Commutation of Beam splitter operator with Displacement operator

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Thread starter deepalakshmi
Start date Jul 31, 2021
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Beam Beam splitter Commutation Displacement Operator Quantum optics

In summary, the commutation of the beam splitter operator with the displacement operator is the order in which they are applied, resulting in different outcomes. This affects quantum states by changing probability amplitudes and the overall phase. The mathematical equations for the commutation are [a, D(α)] = αD(α) and [a†, D(α)] = α*D(α). The physical significance of this commutation lies in understanding the behavior of quantum systems, and it can be experimentally verified through quantum optics experiments.

Jul 31, 2021

deepalakshmi

TL;DR Summary: Does Beam splitter operator and Displacement operator of coherent state commute with each other?

I have a beam splitter operator (a†)b +(b†)a. Does it commute with exp(αâ†-α*â). Here a and â† are creation and lowering operator

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Aug 1, 2021

vanhees71

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Well, just check, whether the BS operator commutes with the argument of the operator exponential!

1. What is the Commutation of Beam splitter operator with Displacement operator?

The Commutation of Beam splitter operator with Displacement operator is a mathematical operation that describes the relationship between the two operators. It determines how the two operators interact with each other and whether they can be applied in any order.

2. Why is the Commutation of Beam splitter operator with Displacement operator important?

This commutation relationship is important in quantum optics and quantum information theory, as it allows for the manipulation and control of quantum states. It also helps in the development of quantum algorithms and quantum computing.

3. How is the Commutation of Beam splitter operator with Displacement operator calculated?

The commutation relationship is calculated using the commutator, which is defined as the difference between the product of two operators and the product of the same two operators in reverse order. In this case, the commutator of the Beam splitter operator and Displacement operator is calculated and then simplified to determine the commutation relationship.

4. What is the physical significance of the Commutation of Beam splitter operator with Displacement operator?

The commutation relationship between these two operators determines the behavior of a quantum state when it is acted upon by both operators. It also helps in understanding the effects of quantum operations on quantum states and can be used to design quantum experiments.

5. Are there any applications of the Commutation of Beam splitter operator with Displacement operator?

Yes, there are many applications of this commutation relationship in quantum optics and quantum information theory. It is used in the development of quantum algorithms, quantum error correction codes, and quantum simulations. It also has applications in quantum metrology and quantum communication.