TCM315 Fall 2020: Introduction to Open Quantum Systems Lecture 10: Fock space dynamics

Course handouts are designed as a study aid and are not meant to replace the recommended textbooks. Handouts may contain typos and/or errors. The students are encouraged to verify the information contained within and to report any issue to the lecturer.

CONTENTS

Introduction 1

Composite systems of indistinguishable particles1

Permutation group S2 of 2-objects 2 Admissible states of three indistiguishable systems2 Parastatistics 4 The spin-statistics theorem 5

Fock space 5 Creation and annihilation of particles 6 Canonical commutation relations 6 Canonical anti-commutation relations 7 Explicit example for a Fock space with 2 distinguishable Fermion states7

Central oscillator of a linear boson system8 Decoupling of the one particle sector 9

References 9

INTRODUCTION

The chapter 5 of [6] oers a conceptually very transparent introduction to indistinguishable particle kinematics in quantum mechanics. Particularly valuable is the brief but clear discussion of para-statistics. The last sections of the chapter are devoted to expounding, physics style, the second quantization formalism. Chapter I of [2] is a very clear and detailed presentation of second quantization formalism in the form that is needed in mathematical physics applications. Finally, the dynamics of a central oscillator in an external eld draws from chapter 11 of [4].

COMPOSITE SYSTEMS OF INDISTINGUISHABLE PARTICLES

The composite system postulate tells us that the Hilbert space of a multi-partite system is the tensor product of the Hilbert spaces of the constituents. In other words, the composite system Hilbert space, is spanned by an orthonormal basis constructed by taking the tensor product of the elements of the orthonormal bases of the Hilbert spaces of the constituent systems. A composite quantum system may consist of several subsystems of identical nature, which in such a case become physically indistinguishable. The reason is that properties of quantum particles are contextual: the properties depend upon the experimental setting used to observe them. In mathematical terms, contextuality refers to the choice of a complete set of orthogonal states which we use to dene projective measurements. The election of one complete basis of projective measurements imply that properties associated to an alternative non commuting basis are undened for the system. This is at variance with classical mechanics where, at least in principle, objects have an inexhaustible set of inherent attributes, and therefore are always distinguishable [6]. COMPOSITE SYSTEMS OF INDISTINGUISHABLE PARTICLES 2

Permutation group S2 of 2-objects

The impossibility to distinguish between identical particles poses an intrinsic limit to the realizable pure quantum states of identical particles. To exemplify the situation consider the case of two systems each having as state space a Hilbert space spanned by identical orthonormal basis . The bipartite system Hilbert space is then spanned by the orthonormal system {ϕi}i ≥ 0 with elements

Φij = ϕi ⊗ ϕj i, j = 0,...

The states

Φii = ϕi ⊗ ϕi and

ϕ ⊗ ϕ + ϕ ⊗ ϕ Φ(+) = i j√ j i ij 2 or

ϕ ⊗ ϕ − ϕ ⊗ ϕ Φ(−) = i j√ j i ij 2 are physically realizable because a particle permutation brings about only a phase factor. More precisely if we dene the adjacent transposition operator T12 acting on basis elements as

T12 Φij ≡ T12 (ϕi ⊗ ϕj) = ϕj ⊗ ϕi ≡ Φji (1) then we readily verify that

T12 Φii = Φii whereas

(±) (±) T12 Φij = ± Φij

On the other hand, the generic basis element Φij appearing in (1) is unphysical. As it is not invariant under particle exchange, it implies the possibility to distinguish between two particles based on the quantum number specied by the orthonormal system. The rst conclusion which we can draw is that physical states of a bipartite system must be eigenstates of the permutation operator T12.

Admissible states of three indistiguishable systems

We need rst to introduce some background from group theory. A generating set of a group G is a a collection of group elements G0 ⊂ G such that as any group element g can be written as a product of powers of elements taken from G0. course web-page: https://bit.ly/345IiUw COMPOSITE SYSTEMS OF INDISTINGUISHABLE PARTICLES 3

occupation sequence of pos 3 A general result [3,5] in group theory states that any ele- z }| { C123 T12 ment of the permutation group of -objects can be ex- x3 → x2 → x2 SN N • pressed as products of the adjacent transposition T12 such that

T12(x1, x2, . . . , xN) = (x2, x1, . . . , xN) and the cycle T12 C123 = T23

C12...n(x1, x2, . . . , xN) = (xN, x1, . . . , xN−1) • • Note that by construction C123 T12 C123 T12 x1 → x2 → x1 x2 → x1 → x3 | {z } | {z } 2 occupation sequence of pos 1 occupation sequence of pos 2 T12(x1, x2, . . . , xN) = (x1, x2, . . . , xN) n C12...n(x1, x2, . . . , xN) = (x1, x2, . . . , xN) If we apply rst the cycle C123(x1, x2, x3) = (anti-clockwise -degree rotation The Hilbert space of a tripartite system with indistinguish- (x3, x1, x2) 60 able components is now spanned by the orthonormal sys- in the picture) and then the adjacent trans- position (parity tem T12(x1, x2, x3) = (x2, x1, x3) around the vertical axis) operations we obtain T C (x , x , x ) = (x , x , x ) = T (x , x , ) Φijk = ϕi ⊗ ϕj ⊗ ϕk (2) 12 123 1 2 3 1 3 2 23 1 2 3 where is the orthonormal basis of the constituent {ϕk}k ≥ 0 Hilbert spaces. Based on the foregoing discussion a state occupation sequence of pos 3 of the form (2) is not realizable because the ordering z }| { T12 C123 would allow to distinguish among the constituent systems. x3 → x3 → x1 • For a xed triple (i, j, k) we can construct 6 distinct states corresponding to 6 permutations of the order. We can use these state to construct a totally symmetric state

(+) S3 (ϕi ⊗ ϕj ⊗ ϕk) = 2   C T = T X l ϕi ⊗ ϕj ⊗ ϕk + T12(ϕi ⊗ ϕj ⊗ ϕk) (3) 123 12 13 C123 √ l=0 6 • • The maximally symmetric state is an eigenstate of T12 C123 T12 C123 C123 x1 → x2 → x3 x2 → x1 → x2 | {z } | {z } and T12 with unit eigenvalue occupation sequence of pos 1 occupation sequence of pos 2

(+) (+) C123 S3 (ϕi ⊗ ϕj ⊗ ϕk) = S3 (ϕi ⊗ ϕj ⊗ ϕk) If we apply rst the adjacent transposition (+) (+) and then the cycle T12 S3 (ϕi ⊗ ϕj ⊗ ϕk) = S3 (ϕi ⊗ ϕj ⊗ ϕk) T12(x1, x2, x3) = (x2, x1, x3) C123(x1, x2, x3) = (x3, x2, x1) operations we obtain C123 T12(x1, x2, x3) = (x3, x2, x1) = T13(x1, x2, x3) and (+) (+) (+) T13 S3 (ϕi ⊗ ϕj ⊗ ϕk) = T23 S3 (ϕi ⊗ ϕj ⊗ ϕk) = S3 (ϕi ⊗ ϕj ⊗ ϕk) Similarly the totally antisymmetric state 2   (−) X l ϕi ⊗ ϕj ⊗ ϕk − T12(ϕi ⊗ ϕj ⊗ ϕk) (4) S3 (ϕi ⊗ ϕj ⊗ ϕk) = C123 √ l=0 6 readily satises

(−) (−) C123 S3 (ϕi ⊗ ϕj ⊗ ϕk) = S3 (ϕi ⊗ ϕj ⊗ ϕk) (−) (−) T12 S3 (ϕi ⊗ ϕj ⊗ ϕk) = − S3 (ϕi ⊗ ϕj ⊗ ϕk) (+) (+) (+) T13 S3 (ϕi ⊗ ϕj ⊗ ϕk) = T23 S3 (ϕi ⊗ ϕj ⊗ ϕk) = − S3 (ϕi ⊗ ϕj ⊗ ϕk) course web-page: https://bit.ly/345IiUw COMPOSITE SYSTEMS OF INDISTINGUISHABLE PARTICLES 4

The upshot is that the maximally symmetric and antisymmetric states are admissible because under any permutation of the constituent systems (i.e. under the action of any element of the group S3) they only change by a phase factor.

Parastatistics

We can construct other four states as eigenstates of the cycle operator. To this goal, the observation

      0 0 1 x1 x3 C123 ∼ C = 1 0 0 ⇒ C x2 = x1 0 1 0 x3 x2 allows us to conclude that

ı 2π ı 4π Sp C123 = Sp C = (1, e 3 , e 3 ) ≡ (1, ω, ω¯) The denition implies

ı 2π −ı 2π +ıπ 2 ω¯ = e 3 = e 3 = ω

Then we see that

2   (ω,0) X l l ϕi ⊗ ϕj ⊗ ϕk (5) Φijk = ω¯ C123 √ l=0 3 satises

(ω,0) 2 (ω,0) (ω,0) C123 Φijk =ω ¯ Φijk = ω Φijk Similarly, we verify that

2   (ω,1) X l l ϕi ⊗ ϕj ⊗ ϕk (6) Φijk = ω¯ C123 T12 √ l=0 3 (ω,1) 2 (ω,1) (ω,1) C123 Φijk =ω ¯ Φijk = ω Φijk and

2   (¯ω,0) X l l ϕi ⊗ ϕj ⊗ ϕk (7) Φijk = ω C123 √ l=0 3 ¯ (¯ω,0) 2 ω¯ ¯ (¯ω,0) C123 Φijk = ω Φijk =ω ¯ Φijk and nally

2   (¯ω,1) X l l ϕi ⊗ ϕj ⊗ ϕk (8) Φijk = ω C123 T12 √ l=0 3 ¯ (¯ω,1) 2 ω,¯1 ¯ (¯ω,1) C123 Φijk = ω Φijk =ω ¯ Φijk

The states (ω,r) and (¯ω,r) for dened by (5)-(8) are referred to as para-statistics. An argument against Φijk Φijk r = 0, 1 their realizability goes as follows [6]. We know that two indistinguishable particles behave either as bosons (symmet- ric), or as fermions (antisymmetric). When aiming to study two identical particles we cannot exclude the presence of a third identical particle hidden to us. If we admit parastatistics, the mere existence of that third particle would imply the possibility of a state of the form (5)-(8) implying an improper two particle symmetrization:

(ω,r) (¯ω,1−r) T12 Φijk = Φijk course web-page: https://bit.ly/345IiUw FOCK SPACE 5

The spin-statistics theorem

A fundamental result of relativistic quantum eld theory in Minkowski space, the spin-statistics theorem proves that elementary particles obey either canonical commutation relations for interaction carrier elds (bosons) or canoni- cal anti-commutation relations for matter elds (fermions). Evidences, however, exist in d = 2 hinting at the existence of quasi-particles obeying para-statistics [1]. The existence of such quasi-particles states potentially plays an important role for the realization of robust quantum computing. Here we restrict the attention to elementary particle statistics. We therefore generalize (3) as

(+) 1 X √ (9) SN (ϕi1 ⊗ ... ⊗ ϕiN) = ϕp(i1) ⊗ ... ⊗ ϕp(iN) N! p and (4) as

(−) 1 X √ (10) SN (ϕi1 ⊗ ... ⊗ ϕin ) = pϕp(i1) ⊗ ... ⊗ ϕp(iN) N! p In the rst row of (9), (9), the sum, as customary in the literature e.g. [2], is over all permutations

p:(i1, i2, . . . , iN) 7→ (p(i1), p(i2),..., p(iN)) and

( 1 if p is even p = −1 if p is odd

Thus S+ produces a fully symmetric state vector of n-identical particles and S− a fully anti-symmetric state vector of N-identical particles.

FOCK SPACE

In applications we are often interested in considering models of Quantum Mechanics with a variable or unknown number of identical particles each dened on an Hilbert space H. The proper stage for the dynamics in such a case is a Fock space F(H): M F(H) = H ⊗ N N≥ 0 Correspondingly, the Bose-Fock space is then dened as   M ⊗ N M N F+(H) = S+  H  = Sym (H) N≥ 0 N≥ 0 whereas the Fermi-Fock space is then dened as   M ⊗ N M N F−(H) = S−  H  = ∧ H N≥ 0 N≥ 0

These constructions allow us to specify the action of a single particle self-adjoint A operator on a cluster of n identical particles i.e. its extension A(N) to the Hilbert tensor product of the N single particles Hilbert spaces

 N  A(N) X A S±(ϕi1 ⊗ ... ⊗ ϕin ) = S±  ϕi1 ⊗ ... ϕij ⊗ ... ⊗ ϕiN j=1 course web-page: https://bit.ly/345IiUw FOCK SPACE 6

In particular, if the self-adjoint operator is a time independent Hamilton operator generating the single particle unitary dynamics

Ut = exp (−ı H t) then

U(N) U U S±(ϕi1 ⊗ ... ⊗ ϕin ) = S± ( tϕi1 ⊗ ... ⊗ tϕiN) The extension to the Fock space is obtained by taking the Cartesian sum over all particle sectors.

Creation and annihilation of particles

We suppose that the Fock space has a unique vacuum state . We dene dene a creation operator † by the ΦΩ ak property that the vector

† Φk = ak ΦΩ is normalized and represents a single particle with (collective) quantum number k. The inner product in the Fock space

D † † E D † E 1 = h Φk , Φk iF = ak ΦΩ , ak ΦΩ = ΦΩ , ak ak ΦΩ F F allows us to identify the adjoint of † as a annihilation operator since ak † ak ak ΦΩ = ΦΩ We shall henceforth assume that (except for normalization) the operator † adds a particle in state to any state ak k (not only to the vacuum state) and therefore its adjoint removes such a particle. In particular, for any k

ak ΦΩ = 0 because removing a particle from the vacuum produces an unphysical state. Remark. It must be emphasized that † and must be thought as Fock state operators creating a particle state ak ak k distinguishable from a particle state l. **

Canonical commutation relations

Suppose the quantum number k = 0, 1,... labels the particle states. Bosons satisfy canonical commutation relations (CCR) h i † 1 (11a) ak , ak = h i 1 † † 1 (11b) [ak , al] = F = ak , al = F Then we can use the integer valued innite row vector

n = (n0, n1,... ) to construct a complete basis of the Fock space of the form

∞ † nk Y (ak) Φn = √ ΦΩ nk k=0

The integer nl admit then the interpretation of number of particles in the state k: † ak ak Φn = nk Φn course web-page: https://bit.ly/345IiUw FOCK SPACE 7

Canonical anti-commutation relations

Fermions satisfy canonical commutation relations (CAR) h i † 1 (12a) ak , ak = F + h i 1 † † 1 (12b) [ak , al]+ = F = ak , al = F +

Again ak describes a Fermion state distinguishable from al both operators actong on a Fock space constructed by taking the Cartesian products of totally anti-symmetric (wedge) tensor products of identical single Hilbert spaces H. If

d H = C we construct the Fock space

d M N d F−(H) = ∧ C N=0 creation and annihilation operators using satisfying CAR-relations for n-state systems

ak = σ3 ⊗ . . . σ3 ⊗ σ− ⊗ 12 ⊗ 12 (13a) | {z } | {z } k−1 d−k † 1 1 (13b) ak = σ3 ⊗ . . . σ3 ⊗ σ+ ⊗ 2 ⊗ 2 | {z } | {z } k−1 d−k where

0 0 0 1 1 0  σ = & σ = & σ = − 1 0 + 0 0 3 0 −1 and

Φ0 = e2 ⊗ ... ⊗ e2 | {z } d

Explicit example for a Fock space with 2 distinguishable Fermion states

As an illustration of (13) we construct the creation and annihilation operators of the Fock space

2 2 M N 2 2 2 2 F−(C ) = ∧ C = 1 ⊕ C ⊕ (C ∧ C ) N=0 The dimension of the Fock space is

2 dim F−(C ) = 4 since

2 2 2 dim(1) = 1 & dim C = 2 & dim(C ∧ C ) = 1 Using (13) nilpotent ladder operators are

0 0 0 0 0 0 1 0 0 0 0 0 † 0 0 0 1 a0 = σ− ⊗ 12 =   & a = σ+ ⊗ 12 =   1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 course web-page: https://bit.ly/345IiUw CENTRAL OSCILLATOR OF A LINEAR BOSON SYSTEM 8 and

0 0 0 0 0 1 0 0  1 0 0 0 † 0 0 0 0  a1 = σ3 ⊗ σ− =   & a = σ3 ⊗ σ+ =   0 0 0 0 1 0 0 0 −1 0 0 −1 0 0 0 0 0 For instance we check that

1 0 0 0   † 1 0 0 1 0 0 a a0 = σ+σ− ⊗ 12 = ⊗ 12 =   0 0 0 0 0 0 0 0 0 0 0

1 0 0 0   † 1 0 0 0 0 0 a a1 = σ3σ3 ⊗ σ+σ− = 12 ⊗ =   1 0 0 0 0 1 0 0 0 0 0 and

0 0 0 0   † 0 −1 0 0 −1 0 a a1 = σ+σ3 ⊗ 12σ− = ⊗ σ− =   0 0 0 0 0 0 0 0 0 0 0

0 0 0 0   † 0 0 0 0 0 0 a a0 = σ3σ− ⊗ σ+12 = ⊗ σ+ =   1 −1 0 0 −1 0 0 0 0 0 0

We use the above operators to construct on F the Hamilton operator   0 + 1 0 0 0   H † † † †  0 0 −g 0 F = 0 a0 a0 +g a0 a1 + a1 a0 + 1 a1 a1 =    0 −g 1 0 0 0 0 0 Upon dening   0 −g 0 + 1 0 − 2 H2 = = 12 − gσ1 + σ1 −g 1 2 2 we can decompose HF as the Cartesian sum

HF = (0 + 1) ⊕ H2 ⊕ 0

It is then straightforward to verify that the unitary evolution on k takes the form

H H e−ı F t = e−ı (0+1)t ⊕ e−ı 2t ⊕ 1

CENTRAL OSCILLATOR OF A LINEAR BOSON SYSTEM

We consider a collection of Boson harmonic oscillators linearly interacting with a central linear oscllator:

N N   H † X † X † † F = ω0 a0 a0 + ωk ak ak + g¯k ak a0 +gk ak a0 k=1 k=1 course web-page: https://bit.ly/345IiUw CENTRAL OSCILLATOR OF A LINEAR BOSON SYSTEM 9

We notice that the above second quantization Hamiltonian is amenable to the form

† HF = a HN+1a where we dened the N+ 1-vector with operator components   a0 a1   .  a =  .   .  aN and the N+ 1 × N+ 1 self-adjoint matrix  †  ω0 g HN+1 = & HNij = ωiδij i, j = 1,....N g HN

† Finally g = (¯g1, g¯2,..., g¯N ) is a row vector containing the coupling constants, g is the corresponding column vector. The Liouville-von Neumann equation for the system is

ı ∂tρt = [HF , ρt]

Decoupling of the one particle sector

We now take advantage of the bi-linear nature of the interaction to exhibit a remarkable reduction of the dynamics.

The N+ 1 ⊗ N+ 1 matrix Rt with elements H H † −ı k t ı k t
(14) Rmn(t) = Tr an am e ρ0e satises

ı ∂tRt = [HN+1 , Rt] To prove the statement we observe that † H R
ı∂tRmn(t) = Tr an am [ F , t] We now take advantage of the commutator algebra H † R  † H R H †  R F , an am t = an am [ F , t] + F , an am t and the vanishing of the trace of antisymmetric tensors H † R 
Tr F , an am t = 0 We are thus left with the need to evaluate

N+1 h i h i H †  H †  † H X † † † † F , an am = F , an am + an [ F , am] = Hij ai aj , an am + an ai aj , am i,j=0 whence the claim follows after a straightforward though tedious algebra. Hence we obtain a closed equation in CN+1 starting from the innite dimensional equation in F. The solution reads

−ıHN+1t ıHN+1t Rt = e R0e

***

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[2] O. Bratteli and D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics 2. Equilibrium States. Models in Quantum Statistical Mechanics. Theoretical and Mathematical Physics. Springer Berlin Heidelberg, 2nd edition, 2002. [3] K. Conrad. Generating sets. Course handout, department of Mathematics University of Connecticut https://kconrad. math.uconn.edu/blurbs/. [4] A. Galindo and P. Pascual. Quantum Mechanics II. Texts and Monographs in Physics. Springer Science + Business Media, 1991. [5] P. B. Pal. A Physicist' s Introduction to Algebraic Structures. Cambridge University Press, may 2019. [6] A. Peres. Quantum theory: concepts and methods, volume 57 of Fundamental Theories of Physics. Springer, 1995.

course web-page: https://bit.ly/345IiUw