[Paper Review] Conditional quantum state engineering at beam splitter arrays

Jens Clausen, Mohammed Dakna+2 more|arXiv (Cornell University)|May 25, 1999

Photonic and Optical Devices3 citations

TL;DR

This paper proposes a method for conditional quantum state engineering using beam splitter arrays, enabling the preparation of arbitrary single-mode quantum states from vacuum via coherent displacement and photon addition, along with overlap measurement with target states. The key contribution is a non-unitary transformation operator derived as a product of s-ordered operators per beam splitter, with transmittance/reflectance determining the s-ordering, demonstrated for Schrödinger-cat-like states.

ABSTRACT

The generation of arbitrary single-mode quantum states from the vacuum by alternate coherent displacement and photon adding as well as the measurement of the overlap of a signal with an arbitrarily chosen quantum state are studied. With regard to implementations, the transformation of the quantum state of a traveling optical field at an array of beam splitters is considered, using conditional measurement. Allowing for arbitrary quantum states of both the input reference modes and the output reference modes on which the measurements are performed, the setup is described within the concept of two-port non-unitary transformation, and the overall non-unitary transformation operator is derived. It is shown to be a product of operators, where each operator is assigned to one of the beam splitters and can be expressed in terms of an s-ordered operator product, with s being determined by the beam splitter transmittance or reflectance. As an example we discuss the generation of and overlap measurement with Schroedinger-cat-like states.

Motivation & Objective

• To develop a scheme for generating arbitrary single-mode quantum states from vacuum using coherent displacement and photon addition.
• To enable conditional measurement-based state engineering at beam splitter arrays for arbitrary input and output reference states.
• To derive a general non-unitary transformation operator for beam splitter networks, parameterized by transmittance and reflectance.
• To demonstrate the method's feasibility for creating and measuring Schrödinger-cat-like states.
• To provide a theoretical framework for implementing conditional quantum state engineering in linear optical networks.

Proposed method

• The method employs a beam splitter array to transform the quantum state of a traveling optical field through conditional measurements.
• Each beam splitter contributes an operator to the overall transformation, expressed as an s-ordered operator product where s is determined by the beam splitter's transmittance or reflectance.
• The overall transformation is modeled as a product of individual beam splitter operators, forming a non-unitary transformation operator.
• The approach allows for arbitrary input reference modes and output reference modes, enabling flexible state engineering.
• The formalism is applied to generate and measure Schrödinger-cat-like states via conditional measurement on the output.
• The derivation is grounded in two-port non-unitary transformation theory, with explicit operator expressions derived for each beam splitter stage.

Experimental results

Research questions

• RQ1 How can arbitrary single-mode quantum states be engineered from vacuum using only coherent displacement and photon addition?
• RQ2 What is the mathematical structure of the non-unitary transformation arising from beam splitter arrays with conditional measurements?
• RQ3 How does the beam splitter transmittance or reflectance affect the s-ordering of the resulting operator in the transformation?
• RQ4 Can the proposed method generate and measure Schrödinger-cat-like states with high fidelity?
• RQ5 What is the general form of the transformation operator for a beam splitter network under conditional measurement?

Key findings

• The overall non-unitary transformation operator is derived as a product of s-ordered operators, with s determined by beam splitter transmittance or reflectance.
• The method enables conditional preparation of arbitrary single-mode quantum states from vacuum using coherent displacement and photon addition.
• The formalism supports arbitrary input and output reference states, allowing flexible state engineering.
• The approach is demonstrated to be applicable to the generation and overlap measurement of Schrödinger-cat-like states.
• The transformation is expressed within a two-port non-unitary framework, providing a systematic method for state engineering in linear optical networks.
• The derived operator structure allows for precise control over quantum state evolution through beam splitter arrays using conditional measurements.