Experimental realization of the Bose-Einstein condensation of trapped weakly interacting atomic gases has renewed interest in various concepts of quantum mechanics of many body systems at ultra-low temperatures. Thermodynamics of an ideal gas of massive bosons shows that below some critical temperature most of particles accumulate in the ground state. A folk wisdom directly associates the condensate with macroscopic occupation of this one-particle ground state and such a scenario serves as the paradigm of the Bose-Einstein condensation. It is also well known that the above picture cannot be directly applied to an interacting system. One-particle states of a trapping potential lose their physical meaning for interacting system. Instead one has to consider a many body system, and it is not clear which is this macroscopically occupied state. Nevertheless, at least in theory, the condensate can be described by a macroscopic wave function (order parameter) with a given amplitude and phase. The condensate is believed to behave like a “giant matter wave” which shows long-range order. Coherence of such a macroscopic matter wave is one of the main attributes of Bose condensates, [1]. So far, mainly the first order coherence of condensed Bose gases at very low temperatures as well as at the phase transitions has been measured experimentally[2, 3, 4, 5].

The relation of the order parameter, off-diagonal long-range order, and phase to coherence of a Bose condensate is widely discussed in the literature and it seems to be a somewhat controversial issue [6, 7, 8, 9, 10, 11]. The reason is that all above mentioned characteristics can be studied in their whole extend only in the case of ideal gas. Analysis of their significance in a realistic situation of an interacting system requires an exact solution of the many body system. This, however, seems to be impossible for real forces.

The aim of this paper is to contribute to the discussion on the fundamental concepts related to the interacting condensate. To this end we analyze the ground state of an interacting Bose system using exactly soluble model[12]. This model has, of course, some unrealistic features, but it has also many others which are generic for all experimental realizations of atomic condensates of dilute gases.

Real condensates are created inside a magnetic trap providing harmonic trapping potential. The condensate exists in such a trap for a time of the order of a minute. Finite lifetime indicates that the existing condensates are, strictly speaking, open systems and therefore they should be in a highly entangled state of particles forming the condensate and the environment. On the other hand atomic condensates are isolated from the rest of the world much better than anything else and on the time scale much shorter than the condensate lifetime (which is long enough for practical purposes). Therefore we may safely assume that condensates are closed systems in a pure N-body state with a fixed energy and particle number.

We consider the following N-particle wave function which corresponds to the ground state of bosonic system:

where x_i is the position of the i – th particle and x_CM =${\sum }_{i=1}^N$ x _i /√N is the collective variable proportional to the center-off-mass coordinate. Let us list some important properties of this wave function:

1. It is totally symmetric.

2. It in not an N-fold product (unless ω=1) of one-particle states. This feature signifies an interacting system.

3. It is a product of the center of mass function (the first term) and the function depending on N – 1 independent internal degrees of freedom only (the second term). The relative-coordinates wave function is transitionally invariant. These features are generic for any two-body central forces.

4. The center-of-mass wave function describes the ground state of the trapping harmonic potential. Note that an unconventional normalization of the center-of-mass coordinate is adopted.

5. The function in Eq.(1) involves two different length scales. ℓ_CM =1 is the characteristic spatial extension of the center-of-mass state and characterizes the trapping potential, ℓ_int =1/√ω is the length scale corresponding to the spatial extension of the internal motion. The later length scale is related to the two-body interactions.

6. ω=1 signifies the ideal gas case and the function given in Eq.(1) becomes an N-fold product of identical gaussian functions. For attractive forces ω>1 and the center-off-mass extension is larger than the spatial extension of the internal coordinates. Repulsive forces are characterized by 0<ω<1. In such a case the center-off-mass length scale is smaller than the length scale of the relative motion.

We want to stress that the form of the “internal” wave function has been chosen because of its simplicity. In particular in the case of realistic interactions one can expect that each of N – 1 normal modes has its own characteristic length scale. In addition, it is easy to check that this wave function is the ground state of the Hamiltonian with quadratic two-body interactions, i.e. it corresponds to a long distance forces. The solution to this many body problem as well as analysis of some properties of the Bose condensate interacting by two-body harmonic forces are presented in [12, 13].

The meaning of the Bose condensation for an interacting system has been clarified by Penrose and Onsager[6, 7] who showed that Bose-Einstein condensation is evidenced by the occurrence of the off diagonal long range order in the first reduced density matrix. The off-diagonal long range order, however, is not a quantity which can be measured directly. In our opinion the best practical definition of the Bose condensated system can be formulated in the spirit of Feynman–s[14] description of superconductivity. Following his reasoning by Bose condensate we mean a bosonic many-body system under such a special situation in which quantum mechanics will produce its own characteristic effects on a large or “macroscopic” scale - the scale of a spatial extension of the system. Being more specific we may say that coherence is the feature which, similarly as in the case of laser light, proves (or disproves) quantum character of a Bose system.

It is convenient to introduce reduced density matrices. The one-particle density matrix is defined as: ρ1(x; y)=∫dr₂ ,…, r_NΨ*(x, r ₂,…, r _N)Ψ(y, r ₂,…r _N). Similarly, one can define s-particle reduced matrices for s larger than one. Although direct expressions for these matrices can be easily obtained [13] we are not going to invoke here their explicit form. Reduced matrices have direct physical interpretation. For example the diagonal part of ρ1(r; r) is equal to a one-particle density. Similarly, the ρ2(r ₁, r ₂; r ₁, r ₂) is the probability density of a joint detection of one particle at r ₁ and the other at r ₂.

Spectral decomposition of reduced density matrices:

uniquely determines the set of eigenstates ${\mathrm{\Phi }}_i^{\left(s\right)}$ (x₁,…x _s ) and eigenvalues ${\mathrm{\lambda }}_i^{\left(s\right)}$ . These states define natural s-particle “orbitals” and their mean populations. The s-particle subsystem is an open system because it interacts with N – s particle “reservoir”. Therefore it is typically in a mixed state. However, there are some particular situations when the s-particle subsystem is in a pure state, i.e ${\mathrm{\lambda }}_0^{\left(s\right)}$=1.

Let us consider the case of single particle density matrix. If ${\mathrm{\lambda }}_0^{\left(1\right)}$=1 then, from the point of view of a one-particle observables, the system may be viewed as if all particles occupied the same quantum state – the state corresponding to the dominant eigenvalue of the one-particle density matrix. Standard approaches to the description of the interacting condensate associate therefore the Bose condensate with a macroscopic occupation of the “ground state” of the one-particle density matrix, ρ1(x; y)≈[${\mathrm{\Phi }}_0^{\left(1\right)}$ (x)]*${\mathrm{\Phi }}_0^{\left(1\right)}$ (y). The function ${\mathrm{\Phi }}_0^{\left(1\right)}$ is the order parameter which is also called the condensate wave function – a complex field with well defined amplitude and phase. This is just this wave function which in an approximate approach is given by the solution of the Gross-Pitaevskii equation[15]. It is rather obvious that existence of a non-vanishing order parameter implies the off-diagonal long-range order in the first reduced density matrix, i.e. on the distance equal to the characteristic spatial extension of the condensate wave function we have: ρ ₁(x; y)≠0

In the typical case we can distinguish two length scales: (i) ${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(1\right)}$ which determines the spatial extension of the diagonal elements ρ1(x; x), (ii) and ${\mathit{\ell }}_{\mathit{\text{off}}}^{\left(1\right)}$ which characterizes the extension of the off-diagonal part of ρ ₁(x; y). The first scale is approximately equal to larger of the two: ℓ_CM and ℓ_int . It determines the region (around the trap center) where an individual particle can be effectively detected. The second scale is associated with an extension of region of a “one-particle phase correlation”. The off-diagonal long-range order or, in other words, the long range coherence is present in the system if the spatial extension of the off-diagonal elements is at least of the size of the spatial extension of the particle density, i.e. if ${\mathit{\ell }}_{\mathit{\text{off}}}^{\left(1\right)}$ ≥${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(1\right)}$ .

Existence of the macroscopically occupied state, ${\mathrm{\lambda }}_0^{\left(1\right)}$≈1, (order parameter) implies the off-diagonal order in the system, ${\mathit{\ell }}_{\mathit{\text{off}}}^{\left(1\right)}$ ${\mathit{\ge }}_{\mathit{\text{diag}}}^{\left(1\right)}$ . In the model studied in this paper (for N≫1) the one-particle state is macroscopically occupied if |κ|=|log ω|/logN<1. The latter inequality defines the weak interaction limit of our model. Interestingly, in the opposite case of strong interactions the one-particle density matrix has a form characteristic for a high temperature thermal state. Note however, that the total system is in its ground state given by a wave function, see Eq.(1). Let us add that in the case of trapped atomic gases a relative occupation of the condensate at zero temperature is estimated to be about ${\mathrm{\lambda }}_0^{\left(1\right)}$=0.99 [15]. On the other hand for strongly interacting system of liquid ⁴He the relative occupation of the condensate is only of the order of a few percent [16].

For the inhomogeneous system the long distance behavior of the off-diagonal order can be qualitatively described by the first order coherence function

Quantum optics teaches us that coherent laser field cannot be distinguished from a filtered chaotic field of a thermal source by measuring the first order coherence function. In Young’s two-slit interference experiment one sees the some interference patterns from a laser as from a conventional but filtered light source. This should also be true for massive particles. Therefore the distinction between the classical and quantum-mechanical systems requires the study of higher order correlation functions, [17, 18]. We will concentrate on the second order correlation function. It is related to the 2-particle reduced density matrix and therefore corresponds to a probability density of a simultaneous measurement of two particles:

Detection of the second order coherence has its analogy in the intensity-intensity correlation of the Hanbury-Brown and Twiss experiment. According to Glauber’s [19] theory of coherence of light the coherence of the system means factorization:

For the studied system the above condition is fulfilled in the weak interaction limit |κ|<1. For the coherent state of electromagnetic field similar equalities hold for all orders of coherence functions. This cannot be true for Bose systems of particles with mass. Such systems are in a Fock state (state with a fixed number of particles). Superselection rules forbid to superpose states with different particle numbers. In the case of noninteracting system equality of the type of the Eq.(5) holds up to the N-th order of coherence[20].

We show that this, however, cannot be true for interacting systems. We argue that interactions introduce quite different elements into the analysis of coherence. At this point we want to address the question if the second order coherence can exist in a interacting Bose system which does not show the first order coherence over its total spatial extension.

For example, the function g2 introduces, similarly as g1, at least two characteristic length scales. The first one ${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(2\right)}$ defines separation at which joint detection of two particles is possible. If particles interact this length scale is typically different than ${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(1\right)}$. In the case of attraction particles are expected to be close to each other, and therefore ${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(2\right)}$ <${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(1\right)}$ . Due to this fact even when the interacting system does not show the first order coherence it may exhibit the second order coherence – it will produce interference fringes in the Hanbury-Brown and Twiss - like experiment when ${\mathit{\ell }}_{\mathit{\text{off}}}^{\left(2\right)}$ >${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(2\right)}$ .

 

Fig. 1. Probability distributions: (i) of a one particle detection – red line; (ii) conditional probability density for detection of the second particle provided that the first one has been found at position x=0.002 – green line. The total number of particles is N=100000 and the interaction parameter κ=1.25.

Download Full Size | PDF

This fact has a very simple explanation. The attractive system is correlated. Although the position of one particle is very uncertain (smeared over the entire region of the trap) but if detected it significantly reduces the uncertainty of the position of the second particle. The later has to be “very close” to the first one. And over this small distance over which particles cluster the system can show coherence. This is nothing else but the off-diagonal long range order of the 2-particle density matrix. The difference from previously studied first order coherence is in the notion of the “long range order” – now it is the spatial extension of the 2-particle correlation function. This is illustrated in Fig. 1. A suggestion that nonexistence of the dominant eigenvalue in the one particle reduced density matrix not necessarily implies lack of coherence in the attractive Bose system has been made recently by Pitaevskii and Pethick [21].

For the repulsive interactions we have ${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(2\right)}$ ≥${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(1\right)}$ . As particles tend to avoid each other a particle is not very likely to be detected “close” to another. Because of this inequality if the repulsive system does not exhibit first order coherence, ${\mathit{\ell }}_{\mathit{\text{diag}}}^{\left(1\right)}$ >${\mathit{\ell }}_{\mathit{\text{off}}}^{\left(1\right)}$ it also does not show higher order coherence.

Finally, we want to make a more general remark. In our arguments we supported our point with results obtained within the exactly soluble model. This model, however, has some unrealistic features – in fact it corresponds to the forces of infinite range. It is important therefore, to distinguish which of above mentioned results are of general character and which are model dependent. Our line of reasoning went along arguments based on distinction of different length scales. We believe that this is the proper way to get model-independent conclusions. Using these argumentation we want to make a conjecture that repulsive interactions destroy higher order coherence in the ground state of an isolated Bose system. In the limit of very weak interactions when the total number of particles becomes sufficiently large higher order coherence can extend only over a distance smaller than the spatial extension of the system. To support our conjecture let us assume that a well defined order parameter exists, being a solution of the Gross-Pitaevskii equation Ψ(r). The spatial extension R_N of this one-particle wave function grows with N to infinity and in the Thomas-Fermi limit it can be approximated by R_N =(15Na_s )^1/5 [15] where a_s is the scattering length measured in units of the external harmonic trap extension ℓ_CM . Then the N-particle system can be described as the N-fold product of the order parameter and the central limit theorem leads to the following probability distribution of the center-of-mass variable x_CM =1/√N∑N _i=1 r _i :

where $\delta R=\sqrt{3⁄7}{R}_N\to \infty$ is a spatial dispersion of the condensate wave function. On the other hand the exact N-body solution gives N-independent size of the center-of-mass probability distribution

This is a totally different result than the single particle Gross-Pitaevskii theory. Thus the single particle approach to the interacting many body system cannot be adequate if one is interested in higher order correlations which depend on many particles variables. This fact means that the interacting Bose system (in the limit of large N) cannot exhibit high order coherences over its total spatial extension. High order coherence may exists on a smaller scale only. Such system cannot be easily distinguished from the filtered (i.e. monochromatic) beam of atoms produced by a chaotic source and the notion of the Bose-Einstein condensate can be ambiguous.

Acknowledgements This work is supported by Polish KBN grant no 2 P03B 130 15.