Applications

Abstract

Applications of the formalism of finite quantum systems, to angle and angular momentum operators, interferometry, orbital angular momentum states, etc, are briefly discussed.

In this chapter we discuss applications of the formalism into the area of Quantum Optics and Quantum Information, and also into other areas. Each of these applications is a subject in its own right, and here we briefly define the basic quantities and guide the reader through the literature.

7.1 Angle States and Angular Momentum States

In this section we apply the general formalism of finite quantum systems, to a system with angular momentum j. In this case $d=2j+1$ where j is an integer (‘Bose case’), and the variables take values in ${\mathbb Z}(2j+1)$. The relevant Hilbert space is $H[{\mathbb Z}(2j+1)]$, which in this chapter we denote for simplicity $H(2j+1)$.

The analogue of the momentum states are here the usual angular momentum states, which we denote as $|J;j\; m\rangle$. The extra J to the usual notation is not a variable, but it simply indicates angular momentum states. The analogue of position states are the angle states [1], which we denote as $|\theta ;j\; m\rangle$, and which are defined through Fourier transform below.

The angular momentum operators $J_z$, $J_+$, $J_-$, form the SU(2) algebra

$$\begin{aligned}{}[J_z,J_+]=J_+;\;\;\;\;\;[J_z,J_-]=-J_-;\;\;\;\;\;[J_+,J_-]=2J_z. \end{aligned}$$

(7.1)

The Casimir operator is

$$\begin{aligned} J^2=J_z^2+\frac{1}{2}(J_+J_-+J_-J_+)=j(j+1)\mathbf{1}. \end{aligned}$$

(7.2)

Then

$$\begin{aligned} J_+|J;j\;m\rangle= & {} [j(j+1)-m(m+1)]^{1/2}|J;j\;m+1\rangle \nonumber \\ J_-|J;j\;m\rangle= & {} [j(j+1)-m(m-1)]^{1/2}|J;j\;m-1\rangle \nonumber \\ J_z|J;j\;m\rangle= & {} m|J;j\;m\rangle \nonumber \\ J^2|J;j\;m\rangle= & {} j(j+1)|J;j\;m\rangle . \end{aligned}$$

(7.3)

The Fourier transform in the present context is

$$\begin{aligned} F=\frac{1}{\sqrt{2j+1}}\sum _{m,n} \omega (mn)|J;j\; m\rangle \langle J;j\; n|;\;\;\; F^4=\mathbf{1}. \end{aligned}$$

(7.4)

Acting with it on the angular momentum states, we get angle states:

$$\begin{aligned} |\theta ;j\;m\rangle =F^{\dagger }|J;j\;m\rangle =\frac{1}{\sqrt{2j+1}}\sum _{n} \omega (-mn)|J;j\;n\rangle \end{aligned}$$

(7.5)

Also acting with it on the angular momentum operators we get the angle operators

$$\begin{aligned} F^{\dagger }J_zF=\theta _z;\;\;\;F^{\dagger }J_+F=\theta _+;\;\;\;F^{\dagger }J_-F=\theta _- \end{aligned}$$

(7.6)

which form the SU(2) algebra

$$\begin{aligned}{}[\theta _z,\theta _+]=\theta _+;\;\;\;[\theta _z,\theta _-]=-\theta _-;\;\;\;[\theta _+,\theta _-]=2\theta _z \end{aligned}$$

(7.7)

The corresponding Casimir operator is

$$\begin{aligned} \theta ^2=\theta _z^2+\frac{1}{2}(\theta _+\theta _-+\theta _-\theta _+)=j(j+1)\mathbf{1}. \end{aligned}$$

(7.8)

Relations analogous to Eqs. (7.3), also hold for angle operators and angle states (because we have performed a Fourier transform, which is a unitary transform):

$$\begin{aligned} \theta _+|\theta ;j\;m\rangle= & {} [j(j+1)-m(m+1)]^{1/2}|\theta ;j\;m+1\rangle \nonumber \\ \theta _-|\theta ;j\;m\rangle= & {} [j(j+1)-m(m-1)]^{1/2}|\theta ;j\;m-1\rangle \nonumber \\ \theta _z|\theta ;j\;m\rangle= & {} m|\theta ;j\;m\rangle \nonumber \\ \theta ^2|\theta ;j\;m\rangle= & {} j(j+1)|\theta ;j\;m\rangle \end{aligned}$$

(7.9)

We next introduce a polar decomposition of the ‘Cartesian operators’ $J_+$ and $J_-$ in terms of the ‘radial operator’ $J_r$ and the ‘exponential of the phase operator’ Z:

$$\begin{aligned}&J_+=J_rZ;\;\;\;\;\;J_-=Z^{\dagger }J_r\nonumber \\&J_r=(J_+J_-)^{1/2}=[j(j+1)\mathbf{1}-J_z^2+J_z]^{1/2};\;\;\;[J_r,J_z]=0\nonumber \\&Z=\sum _m |J;j\;m+1\rangle \langle J;j\;m| \end{aligned}$$

(7.10)

The dual relations to them are

$$\begin{aligned}&\theta _+=\theta _rX;\;\;\;\;\;\theta _-=X^{\dagger }\theta _r\nonumber \\&\theta _r=(\theta _+\theta _-)^{1/2}=[j(j+1)\mathbf{1}-\theta _z^2+\theta _z]^{1/2};\;\;\;[\theta _r,\theta _z]=0\nonumber \\&X=\sum _m |\theta ;j\;m+1\rangle \langle \theta ;j\;m|. \end{aligned}$$

(7.11)

We can show that the X, Z obey Proposition 4.2, with the following correspondence:

$$\begin{aligned} |X;m\rangle \;\rightarrow \; |\theta ;j\;m\rangle ;\;\;\;|P;m\rangle \;\rightarrow \; |J;j\;m\rangle . \end{aligned}$$

(7.12)

Also the analogue of Eq. (4.18), is here

$$\begin{aligned} X=\exp \left[ -i \frac{2\pi }{d}J_z \right] ;\;\;\; Z=\exp \left[ i \frac{2\pi }{d}\theta _z \right] . \end{aligned}$$

(7.13)

Therefore all the formalism in Chap. 4, can be used here also.

7.1.1 The Schwinger Representation

We consider a two-mode harmonic oscillator with Hilbert space ${\mathscr {H}}_1\times {\mathscr {H}}_2$. Let $a_1^{\dagger }, a_1$ and $a_2^{\dagger }, a_2$ be the creation and annihilation operators for the two modes, and $|N_1,N_2\rangle$ the number eigenstates:

$$\begin{aligned} a_1^{\dagger }a_1 |N_1,N_2\rangle =N_1|N_1,N_2\rangle ;\;\;\;a_2^{\dagger }a_2 |N_1,N_2\rangle =N_2|N_1,N_2\rangle . \end{aligned}$$

(7.14)

In the Schwinger representation of SU(2) [2], the angular momentum operators are expressed as

$$\begin{aligned} J_+= & {} a_1^{\dagger }a_2;\;\;\;\;\;\;\;J_-=a_1a_2^{\dagger };\;\;\;\;\;\;\; J_z=\frac{1}{2}\left( a_1^{\dagger }a_1-a_2^{\dagger }a_2\right) . \end{aligned}$$

(7.15)

The Casimir operator is

$$\begin{aligned}&J^2=n_s\left( n_s+1\right) ;\;\;\; n_s=\frac{1}{2}(a_1^{\dagger } a_1+a_2^{\dagger } a_2)\nonumber \\&[n_s,J_+]=[n_s,J_-]=[n_s,J_z]=0. \end{aligned}$$

(7.16)

The number eigenstates play the role of the angular momentum states, as follows:

$$\begin{aligned} |N_1,N_2\rangle \;\leftrightarrow \;|J;j\;m\rangle ;\;\;\; j=\frac{1}{2} (N_1+N_2);\;\;\;m=\frac{1}{2} (N_1-N_2) \end{aligned}$$

(7.17)

With this correspondence, we can easily show that the standard angular momentum relations in Eq. (7.3) hold. Here the $(2j+1)$-dimensional Hilbert space $H(2j+1)$, contains superpositions of the states

$$\begin{aligned} H(2j+1)=\{|N,2j+1-N\rangle \;|\;N=0,...,2j+1\} \end{aligned}$$

(7.18)

Then the Hilbert space ${\mathscr {H}}_1\times {\mathscr {H}}_2$ can be written as the direct sum:

$$\begin{aligned}&{\mathscr {H}}_1\times {\mathscr {H}}_2={\mathscr {H}}_B\oplus {\mathscr {H}}_F\nonumber \\&{\mathscr {H}}_B=\bigoplus _j {H}(2j+1);\;\;\;j=0,1,2,...\nonumber \\&{\mathscr {H}}_F=\bigoplus _j {H}(2j+1);\;\;\;j=\frac{1}{2} ,\frac{3}{2},... \end{aligned}$$

(7.19)

${\mathscr {H}}_B$ is the Bose Hilbert space (the direct sum of spaces with integer j), and ${\mathscr {H}}_F$ is the Fermi Hilbert space (the direct sum of spaces with half-integer j). ${\mathscr {H}}_B$ is spanned by number eigenstates with an odd total number of photons in the two modes. ${\mathscr {H}}_F$ is spanned by number eigenstates with an even total number of photons in the two modes.

As an application of this we consider a two-mode system described by the following Hamiltonian, which is used for the description of frequency converters in Quantum Optics:

$$\begin{aligned} {\mathfrak H}= & {} E _1a_1^{\dagger }a_1+E _2a_2^{\dagger }a_2+\lambda a_1^{\dagger }a_2+\lambda ^*a_1a_2^{\dagger }\nonumber \\= & {} (E_1+E_2) n_s+(E_1-E_2) J_z+\lambda J_++\lambda ^*J_- \end{aligned}$$

(7.20)

Systems with this Hamiltonian can be studied with the above formalism.

7.1.2 Angle States and Angular Momentum States in ${\mathscr {H}}_B$

Let $\alpha , \beta$ be spherical coordinates describing the points on a two-dimensional sphere $S_2$, with radius one. We define the following angular momentum states in ${\mathscr {H}}_B$:

$$\begin{aligned} |J;\alpha , \beta \rangle =\sum _{j,m}Y_{jm}^*(\alpha , \beta )|J;j\ m\rangle ;\;\;\; 0\le \alpha \le \pi ;\;\;\;0\le \beta <2\pi \end{aligned}$$

(7.21)

$Y_{jm}(\alpha , \beta )$ are the usual spherical harmonics. We also introduce the ‘dual spherical harmonics’ [3] which are related to the usual spherical harmonics through a finite Fourier transform:

$$\begin{aligned} X_{jn}(\alpha , \beta )=\frac{1}{\sqrt{2j+1}}\sum _m Y_{jm}(\alpha , \beta )\omega (nm) \end{aligned}$$

(7.22)

We define angle states in ${\mathscr {H}}_B$, as:

$$\begin{aligned} |\theta ;\alpha , \beta \rangle =\sum _{j,m}Y_{jm}^*(\alpha , \beta )|\theta ;j\ m\rangle = \sum _{j,m}X_{jm}^*(\alpha , \beta )|J;j\ m\rangle . \end{aligned}$$

(7.23)

The states $|\theta ;\alpha , \beta \rangle$ and also the states $|J;\alpha , \beta \rangle$ form orthonormal bases in ${\mathscr {H}}_B$.

$$\begin{aligned} \int |\theta ;\alpha , \beta \rangle \langle \theta ;\alpha , \beta |d\cos \alpha d\beta =\int |J ;\alpha , \beta \rangle \langle J ;\alpha , \beta |d\cos \alpha d\beta =\mathbf{1}. \end{aligned}$$

(7.24)

An arbitrary state $|f\rangle$ in ${\mathscr {H}}_B$, can be represented with the functions

$$\begin{aligned} f_J(\alpha , \beta )=\langle J;\alpha , \beta |f\rangle ;\;\;\;f_{\theta }(\alpha , \beta )=\langle \theta ;\alpha , \beta |f\rangle . \end{aligned}$$

(7.25)

7.1.3 Area Preserving Diffeomorphisms on a Sphere

Above we discussed angle and angular momentum operators based on the SU(2) group. The SU(2) is locally isomorphic to SO(3) which describes rotations of a solid sphere.

A more general group is the $SDiff(S_2)$ of area preserving diffeomorphisms on a sphere $S_2$. They describe general transformations of a perfect liquid on a sphere. Since rotations of a solid sphere are a very special case of these transformations, we expect that this more general formalism will lead to the standard angular momentum operators plus many other operators. Such groups for a sphere and also other surfaces, have been studied in the context of string theory [4,5,6,7,8,9,10].

We consider the following transformations from $(\cos \alpha , \beta )$ to

$$\begin{aligned}&\cos \gamma ={\mathscr {A}}(\cos \alpha , \beta );\;\;\;\delta ={\mathscr {B}}(\cos \alpha , \beta )\nonumber \\&\frac{\partial (\cos \gamma , \delta )}{\partial (\cos \alpha , \beta )}=\frac{\partial \cos \gamma }{\partial \cos \alpha }\frac{\partial \delta }{\partial \beta }-\frac{\partial \delta }{\partial \cos \alpha }\frac{\partial \cos \gamma }{\partial \beta }=1. \end{aligned}$$

(7.26)

Since the Jacobian is equal to one, the area is preserved under these transformations.

An infinitesimal version of these transformations is

$$\begin{aligned}&\cos \gamma =\cos \alpha +{A}(\cos \alpha , \beta )\varepsilon ;\;\;\;\delta =\beta +{B}(\cos \alpha , \beta ) \varepsilon \nonumber \\&\frac{\partial A}{\partial \cos \alpha }+\frac{\partial B}{\partial \beta }=0. \end{aligned}$$

(7.27)

$\varepsilon$ is an infinitesimal parameter. The last equation comes from the fact that the Jacobian is equal to one, and for topologically trivial manifolds like a sphere, implies the existence of a function $g(\alpha , \beta )$ such that

$$\begin{aligned} A=-\frac{\partial g}{\partial \beta };\;\;\;B=\frac{\partial g}{\partial \cos \alpha }. \end{aligned}$$

(7.28)

We consider two bases $|J;\alpha , \beta \rangle$ and $|J;\gamma , \delta \rangle$, where $\gamma , \delta$ are related to $\alpha , \beta$ through the infinitesimal transformations in Eq. (7.27). We represent an arbitrary state $|f\rangle$ in ${\mathscr {H}}_B$, with the functions

$$\begin{aligned} f(\alpha , \beta )=\langle J;\alpha , \beta |f\rangle ;\;\;\;f(\gamma , \delta )=\langle J;\gamma , \delta |f\rangle . \end{aligned}$$

(7.29)

Then

$$\begin{aligned} \frac{f(\gamma , \delta )-f(\alpha , \beta )}{\varepsilon }\approx \frac{\partial (g(\alpha , \beta ),f(\alpha , \beta ))}{\partial (\cos \alpha , \beta )}. \end{aligned}$$

(7.30)

This leads to the following definition.

Definition 7.1

The operator $J_g$ acts on $f_J(\alpha , \beta )$, as follows:

$$\begin{aligned} J_gf(\alpha , \beta )=\langle J;\alpha , \beta |J_g|f\rangle = \frac{\partial (g(\alpha , \beta ),f(\alpha , \beta ))}{\partial (\cos \alpha , \beta )}. \end{aligned}$$

(7.31)

In analogous way we define the operators $\theta _g$. The following proposition describes some properties of $J_g$.

Proposition 7.1

• (1) The commutator of $J_g$ and $J_h$, is given in terms of the Poisson bracket of g, h (with respect to $\cos \alpha , \beta$), by

  $$\begin{aligned}{}[J_{g},J_{h}]=J_{\{g,h\}};\;\;\;\{g,h\}=\frac{\partial g}{\partial \cos \alpha }\frac{\partial h}{\partial \beta }-\frac{\partial h}{\partial \beta }\frac{\partial g}{\partial \cos \alpha }. \end{aligned}$$

  (7.32)

• (2) $J_g$ acts on the sum of two functions as follows:

  $$\begin{aligned} J_{g}[\mu _1f_1(\alpha , \beta )+\mu _2f_2(\alpha , \beta )]=\mu _1J_{g}f_1(\alpha , \beta )+\mu _2J_{g}f_2(\alpha , \beta ). \end{aligned}$$

  (7.33)

• (3) $J_g$ acts on the product of two functions as follows:

  $$\begin{aligned} J_{g}[f_1(\alpha , \beta )f_2(\alpha , \beta )]=f_1(\alpha , \beta )J_{g}f_2(\alpha , \beta )+f_2(\alpha , \beta )J_{g}f_1(\alpha , \beta ). \end{aligned}$$

  (7.34)

• (4) The exponential of $J_g$ acts on the sum of two functions as follows:

  $$\begin{aligned} \exp (\lambda J_{g})[\mu _1f_1(\alpha , \beta )+\mu _2f_2(\alpha , \beta )]= & {} \mu _1\exp (\lambda J_{g})f_1(\alpha , \beta )\nonumber \\ {}+ & {} \mu _2\exp (\lambda J_{g})f_2(\alpha , \beta ). \end{aligned}$$

  (7.35)

• (5) The exponential of $J_g$ acts on the product of two functions as follows:

  $$\begin{aligned} \exp (\lambda J_{g})[f_1(\alpha , \beta )f_2(\alpha , \beta )]=[\exp (\lambda J_{g})f_1(\alpha , \beta )][\exp (\lambda J_{g})f_2(\alpha , \beta )]. \end{aligned}$$

  (7.36)

Proof

For the proof we refer to Ref. [11].

We expand the function $g(\alpha , \beta )$ in terms of spherical harmonics, as

$$\begin{aligned} g(\alpha , \beta )=\sum _{j,m}g_{jm}Y_{jm}(\alpha , \beta ). \end{aligned}$$

(7.37)

Then

$$\begin{aligned} J_g=\sum _{j,m}g_{jm}J_{jm};\;\;\; J_{jm}f(\alpha , \beta ) =\frac{\partial (Y_{jm}(\alpha , \beta ),f(\alpha , \beta ) )}{\partial (\cos \alpha , \beta )}. \end{aligned}$$

(7.38)

In particular

$$\begin{aligned} J_{jm}Y_{\ell n}(\alpha , \beta ) =\langle J;\alpha , \beta |J_{jm}|J; \ell \;n\rangle = \frac{\partial (Y_{jm}(\alpha , \beta ),Y_{\ell n}(\alpha , \beta ) )}{\partial (\cos \alpha , \beta )}. \end{aligned}$$

(7.39)

The Poisson bracket of $Y_{j_1m_1}(\alpha , \beta )$ and $Y_{j_2m_2}(\alpha , \beta )$, is given by

$$\begin{aligned} \{Y_{j_1 m_1},Y_{j_2 m_2}\}=\sum _{\ell , n}\tau (j_1 ,m_1; j_2, m_2|\ell , n)Y_{\ell n}. \end{aligned}$$

(7.40)

The structure constants $\tau (j_1 ,m_1; j_2, m_2|\ell , n)$ are given in [5]. Consequently

$$\begin{aligned}{}[J_{j_1 m_1},J_{j_2 m_2}]=\sum _{\ell , n}\tau (j_1 ,m_1; j_2, m_2|\ell , n)J_{\ell n}. \end{aligned}$$

(7.41)

The $J_{jm}$ are generalizations of the angular momentum operators. The $J_{1m}$ are simply the standard angular momentum operators $J_+, J_z, J_-$ (with a different normalization).

This formalism has been used in string theory, but it might also be useful in the general area of quantum optics and quantum information, because it generalizes the angular momentum formalism.

7.2 Interferometry in Multimode Systems

In this section we use the formalism of finite quantum systems, in the context of interferometry that involves d harmonic oscillators. The overall Hilbert space in this problem is $H_\mathrm{osc}\otimes ...\otimes H_\mathrm{osc}$, where $H_\mathrm{osc}$ is the infinite-dimensional Hilbert space of the harmonic oscillator. The mode index is the ‘position’ in this problem, and it takes values in ${\mathbb Z}(d)$. Through a finite Fourier transform of the d modes, we get a dual mode index which plays the role of ‘momentum’, and which also takes values in ${\mathbb Z}(d)$. So in this context, the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ is a ‘mode phase space’.

The formalism has important applications in metrology, because it leads to resolutions below the standard quantum limit [12]. It has been studied extensively both with photons and also with Bose-Einstein condensates. Here we present briefly the link between this area, and the formalism of finite quantum systems studied in Chap. 4. We refer to the literature for more details, and for practical applications of these devices [13,14,15,16,17,18,19,20,21,22,23].

We consider a system comprised of d harmonic oscillators. The creation and annihilation operators corresponding to the m-th mode, are:

$$\begin{aligned}&a_m^{\dagger }=\mathbf{1}\otimes ...\otimes a^{\dagger }\otimes ...\otimes \mathbf{1};\;\;\;a_m=\mathbf{1}\otimes ...\otimes a\otimes ...\otimes \mathbf{1} \nonumber \\\;&[a_m, a_n^{\dagger }] =\delta (m,n);\;\;\;m,n\in {\mathbb Z}(d). \end{aligned}$$

(7.42)

Let $\varLambda$ be a $d\times d$ Hermitian matrix, and U the unitary operator

$$\begin{aligned} U=\exp \left[ i\sum _{m,n}a_m^{\dagger }\varLambda _{mn}a_n\right] . \end{aligned}$$

(7.43)

It is known (e.g. [24]) that

$$\begin{aligned}&b_m=Ua_mU^{\dagger }=\sum _n V_{mn}a_n;\;\;\;b_m^{\dagger }=Ua_m^{\dagger }U^{\dagger }=\sum _n V_{mn}^*a_n^{\dagger }\nonumber \\&V=\exp (-i\varLambda );\;\;\;\;\;\;VV^{\dagger }=\mathbf{1}. \end{aligned}$$

(7.44)

The vacuum state remains invariant under these transformations. Also the total average number of photons in a state remains invariant under the U transformations:

$$\begin{aligned} U|0,...,0\rangle =|0,...,0\rangle ;\;\;\; \sum _mb_m^{\dagger }b_m =\sum _ma_m^{\dagger }a_m. \end{aligned}$$

(7.45)

7.2.1 Fourier Interferometry and Applications to Metrology

A special case of the formalism above, is the Fourier transform of the modes:

$$\begin{aligned} U_F=\exp \left[ i\sum _{m,n}a_m^{\dagger }\varLambda _{mn}a_n\right] ;\;\;\;\varLambda =i\ln F;\;\;\;;\;\;\;(U_F)^4=\mathbf{1}, \end{aligned}$$

(7.46)

where F is the $d\times d$ Fourier matrix, in Eq. (4.2). Then

$$\begin{aligned}&b_m= U_Fa_mU_F^{\dagger }=\frac{1}{\sqrt{d}}\sum _n \omega (mn) a_n\nonumber \\&b_m^{\dagger }= U_Fa_m^{\dagger }U_F^{\dagger }=\frac{1}{\sqrt{d}}\sum _n \omega (-mn) a_n^{\dagger } \end{aligned}$$

(7.47)

The dual mode index related to $b_m, b_m^{\dagger }$ plays the role of momentum. So in the present context position and momentum is the mode index related to the $a_m, a_m^{\dagger }$ and $b_m, b_m^{\dagger }$, correspondingly. Experiments that use beam splitters to implement these transforms have been discussed in [14]. The use of the factorization in Sect. 4.9 reduces the number of beam splitters, as discussed in [23].

There are various applications of these devices. As an example, we consider the case where the input is a number state with N photons in the m-th mode, and vacuum in the other modes:

$$\begin{aligned} |s\rangle =|0,...,0,N,0,...0\rangle \end{aligned}$$

(7.48)

Then in the large d limit, the phase uncertainty in the m-th output is [20]

$$\begin{aligned} \varDelta \theta _m\sim \frac{\sqrt{d}}{N}. \end{aligned}$$

(7.49)

This is below the standard quantum limit and can have applications in metrology.

It is seen that the formalism of finite quantum systems presented in this monograph, can also be used for the study of interferometry in multimode systems (with a finite number of modes).

7.2.2 Other Types of Interferometry

Here we consider other special cases of the general operators U in Eq. (7.43). The first one, is:

$$\begin{aligned} U_X=\exp \left[ i\sum _{m,n}a_m^{\dagger }\varLambda _{mn}a_n\right] ;\;\;\;\varLambda =i\ln X;\;\;\;(U_X)^d=\mathbf{1}. \end{aligned}$$

(7.50)

where X is the $d\times d$ matrix, in Eq. (4.19). Then

$$\begin{aligned}&b_m= U_Xa_mU_X^{\dagger }=a_{m+1}\nonumber \\&b_m^{\dagger }= U_Xa_m^{\dagger }U_X^{\dagger }=a_{m+1}^{\dagger } \end{aligned}$$

(7.51)

This shifts the modes by one place (and the last mode becomes first). In other words, it shifts the modes in the ‘mode-position’ direction, in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ mode phase space.

Another special case is

$$\begin{aligned} U_Z=\exp \left[ i\sum _{m,n}a_m^{\dagger }\varLambda _{mn}a_n\right] ;\;\;\;\varLambda =i\ln Z;\;\;\;(U_Z)^d=\mathbf{1}. \end{aligned}$$

(7.52)

where Z is the $d\times d$ matrix, in Eq. (4.19). Then

$$\begin{aligned}&b_m= U_Za_mU_Z^{\dagger }=a_{m}\omega (m)\nonumber \\&b_m^{\dagger }= U_Za_m^{\dagger }U_Z^{\dagger }=a_{m}\omega (-m). \end{aligned}$$

(7.53)

This multiplies each mode $a_{m}$ by $\omega (m)$, i.e., it shifts the modes in the ‘mode-momentum’ direction, in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ mode phase space.

We next divide the Hilbert space $H_\mathrm{osc}\otimes ...\otimes H_\mathrm{osc}$, into d ‘sectors’:

$$\begin{aligned}&H_\mathrm{osc}\otimes ...\otimes H_\mathrm{osc}=\bigoplus _{n=0}^{d-1} {\mathscr {H}}_n\nonumber \\&{\mathscr {H}}_n=\mathrm{span}\{|N_0,...,N_{d-1}\rangle \;|\;N_0+...+N_{d-1}=n(\mathrm{mod}\;d)\};\;\;\;n\in {\mathbb Z}(d). \end{aligned}$$

(7.54)

The sector ${\mathscr {H}}_n$ is spanned by number eigenstates, with a total number of photons equal to $n(\mathrm{mod}\;d)$. We call $\pi _n$ the projector to ${\mathscr {H}}_n$. It can be shown that $\pi _n$ commutes with both $U_X, U_Z$, and we define the:

$$\begin{aligned} U_{Xn}=U_X\pi _n;\;\;\;U_X=\sum _{n=0}^{d-1}U_{Xn};\;\;\;[U_X,\pi _n]=0\nonumber \\ U_{Zn}=U_Z\pi _n;\;\;\;U_Z=\sum _{n=0}^{d-1}U_{Zn};\;\;\;[U_Z,\pi _n]=0. \end{aligned}$$

(7.55)

Then the $U_{Xn}, U_{Zn}$ form a Heisenberg-Weyl group within ${\mathscr {H}}_n$, which has been studied in [21]:

$$\begin{aligned} U_{Xn}^\alpha U_{Zn}^\beta =U_{Zn}^\beta U_{Xn}^\alpha \omega (-n\alpha \beta );\;\;\;\alpha , \beta \in {\mathbb Z}(d). \end{aligned}$$

(7.56)

So apart from the Fourier interferometry devices, there are many other devices which can have various applications in Quantum Optics and Quantum Information.

7.3 Orbital Angular Momentum States

The paraxial wave equation in cylindrical coordinates, leads to the Laguerre-Gauss modes

$$\begin{aligned} u_{nm}(r,\phi )\sim r^{|m|}L_n^{|m|}\left( \frac{2r^2}{w^2}\right) \exp \left( -\frac{r^2}{w^2}\right) \exp (-im\phi ) \end{aligned}$$

(7.57)

Here $L_n^{|m|}$ are Laguerre polynomials, and n, m are the radial quantum number, and the orbital angular momentum quantum number, correspondingly. The physical meaning of the radial quantum number n is discussed in [25]. w describes the width of the beam. Photons in these beams have angular momentum m.

These solutions describe the orbital angular momentum states or twisted light [26,27,28,29], and they are an important tool in modern quantum optical technologies. They are created experimentally by imposing $\exp (im \phi )$ phase structure on a laser beam. There is currently much work on the generation of orbital angular momentum states and their applications (e.g., [30,31,32,33,34]). They are robust in noisy environments (e.g., [35]), and therefore important for quantum communications.

In our context, they are important because they provide an experimental implementation of a quantum system with a finite dimensional Hilbert space. The whole formalism of this monograph can be used in the context of orbital angular momentum states. Mutually unbiased bases with orbital angular momentum states have been studied in [36, 37], and entanglement in [38]. Applications to quantum cryptography have been discussed in [39].

7.4 Other Applications

We discussed above applications in the area of quantum optics and quantum information. Applications in other areas include quantum maps [40,41,42,43,44,45], two-dimensional electron system in a uniform magnetic field and the magnetic translation group [46,47,48,49,50], and the quantum Hall effect [51, 52].

All these ideas are also used in the context of Signal Processing, where the dual variables position and momentum become time and frequency [54, 55]. For example, the factorization discussed in Sect. 4.9, is inspired by Ref. [56] on fast Fourier transforms, in the context of Signal Processing.

Work related to the formalism of finite quantum systems, in the context of Applied Mathematics is summarized in [57].