Calculating the Pauli Matrix Equivalent for Spin-1 Particles and Further

Home , Matrix exponential, Pauli matrices

Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System Rajdeep Tah

To cite this version:

Rajdeep Tah. Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System. 2020. hal-02909703

HAL Id: hal-02909703 https://hal.archives-ouvertes.fr/hal-02909703 Preprint submitted on 31 Jul 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés.

Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License Calculating the Pauli Matrix equivalent for Spin-1 Particles and further implementing it to calculate the Unitary Operators of the Harmonic Oscillator involving a Spin-1 System

Rajdeep Tah1, ∗ 1School of Physical Sciences, National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni, P.O.-752050, Odisha, India (Dated: July, 2020) Here, we derive the Pauli Matrix Equivalent for Spin-1 particles (mainly Z-Boson and W-Boson). 1 Pauli Matrices are generally associated with Spin- 2 particles and it is used for determining the 1 properties of many Spin- 2 particles. But in our case, we try to expand its domain and attempt to implement it for calculating the Unitary Operators of the Harmonic oscillator involving the Spin-1 system and study it.

I. INTRODUCTION discuss its analytical solution along with its features. In Section IV, we involve with the transformation of Pauli Matrices are a set of three 2 × 2 complex matri- Unitary Operators and after that in Section V, we ces which are Hermitian and Unitary in nature and they involve the implementation of the Spin-1 system into occur in the Pauli Equation which takes into account the Quantum Harmonic Oscillator with relevant trans- the interaction of the spin of a particle with an exter- formations and generalization which ultimately leads to nal electromagnetic ﬁeld. In Quantum Mechanics, each the derivation of the Hamiltonian of our system. Then Pauli matrix is related to an angular momentum opera- in Section VI, we derive the Unitary Operators of our tor that corresponds to an observable describing the spin system using the informations from the previous sections 1 and then in its following subsection we represent the of a Spin- 2 particle, in each of the three spatial direc- tions. But we seldom need to deal with particles which 16 × 16 Matrix of the diﬀerent Unitary Operators. At 1 last, in Section VII, we discuss the Results of our project are having spin more than 2 i.e. Spin-1 Particles; Spin- 3 followed by the Conclusion and Acknowledgement. 2 Particles; Spin-2 Particles; etc. and for that we need to search for matrices which perform similar function to 1 that of Pauli Matrices (in case of Spin- 2 particles). In 1 II. MATHEMATICAL MODELLING Spin- 2 particles, the Pauli Matrices are in the form of:

0 1 Let us assume that we have a Spin-s system for which X = σ = (1) 2 x 1 0 the Eigenvalue S is given by: 2 2 S = s(s + 1)~ (4) 0 −i Or Y = σy = (2) i 0 p S = s(s + 1)~ The eigenvalues of S are written s , where s is al- 1 0 z z ~ z Z = σ = (3) lowed to take the values s, s−1, ··· , −s+1, −s i.e. there z 0 −1 are 2s + 1 distinct allowed values of sz. We can represent the state of the particle by (2s + 1) diﬀerent Wave- Also sometimes the Identity Matrix, I is referred to 0 functions which are in-turn denoted as ψsz (x ). Here as the ‘Zeroth’ Pauli Matrix and is denoted by σ0. 0 1 ψsz (x ) is the Probability density for observing the parti- The above denoted Matrices are useful for Spin- 2 0 particles like Fermions (Proton, Neutrons, Electrons, cle at position x with spin angular momentum sz~ in the Quarks, etc.) and not for other particles. So, in this z- direction. Now, by using the extended Pauli scheme, paper we will see how to further calculate the equiv- we can easily ﬁnd out the Momentum operators and the alent Pauli Matrix for Spin-1 particles and implement Spin operators. The Spin Operator comes out in the it to calculate the Unitary Operators of the Quan- form: tum Harmonic Oscillator involving a Spin-1 system. hs, j|Sk|s, li (σk)jl = (5) s~ Structure: Where, j and l are integers and j, l ∈ (−s, +s). In Section II, we do the Mathematical Modelling Now, to make our calculations easier, we continue with for the Equivalent Matrices and discuss the results for σ matrix. We know that: higher spin particles. Then, in Section III we revisit the z harmonic oscillator in-context of our quantum world and Sz|s, ji = j~|s, ji (6) 2

∴ We can write: oscillator in general can be written as: hs, j|S |s, li j (σ ) = k = δ (7) 3 jl s s ij pˆ2 1 pˆ2 1 ~ Hˆ = + mω2xˆ2 = + kxˆ2 (16) 2m 2 2m 2 Here we have used the Orthonormality property of |s, ji. Thus, σz is the suitably normalized diagonal matrix of where Hˆ is the Hamiltonian of the System, m is the eigenvalues of Sz. The elements of σx and σy are most easily obtained by considering the ladder operators: the mass of the particle, k is the bond stiﬀness (which is analogous to spring constant in clas-

S± = Sx ± iSy (8) sical mechanics),x ˆ is the position operator and

Now, according to Eq.(5)-(8), we can write9:

1/2 ∂ S+ |s, ji = [s (s + 1) − j (j + 1)] ~ |s, j + 1i (9) pˆ = −i (17) ~∂x and is the momentum operator where ~ − 1/2 is the reduced Plank’s constant. S |s, ji = [s (s + 1) − j (j − 1)] ~ |s, j − 1i. (10) Now, by combining all the conditions and equations, we The analytical solution of the Schrodinger 1 have: wave equation is given by Ref. :

1/2 [s (s + 1) − j (j − 1)] ∞ ∞ (σ ) = δ 1/2 2 2 1 j l j l+1 X X 1 mω ζ β 2 s − 2 − 2 (11) Ψ = n e e Hnx (ζ)Hny (β)U(t) [s (s + 1) − j (j + 1)]1/2 2 n! π~ + δ nx=0 ny =0 2 s j l−1 (18)

and Where; r r [s (s + 1) − j (j − 1)]1/2 mω mω (σ2)j l = δj l+1 ζ = x and β = y (19) 2 i s (12) ~ ~ [s (s + 1) − j (j + 1)]1/2 − δj l−1 Here Hn is the nth order Hermite polynomial. U(t) is the 2 i s unitary operator of the system showing its time evolution and is given by: ∴ According to Eq.(7), Eq.(11) and Eq.(12), we have:

−itE −itEn 0 1 0 n 1 U(t) = exp( ) = e ~ (20) σ1 = √ 1 0 1 (13) ~ 2 0 1 0 where En are the allowed energy eigenvalues of the particle and are given by: 0 −i 0  1 1 1 En = (nx + )~ω + (ny + )~ω = (nx + ny + 1)~ω (21) σ2 = √ i 0 −i (14) 2 2 2 0 i 0 And the states corresponding to the various energy eigenvalues are orthogonal to each other and satisfy: 1 0 0  Z +∞ σ = 0 0 0 (15) 3   ψjψxdxi = 0 : ∀ xi (22) 0 0 −1 −∞ A much simpler approach to the harmonic oscil- Where, σ1, σ2, σ3 are the Pauli Matrix equivalents for Spin-1 Particles. lator problem lies in the use of ladder operator method where we make use of ladder operators i.e. the creation and annihilation opera- III. HARMONIC OSCILLATOR IN BRIEF tors (ˆa, aˆ†), to ﬁnd the solution of the problem.

The most common and familiar version of ** Herea ˆ† denotes the ‘Creation’ operator anda ˆ the Hamiltonian of the quantum harmonic denotes the ‘Annihilation’ operator in Spin-1 system. 3

We can also the Hamiltonian in terms of the creation V. IMPLEMENTATION ON A SPIN-1 SYSTEM and annihilation operators (â, aˆ†)7: 4 1 1 The Hamiltonian of the full system is given by : Hˆ = ω(âaˆ † − ) = ω(âaˆ † + ) ~ 2 ~ 2 Hˆ = Hˆfield + Hâtom + Hînt Now the Hamiltonian for “a discrete quantum harmonic oscillator” is given by: where Hˆfield is the free Hamiltonian, Hâtom is the atomic excitation Hamiltonian and Hˆ is the interaction Hamil- (ˆpd)2 (ˆxd)2 + (ˆyd)2 int Hˆ = + (23) tonian. 2 2 wherep ˆd is the discrete momentum operator andx ˆd and yˆd are the discrete position operators in in x and y spatial A. MODEL dimension respectively. Alsop ˆd can be expressed as: We have modeled our system using Rabi Hamiltonian5. d d −1 d d pˆ = (F ) · xˆ · (F ) (24) However, in our case we will be using somewhat modified 6 where F d is the standard discrete Quantum Fourier version of Rabi Hamiltonian : 3 d Transform matrix . Each element of F can be expressed 2 2 X ω0 X as: H = ω a† a + σ + g (eiθk a + e−iθk a† )σ s k k k 2 3 k k k 1 k=1 k=1 d exp(2iπjk/N) [F ]j,k = √ (25) (30) N Where ω0 is the frequency of the main oscillator, ωk † N N is the frequency of the k-th environment oscillator; a Where j, k ∈ [− 2 , ...... , 2 − 1] and j= no. of rows in the k matrix and k= no. of columns in the matrix. and ak are the creation and annihilation operators of the main system and the k-th environmental oscillator respectively. Whereas gk’s are the coupling constant for IV. UNITARY OPERATOR the interaction between the k-th environment oscillator TRANSFORMATIONS and the main quantum oscillator. We set k=1 from now to prevent us from complicating the process. For the sake of reducing mathematical complexity, let us assume ~, ω and m is unity (i.e.all are having value 1). So, we can write the Schrodinger equation as: For simplicity, we will consider the simplest case of our model and substitute k=1 in our original Hamiltonian ∂Ψ i = Hˆ Ψ (26) [in Eq.(30)] to obtain the special case of our Hamiltonian ∂t which will be our working Hamiltonian from now: which further implies: ω † 0 iθ1 −iθ1 † H = ω1a1a1 + σ3 + g1(e a1 + e a1)σ1 Ψ(t) = Ψ(0)exp(−iHtˆ ) 2

From the above, it is vivid that the unitary operator For simplicity we will drop the sub-script 1 from d d to be computed is U(t) = exp(−iHˆ t) where Hˆ is the our Hamiltonian and obtain: discretized Hamiltonian operator mentioned in Equation ω (3). So. the unitary operator is given by: H = ωa†a + 0 σ + g(eiθa + e−iθa†)σ (31) 2 3 1 (ˆpd)2 (ˆxd)2 + (ˆyd)2 U(t) = exp(−it( + )) (27) 2 2 B. RELEVANT TRANSFORMATION AND Or if we consider the X-dimension only, then we get the GENERALIZATION unitary operator as: −it Now, as our system involves Spin-1 particles, so the U (t) = exp( ((F d)−1 · ˆ(xd)2 · (F d) + (ˆxd)2)) (28) following commutation relations uphold: xˆ 2 † † † Due to the discretization of space; the position operator [ai, aj] ≡ aiaj − ajai = δij (32) [ˆxd], being a diagonal matrix, can be expanded by using the concept of Matrix exponential as Ref.2: † † ∞ [ai , aj] = [ai, aj] = 0 (33) it X it [A]m exp(− [A]) = + (− )m (29) 2 I 2 m! m=1 Here δij is known as ‘Kronecker delta’. Here A is the corresponding Operator Matrix. 4

The operators used in the Hamiltonian can be trans- Similarly, formed according to Holstein-Primakoff transformations8     (i.e. it maps spin operators for a system of spin-S mo- 1 0 0 1 0 0 ω0 ω0 ments on a lattice to creation and annihilation operators) ⊗ Sz = 0 1 0 ⊗ 0 0 0 2 I 2     as: 0 0 1 0 0 −1 q ˆ+ Sj = (2S − nˆj)âj (34)  ω0  2 0 0 0 0 0 0 0 0 q ˆ− †  0 0 0 0 0 0 0 0 0  Sj =a ˆj (2S − nˆj) (35)  ω0   0 0 − 2 0 0 0 0 0 0   0 0 0 ω0 0 0 0 0 0  wherea ˆ† (â ) is the creation (annihilation) operator at ω  2  j j ⇒ 0 ⊗ S =  0 0 0 0 0 0 0 0 0  site j that satisfies the commutation relations mentioned I z   2  ω0  † 0 0 0 0 0 − 2 0 0 0 above andn ˆ =a ˆ aˆ is the “Number Operator”. Hence  ω  j j j  0 0 0 0 0 0 0 0 0  we can generalize the above equations as:  2   0 0 0 0 0 0 0 0 0  q 0 0 0 0 0 0 0 0 − ω0 S+ = (2S − a†a)a (36) 2 (41) q Finally, S− = a† (2S − a†a) (37)  0 eiθ 0  0 1 0 iθ −iθ † g −iθ iθ Where; g(e b + e b ) ⊗ Sx = √ e 0 e  ⊗ 1 0 1 2 0 e−iθ 0 0 1 0 S+ ≡ Sx + iSy and S− ≡ Sx − iSy Where;  iθ  Sx (= σ1), Sy (= σ2), Sz (= σ3) are the Pauli matrices 0 0 0 0 e 0 0 0 0 for Spin-1 system (as mentioned in the previous section).  0 0 0 eiθ 0 eiθ 0 0 0     0 0 0 0 eiθ 0 0 0 0    Now by using the above transformations; we can write  0 e−iθ 0 0 0 0 0 eiθ 0  g   our creation and annihilation operators in terms of Ma- ⇒ √ e−iθ 0 e−iθ 0 0 0 eiθ 0 eiθ 2  −iθ iθ  trices as:  0 e 0 0 0 0 0 e 0       −iθ  0 0 0 0 1 0  0 0 0 0 e 0 0 0 0  †  −iθ −iθ  a = 1 0 0 and a = 0 0 1 (38)  0 0 0 e 0 e 0 0 0  0 1 0 0 0 0 0 0 0 0 e−iθ 0 0 0 0 (42) Substituting the above values in Eq.(39), we Now the Hamiltonian for our coupled quantum har- get the value of H (a 9 × 9 matrix) as: monic oscillator in Eq.(31) can be decomposed as: ω H = ωa†a ⊗ + 0 ⊗ σ + g(eiθa + e−iθa†) ⊗ σ I 2 I 3 1 Or the above equation can be written as: ω H = ωa†a ⊗ + 0 ⊗ S + g(eiθa + e−iθa†) ⊗ S (39) I 2 I 3 1 Now, we will evaluate each term to simplify the expression of the Hamiltonian in the form of matrix. Here, 0 0 0 1 0 0 † ωa a ⊗ I = ω 0 1 0 ⊗ 0 1 0 0 0 1 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0   0 0 0 ω 0 0 0 0 0 †   ⇒ ωa a ⊗ I = 0 0 0 0 ω 0 0 0 0 (40) 0 0 0 0 0 ω 0 0 0   0 0 0 0 0 0 ω 0 0   0 0 0 0 0 0 0 ω 0 0 0 0 0 0 0 0 0 ω 5

 iθ  ω0 0 0 0 ge√ 0 0 0 0 2 2  iθ iθ   0 0 0 ge√ 0 ge√ 0 0 0   2 2   iθ   0 0 − ω0 0 ge√ 0 0 0 0   2 2   −iθ iθ   0 ge√ 0 (ω + ω0 ) 0 0 0 ge√ 0   2 2 2   −iθ −iθ iθ iθ  H =  ge√ 0 ge√ 0 ω 0 ge√ 0 ge√   2 2 2 2   −iθ iθ   0 ge√ 0 0 0 (ω − ω0 ) 0 ge√ 0   2 2 2   −iθ   0 0 0 0 ge√ 0 (ω + ω0 ) 0 0   2 2   −iθ −iθ   0 0 0 ge√ 0 ge√ 0 ω 0   2 2   −iθ  0 0 0 0 ge√ 0 0 0 (ω − ω0 ) 2 2

VI. DERIVATION OF UNITARY OPERATORS Thus we have,

−iXt −iY t Clearly, we know that for a system with Hamiltonian U = e .e H, the unitary operator is given by: =⇒ U = U (t).U (t) U = e−iHt (43) x y where U (t) = e−iXt and U (t) = e−iY t. First we will Where H is the Hamiltonian of the system derived in the x y compute U (t), then U (t). We can see that U (t) can previous section. y x y be expanded using Taylor series of expansion of the ex- But to ﬁnd the unitary operator compatible, we need ponential function as: to change the form of our Hamiltonian and write it as a sum of two matrices whose corresponding unitary oper- ∞ m X m Y ators are relatively easier to compute: Uy(t) = exp(−itY ) = + (−it) I m! m=1 H = X + Y Y Y 2 Y 3 Where, =⇒ U (t) = + (−it)1 + (−it)2 + (−it)3 y I 1! 2! 3! ω  0 0 0 0 0 0 0 0 0  2 4 5 0 0 0 0 0 0 0 0 0 4 Y 5 Y ω 0 0 − 0 0 0 0 0 0 0 +(−it) + (−it) + ......  2 ω  0 0 0 (ω + 0 ) 0 0 0 0 0 4! 5!  2   0 0 0 0 ω 0 0 0 0  X = ω 0  0 0 0 0 0 (ω − 0 ) 0 0 0  √g 2 ω Now, for simplicity, let us denote = g  0 0 0 0 0 0 (ω + 0 ) 0 0  2  2  0 0 0 0 0 0 0 ω 0 ω 0 0 0 0 0 0 0 0 (ω − 0 ) 0 0 2 (−itg )2 (−itg )4 =⇒ U (t) = [1 + + + ...] y 2! 4! I geiθ 0 0 0  0 0 0 0 √ 0 0 0 0  3 5 2 (−itg ) (−itg ) (−itg ) geiθ geiθ +[ + + + ...]M 0 0 0 √ 0 √ 0 0 0  2 2  1! 3! 5! geiθ  0 0 0 0 √ 0 0 0 0   2  ge−iθ geiθ Where;  0 √ 0 0 0 0 0 √ 0   2 2   ge−iθ ge−iθ geiθ geiθ  iθ √ 0 √ 0 0 0 √ 0 √   Y =  2 2 2 2  0 0 0 0 e 0 0 0 0  ge−iθ geiθ  iθ iθ 0 √ 0 0 0 0 0 √ 0 0 0 0 e 0 e 0 0 0  2 2     ge−iθ  iθ  0 0 0 0 √ 0 0 0 0   0 0 0 0 e 0 0 0 0  2  −iθ iθ   ge−iθ ge−iθ  0 0 0 √ 0 √ 0 0 0  0 e 0 0 0 0 0 e 0   2 2   −iθ −iθ iθ iθ ge−iθ M = e 0 e 0 0 0 e 0 e 0 0 0 0 √ 0 0 0 0   2  −iθ iθ   0 e 0 0 0 0 0 e 0   −iθ   0 0 0 0 e 0 0 0 0   0 0 0 e−iθ 0 e−iθ 0 0 0  0 0 0 0 e−iθ 0 0 0 0

(** We can observe that [Y 2, Y 4, Y 6,....] will give Iden- tity matrices whereas [Y 1, Y 3, Y 5,...] will give the same 6

ω0 Qubit states Results after Uy(t)acts Uxˆ(t)[1, 1] exp(-( 2 )it) gt gt −iθ |0000i (cos √ |0000i − i sin √ e |0100i) Uxˆ(t)[2, 2] exp(-(0)it) 2 2 ω gt gt −iθ U (t)[3, 3] exp(( 0 )it) |0001i (cos √ |0001i − i sin √ e (|0011i + 0101)) xˆ 2 2 2 ω0 Uxˆ(t)[4, 4] exp(-(ω + )it) |0010i (cos √gt |0010i − i sin √gt e−iθ |0100i) 2 2 2 Uxˆ(t)[5, 5] exp(-(ω)it) gt gt −iθ (cos √ |0011i − i sin √ e |0001i − ω0 2 2 Uxˆ(t)[6, 6] exp(-(ω − 2 )it) gt iθ ω0 |0011i i sin √ e |0111i) Uxˆ(t)[7, 7] exp(-(ω + )it) 2 2 (cos √gt |0100i − i sin √gt e−iθ(|0110i + Uxˆ(t)[8, 8] exp(-(ω)it) 2 2 ω0 Uxˆ(t)[9, 9] exp(-(ω − )it) |0100i |1000i) − i sin √gt eiθ(|0000i + |0010i)) 2 2 (cos √gt |0101i − i sin √gt e−iθ |0111i − 2 2 |0101i i sin √gt eiθ |0001i) 2 |0110i (cos √gt |0110i − i sin √gt eiθ |0100i) In this case also we will consider a 16 × 16 matrix 2 2 |0111i (cos √gt |0111i − i sin √gt eiθ(|0011i + 0101)) (in place of a 9 × 9 matrix) because of same reason 2 2 gt gt iθ mentioned before and also we will construct the matrix |1000i (cos √ |1000i − i sin √ e |0100i) 2 2 in the same pattern as mentioned in case of Uy(t) operator. It is easy to observe as X is a diagonal TABLE I. Operator Uy(t) acting on Qubit States. matrix, each diagonal element of Ux(t) makes an exact Taylor expansion of the exponential function matrix which is given above as M. So we diﬀerentiate (**The 16 × 16 matrix for both Uy(t) and Ux(t) them in two groups.) operators are mentioned in the next sub-section.)

0 0 =⇒ Uy(t) = cos g tI − iM sin g t Again, we operate this operator on diﬀerent 4-qubits gt gt states (in our situation we need only nine of the sixteen =⇒ Uy(t) = cos √ I − iM sin √ (44) 4-qubit states because for the other seven states we 2 2 will get the same Unitary matrix as result.) and then Now for Spin-1 particles, we need to use a 4-qubit study the results for the same given in Table(IV): system but for implementing a 4-qubit system we must Qubit states Results after Ux(t)acts require a 16 × 16 matrix because any matrix of order ω (− 0 )it N × N must satisfy the condition N = 2n (where |0000i e 2 |0000i n= number of qubits). But we can express the above |0001i e(0)it |0000i ω ( 0 )it equation in form of a 16 × 16 matrix (which we have |0010i e 2 |0000i ω −(ω+ 0 )it shown in the next sub-section), instead of a 9 × 9 |0011i e 2 |0011i matrix, by adding 1 diagonally seven times and placing |0100i e(−ω)it |0100i ω −(ω− 0 )it 0 in other positions. In our situation we need only |0101i e 2 |0101i ω nine of the sixteen 4-qubit states (mentioned in Table −(ω+ 0 )it |0110i e 2 |0110i (I)) because for the other seven states we will get the |0111i e(−ω)it |0111i ω same Unitary matrix as result (i.e. without any change). −(ω− 0 )it |1000i e 2 |1000i

TABLE II. Operator Ux(t) acting on Qubit States. In the above segment, we computed the Uy(t) operator. In order to compute U (t) which is equal to e−iXt, we first x From the above table we can see the effect of U (t) expand the expression using the Taylor expansion of the x operator acting on the different 4-qubit states. exponential function just like we did in earlier case as:

∞ Now, we know how to implement both the parts of X Xm U (t) = exp(−itX) = + (−it)m our Unitary operator and the complete unitary matrix x I m! m=1 (16 × 16) can be implemented by by operating both the operations in series. In this way we can easily calculate

2 3 our Unitary operators for Spin-1 system and also verify 1 X 2 X 3 X the eﬀectiveness of Pauli Matrices equivalent to Spin-1 =⇒ Ux(t) = I + (−it) + (−it) + (−it) 1! 2! 3! system/ particles. X4 X5 +(−it)4 + (−it)5 + ...... 4! 5! Therefore by using the above equation, we can express Ux(t) in terms of e as: 7

UNITARY OPERATOR MATRIX VII. RESULTS REPRESENTATIONS In first part of our paper, we extend the idea The 16 × 16 Matrix representation of the Unitary op- 1 of Pauli Matrices from Spin- 2 particles to Spin-1 erators Uy(t) and Ux(t) are: particles and see the implementation of the equivalent matrices. We can also see that equivalent   A 0 0 0 B 00000000000 Pauli Matrices can also be found for higher spin  0 A 0 B 0 B 0 0 0 0 0 0 0 0 0 0 particles like Spin- 3 particles, Spin-2 particles etc.  0 0 A 0 B 00000000000 2    0 C 0 A 0 0 0 B 0 0 0 0 0 0 0 0   In the final part, we introduce a coupled quantum C 0 C 0 A 0 B 0 B 0 0 0 0 0 0 0   harmonic oscillator to a Spin-1 system (Z-Bosonic/ W-  0 C 0 0 0 A 0 B 0 0 0 0 0 0 0 0   Bosonic system etc.) and try to implement its unitary op-  0 0 0 0 C 0 A 0 0 0 0 0 0 0 0 0   erator to the system using our previous section’s knowl-  0 0 0 C 0 C 0 A 0 0 0 0 0 0 0 0 Uy(t) =   edge.  0 0 0 0 C 0 0 0 A 0 0 0 0 0 0 0    0000000001000000    0000000000100000 CONCLUSION    0000000000010000  0000000000001000   Now, we understand that we can further implement  0000000000000100   the idea of Pauli equivalent matrices on higher spin par-  0000000000000010 ticles and derive the unitary operators of the Quantum 0000000000000001 Harmonic Oscillators using those informations in a much Where, simpler yet effective manner. We conclude with one last remark that there are various processes for finding the A = cos( √gt ); B = −i sin( √gt )eiθ and C = −i sin( √gt )e−iθ. Pauli equivalent matrices for higher spin particles but 2 2 2 we have used a much simpler yet effective process to find the values and implement it further onto a Quantum Harmonic Oscillator for finding its Unitary Operators.

S 000000000000000 Also it will be better to notice that I didn’t disentan- 0100000000000000   gle the qubit states in the results which I got after Uy(t) 0 0 1 0000000000000  S  acts on the qubits because it would be necessary in case 0 0 0 P 000000000000   of Quantum simulation of a circuit which we are not in- 0 0 0 0 Q 00000000000   volving and we are keeping all the results in the entangled 0 0 0 0 0 R 0 0 0 0 0 0 0 0 0 0   state. 0 0 0 0 0 0 P 0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 Q 0 0 0 0 0 0 0 0 Ux(t) =   0 0 0 0 0 0 0 0 R 0 0 0 0 0 0 0 ACKNOWLEDGEMENTS   0000000001000000   0000000000100000 I would like to thank School of Physical Sciences (SPS),   0000000000010000 NISER where I got the opportunity to interact with won-   0000000000001000 derful members and professors who helped me a lot with   0000000000000100 the basics of Quantum Mechanics. I acknowledge the 0000000000000010 support of my parents who constantly motivated me and 0000000000000001 guided me through the project during the COVID-19 Where, Pandemic and didn’t let my morale down. −(ω+ ω0 )it (−ω)it −(ω− ω0 )it P = e 2 ; Q = e ; R = e 2 ; (− ω0 )it 1 1 ( ω0 )it S = e 2 and = ω = e 2 s (− 0 )it e 2

DATA AVAILABILITY

Further information regarding process of implementing the Unitary Operators on higher spin particles like Spin- 3 2 particles, Spin-2 particles etc. can be made available upon reasonable number of requests. 8

∗ [email protected] 5 S. Agarwal, S. M. H. Rafsanjani and J. H. Eberly, Tavis- 1 D. J. Griﬀths, Introduction to Quantum Mechanics, Pear- Cummings model beyond the rotating wave approximation: son Prentice Hall (2004). Quasi-degenerate qubits, Phys. Rev. A 85, 043815 (2012). 2 V. K. Jain, B. K. Behera, and P. K. Pani- 6 B. Militello, H. Nakazato, and A. Napoli1, 2 : Synchro- grahi, Quantum Simulation of Discretized Har- nizing Quantum Harmonic Oscillators through Two-Level monic Oscillator on IBMQuantum Computer, DOI: Systems, Phys. Rev. A 96, 023862 (2017). 10.13140/RG.2.2.26280.93448(2019) 7 Relation between Hamiltonian and the Operators, 3 Quantum Fourier Transform, URL: https://en. URL: https://en.wikipedia.org/wiki/Creation_and_ wikipedia.org/wiki/Quantum_Fourier_transform annihilation_operators 4 Jaynes-Cummings model’URL: https://en.wikipedia. 8 Holstein-Primakoﬀ transformation, URL: https://en. org/wiki/Jaynes%E2%80%93Cummings_model wikipedia.org/wiki/Holstein%E2%80%93Primakoff_ transformation 9 Eigen Values, URL: https://farside.ph.utexas.edu/ teaching/qm/Quantum/node40.html#e5.44a