In this thesis we discuss the quantum coherent dynamics of the two-component Bose-Einstein condensates and some applications in researches of analogue gravity.

Firstly, we theoretically study binary Bose-Einstein condensates trapped in a single-well harmonic potential to probe the dynamics of collective atomic motion. The idea is to choose tunable scattering lengths through Feshbach resonances such that the ground-state wave function for two types of the condensates are spatially immiscible where one of the condensates, located at the center of the potential trap, can be effectively treated as a potential barrier between bilateral condensates of the second type of atoms. The condensate in the middle can be approximated by a Gaussian wave function with the displacement of the condensate center $\xi$, while the bilateral condensates in two-mode approximation can be characterized by population imbalance $z$ and phase difference $
hi$. As driven by the time-dependent displacement of the central condensate, we find the Josephson oscillations of the collective atomic motion between bilateral condensates as well as their anharmonic generalization of macroscopic self-trapping effects. In addition, with the increase in the wave-function overlap of bilateral condensates by properly choosing tunable atomic scattering lengths, the chaotic oscillations are found if the system departs from the state of an unstable fixed point. The Melnikov approach with a homoclinic solution of the derived $z$, $
hi$, and $\xi$ equations can successfully justify the existence of chaos.

Then, we turn our attention to the study of ``analogue gravity'' in this condensate system.
By considering the Rabi transition between atomic hyperfine states, we examine the effect of quantum fluctuations in a tunable quantum gas on phonon propagation. We then further trace out the gapped modes to give an effective purely phononic theory using closed-time-path formalism. In particular, we are interested in the sound cone fluctuations due to the variation of the speed-of-sound acoustic metric, induced by quantum fluctuations of the gapped modes. These fluctuations can be interpreted as inducing a stochastic space-time, and thus are regarded as analogue phenomena of light cone fluctuations presumably arising from quantum gravity effects. The effects of fluctuations can be displayed in the variation in the travel time of sound waves. We suggest the relevant experiments to discuss the possibility of experimental observations.

Lastly, another aspect in analogue gravity interests us is the acoustic black hole. We consider the abilities of spatially tuning the scattering length in experiments, to study propagation of the gapless/gapped modes in the spatial steplike discontinuities of sound speed profile, which consequently emit negative frequency mode toward inner acoustic black hole.
The analytical expressions of scattering coefficients for massive case are presented and show significant difference from the massless case. The influence from the gapped excitations to the quantum entanglement of the Hawking mode and its partner of the gapless excitations is also studied according to the Peres-Horodecki-Simon (PHS) criterion. It is found that the presence of the gapped excitations will deteriorate the quantumness of the pair modes of the gapless excitations when the frequency of the pair modes in particular is around $\omega\sim\omega_\text{min}$. On top of that, when the coupling constant between the gapless and gapped excitations becomes large enough, the huge particle density of the gapped excitations in the small $\omega$ regime will significantly disentangle the pair modes of the gapless excitations. The detailed time-dependent PHS criterion will be discussed.

1 Introduction 1
1.1  MixturesofBose-Einsteincondensates 1
1.2  FortheDynamicsAspect 1
1.3  FortheAnalogueGravityAspect 3
1.3.1 Sound-conefluctuation 3
1.3.2 Analogue Hawking radiation for gapped excitations 4
1.4 Summary 6
2  The quantum coherent dynamics in two-component Bose-Einstein condensates 9
2.1  Variational approach and analogous coupled pendulums dynamics 10
2.2  Equilibrium Solutions and Stability analysis 16
2.3  Regularmotion 19
2.3.1  General solutions for small amplitude oscillations 19
2.3.2  Theeectivepotential 22
2.3.3  Results and Discussions 23
lambda_eff/2k > 2 (Josephson oscillation, running phase and fi-mode self trapping) 23
1 < lambda_eff/2k < 2 (Josephson oscillation and fi-mode self trapping) 29
02.4  Chaoticdynamics 31
2.5  Conclusions 36
3  Analogue gravity phenomena: Sound cone fluctuations 39
3.1  Themodelandbackgroundcondensates 39
3.2  the dispersion relation for collective excitations 45
3.3  Langevin equation and induced stochastic sound cone metric 54
3.4  Time-Of-Flightvariance 58
3.4.1 $c_
hi=c_\chi$ 60
3.4.2 $c_
hi>c_\chi$ 62
3.4.3 $c_
hi3.5  Conclusions 67
4 Analogue gravity phenomena: Analogous Hawking radiation and quantum entanglement 69
4.1 Themodel 69
4.1.1 Plane wave solutions and dispersion relations 72
4.1.2 Gappedexcitationmodes 76
4.2  Matching of the mode functions and construction of the S matrix 79
4.2.1 Matchingofmodefunctions 79
4.2.2 ConstructionoftheS-matrix 80
4.2.3 urout-goingchannel 83
4.2.4 ulout-goingchannel 86
4.2.5 vlout-goingchannel 88
4.3  Density-densitycorrelations 90
4.3.1  The quadrant of correlation function within x < 0,xÕ > 0 or x>0,xÕ<0 92
4.3.2  The quadrant of correlation function within x < 0, xÕ < 0 94
4.4  Eects from gapped modes to quantum entanglement of gapless modes 95
4.4.1  ThePeres-Horodecki-Simoncriterion 95
4.4.2  Inhomogeneousequations 96
4.4.3  Nonseparability of Hawking-partner pairs 101
4.5  Conclusion 103
5 Conclusions and Further work 107
5.1 Conclusions 107
5.2 Furtherwork 108
A  Publications not included in the thesis 113 

B  Variational principle 115

C  Subsonic-Subsonic Configuration 119
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