玻色凝聚

������ Many�body phenomena in condensed matter and atomic physics Last modi�ed� September ��� ���� � Lectures �� �� Bose condensation� Symmetry� breaking and quasiparticles� In an ideal Bose gas� at su�ciently low temperature� the lowest energy state becomes occupied by a macroscopic numb e r of particles �Bose�Einstein condensate�� The density of a gas of free bosons� given by a s u m of occupancies of di�erent momentum states� has the form Z � � � h� �3 (reg ) (reg ) n � n 0 ��p� � � � n p d 3 p � n p � e �(p 2 �2m��) � � ��� with m the particle mass and � the chemical potential� At high temperature T � T BEC � with h 2 � T BEC � � n 2�3 � � � � � ��� ��� � � m � 2�3 ��� � at density n� there is no condensate� n 0 � �� � �� On the other hand� at low temperature T T BEC � there is a macroscopic numb e r of particles in the ground state� while the chemical potential is zero� In this case� the condensate density is � � � 3�2 n 0 � n � n c � n � � �T �T ��� BEC The question we shall discuss below is how this b e h a vior is modi�ed by the presence of interatomic interaction� We shall focus on the problem of weakly nonideal Bose gas� This problem� due to the existence of a simple analytical method� serves well to illustrate the new features of Bose condensation of interacting particles� spontaneous symmetry breaking� the o��diagonal long�range order� and collective excitations� ��� Spontaneous symmetry breaking Weakly interacting Bose gas with a short�range interaction� � � Z h 2 � + � + + � �H � � � �x� r 2 �x� � � �x� � �x� �x� �x� dx � � m x where x � r in D � �� The coupling constant is the two�particle scattering amplitude in the Born approxi� R � h 2 mation� � � U k�0 � U �x � x � �dx� A more accurate formula� � � � a�m� where a is the s�wave scattering length� to be discussed b e l o w� � The ground state at T � � is characterized by large occupation numb e r of the k � � state� In number representation� the BEC state of N particles is jBEC i � jN k�0 � �� �� ��� i� i�e� � a k�0 jB p N jBEC N �1 i� k � � EC N i � ��� a �� k �� � This formula� at large N � suggests to replace the numb e r state by a coherent state� 0 jBEC i � p N jBEC i� which is equivalent to replacing the operator � p N � a 0 by a c�numb e r This can be achieved if the BEC state is understood as a coherent state� which requires considering the problem � � in the �big� space with all particle numb e r s allowed� Such an approach gives results equivalent to that of the problem with �xed particle numb e r N in the limit N � �� since for a coherent state h�N 2 i 1�2 � N 1�2 � N � Upon such a P replacement� the �eld operator � � V �1�2 k a k e ikr turns into a classical �eld � q N �V � where V is system volume� To that end� we are led to consider the coherent states � m p V � � p � � �� X V V j�j+ � a 0 �i � e � � 1 2 2 j i � exp �a j jmi ��� p m� 0 m�0 �which have the desired property j i � j i� These states do not correspond to any speci�c numb e r of particles� in fact they are characterized by a distribution of particle � numb e r s � Accordingly� the states j i are not invariant under the numb e r operator N � P � + k a k a k � while the hamiltonian � � commutes with N � One has to understand why the BEC state apparently does not respect the particle numb e r conservation� We start by noting that e i� ^ N j i � je i� i � e i� ^ N He �i� ^ N � H ��� ^ i�N � applied to i�e� the operator e j i� produces a state of the same energy� with a phase � 0 ��j Since the overlap of coherent states obeys jh � any two di�erent states j i� j � i are orthogonal in the limit V � �� with di�erent phase factors� j i� demonstrates that the BEC states form a degenerate manifold parameterized by a phase ij 2 � e �V j Thus the states 2 of shifted by �� j � i� i� are macroscopically distinct� j This observation e variable � � �� To clarify the origin of this degeneracy� let us �nd the states j i that provide minimum to the energy � �� Taking minimum at �xed particle density can b e achieved by adding � � to H a term proportional to N � H � H� �N � Taking the expectation value� obtain � 2 � U � � � h jH � �N j i � j j 4 � �j j ��� � the so�called Mexican hat potential� The energy minima are found on the circle j j 2 � ���� i�e� the phase of is arbitrary� while the modulus j j is �xed� thereby giving a relation between the density and chemical potential� � � �n� 1 From the symmetry point of view� the systuation is quite interesting� The microscopic hamiltonian � � has global U ��� symmetry� since it is invariant under adding a constant � � e i� �phase factor to the wavefunction of the system� � The ground states� however� do not possess this symmetry� adding a phase factor to the state j i produces a di�erent ground state� This phenomenon� called spontaneous symmetry breaking� is absent in the noninteracting Bose gas� In the interacting system� the U ��� symmetry breaking has a very fundamental consequence� it leads to super�uidity� There is yet another way to understand the phenomenon of U ��� symmetry breaking� due to Penrose and Onsager� that does not require to consider the states with �uctuating particle number� One can instead start with the density matrix of the Bose gas � � ground state j� � i in the coordinate representation� + �R�x� x � � � h� � j � �x � � �x�j� � i ��� Using the translational invariance� one expects that the quantity ��� will depend only on the distance x � x � b e t ween the two p o i n ts� By going to Fourier representation� one can transform ��� to the form R�x� x � � � Z e �ik(x�x 0 ) n k d 3 k � �� 3 � n k � h� � j�a + k �a k j� � i ���� In a B o s e condensate� the particle distribution n k has singularity a t k � � � n k � n 0 � �� 3 ��k� � f �k� ���� where f �k� i s a smooth function� Accordingly� t h e density matrix ���� has two terms� Z d 3 k � � R�x� x � � � n 0 � f �x � x � � � f �x � x � � e �ik(x�x 0 ) f k � �� 3 �� � � the constant n 0 � independent of p o i n t separtion� and the second part� f �x � x � �� that vanishes at large jx � x � j� A density matrix that does not vanish at large point separtion represents an anomaly �recall that in any ordinary liquid all correlations vanish at several interatomic distances�� 0 � � � �The �nite limit n 0 � lim jx�x j�� h� � j + �x � �x�j� � i suggests that the quantities �x�� � � �x�i � e i� p n 0 � h �e sen se + �x � i n s o m h a ve �nite expectation values� h � + �x�i � e �i� p n 0 � with �xed modulus� but an undetermined phase� The name O� diagonal Long range Order� or ODLRO� associated with this phenomenon� expresses the fact that in the density matrix the ordering is revealed by the behavio of the o��diagonal component R�x� x � � jx�x j�� � 0 1 In a �nite� but large system� with �xed particle number� the true ground state �TGS� of a quantum� mechanical hamiltonian is nondegenerate� This TGS is isotropic in �� due to boundary e�ects that split the circular manifold� The statement about the absence of degeneracy of TGS in a �nite system is formally coirrect� but misleading� since this TGS is not �pure�� Typically� at any moment of time the state is characterized by a global phase� changing slowly as a function of time� �In D � the mixing of ��s with di�erent phases results from vortices passing across the system� from one boundary to another�� � ��� Quasiparticles To s t u d y the excitations above the ground state� we substitute � � p N � a c�number� in a 0 the hamiltonian � �� and keep quadratic terms� � X X � � � � + + + � H� �N � �n 2 V � � (0) � � � �n a k a k � � �n a k a �k � a k a �k ���� k �0 k�k� �0 � � � X � (0) + + + + � �n 2 V � k �a k a k � a �k a �k � � �n�a k � a �k � + �a k � a �k � �� � (k��k) where the sum is taken over pairs �k� �k� Here we used the value � � �n obtained above� 1 + At this stage� it is convenient to introduce the quantities q� k � p 2 �a k � a �k �� p� k � + p i 2 �a �k � a k �� These operators are non�hermitian� q� + � q� �k � p� + � p� �k � but obey the k k standard p� q algebra� � q� k � p� k 0 � � � kk 0 � which allows to treat them as coordinate and momentum� In terms of the operators p� k � q� k the hamiltonian is represented as a sum of + + + independent harmonic oscillators� Indeed� since a k a k � a �k a �k � p� + p� k � �q k q� k � we can k rewrite the hamiltonian as follows� � � � � � X � (0) p� + p� k � � (0) � �n q + �H � �n 2 V � k k k k q k ���� (k��k) This hamiltonian� quadratic in �q k � p� k � can b e brought to the normal form by a rescaling �squeezing� transformation � (0) � � 4� k k k k k k q� k � e � q� � p� k � e �� p� � e � � (0) ���� � �n k which acts on the operators a k � a + as k + a k � cosh k b k � sinh k b + � a �k � cosh k b + �k �k � sinh k b k ���� �see Lecture �� The transformation ����� called Bogoliubov transformation� can be shown to preserve the canonical commutation relations� �b k � b + � � �� k The hamiltonian is now reduced to � � � X � X b + b k � b + k H � �n 2 V � E k k �k b �k � �n 2 V � E k b + b k ���� �0 (k��k) k� describing a gas of Bogoliubov quasiparticles� the noninteracting bosons created and an� nihilated by the operators b + k � b k � having energy r � 2 E k � � (0) � �n � 2 � ��n� ���� k The new ground state is annihilated by all the b k � Since for the ground state of the ideal Bose gas a k j� 0 i � �� and the transformation ���� can b e represented as b k � Ua k U �1 � + b + � Ua k U �1 � with k � � � � X U � exp � k a k a �k � a k a �k � + + A � �� k��0 �see Lecture �� one can write the new ground state as j� � i � U j� 0 i� The dispersion relation ����� for small k� is linear� E k � hcjkj � c � q �n�m � �� which is characteristic for sound waves in a �uid� For higher values of k� the dispersion h 2 takes the form of a usual free�particle expression E k � k 2 � m � �n� Remarkably� both the collective modes� sound waves� and the single�particle excita� tions appear on the same dispersion curve� gradually blending into one another at the energy ca E k � �n� One can gain some insight i n to the di�erence of the modes at large and small k by considering the �eld equations h 2 i � � � � �H� �N �� � � r � � � 2 � � � � �h� t � � 2 � + � h m 2 h� t � � + + + � 2 � + � ���i + � �� H� �N �� � � � � r 2 � � �� � � � � m It is instructive to treat these equations as a classical dynamics problem� linearizing near stationary solution� � 0 � �� where � q ���� The linearized equation has solution of the form be �ikr+i�t � � 0 ��r� t � � ae ikr�i�t � r � 2 with k h� � � � (0) � �n � 2 � ��n� the same as Eq������ In other words� one can con� sider condensate with �uctuating amplitude and phase� and show that these �uctuations propagate in just the same way as the collective m o d e s ����� In such an approach� the di�erence between small and large k follows from the relation b e t ween the amplitudes a and b obtained from the dynamical equation� At small k� the sum a � b is much smaller than the di�erence a � b� This means that the oscillations are predominantly in the phase of the �eld � not in the modulus� just as one expects from Goldstone theorem �and the above Mexican hat picture�� At large k� however� the normal modes have a � b or a � b nearly equal in magnitude� which means that the oscillation follows a small circle in the complex plane� i�e� the phase and the modulus of participate in the collective oscillations roughly equally� We can use the above results to estimate the e�ect of condensate depletion due to interactions� The total density of all particles in the system can be written as X X + + n � h� � ja 0 a 0 � a k a k j� � i � n 0 � sinh 2 k h� � jb k b + j� � i � �� k �0 k�k� �0 � � � X B � (0) � �n C � n 0 � B r k � � C � �� � � A k� 2 �0 � (0) � �n � 2 � ��n� k Estimating the sum as O�� 3�2 �� we �nd that the condensate depletion is a small e�ect� In contrast� in superfuid 4 He only few percent of the helium atoms are in the single�particle ground state� � << /ASCII85EncodePages false /AllowTransparency false 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