Multiphoton processes by conditional measurements in the atom-field interaction

TL;DR

This paper demonstrates how conditional measurements on a two-level atom interacting with a cavity field can induce multiphoton addition or subtraction, enabling control over the photon number in the cavity.

Contribution

It introduces a method to add or remove multiple photons from a cavity field via entanglement and measurement of a two-level atom, expanding quantum control techniques.

Findings

Multiphoton addition and subtraction are achievable through conditional measurements.

Entanglement between atom and field enables control of photon number.

The process allows for manipulating cavity fields beyond single-photon transitions.

Abstract

We show that it is possible to add or subtract many photons from a cavity field by interacting it resonantly with a two-level atom. The atom, after entangling with the field inside the cavity and exiting it, may be measured in one of the Schmidt states, producing a multiphoton process (in the sense that can add or annihilate more photons than a single transition allows), i.e., adding or subtracting several photons from the cavity field.

Figures

Equations

$$H_I=\lambda (\hat{a}{\sigma }_{+}+{\hat{a}}^{†}{\sigma }_{-}),$$

$$\displaystyle\hat{H}_{I}^{2k}=\lambda^{2k}\left(\begin{array}[]{ll}(\sqrt{\hat{a}\hat{a}^{\dagger}})^{2k}&0\\ 0&(\sqrt{\hat{a}^{\dagger}\hat{a}})^{2k}\end{array}\right),$$

$$\displaystyle\hat{H}_{I}^{2k+1}=\lambda^{2k+1}\left(\begin{array}[]{ll}0&(\sqrt{\hat{a}\hat{a}^{\dagger}})^{2k+1}\hat{V}\\ \hat{V}^{\dagger}(\sqrt{\hat{a}\hat{a}^{\dagger}})^{2k+1}&0\end{array}\right),$$

$$\displaystyle\hat{U}_{I}(t)=\left(\begin{array}[]{ll}\cos\left(\lambda t\sqrt{\hat{a}\hat{a}^{\dagger}}\right)&-i\sin\left(\lambda t\sqrt{\hat{a}\hat{a}^{\dagger}}\right)\hat{V}\\ -i\hat{V}^{\dagger}\sin\left(\lambda t\sqrt{\hat{a}\hat{a}^{\dagger}}\right)&\cos\left(\lambda t\sqrt{\hat{a}^{\dagger}\hat{a}}\right)\end{array}\right).$$

$$|\alpha ⟩=e^{-\frac{|\alpha {|}^2}{2}}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}|n⟩,$$

$$\displaystyle|\psi(0)\rangle=\left(\begin{array}[]{l}|\alpha\rangle\\ 0\end{array}\right),$$

$$|\psi (t)⟩=|c⟩|e⟩+|s⟩|g⟩,$$

$$|c⟩=\cos \left(\lambda t\sqrt{\hat{a}{\hat{a}}^{†}}\right)|\alpha ⟩,|s⟩=-i{V}^{†}\sin \left(\lambda t\sqrt{\hat{a}{\hat{a}}^{†}}\right)|\alpha ⟩.$$

$$|+⟩=\cos \theta |e⟩+e^{-i\varphi }\sin \theta |g⟩,|-⟩=e^{i\varphi }\sin \theta |e⟩-\cos \theta |g⟩,$$

$$|\psi ⟩=\sqrt{{\lambda }_{+}}|+⟩|{\psi }_{+}⟩+\sqrt{{\lambda }_{-}}|-⟩|{\psi }_{-}⟩,$$

$$|{\psi }_{+}⟩=\frac{1}{\sqrt{{\lambda }_{+}}}\left(\cos \theta |c⟩+e^{i\varphi }\sin \theta |s⟩\right),|{\psi }_{-}⟩=\frac{1}{\sqrt{{\lambda }_{-}}}\left(\sin \theta e^{-i\varphi }|c⟩-\cos \theta |s⟩\right).$$

$$⟨c|s⟩=r{e}^{-i\varphi },2W=⟨c|c⟩-⟨s|s⟩,$$

$$\theta =\frac{1}{2}{\tan }^{-1}\frac{r}{W}.$$

$$Q=\frac{⟨{\psi }_{+}|({\hat{a}}^{†}\hat{a}{)}^2|{\psi }_{+}⟩-⟨{\psi }_{+}|{\hat{a}}^{†}\hat{a}|{\psi }_{+}{⟩}^2}{⟨{\psi }_{+}|{\hat{a}}^{†}\hat{a}|{\psi }_{+}⟩}-1$$

Full text

1 Introduction

Conditional measurements in the atom field interaction [1, 2, 3, 4, 5], i.e. projections of the entangled state onto a selected atomic state once the atom and field have interacted and the atom has exit the cavity, carrying the information of the field inside it, have been used to produce nonclassical states of light such as Schrödinger cat states [1] or more complex superpositions of coherent states [2, 3, 4] that may lead to generation of numbers states.

Gosh and Gerry [6] have shown that a class of nonclassical states may be produced by such conditional measurements which Luis [7] called paradoxical, as he showed that a measurement leading to absorption of light produced an increase of the number of photons. More recently, Valverde and Baseia [8] have studied such paradoxical generation of nonclassical states in models of interpolating Hamiltonians, in particular of the (nonlinear) Jaynes-Cummings type.

Here we show that conditional measurements can lead efficiently to multiphoton processes [9], and in this way, a resonant interaction that allows single-photon exchange of energy between atom and field [10] may generate, via highly probable conditional measurements, the creation or annihilation of more than one photons in the cavity.

Because the atom-field interaction may be modeled in other type of systems, namely ion-laser interactions [11, 12], the results shown here may be translated to such systems, where one could use this effect to cool trapped ions.

2 Conditional measurements

Consider the interaction Hamiltonian for the on-resonance interaction of an atom and a quantized field ($\hbar$ is set to one)

[TABLE]

the operators $\hat{a}$ and $\hat{a}^{\dagger}$ are the annihilation and creation operators, the parameter $\lambda$ is the so-called Rabi frequency and the atomic operators, $\sigma_{-}$ and $\sigma_{+}$, are the lowering and raising Pauli matrices, respectively. We can obtain the evolution operator, $\hat{U}_{I}(t)=e^{-i\hat{H}_{I}t}$, by expanding the exponential in Taylor series; the even and odd powers may be written as (we use the Pauli matrices in their $2\times 2$ matrix notation) [13]

[TABLE]

and

[TABLE]

with $\hat{V}=\frac{1}{\sqrt{\hat{a}\hat{a}^{\dagger}}}\hat{a}$ the London phase operator [14, 15]. So, we may obtain the evolution operator in the form

[TABLE]

If we consider an initial coherent state for the field, $|\alpha\rangle$, with its distribution in terms of number states given by

[TABLE]

where $|n\rangle$ are the Fock states, and an excited state for the atom, the initial wavefunction is given by

[TABLE]

we obtain the evolved wavefunction as $|\psi(t)\rangle=\hat{U}_{I}(t)|\psi(0)\rangle$, that in the (complete) ket notation is written as

[TABLE]

where the unnormalized wavefunctions $|c\rangle$ and $|s\rangle$ are given by

[TABLE]

We now write the above wavefunction in the rotated atomic basis

[TABLE]

such that $|\psi\rangle$ may be written as a Schmidt decomposition [16]

[TABLE]

where $\lambda_{\pm}=\frac{1}{2}\pm\sqrt{W^{2}+r^{2}}$ are the eigenvalues of the atomic density matrix an the orthonormal field wavefunctions are written as

[TABLE]

The quantities $r$, $W$, $\phi$ and $\theta$ are related as follows

[TABLE]

and

[TABLE]

By measuring the atom as it exits the cavity in the $|+\rangle$ state, i.e. doing a conditional measurement of the atom in that specific Schmidt state, the field is projected to the normalized state $|\psi_{+}\rangle$. From it, we can calculate the average number of photons, $N=\langle\psi_{+}|a^{\dagger}a|\psi_{+}\rangle$. We plot this quantity as a function of $\lambda t$ in Figure 1, where it is shown that an increase (after a severe decrease) of the average number of photons would occur for certain times (where the maximums happens). Of course, although it may be seen as a surprising effect such nonlinear generation of photons in the cavity, it may be understood by the fact that a distribution of photons is initially considered as well as the fact that the conditional measurement is in itself a non-unitary process.

In Figure 2, we plot (dashed line) the probability of such events occurring. We see that for the first maximum of the solid line in Figure 1, $\lambda t\approx 1.75$, the probability shown in Figure 2 is a minimum yielding a probability of approximately $3$ percent. It may be also seen that all the maximums happen with minimum probability, although this probability increases such that the second relevant “multiphoton process”has a probability to occur greater than 20 percent.

In Figure 3, we plot the Mandel-$Q$ parameter [17]

[TABLE]

as a function of $\lambda t$ where the generation of nonclassical states may be seen for times for which the Mandel-$Q$ parameter is less than zero.

In Figure 4 (b), we plot the photon distribution, $P_{n}=|\langle\psi_{+}|n\rangle|^{2}$ as a function of $n$ for $\alpha=4$ and $\lambda t\approx 1.75$. A deviation from the original distribution, Figure 4 (a), centered at $n=16$ may be seen, as the one obtained from a conditional measurement may be seen centered at about $n=20$. Therefore, an increase of the average number of photons greater than one, i.e. a multiphoton process, takes place. This figure resembles that of a squeezed state [18]. In fact, squeezed light has been predicted in the Jaynes-Cummings model by Kuklinski and Madajczyk [19] and its interaction with two-level atoms studied by Satyanarayana et al. [20] and Moya-Cessa and Vidiella-Barranco [21].

3 Conclusions

We have shown that, after an atom resonantly interacts with a field, the average number of photons inside the cavity is severely affected by measuring it when it exits it. We call this a multiphoton process as the atom-field interaction allows the exchange of only single photons, but the conditional measurement generates the creation or absorption of several photons in the cavity. We have shown, numerically, that such effects are related to the initial number of photons in the cavity for both coherent states and thermal distributions giving rise to the addition or subtraction proportional to the square of the initial average number of photons.