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The role of positronium decoherence in positron annihilation in matter
Physics Letters A 375 (2011) 3872–3876
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The role of positronium decoherence in positron annihilation in matter M. Pietrow a,∗ , P. Słomski b a b
Institute of Physics, M. Curie-Skłodowska University, ul. Pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland Geographic Information Systems Development Company Martinex, ul. Mełgiewska 95, 21-040 S´widnik, Poland
a r t i c l e
i n f o
Article history: Received 23 June 2011 Received in revised form 1 September 2011 Accepted 12 September 2011 Available online 17 September 2011 Communicated by P.R. Holland Keywords: Positronium Decoherence Positron annihilation lifetime spectroscopy (PALS)
a b s t r a c t A small difference between the energies of the para-positronium (p-Ps) and ortho-positronium (o-Ps) states suggests the possibility of the superposition of p-Ps and o-Ps during the formation of positronium (Ps) from pre-Ps, terminating its migration in the matter in a void. It is shown that such a superposition decoheres in the basis of p-Ps and o-Ps. The decoherence time scale estimated here motivates a correction in the precise analysis of the positron annihilation lifetime spectra. More generally, the superposited Ps state should contribute to the theory of the evolution of positronium in matter. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The most powerful model describing the formation of the positronium atom in matter was developed by Stepanov and Byakov [1]. A positron from a radioactive source passing through a sample creates products including electrons, ions, and radicals, losing its own energy in the process. The positron comes to rest in a spreading cloud of ionised particles (the blob), with its energy lower than that of the ionisation level. At the thermalisation energy level, the production of Ps atom becomes a profitable process. Before the formation of a positronium atom, a positron and an electron can leave the blob as a weakly interacting, delocalised system (pre-Ps), migrating to the near void (‘free volume’), where they form a localised Ps atom. It is generally assumed that at this time, the entire population of Ps consists of p-Ps (spin S = 0) and o-Ps (S = 1), such that the spin value for each particle is definite. Such a Ps can annihilate in three different ways: 1) p-Ps can annihilate, which produces two gamma quanta, 2) o-Ps can annihilate, which produces three gamma quanta, or 3) both Ps substates can annihilate with an electron captured from the environment (the pick-off process). The probability of pick-off annihilation increases with the local electron density and dominates in the case of annihilation in condensed matter. The annihilation of o-Ps via pick-off is a twoquantum process. However, when the pre-Ps does not reach the free volume, the positron annihilates as a quasi-free particle with one of electrons from the bulk.
*
Corresponding author. Tel.: +48 815376286, fax: +48 815376191. E-mail address: [email protected] (M. Pietrow).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.09.012
Positron annihilation is widely used in studies of the properties of matter, such as the free volume distribution [2], and of the trapping electrons produced during irradiation processes [3]. One of the most common positronic measurement techniques is Positron Annihilation Lifetime Spectroscopy (PALS) [4], which measures the time between the creation and annihilation of a positron (and its bound state, positronium). The lifetime of each positron fraction is related in some way to the properties of the sample where the annihilation takes place; in particular, the lifetime of positronium is related to the local electron density. In this way, the time spectra recorded by PALS can provide information about the properties of matter. The most useful mode of positron annihilation in matter is the o-Ps annihilation, which occurs by pick-off. To separate the data for this long-living fraction from the lifetime spectra, these spectra must be decomposed into assumed elementary channels of annihilation (free positron annihilation, o-Ps and p-Ps annihilation, etc.). The exponential form of the decay function for each channel is commonly dictated. 2. Motivation The papers exploiting the blob model assume that the spin number has a definite value for pre-Ps and Ps (see positronium formation formulae in [5]). However, as the energy difference between p-Ps and o-Ps states in a vacuum is approximately 8.4 · 10−4 eV [6], the spin value could be expected to change during the migration of a pre-Ps through the sample due to intermolecular interactions. The consideration of only the definite spin value seems to be an oversimplification (e.g., only ortho-pre-Ps). Instead
M. Pietrow, P. Słomski / Physics Letters A 375 (2011) 3872–3876
of this a priori assumption, it should be assumed that quasi-free Ps enters a free volume as a superposition of the para and ortho states. The spin value plays an important role in the further analysis of the life of Ps. The annihilation of Ps, as an electromagnetic process, should obey the principle of charge parity conservation. For Ps in its ground state with the spin value S, the charge parity is πC = (−1) S , whereas for n photons, it is πC = (−1)n [7]. Until the spin number is well defined, the system cannot be said to decay with the emission of any definite n photons, which are an identifying signal of the state. This consideration implies that one cannot simply decompose the PALS spectra into the exponential curves for p-Ps and o-Ps, neglecting their superposition. In our opinion, as long as a population of Ps exists in a superposition of spin states, one should make a corresponding correction when decomposing the PALS spectra (the analysis of the correction method is not the subject of this Letter; however, an example of how the correction can be applied is shown in Appendix A). In this Letter, we show that Ps is expected to decohere in the basis defined by p-Ps and o-Ps, and we estimate the time of decoherence. 3. Rate of decoherence. Method of calculation
N +2 h¯ 2 (2) H= J σ¯ ◦ σ¯ (i) , 4
(1)
where J is the coupling constant, σ (2) are the Pauli matrices for the electron in Ps, and σ (i ) are the Pauli matrices for the ith (of N) electron from an environment; the sign ‘◦’ denotes a scalar product. We assume the existence of free volumes in which Ps could live a sufficiently long time (e.g., in molecular crystals, polymers, or bubbles present in a liquid state). We suppose that the atoms from the wall have a certain electron density around (inside) the free volume. These valence electrons can interact with an electron from Ps by the exchange interaction, resulting in the Ps pick-off process. The radii of the free volumes are in the range considered by PALS (from approximately 1 Å to hundreds of nanometres), while the wavelength of the valence electrons lies in the range of angstroms. One can expect in general that Ps in the free volume interacts with the electrons only momentarily. Generally, it is not obvious how many (N is variable in this work) electrons interact simultaneously with Ps. Suppose the initial (just after the localisation of Ps in the free volume) state of the entire system is ρ0 ≡ |Ψ0 Ψ0 |, where N +2 i =3
|si ≡ |Ps0 | S
and
1 2
|0, 0 + |1, −1 + |1, 0 + |1, +1 .
(3)
The ket | S denotes the state of the environment electrons, and the numbers in the kets are the spin s and its zth projection s z for Ps, respectively. The latter equation expresses the supposition that after interacting with the bulk and entering into the free volume, the particular Ps states are indistinguishable. The time evolution gives the following expression for the whole state at the time t
ρ (t ) = |Ψt Ψt |,
(4)
where |Ψt = e −i Ht /¯h |Ψ0 . For the positronium only, we calculate the state ρPs (t ) tracing ρ (t ) over the environmental subspace. To estimate the decoherence, we set the auxiliary expression
S (t ) =
+1 s = 0|ρPs (t )|s = 1, s z 2 ,
(5)
s z =−1
i =3
|Ψ0 = |Ps0
Fig. 1. |ρ[0,0;1,+1] (t )|2 calculated for a random initial state vector (2). The interaction of Ps with N = 2 electrons from the wall was assumed, whereas n in the expansion of |Ψt was set equal to 10. The time to the minimal value of the function is regarded as the decoherence time.
|Ps0 =
The most convenient Hamiltonian for considering the spin system interacting with its spin environment (bath) is the Heisenberg Hamiltonian [8], which is the phenomenological expression that incorporates an overlap of the electron wave functions of a system and its environment [9,10]. The problem of the decoherence of certain spin systems, in particular, those systems consisting of an electron pair interacting with a bath, was considered in [11–13], which supports our calculation method. The novel aspect of this Letter is the application of this calculation method to Ps (which is, in contrast to the systems mentioned above, a bound system with the fine structure, moreover the e + cannot add to the exchange interaction) and the demonstration of the consequences of interpreting the measurements. Let us consider the Hamiltonian for the interaction of Ps and the molecules of the wall of the free volume as
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(2)
which, in our case, is expected to collapse with time. Such a definition of S (t ) refines the nondiagonal terms related to the transition between s = 0 and s = 1. Decoherence occurs when all elements of this sum decrease considerably; thus, any state that contains an admixture of one spin state and the other disappears. We define the time when S reaches the minimal value as the time of the decoherence, T d . Because of the mathematical problems with calculating the entire ρ (t ) analytically, we truncated the Taylor expansion of |Ψt at the nth term. This truncation at an appropriate n still allows us to approximate T d sufficiently for our aims (see Appendix B). 4. Results and discussion The numerical calculation of (5), using |Ψt defined above, results in the following statements: 1. Indeed, one can observe the decrease in S (t ) (and each of its elements) in the basis of the p-Ps and o-Ps states (Fig. 1), but the nondiagonal terms do not disappear completely and permanently. 2. After the decrease in S (t ) shown in Figs. 1 and 2, oscillations
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M. Pietrow, P. Słomski / Physics Letters A 375 (2011) 3872–3876
Fig. 2. S (t ) averaged over 100 randomly chosen samples of the initial state vector for different electrons interacting with a Ps atom (N). Comparison of calculations with the evolution operator truncated at n = 3 and at n = 10. The results are shown for N = 1 (n = 3 and n = 10 are represented by the solid and dashed line, respectively) and for N = 4 (n = 3 and n = 10 are represented by the dotted and dash-dotted line, respectively).
3.
4.
5.
6.
7.
were obtained1 that are consistent with the results of the decoherence calculations for other complex systems in the spin bath [12,13]. For N = 2, T d is approximately 29 ps (figure not shown) which means that the effect of the existence of the superposition of states is detectable on the PALS timescale and that one should take the effect into account when decomposing the PALS spectra. We consider this statement to be the most significant conclusion of this work. Higher orders of the Hamiltonian in the evolution operator do not substantially affect the value of the decoherence time (see ‘Supplementary material 1’ and Fig. 2). The decoherence time depends on a coupling constant J , which was set with the assumption that the interaction energy of Ps and the electrons from the wall is approximately 10−4 eV. This energy could convert the ortho and para states and cause the initial superposition state in the bulk. However, such a large value of the Ps interaction energy in the free volume seems to be reasonable only for some specific substances (e.g., when the ortho–para conversion takes place [14]). If the energy of the interaction is smaller, then one can expect that the decoherence time will increase (e.g., if the interaction energy is set to 10−5 eV, then T d ∼ 150 ps for N = 3 instead of T d = 18 ps as estimated above). The value of T d decreases with the number N of electrons in the environment, reaching an asymptotic value of 12.2 ps (Fig. 3). The influence of the magnetisation of the environment was investigated. These results are described in ‘Supplementary material 2’. The Heisenberg Hamiltonian shows that the decoherence is not complete in the basis of p-Ps and o-Ps (there exists at any time a population of Ps atoms in the superposition state, with the rate 1 − (| p-Ps|Ps(t )|2 + |o-Ps|Ps(t )|2 )). We suppose that the existence of this residue may be an artefact of the Heisenberg Hamiltonian because it is the simplest treatment of the realistic case. We intend to follow these calculations with a more complicated Hamiltonian, in which J is time dependent
Fig. 3. The decoherence time T d as a function of the number N of electrons interacting with the Ps. The value is averaged over 20 randomly chosen initial state vectors for n = 7.
(the Ps atom approaches or recedes from the wall so that the interaction strength changes). 5. Conclusions Our theoretical discussion of the decoherence of Ps in matter indicates that coherent Ps transforms into well-defined spin states and that the decoherence lasts long enough (picoseconds or more) to measurably modify the positron lifetime spectra and momentum distributions; additionally, the decoherence should be taken into account when the results of positron studies are precisely interpreted. We have demonstrated a method to modify the formulas used for the decomposition of PALS spectra. The decoherence time decreases exponentially with the number of electrons interacting with Ps in the free volume. We also examined the expected influence of magnetisation on the decoherence process. Our calculations add a new aspect to the recent theory of Ps formation and its evolution in matter. Acknowledgements The authors want to thank Dr. T. Paterek (from the Centre for Quant. Technol. of the National University of Singapore) for the introductory discussions and Dr. M. Turek (Inst. of Phys., M. CurieSkłodowska University, Poland) for encouraging words. We thank also M. Opala (computer farm administrator at the Dept. of Theoretical Physics, UMCS) for help in adjustment of our work on the accessible computer hardware. Appendix A The influence of the hypothetical coherence of Ps on PALS spectra seems to be a complex problem that requires a separate study. We propose the following example in which this influence can be observed. To interpret PALS spectra, one considers the formula for coincidence (2- and 3-quantum) rate (N γ ) resulting from the formula (see also [5]) (λe )
dN γ ∼ −d P e 1
Oscillations are visible when calculated with a sufficiently large n to obtain reliable outcomes for t > T d . Here, maximally n = 10 was applied, which is enough to calculate reliably the evolution up to approximately 30 ps, so reliable oscillations are shown for the settings with enough small T d . As an example, see the calculation for N = 4, n = 10 shown in Fig. 2 and in Figs. 1 and 2 of ‘Supplementary material 2’.
(λe )
− dS o
− d(oPs)(λoPs ) ,
(λ )
− dS p e − d(pPs)(λpPs ) (A.1)
in which quantities are defined as follows: P e —the number of free positrons, S p , S o —the same for pre-para-Ps and pre-ortho-Ps, pPs,
M. Pietrow, P. Słomski / Physics Letters A 375 (2011) 3872–3876
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Fig. 4. The boundary value of the absolute of nth term (normalised to the maximal value) related to the evolution operator expansion showing its contribution at t = 10 ps and t = 20 ps to approximate the exact result in the region of the calculated T d .
oPs—the number of para- and ortho-Ps, respectively; λs are the decay rates for a particular type of particles. Furthermore, one assumes that P e , S p , etc. obey the following equations:
⎧ d P e (t ) = −(λe + ν ) P e (t ) dt , ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ dS o (t ) = ν P e (t ) dt − (λe + K ) S o (t ) dt , ⎪ ⎪ 4 ⎨
H =
n
σ
(2 )
◦σ
( p)
n-fold sum
σ (2 ) ◦ σ ( p ) · σ (2 ) ◦ σ ( s ) · . . . .
(B.1)
p ,s,t ,...
P e (0) = 1,
Each superscript changes in the range (3, N + 2), so we have N n different terms with fixed index values. Because σ (2) ◦ σ ( p ) =
( p) (2) 3 · σi we have 3n different values of each term summed i =1 σi in (B.1) (for fixed p, s, t, . . .). Each of these terms can give the maximal value 1 on any state,2 so the resultant coefficient related to the nth power of the Hamiltonian in (5) is ℵ/n! · (−i J h¯ /4 · t )n , where ℵ = (3N )n . Substituting the assumed value of J into this coefficient, we can see the contribution of the nth term as a function of t. This contribution is shown in Fig. 4 for fixed t = 10 ps and t = 20 ps (the region of T d for several N). If one knows the contribution of the skipped terms in the evolution operator at any t, one can estimate the expected shift of the minimum (T d ) of the polynomial that we obtained from the truncation at smaller n and that one expected when more accurate calculations could be performed. Appendix C. Supplementary material Supplementary material related to this article can be found online at doi:10.1016/j.physleta.2011.09.012.
( S )(0) = 0, (oPs)(0) = 0,
References
(pPs)(0) = 0 (A.3)
and
− d( S )(λ S ) − d(pPs)(λpPs ) − d(oPs)(λoPs )
− d(MPs)(λMPs ) .
=
(A.2)
where ν , λe , etc. are the constants describing the creation and decay rates of each population. If one considers an annihilation in matter, where the pick-off process dominates, (A.1) relates mainly to the 2-quantum annihilation process, and both (A.1) and (A.2) allow the calculation of the annihilation intensity rate for a particular set of channels (e.g., for P e , (oPs), etc.). The decomposition of PALS spectra is a direct consequence of the use of the formulas above, which are implemented in the numerical programs performing the decomposition. If one supposes the existence of the coherence of the states, one does not distinguish the spin value for pre-Ps, and one assumes the existence of pre-Ps (S), and also Ps (MPs), in the superposition state, which annihilate at the rate λ S and λMPs , respectively. Thus, one can replace (A.2) and (A.1) by
(λe )
N +2 p =3
S o (0) = 0,
dS p (t ) = ν P e (t ) dt − (λe + K ) S p (t ) dt , S p (0) = 0, ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ d ( oPs )( t ) = K S o (t ) dt − λoPs (oPs)(t ) dt , (oPs)(0) = 0, ⎪ ⎩ d(pPs)(t ) = K S p (t ) dt − λpPs (pPs)(t ) dt , (pPs)(0) = 0,
dN γ ∼ −d P e
The H n operator can be written as n
P e (0) = 1,
1
⎧ d P e (t ) = −(λe + ν ) P e (t ) dt , ⎪ ⎪ ⎪ ⎨ dS (t ) = ν P e (t ) dt − (λe + K )( S )(t ) dt , ⎪ d(oPs)(t ) = K 1 ( S )(t ) dt − λoPs (oPs)(t ) dt , ⎪ ⎪ ⎩ d(pPs)(t ) = K 2 ( S )(t ) dt − λpPs (pPs)(t ) dt ,
Appendix B
(A.4)
(K
Additionally the equations d(MPs)(t ) = K ( S )(t ) dt − + λMPs ) × (MPs)(t ) dt, where K 1 + K 2 = K and (MPs)(0) = 0 for the super-
posited Ps are needed. The formula (A.4) is not a simple replacement of (A.1) because its last term describes decays in which the number of photons per annihilation is not fixed. Because of this term, 3-quantum decay cannot be neglected here (there is no reason to neglect it, as it would be a significantly improbable process). The simplest consequence of our hypothesis is the increase of 3-quantum annihilation as long as the cohered state exists (picoseconds).
[1] S.V. Stepanov, V.M. Byakov, in: Y.C. Jean, P.E. Mallone, D.M. Schrader (Eds.), Principles and Applications of Positrons and Positronium Chemistry, World Scientific, 2003, p. 117. ´ [2] T. Goworek, K. Ciesielski, B. Jasinska, J. Wawryszczuk, Radiat. Phys. Chem. 58 (2000) 719. [3] T. Hirade, in: Y. Hatano, Y. Katsumura, A. Mozumder (Eds.), Charged Particle and Photon Interactions with Matter. Recent Advances, Applications, and Interfaces, Taylor and Francis Group, 2011, pp. 137–167. [4] P.G. Coleman, in: Y.C. Jean, P.E. Mallone, D.M. Schrader (Eds.), Principles and Applications of Positrons and Positronium Chemistry, World Scientific, 2003, p. 49. [5] C. He, V.P. Shantarovich, T. Suzuki, S.V. Stepanov, R. Suzuki, M. Matsuo, J. Chem. Phys. 122 (2005) 214907.
2
We assumed a very coarse approximation: if one of ( p)
σi( p) returned 1 in the ex-
treme case, then none of σ j for i = j could reach this value. In spite of this, such an extreme value is applied for simplicity. Because of this simplification the approximation is overestimated.
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O.E. Mogensen, Hyperfine Interact. 84 (1994) 377. W. Greiner, J. Reinhardt, Field Quantization, Springer-Verlag, 1996. F.M. Cucchietti, J.P. Paz, W.H. Zurek, Phys. Rev. A 72 (2005) 052113. A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer, 1994. W. Nolting, A. Ramakanth, Quantum Theory of Magnetism, Springer, 2009. V.V. Dobrovitski, H.A. De Raedt, M.I. Katsnelson, B.N. Harmon, Phys. Rev. Lett. 90 (2003) 210401.
[12] A. Melikidze, V.V. Dobrovitski, H.A. De Raedt, M.I. Katsnelson, B.N. Harmon, Phys. Rev. B 70 (2004) 014435. [13] H. De Raedt, V.V. Dobrovitski, in: D.P. Landau, et al. (Eds.), Comp. Simul. Stud. in Cond.-Matt. Phys. XVI, Springer Proceedings in Physics, 2004, quantph/0301121. [14] S.C. Sharma, in: D.M. Schrader, Y.C. Jean (Eds.), Positron and Positronium Chemistry, Elsevier, 1988, p. 193.
Contents lists available at SciVerse ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The role of positronium decoherence in positron annihilation in matter M. Pietrow a,∗ , P. Słomski b a b
Institute of Physics, M. Curie-Skłodowska University, ul. Pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland Geographic Information Systems Development Company Martinex, ul. Mełgiewska 95, 21-040 S´widnik, Poland
a r t i c l e
i n f o
Article history: Received 23 June 2011 Received in revised form 1 September 2011 Accepted 12 September 2011 Available online 17 September 2011 Communicated by P.R. Holland Keywords: Positronium Decoherence Positron annihilation lifetime spectroscopy (PALS)
a b s t r a c t A small difference between the energies of the para-positronium (p-Ps) and ortho-positronium (o-Ps) states suggests the possibility of the superposition of p-Ps and o-Ps during the formation of positronium (Ps) from pre-Ps, terminating its migration in the matter in a void. It is shown that such a superposition decoheres in the basis of p-Ps and o-Ps. The decoherence time scale estimated here motivates a correction in the precise analysis of the positron annihilation lifetime spectra. More generally, the superposited Ps state should contribute to the theory of the evolution of positronium in matter. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The most powerful model describing the formation of the positronium atom in matter was developed by Stepanov and Byakov [1]. A positron from a radioactive source passing through a sample creates products including electrons, ions, and radicals, losing its own energy in the process. The positron comes to rest in a spreading cloud of ionised particles (the blob), with its energy lower than that of the ionisation level. At the thermalisation energy level, the production of Ps atom becomes a profitable process. Before the formation of a positronium atom, a positron and an electron can leave the blob as a weakly interacting, delocalised system (pre-Ps), migrating to the near void (‘free volume’), where they form a localised Ps atom. It is generally assumed that at this time, the entire population of Ps consists of p-Ps (spin S = 0) and o-Ps (S = 1), such that the spin value for each particle is definite. Such a Ps can annihilate in three different ways: 1) p-Ps can annihilate, which produces two gamma quanta, 2) o-Ps can annihilate, which produces three gamma quanta, or 3) both Ps substates can annihilate with an electron captured from the environment (the pick-off process). The probability of pick-off annihilation increases with the local electron density and dominates in the case of annihilation in condensed matter. The annihilation of o-Ps via pick-off is a twoquantum process. However, when the pre-Ps does not reach the free volume, the positron annihilates as a quasi-free particle with one of electrons from the bulk.
*
Corresponding author. Tel.: +48 815376286, fax: +48 815376191. E-mail address: [email protected] (M. Pietrow).
0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.09.012
Positron annihilation is widely used in studies of the properties of matter, such as the free volume distribution [2], and of the trapping electrons produced during irradiation processes [3]. One of the most common positronic measurement techniques is Positron Annihilation Lifetime Spectroscopy (PALS) [4], which measures the time between the creation and annihilation of a positron (and its bound state, positronium). The lifetime of each positron fraction is related in some way to the properties of the sample where the annihilation takes place; in particular, the lifetime of positronium is related to the local electron density. In this way, the time spectra recorded by PALS can provide information about the properties of matter. The most useful mode of positron annihilation in matter is the o-Ps annihilation, which occurs by pick-off. To separate the data for this long-living fraction from the lifetime spectra, these spectra must be decomposed into assumed elementary channels of annihilation (free positron annihilation, o-Ps and p-Ps annihilation, etc.). The exponential form of the decay function for each channel is commonly dictated. 2. Motivation The papers exploiting the blob model assume that the spin number has a definite value for pre-Ps and Ps (see positronium formation formulae in [5]). However, as the energy difference between p-Ps and o-Ps states in a vacuum is approximately 8.4 · 10−4 eV [6], the spin value could be expected to change during the migration of a pre-Ps through the sample due to intermolecular interactions. The consideration of only the definite spin value seems to be an oversimplification (e.g., only ortho-pre-Ps). Instead
M. Pietrow, P. Słomski / Physics Letters A 375 (2011) 3872–3876
of this a priori assumption, it should be assumed that quasi-free Ps enters a free volume as a superposition of the para and ortho states. The spin value plays an important role in the further analysis of the life of Ps. The annihilation of Ps, as an electromagnetic process, should obey the principle of charge parity conservation. For Ps in its ground state with the spin value S, the charge parity is πC = (−1) S , whereas for n photons, it is πC = (−1)n [7]. Until the spin number is well defined, the system cannot be said to decay with the emission of any definite n photons, which are an identifying signal of the state. This consideration implies that one cannot simply decompose the PALS spectra into the exponential curves for p-Ps and o-Ps, neglecting their superposition. In our opinion, as long as a population of Ps exists in a superposition of spin states, one should make a corresponding correction when decomposing the PALS spectra (the analysis of the correction method is not the subject of this Letter; however, an example of how the correction can be applied is shown in Appendix A). In this Letter, we show that Ps is expected to decohere in the basis defined by p-Ps and o-Ps, and we estimate the time of decoherence. 3. Rate of decoherence. Method of calculation
N +2 h¯ 2 (2) H= J σ¯ ◦ σ¯ (i) , 4
(1)
where J is the coupling constant, σ (2) are the Pauli matrices for the electron in Ps, and σ (i ) are the Pauli matrices for the ith (of N) electron from an environment; the sign ‘◦’ denotes a scalar product. We assume the existence of free volumes in which Ps could live a sufficiently long time (e.g., in molecular crystals, polymers, or bubbles present in a liquid state). We suppose that the atoms from the wall have a certain electron density around (inside) the free volume. These valence electrons can interact with an electron from Ps by the exchange interaction, resulting in the Ps pick-off process. The radii of the free volumes are in the range considered by PALS (from approximately 1 Å to hundreds of nanometres), while the wavelength of the valence electrons lies in the range of angstroms. One can expect in general that Ps in the free volume interacts with the electrons only momentarily. Generally, it is not obvious how many (N is variable in this work) electrons interact simultaneously with Ps. Suppose the initial (just after the localisation of Ps in the free volume) state of the entire system is ρ0 ≡ |Ψ0 Ψ0 |, where N +2 i =3
|si ≡ |Ps0 | S
and
1 2
|0, 0 + |1, −1 + |1, 0 + |1, +1 .
(3)
The ket | S denotes the state of the environment electrons, and the numbers in the kets are the spin s and its zth projection s z for Ps, respectively. The latter equation expresses the supposition that after interacting with the bulk and entering into the free volume, the particular Ps states are indistinguishable. The time evolution gives the following expression for the whole state at the time t
ρ (t ) = |Ψt Ψt |,
(4)
where |Ψt = e −i Ht /¯h |Ψ0 . For the positronium only, we calculate the state ρPs (t ) tracing ρ (t ) over the environmental subspace. To estimate the decoherence, we set the auxiliary expression
S (t ) =
+1 s = 0|ρPs (t )|s = 1, s z 2 ,
(5)
s z =−1
i =3
|Ψ0 = |Ps0
Fig. 1. |ρ[0,0;1,+1] (t )|2 calculated for a random initial state vector (2). The interaction of Ps with N = 2 electrons from the wall was assumed, whereas n in the expansion of |Ψt was set equal to 10. The time to the minimal value of the function is regarded as the decoherence time.
|Ps0 =
The most convenient Hamiltonian for considering the spin system interacting with its spin environment (bath) is the Heisenberg Hamiltonian [8], which is the phenomenological expression that incorporates an overlap of the electron wave functions of a system and its environment [9,10]. The problem of the decoherence of certain spin systems, in particular, those systems consisting of an electron pair interacting with a bath, was considered in [11–13], which supports our calculation method. The novel aspect of this Letter is the application of this calculation method to Ps (which is, in contrast to the systems mentioned above, a bound system with the fine structure, moreover the e + cannot add to the exchange interaction) and the demonstration of the consequences of interpreting the measurements. Let us consider the Hamiltonian for the interaction of Ps and the molecules of the wall of the free volume as
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(2)
which, in our case, is expected to collapse with time. Such a definition of S (t ) refines the nondiagonal terms related to the transition between s = 0 and s = 1. Decoherence occurs when all elements of this sum decrease considerably; thus, any state that contains an admixture of one spin state and the other disappears. We define the time when S reaches the minimal value as the time of the decoherence, T d . Because of the mathematical problems with calculating the entire ρ (t ) analytically, we truncated the Taylor expansion of |Ψt at the nth term. This truncation at an appropriate n still allows us to approximate T d sufficiently for our aims (see Appendix B). 4. Results and discussion The numerical calculation of (5), using |Ψt defined above, results in the following statements: 1. Indeed, one can observe the decrease in S (t ) (and each of its elements) in the basis of the p-Ps and o-Ps states (Fig. 1), but the nondiagonal terms do not disappear completely and permanently. 2. After the decrease in S (t ) shown in Figs. 1 and 2, oscillations
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Fig. 2. S (t ) averaged over 100 randomly chosen samples of the initial state vector for different electrons interacting with a Ps atom (N). Comparison of calculations with the evolution operator truncated at n = 3 and at n = 10. The results are shown for N = 1 (n = 3 and n = 10 are represented by the solid and dashed line, respectively) and for N = 4 (n = 3 and n = 10 are represented by the dotted and dash-dotted line, respectively).
3.
4.
5.
6.
7.
were obtained1 that are consistent with the results of the decoherence calculations for other complex systems in the spin bath [12,13]. For N = 2, T d is approximately 29 ps (figure not shown) which means that the effect of the existence of the superposition of states is detectable on the PALS timescale and that one should take the effect into account when decomposing the PALS spectra. We consider this statement to be the most significant conclusion of this work. Higher orders of the Hamiltonian in the evolution operator do not substantially affect the value of the decoherence time (see ‘Supplementary material 1’ and Fig. 2). The decoherence time depends on a coupling constant J , which was set with the assumption that the interaction energy of Ps and the electrons from the wall is approximately 10−4 eV. This energy could convert the ortho and para states and cause the initial superposition state in the bulk. However, such a large value of the Ps interaction energy in the free volume seems to be reasonable only for some specific substances (e.g., when the ortho–para conversion takes place [14]). If the energy of the interaction is smaller, then one can expect that the decoherence time will increase (e.g., if the interaction energy is set to 10−5 eV, then T d ∼ 150 ps for N = 3 instead of T d = 18 ps as estimated above). The value of T d decreases with the number N of electrons in the environment, reaching an asymptotic value of 12.2 ps (Fig. 3). The influence of the magnetisation of the environment was investigated. These results are described in ‘Supplementary material 2’. The Heisenberg Hamiltonian shows that the decoherence is not complete in the basis of p-Ps and o-Ps (there exists at any time a population of Ps atoms in the superposition state, with the rate 1 − (| p-Ps|Ps(t )|2 + |o-Ps|Ps(t )|2 )). We suppose that the existence of this residue may be an artefact of the Heisenberg Hamiltonian because it is the simplest treatment of the realistic case. We intend to follow these calculations with a more complicated Hamiltonian, in which J is time dependent
Fig. 3. The decoherence time T d as a function of the number N of electrons interacting with the Ps. The value is averaged over 20 randomly chosen initial state vectors for n = 7.
(the Ps atom approaches or recedes from the wall so that the interaction strength changes). 5. Conclusions Our theoretical discussion of the decoherence of Ps in matter indicates that coherent Ps transforms into well-defined spin states and that the decoherence lasts long enough (picoseconds or more) to measurably modify the positron lifetime spectra and momentum distributions; additionally, the decoherence should be taken into account when the results of positron studies are precisely interpreted. We have demonstrated a method to modify the formulas used for the decomposition of PALS spectra. The decoherence time decreases exponentially with the number of electrons interacting with Ps in the free volume. We also examined the expected influence of magnetisation on the decoherence process. Our calculations add a new aspect to the recent theory of Ps formation and its evolution in matter. Acknowledgements The authors want to thank Dr. T. Paterek (from the Centre for Quant. Technol. of the National University of Singapore) for the introductory discussions and Dr. M. Turek (Inst. of Phys., M. CurieSkłodowska University, Poland) for encouraging words. We thank also M. Opala (computer farm administrator at the Dept. of Theoretical Physics, UMCS) for help in adjustment of our work on the accessible computer hardware. Appendix A The influence of the hypothetical coherence of Ps on PALS spectra seems to be a complex problem that requires a separate study. We propose the following example in which this influence can be observed. To interpret PALS spectra, one considers the formula for coincidence (2- and 3-quantum) rate (N γ ) resulting from the formula (see also [5]) (λe )
dN γ ∼ −d P e 1
Oscillations are visible when calculated with a sufficiently large n to obtain reliable outcomes for t > T d . Here, maximally n = 10 was applied, which is enough to calculate reliably the evolution up to approximately 30 ps, so reliable oscillations are shown for the settings with enough small T d . As an example, see the calculation for N = 4, n = 10 shown in Fig. 2 and in Figs. 1 and 2 of ‘Supplementary material 2’.
(λe )
− dS o
− d(oPs)(λoPs ) ,
(λ )
− dS p e − d(pPs)(λpPs ) (A.1)
in which quantities are defined as follows: P e —the number of free positrons, S p , S o —the same for pre-para-Ps and pre-ortho-Ps, pPs,
M. Pietrow, P. Słomski / Physics Letters A 375 (2011) 3872–3876
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Fig. 4. The boundary value of the absolute of nth term (normalised to the maximal value) related to the evolution operator expansion showing its contribution at t = 10 ps and t = 20 ps to approximate the exact result in the region of the calculated T d .
oPs—the number of para- and ortho-Ps, respectively; λs are the decay rates for a particular type of particles. Furthermore, one assumes that P e , S p , etc. obey the following equations:
⎧ d P e (t ) = −(λe + ν ) P e (t ) dt , ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ dS o (t ) = ν P e (t ) dt − (λe + K ) S o (t ) dt , ⎪ ⎪ 4 ⎨
H =
n
σ
(2 )
◦σ
( p)
n-fold sum
σ (2 ) ◦ σ ( p ) · σ (2 ) ◦ σ ( s ) · . . . .
(B.1)
p ,s,t ,...
P e (0) = 1,
Each superscript changes in the range (3, N + 2), so we have N n different terms with fixed index values. Because σ (2) ◦ σ ( p ) =
( p) (2) 3 · σi we have 3n different values of each term summed i =1 σi in (B.1) (for fixed p, s, t, . . .). Each of these terms can give the maximal value 1 on any state,2 so the resultant coefficient related to the nth power of the Hamiltonian in (5) is ℵ/n! · (−i J h¯ /4 · t )n , where ℵ = (3N )n . Substituting the assumed value of J into this coefficient, we can see the contribution of the nth term as a function of t. This contribution is shown in Fig. 4 for fixed t = 10 ps and t = 20 ps (the region of T d for several N). If one knows the contribution of the skipped terms in the evolution operator at any t, one can estimate the expected shift of the minimum (T d ) of the polynomial that we obtained from the truncation at smaller n and that one expected when more accurate calculations could be performed. Appendix C. Supplementary material Supplementary material related to this article can be found online at doi:10.1016/j.physleta.2011.09.012.
( S )(0) = 0, (oPs)(0) = 0,
References
(pPs)(0) = 0 (A.3)
and
− d( S )(λ S ) − d(pPs)(λpPs ) − d(oPs)(λoPs )
− d(MPs)(λMPs ) .
=
(A.2)
where ν , λe , etc. are the constants describing the creation and decay rates of each population. If one considers an annihilation in matter, where the pick-off process dominates, (A.1) relates mainly to the 2-quantum annihilation process, and both (A.1) and (A.2) allow the calculation of the annihilation intensity rate for a particular set of channels (e.g., for P e , (oPs), etc.). The decomposition of PALS spectra is a direct consequence of the use of the formulas above, which are implemented in the numerical programs performing the decomposition. If one supposes the existence of the coherence of the states, one does not distinguish the spin value for pre-Ps, and one assumes the existence of pre-Ps (S), and also Ps (MPs), in the superposition state, which annihilate at the rate λ S and λMPs , respectively. Thus, one can replace (A.2) and (A.1) by
(λe )
N +2 p =3
S o (0) = 0,
dS p (t ) = ν P e (t ) dt − (λe + K ) S p (t ) dt , S p (0) = 0, ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ d ( oPs )( t ) = K S o (t ) dt − λoPs (oPs)(t ) dt , (oPs)(0) = 0, ⎪ ⎩ d(pPs)(t ) = K S p (t ) dt − λpPs (pPs)(t ) dt , (pPs)(0) = 0,
dN γ ∼ −d P e
The H n operator can be written as n
P e (0) = 1,
1
⎧ d P e (t ) = −(λe + ν ) P e (t ) dt , ⎪ ⎪ ⎪ ⎨ dS (t ) = ν P e (t ) dt − (λe + K )( S )(t ) dt , ⎪ d(oPs)(t ) = K 1 ( S )(t ) dt − λoPs (oPs)(t ) dt , ⎪ ⎪ ⎩ d(pPs)(t ) = K 2 ( S )(t ) dt − λpPs (pPs)(t ) dt ,
Appendix B
(A.4)
(K
Additionally the equations d(MPs)(t ) = K ( S )(t ) dt − + λMPs ) × (MPs)(t ) dt, where K 1 + K 2 = K and (MPs)(0) = 0 for the super-
posited Ps are needed. The formula (A.4) is not a simple replacement of (A.1) because its last term describes decays in which the number of photons per annihilation is not fixed. Because of this term, 3-quantum decay cannot be neglected here (there is no reason to neglect it, as it would be a significantly improbable process). The simplest consequence of our hypothesis is the increase of 3-quantum annihilation as long as the cohered state exists (picoseconds).
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2
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σi( p) returned 1 in the ex-
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