Cognitive abilities are responsible for performing various simple and complex activities that affect a person's mental performance. These are also responsible for different day-to-day actions in human life. In the past few years, studies on cognitive ability, mental performance, and mindfulness meditation have been seen more frequently. The electroencephalogram (EEG) is an effective technique to study brain dynamics while executing any cognitive task and leads to new possibilities in the brain-computer interface (BCI) field. In this study, twenty-seven (27) healthy subjects performed a designed cognitive task having three different states (i.e., rest, meditation, and arithmetic) to stimulate the brain's cognitive functions. BIOPAC-MP-160 has been used for the EEG signal acquisition of the designed cognitive task according to the international 10-20 data acquisition system. The EEGLAB has been used to visualize, pre-process, filter, and removal of noise from the data. Then phase-amplitude coupling is performed to extract the features. After completing the feature extraction, the classification has been performed by three different deep learning approaches, i.e., sequential convolutional network (SCN), multi-branch convolutional network (MBCN), and multi-branch convolutional network-bidirectional long short-term memory network (MBCN-Bi-LSTM). The performance of the different classifications model has been estimated in terms of accuracy, precision, F1 score, and recall. The results demonstrated that MBCN-Bi-LSTM performs better than the SCN and MBCN, with a significant improvement in accuracy of 97.99%. The comparative analysis of the previously used deep learning and machine learning approaches to classify the EEG signal of different brain states substantially indicates that the proposed MBCN-Bi-LSTM model performs better in terms of accuracy and error rate. Also, the computational execution time of the proposed MBCN-Bi-LSTM is found to be less than the previous methods. The proposed classification approach may be utilized in future research to classify the various physiological signals.