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arXiv:2410.21907 · APL Quantum

Adding or subtracting a single photon is the same for pure squeezed vacuum states

Ole Steuernagel, Shao-Hua Hu, Ray‐Kuang Lee

Published Feb 12, 2025• 1 citation
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Abstract

The addition of a single photon to a light field can lead to exactly the same outcome as the subtraction of a single photon: starting from the same initial state, both procedures can generate the same final quantum state. We prove that this identity-of-outcome is true for pure squeezed vacuum states of light, and in some sense only for those. We show that mixed states can show this identity-of-outcome for the addition or subtraction of a photon if they are generated from the incoherent sums of pure squeezed vacuum states with the same squeezing. We point out that our results give a reinterpretation to the fact that pure squeezed vacuum states, with squeezing e−z, are formally annihilated by Bogoliubov-transformed annihilation operators: âz=âcosh(z)−â†sinh(z).

DOI
10.1063/5.0246230

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The addition of a single photon to a light field can lead to exactly the same outcome as the subtraction of a single photon: starting from the same initial state, both procedures can generate the same final quantum state. We prove that this identity-of-outcome is true for pure squeezed vacuum states of light, and in some sense only for those. We show that mixed states can show this identity-of-outcome for the addition or subtraction of a photon if they are generated from the incoherent sums of pure squeezed vacuum states with the same squeezing. We point out that our results give a reinterpretation to the fact that pure squeezed vacuum states, with squeezing e−z, are formally annihilated by Bogoliubov-transformed annihilation operators: âz=âcosh(z)−â†sinh(z).
The addition of a single photon to a light field can lead to exactly the same out-comeas the subtraction of a single photon. We prove that this identity-of-outcome is true for pure squeezed vacuum states of light, and in some sense only for those. We show that mixed states can show this identity-of-outcome for addition or subtraction of a photon if they are generated from incoherent sums of pure squeezed vacuum states with the same squeezing. Our results gives a reinterpretation to the fact that pure squeezed vacuum states are formally annihilated by Bogoliubov-transformed annihilation operators: â z = âcosh(z) − ↠sinh(z).
In atomic-scale or quantum physics, an electromagnetic field is composed of photons, optical packets so small that a typical laser pointer with 1mW of power emits billions of them each second. Being able to take away or add a single photon in a light field at will would be useful for accurate engineering of quantum states. Such pure single photons are ideal for carrying and manipulating information in emerging quantum technologies, but generating them still is very challenging. In quantum physics, subtraction or addition of a photon is not a deterministic process. Instead, the probability is proportional to the number of photons originally in the electromagnetic field. This is described by the equation â|n〉 = √ n|n − 1〉 and â†|n〉 = √ n+ 1|n + 1〉, where â and ↠are annihilation and creation operators, respectively, representing the action of subtracting and adding a photon in a field composed of n photons.1 Such single-photon operations were limited to pure theoretical discussions until 2004, when Grangier and coworkers found a simple way to subtract a photon2 from a field. Passing through a beam splitter of high transmittivity, some photons can be split from the initial field. A ‘click’ detected in the photodetector in the reflected channel means one photon has been subtracted from the field: see Figure 1(a). When the initial field is composed of a small number of photons, the effect is dramatic and may even completely change the characteristics of the field. For example, let us take an initial single-mode squeezed state generated by ‘degenerate’ parametric down-conversion, which is an optical nonlinear process that converts a pump photon into two twin photons with half the pump photon energy. (This state is well known for its quantum noise squeezed into one quadrature at the expense of expanded noise in the other quadrature.) In order to visualize the noise properties and characteristics of the squeezed state, a probability-like function, such Figure 1. Setups to subtract (a) and add (b) a single photon from/to a light field. ρin and ρout denote the density operators of the input and output fields; BS is a low-reflectivity beam splitter; PDC is a nonlinear crystal where parametric down-conversion takes place; â and ↠denote on/off photodetectors.
We introduce states a † m | ξ ⟩ and a m | ξ ⟩, where| ξ ⟩ is a two-mode squeezed vacuum state. We examine the properties of these states by analytic derivation. Because of the entanglement between the modes of the field, photon adding (or substracting) for one of the modes results in interesting outcomes. For a two-mode squeezed vacuum state, the effect of annihilation of m a-photons is just the same as creation of m b-photons. Annihilation of photons of mode a (or mode b ) actually increases the mean number of photons of mode a and mode b . We also calculate Mandel Q parameter, cross-correlation function and squeezing of the field.
We investigate the entanglement enhancement of the two-mode squeezed vacuum state using various combinations of photon additions and photon subtractions. We suppose that a sequence of N operations are performed on each mode and on a single mode of the two-mode squeezed vacuum state, each operation being photon addition or photon subtraction; then there are 2N possible combinations of operations in total. We calculate the entanglement for all the photon-added/subtracted states generated by all possible combinations for N=2, N=3, and N=4, respectively, and deduce that the combination of operations alternating between photon addition and photon subtraction enhances the entanglement most effectively, while adding or subtracting N photons repeatedly enhances the entanglement least effectively.
We study how photon addition and subtraction can be used to generate squeezed superpositions of coherent states (SSCSs) in free-traveling fields with high fidelities and large amplitudes. It is shown that an arbitrary NNN-photon subtraction results in the generation of a SSCS with nearly the perfect fidelity (F>0.999)(F>0.999)(F>0.999) regardless of the number of photons subtracted. In this case, the amplitude of the SSCS increases as the number of the subtracted photons gets larger. For example, two-photon subtraction from a squeezed vacuum state of 6.1dB6.1\phantom{\rule{0.3em}{0ex}}\mathrm{dB}6.1dB can generate a SSCS of $\ensuremath{\alpha}=1.26$, while in the case of the four-photon subtraction a SSCS of a larger amplitude $\ensuremath{\alpha}=1.65$ is obtained under the same condition. When a photon is subtracted from a squeezed vacuum state and another photon is added subsequently, a SSCS with a lower fidelity $(F\ensuremath{\approx}0.96)$ yet higher amplitude $(\ensuremath{\alpha}\ensuremath{\approx}2)$ can be generated. We analyze some experimental imperfections including inefficiency of the detector used for the photon subtraction.
Recently Parigi et al. [Science 317, 1890 (2007)] implemented experimentally the photon subtraction and addition processes from and to a light field in a conditional way, when the required operations were produced successfully only upon the positive outcome of a separate measurement. It was verified that for a low-intensity beam (quantum regime) the bosonic annihilation operator aaa does indeed describe a single photon subtraction, while the creation operator ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ describes a photon addition. Nonetheless, the exact formal expressions for these operations do not always reduce to these simple identifications, and in this connection here we deduce the general superoperators for multiple photon subtraction and addition processes and analyze the statistics of the resulting states for classical field states having an arbitrary intensity. We obtain closed analytical expressions and verify that for classical fields with high intensity (classical regime) the operators that describe photon subtraction and addition processes deviate significantly from simply aaa and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. Complimentarily, we analyze in detail such a smooth quantum-classical transition as a function of beam intensity for both processes.