**5.3** *SU*ð Þ **1, 1 quantum systems and a two-level atom**

A two-level atom is considered which interacts with two coupled quantum systems that can be represented in terms of a *su*ð Þ 1, 1 Lie algebra. When for example mixed four-waves are injected into a cavity containing a single two level-atom an interaction occurs between the four waves and the atom that is electromagnetic radiation and matter. The Hamiltonian has the form

$$\frac{1}{\hbar}H = \sum\_{i=1}^{2} \alpha\_i \left(a\_i^\dagger a\_i + \frac{1}{2}\right) + \frac{1}{2}\alpha\_0 \sigma\_\mp + \lambda \left(a\_1^2 a\_2^2 \sigma\_+ + a\_1^{\dagger 2} a\_2^{\dagger 2} \sigma\_-\right). \tag{80}$$

It is similar to **5.1**, so we sketch the physical situation. The *σ*�, *σ<sup>z</sup>* are raising lowering and inversion operators which satisfy the commutation relations ½ �¼ *<sup>σ</sup>z*, *<sup>σ</sup>*� <sup>2</sup>*σ*�, ½ �¼ *<sup>σ</sup>*þ, *<sup>σ</sup>*� *<sup>σ</sup>z*, whereas the *<sup>a</sup>*† *<sup>i</sup>* , *ai* are basic creation and annihilation operators with *a <sup>j</sup>*, *a*† *j* h i <sup>¼</sup> *<sup>δ</sup>ij*. The interaction term in (80) can be thought of as the interaction between two different second harmonic modes. This can be cast in terms of three *su*ð Þ 1, 1 Lie algebra generators *K*þ, *K*� and *Kz* which satisfy the commutation relations,

$$[K\_x, K\_\pm] = \pm K\_\pm, \qquad [K\_-, K\_+] = 2K\_x. \tag{81}$$

The corresponding Casimir *K* which has eigenvalue *k k*ð Þ � 1 given by

$$K^2 = K\_x^2 - \frac{1}{2}(K\_+K\_- + K\_-K\_+). \tag{82}$$

Given that this is the Lie algebra, it can be said that the Fock space is spanned by the set of vectors *<sup>V</sup><sup>F</sup>* : f g <sup>j</sup>*m*; *<sup>k</sup>*<sup>i</sup> and the operators in (81) act on these states as follows,

$$\begin{aligned} K\_{\pi}|m;k\rangle &= (m+k)|m;k\rangle, & K^{2}|m;k\rangle &= k(k-1)|m;k\rangle, \\ K\_{+}|m;k\rangle &= \sqrt{(m+1)(m+2k)}|m+1;k\rangle, & K\_{-}|m;k\rangle &= \sqrt{m(m+2k-1)}|m-1;k\rangle. \end{aligned} \tag{83}$$

It is the case that *K*�∣0; *m*i ¼ 0 so this is the lowest level state. The *su*ð Þ 1, 1 Lie algebra can be realized in terms of boson annihilation and creation operators and it is isomorphic to the Lie algebra of the non-compact *SU*ð Þ 1, 1 group. For the Hamiltonian (80) define operators *K*ð Þ*<sup>i</sup>* � and *<sup>K</sup>*ð Þ*<sup>i</sup> <sup>z</sup>* as

*Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos DOI: http://dx.doi.org/10.5772/intechopen.94491*

$$K\_+^{(i)} = \frac{1}{2}a\_i^{\dagger^2}, \quad K\_-^{(i)} = \frac{1}{2}a\_i^2, \quad K\_x^{(i)} = \frac{1}{2}\left(a\_i^\dagger a\_i + \frac{1}{2}\right), \qquad i = 1, 2,\tag{84}$$

where the Bargmann index *k* is either 1*=*4 for the even parity states while 3*=*4 applies to the odd-parity states.

Using these operators, (80) is written in terms of *su*ð Þ2 and *su*ð Þ 1, 1 operators such that it has the form,

$$\frac{1}{\hbar}H = \sum\_{i=1}^{2} \eta\_i K\_x^{(i)} + \frac{\alpha}{2} \sigma\_x + \lambda \left( K\_+^{(1)} K\_+^{(2)} \sigma\_- + K\_-^{(1)} K\_-^{(2)} \sigma\_+ \right). \tag{85}$$

The Heisenberg equations of motion obtained from (85) gives

$$\begin{split} i\frac{d}{dt}K\_x^{(1)} &= \lambda \Big( K\_+^{(1)}K\_+^{(2)}\sigma\_- - K\_-^{(1)}K\_-^{(2)}\sigma\_+ \Big), \quad i\frac{d}{dt}K\_x^{(2)} = \lambda \Big( K\_+^{(1)}K\_+^{(1)}\sigma\_- - K\_-^{(1)}K\_-^{(2)}\sigma\_+ \Big), \\ i\frac{d}{dt}\sigma\_x &= \lambda \Big( K\_-^{(1)}K\_-^{(2)}\sigma\_+ - K\_+^{(1)}K\_+^{(1)}\sigma\_- \Big). \end{split} \tag{86}$$

The following two operators *N*<sup>1</sup> and *N*<sup>2</sup> are constants of the motion

$$N\_1 = K\_x^{(1)} + \sigma\_x, \qquad N\_2 = K\_x^{(2)} + \sigma\_x. \tag{87}$$

Hamiltonian (80) can now be put in the equivalent form

$$\frac{1}{\hbar}H = N + C + I,\tag{88}$$

where *I* is the identity operator and *N* and *C* are the operators

$$N = \sum\_{i=1}^{2} \eta\_i N\_i, \qquad \mathbf{C} = \Delta \sigma\_x + \lambda \left( K\_+^{(1)} K\_+^{(2)} \sigma\_- + K\_-^{(1)} K\_-^{(2)} \sigma\_+ \right). \tag{89}$$

The constant Δ is the detuning parameter defined as

$$
\Delta = \frac{\alpha}{2} - \eta\_1 - \eta\_2. \tag{90}
$$

As *N* and *C* commute, each commutes with the Hamiltonian *H* so *N* and *C* are constants of the motion. The time evolution operator *U t*ð Þ is given by

$$U(t) = \exp\left(-i\frac{H}{\hbar}t\right) \cdot \exp\left(-iNt\right) \cdot \exp\left(iCt\right). \tag{91}$$

In the space of the two-level eigenstates

$$e^{-iNt} = \begin{pmatrix} e^{-iW\_1t} & \mathbf{0} \\ \mathbf{0} & e^{-iW\_2t} \end{pmatrix}. \tag{92}$$

The operators *Wi*, *<sup>i</sup>* <sup>¼</sup> 1, 2 are defined by *<sup>W</sup>*<sup>1</sup> <sup>¼</sup> *<sup>η</sup>*1*K*ð Þ<sup>1</sup> *<sup>z</sup>* <sup>þ</sup> *<sup>η</sup>*2*K*ð Þ<sup>2</sup> *<sup>z</sup>* þ 1 and *W*<sup>2</sup> ¼ *η*1*K*ð Þ<sup>1</sup> *<sup>z</sup>* <sup>þ</sup> *<sup>η</sup>*2*K*ð Þ<sup>2</sup> *<sup>z</sup>* � 1. The second exponential on the right of (91) takes the form,

#### *A Collection of Papers on Chaos Theory and Its Applications*

$$\exp\left(-i\mathbf{C}t\right) = \begin{pmatrix} \cos\tau\_t t - \frac{i\Delta}{\tau} \sin\tau\_1 t & -i\lambda \frac{\sin\tau\_1 t}{\tau\_1} K\_-^{(1)} K\_-^{(2)} \\\\ -i\lambda K\_+^{(1)} K\_+^{(2)} \frac{\sin\tau\_1 t}{\tau\_1} & \cos\tau\_2 t - \frac{i\Delta}{\tau\_2} \sin\tau\_2 t \end{pmatrix} \tag{93}$$

where *τ*<sup>2</sup> *<sup>j</sup>* <sup>¼</sup> <sup>Δ</sup><sup>2</sup> <sup>þ</sup> *<sup>ν</sup> <sup>j</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2 and

$$
\pi\_1 = \lambda^2 K\_-^{(1)} K\_+^{(1)} K\_-^{(2)} K\_+^{(2)}, \qquad \pi\_2 = \lambda^2 K\_+^{(1)} K\_-^{(1)} K\_+^{(2)} K\_-^{(2)} \tag{94}
$$

The coherent atomic state ∣*ϑ*, *φ*i is considered to be the initial state that contains both excited and ground states and has the structure,

$$|\mathfrak{G},\mathfrak{g}\rangle = \cos\left(\frac{\mathfrak{g}}{2}\right)|e\rangle + \sin\left(\frac{\mathfrak{g}}{2}\right)e^{-i\mathfrak{g}}|\mathfrak{g}\rangle. \tag{95}$$

where *ϑ* is the coherence angle, *φ* the relative phase of the two atomic states. The excited state is attained by taking *ϑ* ! 0, while the ground state of the atom is derived from the limit *ϑ* ! *π*. The initial state of the system that describes the two *su*ð Þ 1, 1 Lie algebras is assumed to be prepared in the pair correlated state ∣*ξ*, *q*i defined by

$$K\_-^{(1)}K\_-^{(2)}|\xi,q\rangle = \xi|\xi,q\rangle,\qquad \left(K\_x^{(1)} - K\_x^{(1)}\right)|\xi,q\rangle = q|\xi,q\rangle.\tag{96}$$

Since the operators *K*ð Þ<sup>1</sup> � *<sup>K</sup>*ð Þ<sup>2</sup> � and *<sup>K</sup>*ð Þ<sup>1</sup> *<sup>z</sup>* � *<sup>K</sup>*ð Þ<sup>2</sup> *z* � � commute, <sup>∣</sup>*ξ*, *<sup>q</sup>*<sup>i</sup> can be introduced which is simultaneously an eigenstate of both operators,

$$|\xi, q\rangle = \sum\_{n=0}^{\infty} \mathcal{C}\_n |q + n + k\_2 - k\_1; k\_1; n, k\_2\rangle. \tag{97}$$

Then applying *K*ð Þ<sup>2</sup> � and then *<sup>K</sup>*ð Þ<sup>1</sup> � we obtain,

$$\begin{aligned} &K\_{-}^{(1)}K\_{-}^{(2)}|\xi,q\rangle\\ &=\sum\_{n=0}^{\infty}\mathcal{C}\_{n}\sqrt{n(n+2k-1)(q+n+k\_{2}-k\_{1})(q+n+k\_{2}-k\_{1}+2k\_{1}-1)}q+n+k\_{2}-k\_{1}-1,k\_{1};n-1,k\_{2}\rangle,\\ &=\sum\_{n=0}^{\infty}\mathcal{C}\_{n+1}\sqrt{(n+1)(n+2k)(q+n+k\_{2}-k\_{1}+1)((q+n+k\_{2}-k\_{1}+2k\_{1}))}q+n+k\_{2}-k\_{1},k\_{1};n,k\_{2}\rangle. \end{aligned} \tag{98}$$

This calculation implies that the normalization constant *Cn* can be obtained by solving

$$\sqrt{(n+1)(n+2k)(q+n+k\_2-k\_1+1)(n+q+k\_1+k\_2)}C\_{n+1} = \xi C\_0. \tag{99}$$

The new state is of the form,

$$|\xi, q\rangle = N\_q \sum\_{n=0}^{\infty} \mathcal{C}\_n |q + n + k\_2 - k\_1, k; n, k\_2\rangle, \qquad N\_q^{-2} = \sum\_{n=0}^{\infty} |\mathcal{C}\_n|^2. \tag{100}$$

If it is assumed that at *t* ¼ 0 the wave function of the system is ∣*ψ*ð Þi ¼ 0 ∣*ϑ*, *φ*i ⊗ ∣*ξ*, *q*i, using (91) on ∣*ψ*ð Þi 0 , the state can be calculated for *t*>0 can be determined

*Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos DOI: http://dx.doi.org/10.5772/intechopen.94491*

$$|\psi(t)\rangle = e^{-iW\_{\text{f}}t} \left[ (\cos\tau\_{\text{f}}t) - i\frac{\Delta}{\tau\_1}\sin\tau\_{\text{f}}t \right) \cos\left(\frac{\theta}{2}\right) - i\frac{\lambda}{\tau\_1}\sin\left(\tau\_{\text{f}}t\right)K\_{-}^{(1)}K\_{-}^{(2)}e^{-i\varphi}\sin\frac{\theta}{2} \right]|e\rangle \otimes |\xi, q\rangle. \tag{10.10}$$

$$+ e^{iW\_{\text{f}}t} \left[ \left(\cos\tau\_{2}t + i\frac{\Delta}{\tau\_2}\sin\tau\_{2}t\right)e^{-i\varphi}\sin\frac{\theta}{2} - i\frac{\lambda}{\tau\_2}\sin\tau\_2 tK\_{+}^{(1)}K\_{+}^{(2)}\cos\frac{\theta}{2} \right]|\xi\rangle \otimes |\xi, q\rangle. \tag{10.11}$$

The reduced density matrix is constructed from this

$$\rho\_f(t) = |\mathbf{T} \mathbf{r}\_{atom}| \psi(t) / \langle \psi(t) |. \tag{102}$$
