# Binding energies of composite boson clusters using the Szilard engine

###### Abstract

We evaluate the binding energies of systems of bosonic and fermionic particles on the basis of the quantum Szilard engine, which confers an energetic value to information and entropy changes. We extend treatment of the quantum information thermodynamic operation of the Szilard engine to its non-trivial role in Bose-Einstein condensation of the light mass polariton quasiparticle, and binding of large multi-excitonic complexes, and note the same order of magnitudes of exchange and extraction energies in these disparate systems. We examine the gradual decline of a defined information capacitive energy with size of the boson cluster as well as the influence of confinement effects in composite boson systems. Can quantum informational entropy changes partly explain the observations of polariton condensates? We provide energy estimates using the system of polariton condensates placed in a hypothetical quantum Szilard engine, and discuss the importance of incorporating entropy changes introduced during quantum measurements, and in the interpretation of experimental results.

## I Introduction

Recently, studies of deviations from perfect bosonic behavior in composite bosons or “cobosons” Combe1 ; Combe2 ; Combe3 , and from which fermionic features emerge by virtue of the Pauli exclusion principle has received increased attention Law ; Woot ; Rama ; Tichy1 ; Tichy2 ; bobby ; thilamc , partly due to the quantum correlated electronics of such systems. While a system of ideal bosons possess enhanced entanglement features as quantified by the Schmidt number measure, the degree of correlations between the particles diminish with appearance of fermionic attributes. There is loss in information that occurs when composite bosons are formed, as illustrated by the inaccessibility of spin configuration variables, which otherwise is explicitly known in the constituent fermions. There are subtleties to the degree of loss of information which occurs, for instance, in a system of two or more bosons: two fermions constituting a single boson may be distinguished if the separate bosons occupy distinct external states (position or momentum of center of mass wise), and entanglement involves pairs of different fermions Rama . The intrinsically quantum feature of distinguishability therefore underpins quantum processing attributes of composite bosons.

An area that has been gathering great interest in recent years, and of relevance to composite boson systems, are the links between thermodynamic principles and quantum information entities laud ; Szilard ; bied ; Wook ; Yao ; Saga1 ; Saga2 ; ved1 ; ved2 . Such links were first examined on the basis of computation energy cost by Landauer laud . Landauer considered that the irreversible manipulation of information is accompanied by a corresponding increase in the tangible degrees of freedom of system, or the environmental reservoir at the receiving end of the coded information. This scheme establishes a vital link between information exchanges and a physical mechanism that may take a specific form. In an earlier work, Szilard employed the Maxwell’s demon model to show that of work can be extracted from a thermodynamic cycle, and highlighted that a positive entropy production in measurement compensates for the work gain Szilard . This ensures that the second law of thermodynamics is left intact, by virtue of being governed by statistical uncertainties.

The hypothetical Szilard engine (shown below in Fig.1) consists of an atom confined in box, with the thermodynamic engine cycle involving the main steps of: (1) Insertion of a partition to divide the engine box into two disconnected parts, (2) Uncertainty in actual position of the atom is acknowledged, (3) Measurement procedure to determine the exact location of the atom, (4) Extraction of work () via isothermal expansion by moving the partition quasi-statically while in contact with a thermal reservoir of temperature , and (5) Removal of the partition completing the engine cycle. While the classical engine ignores entropy changes due to measurement procedures, the entropy that is created by the process of observation bied forms a key element of the quantum version of the Szilard engine, highlighting the critical role of monitoring instruments in the quantum regime.

This work aims to examine composite boson systems in the context of the energy-information link, and the application of energetic value of information to the evaluation of the binding energies of exotic systems such as boson and fermion condensates, which appears not to have been examined in greater details in earlier works. Excitons which are correlated electron-hole quasi-particles possess high binding energies when confined in semiconductor two-dimensional quantum wells chem ; ding ; masu ; taka ; jai ; oh where motion is quantized in the direction perpendicular to the well. A strong-coupling interaction occurs between excitons and photons when the quantum well is positioned to coupled with the optical modes of microcavities, giving rise to polaritons with integer-spins weis ; yama ; butov . Polaritons are composed of interacting exciton-photon quasi-particles, which provide an ideal example of a system capable of forming composite boson condensates. Due to the photonic contribution (with polariton resonances around 800 nm in one work amo ), the polaritons possess greatly reduced effective masses of about 10 where is the free-electron mass. The small mass results in large Broglie wavelengths which exceed the mean average separation of the bosonic polaritons, a condition needed for Bose-Einstein condensation. Experimental observations have been reported at lattice temperatures as high as a few kelvins in several works lag ; bali , these temperatures in general are higher than those obtainable for the pure exciton or cold atoms bosecom ; moska ; davy . It has been shown lag that condensates can preserve spatial coherences over distances much larger than the polariton De Broglie wavelength timo , hence quantum properties are remarkably revealed on classical scales of time and length below a certain transition temperature. As a consequence, the condensates exhibit superfluidity amo where quantum fluid appear to move without great friction, and composite bosons exhibit superconducting properties carbo . The phase transitions that occur in Bose-Einstein condensates are yet to be examined from the context of the deep links between energy and information. In this work, we use both these experimental values (mass, temperature) to make comparison of energies of boson systems derived via the Szilard engine.

We point out other excitonic systems, that can be examined from the Szilard engine perspective. For instance, it is well known that large exciton complexes (), with number of neutral excitons bound to an electron are known to exist in two-dimensional systems of fermions (electrons, holes) in the presence of strong magnetic fields 2d0 ; 2d1 ; 2d2 ; 2d3 . Such complexes may be present in a multi-component plasma systems containing both electrons and complexes. The appearance of these large complexes, may partly be explained by a model of localized system of distinguishable particles with negligible wavefunction overlap, that interact more significantly with an external field than with each other. Under these conditions, the -body problem may be reduced to an equivalent two-body problem hall ; silve . In similar light, the coalescing of the properties of a system of two excitons into one exciton has produced good agreement with experimental results for the singly charged exciton thilex and biexciton jai1 ; jai2 .

In the context of the Szilard engine, it would be worthwhile to seek clarification on whether the energies of large Coulombic systems such as multi-excitonic systems are partly due to quantum informational entropy exchanges. The dependence of binding energies on factors such as the number of charge carriers, degree of fermionic attributes and characteristics (shape,size) of the confining potential, may provide clues to the actual contribution of quantum information and entropy measures to their overall binding. The role of quantum information exchanges in complex systems in the creation of new forms elementary excitations remains unsolved, and is worthy of further investigations. As far as we are aware, all known works on the composite boson system, such as that represented by multi-excitonic systems, have excluded the binding forces that may arise from the energetic value of information. The obvious reason for this being that any information exchanges arising from Pauli-based interactions have been considered “non-physical”, and consequently the associated retrieval of an energy-like quantity from specific interactions have not been given much attention.

## Ii Binding energy of two composite bosons due to the Pauli exchange interactions

In order to demonstrate the evaluation of binding energy of two composite bosons, we revisit the recent work of Kim et. al. Wook in which the quantum version of the Szilard engine was considered. Kim et. al. Wook showed that the crossover from indistinguishability (true bosonic) to distinguishability (which we interprete as loss in bosonic quality) occurs as the temperature increases. We thus consider that bosons assume fermionic features with change in environmental conditions. We first examine results of the toy model of a system of two bosons confined in a symmetric potential box of size , with a wall inserted at = at the start. The quantum Szilard engine is based on the relation Wook

(1) |

where is the total work arising from the wall insertion, measurement and adiabatic wall movements (Fig. 1) performed by the engine during a single cycle. is the Boltzmann constant, is the temperature of the heat bath and is the probability of measuring particles to the left of the partition. The term where is the partition function of the system with particles on the left, at the equilibrium position denoted by . This equilibrium position is determined by a delicate balance of forces on the partition Wook :

(2) |

where is an external parameter and is the mean occupation number of the th eigenstate with energy . It is assumed that the system is in thermal equilibrium throughout the engine operation, and includes the possibility that the particles may in tunnel into the other side of the wall. For the one particle (=1) case, the total work performed by the quantum Szilard engine during one cycle is obtained using ==, ==1 and Eq. 1 as = ln2.

For the two indistinguishable bosons that can be present at two possible locations of the box, using =, ==1 (when the partition is pushed to the end), =, the total work that can be extracted was obtained as Wook ,

(3) | |||||

(4) |

where is a function of the temperature, . Here however, we consider the decrease in (with increase in ) as loss in quality of the bosonic character linked to “degree of indistinguishability”. We note the importance role of temperature which is associated with thermal fluctuations: it determines whether a boson particle acquires distinguishable features, hence with the availability of a range of electronic states at higher temperatures, bosons become increasingly fermionic and distinguishable. The importance of the parameter provided by the approximate separation between energy levels, should be noted. The high temperature regime refers to , while the reverse holds valid for low temperatures.

The acquisition of fermionic features with increase in temperature appears similar in context to the loss in “purity” used in earlier works on composite bosons Law ; Woot ; Rama ; Tichy1 ; Tichy2 ; bobby ; thilamc . A system of bosons which are indistinguishable possess a large Schmidt number , and hence can be considered a highly correlated entangled system. Any deviations from this arrangement due to changes in the external environment, leads to irretrievable loss in bosonic indistinguishability. One can appreciate that an increasing temperature also has the same effect as increased confinement of a system of ideal bosons. For each boson that constitutes two fermions, as in the case of the exciton, a display of the crossover from indistinguishability to distinguishability for bosons with change in external parameters (temperature, confinement) becomes evident. We therefore quantify the quality of the bosonic indistinguishability using as it appears in Eq. 3. At =1, the true bosonic state yields , and at the extreme limit of =0, where the boson exhibits fermionic features, .

In the low-temperature limit, the ground state is predominantly occupied as higher excitation levels are suppressed. The possibility that particles tunnels from high to low energy levels as the barrier is shifted, during the operation of the Szilard engine, is minimized at low temperatures. Such tunneling effects are expected to dominate at higher temperatures. The importance of including the work gained during the removal of the barrier, during which tunneling occurs in the event of non-equilibrium partition position, was discussed in Ref.ved1 . The same authors highlighted that work extracted is based mainly on the information gain of the initial measurement ved2 , which ultimately, is dependent on the nature (e.g. distinguishable or indistinguishable partition, boson or fermion) of the system being examined.

Fig. 2 shows the arrangement of 20 indistinguishable particles using the Ferrers diagram, which depicts the array of left-justified solid circles (e.g, bosons) for a number of partition possibilities. As the temperature approaches zero, the possible arrangements converge to one partition where all particles occupy one level state. The incorporation of entropy changes associated with tunneling effects based on the Ferrers diagram (see Fig. 2) which shows the different occupation possibilities, is therefore appropriate at higher temperatures, in more realistic treatments of the Szilard engines.

Bearing in mind that spin attributes were neglected in the original formulation Wook of Eq. 1, we define the composite boson binding energy due to Pauli interactions as

(5) |

where the superscript “b” correspond to a particle with a higher measure of bosonic quality and superscript “f” correspond to one that exhibits greater fermionic feature. This defined binding energy removes, though not entirely, the tunneling effects (discussed in Ref.ved1 ) that is common to the terms on the right hand side of Eq. 5. We note that in excitonic systems, the binding energy is the difference between the coboson energies and the optical gap thilex ; jai1 . Fig. 3 shows the monotonic increase of the binding energy of the boson with increase in the bosonic quality, . As is well known, more work can be extracted from the “information-rich” boson which has the highest quality measure of unity.

## Iii The distinguishable particle systems

The case of the general distinguishable particles enclosed in the Szilard engine, has been examined in earlier works Yao ; ved2 , however we reiterate some salient features of generalized systems for the purpose of examining the binding and capacitive energies, which remains largely unexplored. For instructive purposes, we briefly examine the case of three particles in the Szilard engine. In Fig. 4, we show the number of ways that three distinguishable (and indistinguishable) particles can be placed within two distinguishable partitioned spaces. For the general particles, the total number of microstates is = since each particle can occupy one of two possible locations, we thus obtain =8. For probable number of particles on the left partition, we obtain the total multiplicity,

(6) |

A general expression for is given by

(7) |

At the equilibrium point, the total number of microstates is determined by the probability of particles on the left and on the right side of the engine, hence

(8) |

(Eq. 16) and (Eq. 8) applies to boson and fermions in the high temperature limit, due to increased availability of extra energy states. In the case of three distinguishable particles, we obtain = =, ==. At equilibrium, it is easily shown that ==1, while at high temperatures, the value of == can be obtained using Eq. 2.

By setting , and using Eq. 6 and the Stirling formula , we obtain for a small , or

(9) |

where =. Hence the distribution, becomes Gaussian in the limit of very large , a result that is well known in the standard statistical physics text sbook . Eq. 9 indicates that it is more likely that the particles are equally distributed on either side of the box, (i.e. = correspond to the most probable microstate), and . At equilibrium position, as well, thus the total work that can be extracted at is

(10) |

where we neglect the small population difference in the case of an odd number of particles. Eq. 10 is the result of minimal changes in population (and information) imbalances at start and end of the engine cycle, and also due to the choice of an initial measurement in which the partition is inserted in the middle of the Szilard engine. Thus far in the analysis involving various entropy changes, we have employed a simplified model where information content associated with tunneling of the particles within the various energy levels (see Ferrers diagram in Fig. 2) was neglected.

### iii.1 Biased Szilard machine

Here we consider the existence of a biased distribution of quantum particles at the beginning of the cycle. This may occur as a result of a preference for particles to be located in the left side of the engine, as compared to the right side, due to variations in available energy levels. We thus rewrite Eq. 16 as

(11) |

where the superscript denotes a biased Szilard machine. Assuming the presence of large number of participants, and a equilibrium state where the biased arrangement is absent, we obtain, using the Stirling formula and Eq. 1,

(12) |

where = , and we set at the equilibrium point of the engine. The non-zero work extracted here arises due to the biased distribution at the start of the cycle, and is a result of energy that is transferred from the “observation energy” that was expanded in order to introduce a preferred distribution of particles.

### iii.2 Szilard engine with indistinguishable partitions

In the case of a quantum szilard machine, where the partitions are indistinguishable, i.e, we are unable to tell the left box from the right box, the total number of possible microstates is decreased further. For instance, the possibilities of arrangements numbered 1 and 8 in Fig. 4 become equivalent (or = =), as the total number of partitions is 4. Likewise those numbered 2 to 4 can be merged with options 5 to 7, and we get = =. For the general case, where distinguishable balls distributed into the 2 indistinguishable sides of the Szilard engine, the total number of distribution possibilities is given by

(13) |

where is the Stirling numbers of the second kind stir1 ; stir2 .

## Iv indistinguishable particle systems: Information Capacitive Energies

Unlike in Section III where we considered distinguishable particle systems, here we obtain some estimates of the information capacitive (or addition) energies of indistinguishable particles such as bosons. Following the relations for capacitive energies in Ref.ind , we write

(14) | |||||

(15) |

, the work extracted from a system of bosons, at low temperatures, is obtained by considering the case of indistinguishable particles placed in two distinguishable partitions where the total number of microstates is given by , with

(16) |

As shown in Fig. 4b, the total number of possible arrangements in the case of =3, is given by =4, with == ==. is dependent on the shape of the confining potential, and has been evaluated for the infinite potential well as Yao

(17) |

for , and where the initial location of the partition is set at . It is obvious that ==1. ,

We consider the one-dimensional cluster of bosons trapped in the potential that is approximated by the harmonic oscillator well, noting that aside its relevance to fundamental studies, one-dimensional systems can be fabricated by restricting the boson degrees of freedom in the transverse or radial directions. The radial confinement energy then becomes dominant compared to other system energies such as and the axial confinement trapping energy . Experimental state of condensates (of about 10 atoms) have been realized using elongated magnetic traps exptharm with respective radial and axial trapping frequencies of 360 Hz and 3.5 Hz.

In the harmonic oscillator well, trap length of the Szilard machine equals

(18) |

where is the mass of bosons and is the axial confinement frequency. Based on the convergence of all possible arrangements of particles in different energy levels to one partition of particle occupation (example of 20 particles shown in Ferrers diagram of Fig. 2), we obtain using Eq. 2, a simple expression for the equilibrium length,

(19) |

Using Eqs.1, 16 and 19, and an analogous expression to (Eq. 17) for the harmonic oscillator well, we obtain (in units of ) the total work that can be extracted from a condensed state of bosons at very low temperatures as

(20) |

where = 1 () for odd (even) number of bosons, and = (1) for odd (even) number of bosons. The dimensionless quantity, , hence Szilard engines with small trap lengths give rise to higher confinement energies, resulting in larger values of . It is obvious that for weakly confined particles (large trap lengths), the second term in Eq. 20 can be neglected, and the total work that can be extracted increases monotonically with . The basis for the slight differences between the odd and even number of particles is due to the asymmetrical partitioning in the vicinity of for the odd number of bosons. This attribute has been examined as a parity effect in a recent work by Lu et. al. Yao for the -particle quantum Szilard engine, for bosons and fermions, using the infinite potential well. As noted earlier in Eq. 10, the difference due to the odd and even number of particles in the Szilard engine become negligible in the limit of very large .

In Fig.5, we plot the information capacitive energies using Eqs.14, 15 and in Eq. 20 as function of boson number . There is gradual decline of the capacitive energy with , which indicates that the information content associated with an additional boson diminishes with the size of the boson cluster, as quantified by . Hence the work that can be extracted from an extra boson decreases with the number of bosons already contained within the Szilard engine. In the presence of confinement effects (non-zero in Eq. 20), the extractable work vanishes at a critical , due to boson particles acquiring fermionic features . Hence the confinement effects introduces a limit to the size of the total number of boson in the Szilard engine, from which work can be extracted. It is to be noted that these results are obtained in the low temperature regime, and are therefore not representative of trends at high temperatures.

We note that the defined information energies in Eqs14, 15 are not unique, as analogous definitions can be obtained for systems that consist of bosonic particles and fermions. If is even, then total energy that can be extracted depends, solely on bosons within the confines of simplifying assumptions at low temperatures. In the case where is odd, an additional energy, T, will need to be accounted for, this will be discussed in the next Section.

## V Confinement effects on the binding energies of bosons

Following the results presented in Section III, we now examine the binding energies of a system of boson at low temperature by extending Eq. 5 to an -particle system

(21) |

where appears in Eq. 20, and only needs to be obtained to evaluate the binding energy, . As in the case of for bosons (see Eq. 20), the extractable work, depends on whether it is an odd or even number of fermions present in the Szilard engine. Due to prohibition of occupation of the same state by two fermions by virtue of the exclusion principle, even number of fermions are more likely to be distributed equally over the two sides of the partition. In this case, no work can be extracted from an even number of fermions at low temperatures, as isothermal expansion does not occur with the same number of particles on either side of the wall. As is well known, and also indicated by Eq. 20, bosons are not subject to these same rules.

The low temperature characteristics of fermions have been highlighted by Kim et. al. kim2 using the third law of thermodynamics, which predicts that the entropy becomes zero in the absence of degeneracy in the ground state. Otherwise a non-zero residual entropy is retained, as is the case for the odd number of fermions. In the latter case, there exists some degree of uncertainty with respect to the unpaired fermion, and there appears ground state degeneracies at low temperatures. The work performed during a single engine cycle for the odd number of fermions is independent of

(22) |

In Fig.6, we plot the binding energies computed using Eqs.21, 20 and 22 as a function of for various values of the confinement parameter, (see Eq. 20). The difference in the binding energies due to the odd and even number of particles, is attributed to the residual entropy retained by odd number of fermions, as discussed earlier. An increasing confinement effect obviously results in a decrease of the binding energy of the system of bosons, as it acquires greater fermionic characteristic features. In other words, increasing confinement effects results in decrease in the bosonic quality (similar to the decrease in the measure in Eq.3)

The results obtained in Fig. 6 may be compared to the binding of excitonic systems in quantum dot structures triex . Recently, several sharp peaks, indicating the multi-excitonic complexes (including the charged triexciton) have been observed in in the photoluminescence spectra of self-assembled GaAs quantum dot systems ara . The triexciton, consisting of three electrons and four holes, as well as singlet and triplet states of charged biexcitons have been identified in the photoluminescence spectra ara , showing typical exchange splittings of 120 eV (0.12 meV). The micro-photoluminescence procedure was performed on quantum dots with base size of 30 nm, which were kept at 8 K in a cryostat. If we assume each quantum dot to be a miniature Szilard engine, and taking 2 to 5 (which gives binding energy of order of in Fig. 6), one obtains meV at 10 K. This energy estimate is obtained by neglecting confinement effects associated with the base structure, nevertheless it is of the same order of magnitude as the experimentally observed exchange splitting energies. This indicates that there may be non-negligible contributions of energy like quantities arising from information exchanges during quantum measurements.

Based on the results obtained in Section VII, we note that energies of the order of 0.1 meV can be associated with the residual entropy of odd number of fermions at low enough temperatures. This is about the same order of interaction energies noted in quantum dot excitonic systems ara . Hence, it not unlikely that exchange interactions of multi-excitonic complexes may include a non-trivial component of quantum information theoretic entities. To differentiate various energies involved is however challenging, in view of the resolution of existing technologies needed to perform this task. To this end, there is need to reexamine the role of the monitoring instruments during measurements of exchange interaction energies, for accurate interpretation of experimental results. Further investigations of Pauli exchange interactions of multi-excitonic complexes will contribute to understanding of various recombination mechanisms and the polarization properties of photons, essential for quantum information and processing applications niel .

## Vi Bose-Einstein condensate in a Szilard engine

We explore the feasibility of constructing a quantum Szilard engine using one-dimensional Bose-Einstein condensates systems in this Section. Currently, Bose-Einstein condensation is only possible in three-dimensional homogeneous systems, with no definite transition noted in unconfined one-dimensional systems kos . In the case of trapped one-dimensional boson systems however, it was proposed that ground state condensates may exist at enough high densities kette ; reca , although to date, there has been no convincing experimental evidence in these low-dimensional systems. Strongly repulsive one-dimensional bosonic systems can be mapped to a noninteracting fermionic system. Hence at lower densities, a correlated state (Tonks-Girardeau gas) tonk ; lieb appears on the basis of the Lieb-Liniger model of impenetrable bosons, and the repulsive interactions between bosons appear to simulate the Pauli exclusion forces between fermions. The dependence of the Bose system on dimensions of the confining potential, shows the important role of the dimensionality of the Szilard engine. In order to compare results obtained in earlier sections, we closely follow the one-dimensional Bose-Einstein condensate case for evaluation of some energy estimates.

During boson condensation, there is rapid accumulation of a substantial fraction of particles into the ground state, below a finite critical temperature. Bose condensation becomes favorable when the density of particles reaches a point when there is at least one particle per de Broglie wavelength. In general, there occurs insignificant fluctuations about the average condensate population at temperature below a finite value. A microcanonical approach to the population fluctuations, of ideal bosons trapped in a one-dimensional harmonic potential shows that gross1 ; gross2 . If we model a Bose condensate system as a quantum Szilard engine, fluctuations in particle number will be kept minimal, at low operating temperatures. In the limit of large N, an expression linking the number of atoms and the transition temperature in one-dimensional systems has been obtained as kette for

(23) |

Setting the trap-length, , of the harmonic well Szilard machine at 100 m, we can evaluate using Eq. 18. Using the typical excitonic polariton mass as 1 where is the free electron mass, we obtain using Eq. 23, for given , a transition temperature of that occurs at 10K. The inter-particle distance is computed as 0.1 m.

There is formation of a collective state of Bose-Einstein condensate when the de Broglie wavelengths of particles given by

(24) |

becomes comparable or exceed the inter-particle distance, , where is the mass of the boson, and is the particle number density. For the excitonic polariton at 10 K, we obtain using Eq. 24, 5 m at 10K, which increases to about 17 m at 1K. We thus obtain an estimate of , that far exceeds the inter-particle distance (0.1 m), noted earlier for particles in a Szilard engine of trap-length, = 100 m. These results, even if it is considered for one set of model parameters, highlight the favorable conditions for Bose condensation to occur, especially in highly anisotropic micro-cavities traps where the degrees of freedom are essentially erased in two possible direction, and in which the polariton species is introduced.

It is instructive to compare the results obtained here with the information capacitive energies computed (in units of ), at low temperatures, as shown in Fig. 5. We note that the Szilard engine of trap-length, = 100 m would yield the confinement parameter (see Eq. 15) values of about 0.04 (at 1 K) and 0.004 (at 10K). These small confinement measures indicate that the condensate would experience non-zero capacitive energies of about 0.008 meV, and binding energies (Fig. 6) that arise due to quantum informational entropy changes. We point out that small trap-lengths will result in confinement effects reducing binding effects (Fig. 6), thus there is a lower limit to the size of the length, , if such one dimensional Szilard-model systems are to be realized in future experiments. It is expected that observations can be made at the cryogenic temperatures of a few Kelvins, which is already accessible with modern technologies.

On the basis of the results obtained so far, we may make some pertinent queries, with respect to the model system of polariton condensate in a Szilard engine. Can quantum informational entropy changes and quantum mechanical effects partly (or even substantially) explain the observations of polariton condensates? It may seem that energy derived from the Szilard engine via the process of observation, may help sustain at least a fraction of the condensation process. To what degree do such changes, if any, are introduced during observations during quantum measurements and contribute to the Bose-Einstein condensation of material systems? Is there an interference effect that arises due to the monitoring apparatus? The answers to these questions are not immediately clear and needs further scrutiny on the delicate roles played by monitoring instruments, and are best answered via direct experimental verification.

It is not certain whether quantum information that manifest in the form of entanglement can provide unification between condensation phenomena and correlation entities. In a recent work, the generality of Landauer s principle to quantum situations was considered lidia , in which the role of the observer was emphasized. In the event that the observer and system are entangled, thermodynamic entities linked to conditional entropies will arise, giving rise to a cooling effect on the environment. When applied to the condensation process taking place within the Szilard engine, this suggests, that any increased heat generated due to “bunching” of a large number of particles into a single quantum state, may be reduced by the presence of an entangled observer. To this end, there is ample scope to examine the role of irreversibility during Bose condensation, and to probe excitations that arise due to the measurement effect in future investigations.

Lastly, we note that unlike the polariton species, the exciton gas whose constituent possess a higher mass, appears to have less likelihood of achieving the condensed state at comparable (few Kelvins) temperatures. Till now, the exciton Bose-Einstein condensate has not been directly observed due to the critical conditions for the condensation process timo . As pointed out in Ref.bosecom , excitons exhibit Bose-Einstein condensation only if these quasiparticles have long lifetimes, and exciton-exciton interactions become repulsive to prevent the formation of larger exciton systems such as biexcitons. While the long exciton lifetimes is achievable in highly confined systems, it is not certain whether biexciton formation can be avoided. For the latter reason, the formation of a biexciton-polariton quasiparticule is also likely, and may act as an obstacle to condensation phenomenon involving the pure (and ideal) exciton gas system. Hence the conditions which favor the depletion of condensates due to formation excitonic complexes have to be eradicated before possible observations of the condensed phase of exciton gas can be contemplated.

## Vii Conclusion

In summary, the concept of the Szilard engine which extracts work from a thermal reservoir, via quantum measurements of the particle position, is used to define the binding energies of composite bosons that possess some degree of fermionic features. The significance of the quantum Szilard engine is that it allows the evaluation of binding energies that can be associated with quantum information linked to the indistinguishable and/or distinguishable nature of a system of particles. The non-trivial role of the quantum information thermodynamic operation of the Szilard engine during Bose-Einstein condensation of the light mass polariton quasiparticle has been examined, along with applications related to the binding energies of large multi-excitonic complexes in quantum dot systems. We allude to the possibility that quantum informational entropy changes may contribute to the formation of polariton condensates, however further advancement in instrumentation techniques, with ability to differentiate different forms of energies, is needed to verify these extensions.

Although in this work, we have only focused on two specific systems: Multiexcitonic quasiparticles and Bose-Einstein condensates, the developed methodology is general and might be useful for improving density functional techniques of strategic material systems. The link between information and energy may be extended to improve methods based on Kohn-Sham density functional theories (DFT) kohn ; parr ; BLYP ; PBE . While it is known that density functional theories provide accurate results in systems with weak electronic correlations, many density functionals fail to provide adequate description of strongly correlated electronic systems fail . The use of the quantum Szilard model in the latter systems may reveal new forms of elementary excitations that arise due to information processing. These extensions are expected to contribute to novel quantum processes that underpins superior properties of strongly correlated systems, with potential applications in optical devices and quantum information processing.

## Viii Acknowledgments

The author gratefully acknowledges the support of the Julian Schwinger Foundation Grant, JSF-12-06-0000. The author would like to thank Monique Combescot for earlier discussions related to the binding energies that arise from Pauli interactions, and Malte Tichy for useful correspondences regarding specific properties of composite bosons, and for pointing out the use of Szilard engine model in Ref.bobby , at the time of preparation of this manuscript.

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