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Independent pair approximation for attractive bose condensates
Physics Letters A 371 (2007) 7–10 www.elsevier.com/locate/pla
Independent pair approximation for attractive bose condensates George E. Cragg a,∗ , Arthur K. Kerman b a Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 1 February 2007; received in revised form 20 May 2007; accepted 24 May 2007 Available online 12 June 2007 Communicated by C.R. Doering
Abstract A bosonic analog of the independent pair approximation is developed in the context of a zero-temperature assembly with a positive s-wave scattering length, but attractive interparticle interactions. By preferentially scattering any pair back into the condensate, we account for the effect that all other particles have on the two-body state inside the medium. Instead of the energy of a pair, the poles of the resulting T matrix now give the chemical potential of the system. One pole reveals a collapsing many-body ground state whereas the other gives a complex valued chemical potential, where the imaginary part quantifies the decay rate of the condensate. Both solutions are in close agreement with those obtained in a variational calculation. © 2007 Elsevier B.V. All rights reserved. PACS: 03.75.Nt; 03.65.Nk; 05.30.Jp
Originally developed for the description of nuclear matter, the independent pair approximation treats a pair of particles as if the interaction between them is modified by the surrounding particles in which the pair is embedded [1–4]. In this way, finding the energy of the many-body collective reduces to the solution of an effective scattering problem. Especially important applications of this approach include cases in which perturbation expansions cannot be applied, such as inside nuclei where the nucleon–nucleon force was once thought to be accurately modeled by an infinitely repulsive core. Unlike the case of fermions where the medium effects are accounted for by excluding the scattering into occupied levels, we discuss the bosonic analog where the scattered pair preferentially fills a single state. Specifically, an independent pair model is sought for a system of attractive bosons that can also pair together into what we call a molecular Feshbach state [5]. First, we examine the two-body scattering problem in vacuum, where the inclusion of the molecular state is dealt with by deciding how its presence alters the interatomic interac-
* Corresponding author.
E-mail address: [email protected] (G.E. Cragg). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.05.114
tion. This result is then used to define the effective potential between pairs. Denoting the particle wave numbers by k1 and k2 , it is convenient to define the relative and total momenta by q = (k1 − k2 )/2 and P = k1 + k2 , respectively. In using a separable form for the two-body potential, q|V |q = λf (q)f (q ),
(1)
the parameter λ is identified as the strength whereas f (q) is the form factor for which f (0) ≡ 1. Both the Feshbach wave function, φ, and the scattering wave function, ψ(q), are self consistently described by the coupled Schrödinger equations 2 2q − E ψ(q) + λf (q) f (q )ψ(q ) + λαf (q)φ = 0, (2a) q
( − E)φ + λα
f (q )ψ(q ) = 0,
(2b)
q
where the interaction (1) has been used in the ψ equation. On the scale of h¯ 2 /2m, E is the energy of the pair and is an energy difference or detuning of the Feshbach state relative to the free atoms. Since these two equations must become independent in the limit where the interaction vanishes, the coupling strength
8
G.E. Cragg, A.K. Kerman / Physics Letters A 371 (2007) 7–10
is represented by the product λα. Furthermore, the momentum dependence of the coupling is conveniently accounted for by the form factor, f (q). Elimination of the molecular state in (2b) results in a single scattering equation from which the effective two-body interaction strength is given by λ→λ−
λ2 α 2 . −E
(3)
Therefore, in all further expressions, the presence of molecules can easily be incorporated by shifting the strength according to (3). An important parameter in low-energy scattering is the s-wave scattering length which is found from its relationship to the T matrix, |q |T |q|q =q=0 = 8πa(). Relating T to the potential requires the Lippmann–Schwinger equation, T = V + V GT ,
(4)
which contains the single-particle Green’s function given by G = (E − H0 − 2iε)−1 . As usual, H0 is the free Hamiltonian whereas ε is a parameter which is taken to zero from the positive side of the real axis. Taking V as the two-body interaction (1) along with the shifted strength (3), a separable T matrix, q |T |q = tf (q )f (q), solves (4), thus obtaining the effective scattering length as 1 1 1 + . = 2 2 8πa() λ − λ α / b
(5)
The range, b, is defined by b−1 = q f (q)2 /2q 2 , where the no tation q is shorthand for (2π)−3 d 3 q. As treated here, an attractive interaction can be simplified by taking b → 0 and λ → 0− such that the effective scattering length is positive. Correspondingly, this choice of parameters is relevant to atomic condensation experiments in 85 Rb [6].1 Eqs. (1), (3) and (5) account for the presence of a molecular state when constructing an independent pair model for bosons. Consider a collection of particles interacting with each other through the two-body separable potential (1) with the strength given by (3). From the independent particle perspective, the collective is viewed as an effective potential through which each of the constituents move. In a uniform medium, this single-particle potential is independent of spatial location, but does depend on the momentum. To second order in the relative momentum, q, the effective interaction is m − 1 q 2, W (q) = W0 + (6) m∗ where the coefficient of the quadratic term is expressed in terms of the particle mass, m, and the effective mass, m∗ .2 As a result of the medium’s interaction, the energy of a pair, E, is in general shifted from the value that would be obtained in vacuum, E0 . Let i and j label two particles in some pair. Summing 1 In the absence of molecular coupling (α = 0), Eq. (5) indicates that the negative background scattering length of 85 Rb implies an attractive interaction. 2 For further details on the relationship between the effective mass and the single-particle potential, see [1,4].
the shifts over all potential partners {j } produces the total energy shift on the arbitrarily chosen i. Equating this result to the single-particle potential obtains a self-consistent value for the effective mass. Quantitatively, the energy shift of a pair is given by E = E − E0 = ϕ0 |V |ψ, which follows from the normalization, φ0 |ψ = 1, of the center of mass wave function |ψ = |ϕ0 +
|ϕn
n=0
1 ϕn |V |ψ. E − En
(7)
This equation is simply the projection of the Schrödinger equation onto the free particle eigenstates, {|ϕn }. Using the separable potential along with the strength that accounts for the Feshbach coupling, the continuum energy shift is λ2 α 2 f (q) f (q )ψ(q ). E = q|V |ψq = λ − (8) −E q
Hence, in the zero-range limit, λ → 0− , the shift vanishes whereby the effective mass assumes its value in vacuum: m∗ −→ m.
(9)
λ→0−
Although going to zero range simplifies the problem to a collection of free pairs, the medium nonetheless affects the interaction within each pair. To arrive at the bosonic form of this effect, it is helpful to first recall that in the fermionic case the scattering state is precluded from all occupied momenta, resulting in a “Fermi force” arising due to the presence of the medium. However, bosons have no limit to the occupation of each state, implying that the interaction should be weighted by the sum of the occupation numbers for the two particles in the pair. Therefore, the bosonic projection operator has the center of mass matrix element q |QB |q = (1 + nq+P/2 + nq−P/2 )δ(q − q )
−→ 1 + 2ρa δ(q) δ(q − q ), T →0
(10)
where q ± P/2 are the particle momenta in the laboratory. Also, the occupation numbers, nq±P/2 are bose distributions depending on temperature in the normal way. At zero temperature, the momenta vanish, reducing the occupation number sum to twice the atomic density, ρa , times a delta function, as given by Eq. (10). Since this projection weights the interaction, its action is on V |ψ, thus giving the Bethe–Goldstone equation for the two-body state: |ψ = |φ + GQB V |ψ.
(11)
Alternatively, we may use the separable potential along with its shifted strength (3) to arrive at the integro-differential equation for the wave function [7] 2 λ2 α 2 f (r) + 2ρa d 3 r f (r ) 2∇ + E ψ(r) = λ − −E × d 3 r f (r )ψ(r ), (12)
G.E. Cragg, A.K. Kerman / Physics Letters A 371 (2007) 7–10
9
Fig. 1. For the collapsing lower state of 85 Rb, the independent pair model of Eq. (17) (solid curve) is compared with the variational many-body result (dashed curve). Plot (a) shows the case for an applied magnetic field of 162.3 G whereas part (b) shows an expanded comparison of the two curves. In each case, the dotted vertical line ending at the single point marks the quantum phase transition found in the many-body analysis. Decreasing the applied field shifts this point to lower density. Hence, for the same range of densities as in (a), a more global comparison is achieved for B = 159 G.
in which f (r) is the Fourier transform of f (q).3 Going to zero range, we let f (q) = 1, thus simplifying the right-hand side of (12) to [λ − λ2 α 2 /( − E)][δ(r) + 2ρa ]ψ(0). However, for our present purposes it is simpler to work with the form in (11) which, upon operating on both sides by V , gives the Lippmann– Schwinger equation T = V + V GQB T ,
(13)
where the effective T matrix is identified according to T |φ = V |ψ. By solving (13) as was done in (4), the poles of the T matrix are 2ρa f (q)2 λ2 α 2 − = 0. 1+ λ− (14) −E E 2q 2 − E + 2iε q
To relate this expression to the two-body parameters, the integrals are decomposed as a sum of an inverse range plus a radial integral: q
1 f (q)2 E = + 2 2q − E + 2iε b 8π 2
∞ 0
f (q)2 dq. q 2 − E/2 + iε
(15)
With f (q) = 1, we find that √ despite the sign of E, the integral on the right reduces to π/ −2E. Because these equations describe scattering within the medium, the two-body energy must be appropriately generalized to twice the chemical potential, E = 2μ. Through this identification, the matrix pole becomes a relationship between μ and the atomic density, ρa =
1 1 μ+ (−μ)3/2 . 8πa( − 2μ) 8π
(16)
On the right-hand side, the first term is the inverse scattering length as defined in (5), but where the detuning has been shifted by twice the chemical potential. Solving Eq. (16) for μ obtains the chemical potential from which other thermodynamic quantities may be derived, all as functions of the atomic density. 3 For a local interaction, V (r), the right side of (12) would be [λ − λ2 α 2 /( − E)][V (r)ψ(r) + 2ρa d 3 r V (r )ψ(r )]. Using a separable potential, this generalizes according to V (r)ψ(r) → f (r) d 3 r f (r )ψ(r ).
According to the second term on the right side of (16), a negative chemical potential must be chosen to avoid having a complex density. Therefore, we first consider the energy per particle for the case where μ < 0. At zero temperature, the Helmholtz free energy (A) is equal to the internal energy (U ) which, given N atoms per volume V , may be expressed in terms of the energy density (u) as U = uV . From the thermodynamic relationship μ = (∂A/∂N )V ,T , the chemical potential is equal to the derivative of the energy density with respect to ρ = N/V . Since the molecular density is a small fraction of the total (ρa ρ), the energy per particle, e = u/ρ, may finally be approximated as e
1 ρa
μ(ρa ) dρa .
(17)
In previous work [8] a variational many-body calculation shows that the μ < 0 solution yields a two-piece energy per particle with a negative slope for all values of the density. Since the pressure acquires the same sign as de/dρ, we identify this solution as the collapsing ground state of the system. Fig. 1 demonstrates that the independent pair solution given by (16) and (17) compares favorably with the variational result, as there is only ∼15% discrepancy between the two. Because of the attractive microscopic interactions, the collapsing solution makes physical sense but seems to contradict the expected result where the lowest order energy per particle is proportional to the product of density with the scattering length, e 4πa()ρa . As demonstrated experimentally, the expected solution is stable against collapse due to the positive scattering length. Aside from the collapsing state, the only other solution to (16) is to allow for a complex chemical potential. Recognizing μ as the phase of the order parameter, the imaginary part acquires a physical meaning as a quantum mechanical decay rate for the condensate. A similar interpretation arises in the description of collective modes, where a dispersion relation that gives rise to complex frequencies signifies the presence of damping [9,10]. In contrast, the effect illustrated by the complex chemical potential indicates that even in the absence of excitations, sufficiently attractive bose systems are themselves inherently unstable despite the positivity of the scattering length [11].
10
Expanding to (16) is
G.E. Cragg, A.K. Kerman / Physics Letters A 371 (2007) 7–10
√
μ in powers of
√ ρa the low-density solution
√ 3/2 μ = 8πa()ρa − i16 2π 3/2 a()5/2 ρa + · · · ,
(18)
which gives the correct low density dependence but is complex at higher order. Comparing this with the many-body treatment, the density equation differs √ from (16) only in the coefficient of (−μ)3/2 , which is 2/(3π 2 ) in the former calculation [8]. Accordingly, the lowest order decay rate obtained √ 3/2 from (18), Γ = (h¯ /m)16 2π 3/2 a()5/2 ρa , is 17% smaller than its many-body counterpart. Under the conditions of the 85 Rb experiments, the full scattering length is 193 Bohr radii, corresponding to 104 atoms within a cloud of radius 25 µm. Using the average density, the decay time, τ = 1/Γ , is found to be approximately 17 s, which supports the earlier result of 14 s. Both calculations are in qualitative agreement with the observed 10 s lifetime of the 85 Rb condensate [6]. Using a small oscillation analysis, it is possible to show that the excited solution decays into collective excitations of the collapsing lower state.4 Hence, although there is a solution that is stable against collapse, it has an inherent instability which destroys the condensate by heating. By considering the scattering of a pair embedded inside the condensate of all other particles, we have deduced the manybody equations of state for a system of attractive bosons. Due to the bosonic character of the assembly, the pair interaction was effectively altered in that particles preferentially scattered into the bulk condensate. Defining a projection operator that weights the two-body potential by the atomic density, we derived the effective T matrix, the poles of which gave the chemical potential of the system. One solution gave rise to a negative pressure, thus indicating a collapsing state despite the positive scattering length. The other solution was stable against collapse, but manifested a new instability in the form of a complex valued
4 The small oscillation analysis will be presented in a future publication by the current authors.
chemical potential, where the imaginary part quantified the decay rate of the macroscopic quantum state. Since these results are in quantitative agreement with earlier many-body calculations, we conclude that the bosonic independent pair approximation can provide both an intuitive and accurate description of systems such as the one considered. Acknowledgement This work was performed under the auspices of the US Department of Energy at the Los Alamos National Laboratory managed by Los Alamos National Security LLC. Support for this work came from the LANL Laboratory Directed Research and Development (LDRD) Program. References [1] A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, Dover, New York, 2003, p. 357. [2] K.A. Brueckner, C.A. Levinson, Phys. Rev. 97 (1954) 1344. [3] H.A. Bethe, J. Goldstone, Proc. R. Soc. A 238 (1957) 551. [4] L.C. Gomes, J.D. Walecka, V.F. Weisskopf, Ann. Phys. (N.Y.) 3 (1958) 241. [5] H. Feshbach, Ann. Phys. (N.Y.) 281 (2000) 519, treats coupled channels scattering in the context of nuclear reactions, from which the present analysis is adopted. [6] S.L. Cornish, N.R. Claussen, J.L. Roberts, E.A. Cornell, C.E. Wieman, Phys. Rev. Lett. 85 (2000) 1795. [7] S. Fantoni, T.M. Nguyen, S.R. Shenoy, A. Sarsa, Phys. Rev. A 66 (2002) 033604, describes a variational treatment where an independent pair ansatz is taken for the many-body wave function. [8] G.E. Cragg, A.K. Kerman, Phys. Rev. Lett. 94 (2005) 190402. [9] S.T. Beliaev, Sov. Phys. JETP 34 (1958) 299. [10] L. Landau, J. Phys. (Moscow) 10 (1946) 25. [11] A close analogy to our result occurs for the case of a complex action which signifies the decay of a constant, uniform electric field, as discussed by J. Schwinger, Phys. Rev. 82 (1951) 664.
Independent pair approximation for attractive bose condensates George E. Cragg a,∗ , Arthur K. Kerman b a Los Alamos National Laboratory, Los Alamos, NM 87545, USA b Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 1 February 2007; received in revised form 20 May 2007; accepted 24 May 2007 Available online 12 June 2007 Communicated by C.R. Doering
Abstract A bosonic analog of the independent pair approximation is developed in the context of a zero-temperature assembly with a positive s-wave scattering length, but attractive interparticle interactions. By preferentially scattering any pair back into the condensate, we account for the effect that all other particles have on the two-body state inside the medium. Instead of the energy of a pair, the poles of the resulting T matrix now give the chemical potential of the system. One pole reveals a collapsing many-body ground state whereas the other gives a complex valued chemical potential, where the imaginary part quantifies the decay rate of the condensate. Both solutions are in close agreement with those obtained in a variational calculation. © 2007 Elsevier B.V. All rights reserved. PACS: 03.75.Nt; 03.65.Nk; 05.30.Jp
Originally developed for the description of nuclear matter, the independent pair approximation treats a pair of particles as if the interaction between them is modified by the surrounding particles in which the pair is embedded [1–4]. In this way, finding the energy of the many-body collective reduces to the solution of an effective scattering problem. Especially important applications of this approach include cases in which perturbation expansions cannot be applied, such as inside nuclei where the nucleon–nucleon force was once thought to be accurately modeled by an infinitely repulsive core. Unlike the case of fermions where the medium effects are accounted for by excluding the scattering into occupied levels, we discuss the bosonic analog where the scattered pair preferentially fills a single state. Specifically, an independent pair model is sought for a system of attractive bosons that can also pair together into what we call a molecular Feshbach state [5]. First, we examine the two-body scattering problem in vacuum, where the inclusion of the molecular state is dealt with by deciding how its presence alters the interatomic interac-
* Corresponding author.
E-mail address: [email protected] (G.E. Cragg). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.05.114
tion. This result is then used to define the effective potential between pairs. Denoting the particle wave numbers by k1 and k2 , it is convenient to define the relative and total momenta by q = (k1 − k2 )/2 and P = k1 + k2 , respectively. In using a separable form for the two-body potential, q|V |q = λf (q)f (q ),
(1)
the parameter λ is identified as the strength whereas f (q) is the form factor for which f (0) ≡ 1. Both the Feshbach wave function, φ, and the scattering wave function, ψ(q), are self consistently described by the coupled Schrödinger equations 2 2q − E ψ(q) + λf (q) f (q )ψ(q ) + λαf (q)φ = 0, (2a) q
( − E)φ + λα
f (q )ψ(q ) = 0,
(2b)
q
where the interaction (1) has been used in the ψ equation. On the scale of h¯ 2 /2m, E is the energy of the pair and is an energy difference or detuning of the Feshbach state relative to the free atoms. Since these two equations must become independent in the limit where the interaction vanishes, the coupling strength
8
G.E. Cragg, A.K. Kerman / Physics Letters A 371 (2007) 7–10
is represented by the product λα. Furthermore, the momentum dependence of the coupling is conveniently accounted for by the form factor, f (q). Elimination of the molecular state in (2b) results in a single scattering equation from which the effective two-body interaction strength is given by λ→λ−
λ2 α 2 . −E
(3)
Therefore, in all further expressions, the presence of molecules can easily be incorporated by shifting the strength according to (3). An important parameter in low-energy scattering is the s-wave scattering length which is found from its relationship to the T matrix, |q |T |q|q =q=0 = 8πa(). Relating T to the potential requires the Lippmann–Schwinger equation, T = V + V GT ,
(4)
which contains the single-particle Green’s function given by G = (E − H0 − 2iε)−1 . As usual, H0 is the free Hamiltonian whereas ε is a parameter which is taken to zero from the positive side of the real axis. Taking V as the two-body interaction (1) along with the shifted strength (3), a separable T matrix, q |T |q = tf (q )f (q), solves (4), thus obtaining the effective scattering length as 1 1 1 + . = 2 2 8πa() λ − λ α / b
(5)
The range, b, is defined by b−1 = q f (q)2 /2q 2 , where the no tation q is shorthand for (2π)−3 d 3 q. As treated here, an attractive interaction can be simplified by taking b → 0 and λ → 0− such that the effective scattering length is positive. Correspondingly, this choice of parameters is relevant to atomic condensation experiments in 85 Rb [6].1 Eqs. (1), (3) and (5) account for the presence of a molecular state when constructing an independent pair model for bosons. Consider a collection of particles interacting with each other through the two-body separable potential (1) with the strength given by (3). From the independent particle perspective, the collective is viewed as an effective potential through which each of the constituents move. In a uniform medium, this single-particle potential is independent of spatial location, but does depend on the momentum. To second order in the relative momentum, q, the effective interaction is m − 1 q 2, W (q) = W0 + (6) m∗ where the coefficient of the quadratic term is expressed in terms of the particle mass, m, and the effective mass, m∗ .2 As a result of the medium’s interaction, the energy of a pair, E, is in general shifted from the value that would be obtained in vacuum, E0 . Let i and j label two particles in some pair. Summing 1 In the absence of molecular coupling (α = 0), Eq. (5) indicates that the negative background scattering length of 85 Rb implies an attractive interaction. 2 For further details on the relationship between the effective mass and the single-particle potential, see [1,4].
the shifts over all potential partners {j } produces the total energy shift on the arbitrarily chosen i. Equating this result to the single-particle potential obtains a self-consistent value for the effective mass. Quantitatively, the energy shift of a pair is given by E = E − E0 = ϕ0 |V |ψ, which follows from the normalization, φ0 |ψ = 1, of the center of mass wave function |ψ = |ϕ0 +
|ϕn
n=0
1 ϕn |V |ψ. E − En
(7)
This equation is simply the projection of the Schrödinger equation onto the free particle eigenstates, {|ϕn }. Using the separable potential along with the strength that accounts for the Feshbach coupling, the continuum energy shift is λ2 α 2 f (q) f (q )ψ(q ). E = q|V |ψq = λ − (8) −E q
Hence, in the zero-range limit, λ → 0− , the shift vanishes whereby the effective mass assumes its value in vacuum: m∗ −→ m.
(9)
λ→0−
Although going to zero range simplifies the problem to a collection of free pairs, the medium nonetheless affects the interaction within each pair. To arrive at the bosonic form of this effect, it is helpful to first recall that in the fermionic case the scattering state is precluded from all occupied momenta, resulting in a “Fermi force” arising due to the presence of the medium. However, bosons have no limit to the occupation of each state, implying that the interaction should be weighted by the sum of the occupation numbers for the two particles in the pair. Therefore, the bosonic projection operator has the center of mass matrix element q |QB |q = (1 + nq+P/2 + nq−P/2 )δ(q − q )
−→ 1 + 2ρa δ(q) δ(q − q ), T →0
(10)
where q ± P/2 are the particle momenta in the laboratory. Also, the occupation numbers, nq±P/2 are bose distributions depending on temperature in the normal way. At zero temperature, the momenta vanish, reducing the occupation number sum to twice the atomic density, ρa , times a delta function, as given by Eq. (10). Since this projection weights the interaction, its action is on V |ψ, thus giving the Bethe–Goldstone equation for the two-body state: |ψ = |φ + GQB V |ψ.
(11)
Alternatively, we may use the separable potential along with its shifted strength (3) to arrive at the integro-differential equation for the wave function [7] 2 λ2 α 2 f (r) + 2ρa d 3 r f (r ) 2∇ + E ψ(r) = λ − −E × d 3 r f (r )ψ(r ), (12)
G.E. Cragg, A.K. Kerman / Physics Letters A 371 (2007) 7–10
9
Fig. 1. For the collapsing lower state of 85 Rb, the independent pair model of Eq. (17) (solid curve) is compared with the variational many-body result (dashed curve). Plot (a) shows the case for an applied magnetic field of 162.3 G whereas part (b) shows an expanded comparison of the two curves. In each case, the dotted vertical line ending at the single point marks the quantum phase transition found in the many-body analysis. Decreasing the applied field shifts this point to lower density. Hence, for the same range of densities as in (a), a more global comparison is achieved for B = 159 G.
in which f (r) is the Fourier transform of f (q).3 Going to zero range, we let f (q) = 1, thus simplifying the right-hand side of (12) to [λ − λ2 α 2 /( − E)][δ(r) + 2ρa ]ψ(0). However, for our present purposes it is simpler to work with the form in (11) which, upon operating on both sides by V , gives the Lippmann– Schwinger equation T = V + V GQB T ,
(13)
where the effective T matrix is identified according to T |φ = V |ψ. By solving (13) as was done in (4), the poles of the T matrix are 2ρa f (q)2 λ2 α 2 − = 0. 1+ λ− (14) −E E 2q 2 − E + 2iε q
To relate this expression to the two-body parameters, the integrals are decomposed as a sum of an inverse range plus a radial integral: q
1 f (q)2 E = + 2 2q − E + 2iε b 8π 2
∞ 0
f (q)2 dq. q 2 − E/2 + iε
(15)
With f (q) = 1, we find that √ despite the sign of E, the integral on the right reduces to π/ −2E. Because these equations describe scattering within the medium, the two-body energy must be appropriately generalized to twice the chemical potential, E = 2μ. Through this identification, the matrix pole becomes a relationship between μ and the atomic density, ρa =
1 1 μ+ (−μ)3/2 . 8πa( − 2μ) 8π
(16)
On the right-hand side, the first term is the inverse scattering length as defined in (5), but where the detuning has been shifted by twice the chemical potential. Solving Eq. (16) for μ obtains the chemical potential from which other thermodynamic quantities may be derived, all as functions of the atomic density. 3 For a local interaction, V (r), the right side of (12) would be [λ − λ2 α 2 /( − E)][V (r)ψ(r) + 2ρa d 3 r V (r )ψ(r )]. Using a separable potential, this generalizes according to V (r)ψ(r) → f (r) d 3 r f (r )ψ(r ).
According to the second term on the right side of (16), a negative chemical potential must be chosen to avoid having a complex density. Therefore, we first consider the energy per particle for the case where μ < 0. At zero temperature, the Helmholtz free energy (A) is equal to the internal energy (U ) which, given N atoms per volume V , may be expressed in terms of the energy density (u) as U = uV . From the thermodynamic relationship μ = (∂A/∂N )V ,T , the chemical potential is equal to the derivative of the energy density with respect to ρ = N/V . Since the molecular density is a small fraction of the total (ρa ρ), the energy per particle, e = u/ρ, may finally be approximated as e
1 ρa
μ(ρa ) dρa .
(17)
In previous work [8] a variational many-body calculation shows that the μ < 0 solution yields a two-piece energy per particle with a negative slope for all values of the density. Since the pressure acquires the same sign as de/dρ, we identify this solution as the collapsing ground state of the system. Fig. 1 demonstrates that the independent pair solution given by (16) and (17) compares favorably with the variational result, as there is only ∼15% discrepancy between the two. Because of the attractive microscopic interactions, the collapsing solution makes physical sense but seems to contradict the expected result where the lowest order energy per particle is proportional to the product of density with the scattering length, e 4πa()ρa . As demonstrated experimentally, the expected solution is stable against collapse due to the positive scattering length. Aside from the collapsing state, the only other solution to (16) is to allow for a complex chemical potential. Recognizing μ as the phase of the order parameter, the imaginary part acquires a physical meaning as a quantum mechanical decay rate for the condensate. A similar interpretation arises in the description of collective modes, where a dispersion relation that gives rise to complex frequencies signifies the presence of damping [9,10]. In contrast, the effect illustrated by the complex chemical potential indicates that even in the absence of excitations, sufficiently attractive bose systems are themselves inherently unstable despite the positivity of the scattering length [11].
10
Expanding to (16) is
G.E. Cragg, A.K. Kerman / Physics Letters A 371 (2007) 7–10
√
μ in powers of
√ ρa the low-density solution
√ 3/2 μ = 8πa()ρa − i16 2π 3/2 a()5/2 ρa + · · · ,
(18)
which gives the correct low density dependence but is complex at higher order. Comparing this with the many-body treatment, the density equation differs √ from (16) only in the coefficient of (−μ)3/2 , which is 2/(3π 2 ) in the former calculation [8]. Accordingly, the lowest order decay rate obtained √ 3/2 from (18), Γ = (h¯ /m)16 2π 3/2 a()5/2 ρa , is 17% smaller than its many-body counterpart. Under the conditions of the 85 Rb experiments, the full scattering length is 193 Bohr radii, corresponding to 104 atoms within a cloud of radius 25 µm. Using the average density, the decay time, τ = 1/Γ , is found to be approximately 17 s, which supports the earlier result of 14 s. Both calculations are in qualitative agreement with the observed 10 s lifetime of the 85 Rb condensate [6]. Using a small oscillation analysis, it is possible to show that the excited solution decays into collective excitations of the collapsing lower state.4 Hence, although there is a solution that is stable against collapse, it has an inherent instability which destroys the condensate by heating. By considering the scattering of a pair embedded inside the condensate of all other particles, we have deduced the manybody equations of state for a system of attractive bosons. Due to the bosonic character of the assembly, the pair interaction was effectively altered in that particles preferentially scattered into the bulk condensate. Defining a projection operator that weights the two-body potential by the atomic density, we derived the effective T matrix, the poles of which gave the chemical potential of the system. One solution gave rise to a negative pressure, thus indicating a collapsing state despite the positive scattering length. The other solution was stable against collapse, but manifested a new instability in the form of a complex valued
4 The small oscillation analysis will be presented in a future publication by the current authors.
chemical potential, where the imaginary part quantified the decay rate of the macroscopic quantum state. Since these results are in quantitative agreement with earlier many-body calculations, we conclude that the bosonic independent pair approximation can provide both an intuitive and accurate description of systems such as the one considered. Acknowledgement This work was performed under the auspices of the US Department of Energy at the Los Alamos National Laboratory managed by Los Alamos National Security LLC. Support for this work came from the LANL Laboratory Directed Research and Development (LDRD) Program. References [1] A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, Dover, New York, 2003, p. 357. [2] K.A. Brueckner, C.A. Levinson, Phys. Rev. 97 (1954) 1344. [3] H.A. Bethe, J. Goldstone, Proc. R. Soc. A 238 (1957) 551. [4] L.C. Gomes, J.D. Walecka, V.F. Weisskopf, Ann. Phys. (N.Y.) 3 (1958) 241. [5] H. Feshbach, Ann. Phys. (N.Y.) 281 (2000) 519, treats coupled channels scattering in the context of nuclear reactions, from which the present analysis is adopted. [6] S.L. Cornish, N.R. Claussen, J.L. Roberts, E.A. Cornell, C.E. Wieman, Phys. Rev. Lett. 85 (2000) 1795. [7] S. Fantoni, T.M. Nguyen, S.R. Shenoy, A. Sarsa, Phys. Rev. A 66 (2002) 033604, describes a variational treatment where an independent pair ansatz is taken for the many-body wave function. [8] G.E. Cragg, A.K. Kerman, Phys. Rev. Lett. 94 (2005) 190402. [9] S.T. Beliaev, Sov. Phys. JETP 34 (1958) 299. [10] L. Landau, J. Phys. (Moscow) 10 (1946) 25. [11] A close analogy to our result occurs for the case of a complex action which signifies the decay of a constant, uniform electric field, as discussed by J. Schwinger, Phys. Rev. 82 (1951) 664.