BEYOND MEAN-FIELD DYNAMICS OF TWO-MODE BOSE-HUBBARD MODEL WITH LINEAR COUPLING RAMPING CHENG KOK CHEONG NATIONAL UNIVERSITY OF SINGAPORE 2014 BEYOND MEAN-FIELD DYNAMICS OF TWO-MODE BOSE-HUBBARD MODEL WITH LINEAR COUPLING RAMPING CHENG KOK CHEONG (B.Sc (Hons), NUS ) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2014 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Cheng Kok Cheong June 4, 2014 Name : Degree : Supervisor(s) : Department : Thesis Title : : Cheng Kok Cheong Master of Science Associate Professor Gong Jiangbin Department of Physics Beyond Mean-Field Dynamics of Two-Mode Bose-Hubbard Model with Linear Coupling Ramping Summary The mean-field Hamiltonian of two-mode Bose-Hubbard model with real and imaginary coupling constants demonstrates pitchfork bifurcation in its phase-space structure within certain interval of real coupling constant Its mean-field dynamics have been previously studied by Zhang et al [11] It was shown therein that when the real coupling constant is ramped adiabatically towards the pitchfork bifurcation critical point, the classical intrinsic dynamical fluctuations assist in the selection between the two stable stationary points Based on this finding, we set out to study the corresponding quantum Hamiltonian with the real coupling constant ramped linearly At very slow ramping, the quantum system is able to resolve the energy difference of the two nearly degenerate lowest energy states Therefore, it no longer demonstrates self-trapping as what is predicted by the mean-field dynamics Such breakdown of mean-field within dynamical instability is an example of incommutability between semiclassical and adiabatic limit To extend beyond the mean-field level, we employ the Bogoliubov backreaction method and the semiclassical phase space method to understand how the second and higher order quantum fluctuations alter the system dynamics It turns out that both approaches yield good prediction on the dynamics of population imbalance between the two modes and the fraction of non-condensed atoms at fast and very slow ramping Acknowledgment I am very grateful to Associate Professor Gong Jiangbin for his insightful and inspiring supervision throughout this master project Every discussion with him taught me new physical ideas and cleared my doubts on certain issues His insights are always refreshing, and his sensitivity to physical fallacy is unquestionable Also, his generosity in letting students explore physics on their own, yet not too much that they fall, has allowed me to grow and continuously challenge myself with the right amount of support I also immensely appreciate his caring and forgiving nature as he is always concerned about the future undertaking of his students and showed his understanding when I lost focus on my project at a certain point Without him, I would not have realized how much I can accomplish throughout this journey I also want to thank Professor Zhang Qi for all his meaningful and useful input on the subject without which the general objective of the project would not have been formed Last but not least, I will never leave out all physics department personnel who have unconditionally given me advice on all the administrative procedures with which I was not familiar These include different requirements for graduation, access to central printing system and most importantly access to the CSE high performance computer which allows me to carry out various heavy numerical computations All your help has tremendously shed my burden along the way My dearest family and friends, without your emotional support along my long educational journey, I would not have become a better person With your ever-lasting love and support, I am still getting better June 4, 2014 Contents Introduction Quantum and Mean-Field Dynamics of Bose-Hubbard Model 2.1 The Bose-Hubbard Hamiltonian 2.1.1 Quantum Energy Spectrum and Eigenstates 2.1.2 Time-Dependent Quantum Dynamics 11 2.1.3 Non-Abelian Geometry Phase 13 2.2 Mean-Field Correspondence of Bose-Hubbard Hamiltonian 17 2.2.1 Classical Stationary Points and Energy Spectrum 18 2.2.2 Mean-field Classical Dynamics and Intrinsic Dynamical Fluctuation 20 Beyond Mean-Field Dynamics 3.1 Second Order Dynamics 3.2 Effects of Higher Order Moments 3.2.1 SU(2) Coherent States 3.2.2 Formulation of Method 3.2.3 Simulation Result and Discussion 25 25 33 34 37 43 Future Work and Development 52 Conclusion 62 Appendices 65 A Classical-Quantum Correspondence For Multilevel A.1 Generalized Coherent States A.2 Multimode Lie Algebra A.3 Multilevel Lie Algebra and its Coherent States A.4 Multimode to Multilevel Mapping A.4.1 Coherent States Mapping System 66 66 67 69 71 72 A.4.2 Projector Mapping 73 A.4.3 Multimode and Multilevel Differential Algebra Mapping 74 A.5 Husimi Distribution and Equation of Motion 77 List of Figures 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Energy spectrum of Hamiltonian (2.1) as a function of R for N = 10 and 20, ∆ = 0.1 and c = 0.2 The right panel includes only up to the 6-th lowest eigenvalues Energy difference in ground and first excited state versus N for different R ∆ = 0.1 and c = 0.2 Modulus square of the components of ground (blue, square, solid line) and first excited states (red, circle, dashed line) in Fock basis for different R ∆ = 0.1 , c = 0.2 and N = 10 Quantum evolution of population imbalance expectation value for different high ramping speeds α α = 0.001 (black, solid line), α = 0.01 (red, dashed line) and α = 0.1 (blue, dotted line) R0 = −0.18, ∆ = 0.1, N = 10 and c = 0.2 Quantum evolution of population imbalance expectation value for different low ramping speeds α α = 0.000001 (black, solid line), α = 0.00001 (red, dashed line) and α = 0.0001 (blue, dotted line) R0 = −0.18, ∆ = 0.1, N = 10 and c = 0.2 Evolution of projection values |c01 |2 and |c02 |2 for N = 10 Blue solid line - ground state; Red dashed line - first excited state From top to bottom: theoretical non-Abelian geometry phase simulation, actual quantum simulation for α=0.01, 0.001, 0.0001 and 0.00001 ∆ = 0.1, and c = 0.2 Stationary mean-field energy spectrum for different R Top √ √ R2 + ∆2 ; Bottom solid line: − R2 + ∆2 ; Bottom line: +∆2 dashed line: − 2c − R 2c ∆ = 0.1, c = 0.2 The dashed line is only valid between -0.173 and 0.173 Phase space structure of Hamiltonian (2.22) for different R Top left: R = −0.2; Top right: R = −0.1; Bottom left: R = 0.13; Bottom right: R = 0.19 ∆ = 0.1, c = 0.2 8 10 12 12 16 20 21 2.9 Dynamics of population imbalance q against R for different fast ramping speeds Black solid line: q-coordinates of mean-field stationary points; Blue dashed line: α = 0.1; Red dotted line: α = 0.01; Green dash-dotted line: α = 0.001 ∆ = 0.1, c = 0.2 23 2.10 Dynamics of population imbalance q against R for different slow ramping speeds Black solid line: q-coordinates of mean-field stationary points; Blue dashed line: α = 0.0001; Red dotted line: α = 0.00001; Green dash-dotted line: α = 0.000001 ∆ = 0.1, c = 0.2 24 3.1 3.2 3.3 3.4 3.5 3.6 Mean-field trajectories for two different R at various initial points Left: R = −0.2; Right: R = −0.1 ∆ = 0.1, c = 0.2 Dynamics of population imbalance sz at different ramping speeds Black solid line: mean-field stationary points; Blue dashed line: α = 0.01; Red dotted line: α = 0.001; Green dash-dotted line: α = 0.0001 ∆ = 0.1, c = 0.2 and N = 10 Dynamics of population imbalance sz at for different N at equal ramping speeds Black solid line: mean-field stationary points; Blue dashed line: N = 10; Red dotted line: N = 30; Green dash-dotted line: N = 100 ∆ = 0.1, c = 0.2, α = 0.00001 Evolution of relative population imbalance against R for different fast ramping speeds Solid black line: Exact quantum evolution; Blue dashed line: Backreaction-type evolution; Red dotted line: mean-field evolution Left to right: α=0.1, 0.01 and 0.001; ∆ = 0.1, c = 0.2, N = 10, R0 = −0.18 Evolution of population imbalance against R for different slow ramping speeds Solid black line: Exact quantum evolution; Blue dashed line: Backreaction-type evolution; Red dotted line: mean-field evolution Left to right: α = 10−4 , 10−5 and 10−6 ; ∆ = 0.1, c = 0.2, N = 10, R = −0.18 Exact quantum dynamics of lowest SPDM eigenvalues for different ramping speeds Bottom solid line: α = 0.1; Blue dashed line: α = 0.01; Red dotted line: α = 0.001; Green dash-dotted line: α = 0.0001; Top thick black solid line: α = 0.00001 ∆ = 0.1, c = 0.2, N = 10 26 27 27 30 31 32 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 Backreaction dynamics of lowest SPDM eigenvalues for different ramping speeds Bottom solid line: α = 0.1; Blue dashed line: α = 0.01; Red dotted line: α = 0.001; Green dash-dotted line: α = 0.0001 ∆ = 0.1, c = 0.2, N = 10 The evolution for α = 0.00001 is not indicated above as its plot is excessively oscillating Mean-field phase space structure and Husimi Distribution for R = −0.2 Top left: Mean-field phase space; Top right: Husimi distribution of ground state; Bottom left and right: Husimi distribution of fifth and tenth excited states ∆ = 0.1, c = 0.2 and N = 10 Mean-field phase space structure and Husimi Distribution for R = Top left: Mean-field phase space; Top right: Husimi distribution of ground state; Bottom left and right: Husimi distribution of fifth and tenth excited states ∆ = 0.1, c = 0.2 and N = 10 Husimi Distribution for different N at R = −0.2 Top left: N = 10; Top right: N = 20; Bottom left: N = 30; Bottom right: N = 50 ∆ = 0.1, c = 0.2 Evolution of Husimi distribution and classical ensemble at different R for α = 0.1 Top: Husimi distribution; Bottom: Classical ensemble ∆ = 0.1, c = 0.2, M = 200 Evolution of Husimi distribution and classical ensemble at different R for α = 0.001 Top: Husimi distribution; Bottom: Classical ensemble ∆ = 0.1, c = 0.2, M = 200 A qualitative pictorial understanding of IDF, assuming that the stable stationary point moves downward in phase-space at speed of vF The orbits around the stationary points are assumed to rotate in the clockwise direction Top: before bifurcation Bottom: once after bifurcation Dynamics of population imbalance and SPDM lowest eigenvalues f for exact quantum evolution and classical Liouvillian dynamics Left column: evolution of population imbalance; Right column: evolution of lowest SPDM eigenvalues From top to bottom: α=0.1, 0.01, 0.001, 0.0001 and 0.00001 Black solid line: exact quantum calculation; Blue dashed line: classical Liouvillian calculation 10 33 43 44 45 46 47 48 51 Then, any element in G can be decomposed into the product of one element h from the isotropy subgroup and another Ω from the coset space G/H The one-to-one mapping between the element Ω(g) of the coset space and the coherent state |Ω ≡ |ψΩ implies that coherent state stays coherent if it is acted upon by element linear in the generator of the dynamical group Also, coherent states are over-complete in the following sense: |Ω Ω|dµ(Ω) = I (A.3) G/H where dµ(Ω) is the Haar measure on the coset space A.2 Multimode Lie Algebra For a system of r independent oscillator modes, each mode can be described by its annihilation operators a ˆj and creation operators a ˆ†j , where j = 1, · · · , r The operators a ˆj , a ˆ†j and I span the r-mode Lie algebra, coupled with the commutation relations [ˆ aj , a ˆ†k ] = Iδjk ; ˆ†k ] = 0; [ˆ aj , a ˆk ] = [ˆ a†j , a (A.4) [ˆ aj , I] = The coherent states of the r-modes Lie algebra are the products of r single mode coherent state The coset representative of a single mode coherent state is parametrized by a complex number αj : r r-mode coherent states: |¯ α = |αj j=1 r exp(αj a ˆ†j − αj∗ a ˆj )|0 = j=1 where |0 is the vacuum state for each mode Apply the Campbell-Hausdorff formula exp(αj a ˆ†j − ˆj ) αj∗ a αj αj∗ = exp − exp(αj a ˆ†j ) exp(−αj∗ a ˆj ) (A.5) and the fact that exp(−αj∗ a ˆj )|0 = |0 67 (A.6) the r-mode coherent state can also be rewritten as r |¯ α = exp − j=1 αj αj∗ exp(αj a ˆ†j ) |0 +∞ +∞ ∗ ··· = exp(−¯ α α ¯ /2) m1 =0 mr (α1 )m1 · · · (αr )mr × |m1 , · · · , mr (m1 ! · · · mr !)1/2 =0 (A.7) where we have made use of short forms below α ¯ = (α1 , · · · , αr ); (A.8) r α ¯∗α ¯= αj∗ αj ; (A.9) j=1 We can also extend the ket description to the projector description of the coherent state The result is direct and given below +∞ +∞ m1 =0 r −|αj |2 ··· ··· |¯ α α ¯| = +∞ +∞ e nr =0 j=1 mr =0 n1 =0 (αj )mj (αj∗ )nj nj | (A.10) |mj (mj !nj !)1/2 f (αj ,α∗j ) Now, take a look if we apply a differential operator on the function f (αj , αj∗ ) (ignoring the density operator for the moment) ∂ + αj∗ f (αj , αj∗ ) ∂αj ✘ ✘∗✘ (αj )mj −1 (αj∗ )nj √ (α )mj (α )nj −|αj |2 ∗ −|αj |2 ✘j✘✘ j + e mj = −αj e ✘✘✘ (mj !nj !)1/2 ((mj − 1)!nj !)1/2 ✘✘✘ ✘ ∗ n✘ j ) j +1 (αj )m✘ (α ✘j✘ −|αj |2 ✘ + e ✘✘✘ (mj !nj !)1/2 ✘✘ Thus, ∂ + αj∗ |¯ α α ¯| ∂αj +∞ +∞ ··· = m1 =0 +∞ mr =0 n1 =0 +∞ e−|αj | |mj + +∞ nr =0 j=1 +∞ ··· m1 =0 a ˆ†j |¯ α r ··· =a ˆ†j = +∞ +∞ r e−|αj | |mj ··· mr =0 n1 =0 nr =0 j=1 α ¯| (αj )mj (αj∗ )nj nj | mj + (mj !nj !)1/2 (αj )mj (αj∗ )nj nj | (mj !nj !)1/2 (A.11) We see that when the j-th creation operators act on the density operator 68 from the left, its effect is the same as applying the differential operator ∂ + αj∗ on the function f (α, α∗ ) Doing the same for the annihilation op∂αj erators, we find that the j-th annihilation operator can be mapped into the differential operator αj The differential operators being mapped into depend on whether the creation or annihilation operators acts on the density operator from the left or right All the results are summarized below Dl (ˆ aj ) = αj = Dr∗ (ˆ a†j ) ∂ + αj∗ = Dr∗ (ˆ aj ) Dl (ˆ a†j ) = ∂αj (A.12) (A.13) Dl and Dr refer to the differential operators being mapped into from the relevant bosonic operators acting from the left or right of the density opˆ are any elements in the multimode Lie algebra, while erator If Aˆ and B c1 and c2 are any random complex numbers, then we have the following properties ˆ = c1 Dl (A) ˆ + c2 Dl (B) ˆ Dl (c1 Aˆ + c2 B) ˆ = Dl (B)D ˆ l (A) ˆ Dl (AˆB) ˆ B]) ˆ = [Dl (B), ˆ Dl (A)] ˆ Dl ([A, ˆ = Dr∗ (Aˆ† ) Dl (A) (A.14) All the properties above conserve the algebraic structure and constitute an isomorphism between the multimode Lie algebra and its corresponding differential algebra A.3 Multilevel Lie Algebra and its Coherent States We consider a single system where there are r internal degrees of freedom |j , j = 1, · · · r We define the operator Eˆjk which effects transition from the |k to the |j state The operators obey the commutation relations [Eˆjk , Eˆmn ] = Eˆjn δkm − Eˆmk δnj (A.15) For a single atom with r-level , the ground state of the system can be written as |1 = (0, 0, · · · , 0, 1)t Any unitary transformation is in the form 69 of an r × r matrix Its general form can be written as r r (yi1 Eˆi1 + y1i Eˆ1i ) exp r (yij Eˆij + y11 Eˆ11 ) × exp i=2 (A.16) i=2 j=2 The argument of the exponential has to be anti-hermitian to ensure uni† ∗ , we must have yjk = −ykj tarity of the transformation Since Eˆjk = Eˆkj The exponential can be cast in an explicit product of two matrices when acting on the ground state  [Ir−1 − x¯x¯† ]1/2 x¯ U (r − 1) −¯ x† x1 eiφ coset element          =       isotropical element  xr    eiφ (A.17)  x2  x1 where x¯ is the column matrix (xr , · · · , x2 )t The xj are related to yj1 through xj = yj1 sin(¯ y † y¯)1/2 (¯ y † y¯)1/2 r ∗ where y¯† y¯ = j=1 yj1 yj1 Unitarity is also translated into x1 = (1 − r ∗ 1/2 It is then evident we can define the vector (xr , · · · , x2 , x1 )t j=2 xj xj ) as the coherent state of the single r-level system by viewing |1 as our reference state Hence, the coherent state can be mapped one-to-one onto the 2(r − 1) dimensional sphere parametrized independently by x¯ = (xr , · · · , x2 )t and its complex conjugate with the unitarity constraint: r x21 x∗j xj = + (A.18) j=2 Now, we extend the formulation to N number of r-level bosonic systems The bosonic symmetrical many-body state with mj particles in the j-th level can be expressed as |m1 , m2 , · · · , mr = √ |1 ⊗ · · · ⊗ |1 ⊗ · · · ⊗ |r ⊗ · · · ⊗ |r N ! m1 ! · · · mr ! all P m1 times =√ ⊗|1 N ! m1 ! · · · mr ! all P m1 · · · |r mr times mr where P refers to the N ! permutations of the level labelling 70 (A.19) The N -body coherent states can be written as the N -tensor product of an identical single r-level system coherent state Suppose that such identical single r-level coherent state can be expressed in terms of a unitary transformation applied on the ground state |1 , then the corresponding N body coherent state is: N ⊗ [Uj (¯ x)|1 j ] j=1 N r (j) ⊗ = j=1 xk |k j k=1 = m1 +···+mr =N mr xm · · · xr m1 ! · · · mr ! ⊗|1 N! m1 ! · · · mr ! m1 +···+mr =N C(m1 , · · · mr ) xm = ⊗ · · · ⊗ |r mr all P mr xm · · · xr = m1 1/2 √ all P r · · · xm r |m1 , · · · ⊗ |1 N ! m1 ! · · · mr ! , mr m1 · · · |r (A.20) m1 +···+mr =N where C(m1 , · · · , mr ) = N! m1 ! · · · mr ! 1/2 For a two-level system, we recover the well-known SU(2) coherent state in terms of two-mode Fock basis: N |x1 , x2 = n=0 N n 1/2 −n xn1 xN |n, N − n (A.21) From this point onwards, we will denote the N -body r-level coherent states parametrized by vector x¯ as |N ; x¯ It is interesting to note that the the single multilevel algebra can be realized in terms of product of boson creation and annihilation operator through the identification Eˆjk = a ˆ†j a ˆk In fact, there is a deep connection between the multimode and multilevel Lie algebra A.4 Multimode to Multilevel Mapping The multimode Lie algebra has various interesting subalgebras One example is the subalgebra spanned by the number-conserving operators a ˆ†j a ˆk , which is isomorphic to the multilevel Lie algebra In fact, there is a homomorphism between the multimode Lie algebra with the number-conserving 71 mr subalgebra: h(ˆ a†j a ˆk ) = a ˆ†j a ˆk h(ˆ a†j a ˆ†k ) = h(ˆ aj a ˆk ) = (A.22) h(ˆ a†j ) = h(ˆ aj ) = h(I) = Under such homomorphism, we can expect the following: (a) irreducible representation of the multimode Lie algebra to be decomposed into the direct sum of irreducible representation of the multilevel Lie algebra (b) multimode coherent states to be decomposed into direct sum of multilevel coherent states (c) multimode projector provides a generating function for the multilevel projector (d) multimode differential algebra being mapped homomorphically onto the multilevel differential algebra The subject of this section is to give the mathematical formulation of the statements above A.4.1 Coherent States Mapping The mapping between multimode and multilevel Lie algebra can be carried out by the following change of variables: αj = xj αeiφ α = (¯ α† α ¯ )1/2 (A.23) eiφ = α1 /|α1 | The idea is to express the r independent coset complex numbers parametrizing the r-mode multimode coherent states in terms of the r − independent complex parameters xj , where j = 2, · · · , r α and the global phase eiφ are to make such transformation one-to-one With these definitions of the transformation, x1 is automatically real and dependent as x1 = (1 − rj=2 x∗j xj )1/2 Applying equation (A.7) and transformation 72 above, we obtain +∞ +∞ |¯ α = exp(−¯ α∗ α ¯ /2) ··· m1 =0 +∞ = = e −α2 /2 (αe iφ N ) √ N! N =0 m=N ¯ +∞ iφ N −α2 /2 (αe ) √ e mr (α1 )m1 · · · (αr )mr × |m1 , · · · , mr 1/2 (m ! · · · m !) r =0 N =0 N! N! m1 ! · · · mr ! 1/2 × (x1 )m1 · · · (xr )mr |m1 , · · · , mr |N ; x¯ (A.24) It is clear that the multimode coherent state is a generating function for the multilevel coherent states The homomorphism from the multimode to multilevel coherent state can be given as below upon direct calculation: lim e ∂ ∂α −iN φ α→0 A.4.2 N eα /2 √ |¯ α = |N ; x¯ N! (A.25) Projector Mapping Utilizing result (A.24), we can obtain the following +∞ +∞ e−α |¯ α α ¯| = M =0 N =0 αM +N (eiφ )M −N |M ; x¯ N ; x¯| (M !N !)1/2 (A.26) We then take the average over the variable φ: 2π +∞ |¯ α α ¯ |dφ = −α2 (α e N =0 N ) |N ; x¯ N ; x¯| N! (A.27) The mapping from a multimode projector to a multilevel projector can be given as: lim α2 →0 ∂ ∂α2 N eα 2π |¯ α α ¯ |dφ = |N ; x¯ N ; x¯| 73 (A.28) A.4.3 Multimode and Multilevel Differential Algebra Mapping Any multilevel number-conserving operator Eˆjk can be mapped into or identified as a ˆ†j a ˆk Thus, Dl (Eˆjk ) = Dl (ˆ a†j a ˆk ) = Dl (ˆ ak )Dl (ˆ a†j ) = αk ∂ + αj∗ ∂αj (A.29) Instead of viewing αj and αj∗ (j = 1, · · · , r) as the independent variables, we want α, xk , x∗k (k = 2, · · · r), and φ as the independent variables This means that we have to rewrite the multimode product of differential operators above in terms of the new transformed independent variables First, note that by chain rule we have r r ∂ ∂xk ∂ ∂x∗k ∂ ∂α ∂ ∂φ ∂ = + + + ∂αj ∂αj ∂α ∂αj ∂φ k=2 ∂αj ∂xk k=2 ∂αj ∂x∗k Since α2 = r p=1 (A.30) αp∗ αp , by deriving the one-form of both sides, we have r (αp∗ dαp + αp dαp∗ ) 2αdα = (A.31) p=1 αp∗ ∂α ∂α αp and = The other derivatives can be = ∗ ∂αp 2α ∂αp 2α derived similarly by starting from the equations Then, we have αk ; α eiφ α1 eiφ = = |α1 | xk = α1 ; α1∗ A lengthy algebra will then yield ∂xk ∂α1 ∂xk ∂α1∗ ∂xk ∂αj ∂xk ∂αj∗ α1∗ xk − xk 2α 2α1 α1 = − xk + ∗ xk 2α 2α1 ∗ αj δjk = iφ − xk αe 2α αj = − xk 2α =− 74 (A.32) (A.33) (A.34) (A.35) Next, we are going to use the short forms below for a more compact form of the final result r ¯ = x¯ · ∇ ∂ ; ∂xk xk k=2 r ¯∗ = x¯∗ · ∇ x∗k k=2 ∂ ∂x∗k Then, we have ∂ ∂ x1 e−iφ ¯ + x¯∗ · ∇ ¯ ∗) = x∗1 e−iφ − (¯ x·∇ ∂α1 ∂α 2α ∂ ¯ + x¯∗ · ∇ ¯∗ − x¯ · ∇ + iφ 2x1 αe i ∂φ (A.36) and ∂ ∂ ∗ −iφ ∂ x∗k e−iφ ¯ + x¯∗ · ∇ ¯ ∗) + iφ (¯ x·∇ = xk e − ∂αk ∂α αe ∂xk 2α (A.37) for k = 2, · · · , r Plugging in all results above into expression (A.29), we obtain for whatever values of j and k, Dl (Eˆjk ) = xk ∂ + xk x∗j ∂xj α ∂ ¯ + x¯∗ · ∇ ¯ ∗ ) (A.38) + α2 − xk x∗j (¯ x·∇ ∂α Bear in mind that x1 is not an independent variable, so the meaning of its first derivative in expression above has to be redefined as ∂ ≡ ∂x1 2x1 ∂ ¯ + x¯∗ · ∇ ¯∗ − x¯ · ∇ i ∂φ ≡− ∂ ∂x∗1 (A.39) Since ∂ ∂α2 = +∞ M ∂ ∂α2 e α2 e−α N =0 M +∞ N =M (α2 )N N! (α2 )N N! (A.40) Thus, lim α →0 ∂ ∂α2 M +∞ N =M 75 (α2 )N =1 N! (A.41) On the other hand, M ∂ ∂α2 eα = ∂ ∂α2 M ∂ ∂α2 M = +∞ α ∂ + α2 ∂α e−α N =0 +∞ N +∞ (α2 )N (N − 1)! (α2 )N N! (α ) (N − 1)! N =0 N =M (A.42) Thus, lim α →0 ∂ ∂α2 +∞ M N =M (α2 )N =M N! (A.43) Substituting all the limiting results into the expression of projector mapping with the action of Eˆjk : ∂ ∂α2 lim α →0 = lim α →0 = xk M eα ∂ ∂α2 dφ Dl (Eˆjk )|¯ α α ¯| 2π +∞ M α2 e D (Eˆjk ) e−α l N =0 (α2 )N |N ; x¯ N ; x¯| N! ∂ ¯ + x¯∗ · ∇ ¯ ∗ )] |M ; x¯ M ; x¯| + xk x∗j [M − (¯ x·∇ ∂xj (A.44) where the definition of partial derivative with respect to x1 dependent variable is still ∂ ¯ + x¯∗ · ∇ ¯∗ ≡ − ∂ ≡ −¯ x·∇ ∂x1 2x1 ∂x∗1 This result shows that the action of the operator Eˆjk on the multilevel projector from the left can be expressed in terms of the differential algebra as Eˆjk |M ; x¯ M ; x¯| = Dl (Eˆjk )|M ; x¯ M ; x¯| ≡ xk ∂ ¯ + x¯∗ · ∇ ¯ ∗ )] |M ; x¯ M ; x¯| + xk x∗j [M − (¯ x·∇ ∂xj (A.45) As the expression (A.45) is the manifestation of the number-conserving operator on the multilevel Lie subalgebra of the multimode Lie algebra, all the properties (A.14) established for the multimode differential operators are automatically verified for the multilevel differential operators 76 A.5 Husimi Distribution and Equation of Motion Knowing how to map the number-conserving operators to their corresponding differential operators is not adequate as the action is solely on the relevant coherent states We still need to know how to describe a general quantum state in terms of the coherent states and describe its dynamics in the new language of differential operators First step to achieve our goal is to introduce the Husimi distribution as below Husimi distribution: Q(N ; x¯; t) = N ; x¯|ˆ ρ(t)|N ; x¯ (A.46) where ρˆ(t) is the density operator of the quantum system Due to overcompleteness of the coherent states, such representation of the quantum state in terms of the multilevel coherent states is always unique Next, we consider the most general form of dynamical equation satisfied by the density operator: dˆ ρ = dt ˆm Cn,m Aˆn ρˆB (A.47) n,m Then, we multiply both sides by the coherent state projector |N ; x¯ N ; x¯| and take the trace of the expression tr[ˆ ρ(t)|N ; x¯ N ; x¯|] i|ˆ ρ(t)|N ; x¯ N ; x¯|i = i∈B N ; x¯|i i|ˆ ρ(t)|N ; x¯ = i∈B = N ; x¯|ˆ ρ|N ; x¯ = Q(N ; x¯, t) ˆm |N ; x¯ N ; x¯|] tr[Aˆn ρˆB ˆm |N ; x¯ N ; x¯|Aˆn ] = tr[ˆ ρB ˆm )Dr (Aˆn )|N ; x¯ N ; x¯|] = tr[ˆ ρ Dl (B 77 (A.48) (A.49) B denotes any set of basis vectors Using the results above, the equation of motion of quantum state expressed in terms of the Husimi distribution is given as: ∂Q(N ; x¯, t) = ∂t ˆm )Dr (Aˆn ) Q(N ; x¯, t) Cn,m Dl (B (A.50) n.m The order of the left and right differential operators is not important as they commute with each other Once the dynamics are contained in the partial differential equation above, the next step would be to write the expectation values of any physˆ in terms of the Husimi distribution ical observable X ˆ = tr(ˆ ˆ X ρX) = ˆ ; x¯ N ; x¯| ]dµx¯ tr[ ρˆX|N = ˆ Q(N ; x¯, t) dµx¯ Dl (X) (A.51) Thus, Eˆjk = N xk x∗j Q(N ; x¯)dµx¯ + xk ∂ Q(N ; x¯)dµx¯ ∂xj first term − ¯ + x¯∗ · ∇ ¯ ∗ )Q(N ; x¯)dµx¯ xk x∗j (¯ x·∇ (A.52) second term The first term and second term can be further simplified using integration by parts and periodic boundary condition first term = xk Q(N ; x¯) − δjk Q(N ; x¯)dµx¯ vanishes under periodic condition = −δjk (A.53) 78 r xk x∗j second term = xp p=2 ∂ ∂ + x∗p ∗ ∂xp ∂xp Q(N ; x¯)dµx¯ r r Q(N ; x¯)(x∗j xp δpk + xk x∗j ) dµx¯ − =− p=2 = −2r Q(N ; x¯)(xk x∗j + x∗p xk δpj )dµx¯ p=2 Q(N ; x¯)xk x∗j dµx¯ (A.54) Hence, we obtain the final expression Eˆjk = (N + r) Q(N ; x¯) xk x∗j dµx¯ − δjk (A.55) With all the ingredients which we have developed so far, all the quantum dynamics can be completely described in terms of the differential algebra of the multilevel Lie algebra 79 Bibliography [1] Franco Dalfovo, Stefano Giorgini, Lev P Pitaevskii, and Sandro Stringari Theory of bose-einstein condensation in trapped gases Rev Mod Phys., 71:463–512, Apr 1999 [2] Anthony Chefles and Stephen M Barnett Quantum theory of twomode nonlinear directional couplers Journal of Modern Optics, 43(4):709–727, 1996 [3] D M ESTERLING and ROBERT V LANGE Degenerate mass operator perturbation theory in the hubbard model Rev Mod Phys., 40:796–799, Oct 1968 [4] W T Vetterling W H Press, S A Teukolsky and B P Flannery Numerical Recipes Cambridge University Press, London, 3rd edition, 2007 [5] S Raghavan, A Smerzi, S Fantoni, and S R Shenoy Coherent oscillations between two weakly coupled bose-einstein condensates: Josephson effects,pi- oscillations, and macroscopic quantum selftrapping Phys Rev A, 59:620–633, Jan 1999 [6] G J Milburn, J Corney, E M Wright, and D F Walls Quantum dynamics of an atomic bose-einstein condensate in a double-well potential Phys Rev A, 55:4318–4324, Jun 1997 [7] A Imamo¯glu, M Lewenstein, and L You Inhibition of coherence in trapped bose-einstein condensates Phys Rev Lett., 78:2511–2514, Mar 1997 [8] A Vardi and J R Anglin Bose-einstein condensates beyond mean field theory: Quantum backreaction as decoherence Phys Rev Lett., 86:568–571, Jan 2001 80 [9] Y Castin and R Dum Low-temperature bose-einstein condensates in time-dependent traps: Beyond the u(1) symmetry-breaking approach Phys Rev A, 57:3008–3021, Apr 1998 [10] A Griffin Conserving and gapless approximations for an inhomogeneous bose gas at finite temperatures Phys Rev B, 53:9341–9347, Apr 1996 [11] Qi Zhang, Jiangbin Gong, and C H Oh Intrinsic dynamical fluctuation assisted symmetry breaking in adiabatic following Phys Rev Lett., 110:130402, Mar 2013 [12] J Denschlag, J E Simsarian, D L Feder, Charles W Clark, L A Collins, J Cubizolles, L Deng, E W Hagley, K Helmerson, W P Reinhardt, S L Rolston, B I Schneider, and W D Phillips Generating solitons by phase engineering of a bose-einstein condensate Science, 287(5450):97–101, 2000 [13] Wilkinson J H The Algebraic Eigenvalue Problem Oxford: Oxford University Press [14] H Koizumi Q Niu J Zwanziger A Bohm, A Mostafazadeh The Geometric Phase in Quantum Systems Springer - Verlag Berlin Heidelberg, 2003 [15] J R Anglin and A Vardi Dynamics of a two-mode bose-einstein condensate beyond mean-field theory Phys Rev A, 64:013605, May 2001 [16] Anthony J Leggett Bose-einstein condensation in the alkali gases: Some fundamental concepts Rev Mod Phys., 73:307–356, Apr 2001 [17] R Gilmore, C M Bowden, and L M Narducci Classical-quantum correspondence for multilevel systems Phys Rev A, 12:1019–1031, Sep 1975 [18] F Trimborn, D Witthaut, and H J Korsch Exact number-conserving phase-space dynamics of the m-site bose-hubbard model Phys Rev A, 77:043631, Apr 2008 [19] B Juli-Daz, T Zibold, M K Oberthaler, M Mel-Messeguer, J Martorell, and A Polls Dynamic generation of spin-squeezed states in bosonic Josephson junctions (arXiv:1205.6756), May 2012 81 ... Master of Science Associate Professor Gong Jiangbin Department of Physics Beyond Mean- Field Dynamics of Two- Mode Bose- Hubbard Model with Linear Coupling Ramping Summary The mean- field Hamiltonian of. . .BEYOND MEAN- FIELD DYNAMICS OF TWO- MODE BOSE- HUBBARD MODEL WITH LINEAR COUPLING RAMPING CHENG KOK CHEONG (B.Sc (Hons), NUS ) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF. .. of two- mode Bose- Hubbard model with real and imaginary coupling constants demonstrates pitchfork bifurcation in its phase-space structure within certain interval of real coupling constant Its mean- field