Thermo Vacuum States For A Variety Of Light Fields Obtained By Virtue Of The Technique Of Integration Within Ordered Product Of Operators

In nature, most light fields are steeped in a thermo reservoir. So the excitation and decay of the system will inevitably suffer the influence of energy exchanging between the fields and thermo reservoir. When field’s temperature is relatively higher, the question belongs to classical optics category, and when the temperature is lower, it belongs to the category of quantum optics. The presence of thermo reservoir provides a certain number of excited quantum for quantum systems. According to quantum mechanics, in a non-zero temperature, the quantum state in the thermo balance can be portrayed by density matrix of a mixed state, and expectation of physical quantity A can be achieved by the calculation of trace of the product of matrix A and p, and yet the procedure is difficult to operate. In order to explore the influence of thermal reservoir on quantum system. Takahashi and Umezawa proposed the thermal field dynamics theory in 1975, by introduction a virtual degrees of freedom and the concept of thermal vacuum state, the expectation calculation of a mix state in a non-zero temperature can be turned into the expectation calculation of pure state, but the price is that the degrees of freedom will be doubled. But Takahashi and Umezawa just derived the thermo vacuum state in a chaotic light field, and the method they used is in primary stage. A new method for constructing the thermo vacuum state is presented in this paper, by using the technique of integration within an ordered product of operators based on the existing method of the partial trace, the corresponding thermal vacuum state can be introduced for variety of complex systems. Its advantage is:1) The ensemble average of mechanical quantity can be turned into the calculation of the pure state’s expectation, it is not only greatly simplifies the quantum statistics calculations, but also convenient for studying the light field properties and the evolution rule in quantum channels.2) Thermo vacuum state can reflect the quantum’s entanglement between system and thermal reservoir, it is convenient for further researching the system’s evolution rule in quantum channels by entangled state representation.3) It benefits to find new light field in theory.Therefore the theory is enriched and developed the theory of thermal field dynamics. The main contents of this paper include:1. This paper gives a brief introduction on the theory of the technique of integration within an ordered product (IWOP) of operators, which is originated by Prof. Fan Hong-Yi. We discussed the technique of integration within the normal product, anti-normal product and the Weyl ordered products of operators, and complete expression of pure gaussian integral form of commonly used quantum mechanics representation and the normal product form of single(double) mode squeezing operators are also derived. We introduced the entangled state representation presented by professor Fan, Wigner operator, Wigner function and its calculation method by the thermo vacuum state.2. Several common light fields and their properties are introduced by using the IWOP technology from brand-new perspective, such as negative binomial light field, binomial and negative binomial joint distribution light field and chaotic light field.3. The method in deriving the corresponding generalized thermal vacuum state of system by IWOP technology and partial trace method is presented in the paper. The thermo vacuum state of the negative binomial state is deduced by this method for the first time. So for complex systems, the ensemble average of mechanical quantity can turns into the calculation of the pure state’s expectation, it is not only greatly simplifies the quantum statistics calculations, but also develops the theory of thermo field dynamics. The thermo vacuum of negative binomial state brings convenience for deriving the average photon numbers, the fluctuation of photon numbers and the Wigner function for negative binomial state.4. We skillfully turn the normal ordering form of density operator for the squeezed chaotic light by the IWOP technology, and the thermo vacuum state of the squeezed chaotic light is derived by the partial trace method. Furthermore, we deduce the average photon numbers, the fluctuation of photon numbers, the second-order coherent degree and the Wigner function for the squeezed chaotic light. Finally, the properties of light field are analyzed.5. Based on the thermal vacuum states of the negative binomial state and the squeezed chaotic light, we study the law of the pure state’s expectation using thermo vacuum state, and find the expectation value principle for these two thermal vacuum states respectively. The thermal average theorem of translation operator is also deduced.6. We study the thermal vacuum state of the photon subtracted squeezed chaotic light field. Then in order to derive the normalized coefficient of the density operator, we first turn the density operator to anti-normal product form, and then convert the normal product form, and finally the normalized coefficient is derived by tracing over the light field under the coherent state representation. Further, the thermal vacuum state, the photon number distribution and the fluctuation of photon number are derived, based on these, we discuss the characteristics of the optical field.7. As another application of the partial trace technique, we explore the temperature influence of a new two-mode optical field which involves both squeezing and chaotic behaviour. A reasonable physical interpretation is given for the temperature effect, when the mean photon number in each mode is calculated.