Abstract
Ahokas et al.
[1] measured the hyperfine frequency shifts in three-dimensional spin-polarized atomic hydrogen by means of ESR. In this Comment, we address their analysis of the interaction energy of the ground-state H atoms in different hyperfine states and show that the quoted difference ∆a = a t − a s between the triplet and singlet scattering lengths derived from the correctly measured shifts is overestimated by a factor of two.
Ahokas et al. observed the transitions a → d and b → c in the presence of the third-state atoms (b and a, respectively) and found the shifts of the corresponding resonance fields to be ∆B ad = C ab n b + C aa n a and ∆B bc = C bb n b + C ba n a with C ab ≈ C ba = 8(2) × 10 −19 cm 3 and C aa , C bb ≪ C ba . To explain this observation, Ahokas et al. expressed the spin states of a pair of atoms in the |S, m S , I, m I basis, that is, in terms of electron and nuclear singlets and triplets. In particular, in the highfield limit, |ab = (|e t , n t + |e t , n s ), |ac = |bd = 1 2 (|e t , n t + |e t , n s + |e s , n t + |e s , n s ). At low temperature, only the symmetric components |e t , n t and |e s , n s contribute to (s-wave) scattering via the triplet V t and singlet V s interatomic potential, respectively. More specifically, the interaction energy of H atoms in different hyperfine states α and β (α, β = a, b, c, d) is given by the second-quantization Hamiltonian (i, j = α, β; i = j) [2]
The wavefunction of two bosons must be symmetric,
where the spatial and spin parts are, respectively,
(|ij ± |ji ). The use of the symmetric form (2) of the diatomic wavefunction instead of simply |ki, qj = ψ k (r 1 )ψ q (r 2 )|ij does not change the sum (1) because the bosonic creation (annihilation) operatorsâ + ki andâ + qj (â ki andâ qj ) with i = j obviously commute. The interaction strength λ of the pseudopotential V (r 2 − r 1 ) = λδ(r 2 − r 1 ) has the eigenvalues λ s or λ t corresponding to the spin states |e s or |e t of the atomic pair. As such, the potential is nearly independent of nuclear spins: e t n s |λ|e t n s = e t n t |λ|e t n t = λ t and e s n t |λ|e s n t = e s n s |λ|e s n s = λ s (here we write only the spin parts of the matrix elements). Instead, Ahokas et al. used e t n s |λ|e t n s = e s n t |λ|e s n t = 0 arguing that the antisymmetric states do not scatter via s-waves. This zeroing is only valid if the matrix elements include the vanishing spatial factor R − |δ(r 2 − r 1 )|R − . Actually, it is this spatial factor, which cancels the contribution of the antisymmetric states to s-wave scattering and to the interaction energy (1). On the contrary, the spatial part of the matrix elements for the symmetric states R + |ij + is doubled. In other words, the atoms of a heterostate symmetric pair behave as identical.
Clearly, |ab + = |e t n t,0 and |ac
(|e t,0 n t,0 +|e s n s ). Consequently, λ
Then, according to Eqs. (1), (2), the interaction energies are E ab = λ t n a n b ,
in agreement with our previous result [3] . Note, that Ahokas et al. consider the atoms of a symmetric heterostate pair as distinguishable and therefore lose a factor of two in the expressions for the shifts. From Eqs. (3) and (4) we find C ba = C ab = γem (a t − a s ) and ∆a = 30(5) pm. This is somewhat lower than the theoretical values ∆a = 42 − 55 pm for the difference in the scattering lengths [4] [5] [6] .

Ahokas et al. [1] measured the hyperfine frequency shifts in three-dimensional spin-polarized atomic hydrogen by means of ESR. In this Comment, we address their analysis of the interaction energy of the ground-state H atoms in different hyperfine states and show that the quoted difference ∆a = a t − a s between the triplet and singlet scattering lengths derived from the correctly measured shifts is overestimated by a factor of two.
Ahokas et al. observed the transitions a → d and b → c in the presence of the third-state atoms (b and a, respectively) and found the shifts of the corresponding resonance fields to be ∆B ad = C ab n b + C aa n a and ∆B bc = C bb n b + C ba n a with C ab ≈ C ba = 8(2) × 10 −19 cm 3 and C aa , C bb ≪ C ba . To explain this observation, Ahokas et al. expressed the spin states of a pair of atoms in the |S, m S , I, m I basis, that is, in terms of electron and nuclear singlets and triplets. In particular, in the highfield limit, |ab = (|e t , n t + |e t , n s ), |ac = |bd = 1 2 (|e t , n t + |e t , n s + |e s , n t + |e s , n s ). At low temperature, only the symmetric components |e t , n t and |e s , n s contribute to (s-wave) scattering via the triplet V t and singlet V s interatomic potential, respectively. More specifically, the interaction energy of H atoms in different hyperfine states α and β (α, β = a, b, c, d) is given by the second-quantization Hamiltonian (i, j = α, β; i = j) [2] 
The wavefunction of two bosons must be symmetric,
where the spatial and spin parts are, respectively,
(|ij ± |ji ). The use of the symmetric form (2) of the diatomic wavefunction instead of simply |ki, qj = ψ k (r 1 )ψ q (r 2 )|ij does not change the sum (1) because the bosonic creation (annihilation) operatorsâ + ki andâ + qj (â ki andâ qj ) with i = j obviously commute. The interaction strength λ of the pseudopotential V (r 2 − r 1 ) = λδ(r 2 − r 1 ) has the eigenvalues λ s or λ t corresponding to the spin states |e s or |e t of the atomic pair. As such, the potential is nearly independent of nuclear spins: e t n s |λ|e t n s = e t n t |λ|e t n t = λ t and e s n t |λ|e s n t = e s n s |λ|e s n s = λ s (here we write only the spin parts of the matrix elements). Instead, Ahokas et al. used e t n s |λ|e t n s = e s n t |λ|e s n t = 0 arguing that the antisymmetric states do not scatter via s-waves. This zeroing is only valid if the matrix elements include the vanishing spatial factor R − |δ(r 2 − r 1 )|R − . Actually, it is this spatial factor, which cancels the contribution of the antisymmetric states to s-wave scattering and to the interaction energy (1). On the contrary, the spatial part of the matrix elements for the symmetric states R + |ij + is doubled. In other words, the atoms of a heterostate symmetric pair behave as identical.
Clearly, |ab + = |e t n t,0 and |ac
(|e t,0 n t,0 +|e s n s ). Consequently, λ + ab ≡ ab|λ|ab + = λ t and λ
Then, according to Eqs. (1), (2), the interaction energies are E ab = λ t n a n b ,
in agreement with our previous result [3] . Note, that Ahokas et al. consider the atoms of a symmetric heterostate pair as distinguishable and therefore lose a factor of two in the expressions for the shifts. From Eqs. (3) and (4) we find C ba = C ab = γem (a t − a s ) and ∆a = 30(5) pm. This is somewhat lower than the theoretical values ∆a = 42 − 55 pm for the difference in the scattering lengths [4] [5] [6] . This work was supported by the Russian Foundation for Basic Research, project no. 09-02-01002-a.

{'paper_id': '13383621', 'title': 'Comment on"Clock Shift in High Field Magnetic Resonance of Atomic Hydrogen"', 's2_url': 'https://api.semanticscholar.org/CorpusID:13383621', 'source': 'S2ORC'}