New schemes for creating large optical Schrödinger cat states using strong laser fields

Abstract

Recently, using conditioning approaches on the high-harmonic generation process induced by intense laser-atom interactions, we have developed a new method for the generation of optical Schrödinger cat states (Lewenstein et al. in Nat Phys, 17 1104–1108, 2021. https://doi/10.1038/s41567-021-01317-w). These quantum optical states have been proven to be very manageable as, by modifying the conditions under which harmonics are generated, one can interplay between kitten and genuine cat states. Here, we demonstrate that this method can also be used for the development of new schemes towards the creation of optical Schrödinger cat states, consisting of the superposition of three distinct coherent states. Apart from the interest these kind of states have on their own, we additionally propose a scheme for using them towards the generation of large cat states involving the sum of two different coherent states. The quantum properties of the obtained superpositions aim to significantly increase the applicability of optical Schrödinger cat states for quantum technology and quantum information processing.

Appendix: Probability of success in the generalization of the conditioning approach

Here, we present a more detailed analysis about the probability of success plotted in Figs. 7 and 8. In the text, we are interested in measuring even or odd number of photons in one of the output modes of the beam splitter, as this operation projects the other mode into a cat state, which differ in a relative phase of $\pi$ depending on the specific measurement that is performed. Thus, the set of positive operator-valued measurements characterizing the operations that we can implement is

$$\begin{aligned} \bigg \{ \mathbbm {1} \otimes {|{0}\rangle }{\langle {0}|}, \mathbbm {1} \otimes \sum _{n=1}^{\infty } {|{2n}\rangle }{\langle {2n}|}, \mathbbm {1} \otimes \sum _{n=0}^{\infty } {|{2n+1}\rangle }{\langle {2n+1}|} \bigg \}, \end{aligned}$$

(32)

which project the second mode into the vacuum state, an even number of photons and an odd number of photons, respectively.

Each of the measurements described in Eq. (32) have a probability associated, so we define the probability of success $\mathcal {P}$ as the probability of successfully performing one of such measurements. For instance, the probability of successfully measuring an even number of photons is given by

$$\begin{aligned} \mathcal {P}_\text {even} = \dfrac{1}{\mathcal {P}_\text {total}} \hbox {tr}{P_\text {even}\rho }, \end{aligned}$$

(33)

where $\rho$ is the density matrix characterizing the state of our system, and $\mathcal {P}_\text {total}$ is given as the sum of all the possible measurements that we can perform, i.e.

$$\begin{aligned} \mathcal {P}_\text {total} = \mathcal {P}_\text {zero} + \mathcal {P}_\text {even} + \mathcal {P}_\text {odd}. \end{aligned}$$

(34)

With all this set, let us consider the architecture presented in the main text used for the generation of enlarged cat states. Using a beam splitter characterized by the mixing angle $\theta$ and a coherent state of the form $\left| (\alpha - \delta \alpha ) \tan (\theta ) \right\rangle$, the final state after the beam splitter is given by

$$\begin{aligned} \begin{aligned} \left| \varPhi _\text {BS} \right\rangle&= \dfrac{1}{\sqrt{N}} \Big [ - \xi _1' \left| {\tilde{\alpha }} + \delta \alpha \cos \theta \right\rangle \left| -\delta \alpha \sin \theta \right\rangle \\& \quad + \left| {\tilde{\alpha }} \right\rangle \left| 0 \right\rangle - \xi _2 \left| {\tilde{\alpha }} - \delta \alpha \cos \theta \right\rangle \left| \delta \alpha \sin \theta \right\rangle \Big ], \end{aligned} \end{aligned}$$

(35)

where ${\tilde{\alpha }} = (\alpha -\delta \alpha )/\cos \theta$. Thus, the measurement of an even number of photons projects the state in Eq. (35) into

$$\begin{aligned} \begin{aligned} \left| \varPhi _\text {even} \right\rangle&= \dfrac{1}{\sqrt{N}} \Big [ -\xi _1' \left| {\tilde{\alpha }}+\delta \alpha \cos \theta \right\rangle - \xi _2 \left| {\tilde{\alpha }}-\delta \alpha \cos \theta \right\rangle \Big ]\\& \quad \otimes \sum _{n=1}^\infty \dfrac{(\delta \alpha \sin \theta )^{2n}}{\sqrt{2n!}} e^ {-\tfrac{|\delta \alpha \sin \theta |^2}{2}} \left| 2n \right\rangle \\&\equiv \left| \phi _\text {even} \right\rangle \otimes \sum _{n=1}^\infty \dfrac{(\delta \alpha \sin \theta )^{2n}}{\sqrt{2n!}} e^{-\tfrac{|\delta \alpha \sin \theta |^2}{2}} \left| 2n \right\rangle ; \end{aligned} \end{aligned}$$

(36)

the measurement of an odd number of photons projects the state into

$$\begin{aligned} \begin{aligned} \left| \varPhi _\text {odd} \right\rangle&= \dfrac{1}{\sqrt{N}} \Big [ \xi _1' \left| {\tilde{\alpha }}+\delta \alpha \cos \theta \right\rangle - \xi _2 \left| {\tilde{\alpha }}-\delta \alpha \cos \theta \right\rangle \Big ]\\& \quad \otimes \sum _{n=0}^\infty \dfrac{(\delta \alpha \sin \theta )^{2n+1}}{\sqrt{2n+1!}} e^{-\tfrac{|\delta \alpha \sin \theta |^2}{2}} \left| 2n+1 \right\rangle \\&\equiv \left| \phi _\text {odd} \right\rangle \otimes \sum _{n=0}^\infty \dfrac{(\delta \alpha \sin \theta )^{2n+1}}{\sqrt{2n+1!}} e^{-\tfrac{|\delta \alpha \sin \theta |^2}{2}} \left| 2n+1 \right\rangle ; \end{aligned} \end{aligned}$$

(37)

and, finally, the measurement of zero number of photons leads to

$$\begin{aligned} \begin{aligned} \left| \varPhi _\text {zero} \right\rangle&= \dfrac{1}{\sqrt{N}} \Big [ -\xi _1' \left\langle 0 \right| \left. -\delta \alpha \sin \theta \right\rangle \left| {\tilde{\alpha }} + \delta \alpha \cos \theta \right\rangle + \left| {\tilde{\alpha }} \right\rangle \\& \quad - \xi _2 \left\langle 0 \right| \left. \delta \alpha \sin \theta \right\rangle \left| {\tilde{\alpha }} - \delta \alpha \cos \theta \right\rangle \Big ]\otimes \left| 0 \right\rangle \\&\equiv \left| \phi _\text {zero} \right\rangle \otimes \left| 0 \right\rangle . \end{aligned} \end{aligned}$$

(38)

Thus, the probability of success for each of the three measurements that we can do is given by

$$\begin{aligned}&\mathcal {P}_\text {even} = \dfrac{\langle {\phi _\text {even}|\phi _\text {even}}\rangle }{\mathcal {P}_\text {total}} e^{-|\delta \alpha \sin \theta |^2} \big [ -1 + \cosh (\delta \alpha \sin \theta ) \big ], \end{aligned}$$

(39)

$$\begin{aligned}&\mathcal {P}_\text {odd} = \dfrac{\langle {\phi _\text {odd}|\phi _\text {odd}}\rangle }{\mathcal {P}_\text {total}} e^{-|\delta \alpha \sin \theta |^2} \sinh (\delta \alpha \sin \theta ), \end{aligned}$$

(40)

and

$$\begin{aligned} \mathcal {P}_\text {zero} = \dfrac{\langle {\phi _\text {zero}|\phi _\text {zero}}\rangle }{\mathcal {P}_\text {total}}. \end{aligned}$$

(41)

Note that the case of $\delta \alpha = 0$ leads to a divergence in the obtained equations. In that case, the applied measurement does not make sense at all because the case of $\delta \alpha = 0$ corresponds to the situation in which we do not have HHG processes taking place. Thus, no optical Schrödinger cat state is being generated.