Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1011.1333 · Physical Review A

Relevance of Bose-Einstein condensation to the interference of two independent Bose gases

Mauro Iazzi, Kazuya Yuasa

Published Mar 14, 2011• 9 citations
arXivHTMLPDF

Abstract

Interference of two independently prepared ideal Bose gases is discussed, on the basis of the idea of measurement-induced interference. It is known that, even if the number of atoms in each gas is individually fixed finite and the symmetry of the system is not broken, an interference pattern is observed on each single snapshot. The key role is played by the Hanbury Brown and Twiss effect, which leads to an oscillating pattern of the cloud of identical atoms. Then, how essential is the Bose-Einstein condensation to the interference? In this work, we describe two ideal Bose gases trapped in two separate three-dimensional harmonic traps at a finite temperature $T$, using the canonical ensembles (with fixed numbers of atoms). We compute the full statistics of the snapshot profiles of the expanding and overlapping gases released from the traps. We obtain a simple formula valid for finite $T$, which shows that the average fringe spectrum (average fringe contrast) is given by the purity of each gas. The purity is known to be a good measure of condensation, and the formula clarifies the relevance of the condensation to the interference. The results for $T=0$, previously known in the literature, can be recovered from our analysis. The fluctuation of the interference spectrum is also studied, and it is shown that the fluctuation is vanishingly small only below the critical temperature ${T}_{c}$, meaning that interference pattern is certainly observed on every snapshot below ${T}_{c}$. The fact that the number of atoms is fixed in the canonical ensemble is crucial to this vanishing fluctuation.

DOI
10.1103/physreva.83.033611

Related papers

Canonical Ensemble of Noninteracting Bosonic Atoms and Interference of Bose-Einstein Condensates(Perspectives of Nonequilibrium Statistical Physics-The Memory of Professor Shuichi Tasaki-)

Kazuya Yuasa

We have discussed the interference of two independently prepared Bose-Einstein condensates. When the number of atoms in each gas is fixed and the phases of the gases are uncertain, no interference is expected in the single-particle distribution. However, the interference is observed in each single snapshot. We have presented tools for studying such snapshot interference, the measurement-induced interference. The idea is to discard the randomness in the snapshot profiles (the spatial offsets of the interference patterns) and to focus on the typical feature in the interference patterns (the fringe spacing). If the sinusoidal patterns with a definite fringe spacing is actually typical, the fringe spectrum with the random offset discarded does not fluctuate, and we are allowed to speak of the snapshot interference through its average. We have described the two gases as canonical ensembles with the numbers of atoms fixed at N individually. We have shown that the covariance of the fringe spectrum over all possible snapshot profiles is vanishingly small below the critical temperature of the Bose-Einstein condensation: the interference pattern whose fringe contrast is characterized by the average fringe spectrum is certainly observed in every snapshot in the presence of condensates. The fringe contrast becomes stronger as the temperature is lowered and the condensation fraction is increased. This clarifies the importance of the Bose-Einstein condensation to the interference of independent Bose-Einstein condensates. The knowledge I acquired from Prof. Tasaki's lecture on how to characterize the canonical ensemble of bosonic atoms has enabled us to carry out this analysis. Like this example, the coherence phenomena of quantum many-body systems in general would be explained on the basis of the idea of the measurement-indued coherence, without resort to the idea of the spontaneous symmetry breaking. This is actually true also for fermionic systems: the relative phase between two independent superconductors is built up by measurement, even though their U(1) symmetries are not broken and their phases are uncertain beforehand. It would be interesting to explore the possibility of explaining various phenomena, in which the symmetry breaking plays an essential role, on the basis of the measurement-induced coherence.

物性研究•Dec 5, 2011•0 citations

INTERFERENCE OF AN ARRAY OF INDEPENDENT BOSE-EINSTEIN CONDENSATES WITH FIXED NUMBER OF ATOMS

SATOSHI ANDO, KAZUYA YUASA, Mauro Iazzi

Interference of an array of independent Bose-Einstein condensates, whose experiment has been performed recently, is theoretically studied in detail. Even if the number of the atoms in each gas is kept finite and the phases of the gases are not well-defined, interference fringes are observed on each snapshot. The statistics of the snapshot interference patterns, i.e. the average fringe amplitudes and their fluctuations (covariance), are computed analytically, and concise formulas for their asymptotic values for long time of flight are derived. Processes contributing to these quantities are clarified and the relationship with the description on the basis of the symmetry-breaking scenario is revealed.

arXiv:1104.0295•International Journal of Quantum Information•Jan 1, 2011•5 citations

Interference and correlations

C. J. Pethick, H. Smith

Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. In Sec. 13.1 we discuss Josephson tunnelling of a condensate between two wells and the role of fluctuations in particle number and phase. The number and phase variables play a key role in the description of the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap (Sec. 13.2). Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. Density fluctuations in a Bose gas are studied in Sec. 13.3, where we relate atomic clock shifts to the two-particle correlation function. The ability to manipulate gaseous Bose–Einstein condensates by lasers has made possible the study of coherent matter wave optics and in Sec. 13.4 we describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves. How to characterize Bose–Einstein condensation in terms of the density matrix is the subject of Sec. 13.5, where we also consider fragmented condensates.

Cambridge University Press eBooks•Sep 11, 2008•0 citations

Interference and correlations

C. J. Pethick, H. Smith

Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of many aspects of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. This field is a vast one, and in this chapter we shall touch briefly on selected topics. In Sec. 13.1 we describe the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap. Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. We shall analyse the reasons for this effect. The marked decrease in density fluctuations in a Bose gas when it undergoes Bose–Einstein condensation is demonstrated in Sec. 13.2. Gaseous Bose–Einstein condensates can be manipulated by lasers, and this has made possible the study of coherent matter wave optics. We describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves in Sec. 13.3. The atom laser and amplification of matter waves is taken up in Sec. 13.4. How to characterize Bose–Einstein condensation microscopically is the subject of Sec. 13.5, where we also consider fragmented condensates. Interference of two condensates One of the striking manifestations of the wave nature of Bose–Einstein condensates is the observation of an interference pattern when two condensed and initially separated clouds are allowed to overlap. An example is shown in Fig. 13.1.

Cambridge University Press eBooks•Nov 29, 2001•0 citations

Interference of two Bose-Einstein condensates

В. А. Алексеев

The visibility of the density interference pattern of two Bose-Einstein condensates, which are produced in traps and overlap after the trapping potential is switched off, is investigated. Coherent wave packets are used to describe the order parameter in a second-quantization formalism. This results in a decrease of the visibility of the interference fringes with increasing time delay between the formation of the condensates and the observation of interference. In the two limiting cases of ideal and very dense gases the correlation time increases, τ →∞, and the result is identical to that obtained using an approach based on the Gross-Pitaevskii equation. Under the conditions of the experiment performed by M. R. Andrews, C. G. Townsend, H. J. Miesner et al. , Science 275 , 6367 (1997), the computed correlation time τ ≈0.2 s is much longer than the confinement time of the condensate, and it is possible to observe the predicted decrease of visibility of the interference fringes of the density of the atoms.

Journal of Experimental and Theoretical Physics Letters•Apr 1, 1999•3 citations

Mesoscopic Effects in Bose-Einstein Condensate Fluctuations of an Ideal Gas in a Box

Konstantin E. Dorfman

The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute\nthe main subject of the present thesis. These effects are the most difficult for\nthe theoretical analysis in the quantum statistical physics since they can’t be seen by\nneither a standard quantum mechanics of the simple microscopic systems of one or\nvery few atoms nor a standard statistical physics of the macroscopic systems that are\ninfinite in the bulk (thermodynamic) limit.\nMost of the experiments on the cold quantum gases performed in the last decade,\nstarting from the first demonstration of BEC in 1995, involve the mesoscopic systems\nof a finite number of atoms. The mesoscopic effects should manifest themselves most\nclearly and easily near a critical temperature of BEC; however, they could be observed\nalso above and below the critical temperature.\nHere I study the quantum and thermal fluctuations of the Bose-Einstein condensate\n(BEC) in a box with the periodic boundary conditions under a particle-number\nconstraint. The above constraint is the only reason for the BEC and is crucial for the\nmesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical\ntemperature in the Bose gas.\nI employ the particle-number conserving operator formalism of Girardeau and\nArnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present\nanalytical formulas and numerical calculations for the central moments of the ground\nstate occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles.\nI present the analysis of the BEC statistics both on a temperature at a fixed\nnumber of particles and on a number of particles at a fixed temperature. Both analyses\nare valid for the purpose of understanding the important mesoscopic effects near the\ncritical temperature. I emphasize the non-Gaussian nature of the fluctuations.\nThe presented formalism can be generalized to the case of a weakly interacting\nBose gas in a box in the framework of the Bogoliubov approximation. The work in\nthis direction is in progress but is not included in the present thesis.

•May 15, 2009•0 citations
Interference of two independently prepared ideal Bose gases is discussed, on the basis of the idea of measurement-induced interference. It is known that, even if the number of atoms in each gas is individually fixed finite and the symmetry of the system is not broken, an interference pattern is observed on each single snapshot. The key role is played by the Hanbury Brown and Twiss effect, which leads to an oscillating pattern of the cloud of identical atoms. Then, how essential is the Bose-Einstein condensation to the interference? In this work, we describe two ideal Bose gases trapped in two separate three-dimensional harmonic traps at a finite temperature TTT, using the canonical ensembles (with fixed numbers of atoms). We compute the full statistics of the snapshot profiles of the expanding and overlapping gases released from the traps. We obtain a simple formula valid for finite TTT, which shows that the average fringe spectrum (average fringe contrast) is given by the purity of each gas. The purity is known to be a good measure of condensation, and the formula clarifies the relevance of the condensation to the interference. The results for T=0T=0T=0, previously known in the literature, can be recovered from our analysis. The fluctuation of the interference spectrum is also studied, and it is shown that the fluctuation is vanishingly small only below the critical temperature Tc{T}_{c}Tc​, meaning that interference pattern is certainly observed on every snapshot below Tc{T}_{c}Tc​. The fact that the number of atoms is fixed in the canonical ensemble is crucial to this vanishing fluctuation.
We have discussed the interference of two independently prepared Bose-Einstein condensates. When the number of atoms in each gas is fixed and the phases of the gases are uncertain, no interference is expected in the single-particle distribution. However, the interference is observed in each single snapshot. We have presented tools for studying such snapshot interference, the measurement-induced interference. The idea is to discard the randomness in the snapshot profiles (the spatial offsets of the interference patterns) and to focus on the typical feature in the interference patterns (the fringe spacing). If the sinusoidal patterns with a definite fringe spacing is actually typical, the fringe spectrum with the random offset discarded does not fluctuate, and we are allowed to speak of the snapshot interference through its average. We have described the two gases as canonical ensembles with the numbers of atoms fixed at N individually. We have shown that the covariance of the fringe spectrum over all possible snapshot profiles is vanishingly small below the critical temperature of the Bose-Einstein condensation: the interference pattern whose fringe contrast is characterized by the average fringe spectrum is certainly observed in every snapshot in the presence of condensates. The fringe contrast becomes stronger as the temperature is lowered and the condensation fraction is increased. This clarifies the importance of the Bose-Einstein condensation to the interference of independent Bose-Einstein condensates. The knowledge I acquired from Prof. Tasaki's lecture on how to characterize the canonical ensemble of bosonic atoms has enabled us to carry out this analysis. Like this example, the coherence phenomena of quantum many-body systems in general would be explained on the basis of the idea of the measurement-indued coherence, without resort to the idea of the spontaneous symmetry breaking. This is actually true also for fermionic systems: the relative phase between two independent superconductors is built up by measurement, even though their U(1) symmetries are not broken and their phases are uncertain beforehand. It would be interesting to explore the possibility of explaining various phenomena, in which the symmetry breaking plays an essential role, on the basis of the measurement-induced coherence.
Interference of an array of independent Bose-Einstein condensates, whose experiment has been performed recently, is theoretically studied in detail. Even if the number of the atoms in each gas is kept finite and the phases of the gases are not well-defined, interference fringes are observed on each snapshot. The statistics of the snapshot interference patterns, i.e. the average fringe amplitudes and their fluctuations (covariance), are computed analytically, and concise formulas for their asymptotic values for long time of flight are derived. Processes contributing to these quantities are clarified and the relationship with the description on the basis of the symmetry-breaking scenario is revealed.
Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. In Sec. 13.1 we discuss Josephson tunnelling of a condensate between two wells and the role of fluctuations in particle number and phase. The number and phase variables play a key role in the description of the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap (Sec. 13.2). Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. Density fluctuations in a Bose gas are studied in Sec. 13.3, where we relate atomic clock shifts to the two-particle correlation function. The ability to manipulate gaseous Bose–Einstein condensates by lasers has made possible the study of coherent matter wave optics and in Sec. 13.4 we describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves. How to characterize Bose–Einstein condensation in terms of the density matrix is the subject of Sec. 13.5, where we also consider fragmented condensates.
Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of many aspects of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. This field is a vast one, and in this chapter we shall touch briefly on selected topics. In Sec. 13.1 we describe the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap. Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. We shall analyse the reasons for this effect. The marked decrease in density fluctuations in a Bose gas when it undergoes Bose–Einstein condensation is demonstrated in Sec. 13.2. Gaseous Bose–Einstein condensates can be manipulated by lasers, and this has made possible the study of coherent matter wave optics. We describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves in Sec. 13.3. The atom laser and amplification of matter waves is taken up in Sec. 13.4. How to characterize Bose–Einstein condensation microscopically is the subject of Sec. 13.5, where we also consider fragmented condensates. Interference of two condensates One of the striking manifestations of the wave nature of Bose–Einstein condensates is the observation of an interference pattern when two condensed and initially separated clouds are allowed to overlap. An example is shown in Fig. 13.1.
The visibility of the density interference pattern of two Bose-Einstein condensates, which are produced in traps and overlap after the trapping potential is switched off, is investigated. Coherent wave packets are used to describe the order parameter in a second-quantization formalism. This results in a decrease of the visibility of the interference fringes with increasing time delay between the formation of the condensates and the observation of interference. In the two limiting cases of ideal and very dense gases the correlation time increases, τ →∞, and the result is identical to that obtained using an approach based on the Gross-Pitaevskii equation. Under the conditions of the experiment performed by M. R. Andrews, C. G. Townsend, H. J. Miesner et al. , Science 275 , 6367 (1997), the computed correlation time τ ≈0.2 s is much longer than the confinement time of the condensate, and it is possible to observe the predicted decrease of visibility of the interference fringes of the density of the atoms.
The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute\nthe main subject of the present thesis. These effects are the most difficult for\nthe theoretical analysis in the quantum statistical physics since they can’t be seen by\nneither a standard quantum mechanics of the simple microscopic systems of one or\nvery few atoms nor a standard statistical physics of the macroscopic systems that are\ninfinite in the bulk (thermodynamic) limit.\nMost of the experiments on the cold quantum gases performed in the last decade,\nstarting from the first demonstration of BEC in 1995, involve the mesoscopic systems\nof a finite number of atoms. The mesoscopic effects should manifest themselves most\nclearly and easily near a critical temperature of BEC; however, they could be observed\nalso above and below the critical temperature.\nHere I study the quantum and thermal fluctuations of the Bose-Einstein condensate\n(BEC) in a box with the periodic boundary conditions under a particle-number\nconstraint. The above constraint is the only reason for the BEC and is crucial for the\nmesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical\ntemperature in the Bose gas.\nI employ the particle-number conserving operator formalism of Girardeau and\nArnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present\nanalytical formulas and numerical calculations for the central moments of the ground\nstate occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles.\nI present the analysis of the BEC statistics both on a temperature at a fixed\nnumber of particles and on a number of particles at a fixed temperature. Both analyses\nare valid for the purpose of understanding the important mesoscopic effects near the\ncritical temperature. I emphasize the non-Gaussian nature of the fluctuations.\nThe presented formalism can be generalized to the case of a weakly interacting\nBose gas in a box in the framework of the Bogoliubov approximation. The work in\nthis direction is in progress but is not included in the present thesis.