D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Entanglement of atoms via cold controlled collisions,” Phys. Rev. Lett.82, 1975–1978 (1999).
[CrossRef]

Cline, R. A.

Cory, D. G.

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys.7, 565–570 (2011).
[CrossRef]

C. A. Ryan, J. S. Hodges, and D. G. Cory, “Robust decoupling techniques to extend quantum coherence in diamond,” Phys. Rev. Lett.105, 200402 (2010).
[CrossRef]

Damodarakurup, S.

S. Damodarakurup, M. Lucamarini, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Experimental inhibition of decoherence on flying qubits via ‘Bang-Bang’ control,” Phys. Rev. Lett.103, 040502 (2009).
[CrossRef]

Davidson, N.

Y. Sagi, I. Almog, and N. Davidson, “Process tomography of dynamical decoupling in a dense cold atomic ensemble,” Phys. Rev. Lett.105, 053201 (2010).
[CrossRef] [PubMed]

M. F. Andersen, A. Kaplan, T. Grünzweig, and N. Davidson, “Suppression of dephasing of optically trapped atoms,” Phys. Rev. A70, 013405 (2004).
[CrossRef]

Di Giuseppe, G.

S. Damodarakurup, M. Lucamarini, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Experimental inhibition of decoherence on flying qubits via ‘Bang-Bang’ control,” Phys. Rev. Lett.103, 040502 (2009).
[CrossRef]

Dotsenko, I.

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Analysis of dephasing mechanisms in a standing-wave dipole trap,” Phys. Rev. A72, 023406 (2005).
[CrossRef]

S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan, V. Gomer, A. Rauschenbeutel, and D. Meschede, “Coherence properties and quantum state transportation in an optical conveyor belt,” Phys. Rev. Lett.91, 213002 (2003).
[CrossRef] [PubMed]

Du, J. F.

J. F. Du, X. Rong, N. Zhao, Y. Wang, J. H. Yang, and R. B. Liu, “Preserving electron spin coherence in solids by optimal dynamical decoupling,” Nature461, 1265–1268 (2009).
[CrossRef] [PubMed]

Evellin, C.

T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett.104, 010502 (2010).
[CrossRef] [PubMed]

Fitch, G.

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys.7, 565–570 (2011).
[CrossRef]

Fixler, J. B.

J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich, “Atom interferometer measurement of the Newtonian constant of gravity,” Science315, 74–77 (2007).
[CrossRef] [PubMed]

Foletti, S.

H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, “Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs,” Nat. Phys.7, 109–113 (2011).
[CrossRef]

Foster, G. T.

J. B. Fixler, G. T. Foster, J. M. McGuirk, and M. A. Kasevich, “Atom interferometer measurement of the Newtonian constant of gravity,” Science315, 74–77 (2007).
[CrossRef] [PubMed]

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H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu, V. Umansky, and A. Yacoby, “Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs,” Nat. Phys.7, 109–113 (2011).
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J. F. Du, X. Rong, N. Zhao, Y. Wang, J. H. Yang, and R. B. Liu, “Preserving electron spin coherence in solids by optimal dynamical decoupling,” Nature461, 1265–1268 (2009).
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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of experimental setup. The quantization axis is defined by a magnetic field along the z-axis. The 87Rb MOT is formed by six cooling lasers (Icool) and an overlap repumping laser (Irep). A laser beam with 830 nm wavelength (Idip) is tightly focused to a waist of 2.1 μm. The resulting optical potential depth is 0.7 mK for a laser power of 7 mW. The fluorescence of single atoms is detected with an avalanche photodiode. The initial state is prepared using an optical pumping laser beam (Ipum) in combination with the repumping beam. The Raman laser beams (IR1, IR2) at 795 nm are coupled into the same polarization maintaining optical fiber and focused onto single atoms. For the state-selective detection, a probe laser beam (Iprob) is applied. (b) Measured fluorescence signals of single atoms within 20 ms time bins. Two steps correspond to either zero or one atom in the trap. (c) Experimental time sequence for dephasing investigation. We depict the MOT magnetic field, compensatory and quantization magnetic fields as BMOT, Bcom and Bquan, respectively. See text for details.

Fig. 2
Fig. 2

Measurement of spin relaxation time. Single atoms is initialized in the state |F = 1〉. After a variable trapping time, the push-out technique is applied to determine the atomic state. The fraction in |F = 1〉 is recorded as a function of the trapping time. Each data is averaged over 200 single atoms. (i) Due to incoherent transitions induced by the background light of the dipole trap laser, the spin relaxation time is T1 = 0.087 ± 0.009 s. (ii) With an interference filter to reduce the background, the obtained time is T1 = 0.83 ± 0.11 s.

Fig. 3
Fig. 3

(a) Rabi oscillations between |F = 1, mF = 0〉 and |F = 2, mF = 0〉 states. The fraction of single atoms in the state F = 1 is measured as a function of Raman pulse length. We observe a sinusoidal variation with high contrast. The Rabi frequency is ΩR = 2π × 130 kHz. This value corresponds to a π/2 rotation time of 1.9 μs. (b) Ramsey spectroscopy recorded with a fixed two-photon detuning from the ground-state hyperfine splitting. The detuning is δ = 2π × 8.6 kHz. We fit the fringes according to the model presented in [14]. The measured dephasing time is T 2 * = 1.4 ± 0.1 ms.

Fig. 4
Fig. 4

Examples of the spin echo and CPMG signals. We plot fringes as a function of the time of second π/2-pulse. (a) For spin echo method, the time of π-pulse is fixed to be τ = 5.0 ms (i), 10.0 ms (ii). (b) For CPMG sequence with n = 2, the first π-pulse is applied at time τ = 5.0 ms (i), 10.0 ms (ii), and thus the second π/2-pulse is at time t = 4τ. These graphs are fitted using the model in [14] and in the text. The initial visibility decreases with increasing time τ.

Fig. 5
Fig. 5

Visibility of the spin echo and CPMG sequences as a function of the total time between two π/2-pulses. The values of spin echo (a) and CPMG sequences with pulse numbers n = 2 (b), 3 (c), 4 (d), 5 (e), and 6 (f) are extracted from the corresponding fringes similar to Fig. 4. We fit all the graphs with a Gaussian depicted in Eq. (10).

Fig. 6
Fig. 6

Dephasing time as a function of number of π-pulses, n. Spin echo or CPMG sequences are applied when n = 1 or n ≥ 2. The homogeneous dephasing time is extracted from the corresponding signals in Fig. 5. These data are shown in Table 1.

Tables (1)

Tables Icon

Table 1 Fit parameters extracted from the signals of spin echo and CPMG sequence of Fig. 5 using Eq. (10). The parameter n represents the number of π-pulses between two π/2-pulses. Spin echo is applied when n = 1.

Equations (10)

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ρ ( δ ls ) = 1 2 k B T ( 2 h ¯ Δ eff k B T ω hfs ) 2 ( δ ls δ 0 ) 2 exp [ 2 h ¯ Δ eff k B T ω hfs ( δ ls δ 0 ) ] ,
Φ π / 2 = ( 1 0 0 0 0 1 0 1 0 ) ,
Φ π = ( 1 0 0 0 1 0 0 0 1 ) ,
Ψ free ( δ , t ) = ( cos ( δ t ) sin ( δ t ) 0 sin ( δ t ) cos ( δ t ) 0 0 0 1 ) .
U CPMG ( t ) = Φ π / 2 × Ψ free ( δ , t ( 2 n 1 ) τ ) × Φ π × Ψ free ( δ , 2 τ ) × Φ π × × Ψ free ( δ , 2 τ ) × Φ π n 1 groups × Ψ free ( δ , τ ) × Φ π / 2 × U 0 .
w CPMG ( t ) = ( 1 ) n cos [ δ ( t 2 n τ ) ] .
w CPMG , inh ( t ) = ( 1 ) n 0 p ( δ l s ) cos [ ( δ set δ B δ ls ) ( t 2 n τ ) ] d δ ls = ( 1 ) n α ( t 2 n τ , T 2 * ) cos [ δ ( t 2 n τ ) + κ ( t 2 n τ , T 2 * ) ] ,
α ( t 2 n τ , T 2 * ) = [ 1 + 0.95 ( t 2 n τ ) 2 / T 2 * 2 ] 3 / 2 ,
κ ( t 2 n τ , T 2 * ) = 3 arctan [ 0.97 ( t 2 n τ ) / T 2 * ] ,
V ( t ) = C 0 exp [ ( t T 2 ) 2 ] .

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