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Iran Journal of Computer Science

, Volume 1, Issue 2, pp 129–132 | Cite as

QuTiP-package applications to five-level atom with one mode

  • Ahmed Salah
Short Communication
  • 355 Downloads

Abstract

In this poster, we apply the Quantum Toolbox in Python (QuTiP) to study the interaction between a five-level atom and a single-mode cavity field. The non-classical statistical aspects such as the Mandel Q parameter, squeezing parameter, and linear entropy are investigated.

Keywords

QuTiP Five-level atom Non-classical aspects 

1 Introduction

The high-performance computers need the high-performance software. Until recently, Fortran and C++ are still a good software to develop high-performance programs but complicated codes. Recently, the high-level programming language Python has emerged as an alternative to compiled languages. Since that, many programs have been built to solve many mathematical problems. The quantum optics and information are the one of these problems which can be solved by Python codes. For this reason, Johansson et al. [1] have presented an object-oriented open-source framework for solving the dynamics of open quantum systems written in Python, which is called Quantum Toolbox in Python (QuTiP). It is open-source software for calculations and numerical simulations of dynamics of open quantum systems. Lately, we [2] study a double \(\Lambda \)-type five-level atom interacting with a single-mode electromagnetic cavity field in the (off) non-resonate case. In this poster, we present a very simple code by QuTiP to investigate the non-classical aspects for the double \(\Lambda \) five-level atom interacting with a single-mode cavity field. The aim of this poster is describe an effective Python code using QuTiP to study the interaction between a five-level atom and a single-mode cavity field. For more information about QuTiP, see the project web page: http://qutip.org/.

To use QuTiP in a Python program, first include the QuTiP module:

2 Setup the Hamiltonian model, operators, and initial state

We consider here a model for the interaction between a five-level atom with a single mode with frequency of the electromagnetic field in an optical cavity, the total Hamiltonian of the model can be written as
$$\begin{aligned} \hat{H}=\hat{H}_0+\hat{H}_I, \end{aligned}$$
(1)
where the free part \(\hat{H}_0\) of Hamiltonian is given by
$$\begin{aligned} \hat{{H}}_0= \Delta _{1} \hat{{\sigma }}_{22} +\Delta _2\hat{{\sigma }}_{33}+(\tilde{\Delta }_3-\Delta _2)\hat{{\sigma }}_{44} +\Delta _{4}\hat{{\sigma }}_{55} \end{aligned}$$
(2)
where \(\hat{{\sigma }}_{ij} =\left| i \right\rangle \left\langle j \right| , ({i=j=1,2, \ldots ,5})\) are the atomic operators for \(i=j\), which they can be written by QuTiP as follows:
and for \(j\ne k\) are the polarization operators which they can be written as
with detuning parameters are defined as
The interaction part \(\hat{{H}}_I\) is defined as
$$\begin{aligned} \hat{{H}}_I=\hat{{a}}({\lambda _{1} \hat{{\sigma }}_{12} +\lambda _{2} \hat{{\sigma }}_{13} +\lambda _{3} \hat{{\sigma }}_{15} +\lambda _{4} \hat{{\sigma }}_{34}}) +\mathrm {h.c.} \end{aligned}$$
(3)
where \(\hat{{a}}^{\dagger }(\hat{{a}})\) is the creation (extinction) operator with the field frequency \(\Omega \) satisfying the commutation relations \(\left[ {\hat{{a}},\hat{{a}}^{\dagger }} \right] =1\), and can be written as follows (Fig. 1):
Fig. 1

Energy-level scheme for a five-level atom with a single-mode cavity

The number of photon \(\hat{{a}}^{\dagger }\hat{{a}}\) is given by
where \(\lambda _i \) is the coupling constants between an atom and a field:
The total Hamiltonian (1) can be written as
with
From the considered initial conditions, the initial state vector of the atom-field system can be written as
$$\begin{aligned} \left| {\psi (t=0)} \right\rangle =\sum _n {q_n \left| n \right\rangle }, \end{aligned}$$
(4)
where \(P_n =\left| {q_n } \right| ^{2}\) is the distribution function for the coherent field. The field will be considered, for example, in a coherent state. Hence, \(q_n \) is given by
$$\begin{aligned} q_n =\mathrm {e}^{-\bar{{n}}/2}\frac{\alpha _0^n }{n!}, \end{aligned}$$
(5)
where \(\bar{{n}}=\left| {\alpha _0 } \right| ^{2}\) is the initial mean photon number. The coherent state by QuTiP is given by
If we consider the field to be in the squeezed state. Hence, \(P_n \) is given by
$$\begin{aligned} q_n = \exp \left( \alpha \hat{a}^\dag -\alpha ^{*} \hat{a}\right) \exp \left( \left. \frac{1}{2} \zeta \hat{a}^{2 \dag }-\zeta ^{*} \hat{a}^2 \right) \right) |0\rangle . \end{aligned}$$
(6)
The squeezed state by QuTiP is given by
At this point, the QuTiP can be calculated the time evolution of the system. In the following example, we use the master equation ODE solver mesolve. In addition to the Hamiltonian, initial state, and the list of collapse operator, we pass a list tlist to the solver that contains the times at which we wish to evaluate the density matrix:

In the following section. we shall investigate numerically the effect of the detuning on the dynamical behavior of collapse and revival, second-order correlation function, the Mandel Q function, the normal squeezing, and the linear entropy.

3 Collapse and revival phenomena

The probability of finding the atom in its state \(\left| j \right\rangle \) is given by
$$\begin{aligned} \left\langle {{{\hat{\sigma }}_{jj}}(t)} \right\rangle = \sum \limits _n {{P_n}{{\left| {{A_j}(n,t)} \right| }^2}}. \end{aligned}$$
(7)
The mean photon number \(\left\langle {\hat{{a}}^{\dagger }(t)\hat{{a}}(t)} \right\rangle \) can be expressed as
$$\begin{aligned} \left\langle {\hat{{a}}^{\dagger }(t)\hat{{a}}(t)} \right\rangle =\bar{{n}}+\left\langle {\hat{{\sigma }}_{22} } \right\rangle +\left\langle {\hat{{\sigma }}_{33} } \right\rangle +2\left\langle {\hat{{\sigma }}_{44} } \right\rangle +\left\langle {\hat{{\sigma }}_{55} } \right\rangle . \end{aligned}$$
(8)
The mean photon number code can be written as (Fig. 2):
Fig. 2

Plots of \(\left( \left\langle {\hat{a}^{\dagger }(t)\hat{a}(t)} \right\rangle \right) \) for the five-level system (lower curves) and \(\left( \left\langle {\hat{a}^{\dagger }(t)\hat{a}(t)} \right\rangle +1\right) \) for the considered four-level system (upper curves) as a function of the scaled time \(\lambda t\); with \((\bar{n}=10)\) and a \((\Delta _s =0)\), b \((\Delta _s =5)\)

4 Calculate second-order correlation function

The second-order correlation function is defined as
$$\begin{aligned} g^{2}(t)= {} \frac{\left\langle {\hat{{a}}^{\dagger 2}(t)\hat{{a}}^{2}(t)} \right\rangle }{\left\langle {\hat{{a}}^{\dagger }(t)\hat{{a}}(t)} \right\rangle ^{2}}= {} \frac{\big \langle {\left( {\hat{{a}}^{\dagger }(t)\hat{{a}}(t)} \right) ^{2}} \big \rangle -\left\langle {\hat{{a}}^{\dagger }(t)\hat{{a}}(t)} \right\rangle }{\left\langle {\hat{{a}}^{\dagger }(t)\hat{{a}}(t)} \right\rangle ^{2}}. \end{aligned}$$
(9)
The pervious expression can be written by QuTiP as follows:

5 Mandel Q function

The sub-Poissonian photon statistics of the state is one of the most remarkable non-classical effect. To analyze such effects, we study the Mandel Q parameter which is given by
$$\begin{aligned} Q(t)=\frac{\langle \hat{a}^{\dag 2}(t) \hat{a}^2(t)\rangle -\langle \hat{a}^{\dag }(t) \hat{a}(t)\rangle }{\langle \hat{a}^{\dag }(t) \hat{a}(t)\rangle ^2}, \end{aligned}$$
(10)
which can be written by Python as (Fig. 3):
Fig. 3

Plot of \(g^{2}(t)\) for the five-level system (lower curves) and of \(g^{2}(t)+0.05\) for the considered four-level system (upper curves) for the same data as in Fig. 2

6 Calculate normal squeezing

Squeezed light is a radiation field without a classical analogue. To study the normal squeezing phenomena, we define
$$\begin{aligned} S_x= & {} \left( {\Delta V_1 } \right) ^{2}-\frac{1}{4},\qquad S_2 =\left( {\Delta V_2 } \right) ^{2}-\frac{1}{4}, \end{aligned}$$
(11)
$$\begin{aligned} \left( {\Delta V_{1}} \right) ^{2}= & {} \frac{1}{4}\left[ \left\langle {\hat{{a}}^{\dagger 2}} \right\rangle +\left\langle {\hat{{a}}^{2}} \right\rangle +2\left\langle {\hat{{a}}^{\dagger }\hat{{a}}} \right\rangle \right. \nonumber \\&+1-\left( {\left\langle {\hat{{a}}^{\dagger }} \right\rangle +\left\langle {\hat{{a}}} \right\rangle } \right) ^{2} \big ], \end{aligned}$$
(12)
and the normal squeezing in y direction is defined as (Fig. 4):
$$\begin{aligned} S_y= & {} \left( {\Delta V_2 } \right) ^{2}-\frac{1}{4},\qquad S_2 =\left( {\Delta V_2 } \right) ^{2}-\frac{1}{4}, \end{aligned}$$
(13)
$$\begin{aligned} \left( {\Delta V_{2}} \right) ^{2}= & {} -\frac{1}{4}\left[ \left\langle {\hat{{a}}^{\dagger 2}} \right\rangle +\left\langle {\hat{{a}}^{2}} \right\rangle +2\left\langle {\hat{{a}}^{\dagger }\hat{{a}}} \right\rangle \right. \nonumber \\&+1-\left( {\left\langle {\hat{{a}}^{\dagger }} \right\rangle -\left\langle {\hat{{a}}} \right\rangle } \right) ^{2} \big ]. \end{aligned}$$
(14)
Fig. 4

Plots of the squeezing parameter S with the same data, as shown in Fig. 2; red curves are for the five-level fan system and blue curves are for the considered four-level system (color figure online)

7 Linear entropy

Linear entropy as a quantitative measure is employed to evaluate the temporal behavior of DEM. However, due to the strong relation between the purity and the linear entropy, we can employ the concept of the purity to measure the degree of entanglement. The linear entropy is defined as
$$\begin{aligned} \mathcal {P}(t)=1-Tr\{\hat{\rho }_f^2(t)\}. \end{aligned}$$
(15)

8 Conclusions

In this poster, the interaction between a five-level atom and a single-mode electromagnetic cavity field has been studied using QuTiP. We investigate the collapse-revival, the anti-bunching, the normal squeezing, and the linear entropy using the Python code when the input field is in a coherent state and squeezing state.

References

  1. 1.
    Johansson, J.R., et al.: QuTiP 2: a Python framework for the dynamics of open quantum systems. Comp. Phys. Commun. 183, 1760 (2012)CrossRefGoogle Scholar
  2. 2.
    Abdel-Wahab, N.H., Salah, A.: Dynamic evolution of double five-level atom interacting with one-mode electromagnetic cavity field. Pramana J Phys 89, 87 (2017)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics and Theoretical Physics Department, Nuclear Research CentreAtomic Energy AuthorityCairoEgypt
  2. 2.Mathematics Department, Faculty of ScienceSouth Valley UniversityQenaEgypt
  3. 3.Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

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