Why the GSI anomaly cannot be explained by Quantum Beats
Alexander Merle ([email protected])
Max–Planck–Institut für Kernphysik Heidelberg ———————– AM: Why a splitting in the final state cannot explain the GSI-Oscillations, to appear soon H. Kienert, J. Kopp, M. Lindner, AM: The GSI anomaly, J. Phys. Conf. Ser. 136, 022049, 2008, arXiv:0808.2389
SFB Tr 27 Meeting, Project C1, Heidelberg, 2009
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 1 / 24 Outline
1 Introduction
2 Quantum Beats
3 One atom of type I
4 One atom of type II
5 Two atoms of type II
6 Conclusions
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 2 / 24 Outline
1 Introduction
2 Quantum Beats
3 One atom of type I
4 One atom of type II
5 Two atoms of type II
6 Conclusions
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 3 / 24 Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
The measurement at GSI
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 The measurement at GSI
Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 The measurement at GSI
Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 The measurement at GSI
Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 The measurement at GSI
Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 The measurement at GSI
Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 The measurement at GSI
Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 The measurement at GSI
Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079
measurement: lifetime of several H-like ions with respect to EC (electron capture) observation: cos-modulation superimposed on the exponential decay law oscillation frequency ∼ 7 sec ⇒ ∼ 10−15 eV!!! Ivanov, Kienle, Lipkin, et al.: ν-oscillations Giunti, we, et al.: reason unknown, but no ordinary ν-oscillations often mentioned in that context: Quantum Beats
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 4 / 24 Outline
1 Introduction
2 Quantum Beats
3 One atom of type I
4 One atom of type II
5 Two atoms of type II
6 Conclusions
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 5 / 24 Type I Type II
Type in “Quantum Beats” into Google!!
,→ This is what you find... But we’re interested in something else!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 6 / 24 Type I Type II
Type in “Quantum Beats” into Google!!
,→ This is what you find... But we’re interested in something else!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 6 / 24 Type in “Quantum Beats” into Google!!
Type I Type II ,→ This is what you find... But we’re interested in something else!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 6 / 24 Type in “Quantum Beats” into Google!!
Type I Type II ,→ This is what you find... But we’re interested in something else!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 6 / 24 Èa\ Èa\ Èb\
Ωab Ωac Ωac Ωbc
Èb\ Èc\ Èc\ Type I Type II
Quantum Beats (Chow et al., Phys. Rev. A, 11, 1380)
usual setting: superposition of three atomic states important: the states |ai, |bi, and |ci correspond to different energy eigenvalues ⇒ They are orthogonal! this orthogonality is not touched by the uncertainty relation BUT: an uncertainty allows for a coherent superposition
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 7 / 24 Quantum Beats (Chow et al., Phys. Rev. A, 11, 1380)
Èa\ Èa\ Èb\
Ωab Ωac Ωac Ωbc
Èb\ Èc\ Èc\ Type I Type II usual setting: superposition of three atomic states important: the states |ai, |bi, and |ci correspond to different energy eigenvalues ⇒ They are orthogonal! this orthogonality is not touched by the uncertainty relation BUT: an uncertainty allows for a coherent superposition
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 7 / 24 Quantum Beats (Chow et al., Phys. Rev. A, 11, 1380)
Èa\ Èa\ Èb\
Ωab Ωac Ωac Ωbc
Èb\ Èc\ Èc\ Type I Type II usual setting: superposition of three atomic states important: the states |ai, |bi, and |ci correspond to different energy eigenvalues ⇒ They are orthogonal! this orthogonality is not touched by the uncertainty relation BUT: an uncertainty allows for a coherent superposition
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 7 / 24 Quantum Beats (Chow et al., Phys. Rev. A, 11, 1380)
Èa\ Èa\ Èb\
Ωab Ωac Ωac Ωbc
Èb\ Èc\ Èc\ Type I Type II usual setting: superposition of three atomic states important: the states |ai, |bi, and |ci correspond to different energy eigenvalues ⇒ They are orthogonal! this orthogonality is not touched by the uncertainty relation BUT: an uncertainty allows for a coherent superposition
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 7 / 24 Quantum Beats (Chow et al., Phys. Rev. A, 11, 1380)
Èa\ Èa\ Èb\
Ωab Ωac Ωac Ωbc
Èb\ Èc\ Èc\ Type I Type II usual setting: superposition of three atomic states important: the states |ai, |bi, and |ci correspond to different energy eigenvalues ⇒ They are orthogonal! this orthogonality is not touched by the uncertainty relation BUT: an uncertainty allows for a coherent superposition
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 7 / 24 Quantum Beats (Chow et al., Phys. Rev. A, 11, 1380)
Èa\ Èa\ Èb\
Ωab Ωac Ωac Ωbc
Èb\ Èc\ Èc\ Type I Type II usual setting: superposition of three atomic states important: the states |ai, |bi, and |ci correspond to different energy eigenvalues ⇒ They are orthogonal! this orthogonality is not touched by the uncertainty relation BUT: an uncertainty allows for a coherent superposition
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 7 / 24 Quantum Beats (Chow et al., Phys. Rev. A, 11, 1380)
Èa\ Èa\ Èb\
Ωab Ωac Ωac Ωbc
Èb\ Èc\ Èc\ Type I Type II usual setting: superposition of three atomic states important: the states |ai, |bi, and |ci correspond to different energy eigenvalues ⇒ They are orthogonal! this orthogonality is not touched by the uncertainty relation BUT: an uncertainty allows for a coherent superposition
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 7 / 24 Outline
1 Introduction
2 Quantum Beats
3 One atom of type I
4 One atom of type II
5 Two atoms of type II
6 Conclusions
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 8 / 24 Èa\ Èb\
Ωac Ωbc
Èc\
Single atom of type I
superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
Èa\ Èb\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ → |A |2 + |B |2 + |C |2 = Ωac Ωbc with: 0 0 0 1 no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower state gets Èc\ populated by photon emission
corresponding state |Ψ(t)i:
A(t)|ai|0iγ+B(t)|bi|0iγ+C(t)|ci|0iγ+C1(t)|ci|1aciγ+C2(t)|ci|1bciγ † † 1-photon state: |1x iγ = a |0iγ, with [a~ , a ] = δ~ ~ 0 δλ,λ0 x k,λ ~k 0,λ0 k,k ,→ ~k: momentum, λ: polarization
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 9 / 24 Single atom of type I
~ P −ikx † +ikx electric field operator: E(~x, t) = ~ ~ a~ e + a e k,λ k,λ k,λ ~k,λ radiated photon intensity:
I ∝ hΨ(t)|E~ 2(~0, t)|Ψ(t)i
effectively:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + bc(1 + 2abcabc) + † i∆t † −i∆t +2acbc(aacabce + abcaace )
† † already used: e.g. γh0| aac aac|...iγ = 0 |{z} 0← photon energy difference: ∆ = ωac − ωbc
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 10 / 24 Single atom of type I
~ P −ikx † +ikx electric field operator: E(~x, t) = ~ ~ a~ e + a e k,λ k,λ k,λ ~k,λ radiated photon intensity:
I ∝ hΨ(t)|E~ 2(~0, t)|Ψ(t)i
effectively:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + bc(1 + 2abcabc) + † i∆t † −i∆t +2acbc(aacabce + abcaace )
† † already used: e.g. γh0| aac aac|...iγ = 0 |{z} 0← photon energy difference: ∆ = ωac − ωbc
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 10 / 24 Single atom of type I
~ P −ikx † +ikx electric field operator: E(~x, t) = ~ ~ a~ e + a e k,λ k,λ k,λ ~k,λ radiated photon intensity:
I ∝ hΨ(t)|E~ 2(~0, t)|Ψ(t)i
effectively:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + bc(1 + 2abcabc) + † i∆t † −i∆t +2acbc(aacabce + abcaace )
† † already used: e.g. γh0| aac aac|...iγ = 0 |{z} 0← photon energy difference: ∆ = ωac − ωbc
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 10 / 24 Single atom of type I
~ P −ikx † +ikx electric field operator: E(~x, t) = ~ ~ a~ e + a e k,λ k,λ k,λ ~k,λ radiated photon intensity:
I ∝ hΨ(t)|E~ 2(~0, t)|Ψ(t)i
effectively:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + bc(1 + 2abcabc) + † i∆t † −i∆t +2acbc(aacabce + abcaace )
† † already used: e.g. γh0| aac aac|...iγ = 0 |{z} 0← photon energy difference: ∆ = ωac − ωbc
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 10 / 24 Single atom of type I
~ P −ikx † +ikx electric field operator: E(~x, t) = ~ ~ a~ e + a e k,λ k,λ k,λ ~k,λ radiated photon intensity:
I ∝ hΨ(t)|E~ 2(~0, t)|Ψ(t)i
effectively:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + bc(1 + 2abcabc) + † i∆t † −i∆t +2acbc(aacabce + abcaace )
† † already used: e.g. γh0| aac aac|...iγ = 0 |{z} 0← photon energy difference: ∆ = ωac − ωbc
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 10 / 24 Single atom of type I
~ P −ikx † +ikx electric field operator: E(~x, t) = ~ ~ a~ e + a e k,λ k,λ k,λ ~k,λ radiated photon intensity:
I ∝ hΨ(t)|E~ 2(~0, t)|Ψ(t)i
effectively:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + bc(1 + 2abcabc) + † i∆t † −i∆t +2acbc(aacabce + abcaace )
† † already used: e.g. γh0| aac aac|...iγ = 0 |{z} 0← photon energy difference: ∆ = ωac − ωbc
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 10 / 24 Single atom of type I
∗ i∆t there are oscillating terms in the intensity I, e.g. C1(t)C2(t)e oscillates and is proportional to:
† † † † γh0|aacaacabcabc|0iγ = γh0|(1 + aac aac)(1 + abc abc )|0iγ = 1 |{z} |{z} 0← →0
⇒ Quantum Beats! intuitively: both upper level decay into the same final state via photon emission, but the energy of the photon cannot be determined by measuring it directly → interference terms
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 11 / 24 Single atom of type I
∗ i∆t there are oscillating terms in the intensity I, e.g. C1(t)C2(t)e oscillates and is proportional to:
† † † † γh0|aacaacabcabc|0iγ = γh0|(1 + aac aac)(1 + abc abc )|0iγ = 1 |{z} |{z} 0← →0
⇒ Quantum Beats! intuitively: both upper level decay into the same final state via photon emission, but the energy of the photon cannot be determined by measuring it directly → interference terms
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 11 / 24 Single atom of type I
∗ i∆t there are oscillating terms in the intensity I, e.g. C1(t)C2(t)e oscillates and is proportional to:
† † † † γh0|aacaacabcabc|0iγ = γh0|(1 + aac aac)(1 + abc abc )|0iγ = 1 |{z} |{z} 0← →0
⇒ Quantum Beats! intuitively: both upper level decay into the same final state via photon emission, but the energy of the photon cannot be determined by measuring it directly → interference terms
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 11 / 24 Single atom of type I
∗ i∆t there are oscillating terms in the intensity I, e.g. C1(t)C2(t)e oscillates and is proportional to:
† † † † γh0|aacaacabcabc|0iγ = γh0|(1 + aac aac)(1 + abc abc )|0iγ = 1 |{z} |{z} 0← →0
⇒ Quantum Beats! intuitively: both upper level decay into the same final state via photon emission, but the energy of the photon cannot be determined by measuring it directly → interference terms
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 11 / 24 Single atom of type I ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino if there is a splitting in the initial state, this can cause oscillations in the decay rate HOWEVER: splitting ∼ 10−15 eV ⇒ tiny, not at all explained... furthermore: preliminary data on β+-decays show no oscillatory behavior (Ivanov et a., 0905.1904) ⇒ if such a splitting were present, it could only be there in the electron energy level, but not in the nucleus
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 12 / 24 Single atom of type I ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino if there is a splitting in the initial state, this can cause oscillations in the decay rate HOWEVER: splitting ∼ 10−15 eV ⇒ tiny, not at all explained... furthermore: preliminary data on β+-decays show no oscillatory behavior (Ivanov et a., 0905.1904) ⇒ if such a splitting were present, it could only be there in the electron energy level, but not in the nucleus
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 12 / 24 Single atom of type I ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino if there is a splitting in the initial state, this can cause oscillations in the decay rate HOWEVER: splitting ∼ 10−15 eV ⇒ tiny, not at all explained... furthermore: preliminary data on β+-decays show no oscillatory behavior (Ivanov et a., 0905.1904) ⇒ if such a splitting were present, it could only be there in the electron energy level, but not in the nucleus
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 12 / 24 Single atom of type I ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino if there is a splitting in the initial state, this can cause oscillations in the decay rate HOWEVER: splitting ∼ 10−15 eV ⇒ tiny, not at all explained... furthermore: preliminary data on β+-decays show no oscillatory behavior (Ivanov et a., 0905.1904) ⇒ if such a splitting were present, it could only be there in the electron energy level, but not in the nucleus
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 12 / 24 Single atom of type I ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino if there is a splitting in the initial state, this can cause oscillations in the decay rate HOWEVER: splitting ∼ 10−15 eV ⇒ tiny, not at all explained... furthermore: preliminary data on β+-decays show no oscillatory behavior (Ivanov et a., 0905.1904) ⇒ if such a splitting were present, it could only be there in the electron energy level, but not in the nucleus
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 12 / 24 Single atom of type I ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino if there is a splitting in the initial state, this can cause oscillations in the decay rate HOWEVER: splitting ∼ 10−15 eV ⇒ tiny, not at all explained... furthermore: preliminary data on β+-decays show no oscillatory behavior (Ivanov et a., 0905.1904) ⇒ if such a splitting were present, it could only be there in the electron energy level, but not in the nucleus
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 12 / 24 Outline
1 Introduction
2 Quantum Beats
3 One atom of type I
4 One atom of type II
5 Two atoms of type II
6 Conclusions
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 13 / 24 Èa\
Ωab Ωac
Èb\ Èc\
Single atom of type II
superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A0|ai|0iγ + B0|bi|0iγ + C0|ci|0iγ 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower states get populated by photon emission
corresponding state |Ψ(t)i:
0 0 A(t)|ai|0iγ +B(t)|bi|0iγ +C(t)|ci|0iγ +B (t)|bi|1abiγ +C (t)|ci|1aciγ
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 14 / 24 Single atom of type II
Èa\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A |ai|0iγ + B |bi|0iγ + C |ci|0iγ Ω Ω 0 0 0 ab ac 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower states get Èb\ Èc\ populated by photon emission
corresponding state |Ψ(t)i:
0 0 A(t)|ai|0iγ +B(t)|bi|0iγ +C(t)|ci|0iγ +B (t)|bi|1abiγ +C (t)|ci|1aciγ
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 14 / 24 Single atom of type II
Èa\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A |ai|0iγ + B |bi|0iγ + C |ci|0iγ Ω Ω 0 0 0 ab ac 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower states get Èb\ Èc\ populated by photon emission
corresponding state |Ψ(t)i:
0 0 A(t)|ai|0iγ +B(t)|bi|0iγ +C(t)|ci|0iγ +B (t)|bi|1abiγ +C (t)|ci|1aciγ
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 14 / 24 Single atom of type II
Èa\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A |ai|0iγ + B |bi|0iγ + C |ci|0iγ Ω Ω 0 0 0 ab ac 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower states get Èb\ Èc\ populated by photon emission
corresponding state |Ψ(t)i:
0 0 A(t)|ai|0iγ +B(t)|bi|0iγ +C(t)|ci|0iγ +B (t)|bi|1abiγ +C (t)|ci|1aciγ
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 14 / 24 Single atom of type II
Èa\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A |ai|0iγ + B |bi|0iγ + C |ci|0iγ Ω Ω 0 0 0 ab ac 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower states get Èb\ Èc\ populated by photon emission
corresponding state |Ψ(t)i:
0 0 A(t)|ai|0iγ +B(t)|bi|0iγ +C(t)|ci|0iγ +B (t)|bi|1abiγ +C (t)|ci|1aciγ
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 14 / 24 Single atom of type II
Èa\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A |ai|0iγ + B |bi|0iγ + C |ci|0iγ Ω Ω 0 0 0 ab ac 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower states get Èb\ Èc\ populated by photon emission
corresponding state |Ψ(t)i:
0 0 A(t)|ai|0iγ +B(t)|bi|0iγ +C(t)|ci|0iγ +B (t)|bi|1abiγ +C (t)|ci|1aciγ
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 14 / 24 Single atom of type II
Èa\ superposition of |ai, |bi, and |ci initially: |Ψ(0)i = A |ai|0iγ + B |bi|0iγ + C |ci|0iγ Ω Ω 0 0 0 ab ac 2 2 2 → with: |A0| + |B0| + |C0| = 1
no photons emitted yet → vacuum |0iγ time-evolution ⇒ lower states get Èb\ Èc\ populated by photon emission
corresponding state |Ψ(t)i:
0 0 A(t)|ai|0iγ +B(t)|bi|0iγ +C(t)|ci|0iγ +B (t)|bi|1abiγ +C (t)|ci|1aciγ
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 14 / 24 Single atom of type II
now we have:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + ab(1 + 2aabaab) + † i∆t † −i∆t +2acab(aacaabe + aabaace ) possibilities: 0-photon state with 0-photon state: e.g. |A(t)|2e±i∆t † † ∝ γh0|aac aab |0iγ = 0 or ∝ γh0|aab aac |0iγ = 0 |{z} |{z} →0 →0 1-photon state with 1-photon state: e.g. |B0(t)|2e±i∆t † † † ∝ γh1ab|aacaab|1abiγ = γh0|aab aac aabaab|0iγ = 0 or |{z} 0← † † † ∝ γh1ab|aabaac|1abiγ = γh0|aabaab aac aab|0iγ = 0 |{z} →0
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 15 / 24 Single atom of type II
now we have:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + ab(1 + 2aabaab) + † i∆t † −i∆t +2acab(aacaabe + aabaace ) possibilities: 0-photon state with 0-photon state: e.g. |A(t)|2e±i∆t † † ∝ γh0|aac aab |0iγ = 0 or ∝ γh0|aab aac |0iγ = 0 |{z} |{z} →0 →0 1-photon state with 1-photon state: e.g. |B0(t)|2e±i∆t † † † ∝ γh1ab|aacaab|1abiγ = γh0|aab aac aabaab|0iγ = 0 or |{z} 0← † † † ∝ γh1ab|aabaac|1abiγ = γh0|aabaab aac aab|0iγ = 0 |{z} →0
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 15 / 24 Single atom of type II
now we have:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + ab(1 + 2aabaab) + † i∆t † −i∆t +2acab(aacaabe + aabaace ) possibilities: 0-photon state with 0-photon state: e.g. |A(t)|2e±i∆t † † ∝ γh0|aac aab |0iγ = 0 or ∝ γh0|aab aac |0iγ = 0 |{z} |{z} →0 →0 1-photon state with 1-photon state: e.g. |B0(t)|2e±i∆t † † † ∝ γh1ab|aacaab|1abiγ = γh0|aab aac aabaab|0iγ = 0 or |{z} 0← † † † ∝ γh1ab|aabaac|1abiγ = γh0|aabaab aac aab|0iγ = 0 |{z} →0
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 15 / 24 Single atom of type II
now we have:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + ab(1 + 2aabaab) + † i∆t † −i∆t +2acab(aacaabe + aabaace ) possibilities: 0-photon state with 0-photon state: e.g. |A(t)|2e±i∆t † † ∝ γh0|aac aab |0iγ = 0 or ∝ γh0|aab aac |0iγ = 0 |{z} |{z} →0 →0 1-photon state with 1-photon state: e.g. |B0(t)|2e±i∆t † † † ∝ γh1ab|aacaab|1abiγ = γh0|aab aac aabaab|0iγ = 0 or |{z} 0← † † † ∝ γh1ab|aabaac|1abiγ = γh0|aabaab aac aab|0iγ = 0 |{z} →0
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 15 / 24 Single atom of type II
now we have:
~ 2 ~ 2 † 2 † E (0, t) = ac(1 + 2aacaac) + ab(1 + 2aabaab) + † i∆t † −i∆t +2acab(aacaabe + aabaace ) possibilities: 0-photon state with 0-photon state: e.g. |A(t)|2e±i∆t † † ∝ γh0|aac aab |0iγ = 0 or ∝ γh0|aab aac |0iγ = 0 |{z} |{z} →0 →0 1-photon state with 1-photon state: e.g. |B0(t)|2e±i∆t † † † ∝ γh1ab|aacaab|1abiγ = γh0|aab aac aabaab|0iγ = 0 or |{z} 0← † † † ∝ γh1ab|aabaac|1abiγ = γh0|aabaab aac aab|0iγ = 0 |{z} →0
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 15 / 24 Single atom of type II
0-photon state with 1-photon state: e.g. B(t)∗B0(t)
† ∝ γh0|1abiγ = γh0| aab |0iγ = 0 or |{z} 0← † † ∝ γh0| aac aac|1abiγ = 0 or ∝ γh0| aab aab|1abiγ = 0 or |{z} |{z} 0← 0← † † ∝ γh0| aac aab|1abiγ = 0 or ∝ γh0| aab aac|1abiγ = 0. |{z} |{z} 0← 0←
⇒ No oscillatory terms left! ⇒ No Quantum Beats!
Intuitively: By waiting long enough, one could determine the photon’s energy by measuring the atomic final state. ⇒ No interferences expected!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 16 / 24 Single atom of type II
0-photon state with 1-photon state: e.g. B(t)∗B0(t)
† ∝ γh0|1abiγ = γh0| aab |0iγ = 0 or |{z} 0← † † ∝ γh0| aac aac|1abiγ = 0 or ∝ γh0| aab aab|1abiγ = 0 or |{z} |{z} 0← 0← † † ∝ γh0| aac aab|1abiγ = 0 or ∝ γh0| aab aac|1abiγ = 0. |{z} |{z} 0← 0←
⇒ No oscillatory terms left! ⇒ No Quantum Beats!
Intuitively: By waiting long enough, one could determine the photon’s energy by measuring the atomic final state. ⇒ No interferences expected!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 16 / 24 Single atom of type II
0-photon state with 1-photon state: e.g. B(t)∗B0(t)
† ∝ γh0|1abiγ = γh0| aab |0iγ = 0 or |{z} 0← † † ∝ γh0| aac aac|1abiγ = 0 or ∝ γh0| aab aab|1abiγ = 0 or |{z} |{z} 0← 0← † † ∝ γh0| aac aab|1abiγ = 0 or ∝ γh0| aab aac|1abiγ = 0. |{z} |{z} 0← 0←
⇒ No oscillatory terms left! ⇒ No Quantum Beats!
Intuitively: By waiting long enough, one could determine the photon’s energy by measuring the atomic final state. ⇒ No interferences expected!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 16 / 24 Single atom of type II
0-photon state with 1-photon state: e.g. B(t)∗B0(t)
† ∝ γh0|1abiγ = γh0| aab |0iγ = 0 or |{z} 0← † † ∝ γh0| aac aac|1abiγ = 0 or ∝ γh0| aab aab|1abiγ = 0 or |{z} |{z} 0← 0← † † ∝ γh0| aac aab|1abiγ = 0 or ∝ γh0| aab aac|1abiγ = 0. |{z} |{z} 0← 0←
⇒ No oscillatory terms left! ⇒ No Quantum Beats!
Intuitively: By waiting long enough, one could determine the photon’s energy by measuring the atomic final state. ⇒ No interferences expected!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 16 / 24 Single atom of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → neutrino, photon → ion (!!!) the neutrino is not expected to interact before losing its coherence 19 40 (estimates: Lcoh . 10 m, mean free path ∼ 10 m in the Galaxy) ones the neutrino interacts, there is no principle problem to determine its mass (e.g., by exploiting the spatial separation of the wave packet components belonging to different mass eigenstates) ⇒ accordingly, no Quantum Beats are expected
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 17 / 24 Single atom of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → neutrino, photon → ion (!!!) the neutrino is not expected to interact before losing its coherence 19 40 (estimates: Lcoh . 10 m, mean free path ∼ 10 m in the Galaxy) ones the neutrino interacts, there is no principle problem to determine its mass (e.g., by exploiting the spatial separation of the wave packet components belonging to different mass eigenstates) ⇒ accordingly, no Quantum Beats are expected
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 17 / 24 Single atom of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → neutrino, photon → ion (!!!) the neutrino is not expected to interact before losing its coherence 19 40 (estimates: Lcoh . 10 m, mean free path ∼ 10 m in the Galaxy) ones the neutrino interacts, there is no principle problem to determine its mass (e.g., by exploiting the spatial separation of the wave packet components belonging to different mass eigenstates) ⇒ accordingly, no Quantum Beats are expected
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 17 / 24 Single atom of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → neutrino, photon → ion (!!!) the neutrino is not expected to interact before losing its coherence 19 40 (estimates: Lcoh . 10 m, mean free path ∼ 10 m in the Galaxy) ones the neutrino interacts, there is no principle problem to determine its mass (e.g., by exploiting the spatial separation of the wave packet components belonging to different mass eigenstates) ⇒ accordingly, no Quantum Beats are expected
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 17 / 24 Single atom of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → neutrino, photon → ion (!!!) the neutrino is not expected to interact before losing its coherence 19 40 (estimates: Lcoh . 10 m, mean free path ∼ 10 m in the Galaxy) ones the neutrino interacts, there is no principle problem to determine its mass (e.g., by exploiting the spatial separation of the wave packet components belonging to different mass eigenstates) ⇒ accordingly, no Quantum Beats are expected
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 17 / 24 Outline
1 Introduction
2 Quantum Beats
3 One atom of type I
4 One atom of type II
5 Two atoms of type II
6 Conclusions
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 18 / 24 Two atoms of type II if the spatial separation of the two atoms is smaller than the wavelength of the photon, one can write down a combined state:
|Ψ(0)i = A0|ai1|ai2|0iγ + B0|bi1|bi2|0iγ + C0|ci1|ci2|0iγ +
+D1,0|ai1|bi2|0iγ + D2,0|bi1|ai2|0iγ + E1,0|ai1|ci2|0iγ +
+E2,0|ci1|ai2|0iγ + F1,0|bi1|ci2|0iγ + F2,0|ci1|bi2|0iγ time evolution |Ψ(t)i (trivial, but lengthy):
A(t)|ai1|ai2|0iγ + B(t)|bi1|bi2|0iγ + C(t)|ci1|ci2|0iγ +
+D1(t)|ai1|bi2|0iγ + D2(t)|bi1|ai2|0iγ + E1(t)|ai1|ci2|0iγ +
+E2(t)|ci1|ai2|0iγ + F1(t)|bi1|ci2|0iγ + F2(t)|ci1|bi2|0iγ +
+G1(t)|bi1|ai2|1abiγ + G2(t)|ai1|bi2|1abiγ +
+H1(t)|ci1|ai2|1aciγ + H2(t)|ai1|ci2|1aciγ +
+I1(t)|bi1|bi2|1abiγ + I2(t)|ci1|ci2|1aciγ +
+J1(t)|bi1|ci2|1abiγ + J2(t)|ci1|bi2|1abiγ +
+K1(t)|bi1|ci2|1aciγ + K2(t)|ci1|bi2|1aciγ A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 19 / 24 Two atoms of type II if the spatial separation of the two atoms is smaller than the wavelength of the photon, one can write down a combined state:
|Ψ(0)i = A0|ai1|ai2|0iγ + B0|bi1|bi2|0iγ + C0|ci1|ci2|0iγ +
+D1,0|ai1|bi2|0iγ + D2,0|bi1|ai2|0iγ + E1,0|ai1|ci2|0iγ +
+E2,0|ci1|ai2|0iγ + F1,0|bi1|ci2|0iγ + F2,0|ci1|bi2|0iγ time evolution |Ψ(t)i (trivial, but lengthy):
A(t)|ai1|ai2|0iγ + B(t)|bi1|bi2|0iγ + C(t)|ci1|ci2|0iγ +
+D1(t)|ai1|bi2|0iγ + D2(t)|bi1|ai2|0iγ + E1(t)|ai1|ci2|0iγ +
+E2(t)|ci1|ai2|0iγ + F1(t)|bi1|ci2|0iγ + F2(t)|ci1|bi2|0iγ +
+G1(t)|bi1|ai2|1abiγ + G2(t)|ai1|bi2|1abiγ +
+H1(t)|ci1|ai2|1aciγ + H2(t)|ai1|ci2|1aciγ +
+I1(t)|bi1|bi2|1abiγ + I2(t)|ci1|ci2|1aciγ +
+J1(t)|bi1|ci2|1abiγ + J2(t)|ci1|bi2|1abiγ +
+K1(t)|bi1|ci2|1aciγ + K2(t)|ci1|bi2|1aciγ A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 19 / 24 Two atoms of type II if the spatial separation of the two atoms is smaller than the wavelength of the photon, one can write down a combined state:
|Ψ(0)i = A0|ai1|ai2|0iγ + B0|bi1|bi2|0iγ + C0|ci1|ci2|0iγ +
+D1,0|ai1|bi2|0iγ + D2,0|bi1|ai2|0iγ + E1,0|ai1|ci2|0iγ +
+E2,0|ci1|ai2|0iγ + F1,0|bi1|ci2|0iγ + F2,0|ci1|bi2|0iγ time evolution |Ψ(t)i (trivial, but lengthy):
A(t)|ai1|ai2|0iγ + B(t)|bi1|bi2|0iγ + C(t)|ci1|ci2|0iγ +
+D1(t)|ai1|bi2|0iγ + D2(t)|bi1|ai2|0iγ + E1(t)|ai1|ci2|0iγ +
+E2(t)|ci1|ai2|0iγ + F1(t)|bi1|ci2|0iγ + F2(t)|ci1|bi2|0iγ +
+G1(t)|bi1|ai2|1abiγ + G2(t)|ai1|bi2|1abiγ +
+H1(t)|ci1|ai2|1aciγ + H2(t)|ai1|ci2|1aciγ +
+I1(t)|bi1|bi2|1abiγ + I2(t)|ci1|ci2|1aciγ +
+J1(t)|bi1|ci2|1abiγ + J2(t)|ci1|bi2|1abiγ +
+K1(t)|bi1|ci2|1aciγ + K2(t)|ci1|bi2|1aciγ A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 19 / 24 Two atoms of type II
∗ −i∆t there can indeed be oscillatory terms, e.g. J1 K1e , which is proportional to:
† † † γh1ab|aabaac|1aciγ = γh0|aabaabaacaac|0iγ = † † = γh0|(1 + aab aab)(1 + aac aac )|0iγ = γh0|0iγ = 1 |{z} |{z} 0← →0
intuitively: If the distance of the atoms is less then the photon’s wavelength, one cannot determine, which atom has emitted the radiation. ⇒ No way to measure the photon’s frequency! ⇒ Quantum Beats!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 20 / 24 Two atoms of type II
∗ −i∆t there can indeed be oscillatory terms, e.g. J1 K1e , which is proportional to:
† † † γh1ab|aabaac|1aciγ = γh0|aabaabaacaac|0iγ = † † = γh0|(1 + aab aab)(1 + aac aac )|0iγ = γh0|0iγ = 1 |{z} |{z} 0← →0
intuitively: If the distance of the atoms is less then the photon’s wavelength, one cannot determine, which atom has emitted the radiation. ⇒ No way to measure the photon’s frequency! ⇒ Quantum Beats!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 20 / 24 Two atoms of type II
∗ −i∆t there can indeed be oscillatory terms, e.g. J1 K1e , which is proportional to:
† † † γh1ab|aabaac|1aciγ = γh0|aabaabaacaac|0iγ = † † = γh0|(1 + aab aab)(1 + aac aac )|0iγ = γh0|0iγ = 1 |{z} |{z} 0← →0
intuitively: If the distance of the atoms is less then the photon’s wavelength, one cannot determine, which atom has emitted the radiation. ⇒ No way to measure the photon’s frequency! ⇒ Quantum Beats!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 20 / 24 Two atoms of type II
∗ −i∆t there can indeed be oscillatory terms, e.g. J1 K1e , which is proportional to:
† † † γh1ab|aabaac|1aciγ = γh0|aabaabaacaac|0iγ = † † = γh0|(1 + aab aab)(1 + aac aac )|0iγ = γh0|0iγ = 1 |{z} |{z} 0← →0
intuitively: If the distance of the atoms is less then the photon’s wavelength, one cannot determine, which atom has emitted the radiation. ⇒ No way to measure the photon’s frequency! ⇒ Quantum Beats!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 20 / 24 Two atoms of type II
∗ −i∆t there can indeed be oscillatory terms, e.g. J1 K1e , which is proportional to:
† † † γh1ab|aabaac|1aciγ = γh0|aabaabaacaac|0iγ = † † = γh0|(1 + aab aab)(1 + aac aac )|0iγ = γh0|0iγ = 1 |{z} |{z} 0← →0
intuitively: If the distance of the atoms is less then the photon’s wavelength, one cannot determine, which atom has emitted the radiation. ⇒ No way to measure the photon’s frequency! ⇒ Quantum Beats!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 20 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Two atoms of type II ↔ GSI-oscillations
Relation to the GSI-experiment: atom → ion, photon → neutrino even in runs where there was only one EC-decay, there might have been more ions in the ring → this possibility has to be considered! the wavelength has to be replaced by the de Broglie wavelength of the neutrino Y. A. Litvinov et al., Phys. Lett. B664, 162 (2008), arXiv:0801.2079 ⇒ Q-value ∼ 1 MeV then: E ∼ 1 MeV ⇒ λ ≈ 2π~c ∼ 10−12 m ν ν Eν c M. Steck et al., Phys. Rev. Lett., 77, 3803 ⇒ typical separation of the ions in the ring ∼ size of the ring ∼ 100 m ⇒ This possibility is excluded for the GSI-experiment!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 21 / 24 Outline
1 Introduction
2 Quantum Beats
3 One atom of type I
4 One atom of type II
5 Two atoms of type II
6 Conclusions
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 22 / 24 → If you have any idea... Phone: +49/6221/516-817 E-Mail: [email protected]
Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing...
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 → If you have any idea... Phone: +49/6221/516-817 E-Mail: [email protected]
Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing...
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 → If you have any idea... Phone: +49/6221/516-817 E-Mail: [email protected]
Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing...
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 → If you have any idea... Phone: +49/6221/516-817 E-Mail: [email protected]
Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing...
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 → If you have any idea... Phone: +49/6221/516-817 E-Mail: [email protected]
Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing...
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 → If you have any idea... Phone: +49/6221/516-817 E-Mail: [email protected]
Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing...
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 Phone: +49/6221/516-817 E-Mail: [email protected]
Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing... → If you have any idea...
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 Conclusions
in principle, Quantum Beats seem to be a tempting possibility for an explanation for the GSI anomaly a single type I model would work, but has its problems a single type II model is claimed to be the solution by some authors, but actually it does not work a double type II model might be okay, but the numbers are wrong by orders a satisfying explanation is still missing... → If you have any idea... Phone: +49/6221/516-817 E-Mail: [email protected]
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 23 / 24 The End
THANK YOU!!!
A. Merle (MPIK) GSI anomaly & Quantum Beats 09.07.2009 24 / 24