- 1Departamento de Fsica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Buenos Aires, Argentina
- 2CONICET-Universidad de Buenos Aires, Instituto de Fsica de Buenos Aires (IFIBA), Ciudad Universitaria, Buenos Aires, Argentina
- 3Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore
We report a scheme for generation of high-order quadrature vortex states using two-mode photon-number squeezed states, generated via the non-linear process of Spontaneous Parametric Down Conversion. By applying a parametric rotation in the quadratures
1 Introduction
In quantum optics, a beam of light is in a squeezed state if its electric field amplitude has a reduced uncertainty, in relation to that of a coherent state. Thus, the term squeezing refers to squeezed uncertainty. In general, for a classical coherent state with N particles, the sensitivity of a measurement is limited by shot noise to the Standard Quantum Limit (SQL
An optical vortex is a singularity or zero point intensity of an optical field. More specific, a generic Laguerre-Gauss beam of order m of the form
In this article, we report a scheme for generation of high-order quadrature vortex states using two-mode photon-number squeezed states generated via the non-linear process of Spontaneous Parametric Down Conversion (SPDC). By applying a parametric rotation in the quadratures
A ϕ converter, also called mode converter, is customarily used in classical optics to convert two orthogonal Hermite-Gauss modes into a Laguerre-Gauss mode. The main motivation of the present work is to explore if an equivalent operation exists that can transform a Hermite-Gauss quadrature representation into a Laguerre-Gauss representation. We found such operation indeed exists. A remarkable feature of this operation is that it can be experimentally realized by using a balanced 50:50 beam splitter. A key application of the scheme reported here is in generation of photon-number squeezed states from quadrature vortex states, by implementation of the inverse protocol.
The article is structured as follows: First, in Section 2 we review the properties of two-mode photon-number squeezed states such as their quadrature representation and photon-number distribution, second in Section 3 we introduce the concept of quadrature rotation. Next, in Section 4, we present the quadrature representation of the rotated states in terms of Laguerre-Gauss modes. In Section 5, we present numerical simulations confirming the creation of N vortices for a squeezed input state containing
2 2-Mode Photon-Number Squeezed State
Consider a truncated two-mode photon-number squeezed state, produced by SPDC, in the Fock state representation of the form [21]:
where
To obtain a quadrature representation of the wavefunction for the state in Eq. 1, we use the standard representation of Fock states (
The quadrature representation of the input state
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/690721/fphy-09-690721-HTML-r3/image_m/fphy-09-690721-g001.jpg)
FIGURE 1. (A) Two-mode photon-number squeezed input state
3 Quadrature Rotation
The photon-number squeezed state depicted in Figure 1 displays a standard Gaussian profile in the quadratures
where
The input state transformed under the unitary operator
which represents a rotation of the quadrature by an angle ϕ. In the Heisenberg picture, considering standard commutation rules for creation and annhiliation operators, we obtain the following expression (see Appendix A):
By a binomial expansion in Eq. 5 we obtain:
where D is the normalization factor. The coefficients in the sums are of the form
In order to observe the action of the rotation
4 Laguerre-Gauss Mode Expansion
The quadrature representation of the rotated state
where
It is well known that Hermite-Gauss (HG) modes with spatial dependence
By choosing the rotation parameter
5 Numerical Results
To explore the resulting mode-profile in the quadrature
where r is the squeezing parameter and the coefficients take the form
We performed numerical simulations in the quadrature for different values of squeezing parameter r, and different values of photon-number N. The results are depicted in Figure 2 and Figure 3. The main result we observe is that, for a sufficiently small squeezing parameter r, the resulting quadrature profile exhibits
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/690721/fphy-09-690721-HTML-r3/image_m/fphy-09-690721-g002.jpg)
FIGURE 2. 3D plots of amplitude
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/690721/fphy-09-690721-HTML-r3/image_m/fphy-09-690721-g003.jpg)
FIGURE 3. Phase profile
5.1 Dependence on Squeezing Parameter r
In order to better understand the impact of the squeezing parameter r in the formation of vortices in the quadrature, we performed numerical simulations for different squeezing parameters, and for different total number of photons N. This is displayed in Figures 2A–F. Figure 2 left column corresponds to
5.2 Dependence on Photon-Number N
To confirm the viability of generation of high-order vortex states in the quadratures we performed numerical simulations for larger total number of photons (
6 Photon-Number Distribution of Quadrature Vortex States
The generation of vortices in the quadrature can be considered an interference effect arising from photon-number fluctuations, therefore it is expected that the photon-number distribution should be modified for quadrature vortex states. To further confirm that photon-number fluctuations are mapped into interference effects in the quadratures, resulting in the emergence of vortices, for a two-mode photon-number squeezed input state, we calculated the photon-number distribution for the resulting vortex states
where
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/690721/fphy-09-690721-HTML-r3/image_m/fphy-09-690721-g004.jpg)
FIGURE 4. Photon-number statistics
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/690721/fphy-09-690721-HTML-r3/image_m/fphy-09-690721-g005.jpg)
FIGURE 5. Numerical simulations of photon-number statistics
In order to further illustrate the photon-number imbalance between the two modes, introduced by the rotation in the quadratures, we performed numerical simulations for the two-mode photon number distribution
7 Discussion
We presented a scheme for generation of high-order quadrature vortex states starting from a two-mode photon-number squeezed state generated via the non-linear process of Spontaneous Parametric Down Conversion (SPDC). By applying a parametric rotation in the quadratures
Vortex states are customarily generated using various tools, such as Dove prisms, spiral plates, fork holograms, or astigmatic mode converters such as a cylindrical lenses. The important distinction is that these operations act on the transverse profile of the input beam. In the context of the present article, the rotation is performed on the quadrature representation of the state, which can be readily implemented in the lab by a balanced beam splitter. A key application of our scheme is in generation of two-mode photon-number squeezed states from two-mode quadrature vortex states, by implementing the inverse protocol.
Our scheme has the potential of exploiting the advantages of optical vortices, such as high dimensionality and topological properties, for quantum applications requiring squeezed uncertainty beyond the SQL limit (
Data Availability Statement
The data and numerical codes are available upon request.
Author Contributions
GP and AB conceived the idea and performed analytical derivations. GP and AB performed numerical simulations. GP wrote the manuscript. All authors provided critical feedback and helped shape the research, analysis and manuscript.
Funding
GP acknowledges financial support via grants PICT Startup 2015 0710 and UBACyT PDE 2017.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Appendix a
The starting point of the derivation is Eq. 5, which defines a
where
The above describes a general two-mode state in the Fock basis with total number of particles N distributed between the two modes. Now Eq. 13 can be written in terms of the mode operators as follows
where it is understood that the operator
Using the Baker-Hausdorff lemma, we can write Eq. 15 as follows
Solving for the commutators, we see that
Now for a
and similarly for
under the effect of the
Keywords: orbital angular momentum, photon-number squeezed states, optical vortices, structured light, spontaneous parametric down conversion
Citation: Puentes G and Banerji A (2021) Generation of High-Order Vortex States From Two-Mode Squeezed States. Front. Phys. 9:690721. doi: 10.3389/fphy.2021.690721
Received: 04 April 2021; Accepted: 31 May 2021;
Published: 24 June 2021.
Edited by:
Antonio Zelaquett Khoury, Fluminense Federal University, BrazilReviewed by:
Rafael Barros, Tampere University, FinlandLuis Sanchez Soto, Complutense University of Madrid, Spain
Copyright © 2021 Puentes and Banerji. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Graciana Puentes, gpuentes@df.uba.ar