Multidimensional quantum solitons with nondegenerate parametric ...

PHYSICAL REVIEW A, VOLUME 61, 063816

Multidimensional quantum solitons with nondegenerate parametric interactions: Photonic and Bose-Einstein condensate environments K. V. Kheruntsyan and P. D. Drummond Department of Physics, University of Queensland, St. Lucia, QLD 4067, Australia 共Received 3 August 1999; published 17 May 2000兲 We consider the quantum theory of three fields interacting via parametric and repulsive quartic couplings. This can be applied to treat photonic ␹ (2) and ␹ (3) interactions, and interactions in atomic Bose-Einstein condensates or quantum Fermi gases, describing coherent molecule formation together with s-wave scattering. The simplest two-particle quantum solitons or bound-state solutions of the idealized Hamiltonian, without a momentum cutoff, are obtained exactly. They have a pointlike structure in two and three dimensions—even though the corresponding classical theory is nonsingular. We show that the solutions can be regularized with a momentum cutoff. The parametric quantum solitons have much more realistic length scales and binding energies than ␹ (3) quantum solitons, and the resulting effects could potentially be experimentally tested in highly nonlinear optical parametric media or interacting matter-wave systems. N-particle quantum solitons and the ground state energy are analyzed using a variational approach. Applications to atomic/molecular BoseEinstein condensates 共BEC’s兲 are given, where we predict the possibility of forming coupled BEC solitons in three space dimensions, and analyze ‘‘superchemistry’’ dynamics. PACS number共s兲: 42.65.Tg, 03.65.Ge, 03.75.Fi, 11.10.St

I. INTRODUCTION

Quantum solitons 关1兴 or bound states of interacting fields are generalizations of nonlinear solitonic solutions of classical wave theory to include quantum fields. Exactly solvable cases include many-body bound states of bosons interacting via a ␦ -function potential in one space dimension. This model 共often called the nonlinear Schro¨dinger model兲 was solved by Lieb, Liniger, McGuire, and Yang 关2兴. Recently it was predicted that this solvable model could lead to experimentally observable quantum effects including quantum squeezing in optical fiber solitons 关3,4兴. This prediction has now been verified experimentally 关5兴. Other examples of exactly soluble models are generally restricted either to one space dimension, or to physically inaccessible systems like the quantum Davey-Stewartson model 关6兴. An exception is Laughlin’s highly innovative theory of a two-dimensional electron gas in an external magnetic field 关7兴, which was able to explain the fractional quantum Hall effect 关8兴. Similar techniques have recently been proposed for treating interacting Bose gases in higher dimensions, in the limit of very weak couplings, leading to an elementary theory of a quantum vortex 关9兴. Experimental success in Bose-Einstein condensation of atomic gases 关10兴 makes it possible that quantum soliton behavior could become observable in ultralow-temperature nonlinear atom optics, as well as with photons. In a recent paper 关11兴, we showed that it is possible to obtain an exact solution in one, two, and three space dimensions, in a nonlinear quantum field theory that includes the most fundamental property that distinguishes quantum mechanics from quantum field theory—that is, the ability to create and destroy particles. The simplest cubic interaction involving two boson fields—the parametric interaction of the ˆ †2 —was analyzed for bound states in higher diˆ 21 ⌿ form ⌿ mensions, resulting in soluble cases with unusual and unex1050-2947/2000/61共6兲/063816共20兲/$15.00

pected properties. This degenerate parametric theory—with similarities to the Friedberg-Lee 关12兴 model of high-T C superconductivity—has bound states in one space dimension 关13,14兴, but is unstable 共like the nonlinear Schro¨dinger model with an attractive ␦ -function potential兲 in higher dimensions. Unlike the nonlinear Schro¨dinger model, the instability does not occur at the classical level. Indeed, classical parametric solitons in higher dimensions are both theoretically predicted 关15–17兴 and observed to exist 关18兴. With the inclusion of an additional 共repulsive兲 quartic interaction term in the Hamiltonian, a rigorous lower bound to the energy was proved to exist, and we demonstrated the existence of exact two-particle bound states in higher dimensions 关11,19兴. These new types of quantum solitons have a finite binding energy, but the corresponding two-particle wave function has a zero radius; the pointlike structure of these bound states can be termed a ‘‘quantum singularity.’’ With a momentum cutoff imposed on the couplings, the bound states develop a finite radius. In the present paper, we extend these earlier results to include the nondegenerate case of parametric interaction 关20兴 of three distinct fields with either Bose or Fermi statistics 共rather than two bosonic fields兲. The results demonstrate the existence of exact two-particle nondegenerate eigenstates in higher dimensions, having a pointlike structure in space, with a finite energy when there is no momentum cutoff. However, typical physical systems that can be experimentally identified as having the requisite three-wave bosonic interactions usually have momentum cutoffs. These cutoffs, of course, provide a spatial extent to the bound states. We therefore provide solutions that include cutoff effects as well. Estimates of typical binding energies and soliton characteristic radii are given for photonic interactions in highly nonlinear optical materials. They appear to be of more realistic magnitudes for possible experiments, as compared to earlier known quantum solitons based on cubic nonlinearities 共see, e.g., 关2,21兴, and 关11,14兴 for comparison兲.

61 063816-1

©2000 The American Physical Society

K. V. KHERUNTSYAN AND P. D. DRUMMOND

PHYSICAL REVIEW A 61 063816

In addition, we discuss the application of the basic model to coherently coupled atomic/molecular Bose-Einstein condensates 共BEC’s兲. This provides the possibility of extending our earlier results 关22兴 on ‘‘superchemistry’’ in degenerate parametric interactions to a larger variety of interacting quantum gases, i.e., to three-species 共two atomic and one molecular兲 BEC systems. We present here a mean-field theory analysis which predicts, at large particle number, a transition to a classical soliton domain, where stable threedimensional BEC solitons can form in certain parameter ranges. II. MODEL

We start by considering the following quantum effective Hamiltonian: ˆ int , ˆ ⫽H ˆ 0 ⫹H H where ˆ 0 ⫽ប H

冕

冉兺 3

d Dx

i⫽1

共1兲

冊

ប ˆ i 共 x兲兩 2 ⫹⌬ ␻ ⌿ ˆ †3 共 x兲 ⌿ ˆ 3 共 x兲 , 兩 “⌿ 2m i 共2兲

(␹) (␬) ˆ int ˆ int ˆ int ⫽H ⫹H , with and H (␹) ˆ int H ⫽ប

冕冕冕

d D xd D yd D z␹ D 共 x,y,z兲

ˆ 1 共 x兲 ⌿ ˆ 2 共 y兲 ⌿ ˆ †3 共 z兲 ⫹H.c.兴 , ⫻关⌿ 3

ប (␬) ˆ int ⫽ H 2 i, j⫽1

兺

冕冕冕冕

oms 共of masses m 1 and m 2 ) convert into diatomic molecules (i j) 共of mass m 3 ⫽m 1 ⫹m 2 ), while the quartic couplings ␬ D would refer to the strength of intra- and interspecies twobody collisions. ˆ 2 ), the coherˆ 1 ⫽⌿ In the case of degenerate couplings (⌿ ent process of dimerization in atomic/molecular BEC interactions and the possibility of formation of coupled atomicmolecular solitons has been studied in 关19,22,23兴. Pure quartic interactions in two-species Bose condensates have been analyzed in 关24兴, while the interplay between parametric and 共attractive兲 quartic interactions in optical soliton propagation, at the classical level, has been studied in 关25兴. An important feature encountered in the treatment of the present nondegenerate parametric interaction is that, although we have specified Bose statistics for all three interˆ i , some of the results obtained here will also acting fields ⌿ be valid if fermionic fields are involved and the corresponding commutators are replaced by anticommutators. An exˆ 2 are ˆ 1 and ⌿ ample of such systems is the case where ⌿ fermionic, as in the ‘‘s-channel’’ model of high-T C superconductivity by Friedberg and Lee 关12兴. Similarly, the case ˆ 3 are fermionic, as in the Lee–Van Hove ˆ 2 and ⌿ where ⌿ model of nuclear interactions 关26兴, is also treatable. To simplify the theory we consider in Sec. III the approximation in which we assume short range interactions and, taking into account translational invariance considerations, replace the interaction potentials by ␦ function pseudopotentials. In this case, the interacting part of the Hamiltonian is

共3兲 ˆ int ⫽ប H

(i j) d D xd D yd D x⬘ d D y⬘ ␬ D 共 x,y,x⬘ ,y⬘ 兲

ˆ †i 共 x兲 ⌿ ˆ †j 共 y兲 ⌿ ˆ i 共 x⬘ 兲 ⌿ ˆ j 共 y⬘ 兲 . ⫻⌿

冉

ˆ 1 共 x兲 ⌿ ˆ 2 共 x兲 ⌿ ˆ †3 共 x兲 d Dx ␹ D关 ⌿

ˆ †1 共 x兲 ⌿ ˆ †2 共 x兲 ⌿ ˆ 3 共 x兲兴 ⫹⌿

共4兲

ˆ 1, ⌿ ˆ 2 , and ⌿ ˆ 3 are three Bose fields with commutaHere ⌿ ˆ i (x),⌿ ˆ †j (x⬘ ) 兴 ⫽ ␦ i j ␦ (x⫺x⬘ ). In addition, tion relations 关 ⌿ m 1 , m 2 , and m 3 are the corresponding effective masses and ⌬ ␻ is the phase mismatch or the bare formation energy of ˆ 3 . Nonlinear interactions are included via the the field ⌿ parametric interaction potential ␹ D describing a particle ˆ 1 and number nonconserving process, in which a pair of ⌿ ˆ 3 quantum is created, while ˆ 2 quanta is destroyed and a ⌿ ⌿ ␬ D(i j) is the particle number conserving potential describing quartic self- and cross-interactions between the fields, in D (D⫽1,2,3) space dimensions. In the case of optical interactions the couplings are due to quadratic and cubic polarizabilities of the nonlinear medium, giving rise to the parametric process of frequency conversion 共sum-frequency generation兲, together with self- and crossphase modulation processes. The above effective Hamiltonian can also be applied to describe nonlinear interactions ˆ2 ˆ 1 and ⌿ of matter-wave fields, such as in coupled atomic (⌿ ˆ 3 field兲 Bose condensates. In this fields兲 and molecular (⌿ case, the parametric coupling ␹ D would refer to the rate of coherent process of atomic dimerization, where pairs of at-

冕

3

⫹

兺

i, j⫽1

1 2

冊

ˆ †i 共 x兲 ⌿ ˆ †j 共 x兲 ⌿ ˆ i 共 x兲 ⌿ ˆ j 共 x兲 . ␬ D(i j) ⌿

共5兲

This is a very idealized model. We note that such models in quantum field theory are usually treated in the context of renormalized perturbation theory, with the understanding that the coupling constants are a function of an implicit momentum cutoff. However, we shall demonstrate a rather unexpected and remarkable result, which is that the above idealized Hamiltonian has an exact ground state with a finite binding energy—even without a cutoff or renormalization (i j) ⬎0 there are no procedure. We emphasize that provided ␬ D energy divergences or collapsing behavior in this idealized cubic-quartic model, unlike the case of a Bose gas with purely quartic attractive ␦ -function interactions. On the other hand, for a Bose gas with purely quartic repulsive ␦ -function interactions the exact eigenvalues in more than one dimension are the same as those for free particles, i.e., the ␦ -function pseudopotential produces no scattering and the ground state energy is the same as for a noninteracting Bose gas 关27兴. Instead, the idealized model we consider gives a nontrivial bound state that has a finite binding energy, but involves a pointlike 共zero-radius兲 structure in more than one space dimension. While physical models typically do have a

063816-2

MULTIDIMENSIONAL QUANTUM SOLITONS WITH . . .


momentum cutoff, the exactly soluble model without cutoff is indicative of behavior with a cutoff, and provides some useful insight. A more sophisticated pseudopotential approach would be to employ the regularized method used by Huang, Yang, and Lee 关28兴. However, for simplicity we choose to start with a simple Dirac ␦ -function interaction which has the advantage of giving a Hermitian Hamiltonian. More careful treatment of the ␦ -function interaction would be to incorporate a momentum cutoff imposed on the nonlinear couplings. This is further treated in Sec. IV, where we obtain a regularized bound state with a finite spatial extent in one, two, and three dimensions.

Operating on 兩 ␸ (2) K 典 , Eq. 共8兲, with the Hamiltonian 共1兲, 共2兲, and 共5兲 gives that the two-particle eigenvalue problem (2) (2) ˆ 兩 ␸ (2) H K 典 ⫽E K 兩 ␸ K 典 is equivalent to the following set of equations:

III. CUTOFF INDEPENDENT RESULTS

where E (2) K is the corresponding energy eigenvalue. To solve these equations we introduce the relative and center-of-mass coordinates according to r⫽x⫺y and R ⫽(m 1 x⫹m 2 y)/(m 1 ⫹m 2 ). With these coordinates we have

To construct the general candidate for the eigenstate to the Hamiltonian given by Eqs. 共1兲, 共2兲, and 共5兲, we note that ˆ2 ˆ 1 and ⌿ the parametric interaction transforms pairs of ⌿ ˆ 3 quanta, and vice versa. That is, the quanta into single ⌿ Hamiltonian does not conserve the corresponding particle numbers. However, it does conserve a generalized particle number, or Manley-Rowe invariant, equal to ˆ 2 ⫹2Nˆ 3 ⫽ ˆ ⫽N ˆ 1 ⫹N N

冕

iប 2

冕

共6兲

d x

兺关 ⌿ˆ †i 共 “⌿ˆ i 兲 ⫺ 共 “⌿ˆ †i 兲 ⌿ˆ i 兴 .

i⫽1

共7兲

m 1m 2 , m 1 ⫹m 2

冊

共8兲

where P and Q are one- and two-particle wave functions, respectively. We note that the quartic terms in the interaction Hamiltonian 共5兲 other than the cross-interaction term between the ˆ 2 fields have no effect on the two-particle eigenˆ 1 and ⌿ ⌿ state. For this reason, we will use a simplified notation

␬ D ⬅ 共 ␬ D(12) ⫹ ␬ D(21) 兲 /2⫽ ␬ D(12) ˆ 2. ˆ 1 and ⌿ for the cross-coupling between the fields ⌿

共13兲

and defined M ⬅m 1 ⫹m 2 . Assuming translational invariance we can seek for P(x) in the form of P(x)⫽ P 0 exp(iK•x), where K is the total momentum. As a consequence, Q(x,x) will be proportional to P(x), and therefore we may look for the general expression for Q(x,y) in a separable form: Q(x,y)⫽g(r) P(R). Substituting this into Eqs. 共10兲 and 共11兲, and dividing the energy into center-of-mass and relative components E (2) K ⫽E c ⫹E r , we then solve the equation for P(R), yielding at P(R)⫽ P 0 exp(iK"R), with K 2 ⫽ 兩 K兩 2 ⫽2M E c /ប 2 , and as a result 共14兲

The remaining equation for the g(r) function is rewritten as ⵜ 2 g 共 r兲 ⫺r ⫺2 0 g 共 r兲 ⫽

ˆ †1 共 x兲 ⌿ ˆ †2 共 y兲兩 0 典 , d D x d D yQ 共 x,y兲 ⌿

共12兲

where we have introduced a reduced mass

ˆ †3 共 x兲 d D xP 共 x兲 ⌿

冕冕

册

2 2 E (2) K ⫽ប K / 共 2m 3 兲 ⫹ប⌬ ␻ ⫹ប ␹ D g 共 0 兲 .

We consider first the two-particle (N⫽2) eigenstate which must have the form of a superposition state:

⫹

冊

m 1 x⫹m 2 y ⫹ ␬ D Q 共 x,y兲 ␦ 共 x⫺y兲 , 共11兲 m 1 ⫹m 2

ប2 2 ប2 2 ប2 2 ប2 2 ⵜ ⫹ ⵜ ⫽ ⵜ , ⵜ ⫹ 2m 1 x 2m 2 y 2M R 2 ␮ r

A. Two-particle eigenvalue equation

冉冕

冋冉

⫽ប ␹ D P

␮⫽

We therefore search for states 兩 ␸ (N) K 典 that are eigenstates ˆ ˆ ˆ of H , N , and K, with energy eigenvalues E (N) K .

兩 ␸ (2) K 典⫽

冊

ប2 2 ប2 2 ⵜ ⫹ ⵜ Q 共 x,y兲 ⫹E (2) K Q 共 x,y 兲 2m 1 x 2m 2 y

ˆ 1兩 2⫹ 兩 ⌿ ˆ 2 兩 2 ⫹2 兩 ⌿ ˆ 3兩 2 兲 . d x 共兩⌿

3

D

冉

D

In addition, the Hamiltonian is translationally invariant, and thus conserves the total momentum given by ˆ ⫽⫺ Pˆ⫽បK

ប2 2 ⵜ P 共 x兲 ⫹ 共 E (2) K ⫺ប⌬ ␻ 兲 P 共 x 兲 ⫽ប ␹ D Q 共 x,x 兲 , 共10兲 2m 3

2␮ 关 ␹ D ⫹ ␬ D g 共 0 兲兴 ␦ 共 r兲 , ប

共15兲

where we have defined a length scale r 0 , according to r ⫺2 0 ⫽⫺

2␮Er ប2

⫽

␮ K 2 2 ␮ E (2) K . ⫺ 2 M ប

共16兲

Together with Eq. 共14兲, this implies that the energy eigenvalue is given by E (2) K ⫽

共9兲

ប2 ប 2K 2 ⫺ , 2M 2 ␮ r 20

共17兲

where r 0 is to be found by solving the following eigenvalue equation: 063816-3


r ⫺2 0 ⫽

2␮ 关 ⌬⫺ ␹ D g 共 0 兲兴 . ប


共18兲

E (2) K ⭓

Here r 0 must be real and positive for a localized bound state or quantum soliton solution. The quantity

E (2) b ⬅

ប2 2 ␮ r 20

⫽ប⌬⫺ប ␹ D g 共 0 兲

共19兲

can be interpreted as the binding energy of the two-particle quantum soliton with the momentum K, and we have defined ⌬⬅

冉

冊

ប K2 K2 ⫺⌬ ␻ . ⫺ 2 M m3

共20兲

Equations 共17兲 and 共18兲 are equivalent to formulating the eigenvalue problem directly in terms of Eq. 共14兲, where g(0) is to be found by solving the following equation: ⵜ 2 g 共 r兲 ⫺

2␮ 2␮ 关 ⌬⫺ ␹ D g 共 0 兲兴 g 共 r兲 ⫽ 关 ␹ D ⫹ ␬ D g 共 0 兲兴 ␦ 共 r兲 . ប ប 共21兲

Thus, the two-particle 共or diboson兲 eigenstate candidate 共8兲 that is a simultaneous eigenstate of the momentum operator takes the following form: 兩 ␸ (2) K 典⫽

冋冕

ˆ †3 共 x兲 ⫹ d D xe iK•x⌿

冉

ˆ †1 R⫹ ⫻⌿

冊冉

冕冕

冊册

共22兲

B. Energy lower bound for two-particle case

The stability of our Hamiltonian in the two-particle sector can be proved by finding a lower bound E l to the Hamil(2) ˆ (2) (2) (2) (2) tonian energy, E (2) K ⫽ 具 ␸ K 兩 H 兩 ␸ K 典 / 具 ␸ K 兩 ␸ K 典 , so that E K (2) ⭓E l . Applying the Hamiltonian to 兩 ␸ K 典 , and using the symmetry property of the two-particle correlation function g(x)⫽g(⫺x), one can find that

冉冕

E (2) K ⫽ 1⫹

冊冉冕 ⫺1

d D rg 2 共 r兲

ប 2K 2 ⫹ 2M

冕

ប2 2␮

冊

ប 2K 2 ⫺ 2M

2 ⫹⌬ ␬ D 兲 ប共 ␹D

冋冕

␬ D 1⫹

d r g 共 r兲 D

2

册

共24兲

,

where ⌬ is defined in Eq. 共20兲. If ( ␹ D ) 2 ⫹⌬ ␬ D ⭐0, then the lower bound E l is given by E l ⫽ប 2 K 2 /(2M )⫽E c . This has a simple interpretation as (2) ⭓E c 共or providing the center-of-mass energy, so that E K (2) E r ⫽E K ⫺E c ⭓0) and no bound states, with E r ⫽E (2) K ⫺E c ⬍0, are possible in this case. If, however, 共 ␹ D 兲 2 ⫹⌬ ␬ D ⬎0,

共25兲

which is the case that we focus on in this paper, then we have E (2) K ⭓ ⫽

ប 2K 2 ប共 ␹D兲2 ⫺ប⌬⫺ 2M ␬D ប 2K 2 ប共 ␹D兲2 ⫹ប⌬ ␻ ⫺ ⬅E l . 2m 3 ␬D

共26兲

2 This implies that E r ⫽E (2) K ⫺E c ⭓ប⌬ ␻ ⫺ប( ␹ D ) / ␬ D , and bound states may become available.

C. Exact diboson solutions

d D r 兩 “g 共 r兲兩 2

g 共 r 兲 ⫽ 兰 d D kG 共 k兲 exp共 ik•r兲 / 共 2 ␲ 兲 D , where r⫽ 兩 r兩 . Expanding the ␦ function into a Fourier integral, we then obtain the Fourier transform equivalent to Eq. 共15兲: 共 k 2 ⫹r ⫺2 0 兲 G 共 k 兲 ⫽⫺q,

共27兲

where k⫽ 兩 k兩 , and we have defined q⬅2 ␮ 关 ␹ D ⫹ ␬ D g 共 0 兲兴 /ប.

共28兲

Solving Eq. 共27兲 for G(k) and substituting it into the expression for g(r) we find g 共 r 兲 ⫽⫺

ប 2K 2 d D r g 2 共 r兲 ⫹ ⫹ប⌬ ␻ ⫹2ប ␹ D g 共 0 兲 2m 3

⫹ប ␬ D g 2 共 0 兲 .

冕

Equations 共15兲 and 共18兲 can easily be analyzed using the Fourier transform method. In this approach we seek a solution to Eq. 共15兲 in the form

d D r d D Re iK•Rg 共 r兲

m 2r † m r ˆ 2 R⫺ 1 兩 0 典 . ⌿ M M

⭓

ប 2 K 2 ប ␬ D g 2 共 0 兲 ⫹2ប ␹ D g 共 0 兲 ⫺ប⌬ ⫹ 2M 1⫹ d D rg 2 共 r兲

q 共2␲兲

D

冕

d Dk

exp共 ik•r兲 k 2 ⫹1/r 20

.

共29兲

1. One-dimensional case (DÄ1)

共23兲

Omitting the first nonnegative term in the square brackets, we arrive at a lower energy that is rigorously bounded from below if ␬ D ⬎0, according to 063816-4

In the one-dimensional case (D⫽1) the integration gives g 共 r 兲 ⫽⫺

q 2␲

冕

⫹⬁

⫺⬁

dk

exp共 ikr 兲 k

2

⫹1/r 20

⫽⫺

qr 0 exp共 ⫺ 兩 r 兩 /r 0 兲 . 2 共30兲



Using this result at r⫽0 and the definition of q, we solve for g(0) and find that g(0)⫽⫺ ␹ 1 关 ␬ 1 ⫹ប/( ␮ r 0 ) 兴 ⫺1 . Correspondingly, the eigenvalue equation 共18兲 for r 0 is now rewritten as a cubic: 2␮2 ប

2

关共 ␹ 1 兲 2 ⫹⌬ ␬ 1 兴 r 30 ⫹

2 ␮ ⌬ 2 ␮␬ 1 r ⫺ r ⫺1⫽0, 共31兲 ប 0 ប 0

where r 0 must be real and positive for a localized bound state. The analysis of this equation shows that if ␬ 1 ⬎0 and ( ␹ 1 ) 2 ⫹⌬ ␬ 1 ⬎0—that is, under the same conditions that we assumed while proving the lower bound, Eq. 共26兲—then there always exists one positive solution for r 0 . This proves the existence of a one-dimensional two-particle quantum soliton, with a characteristic radius r 0 and a binding energy 2 2 of E (2) b ⫽ប /(2 ␮ r 0 ). In the absence of the quartic term ( ␬ 1 ⫽0) and with perfect phase matching ⌬ ␻ ⫽0 and m 3 ⫽M 共so that ⌬⫽0), the equation for r 0 is solved analytically. This gives the following explicit results for the soliton binding energy and the radius: 2 1/3 4/3 E (2) b ⫽ 共 ប ␮ /2 兲共 ␹ 1 兲 ,

r 0⫽共 ប

2

/2␹ 21 ␮ 2 兲 1/3.

共32兲共33兲

2. Higher-dimensional case (DÄ2,3)

The two- and three-dimensional results are qualitatively different. In these cases we evaluate the integrals for g(0), from Eq. 共29兲, in polar 共for D⫽2) and spherical 共for D ⫽3) coordinates. Using the definition of q, we then solve for /(2 ␮ f D ) 兴 ⫺1 , g(0) and obtain g(0)⫽⫺ ␹ D 关 ␬ D ⫹បr D⫺2 0 where we have defined the dimensionless integral f D⫽

1 2 ␲ D⫺1

冕

⬁

0

dx

x D⫺1 1⫹x 2

共 D⫽2,3兲 .

tidimensional solutions is in their structure and dependence on the additional quartic interaction. In one dimension the bound state has finite characteristic size and is available even without a quartic term in the Hamiltonian. In two and three dimensions, the bound states involve a pointlike structure, yet the corresponding binding energy is finite, if ␬ D ⬎0. If, however, ␬ D ⫽0 we obtain an energy collapse: E (2) K →⫺⬁. Thus, while the additional quartic interaction prevents an energy collapse and makes multidimensional quantum solitons possible, these solitons involve a zero-radius relative localˆ 1 and ⌿ ˆ 2 quanta. ization of the ⌿ The diboson solutions can be regarded as a type of dressˆ 3 quanta, which have a lower energy due to the ing of the ⌿ ˆ 2 quanta. We also note ˆ 1 and ⌿ creation of virtual pairs of ⌿ that in a renormalized theory in which ␹ D and ␬ D are regarded as functions of a momentum cutoff k m , the above result implies that ( ␹ D ) 2 / ␬ D must approach a constant value at large k m , in order that the observed binding energy should be cutoff independent. D. Energy lower bound for N-particle case

The zero-radius form of the two-particle bound states in two and three space dimensions simplifies the treatment of the general case of N-particle bound states, so that one can find an exact ground state solution to this quantum manybody system. To show this first we prove a lower bound to the Hamiltonian energy in the N-particle sector. To do so we ˆ kin neglect the non-negative kinetic energy term 兵 H 3 D 2 2 ˆ ⫽ 兰 d x关兺 i⫽1 (ប /2m i ) 兩 “⌿ i 兩兴其 in the Hamiltonian and conˆ ⫺H ˆ kin , such that H ˆ ˆ R ⫽H sider a reduced Hamiltonian H ˆ ⭓H R . Assuming that ␬ D ⬎0, one can show that ˆ R ⭓ប 关 ⌬ ␻ ⫺ 共 ␹ D 兲 2 / ␬ D 兴 H

共34兲

⫹ប

This integral diverges for D⫽2,3. 共A strict treatment of this divergence, as a mathematical limit, is given in Sec. IV, where it is attributed to k m →⬁, with k m being the upper limit in the integral.兲 Therefore we find that g(0) and hence the energy eigenvalue E (2) K from Eq. 共14兲 are given by g 共 0 兲 ⫽⫺ ␹ D / ␬ D , E (2) K ⫽

ប 2K 2 ប共 ␹D兲2 ⫹ប⌬ ␻ ⫺ 2m 3 ␬D

共35兲共 D⫽2,3兲 .

冕

ˆ †3 ⌿ ˆ3 d Dx ⌿

3

d Dx

i⫽1

1 2

(13) ˆ † ˆ ˆ † ˆ ˆ †2 ˆ2 ␬ D(ii) ⌿ i ⌿ i ⫹ ␬ D ⌿ 1⌿ 1⌿ 3⌿ 3

冊

(23) ˆ † ˆ ˆ † ˆ ⫹␬D ⌿ 2⌿ 2⌿ 3⌿ 3 ,

共37兲

ˆR which is simply seen by substituting the expression for H and rewriting this inequality in the form

共36兲

With the above result for g(0) it also follows that q⫽0, and since the integral in Eq. 共29兲 converges for r⫽0, we obtain that g(r)⫽0 if r⫽0. This means that the exact bound-state solution in two and three dimensions have a pointlike 共zero-radius兲 structure, which is in the relative poˆ 2 quanta. ˆ 1 and ⌿ sitions of the ⌿ Thus, the results of this section show that our model Hamiltonian provides quantum solitons or two-particle 共diboson兲 eigenstates in one and more space dimensions. An important difference between the one-dimensional and mul-

冉兺

冕

1 ␬D

冕

ˆ †1 ⌿ ˆ †2 ⫹ ␹ D ⌿ ˆ 3 兩 2 ⭓0. d D x兩 ␬ D ⌿

共38兲

ˆ ⭓H ˆ R and Eq. 共37兲, and Combining now the inequality H assuming that all the other quartic couplings are non(ii) (13) (23) ,␬D ,␬D ⭓0) we arrive at negative ( ␬ D

063816-5

冉

ˆ ⭓ប ⌬ ␻ ⫺ H

共 ␹D兲2

␬D

冊冕

ˆ †3 ⌿ ˆ 3, d Dx ⌿

共39兲



implying that the energy eigenvalue E (N) K (N) (N) ˆ (N) ⫽ 具 ␸ (N) K 兩 H 兩 ␸ K 典 / 具 ␸ K 兩 ␸ K 典 satisfies the following inequality:

冉

E (N) K ⭓ប ⌬ ␻ ⫺

共 ␹D兲2

␬D

冊

共40兲

¯ 3, N

(N) (N) ˆ (N) ¯ i ⬅ 具 ␸ (N) where N K 兩 N i兩 ␸ K 典 / 具 ␸ K 兩 ␸ K 典 . Due to the conservation of the generalized particle numˆ 2 ⫹2N ˆ 3 , we have N ¯ 3 ⭐ 关 N/2兴 , where 关 N/2兴 is ˆ ⫽N ˆ 1 ⫹N ber N the integer part of N/2. Therefore, if

⌬␻⫺

共␹D兲

␬D

2

⬍0,

we obtain, from Eq. 共40兲,

冉

E (N) K ⭓ 关 N/2兴 ប⌬ ␻ ⫺

共41兲

冊

ប共 ␹D兲2 ⬅E (N) . l ␬D

共42兲

to the Hamiltonian energy, This proves the lower bound E (N) l which we note is valid in one, two, and three dimensions. In two and three dimensions the above inequality can be further simplified. Since the expression ប⌬ ␻ ⫺ប( ␹ D ) 2 / ␬ D represents 关see Eq. 共36兲兴 the exact two-particle energy eigenvalue with zero momentum, E (2) 0 , we can rewrite Eq. 共42兲 as (N) (2) E (N) K ⭓E l ⫽ 关 N/2兴 E 0

共43兲

共 D⫽2,3兲 .

E. Exact N-particle ground state „DÄ2,3…

We can now use the lower bound to obtain the zerofor any even particle momentum energy eigenvalue E (N) 0 number N in more than one space dimensions, and without ˆ 3 self-interaction term. In order to understand the the ⌿ physical meaning of these results, we introduce a finite quantization volume V in this section, to give a finite density. The is extremely simple. We will demontechnique to find E (N) 0 strate that there is an upper bound to the Hamiltonian ground state energy, that coincides with the lower bound given (33) ˆ 3 self⫽0 共no ⌿ above, in either the case that ␬ D interaction兲 or the case that V→⬁ 共infinite volume兲. The result in the infinite volume limit is expected, as it corresponds to an infinitely dilute gas of the diboson ( 兩 ␸ (2) 0 典) bound states. However, the same result also holds at finite ˆ 3 self-interaction term. volume provided there is no ⌿ In order to estimate the ground state energy E (N) 0 , in two and three dimensions, we employ a trial wave function that to the energy E (N) gives an upper bound ˜E (N) 0 0 . We use an ansatz that represents N/2 共where we assume N is even兲 independent two-particle quantum solitons or dibosons with K⫽0: 兩˜ ␸ (N) 0 典⫽

冋冕

ˆ †3 共 x兲 ⫹ d D x⌿

冉

ˆ †1 R⫹ ⫻⌿

冕冕

冊冉

d D r d D R g 共 r兲

m 2r † m r ˆ 2 R⫺ 1 ⌿ M M

冊册

N/2

兩0典.

共44兲

Here g(r) is the zero-radius two-particle correlation function found earlier, having the property that 兰 d D r g 2 (r)⫽0 and g(0)⫽⫺ ␹ D / ␬ D , in two and three dimensions. Calculating ˜ (N) ˆ ˜ (N) ˜ (N) ˜ (N) the energy ˜E (N) 0 ⫽ 具 ␸ 0 兩 H 兩 ␸ 0 典 / 具 ␸ 0 兩 ␸ 0 典 with the ansatz 共44兲 gives 共see Appendix A兲 ˜E (N) 0 ⫽

冉

冊冉冊

(33) ប␬D N ប共 ␹D兲2 N N ប⌬ ␻ ⫺ ⫹ ⫺1 2 ␬D 2 2 V

共 D⫽2,3兲 ,

共45兲

where V⫽ 兰 d D x is the integration volume. The self(11) ˆ 2 (⬃ ␬ D ˆ 1 and ⌿ and interaction terms of the fields ⌿ (22) ␬ D ), as well as the cross-interaction terms between the (13) (23) ˆ 3 (⬃ ␬ D ˆ 1,2 and ⌿ and ␬ D ), do not contribute to fields ⌿ (N) ˜ the energy E 0 . The first term in Eq. 共45兲 is simply the energy due to N/2 independent noninteracting dibosons, each having the energy E (2) 0 , Eq. 共36兲. The second term in Eq. ˆ 3 field, which 共45兲 is the self-interaction energy of the ⌿ depends explicitly on the interaction volume V and decreases as V is increased. The above result is easier to understand if we calculate the ˆ ˜ (N) ˜ (N) ˜ (N) ¯ i ⫽ 具 ˜␸ (N) average number of quanta N 0 兩 N i兩 ␸ 0 典 / 具 ␸ 0 兩 ␸ 0 典 ¯ 3 ⫽N/2 and N ¯ 1,2⫽0. This implies in each field, which gives N ˆ ˆ that the ⌿ 1 and ⌿ 2 quanta can only be regarded as virtual, the presence of which is manifested by the finite binding energy of the two-particle bound states. In such a virtual ˆ 1 and ⌿ ˆ 2 quanta can only interact via the parastate, the ⌿ metric ␹ D and the quartic cross-coupling ␬ D within the individual dibosons. This is a consequence of the zero-radius property of the two-particle correlation function g(r) employed in the ansatz 共44兲. Using Eq. 共36兲, we can rewrite Eq. 共45兲 as

冉冊

(33) ប␬D N (2) N N ˜E (N) ⫽ ⫹ E ⫺1 0 2 0 4 2 V

共 D⫽2,3兲 .

共46兲

关Eq. Comparing this result with the lower bound E (N) l (N) (N) ˜ 共43兲兴 we see that the energy E 0 coincides with E l if ␬ D(33) ⫽0. This implies that, in the absence of the quartic ˆ 3 field, our result for ˜E (N) self-interaction of the ⌿ 0 represents the exact ground state energy of this quantum many-body system:

冉

N (2) N ប共 ␹D兲2 E ⫽ ⫽ ប⌬ ␻ ⫺ E (N) 0 2 0 2 ␬D

冊

共 D⫽2,3兲 , 共47兲

and that the ansatz 共44兲 can be regarded as the exact N-particle eigenstate in this case. The N-particle ground state energy diverges as ␬ D →0. This is in contrast to the behavior of the corresponding classical theory, which has rigorous lower bound to the Hamiltonian energy 关15兴. (33) the same result, Eq. 共47兲, for the For a nonzero ␬ D ground state energy would be valid in an infinitely large

063816-6



volume, corresponding to largely separated and effectively ˆ 3 quanta at a vanishing density. noninteracting dressed ⌿ More generally, for a finite interaction volume or a finite density, the above results imply that the ansatz 共44兲 gives an upper bound to the ground state energy. It is possible that the true ground state energy is simply equal to the lower bound E l ⫽ 关 N/2兴 E (2) 0 in this case as well, in analogy with the treatment of a simple single-component Bose gas with a repulsive ␦ -function interaction 关27兴, which reproduces the results of the noninteracting theory.

(␹) ˆ int H ⫽ប

冕冕冕

d D rd D Rd D z␹ D 共 r, ␰兲

冋冉

ˆ †1 R⫹ ⫻ ⌿

冊冉

m 2r m 1r ˆ †2 R⫺ ⌿ m 1 ⫹m 2 m 1 ⫹m 2

册

ˆ 共 R⫹ ␰兲 ⫹H.c. , ⫻⌿ 3

冊共48兲

where we use the same notation ␹ D for the translationally invariant coupling potential. In Fourier space, where

IV. CUTOFF DEPENDENT AND MEAN-FIELD THEORY RESULTS

ˆ i 共 x兲 ⫽ 兰 d D kaˆ i 共 k兲 exp共 ik•x兲 / 共 2 ␲ 兲 D/2 ⌿

The zero-radius behavior of the quantum solitons in two and three dimensions represents a rather unusual situation, since the classical counterpart of the bosonic theory has wellbehaved, stable, multidimensional soliton solutions 关15兴. This leads to a paradox of how such a quantum field theory relates to real physical processes. To resolve this paradox, we note that physical applications usually involve some type of momentum cutoff. In systems with dimension D⬎1 it is known that an effective Hamiltonian of the type we consider here should be renormalized, with a coupling constant that is cutoff dependent, in order to compare the coupling parameters with observable values. Since the exact form of the interaction potentials is not well known, we simply employ a finite bound on the relative momentum. In the case of nonlinear optical parametric interactions, the cutoff originates from the fact that parametric couplings are usually restricted to a finite range of relative momenta of the interacting fields. To estimate the cutoff in this case, we note that the origins of the theory involve rotating-wave and paraxial approximations, and the neglect of higher-order dispersion 关3,29兴. Therefore, in higher dimensions we should include nonparaxial diffraction if the characteristic radius of solutions becomes less than the field carrier wavelengths. To represent this we can introduce a cutoff at k m in the relative ˆ 2 . Since the paraxial apˆ 1 and ⌿ momenta k of the fields ⌿ proximation is valid only for k⬜ Ⰶ2 ␲ /␭ 1 , where ␭ 1 is assumed to be the longest carrier wavelength, then a momentum cutoff of at most k m ⬃2 ␲ /␭ 1 should be imposed on the nonlinear couplings. In the case of atomic BEC interactions 关30兴, the cutoff is usually introduced at the level of inverse s-wave scattering length, and a similar cutoff occurs in cases where fermionic fields are involved 关12,26兴.

and the commutation relations for the operators aˆ i (k) and aˆ †i (k) are 关 aˆ i (k),aˆ †j (k⬘ ) 兴 ⫽ ␦ i j ␦ (k⫺k⬘ ), this transforms into (␹) ˆ int ⫽ 共 2 ␲ 兲 ⫺D/2ប H

⫻aˆ †2

冉

冋

d D Kd D k ˜␹ D 共 k,K兲 aˆ †1

冊

册

冉

m 1K ⫺k m 1 ⫹m 2

m 2K ⫹k aˆ 3 共 K兲 ⫹H.c. . m 1 ⫹m 2

冊

共49兲

We next assume that the Fourier component ˜␹ D (k,K) depends only on k, and impose a momentum cutoff on ˜␹ D (k), such that ˜␹ D (k) vanishes if 兩 k兩 ⬎k m and is a constant, ˜␹ D (k)⫽ ␹ D , for 兩 k兩 ⬍k m . Similar considerations can be applied to the quartic interaction terms in our Hamiltonian, given by Eq. 共4兲. Because (i j) (x,y,x⬘ ,y⬘ ) is written as of translational invariance, ␬ D (i j) ␬ D (r,r⬘ ,R⫺R⬘ ), where r⫽x⫺y, r⬘ ⫽x⬘ ⫺y⬘ , R⫽(m i x ⫹m j y)/(m i ⫹m j ), and R⬘ ⫽(m i x⬘ ⫹m j y⬘ /m i ⫹m j ). Transforming to Fourier space, we assume that the Fourier com(i j) (k,k⬘ ,K), where K⫽ki ⫹k j , does not depend on ponent ˜␬ D (i j) K, and impose a momentum cutoff such that ˜␬ D (k,k⬘ ) (i j) ⫽ ␬ D if 兩 k兩 , 兩 k⬘ 兩 ⬍k m , and is zero otherwise. The final form of the cutoff dependent interaction Hamiltonian can now be written as ˆ int ⫽ 共 2 ␲ 兲 ⫺D/2ប ␹ D H ⫻aˆ †2

冉

3

⫻

A. Hamiltonian with momentum cutoff

To implement a cutoff in the interaction part of our Hamiltonian we first consider the parametric interaction term which is of the form of Eq. 共3兲. Assuming translational invariance we note that ␹ D (x,y,z) can only depend on the relative coordinates, which we choose according to x⫺y⬅r and z⫺R⬅ ␰, where R⫽(m 1 x⫹m 2 y)/(m 1 ⫹m 2 ) is the ˆ 2 fields. That is, ˆ 1 and ⌿ center-of-mass coordinate for the ⌿ ( ␹ ) ˆ int can be written as H

冕冕

km

兩 k兩 ⫽0

d Dk

冊

冕冋冉

m 1K ⫹k m 1 ⫹m 2

d D K aˆ †1

册

冊

m 2K ⫺k aˆ 3 共 K兲 ⫹H.c. ⫹ 共 2 ␲ 兲 ⫺D m 1 ⫹m 2

兺 i, j⫽1

冉冉

冕

(i j) ប␬D

2

冕

km

兩 k兩 ⫽0

d Dk

冊冉冊冉

冕

km

兩 k⬘ 兩 ⫽0

d D k⬘

冊

冕

⫻aˆ †i

m iK m jK ⫹k aˆ †j ⫺k m i ⫹m j m i ⫹m j

⫻aˆ i

m iK m jK ⫹k⬘ aˆ j ⫺k⬘ . m i ⫹m j m i ⫹m j

冊

d DK

共50兲

The noninteracting part of the Hamiltonian, in terms of aˆ i (k), is

063816-7



冕 d Dkk 2aˆ †i 共 k兲aˆ i共 k兲兺 i⫽1 2m i

Here k⫽ 兩 k兩 , K⫽ 兩 K兩 , ⌬ is given by Eq. 共20兲, while q is defined as

3

ˆ 0⫽ H

ប2

⫹ប⌬ ␻

冕

q⬅2 ␮ 关 ␹ D ⫹ ␬ D g 共 0,k m 兲兴 /ប.

共51兲

d D kaˆ †3 共 k兲 aˆ 3 共 k兲 .

In the case of nonlinear optical interactions, the coupling (i j) are proportional to the Bloembergen constants ␹ D and ␬ D second- and third-order susceptibilities of the nonlinear medium 关3,29兴, while in the case of atomic/molecular BEC in(i j) teractions the quartic couplings ␬ D are related to the s-wave scattering amplitudes 关30兴. For example, in the diagonal case is given by ␬ (ii) and in three space dimensions, ␬ (ii) 3 3 ⫽4 ␲ បa ii /m i , where a ii is the s-wave scattering length within the ith species, while the interspecies couplings are ␬ (i3 j) ⫽ ␬ (3ji) ⫽2 ␲ បa i j / ␮ i j , where a i j is the corresponding cross-scattering length and ␮ i j ⫽m i m j /(m i ⫹m j ) is the reduced mass 关24,30兴. The form of the parametric coupling will depend on the particular mechanism that can be used for atomic dimerization, such as Feshbach resonance or Raman photoassociation 关31,22兴. In addition, we note that in cases where fermionic fields are involved, the corresponding quartic self-interaction terms must be omitted from the Hamiltonian.

The above equations represent the Fourier transform equivalent of Eq. 共15兲 and Eq. 共18兲, except that now they are valid for 兩 k兩 ⬍k m . In order to evaluate the soliton binding energy and the effective radius, we solve these equations for g(0,k m ), and obtain

冉

g 共 0,k m 兲 ⫽⫺ ␹ D ␬ D ⫹

f D共 r 0k m 兲 ⫽

1 共 2␲ 兲D

We can now analyze the eigenvalue problem (2) (2) ˆ H 兩 ␸ (2) K (k m ) 典 ⫽E K (k m ) 兩 ␸ K (k m ) 典 directly, by considering the two-particle eigenstate in Fourier space:

冋

⫻aˆ †1

冉

冊冉

km

兩 k兩 ⫽0

f 3 共 r 0 ,k m 兲 ⫽

冊册

so that the cutoff dependent correlation function is g(r,k m ) k ⫽ 兰兩 km兩 ⫽0 d D kG(k)exp(ik•r)/(2 ␲ ) D . This implies that, due to the cutoff in the nonlinearities, we need only investigate eigenstates for which G(k) satisfies the equation 共 k 2 ⫹r ⫺2 0 兲 G 共 k 兲 ⫽⫺q,

共53兲

if 兩 k兩 ⬍k m and vanishes for 兩 k兩 ⬎k m . The energy eigenvalue E (2) K (k m ) is given by E (2) K 共 km兲⫽

ប2 ប 2K 2 ⫺ , 2M 2 ␮ r 20

共54兲

where the length scale r 0 is to be found by solving the following eigenvalue equation: r ⫺2 0 ⫽

2␮ 关 ⌬⫺ ␹ D g 共 0,k m 兲兴 . ប

f 2共 r 0k m 兲 ⫽

d D k G 共 k兲

m 1K m 2K ⫹k aˆ †2 ⫺k 兩 0 典 , 共52兲 M M

2 ␮ f D共 r 0k m 兲

冊

⫺1

共57兲

.

冕

r0km

d Dx

兩 x兩 ⫽0

1⫹x 2

共58兲

,

and its explicit form in one, two, and three dimensions (D ⫽1,2,3) is given by

B. Exact diboson solutions

冕

បr D⫺2 0

Here the dimensionless cutoff structure function is defined as

1 tan⫺1 共 r 0 k m 兲 , ␲

共59兲

1 2 ln共 1⫹r 20 k m 兲, 4␲

共60兲

关 r 0 k m ⫺tan⫺1 共 r 0 k m 兲兴 .

共61兲

f 1共 r 0k m 兲 ⫽

⫺D/2 ˆ† 兩 ␸ (2) K 共 k m 兲典 ⫽ a 3 共 K兲 ⫹ 共 2 ␲ 兲

共56兲

1 2␲2

This result clearly shows the difference caused by the dimensionality of the space. In one dimension f 1 (r 0 k m ) approaches a constant value if k m →⬁, while in two and three dimensions f D (r 0 k m ) has a logarithmic or linear divergence, respectively. The effect of this divergence depends on whether or not the additional quartic interaction is present. If it is present 共with ␬ D ⬎0), there are exact solutions without cutoff, and g(0,k m →⬁)⫽⫺ ␹ D / ␬ D , so that the energy eigenvalue E (2) K (k m →⬁) takes the form of Eq. 共36兲, and g(r)⫽0 if 兩 r兩 ⬎0. In other words, the solutions in two and three dimensions have a finite energy 共unlike the energy divergence in the nonlinear Schro¨dinger model with an attractive ␦ -function potential兲 but zero radius in the limit of k m →⬁. If, however, ␬ D ⭐0, as in the attractive nonlinear Schro¨dinger model, we must impose a finite cutoff on the couplings to prevent an energy divergence. Simultaneously, a finite cutoff prevents singularities in space. With a finite cutoff, the eigenvalue problem for E (2) K (k m ), Eq. 共54兲, reduces to the solution of the following eigenvalue equation:

共55兲 063816-8

r ⫺2 0 ⫽

冋

冉

បr D⫺2 2␮ 0 ⌬⫹ 共 ␹ D 兲 2 ␬ D ⫹ ប 2 ␮ f D共 r 0k m 兲

冊册 ⫺1

,

共62兲



which is Eq. 共55兲 rewritten in terms of the cutoff structure function f D (r 0 k m ), using g(0,k m ) from Eq. 共57兲. Here r 0 must be real and positive for a localized bound state. Analysis of this equation, using the explicit results for the cutoff structure functions 共59兲–共61兲, shows that under certain conditions a positive solution for r 0 is available. This condition in the cases of one and two dimensions can be written in the form of Eq. 共25兲, while in the three-dimensional case it is modified to ( ␹ 3 ) 2 ⫹⌬ 关 ␬ 3 ⫹ ␲ 2 ប/( ␮ k m ) 兴 ⬎0. In the simplest case of ␬ D ⫽0 共which can be considered only if the cutoff and the couplings are independent of each other兲, ⌬⫽0, and in the limit k m Ⰷr ⫺1 0 the eigenvalue equation is simplified, and even solved analytically in one- and three-dimensional cases. The resulting radii r 0 and binding 2 2 energies E (2) b ⫽ប /(2 ␮ r 0 ) are determined by: r 0 ⯝ 共 ប 2 /2␹ 21 ␮ 2 兲 1/3,

2 1/3 4/3 E (2) b ⯝ 共 ប ␮ /2 兲共 ␹ 1 兲

r 0 ⯝ 共 ␲ /2兲共 ប/ ␹ 2 ␮ 兲关 ln共 r 0 k m 兲兴 1/2

2 2 E (2) b ⫽ប / 共 2 ␮ r 0 兲

N (2) N (2) (N) ˜ (N) E 共 km兲, E (N) l ⫽ E 0 ⭐E 0 共 k m 兲 ⭐E 0 共 k m 兲 ⫽ 2 2 0

共 D⫽1 兲 ;

⫺1/2

, 共63兲

共 D⫽2 兲 ;

r 0 ⯝ 共 ␲ ប/ ␹ 3 ␮ 兲共 2k m 兲 ⫺1/2, 2 2 E (2) b ⯝共 ␹3兲 ␮km /␲

field, is negligible since we have used a free space expansion ˆ i (x) in terms of aˆ i (k). This means that the result for of ⌿ ˜E (N) 0 (k m ) corresponds to an infinite volume or zero particle density, where the contributions due to the quartic interactions other than the ␬ D coupling 共which affects the binding within individual dibosons兲 vanish. The lower bound in this cutoff dependent N-particle problem can also be estimated following previous methods. Since the previous cutoff independent result was obtained by ignoring kinetic energy terms, the lower bound is unchanged from the previous section, Eq. 共43兲. Consequently, for the true cutoff dependent ground state energy E (N) 0 (k m ) we have now the result that

共 D⫽3 兲 .

Here, in the two-dimensional case, the diboson radius r 0 and the binding energy E (2) b can easily be found numerically. The one-dimensional result 共in the limit of k m Ⰷr ⫺1 0 ) reproduces the result of Eqs. 共32兲 and 共33兲 obtained using the cutoff independent treatment.

˜ (N) where E (N) l ⫽E 0 (k m ), unless k m →⬁. Thus, with a finite cutoff the ansatz corresponding to N/2 independent dibosons no longer gives the energy coinciding with the lower bound, and only provides an upper bound to the ground state energy. In other words, it is no longer the exact eigenstate and therefore does not necessarily result in the lowest possible energy. D. N-particle results: Coherent variational ansatz

The second type of ansatz that we employ here is the coherent or mean-field theory 共MFT兲 ansatz: 兩˜ ␸ (N) c 典 ⫽exp

C. N-particle results: Independent diboson ansatz

To estimate the ground state energy, in the cutoff dependent N-particle problem, we use the following momentumspace ansatz, corresponding to N/2 共where we assume N is even兲 independent dibosons:

冉

⫺D/2 ˆ† 兩˜ ␸ (N) 0 共 k m 兲典 ⫽ a 3共 0 兲 ⫹ 共 2 ␲ 兲

⫻aˆ †1 共 k兲 aˆ †2 共 ⫺k兲

冊

冕

km

兩 k兩 ⫽0

d D kG 共 k兲

N/2

兩0典.

共64兲

Operating with the cutoff dependent Hamiltonian on this ansatz we find, for an infinite interaction volume, the following result for the corresponding energy 共see Appendix B兲: ˜E (N) 0 共 km兲⫽

N (2) E 共 km兲, 2 0

冉冕

3

3

d x

兺 ␺ i共 x兲 ⌿ˆ †i 共 x兲

i⫽1

冊

兩0典.

共67兲

The coherent ansatz is equivalent to a mean-field theory description of the system, where the operators are replaced by their mean values and a factorization is assumed. It is an approximate 共semiclassical兲 eigenstate that describes three coupled fields at large N, under broken symmetry conditions. Compared with the previous case of the N/2 independent diboson ansatz, the coherent ansatz can provide a lower energy at large N and for certain parameter values. To show this, we use a variational approach and choose trial functions ␺ i (x) in the form of Gaussians, assuming in addition that ␺ 1 (x)⫽ ␺ 2 (x):

␺ 1 共 x兲 ⫽ ␺ 2 共 x兲 ⫽g 1 exp关 ⫺ 兩 x兩 2 / 共 2w 21 兲兴 , ␺ 3 共 x兲 ⫽⫺g 3 exp关 ⫺ 兩 x兩 2 / 共 2w 23 兲兴 .

共65兲

where E (2) 0 (k m ) is determined by Eqs. 共54兲 and 共62兲. This result represents an upper bound to the ground state energy, and is no longer the exact solution unless k m →⬁. It is expressed in terms of the energy E (2) 0 (k m ) of individual dibosons, and depends only on the parametric coupling ␹ D and the quartic cross-coupling ␬ D . The contribution of the other (33) ˆ3 of the ⌿ quartic terms, including the self interaction ⬃ ␬ D

共66兲

共68兲

Here g i and w i are regarded as free variational parameters, the negative sign for ␺ 3 (x) is to ensure that the coupling energy is negative for ␹ D ⬎0, and the normalization implies that the Gaussian parameters must satisfy 2 ␲ 3/2关 g 21 w 31 ⫹g 23 w 33 兴 ⫽N. We will assume that the coherent ansatz is slowly varying such that the Gaussian width scales are much ⫺1 , allowing the momentum cutoff to be nelarger than k m glected. Substituting these trial functions into the Hamil-

063816-9



tonian 共1兲, 共2兲, and 共5兲, we find that the corresponding variational energy ˜E (N) c , in two and three space dimensions, is given by ˜E (N) c ⫽

冉冊冉 ␲ 2

⫹ ⫺ ⫹ ⫹

D/2

2 (D⫺2)/2Dប 2 2 D⫺2 g 1w 1 2␮

2 (D⫺2)/2Dប 2 2 D⫺2 g 3 w 3 ⫹2 D/2ប⌬ ␻ g 23 w D 3 2m 3 D 2 D⫹1 ប ␹ D g 21 g 3 w D 1 w3

共 w 21 ⫹2w 23 兲 D/2 (33) ប␬D

2

冉

冊

␬ D(11) ␬ D(22) 4 D ⫹ ⫹ប ␬ D ⫹ g 1w 1 2 2

g 43 w D 3

(13) (23) 2 2 D D ⫹␬D 2 D/2ប 共 ␬ D 兲g 1g 3w 1 w 3

2 共 w 21 ⫹w 23 兲 D/2

冊

冉冊冉冊 ប2 2␮

2␮␹D ប

g 1 ⫽g 2 ⯝1.7⫻10⫺4 N 2 共 2 ␮ ␹ 3 /ប 兲 3 and g 3 ⯝1.1g 1 . These, in turn, give the following relation for ¯ i ⫽ 兰 d 3 x兩 ␺ i (x,t) 兩兴 present the average number of particles 关 N ¯ 1,2 /N ¯ 3 ⯝1.21. In two dimenin the fields ␺ 1,2 and ␺ 3 : N sions, the parametric soliton optimum widths and amplitudes are given by

共 D⫽2,3兲 . 共69兲

The result of minimization of ˜E (N) c , under the constraint of 2 ␲ 3/2关 g 21 w 31 ⫹g 23 w 33 兴 ⫽N, is considerably simplified in the region where the parametric coupling is dominant and N is not too large, so that one can neglect the terms due to ⌬ ␻ (i j) and the quartic couplings ␬ D . In this region, and for m 3 ⫽m 1 ⫹m 2 and m 1 ⫽m 2 共so that m 3 ⫽4 ␮ ), we obtain the MFT minimum energy of (6⫺D)/(4⫺D) ˜E (N) c ⫽⫺A D N

herent variational ansatz gives optimum Gaussian parameters which correspond to the approximate analytical form of classical solitons in this pure parametric case, in two and three space dimensions. The optimum length scales corresponding to soliton widths 共for ⌬ ␻ ⫽0, m 1 ⫽m 2 , and m 3 ⫽m 1 ⫹m 2 ) are nearly identical for the three fields and are given by w 1 ⫽w 2 ⯝1.2⫻102 N ⫺1 (2 ␮ ␹ 3 /ប) ⫺2 and w 3 ⯝0.88w 1 , in three space dimensions. The corresponding values of the field amplitudes are determined by

w 1 ⫽w 2 ⯝6.98N ⫺1/2共 2 ␮ ␹ 2 /ប 兲 ⫺1 , w 3 ⯝0.86w 1 , g 1 ⫽g 2 ⯝4.06⫻10⫺2 N 共 2 ␮ ␹ 2 /ប 兲 , ¯ 1,2 /N ¯ 3 ⯝1.24. Clearly the soliton and g 3 ⯝1.05g 1 , yielding N ⫺1 for our use of the cutoff width must be much larger than k m independent Hamiltonian to be justified.

4/(4⫺D)

V. PHYSICAL APPLICATIONS: PHOTONIC INTERACTIONS

共 D⫽2,3兲 ,

共70兲

where A D is a dimensionless constant given by A 2 ⯝ 7.42⫻10⫺3 in two dimensions, and by A 3 ⯝1.2⫻10⫺5 in three dimensions. The energy ˜E (N) scales as N 2 in two dic 3 mensions, and as N in the three-dimensional case. Comparing this with the linear dependence on N of the energy estimate ˜E (N) 0 (k m ) from the independent diboson ansatz, Eq. 共65兲, we conclude that there exists a crossover or a critical boson number N cr beyond which 共i.e., for N⬎N cr ) the coherent variational ansatz becomes more favorable, as ˜E (N) c ˜ (N) ⬍E (k ). The value of N is easily found using the above m cr 0 simple result for ˜E (N) which neglects the role of the quartic c couplings so that all parameter dependences are explicit. For nonzero quartic couplings, the dependence of the minimum is no longer given by such a simple expression. energy ˜E (N) c The minimization does not reveal explicit scaling properties similar to those in Eq. 共70兲 and it must be carried out for different values of N independently. This is further analyzed in Sec. VI as applied to parameter values characteristic of BEC interactions. To conclude our discussion of the results in the case of pure parametric interactions, we note that for the symmetric case under consideration, i.e., for ␺ 1 (x)⫽ ␺ 2 (x), the system can be formally reduced to the model of degenerate parametric interaction which is known to support higher-dimensional classical solitons 关15–17兴. Thus, together with providing a minimum energy to the classical Hamiltonian, the above co-

An important application of the results of our parametric field theory is in optics, where it describes the nonlinear optical process of frequency conversion or sum-frequency generation. Here the parametric coupling ␹ D is due to the second-order nonlinearity of a nonlinear medium, while the ␬ D(i j) terms are due to self- and cross-phase modulation. Straightforward application of the previous results is, however, prevented by the fact that the noninteracting Hamilˆ 0 for the propagating light fields is in general differtonian H ent from Eq. 共2兲. It is defined in a moving reference frame and is asymmetric with respect to the longitudinal 共direction of propagation兲 and transverse directions 关15,29兴: ˆ 0⫽ H

冕

冋兺冉 3

D

d x

i⫽1

ប2 ប2 ˆ i兩 2⫹ ˆ 兩2 兩ⵜ 储⌿ 兩ⵜ ⌿ 2m i 储 2m i⬜ ⬜ i

册

ˆ †3 ⌿ ˆ3 . ⫹ប⌬ ␻ ⌿

冊共71兲

ˆ 3 represent three optical fields with carrier ˆ 1,2 and ⌿ Here ⌿ wave numbers k1,2 , k3 ⫽k1 ⫹k2 and frequencies ␻ i ⫽ ␻ (k i ) (i⫽1,2,3), while ⌬ ␻ ⫽ ␻ 3 ⫺( ␻ 1 ⫹ ␻ 2 ) is the phase mismatch. The longitudinal coordinate (x 储 ) is defined in a moving frame, x 储 ⫽x L ⫺ v t, where x L is the laboratory frame coordinate and v ⫽ ⳵␻ i / ⳵ k is the group velocity, which is assumed equal at all three carrier frequencies. In addition, m i 储⫽ប/ ␻ ⬙i are effective longitudinal masses due to the group

063816-10



velocity dispersion, where ␻ ⬙i ⫽ ⳵ 2 ␻ i / ⳵ k 2 is the dispersion coefficient in the ith frequency band. The transverse masses m i⬜ ⫽ប ␻ i / v 2 are caused by diffraction, and the correspondˆ 0 is only relevant in two and three dimensions. ing term in H (i j) in the interaction HamilThe coupling constants ␹ D and ␬ D tonian are proportional to the Bloembergen second- and third-order nonlinear susceptibilities ( ␹ B(2) and ␹ B(3) 关32兴兲 of the nonlinear medium, respectively 关3,14,29兴:

␹ D⯝

冉

␹ B(2) ប ␻ 1 ␻ 2 ␻ 3 2␧ 0 n3

冊

1/2

1

d

␬ D⯝

, (3⫺D)/2

3ប ␹ B(3) ␻ 1 ␻ 2

,

4␧ 0 n 4 d 3⫺D 共72兲

where n is the refractive index, which we assume is nearly the same at all three frequencies, and d is the effective modal 共waveguide兲 diameter. Our treatment here is similar to a previous theory of degenerate optical parametric interaction 关11兴, except that the present nondegenerate theory has an additional degree of ˆ1 freedom due to the fact that the low-frequency fields (⌿ ˆ and ⌿ 2 ) are different. In practical terms, this gives the possibility of employing either type I or type II phase matching, i.e., the fields can be different either in frequencies or in polarization, or else in both.

f D共 r 0k m , ␮ r 兲 ⫽

E (2) K 共 km兲⫽

ប 2

冉

M储

⫹

K⬜2 M⬜

冊

⫺

ប

2

2 ␮ 储 r 20

共73兲

,

where r 0 is to be found by solving the following eigenvalue equation:

冋

冉

r0km

d Dx

兩 x兩 ⫽0

1⫹x 2储 ⫹x⬜2 / ␮ r

⌬⬅

បr D⫺2 2␮储 0 ⌬⫹ 共 ␹ D 兲 2 ␬ D ⫹ r ⫺2 0 ⫽ ប 2 ␮ 储 f D共 r 0k m , ␮ r 兲

冊册

共 D⫽2,3兲 ,

共75兲

冉

冊冉

冊

K⬜2 K⬜2 ប K 2储 ប K 2储 ⫹ ⫺ ⫹ ⫺⌬ ␻ . 2 M 储 M⬜ 2 m 3 储 m 3⬜

共76兲

In the limit of k m →⬁, which corresponds to the simplified cutoff independent treatment, one can again arrive at the same conclusions as in Sec. III on the pointlike structure of the multidimensional two-particle bound states. The cutoff dependent results are modified due to the dependence of the cutoff structure function 共75兲 on the relation ␮ r ⫽ ␮⬜ / ␮ 储 . The integrations in f D (r 0 k m , ␮ r ) cannot be carried out as easily as in the symmetric case of Sec. IV corresponding to ␮ r ⫽1. Instead, for arbitrary values of ␮ r , the integrals and the resulting binding energies can be evaluated numerically. If, however, 冑␮ r Ⰶ1 and r 0 k m Ⰷ1 one can obtain the following approximate results: f 2共 r 0k m , ␮ r 兲 ⯝

f 3共 r 0k m , ␮ r 兲 ⯝

The asymmetric form of the noninteracting Hamiltonian does not qualitatively change the results of the previous sections. The results are, however, modified quantitatively in two and three dimensions. Omitting the details of the derivation we give only the final expressions for the two-particle eigenvalue problem and the simplest diphoton solutions. First we mention that in one dimension the results of the earlier sections are unchanged, with the understanding that the effective masses m i are interpreted as dispersive ones, m i ⬅m i 储 . In two and three dimensions, the two-particle or diphoton energy is determined by K 2储

共 2␲ 兲D

冕

and the effective detuning ⌬ is now given by

A. Analytic results

2

1

冑␮ r 2␲

␮ rr 0k m 2␲2

共77兲

ln共 2r 0 k m 兲 , 共 1⫺ln冑␮ r 兲 .

共78兲

With these functions and for ␬ D ⬎0, the condition 共25兲 of having a positive solution for r 0 in the eigenvalue equation 共74兲 remains unchanged in two dimensions, while in three dimensions it is transformed into 共 ␹ 3 兲 2 ⫹⌬ 兵 ␬ 3 ⫹ ␲ 2 ប/ 关 ␮ 储 ␮ r k m 共 1⫺ln冑␮ r 兲兴其 ⬎0, 共79兲

with ⌬ given by Eq. 共76兲 in both cases. B. Diphoton binding energies for pure parametric case

In the case of ␬ D ⫽0 and ⌬⫽0 Eqs. 共73兲,共74兲 and 共77兲,共78兲, together with the earlier one-dimensional result 共where we replace ␮ by ␮ 储 ), lead to the following simple expressions for the diphoton soliton radii r 0 and the binding 2 2 energies E (2) b ⫽ប /(2 ␮ 储 r 0 ): r 0 ⯝ 共 ប 2 /2␹ 21 ␮ 2储兲 1/3,

E b(2) ⯝ 共 ប 2 ␮ 储 /2兲 1/3共 ␹ 1 兲 4/3

共 D⫽1 兲 ;

r 0 ⯝ 共 ␲ /2兲 1/2共 ប/ ␹ 2 ␮ 储兲 ␮ r⫺1/4关 ln共 2r 0 k m 兲兴 ⫺1/2,

⫺1

.

E b(2) ⫽ប 2 / 共 2 ␮ 储 r 20 兲

共74兲

Here the diphoton momentum K is decomposed into longitudinal and transverse components so that K 2 ⬅ 兩 K兩 2 ⫽K 2储 ⫹K⬜2 . In addition, we have introduced the longitudinal and transverse reduced masses ␮ 储 ⫽m 1 储 m 2 储 /(m 1 储 ⫹m 2 储 ) and ␮⬜ ⫽m 1⬜ m 2⬜ /(m 1⬜ ⫹m 2⬜ ), and have defined M 储 ⬅m 1 储 ⫹m 2 储 , M⬜ ⬅m 1⬜ ⫹m 2⬜ , and ␮ r ⬅ ␮⬜ / ␮ 储 . The cutoff structure function f D (r 0 k m , ␮ r ) is defined as

共 D⫽2 兲 ;

共80兲

r 0 ⯝ 共 ␲ ប/ ␹ 3 ␮ 储兲共 2 ␮ r k m 兲 ⫺1/2共 1⫺ln冑␮ r 兲 ⫺1/2, E b(2) ⯝ 共 ␹ 3 兲 2 ␮ 储 ␮ r k m 共 1⫺ln冑␮ r 兲 / ␲ 2 共 D⫽3 兲 . To illustrate how large a binding energy might be obtained we consider parameter values characteristic of highly nonlinear parametric materials, such as GaAs asymmetric quantum well structures 关33兴. We note, however, that these

063816-11



types of materials often have practical limitations involving restricted wavelengths, lack of phase matching, or high absorption. For example, high values of ␹ B(2) 共up to ⬃9⫻10⫺7 m/V) observed in the GaAs case were only around the fundamental 共subharmonic兲 wavelength of about ␭ 1 ⬃9.2 ␮ m, and the absorption was high. Other candidates such as organic materials 关34兴 have the advantage of operating at shorter wavelength (␭ 1 ⬃1.3 ␮ m), but the values of ␹ B(2) are smaller ( ␹ B(2) ⬃10⫺10 m/V). Other factors and requirements that may have practical importance are similar to those discussed in Ref. 关11兴 for the case of degenerate parametric interaction. For the present nondegenerate parametric interaction, we summarize the estimates of the diphoton radii and binding energies by considering reference parameter values similar to those of GaAs. These are chosen as follows: ␹ B(2) ⫽ 9⫻10⫺7 m/V, ␭ 1 ⫽9.2 ␮ m, and n⫽3.3. In addition, we assume that ␻ 1 ⯝ ␻ 2 ⯝ ␻ 3 /2 (␭ 1 ⯝␭ 2 ⯝2␭ 3 ), resulting in ␮⬜ ⫽1.3⫻10⫺36 kg, and we choose the dispersion coefficients ␻ ⬙1 ⯝ ␻ 2⬙ such that ␮ r ⫽ ␮⬜ / ␮ 储 ⫽0.01. The waveguide diameter d required to evaluate the value of the coupling constant ␹ D ⫽ ␹ 3 d (D⫺3)/2 (D⫽1,2) in one and two dimensions is chosen as d⫽5␭ 1 . Finally, for the cutoff dependent two- and three-dimensional cases, we choose the cutoff at the inverse of the longest wavelength k m ⫽2 ␲ /␭ 1 , while in the one-dimensional case the cutoff dependences can be neglected as long as r 0 k m Ⰷ1. With these parameter values, the resulting radii r 0 and binding energies E (2) b of the parametric diphotons are given in the following table Coupling ␹ D

r 0 共␮m兲

E b(2) 共eV兲

D⫽1

5.4⫻106 (m1/2/s)

22

5.3⫻10

D⫽2

3.7⫻104 (m/s)

43

D⫽3

2.5⫻102 (m3/2/s)

47

1.4⫻10 1.2⫻10⫺7

⫺7 ⫺7

共81兲 indicating that we expect the higher-dimensional quantum solitons to be less strongly bound and of larger radius than their one-dimensional counterparts. Thus we have shown that nondegenerate parametric interactions can provide diphoton bound states in one, two, and three space dimensions. The diphoton has the form of a quantum superposition of two states one of which contains a ˆ 3 , while the other inphoton of the sum-frequency field ⌿ ˆ1 volves a pair of photons of the lower-frequency fields ⌿ ˆ 2 . The diphoton can be viewed as a photonic analog of and ⌿ a two-quark state model of mesons, and be termed, as in the case of degenerate parametric interaction 关14兴, an ‘‘optical meson.’’ The relatively large binding energy, as compared to quantum solitons based on ␹ (3) nonlinearities 关21兴, combined with low-temperature experimental techniques, could make it feasible to observe this simple quantum soliton in experiment.

VI. BEC INTERACTIONS

Another example of a physical system that can be treated by the Hamiltonian 共1兲,共2兲, and 共5兲 or 共50兲 is a coupled atomic/molecular BEC. Here the parametric coupling represents the coherent process of formation of dimer molecules ˆ 1 and ⌿ ˆ 2 fields兲 either of ˆ 3 field兲 from pairs of atoms (⌿ (⌿ distinct atomic species, or in distinct quantum states. In the case of degenerate parametric interaction this has been considered in Refs. 关19,22,23,35,36兴. Here we extend the basic results to the case of nondegenerate parametric interaction, i.e., to three coupled Bose condensates, thus extending the variety of ultracold molecular gases that could be created via BEC interactions. In this directly applicable case of BEC interactions, m 1,2 and m 3 ⫽m 1 ⫹m 2 ⫽M are the atomic and molecular masses, the coupling constant ␹ D is related to the molecular forma(i j) are the effective intra- and interspecies tion rate, while ␬ D couplings due to s-wave scattering amplitudes a i j 关24,30兴. In addition, ប⌬ ␻ is the bare formation energy of the molecular species. Physical mechanisms that can realize coherent atomic dimerization and produce ultracold molecules include Feshbach resonance and Raman photoassociation 关31,35兴. Feshbach resonances have already been observed 关37兴, while experiments of this type with Raman photoassociation are under way 关39兴 in the case of single-species 共degenerate兲 atomic BEC, the theory of which is given elsewhere 关22兴. The simplest nontrivial objects in such coupled atomic/molecular BEC systems that can be described by our theory, are two-particle 共diboson兲 quantum solitons in three dimensions (D⫽3), i.e. ‘‘dressed’’ molecules, each of which exists in a superposition with a pair of atoms. With a characteristic ␹ 3 value estimate of about ␹ 3 ⯝10⫺6 m3/2/s 关35,39,22兴, the atomic masses m 1 ⫽m 2 ⯝10⫺25 kg, and assuming that the s-wave scattering length a 12⯝5 nm, so that ␮ ⫽m 1 /2⯝0.5⫻10⫺25 kg and ␬ 3 ⫽2 ␲ បa 12 / ␮ ⯝ 6.6⫻10⫺17 m3 /s, Eq. 共36兲 results in a quantum soliton bind(2) 2 ⫺11 eV, for ing energy of E (2) b ⫽⫺E 0 ⫽ប( ␹ 3 ) / ␬ 3 ⯝10 ⌬ ␻ ⫽0. This is the result of the idealized quantum theory without a momentum cutoff 共i.e., k m →⬁), which strictly speaking cannot be applied in a self-consistent way to BEC interactions with a nonzero value of ␬ 3 . If we include the effect of momentum cutoffs and assume that the scattering length a 12 provides a natural cutoff at k m ⯝2 ␲ /a 12 , then k m ⯝4 ␲ 2 ប/( ␬ 3 ␮ ). In this case the energy E (2) 0 (k m ) is found from Eqs. 共54兲 and 共62兲 where the cutoff structure function f 3 (r 0 k m ) is given by Eq. 共61兲, with k m ⯝4 ␲ 2 ប/( ␬ 3 ␮ ). For ⌬ ␻ ⫽0 and assuming k m r 0 Ⰷ1, this gives 2 E (2) 0 共 k m 兲 ⯝⫺4ប 共 ␹ 3 兲 / 共 5 ␬ 3 兲 .

共82兲

(2) The resulting binding energy E (2) b (k m )⫽⫺E 0 (k m ) and the (2) ⫺1/2 corresponding radius r 0 ⫽ប(2 ␮ E b ) , for the parameter values as above and k m ⯝2 ␲ /a 12⫽1.26 nm⫺1 , are ⫺11 eV and r 0 ⫽0.3 ␮ m. Thus, the bindE (2) b (k m )⫽0.8⫻10 ing energy with momentum cutoff is very close to the idealized result from Eq. 共36兲, and its magnitude is comparable to achievable temperatures in current BEC experiments.

063816-12



To give the simplest treatment of coupled atomic/molecular BEC systems, we neglect any loss processes such as three-body inelastic collisions. This may not be easy to realize in practice and will depend on the particular mechanism for atom-molecule coupling. For example, in the case of a Feshbach resonance that couples pairs of atoms to a quasibound excited molecular state, losses due to inelastic atom-molecule collisions can occur at a significant rate 关38兴. This is clearly a disadvantage that reduces the condensate lifetime. The Raman photoassociation mechanism is, in this sense, more promising 关22兴. Here the free-bound Raman transitions are induced by two laser fields that couple pairs of atoms to a bound molecular state through excited intermediate states. This has the advantage that one can tune the coupling to a deeply bound molecular state, in which case the rate of inelastic collisions can be significantly reduced 关39兴. The losses due to collisions with the molecules in the intermediate 共virtual兲 excited states and those due to spontaneous emission can also be reduced by operating in an offresonance regime with respect to the excited levels. A. Quantum gas to ‘‘liquid’’ transition

Of more importance than the simplest two-particle bound states are N-particle eigenstates and the ground state energy of this quantum many-body system, in three space dimensions. While this is a difficult problem, some important conclusions can be made by comparing the results obtained with 共i兲 the ansatz of Sec. III E and IV C, corresponding to N/2 independent dibosons, and 共ii兲 the coherent ansatz employed in Sec. IV D. As discussed in Sec. III and IV, a remarkable result that emerges with the treatment of the first type of ansatz is that, in the limit k m →⬁ and for ␬ (33) 3 ⫽0, it turns into an exact eigenstate and provides the exact ground state energy given by Eq. 共47兲. The ground state energy has no lower bound as ␬ 3 →0. This is in contrast to the mean-field behavior corresponding to the classical Hamiltonian energy, which is known to have a rigorous lower bound and to support classical solitons 关15兴. For the case of nonzero ␬ 3 , this idealized result serves as a lower bound to the true ground state energy with a finite momentum cutoff. For a finite cutoff k m ⯝2 ␲ /a 12⫽4 ␲ 2 ប/( ␬ 3 ␮ ), the ansatz corresponding to N/2 independent dibosons is no longer the exact eigenstate, and therefore does not necessarily result in the lowest possible energy. The corresponding estimate of the energy in three dimensions is obtained from Eqs. 共65兲 and 共82兲, for ⌬ ␻ →0 and k m r 0 Ⰷ1: 2 ˜E (N) 0 共 k m ⫽2 ␲ /a 12 兲 ⯝⫺2Nប 共 ␹ 3 兲 / 共 5 ␬ 3 兲 .

共83兲

This corresponds to a low-density regime of a quantum gas of N/2 independent dibosons or ‘‘dressed’’ molecules. We next address the question of whether the coherent or MFT variational ansatz can give a minimum energy ˜E (N) c , from Eq. 共69兲, that is lower than ˜E (N) (k ). This would corm 0 respond to a liquidlike regime of coupled Bose condensates, where formation of stable localized wave forms or matterwave solitons is more energetically favorable than ‘‘evapo-

FIG. 1. Estimates for the ground state energy per particle ⫺6 m3/2/s, m 1 ⫽m 2 E (N) 0 /N as a function of N for ␹ 3 ⫽10 ⫺25 ⫺25 ⫽10 kg 共so that ␮ ⫽m 1 /2⫽0.5⫻10 kg and m 3 ⫽2m 1 ⫽4 ␮ ⫽2⫻10⫺25 kg), and ⌬ ␻ ⫽0. The upper 共u兲 and lower 共l兲 bounds are for ␬ 3 ⫽6.6⫻10⫺17 m3 /s obtained with a 12⫽5 nm, and the cutoff for 共u兲 is k m ⫽2 ␲ /a 12⯝1.26 nm⫺1 . Curve c corresponds to the coherent variational ansatz and represents the mini(22) mum energy ˜E (N) for the case of ␬ (11) c 3 ⫽ ␬ 3 ⫽0, together with (33) (13) (23) ␬ 3 ⫽ ␬ 3 ⫽ ␬ 3 ⫽0.

ration’’ into a low-density gas of dibosons or ‘‘dressed’’ molecules. To answer this question in the general case of arbitrary values of the relevant parameters is a difficult problem. Additional complications emerge from the need to analyze stability properties of the actual soliton dynamics, with both parametric and repulsive quartic couplings. It is clear that strong quartic repulsion terms destabilize soliton propagation. If, however, these couplings are not too strong compared to the parametric coupling, then the parametric interaction can still act as a ‘‘glue’’ and compensate the interparticle repulsion, so that stable soliton propagation may occur. To proceed with our analysis we note that s-wave scattering amplitudes for atom-molecule and molecule-molecule collisional processes are currently not well known. For this reason and for simplicity we neglect the corresponding cou(13) (23) plings ( ␬ (33) 3 ⫽ ␬ 3 ⫽ ␬ 3 ⫽0) compared to the atom-atom (11) . In addition, we note that couplings ␬ 3 , ␬ 3 , and ␬ (22) 3 employing the symmetric Gaussian ansatz ␺ 1 (x)⫽ ␺ 2 (x) can (22) . We restrict our analysis to only be justified if ␬ (11) 3 ⫽␬3 the cases where 共i兲 the atomic self-interactions due to ␬ (11) 3 and ␬ (22) are negligible compared to the cross-interaction ␬ 3 , 3 (22) so that we can set ␬ (11) 3 ⫽ ␬ 3 ⫽0; 共ii兲 the atomic self- and (22) cross-couplings are all equal to each other, i.e., ␬ (11) 3 ⫽␬3 ⫽2 ␬ 3 . The result of minimization of the variational energy ˜E (N) c , Eq. 共69兲, in case 共i兲 is given in Fig. 1 共curve c, where we plot the estimates for the ground state energy per particle E (N) 0 /N versus N. The horizontal line 共l兲 represents the lower bound to the energy given by the idealized solution E (N) l /N⫽ ⫺ប( ␹ 3 ) 2 /(2 ␬ 3 ), Eq. 共47兲, while the line u is an upper bound ˜E (N) 0 (k m )/N obtained with the cutoff dependent ansatz corresponding to a low-density regime of a quantum gas of N/2 independent dibosons, Eq. 共83兲. The coherent or MFT ansatz gives a lower energy than the diboson ansatz for N⬎N cr ⯝3.5⫻105 , so that transition to a liquidlike regime of local-

063816-13



ized BEC solitons is more favorable in this domain. The relative number of particles in the atomic and molecular soli¯ 1,2 /N ¯ 3 , obtained from the optimum values of the tons N Gaussian parameters, decreases as N increases, implying that the coupled condensates stabilize against the interatomic repulsions by converting a larger fraction of atoms into molecules. For example, for the total number of particles N ¯ 1,2 /N ¯ 3 ⯝0.08. ⫽106 , this fraction is given by N (N) ˜ In case 共ii兲, we find that E c stays above the value of ˜E (N) 0 (k m ) for all N and no crossover occurs, implying that the regime of a low-density quantum gas of independent dibosons is always lower in energy than the coupled soliton regime. Thus, at low particle density, the formation of individual ‘‘dressed’’ molecules 共dibosons兲 is favored, as atoms couple to molecules in a particlelike way. These dressed states have interesting properties, reminiscent of Cooper pairs, but cannot be described by the classical parametric soliton equations. At large density 共but not too large so that s-wave scattering is dominant兲 and for parameter values characteristic of the case 共i兲, the coherent coupling of three entire condensates is dominant. With large enough parametric coupling, and provided other recombination processes are negligible, there are coherent nonlinear wavelike interactions between the atomic and the molecular Bose condensates 共just as in nonlinear optics兲, which make it possible to form stable threedimensional BEC solitons. For large s-wave scattering, case 共ii兲 illustrates a classically stable soliton that is unstable against ‘‘evaporation’’ to a quantum gas of dibosons. As mentioned earlier, loss processes can be detrimental to the above properties of coupled condensates. In practical terms, the time scale for inelastic losses must be much longer than the coupled condensate formation time scale. We have not given any experimental technique for generating the coupled condensate in its ground state. However, a possible method is to employ evaporative cooling while the atommolecule coupling is switched on. B. Coherent BEC soliton dynamics

In performing experiments on coupled atom/molecular BECs, the first signature of the nonlinear interactions we are interested in is likely to be in the dynamical behavior of the coupled condensates. This also allows us to check the stability, at the mean-field level, of the coherent soliton ansatz. We therefore consider (3D⫹1) spatiotemporal dynamics of the coupled condensates, obtained by direct numerical simulation of the MFT equations for the field amplitudes ␺ i (t,x). These are modified Gross-Pitaevskii equations of the form i

⳵␺ j ប 2 ⵜ 2␺ j⫹ ␹ 3␺ 3␺ * ⫽⫺ 3⫺ j ⫹ ␬ 3 兩 ␺ 3⫺ j 兩 ␺ j ⳵t 2m 1 ⫹ ␬ (3j j) 兩 ␺ j 兩 2 ␺ j ⫹ ␬ (3j3) 兩 ␺ 3 兩 2 ␺ j

i

共 j⫽1,2兲 ,

⳵␺3 ប 2 ⵜ 2 ␺ 3 ⫹⌬ ␻ ␺ 3 ⫹ ␹ 3 ␺ 1 ␺ 2 ⫹ ␬ (33) ⫽⫺ 3 兩 ␺ 3兩 ␺ 3 ⳵t 2m 3 (23) 2 2 ⫹ 共 ␬ (13) 3 兩 ␺ 1兩 ⫹ ␬ 3 兩 ␺ 2兩兲 ␺ 3 ,

共84兲

FIG. 2. Mean-field densities n 1,2⫽ 兩 ␾ 1,2(x,t) 兩 2 and n 3 ⫽ 兩 ␾ 3 (x,t) 兩 2 , representing simultaneous atomic 共a兲 and molecular 共b兲 solitary waves, as depending on time t and the radial coordinate r⫽ 兩 x兩 for ¯␬ ⫽6.6⫻10⫺17 m3 /s ( ␬ 3 ⫽6.6⫻10⫺17 m3 /s, ␬ (11) 3 (13) (23) ⫽ ␬ (22) ␬ (33) m 1 ⫽m 2 ⫽10⫺25 kg, ␹ 3 3 ⫽0), 3 ⫽ ␬ 3 ⫽ ␬ 3 ⫽0, ⫽10⫺6 m3/2/s, and N⫽106 . The initial optimum Gaussian parameters are g 1 ⫽g 2 ⯝8.6⫻1010 m⫺3/2, g 3 ⯝4.8⫻1010 m⫺3/2, w 1 ⫽w 2 ⯝0.97 ␮ m, and w 3 ⯝0.71 ␮ m. (21) (12) where we recall that ␬ 3 ⬅( ␬ (12) . 3 ⫹ ␬ 3 )/2⫽ ␬ 3 We consider for simplicity the symmetric case of ␺ 1 (x) (22) (13) (23) and ␬ (33) ⫽ ␺ 2 (x), with ␬ (11) 3 ⫽␬3 3 ⫽ ␬ 3 ⫽ ␬ 3 ⫽0. In these cases Eqs. 共84兲 reduce to

i

⳵␺1 ប 2 2 ¯ ⫽⫺ ⵜ ␺ 1⫹ ␹ 3␺ 3␺ * 1 ⫹ ␬ 兩 ␺ 1兩 ␺ 1 , ⳵t 4␮ i

⳵␺3 ប ⵜ 2 ␺ 3 ⫹⌬ ␻ ␺ 3 ⫹ ␹ 3 ␺ 21 , ⫽⫺ ⳵t 2m 3

共85兲

(22) )/2. where we have defined ¯␬ ⬅ ␬ 3 ⫹( ␬ (11) 3 ⫹␬3 The coupled atomic/molecular soliton dynamics can be studied by direct numerical simulation of the above equations, starting with initial Gaussian atomic and molecular mean fields. The results of simulations are given in Figs. 2 and 3, where we plot the density profiles 兩 ␺ 1,2兩 2 and 兩 ␺ 3 兩 2 as depending on time t and the radial coordinate r⫽ 兩 x兩 . This demonstrates stable propagation of coupled atomic and molecular solitons, for two values of ¯␬ corresponding, respec(22) tively, to previously considered cases: 共i兲 ␬ (11) 3 ⫽ ␬ 3 ⫽0, so (11) (22) that ¯␬ ⫽ ␬ 3 , and 共ii兲 ␬ 3 ⫽ ␬ 3 ⫽2 ␬ 3 , so that ¯␬ ⫽3 ␬ 3 . The graphs represent a phase matched (⌬ ␻ ⫽0) parametric inter-

063816-14



FIG. 3. Same as in Fig. 2 but for the value of ¯␬ ⫽ (22) ⫺17 19.8⫻10⫺17 m3 /s ( ␬ (11) m3 /s). The 3 /2⫽ ␬ 3 /2⫽ ␬ 3 ⫽6.6⫻10 initial optimum Gaussian parameters are g 1 ⫽g 2 ⯝3.13 ⫻1010 m⫺3/2, g 3 ⯝1.92⫻1010 m⫺3/2, w 1 ⫽w 2 ⯝1.8 ␮ m, and w 3 ⯝1.3 ␮ m.

action, with the initial optimum Gaussian parameters evaluated for N⫽106 . Clearly the Gaussian profile is only an approximate version of the true soliton envelope 共which can be calculated numerically, as in 关16兴兲; hence we observe small in-phase oscillations. We note that although the case of Fig. 2 has higher energy than the low-density regime of a quantum gas of independent ‘‘dressed’’ molecules, nevertheless it appears from the mean-field theory that soliton propagation can be possible as a metastable regime, presumably with quantum evaporation. This soliton propagation behavior leads to the remarkable property that coupled BEC solitons or localized matterwaves could be generated in three space dimensions without an external trapping potential. Similar spatiotemporal solitons have recently been observed with optical fields, but in the degenerate case of parametric interaction 关40兴. The results of the present nondegenerate theory indicate that additional s-wave scatterings 共or phase modulation processes, in the optical case兲 that exist among all three fields would tend to make solitons less stable than in the degenerate case. C. ‘‘Superchemistry’’ behavior

Another possible experimental approach to generating the coupled condensates is by first cooling an atomic vapor to a BEC, and then switching on the atom-molecule coupling. This can lead to the formation of the molecular condensate

and a ‘‘superchemistry’’ dynamics, where the condensate interconversion is dominated by the coherent stimulated emission of bosonic atoms or molecules into their respective condensates. The phenomenon would be the matter-wave analog to optical frequency summation. We note that similar behavior, in the case of degenerate parametric interaction, has recently been studied for a Feshbach resonance coupling of single atomic and molecular condensates 关41兴. Assuming uniform condensate wave functions, the system was analyzed in the context of quantum tunneling emerging from the oscillatory behavior of the number of atoms and molecules in their respective condensates. The oscillatory dynamics was in response to a sudden change of the detuning of the resonance, applied to a homogeneous atomic/molecular BEC that was initially in equilibrium. For the case of Raman photoassociation coupling, and a trapped atomic BEC as the initial condition, the nonlinear dynamics of the coupled condensates was studied in 关22兴 by direct simulation of the resulting degenerate MFT equations. This gave further insights into the rich variety of dynamical behavior and a theoretical prediction of the possibility of coherent chemistry or ‘‘superchemistry’’ behavior in coupled BEC systems. Here we extend this study to the case of two atomic and one molecular condensates, and analyze the nondegenerate MFT equations modified by the trap potential terms. The trap terms are of the form V i (x) ␺ i (i⫽1,2,3), to be added on the right-hand sides of Eqs. 共84兲. We consider a rotationally symmetric harmonic trap potential V i (x)⫽m i ␻ 2i 兩 x兩 2 /(2ប), where ␻ i is the trap oscillation frequency for the ith species, and restrict our analysis to the symmetric case of ␺ 1 (x) (22) (33) (13) (23) ⫽ ␺ 2 (x), with ␬ (11) 3 ⫽ ␬ 3 ⫽2 ␬ 3 and ␬ 3 ⫽ ␬ 3 ⫽ ␬ 3 4 ⫺1 ⫽0. In addition, we choose ⌬ ␻ ⫽1.5⫻10 s and the trap frequencies ␻ 1 /2␲ ⫽ ␻ 2 /2␲ ⫽ ␻ 3 /2␲ ⫽100 Hz. We simulate Eqs. 共85兲 together with the trap potential terms in two stages. In the first stage, we assume that the parametric coupling ␹ 3 is switched off, and that only atomic species are present in the trap. The result achieved is the steady state of the Gross-Pitaevskii equations for a twocomponent atomic Bose condensate, which we choose to correspond to an initial total number of atomic particles N ¯ 1 ⫹N ¯ 2 ⬃4.8⫻104 at a concentration of n⬃ ⫽N 19 5⫻10 m⫺3 . This provides the starting condition for the second stage of simulations, where we switch on the coupling ␹ 3 . The results are shown in Fig. 4, where we observe giant collective oscillations between the atomic and molecular condensates, which take place on short time scales. These oscillations are due to the coherent process of stimulated emission into a condensate of molecular dimers, followed by the reverse process of stimulated emission into the atomic condensates. The integrated number of particles in the atomic and molecular condensates as depending on time is shown in Fig. 5. This ‘‘superchemistry’’ is a type of coherent chemical reaction that can take place in BEC systems at ultralow temperatures. It is characterized by Bose-enhanced reaction rates (n˙ j(3) ⬀n j 冑n 3 , j⫽1,2) due to the effect of bosonic stimulated emission. This is in a sharp contrast to the predictions

063816-15


PHYSICAL REVIEW A 61 063816 VII. SUMMARY

FIG. 4. ‘‘Superchemistry’’ oscillations: atomic 共a兲 and molecular 共b兲 condensate densities n i ⫽ 兩 ␾ i (x,t) 兩 2 as depending on time t and the radial distance r⫽ 兩 x兩 from the trap center. The values of (22) parameters are ¯␬ ⫽19.8⫻10⫺17 m3 /s ( ␬ (11) 3 /2⫽ ␬ 3 /2⫽ ␬ 3 ⫽ (33) (13) (23) ⫺17 3 6.6⫻10 m /s), ␬ 3 ⫽ ␬ 3 ⫽ ␬ 3 ⫽0, m 1 ⫽m 2 ⫽10⫺25 kg, ⫺6 ␹ 3 ⫽10 m3/2/s, ⌬ ␻ ⫽1.5⫻104 s⫺1 , and ␻ 1 /2␲ ⫽ ␻ 2 /2␲ ⫽ ␻ 3 /2␲ ⫽100 Hz.

of conventional 共Boltzmann兲 chemical kinetics, where the chemical reaction rates do not depend on the number of product particles and go to zero at low temperatures, according to the Arrhenius law. We emphasize that this type of coherent density dependent oscillation is a signature of the nonlinear parametric coupling, and would represent a first step toward observing the liquid-gas phase transition discussed earlier.

FIG. 5. Total number of particles in the atomic 共1,2兲 and mo¯ i ⫽ 兰 d 3 x兩 ␾ i (x,t) 兩 2 as a function of lecular 共3兲 Bose condensates N time t.

In summary, we have presented quantum soliton or bound-state solutions to a nondegenerate parametric quantum field theory, in one, two, and three space dimensions. As in the degenerate parametric case, the results have quantum pointlike 共zero-radius兲 structures in the eigenstates in more than one space dimension, if there is no momentum cutoff. This is quite different from the behavior of solitons in the corresponding classical theory, and the reason for this is the inherently nonclassical structure of the bound state, which is a quantum superposition state. We note that most previous analyses of quantum solitons treated cases where the quantum soliton was at least qualitatively similar to the corresponding classical theory. This is not the case here. With the inclusion of momentum cutoffs on the nonlinear couplings, the two-particle bound state has a finite radius, even in the simplest case of a pure parametric interaction— i.e., without the quartic interaction term. We can estimate, in the case of nonlinear optical or atomic BEC interactions, that the nonlinear couplings should have a momentum cutoff no higher than an inverse carrier wavelength or inverse scattering length, respectively. These estimates can be improved by more careful treatment of the theory at large relative momenta. Such an improved treatment would be especially appropriate in the three-dimensional case where we obtain a linear divergence with k m →⬁. Most significantly, the quantum solitons form in physically testable regimes. Our estimates for characteristic soliton radii and binding energies, in the case of photonic interactions in highly nonlinear parametric ( ␹ (2) ) media, result in much more realistic values than examples of ␹ (3) solitons, with the required experimental environment being nearly available with current technology. In the case of BEC interactions, we point out the possibility of transition between the quantum 共diboson兲 soliton regime, where atoms couple to form molecules in a local way, to a classical soliton regime. In the classical domain, the coherent coupling of three entire condensates takes the place of the nonlinear optical process of sum-frequency generation. This gives the possibility of simultaneous atom and molecular matter-wave solitons in three space dimensions, and therefore an intense, stable, and nondiverging atom/molecular laser output. The stability properties of these solitons depend on the details of the s-wave scattering lengths between all three species present. We note that earlier examples of matter-wave BEC type solitons 共see, e.g., 关42兴, and references therein兲 were only for a one dimensional geometry. Of even more interest is a type of coherent BEC-enhanced chemical reaction or ‘‘superchemistry’’ behavior at ultralow temperatures, which follows from the underlying dynamics of coupled condensate nonlinear equations. Finally, the bosonic character of the fields is not relevant for the quantum bound-state theory derived here. Exactly the same results would occur if fermionic fields were involved, and we changed the corresponding commutation relations to anticommutators. In this respect, the present theory differs from the degenerate case 关11,23兴, where the results were only applicable to bosonic fields. This suggests that part of

063816-16



these results 共but not the classical soliton theory兲 could be extended to possible atomic fermionic superconductors, in which coupling between fermionic atoms is enhanced by the coherent production of bosonic molecules. Another possible application is to models of mesonlike coupling in mixed fermionic-bosonic systems. ACKNOWLEDGMENTS

N/2

We acknowledge the Australian Research Council for the support of this work, and are grateful to D. Heinzen for stimulating discussions. APPENDIX A

To calculate the energy ˜E (N) ˜ (N) ˆ ˜ (N) ˜ (N) ˜ (N) 0 ⫽具␸0 兩H兩␸0 典/具␸0 兩␸0 典 with the ansatz 共44兲 we first transform to the coordinates x ⫽R⫹m 2 r/M and y⫽R⫺m 1 r/M , and use binomial expansion, so that 兩 ˜␸ (N) 0 典 becomes 兩˜ ␸ (N) 0 典⫽

冉冕

d

D

ˆ †3 共 x兲 ⫹ x⌿

冕冕

⫽

兺 j⫽0

冉冊冉冕 N/2 j

冊

冊

⫻ ⫻

j

冉冕

冊冉冕冕

⫻ 具 0 1 ,02 兩

冉冕冕

ˆ 3 共 x⬘ 兲 d D x⬘ ⌿

冊

N/2

ˆ 3 共 x⬘ 兲 d D x⬘ ⌿

ˆ †3 共 x兲 d D x⌿

冊

N/2

兩 0 3典

N/2⫺ j

冊

冊冉冕冕冊 j⬘

d Dx d Dy

j

ˆ †1 共 x兲 ⌿ ˆ †2 共 y兲兩 0 1 ,02 典 . ⫻g 共 x⫺y兲 ⌿

⫽ 共 N/2兲 !V N/2

N/2 ˆ ˜ (N) 具 ˜␸ (N) 0 兩 H 兩 ␸ 0 典 ⫽ 共 N/2 兲 !V

共 D⫽2,3兲 ,

冋冕

N ប2 2 2␮

d D x 兩 ⵜg 共 x兲兩 2

⫹ប⌬ ␻ ⫹2ប ␹ D g 共 0 兲 ⫹ប ␬ D g 2 共 0 兲 ⫹

冉冊册

(33) ប␬D N ⫺1 . 2 2V

Here the contribution from the kinetic energy part 兰 d D x 兩 ⵜg(x) 兩 2 ⫽⫺ 兰 d D x g(x)ⵜ 2 g(x) can be shown to vanish, using Eq. 共15兲 and the property that 兰 d D x g 2 (x)⫽0 (D⫽2,3). Substituting then g(0)⫽⫺ ␹ D / ␬ D from Eq. 共35兲 we finally obtain ˜E (N) 0 ⫽

N/2⫺ j ⬘

兩 0 3典

d Dx

where V⬅ 兰 d D x. Similar calculations apply to the other averages involved ˆ ˜ (N) in 具 ˜␸ (N) 0 兩 H 兩 ␸ 0 典 , so that one can obtain 共for D⫽2,3)

d Dx d Dy

d D x⬘ d D y⬘ g 共 x⬘ ⫺y⬘ 兲

ˆ 1 共 x⬘ 兲 ⌿ ˆ 2 共 y⬘ 兲 ⫻⌿

j⫽0

冉冕冊冉冕冉冕冊

⫽ 共 N/2兲 !

j

j⬘


兵 . . . 其 j ⬘, j ␦ j ⬘ j ⫽ 兺兵 . . . 其 j ⬘⫽ j .

˜ (N) 具 ˜␸ (N) 0 兩␸0 典

N/2

具 0 3兩

兺兺

j ⬘ ⫽0 j⫽0

N/2

N/2

冉冊

冉冊冉冕 N/2

N/2

The nonvanishing terms in this sum contain no operators and can be further simplified by integrating with respect to ␦ functions from the commutators. The result of these integrations is that all terms, except the one corresponding to j ⫽0, will contain factors of the form 兰兰 d D xd D y g 2 (x⫺y) which vanish due to the zero-radius property of the g(r) function 关兰 d D r g 2 (r)⫽0 兴 in two and three dimensions (D ⫽2,3). The remaining nonvanishing contribution of the term with j⫽0 results in

兩0典


兺兺 j ⫽0 j⫽0 ⬘

兵 . . . 其 j ⬘, j ⫽

N/2

N/2⫺ j

N/2

N/2

⫽ 具 0 3兩

Here the vacuum state 兩 0 典 is defined as 兩 0 典 ⫽ 兩 0 1 典兩 0 2 典兩 0 3 典 , so ˆ i 兩 0 i 典 ⫽0. that ⌿ uses the comCalculating the averages involved in ˜E (N) 0 † ˆ ˆ mutation relations 关 ⌿ i (x),⌿ j (x⬘ ) 兴 ⫽ ␦ i j ␦ (x⫺x⬘ ), and relies on the zero-radius property of the two-particle correlation function g(r), i.e. g(r)⫽0 for r⫽0 and g(0)⫽⫺ ␹ D / ␬ D , in two and three dimensions. We demonstrate this on the ex˜ (N) ample of 具 ˜␸ (N) 0 兩 ␸ 0 典 which is written using the above expansion as N/2

兺兺

D

ˆ †1 共 x兲 ⌿ ˆ †2 共 y兲兩 0 典 . ⫻g 共 x⫺y兲 ⌿

˜ (N) 具 ˜␸ (N) 0 兩␸0 典⫽

N/2

j ⬘ ⫽0 j⫽0

d xd y

ˆ †1 共 x兲 ⌿ ˆ †2 共 y兲 ⫻g 共 x⫺y兲 ⌿ N/2

D

We first simplify the calculation by applying the commutation relations and reordering the operators so that all destruction operators stand on the right. Then all terms with j ⫽ j ⬘ in the above double sum will vanish due to extra factors ˆ †i (x)兴 acting on the vacuum state 兩 0 i 典 from the ˆ i (x) 关or ⌿ ⌿ right 共or on 具 0 i 兩 from the left兲. The remaining terms with j ⬘ ⫽ j are combined into a single sum according to

冉

冊冉冊

(33) ប␬D N ប共 ␹D兲2 N N ប⌬ ␻ ⫺ ⫹ ⫺1 2 ␬D 4 2 V

共 D⫽2,3兲 ,

which is the result given in Eq. 共45兲. For the case of odd values of N the N-particle ansatz ˆ †3 (x) acting on the vacuum contains an extra factor of 兰 d D x⌿ state 兩 0 典 from the left, in Eq. 共44兲, and N/2 is replaced by its integer part 关 N/2兴 . The final result for the energy ˜E (N) 0 in this case has the form of the above equation where the same replacement N/2→ 关 N/2兴 is applied. 063816-17



APPENDIX B

To calculate the N-particle energy ˜E (N) 0 (k m ) (N) (N) (N) ˆ ˜ ˜ ˜ (k ) 兩 H 兩 ␸ (k ) / ␸ (k ) 兩 ␸ (k ) with the ⫽ 具 ˜␸ (N) m m 典具 0 m m 典 0 0 0 cutoff dependent ansatz 共64兲 and the Hamiltonian given by Eqs. 共1兲, 共50兲, and 共51兲, we first use the binomial expansion, so that

冉

⫺D/2 ˆ† 兩˜ ␸ (N) 0 共 k m 兲典 ⫽ a 3共 0 兲 ⫹ 共 2 ␲ 兲

⫻

冕

兩 k兩 ⫽0

N/2

⫽

d D kG 共 k兲 aˆ †1 共 k兲 aˆ †2 共 ⫺k兲

冉冊 N/2

兺

j⫽0

⫻

km

j

冕

km

兩 k兩 ⫽0

冉

冊

具 0 3 兩关 aˆ 3 共 0 兲兴 N/2⫺ j 关 aˆ †3 共 0 兲兴 N/2⫺ j 兩 0 3 典 ⫽

冉冉

兩0典

冊

j

⫽

j

d kG 共 k兲 aˆ †1 共 k兲 aˆ †2 共 ⫺k兲兩 0 典 ,

where we assume N is even, and the vacuum state 兩 0 典 ⫽ 兩 0 1 典兩 0 2 典兩 0 3 典 is defined such that aˆ i 兩 0 i 典 ⫽0. We show the main steps involved in the calculation of ˜E (N) ˜ (N) ˜ (N) 0 (k m ) on the example of 具 ␸ 0 (k m ) 兩 ␸ 0 (k m ) 典 . Using the ˜ (N) above expansion, 具 ˜␸ (N) 0 (k m ) 兩 ␸ 0 (k m ) 典 is expressed as a N/2 N/2 double sum 兺 j ⬘ ⫽0 兺 j⫽0 兵 . . . 其 j ⬘ , j , which is reduced to a single sum 兺 N/2 j⫽0 兵 . . . 其 j ⬘ ⫽ j as the terms with j ⬘ ⫽ j will vanish, after reordering the operators, due to the unequal number of creation and destruction operators acting on the vacuum. The terms that can give nonvanishing contributions are written as

⫽

兺 j⫽0

冉冊 N/2 j

⫻

⫻ 共 2 ␲ 兲 ⫺D/2

1 共 2␲ 兲D

兩 k⬘ 兩 ⫽0

km

兩 k兩 ⫽0

d D k G 共 k⬘ 兲 aˆ 1 共 k⬘ 兲 aˆ 2 共 ⫺k⬘ 兲

冊

冊

j

d D kG 共 k兲 aˆ †1 共 k兲 aˆ †2 共 ⫺k兲兩 0 1 ,02 典

km

兩 k⬘p 兩 ⫽0

km

兩 kp 兩 ⫽0

d D k⬘p d D kp G 共 k⬘p 兲 G 共 kp 兲

兺

关 ␦ 共 k(1) ⫺k1⬘ 兲 ␦ 共 k(2) ⫺k2⬘ 兲 ••• ␦ 共 k( j) ⫺k⬘j 兲兴

兺

关 ␦ 共 k(1) ⫺k⬘1 兲 ␦ 共 k(2) ⫺k⬘2 兲 ••• ␦ 共 k( j) ⫺k⬘j 兲兴 ,

perm

j

冊

共B3兲 where 兺 perm represents summation with respect to permutations referring to the set of bracketed indices 关 (1),(2), •••,( j) 兴 in the product of ␦ functions ␦ (k(1) ⫺k1⬘ ) ␦ (k(2) ⫺k2⬘ ) . . . ␦ (k( j) ⫺k⬘j ). There are j! terms in each of the sums, such as

␦ 共 k1 ⫺k⬘1 兲 ␦ 共 k2 ⫺k⬘2 兲 ••• ␦ 共 k j ⫺k⬘j 兲 ⫹ ␦ 共 k2 ⫺k1⬘ 兲 ⫻ ␦ 共 k1 ⫺k⬘2 兲 ••• ␦ 共 k j ⫺k⬘j 兲 ⫹•••.

具 0 3 兩关 aˆ 3 共 0 兲兴 N/2⫺ j 关 aˆ †3 共 0 兲兴 N/2⫺ j 兩 0 3 典

⫻ 共 2 ␲ 兲 ⫺D/2

冉

km

perm

2

⫻ 具 0 1 ,02 兩

冉冉

兿 p⫽1 ⫻

˜ (N) 具 ˜␸ (N) 0 共 k m 兲兩 ␸ 0 共 k m 兲典 N/2

⬘ 冕冕冕冕

⫻ 共 2 ␲ 兲 ⫺D/2

关 aˆ †3 共 0 兲兴 N/2⫺ j 共 2 ␲ 兲 ⫺D/2

D

N ⫺ j ! ␦ 共 0 兲 N/2⫺ j , 2 共B2兲

where ␦ (0) is to be understood as ␦ (0)⫽ 兰 d D x/(2 ␲ ) ⫽V/(2 ␲ ) in the limit of infinitely large volume V→⬁. Similarly, reordering the operators in 具 0 1 ,02 兩 . . . 兩 0 1 ,02 典 produces nonvanishing terms involving products of ␦ functions, so that

具 0 1 ,02 兩共 2 ␲ 兲 ⫺D/2

N/2

冉冊

冕冕

km

兩 k⬘ 兩 ⫽0

km

兩 k兩 ⫽0

d D k⬘ G 共 k⬘ 兲 aˆ 1 共 k⬘ 兲 aˆ 2 共 ⫺k⬘ 兲

冊

冊

The product of the two sums will contain diagonal terms, i.e., terms in which the permutation arrangement of the bracketed indices 关 (1),(2), . . . ,( j) 兴 is the same in both sets of ␦ function products, so that these terms have the following form:

j

␦ 2 共 k1 ⫺k1⬘ 兲 ␦ 2 共 k2 ⫺k2⬘ 兲 ••• ␦ 2 共 k j ⫺k⬘j 兲

j

⫹ ␦ 2 共 k2 ⫺k1⬘ 兲 ␦ 2 共 k1 ⫺k2⬘ 兲 ••• ␦ 2 共 k j ⫺k⬘j 兲 ⫹•••.

d D kG 共 k兲 aˆ †1 共 k兲 aˆ †2 共 ⫺k兲兩 0 1 ,02 典 . 共B1兲

The remaining terms are the off-diagonal terms in which the arrangement of indices is different, as in a term like

Here the calculation of the averages 具 0 3 兩 . . . 兩 0 3 典 and 具 0 1 ,02 兩 . . . 兩 0 1 ,02 典 uses the commutation relations 关 aˆ i (k),aˆ †j (k⬘ ) 兴 ⫽ ␦ i j ␦ (k⫺k⬘ ) to change the order of creation and destruction operators in the operator products, so that all the destruction operators act on the vacuum from the right, while the creation operators act on the vacuum from the left. This gives vanishing terms and simultaneously generates nonvanishing terms due to the ␦ functions from the commutators. For the case of the average 具 0 3 兩 . . . 兩 0 3 典 the nonvanishing terms give

␦ 共 k1 ⫺k1⬘ 兲 ␦ 共 k2 ⫺k2⬘ 兲 ␦ 共 k3 ⫺k3⬘ 兲 ••• ␦ 共 k j ⫺k⬘j 兲 ⫻ ␦ 共 k2 ⫺k1⬘ 兲 ␦ 共 k1 ⫺k2⬘ 兲 ␦ 共 k3 ⫺k3⬘ 兲 ••• ␦ 共 k j ⫺k⬘j 兲 . The diagonal terms can be combined into a single sum over the number of permutations with respect to the set of bracketed indices:

063816-18

兺

perm

关 ␦ 2 共 k(1) ⫺k1⬘ 兲 ␦ 2 共 k(2) ⫺k2⬘ 兲 ••• ␦ 2 共 k( j) ⫺k⬘j 兲兴 .



The integrations in Eq. 共B3兲 over the ␦ functions involved in these diagonal terms will result in a factor of j! ␦ (0) j . The integrations over the ␦ functions in the off-diagonal terms can produce a factor of ␦ (0) k only with k⬍ j, and therefore the contribution of these terms can be neglected as compared to the contribution of the diagonal terms in the limit of V →⬁. This results in j

兿 p⫽1

冉

1 共2␲兲

⫻

冉兺

冕冕 km

D

perm

兩 k⬘p 兩 ⫽0

km

兩 kp 兩 ⫽0

d D k⬘p d D kp G 共 k⬘p 兲 G 共 kp 兲

冊

关 ␦ 2 共 k(1) ⫺k1⬘ 兲 ␦ 2 共 k(2) ⫺k2⬘ 兲 ••• ␦ 2 共 k( j) ⫺k⬘j 兲兴

⫽ j! ␦ 共 0 兲 j

冉

1 共2␲兲

D

冕

km

兩 k兩 ⫽0

d D kp G 2 共 kp 兲

冊

N/2

⫽ 共 N/2兲 ! ␦ 共 0 兲

⫽ 共 N/2兲 ! ␦ 共 0 兲

N/2

N/2

兺

j⫽0

冉

冉冊冉

1⫹

N/2

1

冊

j

共 2␲ 兲D

1

冕

共 2␲ 兲D

km

冕

兩 k兩 ⫽0

兩 k兩 ⫽0

d k G 共 k兲

d kp G 共 kp 兲 D

g 共 0,k m 兲 ⫽

2

2

冊

冊

N/2

.

Applying similar procedures to other averages involved in and keeping only the leading terms ⬃ ␦ (0) we obtain that

⫽

ˆ ˜ (N) 具 ˜␸ (N) 0 共 k m 兲兩 H 兩 ␸ 0 共 k m 兲典

共 2␲ 兲D

共2␲兲

D

兩 k兩 ⫽0

兩 k兩 ⫽0

d D kG 共 k兲 .

冊

共B5兲

q k ⫹1/r 20 2

,

which allows one to express R(r 0 ,k m ) as q2

1

f D 共 r 0 k m 兲 ⫺ 2 F 共 r 0 ,k m 兲 . r D⫺2 r0 0

冊

˜E (N) 0 共 km兲⫽ km

km

ប2 R 共 r 0 ,k m 兲 ⫹ប⌬ ␻ ⫹2ប ␹ D g 共 0,k m 兲 2␮

G 共 k兲 ⫽⫺

where we have defined

冕

冕

1

In addition, we use the definition of q, Eq. 共56兲, and express g(0,k m ) in terms of f D (r 0 k m ), using Eq. 共57兲. This makes it possible to rewrite Eq. 共B5兲 in the form of Eq. 共62兲, thus (2) proving that ˜E (2) 0 (k m )⫽E 0 (k m ), and therefore

⫹2ប ␹ D g 共 0,k m 兲 ⫹ប ␬ D g 2 共 0,k m 兲 ,

1

冉

R 共 r 0 ,k m 兲 ⫽

N ប2 R 共 r 0 ,k m 兲 ⫹ប⌬ ␻ 关 1⫹F 共 r 0 ,k m 兲兴 ⫺1 2 2␮

F 共 r 0 ,k m 兲 ⬅

d D kk 2 G 2 共 k兲 ,

Thus our final step in proving Eq. 共65兲 consists in showing that Eq. 共B5兲 is equivalent to the eigenvalue equation 共62兲. This equivalence can be shown with the use of the explicit expression for G(k), from Eq. 共53兲,

˜ (N) 具 ˜␸ (N) 0 共 k m 兲兩 ␸ 0 共 k m 兲典

冉

兩 k兩 ⫽0

⫹ប ␬ D g 2 共 0,k m 兲 ⫽⫺ប 2 / 共 2 ␮ r 20 兲 .

j

ˆ ˜ (N) 具 ˜␸ (N) 0 (k m ) 兩 H 兩 ␸ 0 (k m ) 典 N/2 ˜E (N) 0 共 km兲⫽

km

and therefore 关 1⫹F 共 r 0 ,k m 兲兴 ⫺1

D

冕

(2) 2 2 ˜E (2) 0 共 k m 兲 ⫽E 0 共 k m 兲 ⫽⫺ប / 共 2 ␮ r 0 兲 ,

共B4兲

km

共2␲兲

D

and

j

.

1

We next note that for the case N⫽2 the expression for ˜E (N) 0 (k m ) must give the exact two-particle solution for the energy E (2) 0 (k m ), given by Eqs. 共54兲 and 共62兲, i.e.,

Combining Eqs. 共B1兲–共B4兲 we obtain ˜ (N) 具 ˜␸ (N) 0 共 k m 兲兩 ␸ 0 共 k m 兲典

R 共 r 0 ,k m 兲 ⬅

d D kG 2 共 k兲 ,

N (2) E 共 km兲, 2 0

which is the result of Eq. 共65兲.

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