Quantum information approach to Bose-Einstein condensation of composite bosons

We consider composite bosons (cobosons) comprised of two elementary particles, fermions or bosons, in an entangled state. First, we show that the effective number of cobosons implies the level of correlation between the two constituent particles. For the maximum level of correlation, the effective number of cobosons is the same as the total number of cobosons, which can exhibit the original Bose-Einstein condensation (BEC). In this context, we study a model of BEC for indistinguishable cobosons with a controllable parameter, i.e., entanglement between the two constituent particles. We find that bi-fermions behave in a predictable way, i.e., the effective number of the ground state coboson is an increasing function of entanglement between a pair of constituent fermions. Interestingly, bi-bosons exhibit the opposite behaviour - the effective number of the ground state coboson is a decreasing function of entanglement between a pair of constituent bosons.


I. INTRODUCTION
The idea of Bose-Einstein condensation (BEC) was originally introduced for a uniform, non-interacting gas of elementary bosons. In reality, BEC experiments are conducted using potential traps for gases of bosonic particles, like alkali atoms, atomic hydrogen or metastable helium, that are composite particles made of fermions, and for which inter-particle interactions cannot be neglected [1][2][3]. Alternative BEC scenarios also take into account composite systems, e.g., condensation of fermionic pairs in liquid 3 He [4] or excitons (electron-hole pairs) in bulk semiconductors [5][6][7][8]. In addition, these BEC scenarios are closely related to other macroscopic quantum phenomena like superfluidity and superconductivity [9].
In many studies the internal structure of composite particles is neglected. On the other hand, it was noted that in some cases this structure plays an important role [10,11]. Therefore, it is interesting to see how BEC can be affected by the internal structure of composite bosonic particles.
In this work we consider a simple model of BEC with composite bosonic particles. In particular, we assume that neither the composite particles nor their constituents interact, such that the internal structure of composite particles is stable and temperature independent.
Of course, the bound states between constituent particles have to result from their interaction. However, here we assume that once the constituents form a composite particle state, they do not interact anymore. Physically, this may correspond to a dilute gas of composite particles for which energy scales of a binding interaction potential between constituents are much greater than energy scales of the confining trap. As an example, one may think of an atomic hydrogen gas in which ionization temperature is much higher than the standard temperatures required to obtain BEC. Such a simplified model allows us to focus on the fundamental problem of how BEC depends on the internal state of composite particles, while neglecting * Electronic address: phykd@nus.edu.sg other physical properties.
Nowadays, the phenomenology of composite bosons such as excitons, can be explained using the tools developed by quantum information theory. The role of quantum correlations between constituents forming a bound composite particle state can be studied qualitatively and quantitatively using the notion of entanglement. In particular, it was shown that the amount of entanglement between a pair of fermions (bosons) is responsible for their behavior as a single bosonic particle, i.e., only entangled particles behave like a single boson and the more entanglement between them, the more (less) bosonic they are [12]. Here, we raise the question: how is BEC affected by the entanglement between the two constituent fermions or bosons?
Before we start our discussion, let us recall the important results that are relevant to this work. Imagine a pair of distinguishable fermionic or bosonic particles. The system is described by the creation operatorsâ † k andb † l , where the indices k, l = 0, 1, . . . , ∞ label different modes that can be occupied by the two particles. These modes can for example correspond to different energies, or different momentum states. The wave function of the system is of the form ∞ k,l=0 where α k,l is the probability amplitude that particle a is in mode k and particle b is in mode l, and |0 is the vacuum state. Using insights from entanglement theory, the mathematical procedure known as the Schmidt decomposition allows us to rewrite the above state as [12] ∞ m=0 where the modes labeled by m are superpositions of the previous modes k and l and √ λ m are probability amplitudes that both particles occupy mode m. Note that despite the fact thatâ † m andb † m share the same label, physically these modes might be totally different. What is important is that, the modes labeled by m give rise to the internal structure of a composite particle.
We have introduced a composite boson creation operatorĉ † , that creates a pair of particles. Note that this operator resembles the one for Cooper pairs. The entanglement between particles is encoded in the amplitudes √ λ m . In particular, one can introduce a measure of entanglement known as purity For P = 1 the particles are disentangled, whereas in the limit P → 0 the entanglement between particles goes to infinity. The amount of entanglement can be also expressed via the so called Schmidt number K = 1/P . Intuitively, K estimates the average number of modes that are taken into account in the internal structure of a composite boson. The bosonic properties ofĉ † can be studied in many ways. For example, the commutation relation gives [ĉ,ĉ † ] = 1 + ξ λ m (â † mâm +b † mbm ), where ξ = −1 if a and b are fermions, or ξ = +1 if they are bosons. On the other hand, following the approach in [12] one may study the ladder properties of this operator where |n are states of n composite bosons, parameters χ n are normalization factors, such that n|n = 1, and | n are unnormalized states that can result form subtracting a single composite particle from a state |n . The states | n do not correspond to n−1 composite bosons of the same type, but rather to a complicated state of n − 1 pairs of particles a and b. The ladder structure of operatorsĉ † andĉ starts to approach those of ideal bosons if χn+1 χn → 1 for all n. In Ref. [12,13] it has been shown that for a pair of fermions the ratio χn+1 χn can be bounded from above and below by the function of entanglement This result shows that in the limit of large entanglement (P 1/n) the pairs of particles behave like real bosons. Other results on the relation between composite bosons and quantum correlations can be found in [14][15][16][17][18][19][20][21][22][23][24][25].
Assuming that composite bosons are in a thermal state, here we can describe the composite bosons with a Gaussian state which contains coherent, thermal, and squeezed ones. Thus, the Gaussian formula of the composite bosons is represented by the following modified operator that is based on the one studied in [12]  where the double indices refer to internal (m) and to external degrees of freedom (r). The internal index m represents their position values when the proton and electron are strongly correlated [13]. In our case r labels the energy levels of the trap in which the BEC takes place. Moreover, as we assumed in the beginning, the internal structure parameters λ m = (1 − x)x m (for 0 ≤ x < 1) are independent of r. The above operator has desirable properties, since it is possible to analytically evaluate the factors χ n and one can control the entanglement between constituents a and b via the parameter x [12]. For x = 0 the system is separable and in the limit x → 1 entanglement goes to infinity. In addition for a pair of fermions and for a pair of bosons. Finally, the average Schmidt number is The rest of the paper is organized as follows. In the next section we discuss the BEC of composite bosons made of fermionic pairs. We consider two cases, a potential trap with only two levels and the 3D harmonic potential trap with an infinite number of energy states. Next, we repeat the same for the composite bosons made of bosonic pairs. Finally, we analyze our results in the last section.

II. BI-FERMION: A PAIR OF FERMIONS
We consider indistinguishable cobosons in a two-level system and in a multi-level system, where each coboson is comprised of two fermions (bi-fermion). We investigate the case in which indistinguishable cobosons are in a Gaussian state, such that the normalization ratio of the coboson operator is represented by the parameter x [12]. From Eq. (8), x represents the degree of entanglement between a pair of fermions, where x = 0 (x = 1) means that a pair of fermions are separable (maximally entangled).
A. Two-level system: Simplified model First we consider a two-level system with a fixed number of N cobosons, see Fig. 1 (a). Although the fraction of cobosons in the ground state cannot exhibit a BEC phase transition, it is still interesting to compare its thermal behaviour with respect to a two-level system occupied by N cobosons.
The thermal state of this system reads where the total number of cobosons is N and where β = 1/(k B T ) and χ n (χ N −n ) is a normalization constant [12]. E 0 and E 1 denote the energy levels.
To evaluate the condensate fraction n 0 /N for a finite number N , we need to know how to describe the mean occupation number with energy E 0 , i.e., n 0 = ĉ † 0ĉ 0 . We derive the mean occupation number in the ground state as Putting µ = E 0 = 0 and E 1 = 1, the Eq. (4) becomes where the partition function Z is given by For a Gaussian state, the normalization ratio is given by [12]. When a pair of fermions is not entangled (x = 0), the mean occupation number becomes equal to one, and then the condensate fraction n 0 /N is given by 1/N regardless of temperature. When a pair of fermions is maximally entangled (x = 1), the condensate fraction is given by Hence for maximally entangled fermions the condensate fraction converges to one as temperature tends to zero. In this case the cobosons behave like elementary bosons. For near maximal entanglement (K N ) between a pair of fermions, we can derive the analytical result by taking the normalization ratio χ n+1 /χ n ≈ 1 − n/K [12]. As T → 0 (β → ∞), the condensate fraction is given by where the Schmidt number K is represented by the parameter x in Eq. (8). From Eq. (8) and the condition K N , the parameter x has the following range x < 1. For N = 100 we have 0.98 x < 1. When the Schmidt number K goes to infinity, then the condensate fraction goes to one. All the cobosons occupy the ground state energy level E 0 .
In Fig. 2 we plot the condensate fraction as a function of T against the parameter x covering its whole range. These are numerical results. As expected, Fig.  2 (a), the condensate fraction increases with the amount of entanglement as well as with decreasing temperature.
This coincides with the behaviour of an ideal bosonic gas. In Fig. 2 (b), where the amount of entanglement is relatively large, we observe an interesting phenomena. The condensate fraction for T → 0 is still small but it decreases with temperature unlike in Fig. 2 (a). We do not know how to explain this behaviour. In Fig. 2 (c) the amount of entanglement between the constituents fermions is relatively small and we expect to see a significant deviation from pure bosonic behaviour. Indeed, the condensate fraction never exceeds 1/N when T → 0, which is a result of the Pauli exclusion principle. Nevertheless, the condensate fraction increases with decreasing temperature. Figure 3 shows the condensate fraction for T ∼ 0 and a large amount of entanglement. For both two-and multilevel systems (discussed in the next section of the paper) the coboson BEC behaves like an ideal bosonic system. many energy levels of a 3D isotropic harmonic trap, see Fig. 1 (b). We fix the average number of cobosons to be N = N and describe the system via a grand canonical ensemble with a chemical potential µ. In this paper we do not take the proper thermodynamical limit (such a limit cannot be attained in real experiments) and thus we cannot observe a genuine BEC phase transition. Instead, we follow Mullin [27] and investigate the "pseudo-critical" temperature T 0 below which the increase in the chemical potential slows and the number of particles in the ground state begins increasing rapidly (this is also known as the accumulation point).
In the grand canonical ensemble, the mean occupation number of the m-th energy level E m and the total mean occupation number are given by whereN m =ĉ † mĉm and Z m = (1 − e −β(Em−µ) ) −1 .
The energy levels in the 3D isotropic harmonic potential are given by E m = ω(m x + m y + m z + 3/2), where m x , m y , m z = 0, 1, 2... The normalization ratio is given by χ n+1 /χ n = x n (n + 1)(1 − x)/(1 − x n+1 ). When a pair of fermions is not entangled (x = 0), the mean occupation number in the ground state is given by where the energy E 0 has been taken to be zero. In this scenario, the ratio N 0 /N goes to 1/N . It can be explained that only one pair of fermions stay on the ground state energy level E 0 for T = 0 and then they start to move to the excited energy levels E m (m = 0) with T . When a pair of fermions is maximally entangled (x = 1), the mean occupation number of the ground state energy level becomes the same as the Bose-Einstein distribution. In this scenario we perfectly recover the conventional Bose-Einstein condensation results. For all regime of x (0 < x < 1), the condensate fraction can be numerically estimated using the approximations, where S approaches ζ(3) = ∞ p=1 1 p 3 ≈ 1.202 as N → ∞ and x → 1, i.e., for an infinite number of maximally entangled cobosons. The average value N 0 α∼1/N satisfies the boundary conditions, N 0 α∼1/N ≈ 1 at x = 0 (no entanglement) and N 0 α∼1/N ≈ N at x = 1 (maximal entanglement). For near maximal entanglement between the two constituent fermions, the detailed calculations are given in the Appendix.
We plot the condensate fraction as a function of T /T 0 for different x in Fig. 4. The condensate fraction increases with decreasing temperature as well as with increasing entanglement between the two constituent fermions. At T /T 0 ∼ 0, we find that as the entanglement approaches 0, the condensate fraction converges to 1/N . Figure 3 (b) demonstrates that when T /T 0 ∼ 0, the condensate fraction is maximized as a monotonically increasing function of entanglement, which asymptotes to 1. In Fig. 4, we can also see that the transition temperature is an increasing function of entanglement, where we have defined the transition temperature as the point at which there are no cobosons in the ground state. This reflects the fact that the condensate fraction increases with increasing entanglement. Therefore, similarly to the two-level system, we observe that the condensate fraction increases as a function of the entanglement between the two constituent fermions.
As an example, we consider how T 0 (pseudo-critical temperature) and T c (critical temperature) are different in a BEC comprised of atomic hydrogen gas for which T e c (experimental critical temperature) was observed at 50µK [28]. Given the density of the hydrogen BEC (n = 1.8 × 10 20 m −3 ), the corresponding theoretical critical temperature in the thermodynamic limit is obtained as T t c = h 2 2πmk B ( n ζ(3/2) ) 2/3 ≈ 51µK. Since the theoretical critical temperature is derived for ideal BEC, the corresponding pseudo-critical temperature is obtained as For the experimental critical temperature, the pseudo-critical temperature is derived as a function of the degree of entanglement between the proton and electron. Using the purity of the proton P = 33 [13] with the experimental trapping size (b ≈ 9.6 × 10 −8 m), we find that the proton and the electron are maximally entangled. Thus, using the relation (T /T 0 ) 3 ζ(3) = (T /T c ) 3 , the corresponding pseudo-critical temperature is derived as 16µK. Therefore, we see that our pseudo-critical temperature is a good approximation for the critical temperature.

III. BI-BOSON: A PAIR OF BOSONS
We consider cobosons comprised of two bosons (biboson). For a Gaussian state, the normalization ratio is represented by χ n+1 /χ n = (n + 1)(1 − x)/(1 − x n+1 ) [12]. Here x parametrizes the degree of entanglement between a pair of bosons. An example of a coboson is a bi-photon generated by spontaneous parametric down conversion, which exhibits composite behavior even if the two photons are spatially separated [12].
A. Two-level system: Simplified model We consider a two-level system with a fixed number of N cobosons. All the formulas used in the previous section are applied here as well -the only difference is the normalization ratio χ n+1 /χ n . When a pair of bosons is not entangled (x = 0), from Eq. (11) the condensate fraction is given by where β = 1/(k B T ). Hence for separable bosons the condensate fraction converges to N as temperature tends to zero. Although the cobosons are no longer behaving like ideal bosons, the dissociated components of each bi-boson pair will both independently exhibit bosonic behavior. This causes the condensate fraction to increase as the entanglement between the two constituent bosons decresases. We can see this directly from the formula forĉ † in Eq. (5). At x = 0 (no entanglement), the coboson operator is represented byĉ † =â †b † . Since the temperature is zero at the moment, the state of cobosons in the ground state can be described by the coboson number state. So the mean occupation number of the cobosons is given by where a and b represent different modes. When a pair of bosons is maximally entangled (x = 1), the condensate fraction converges to one as temperature goes to zero. For near maximal entanglement (K N ) between a pair of bosons, we can make the approximation, χ n+1 /χ n ≈ 1 + n/K [12]. As T → 0, the condensate fraction approaches where n 0 x=1 /N is given by Eq. (12). If the Schmidt number K goes to infinity, then the condensate fraction goes to one. For all regimes of x (0 < x < 1), we plot the condensate fraction as a function T in Fig. 5. The condensate fraction increases with decreasing temperature, and it decreases with increasing entanglement between the two constituent bosons. At T ∼ 0, the condensate fraction is maximized as a decreasing function of entanglement which ranges from N to 1, as shown in Fig. 6 (a). In contrast to cobosons comprised of fermions, therefore, the condensate fraction decreases with entanglement between the two constituent bosons. We consider a 3D isotropic harmonic trap which contains an average of N cobosons. When a pair of bosons is not entangled (x = 0), the mean occupation number of the lowest energy level in Eq. (14) is given by where z = exp(βµ) is the fugacity and E 0 has been taken to be zero. Compared with the Bose-Einstein (BE) distribution where N 0 = z/(1 − z), the mean occupation number is always greater than the BE distribution one.
Hence it is expected that the corresponding condensate fraction is always greater than 1. When a pair of bosons is maximally entangled (x = 1), the mean occupation number of the ground state is the same as for BE distribution. This reaffirms the potential for BEC of bi-bosons. In Fig. 7 we plot the condensate fraction as a function of T /T 0 , for bi-bosons exhibiting a range of entanglement values. The condensate fraction increases with decreas- ing entanglement between the two constituent bosons. As bi-bosons become less entangled they behave more like a system of two independent bosons. Hence at T /T 0 ∼ 0, the condensate fraction is maximized as a linearly decreasing function of entanglement, as shown in Fig. 6 (b). Using Eq. (19), we derive the maximum condensate fraction at x = 0 as where N is sufficiently large. In Fig. 7, we can also see that the transition temperature decreases with increasing entanglement. This reflects the fact that the condensate fraction decreases as entanglement increases. Therefore, similarly to the two-level system, we observe that the condensate fraction decreases as a function of entanglement between the two constituent bosons. We have never experimentally observed the phenomenon of bi-boson BEC, but a BEC experiment has been conducted using photons in an optical micro-cavity [29]. Based on the techniques used to create a BEC from photons, we look forward to observing future bi-photon condensates in optical cavities.

IV. CONCLUSION
We studied how the deviation from ideal bosonic behavior exhibited by cobosons affects BEC. We specifically consider bi-fermions trapped in a two-level system or a 3D isotropic harmonic potential. We demonstrated that the condensate fraction is an increasing function of entanglement between the two constituent fermions. As the entanglement increases, from fully separable to maximally entangled fermions, the condensate fraction increased for T < T 0 in the 3D isotropic harmonic system. Heuristically the higher the level of entanglement between a pair of fermions the more bosonic the cobosons behave. Correspondingly, we found that the transition temperature for the 3D isotropic harmonic system, i.e., the temperature at which all the cobosons moved to the excited states, is an increasing function of entanglement.
Furthermore, we discussed coboson BEC, where each coboson is a bi-boson. Surprisingly it was shown that the condensate fraction decreases with increasing entanglement between a pair of bosons. As the entanglement parameter (x) increased from 0 to 1, the condensate fraction decreased for T < T 0 in the 3D isotropic harmonic system. The higher the entanglement the more closely the cobosons imitated indivisible bosons. When the entanglement between bi-boson became sufficiently small, the bi-boson pairs dissociated, increasing the effective number particles in the condensate. Correspondingly, the transition temperature for the 3D isotropic harmonic system decreased with increasing entanglement.
Here we leave an open question about the strange behavior of cobosons in the intermediate values of x (entanglement), as shown in Fig. 2 (b). As further work, it would be interesting to study how entanglement between a pair of fermions (bosons) could affect super-radiance in coboson BECs.