Deterministic creation of entangled atom-light Schr\"odinger-cat states

TL;DR

This paper demonstrates a method to deterministically generate entangled atom-light Schrödinger-cat states by reflecting a coherent light pulse from an optical cavity with a single atom, enabling quantum state manipulation and quantum-logic gates.

Contribution

It introduces a novel technique for creating entangled atom-light states with high control, including the implementation of a quantum-logic gate between an atom and a photonic qubit.

Findings

Successful generation of entangled atom-light states with negative Wigner functions.

Demonstration of a quantum-logic gate between an atom and a light pulse.

Quantum states produced are suitable for quantum information processing.

Abstract

Quantum physics allows for entanglement between microscopic and macroscopic objects, described by discrete and continuous variables, respectively. As in Schr\"odinger's famous cat gedanken experiment, a box enclosing the objects can keep the entanglement alive. For applications in quantum information processing, however, it is essential to access the objects and manipulate them with suitable quantum tools. Here we reach this goal and deterministically generate entangled light-matter states by reflecting a coherent light pulse with up to four photons on average from an optical cavity containing one atom. The quantum light propagates freely and reaches a remote receiver for quantum state tomography. We produce a plethora of quantum states and observe negative-valued Wigner functions, a characteristic sign of non-classicality. As a first application, we demonstrate a quantum-logic gate between an atom and a light pulse, with the photonic qubit encoded in the phase of the light field.

Tables

Table 1. Table   S1 : Loss budget listing the loss channels in our experiment that we individually quantified. Uncertainties in the individual losses are on the order of 10% of their respective magnitude.

Source of loss	(effective) loss	$L_i$
Finite cavity reflectivity	19.0%	$L_{\text{eff}}$
Lenses, waveplates, mirrors and NPBS	9.5%	\rdelim}3.5*[$L_{\text{o}}$]	\rdelim}10.8*[$L_{\text{p+d}}$]
Isolator transmission	3.0%
Switch AOD transmission	2.5%
Mode matching with LO	6.0%	$L_{\text{m}}$
Quantum efficiency of photodiodes	1.5%	$L_{\text{d}}$
Detector dark noise	2.5%	\rdelim}4.6*[$L_{\text{n}}$]
LO classical laser noise	1.8%
Electronic high pass 0.7 kHz signal reduction	1.1%
Electronic high pass background noise	0.2%
Total losses	39.3%

Equations

$$|\uparrow ⟩|\alpha ⟩$$

$$|\downarrow ⟩|\alpha ⟩$$

$$\frac{1}{2}\bigl{[}|{{\uparrow}}\rangle\bigl{(}|{\alpha}\rangle-|{-\alpha}\rangle\bigr{)}+|{{\downarrow}}\rangle\bigl{(}|{\alpha}\rangle+|{-\alpha}\rangle\bigr{)}\bigr{]},$$

$$\eta =\frac{{\kappa }_r}{\kappa }\frac{g^2}{g^2+\kappa \gamma }={\eta }_{\text{esc}}\frac{C}{C+1/2}=\mathrm{0.81.}$$

$$L_{\text{eff}}=1-\eta =19\%,$$

$$|{\psi }_{\text{cat}}⟩=\frac{1}{\mathcal{N}}\left(\cos (\xi /2)|\alpha ⟩+e^{i\theta }\sin (\xi /2)|e^{i\varphi }\alpha ⟩\right)$$

$$|\uparrow ,+\alpha ⟩$$

$$|\uparrow ,-\alpha ⟩$$

$$|\downarrow ,+\alpha ⟩$$

$$|\downarrow ,-\alpha ⟩$$

$$r_{\downarrow /\uparrow }$$

$$t_{\downarrow /\uparrow }$$

$$m_{\downarrow /\uparrow }$$

$$a_{\downarrow /\uparrow }$$

$${\alpha }_{\text{out}}=\frac{1}{2}|r_{\uparrow }-r_{\downarrow }|=\frac{{\kappa }_r}{\kappa }\frac{g^2}{g^2+\kappa \gamma }\alpha =\eta \alpha ,$$

$$\eta :=\frac{{\kappa }_r}{\kappa }\frac{g^2}{g^2+\kappa \gamma }(=0.81).$$

$$⟨r_{\uparrow }|r_{\downarrow }⟩=e^{-2{\eta }^2{\alpha }^2},$$

$$⟨l_{\uparrow }|l_{\downarrow }⟩=⟨t_{\uparrow }|t_{\downarrow }⟩⟨m_{\uparrow }|m_{\downarrow }⟩⟨a_{\uparrow }|a_{\downarrow }⟩=e^{-2(1-\eta )\eta {\alpha }^2},$$

$$⟨l_{\uparrow }|⟨r_{\uparrow }|r_{\downarrow }⟩|l_{\downarrow }⟩=e^{-2\eta {\alpha }^2}.$$

$$|{\psi }_{\text{out}}⟩=\frac{|r_{\uparrow }⟩|l_{\uparrow }⟩+e^{i\theta }|r_{\downarrow }⟩|l_{\downarrow }⟩}{\sqrt{2(1+e^{-2\eta {\alpha }^2}\cos \theta )}}$$

$$\rho ={\mathrm{tr}}_l|{\psi }_{\text{out}}⟩⟨{\psi }_{\text{out}}|$$

$$=$$

$$=\frac{|r_{\uparrow }⟩⟨r_{\uparrow }|+e^{-2(1-\eta )\eta {\alpha }^2}(e^{-i\theta }|r_{\uparrow }⟩⟨r_{\downarrow }|+e^{i\theta }|r_{\downarrow }⟩⟨r_{\uparrow }|)+|r_{\downarrow }⟩⟨r_{\downarrow }|}{2(1+e^{-2\eta {\alpha }^2}\cos \theta )}$$

$$=\frac{|r_{\uparrow }⟩⟨r_{\uparrow }|+e^{-2(1-\eta ){\alpha }_0^2}(e^{-i\theta }|r_{\uparrow }⟩⟨r_{\downarrow }|+e^{i\theta }|r_{\downarrow }⟩⟨r_{\uparrow }|)+|r_{\downarrow }⟩⟨r_{\downarrow }|}{2(1+e^{-2{\alpha }_0^2}\cos \theta )}$$

$$W(q,p)$$

$$V=\frac{\sinh (2\eta {\alpha }_0^2)}{\sinh (2{\alpha }_0^2)}=\frac{\sinh (2\left(1-L_{\text{eff}}\right){\alpha }_0^2)}{\sinh (2{\alpha }_0^2)},$$

$$V\simeq \exp (-2L_{\text{eff}}{\alpha }_0^2).$$

$$|{\psi }_{\text{cat}}⟩=\frac{|r_{\uparrow }⟩+e^{i\theta }|r_{\downarrow }⟩}{\sqrt{2(1+e^{-2{\eta }^2{\alpha }^2}\cos \theta )}}$$

$$F=⟨{\psi }_{\text{cat}}|\rho |{\psi }_{\text{cat}}⟩$$

$$=1-\frac{(1-e^{-4\eta {\alpha }_0^2})(1-e^{-2(1-\eta ){\alpha }_0^2})}{2(1+e^{-2\eta {\alpha }_0^2}\cos \theta )(1+e^{-2{\alpha }_0^2}\cos \theta )}.$$

$$1-L=\prod _i(1-L_i).$$

$$|{\psi }_{\text{cat}}⟩=\frac{1}{\mathcal{N}}\left(\cos (\xi /2)|\alpha ⟩+e^{i\theta }\sin (\xi /2)|e^{i\varphi }\alpha ⟩\right).$$

$$r(\Delta )=1-\frac{2{\kappa }_r(2i\pi \Delta +\gamma )}{(2i\pi \Delta +\kappa )(2i\pi \Delta +\gamma )+g^2}.$$

$$\varphi (\Delta )=\arg (r(\Delta ){|}_{g=0})-\arg (r(\Delta ){|}_{g=2\pi \times 7.8\text{MHz}}).$$

Full text

As early as 1935, Schrödinger formulated a gedanken experiment schrodinger1935 with a living cat and a radioactive atom placed inside a box, which is then closed. When the atom decays, it triggers a death mechanism that kills the cat. According to the laws of quantum physics, the decay of the atom occurs at some random time. Consequently, the time of death of the cat is unknown. Mathematically, the situation inside the box is described by an entangled superposition state that is known as a ‘Schrödinger-cat state’. Most remarkable, this state offers a unique access to the atom-cat system, at least in principle. For example, a measurement apparatus that is capable of measuring the atom in a superposition of ‘not decayed’ and ‘decayed’ immediately projects the cat into a superposition of ‘alive’ and ‘dead’. Such observation thus transfers the superposition state of the microscopic quantum object into the macroscopic classical world, something weird for a cat. In contrast to the entangled Schrödinger-cat state, the coherent superposition state of the cat is here denoted as a ‘cat state’.

While notoriously difficult to create glancy2008 , several implementations of Schrödinger-cat states or just cat states have emerged during recent decades. In all these experiments, coherent states with distinguishable phases mimic the two cat states ‘dead’ and ‘alive’. Most prominently, Schrödinger-cat states were explored using a trapped ion wineland2013 ; kienzler2016 , with the vibrational state in the trap taking the role of the cat, and coherent microwave fields confined to superconducting boxes were used in combination with Rydberg atoms deleglise2008 ; haroche2013 and superconducting qubits vlastakis2013 . In the latter experiment, the cat state was also released from the microwave resonator pfaff2017 .

Modern applications in an open quantum-communication and distributed quantum-networking architecture could benefit from cat states that propagate over some distances. As long as superconducting transmission lines exist for short distances only, optical fields propagating through low-loss optical fibres in the (near) visible part of the electromagnetic spectrum are required. With this backdrop, purely optical realizations of Schrödinger-cat states morin2014 ; jeong2014 ; ulanov2017 ; jeannic2018 as well as approximate cat states ourjoumtsev2006 ; ourjoumtsev2007 ; nielsen2006 ; takahashi2008 ; lvovsky2009 ; namekata2010 ; gerrits2010 have been demonstrated, but so far only in a heralded fashion. While great effort is under way to release cat states in a quasi-deterministic manner yoshikawa2013 , our experiment follows a different route. We deterministically create entangled light-matter Schrödinger-cat states by reflecting coherent laser pulses from an optical cavity containing a single trapped atom in a controlled superposition of two spin states wang2005 . We then employ the entanglement to control the flying optical cat state by means of a coherent rotation and subsequent measurement of the atomic spin direction. In principle, this measurement is possible even when the atom and light are far apart.

The experiment reported here integrates the established toolbox of discrete variable qubits, like single photons in a superposition of orthogonal polarization modes, with continuous variable qubits, in our case cat states described by a superposition of coherent states with opposite phases. Such cat states have very promising applications in quantum information processing ralph2003 ; gilchrist2004 , mainly because they allow for loss correction cochrane1999 ; leghtas2013 ; bergmann2016 and possibly for fault-tolerant quantum computing lund2008 . One could therefore envision a hybrid networking architecture where short- to mid-distance communication employs deterministically emitted cat states, and long-distance communication uses probabilistically produced photons in combination with success heralds.

Experimental set-up

We employ a single 87Rb atom trapped at the centre of a high-finesse ($6{\times}10^{5}$) optical cavity (Fig. 1; see Methods). The atom acts as a three-level system consisting of the two ground states, $|{\downarrow}\rangle$ and $|{\uparrow}\rangle$, separated by a microwave transition, and one excited state $|{e}\rangle$. The atom in state $|{\uparrow}\rangle$ is strongly coupled to the fundamental mode of the cavity with parameters $(g,\kappa,\gamma)=2\pi{\times}(7.8,2.5,3.0)\,\mathrm{MHz}$. Here, $g$ denotes the atom-photon coupling constant, $\kappa$ is the cavity field decay rate, and $2\gamma$ is the spontaneous atomic decay rate on the transition $|{e}\rangle\rightarrow|{\uparrow}\rangle$. The cavity is actively stabilized on this transition with a wavelength of $780\,\mathrm{nm}$. It is furthermore single-sided in the sense that the optical losses $\kappa$ are dominated by the transmission of the mirror through which light is coupled into and out of the cavity with rate $\kappa_{r}=2\pi{\times}2.3\,\mathrm{MHz}$.

Protocol

The protocol wang2005 to create cat states is outlined in Fig. 2a. Starting with an atom in the coupling state $|{\uparrow}\rangle$, we apply a $\pi/2$ pulse to generate an equal superposition state $(|{{\uparrow}}\rangle+|{{\downarrow}}\rangle)/\sqrt{2}$. Next, we reflect a coherent pulse $|{\alpha}\rangle=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}(\alpha^{n}/\sqrt{n!})|{n}\rangle$ with an arbitrary amplitude $\alpha$ (and the photonic Fock states $|{n}\rangle$) from the cavity. Here, the mean photon number $\langle{n}\rangle$ is given by $|\alpha|^{2}$. This creates a phase shift that depends on the state of the atom duan2004 ; reiserer2015

[TABLE]

Thus, after the reflection, the state is $(|{{\uparrow}}\rangle|{\alpha}\rangle+|{{\downarrow}}\rangle|{-\alpha}\rangle)/\sqrt{2}$, an entangled state of the atom and the light field. With the initial state of the light field, $|{\alpha}\rangle$, being classical and potentially macroscopic, the entangled atom-light system is our ‘Schrödinger cat’. A subsequent $\pi/2$ spin rotation on the atom produces the state

[TABLE]

where the light is in an odd or even superposition of ‘alive’, $|{\alpha}\rangle$, and ‘dead’, $|{-\alpha}\rangle$, entangled with the $|{{\uparrow}}\rangle$ and $|{{\downarrow}}\rangle$ eigenstates of the atom, respectively. Performing a state detection on the atom, the light part of the state (2) is projected onto either an odd cat state $(|{\alpha}\rangle-|{-\alpha}\rangle)/\mathcal{N}^{-}$ if the atom is observed in $|{{\uparrow}}\rangle$ or an even cat state $(|{\alpha}\rangle+|{-\alpha}\rangle)/\mathcal{N}^{+}$ if the atom is found in $|{{\downarrow}}\rangle$. Here, even and odd refers to the photon number parity of the cat states in Fock space, that is, a state containing only even and odd photon numbers. $\mathcal{N}^{\pm}$ are the respective normalization factors. Of course, the projection of the entangled atom-light state is inherently random, and thus produces either an odd or an even cat state in each trial. Predictably obtaining one specific state would require feedforward from the atomic spin measurement to the cat state using unitary transformations jeong2002 . More complex cat states with arbitrary superposition coefficients can be generated by suitable rotations of the atomic state before the measurement. The whole protocol described above lasts $45\,\mathrm{\mu s}$.

The created cat states propagate in a well-defined optical mode and can be sent through an optical fibre that may be part of a larger quantum network. In our experiment, we send the light to a homodyne detector for characterization lvovsky2009 (see Methods). Each measurement returns an amplitude of the optical field for a given phase value. The full optical state is defined by its distribution in phase space, the Wigner function $W(q,p)$ that we reconstruct from a larger set of measurements. Here, $q$ and $p$ are the field quadratures spanning phase space. Our reconstruction basis is the density matrix $\rho$ in truncated Fock space.

Results

We observe the characteristic Wigner functions (Fig. 2b,c) with two Gaussian peaks and interference fringes in the centre that encode the coherent nature of the superposition state schleich1991 . The even cat state displays a local maximum and the odd cat state a local minimum at the centre of the Wigner distribution. The fidelities of these measured states with ideal even and odd cat states are $59.2(6)\%$ and $51.2(6)\%$, respectively. Applying a correction dariano1995 for $25\%$ combined propagation and detection loss, we obtain $68.0(9)\%$ and $62.8(8)\%$, respectively.

A characteristic feature of cat states, which has no classical explanation, is the existence of negative regions in the Wigner function. We observe a minimum value of $-0.016(4)$ for the odd cat state with $\alpha=1.4$, as shown in Fig. 2b. If we correct for propagation and detection losses, the minimum value becomes $-0.083(5)$ (Fig. 2b,c). An intrinsic property of even cat states is ‘squeezing’, that is, the uncertainty of one of the field quadratures, $\Delta p$, is smaller than for a coherent state buzek1992 . Experimental results are displayed in Fig. 3a that shows the minimal width of the measured quadrature distributions as a function of $\alpha^{2}$. We observe squeezing with a maximal amount of $1.18(3)\,\mathrm{dB}$ at $\alpha^{2}=0.5$ (for details see Supplementary Section S2). The measured values follow the theoretical prediction $\Delta p/\Delta p_{\textnormal{vac}}=\sqrt{1-4(1-L)\alpha^{2}/(1+\exp{2\alpha^{2}})}$ with relative optical losses $L$ and the uncertainty of the vacuum state $\Delta p_{\textnormal{vac}}=1/\sqrt{2}$. With increasing $\alpha^{2}$ the width $\Delta p$ increases due to experimental noise in the optical phase $\phi$ (angle between the two Gaussian distributions in phase space) of $\Delta\phi=0.06\pi$, attributed to residual fluctuations of the cavity and laser frequencies of $0.2\,\mathrm{MHz}$.

Losses. It is now worthwhile to discuss the optical losses. These losses reduce the visibility $V$ of the non-classical coherences spagnolo2009 which appear as fringes in the Wigner function. To quantify this effect, we express the visibility as the difference between even- and odd-cat Wigner functions at the centre of the cat state, $V=\frac{\pi}{2}(W_{\textnormal{even}}(0,0)-W_{\textnormal{odd}}(0,0))$. The theoretical calculation (see Methods) gives $V=\sinh(2(1-L)\alpha^{2})/\sinh(2\alpha^{2})$ for an initially ideal cat state. Measured visibilities are presented in Fig. 3b.

In our experiment, losses stem from the finite cooperativity of the atom-cavity system, $C=g^{2}/(2\kappa\gamma)=4.1$, and the reduced escape efficiency $\eta_{\textnormal{esc}}=\kappa_{r}/\kappa=0.92$ of the cavity, which does not reflect all of the incoming light. For a coherent input of size $\alpha$ the effective size of the output cat state, $\alpha_{\textnormal{out}}=|\alpha_{+}-\alpha_{-}|/2$, that is the phase-space distance between the two coherent contributions, is obtained from input-output theory (see Methods) as $\alpha_{\textnormal{out}}=\eta\alpha$ with

[TABLE]

This corresponds to a loss of $L_{\textnormal{cav}}=1-\eta^{2}=34\%$.

Remarkably, not all these losses reduce the coherence of the created cat state like a classical absorber would do. This is caused by some losses that effectively occur before the cat state is created. Theoretically, we find that the effective coherence-reducing losses $L_{\textnormal{eff}}$ are (see Methods)

[TABLE]

(much less than $L_{\textnormal{cav}}$). Further losses $L_{\textnormal{p+d}}$ occur during propagation and detection of the cat state. These are mainly optical absorption losses $L_{\textnormal{o}}=14\%$. A small mode mismatch between the signal and the local oscillator used in the homodyne set-up can be treated as an additional loss $L_{\textnormal{m}}=6.0\%$. The detector there comprises two photodiodes with $L_{\textnormal{d}}=1.5\%$. The subtracted photodiode signals are fed into a field-programmable gate array (FPGA) for further processing. We identified several more noise sources that add effective losses of $L_{\textnormal{n}}=5.5\%$ in total (Supplementary Section S1). Combining these loss channels according to $1-L_{\textnormal{p+d}}=\prod_{i}(1-L_{i})$ with $i\in\{\textnormal{o},\textnormal{m},\textnormal{d},\textnormal{n}\}$, we obtain $L_{\textnormal{p+d}}=25\%$. The fringe visibilities in our reconstructed states (solid line in Fig. 3b) match the theoretical model with combined total losses $L_{\textnormal{det}}=46(1)\%$. In addition to the losses $L_{\textnormal{eff}}$ and $L_{\textnormal{p+d}}$, the observed visibility suffers from an imperfect atomic state detection as well as laser or cavity phase fluctuations. Yet $L_{\textnormal{det}}$ is well below the threshold of $50\%$, where the Wigner function becomes entirely positive spagnolo2009 . While the protocol puts no intrinsic constraint on the size of produced cat states, in our experiment the size is limited by these losses to $\alpha^{2}\lesssim 4$.

Generalized cats.

The even and odd cat states shown so far are special cases of more general coherent state superpositions of the form

[TABLE]

with parameters $\alpha$, $\phi$, $\theta$ and $\xi$. Our experiment allows the control of all these parameters, the phase and modulus of $\alpha$, the optical phase $\phi$ between the coherent contributions, the superposition phase $\theta$ that determines even or odd cat states and the population fraction of the two coherent contributions $\xi$. This opens the possibility to create more complex states that could be useful for continuous-variable error correction codes cochrane1999 ; leghtas2013 ; bergmann2016 .

Specifically, the angle of the last spin rotation (step 4 in Fig. 2a) controls the relative amplitude of the two coherent state contributions in the generated cat state. Scanning the Raman-pulse area $\xi$ from [math] to $\pi/2$ continuously transforms the optical state (after measuring the atom) from a coherent state into a cat state with equal amplitudes of coherent contributions (Fig. 4a).

The cat state in equation (5) can be continuously tuned from an even cat state into an odd cat state via a change of the phase $\theta$ of the final $\pi/2$ rotation (step 4 in Fig. 2a). This phase is imprinted onto the observed interference fringes in the Wigner function. Figure 4b shows the continuous transition from an even into an odd cat state and vice versa for $0\leq\theta\leq\pi$. The optical phase $\phi$ is varied between [math] and $2\pi$ via a detuning between the impinging light and the cavity resonance. On resonance, $\phi=\pi$. Data showing control over $\alpha$, $\phi$, $\theta$ and $\xi$ in more detail are presented in Supplementary Section S3.

Entanglement. So far we have used the light-matter entanglement provided by the Schrödinger-cat state to produce an optical cat state by making a measurement on the atom. We now show that light and matter are indeed entangled by observing an entanglement witness. To this end, we measure the atomic state in three different detection bases via the application of spin rotation pulses. Representing the spin in terms of Pauli matrices $\sigma_{i}=\{\sigma_{I},\sigma_{x},\sigma_{y},\sigma_{z}\}$ and the corresponding optical state as joint Wigner functions vlastakis2015 $W_{i}=\{W_{I},W_{x},W_{y},W_{z}\}$, which play the role of Stokes parameters, allows us to express the full density matrix of the entire atom-cat system as $\rho_{ac}=\frac{1}{2}\sum_{i}\sigma_{i}\otimes W_{i}$. Here $W_{I}$ is the average photonic density matrix with the atom traced out. $W_{x,y,z}$ are sums of photonic density matrices weighted with the outcome of the atomic spin measurement of $\pm 1$ for outcomes $|{{\uparrow}}\rangle$ and $|{{\downarrow}}\rangle$, respectively, in the measurement bases $x$, $y$ and $z$. Wigner representations of our measured $W_{i}$ are displayed in Fig. 5. The obtained density matrix $\rho_{ac}$ has a negativity vidal2002 $N=0.057(5)$ for an input state with $\alpha=1.4$. The statistically significant positive value witnesses the entanglement.

Quantum gate. The reflection mechanism (equation (1)) that has so far been used to create entanglement is essentially a quantum-logic gate between an atom and a light field. Here, the coherent states $|{\alpha}\rangle$ and $|{-\alpha}\rangle$ form an orthogonal qubit basis as long as $\alpha$ is large enough. For our value, $\alpha=1.4$, the overlap of these basis states is $|\langle{-\alpha|\alpha}\rangle|^{2}=\exp(-4|\alpha|^{2})=3.9{\times}10^{-4}$, small enough for a good qubit. To probe our mechanism as a gate, we employ a basis set of input states $|{{\uparrow},\alpha}\rangle,|{{\uparrow},-\alpha}\rangle,|{{\downarrow},\alpha}\rangle,|{{\downarrow},-\alpha}\rangle$. In this basis, the gate acts as a CNOT nielsen2000 with the truth table

[TABLE]

Here the atom serves as the control qubit that can flip the optical target qubit. We characterize this gate with coherent pulses containing a mean photon number of $|\alpha|^{2}=\langle{n}\rangle\approx 2$, the same value as in the measurements confirming the entanglement between the atom and the light field. The quantum nature of the gate has thus been demonstrated in the context of Fig. 5. The classical truth table is shown in Fig. 6 and exhibits the expected CNOT behaviour. Here, the optical losses change the input states $|{\pm\alpha}\rangle$ to output states $|{\pm\alpha_{\textnormal{det}}}\rangle$ with $\alpha_{\textnormal{det}}=\alpha\sqrt{1-L_{\textnormal{det}}}=1.0$. This change is well-characterized and thus predictable, and the overlap between the two output states is still small ($1.8{\times}10^{-2}$). We calculate the mean fidelity of the observed output states with the expected output states and find values around $86\%$. The reduction from $100\%$ comes mainly from phase noise due to cavity and laser frequency fluctuations that broaden the coherent-state Wigner function of the reflected field in the azimuthal direction. If we discriminate the coherent state qubits simply between negative and positive $q$-quadratures, the fidelity of the truth table becomes $96\%$. Note that the gate acts as a CPHASE in the $|{{\uparrow}}\rangle/|{{\downarrow}}\rangle\otimes|{\textnormal{cat}_{\pm}}\rangle$ basis.

Conclusion

The experiment has demonstrated unprecedented control over all relevant parameters of optical cat states, making them a resource for continuous-variable quantum information processing with coherent-state superpositions as qubits ralph2003 ; gilchrist2004 . The useful size of these qubits depends on the optical losses, elimination of which is the key challenge for future improvements. Optical losses in the cavity (equation (4)) can be reduced with improved cavity parameters. For example, parasitic losses $\kappa-\kappa_{r}$ could be lowered by reducing surface scattering and absorption. Likewise, the atom-cavity coupling rate $g$ could be increased with a smaller mode volume, either by reducing the mirror distance or by decreasing the mirror radius of curvature. Here, microscopic fibre cavities offer improvements by more than an order of magnitude. Propagation losses can be mitigated by loss-correction codes that are possibly similar to those that have already been tested in the microwave domain ofek2016 . Optical cats as qubits could therefore be a promising alternative to single photons for quantum communication in a future quantum internet nielsen2006 ; kimble2008 , at least when losses are not too severe.

It should be emphasized that in contrast to all protocols so far realized with single optical photons as qubits reiserer2015 , the creation and detection of Schrödinger-cat states as implemented here prepares and verifies an entangled light-matter state in each trial, without any post-selection. One could utilize this entanglement in a hybrid Bell test teo2013 ; kwon2013 ; vlastakis2015 that takes advantage of the deterministic nature of the entangling operation as well as the near-unity detection efficiency for both the atomic and the photonic state. Our atomic qubit could also be mapped to the polarization of a single photon through the readout process reported in ref. kalb2015 . This would extend our atom-light entanglement towards purely optical photon-light entanglement andersen2015 , a hybrid entanglement between discrete and continuous variables states of light.

Methods

Experimental set-up. A schematic of the set-up is shown in Fig. 1. We used an optical Fabry-Pérot cavity with a mirror separation of $0.5\,\mathrm{mm}$ and finesse of $F=6{\times}10^{4}$. The relevant cavity quantum electrodynamics parameters were $(g,\kappa,\gamma)=2\pi(7.8,2.5,3.0)\,\mathrm{MHz}$, placing our atom-cavity system in the strong coupling regime. In our notation, $g$ is the atom-photon coupling rate, $\kappa$ the cavity field decay rate and $2\gamma$ the spontaneous decay rate of the excited atomic state $|{e}\rangle$. With this set-up, the atoms are initially trapped in a magneto-optical trap and transferred into the cavity. Once an atom arrives at the centre of the cavity, a three-dimensional standing-wave optical lattice is switched on. We used a red-detuned dipole trap ($1064\,\mathrm{nm}$) perpendicular to the cavity axis along the $x$ direction in Fig. 1 and two blue-detuned ($771\,\mathrm{nm}$) repulsive standing waves in the $y$ and $z$ directions. The blue-detuned traps confine the atom to their standing wave nodes, where the intensity-dependent (and therefore fluctuating) light shift is small. The atom is held in this configuration on average for $10\,\mathrm{s}$. We monitored the atom with a camera and a high-numerical-aperture ($\textnormal{NA}=0.4$) objective. Conditioned on the presence of an atom in the cavity, we started the experimental protocol. Initially, the atom is pumped to the state $|{{\uparrow}}\rangle=|{5^{2}S_{1/2},F{=}2,m_{F}{=}2}\rangle$ via a right-circularly polarized pump laser along the cavity axis on the $F{=}2\leftrightarrow F^{\prime}{=}3$ transition. The pump light is applied for $120\,\mathrm{\mu s}$, and successful pumping of the atom is heralded by a reduction of the cavity transmission thompson1992 (44). After this step, the atom can effectively be considered as a three-level system consisting of the states $|{{\uparrow}}\rangle$, $|{{\downarrow}}\rangle=|{5^{2}S_{1/2},F{=}1,m_{F}{=}1}\rangle$ and $|{e}\rangle=|{5^{2}P_{3/2},F^{\prime}{=}3,m_{F}{=}3}\rangle$. Spin rotations and the generation of superposition states of $|{\uparrow}\rangle$ and $|{\downarrow}\rangle$ are realized with a pair of Raman lasers (green arrow in Fig. 1). The frequency difference between these two orthogonally polarized laser beams corresponds to the frequency difference between the states $|{\downarrow}\rangle$ and $|{\uparrow}\rangle$ of $6.8\,\mathrm{GHz}$. The Raman lasers are $131\,\mathrm{GHz}$ red-detuned from the 87Rb D1 transition at $795\,\mathrm{nm}$ and impinge onto the atoms from the side. Each application of the cat state protocol is followed by Sisyphus cooling, which is applied to the atom for $1700\,\mathrm{\mu s}$. The entire protocol is repeated at a rate of $500\,\mathrm{Hz}$.

Homodyne detection. We characterized the optical states via balanced homodyne detectionlvovsky2009 . In this process, the signal is mixed with a $1.8\,\mathrm{mW}$ local oscillator (LO) beam of a continuous-wave laser on a 50/50 non-polarizing beamsplitter (NPBS). The intensities of the two output beams are measured on separate photodiodes (Hamamatsu S3883 with glass windows removed) and subtracted electronically. This difference signal is proportional to the signal beam amplitude and amplified to a measurable level. We recorded a time trace of each signal pulse with a field-programmable gate array (FPGA), and averaged the amplitude over the known temporal mode of the pulse. This results in one electric field amplitude (quadrature) in each homodyne measurement. The quadrature is a projection of the amplitude in phase space under a projection angle that is given by the relative phase of the LO with respect to the signal beam. We do not interferometrically stabilize this phase in our homodyne measurements. The phase thus changes due to thermal drifts of the employed optics and a $1\,\mathrm{Hz}$ detuning of the LO that ensures uniform sampling over all projection angles. For each homodyne measurement the relative phase between signal and LO is determined separately. To this end, the power of the reflected beam is increased by a factor of $30$ for a sufficient signal-to-noise ratio and the LO phase is swept for $100\,\mathrm{\mu s}$ over $2\pi$ to record a full interference oscillation. In order not to lose the atom from the intra-cavity dipole traps during this measurement, it is pumped to the non-coupling state $|{{\downarrow}}\rangle$, which is almost completely decoupled from the cavity. In each shot, the phase of the reflected beam with respect to the local oscillator can be determined with an accuracy of $2^{\circ}$. Phase drifts within a few $100\,\mathrm{\mu s}$ between the experiment and the phase determination are negligible.

For state reconstruction, we performed many repeated measurements, each resulting in one quadrature value and a corresponding LO phase that determines the projection angle in phase space. We acquire on the order of $10^{4}$ samples for each measured optical state, and each outcome can be considered as one projective measurement of the density matrix $\rho$. The density matrix in truncated Fock space is obtained via maximum likelihood estimation using the iterative $R\rho R$ algorithm lvovsky2004 (45). This ensures a physical density matrix and allows for the correction of optical losses. We obtain an estimate for the uncertainty (covariance matrix) of each reconstructed density matrix using the Hessian of the likelihood function banaszek1999 (46). The density matrix fully characterizes the quantum state and therefore allows us to compute all derived quantities, such as the Wigner function, photon statistics or fringe visibility. Some quantities, such as squeezing, can directly be derived from the raw data and do not depend on the density matrix reconstruction.

Analytic treatment of cavity losses. Our atom-cavity system is well described by input-output theorykuhn2015 (47) in the limit of slowly varying light intensities and low atomic excitation probability. For simplicity, we assume the case where cavity, atom and coherent input light $|{\alpha}\rangle$ are on resonance, and all amplitudes are real. We consider four modes: cavity reflection $|{r}\rangle$, cavity transmission $|{t}\rangle$, mirror losses $|{m}\rangle$ and scattering via the atom $|{a}\rangle$. The modes can be considered coherent with amplitudes

[TABLE]

respectively. Here, $N$ is the number of coupling atoms, $N=0$ for $|{{\downarrow}}\rangle$ and $N=1$ for $|{{\uparrow}}\rangle$, $\kappa_{i}$ are optical field decay rates into the respective modes and $\kappa=\kappa_{r}+\kappa_{t}+\kappa_{m}$.

With a coherent input $|{\alpha}\rangle$ of amplitude $\alpha$, the output will be a superposition of two coherent fields of amplitude $r_{{\downarrow}}$ and $r_{{\uparrow}}$, which we call $|{r_{\downarrow}}\rangle$ and $|{r_{\uparrow}}\rangle$, respectively. If the moduli of the two amplitudes differ, the resulting cat state in phase space will be off-centred, which is easily correctable with a displacement. In our experiment however, the two amplitudes are nearly identical. The total size of the cat state is given by the peak separation

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where we use the definition

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The total losses can be expressed as $L_{\textnormal{cav}}=1-\alpha_{\textnormal{out}}^{2}/\alpha^{2}=1-\eta^{2}\ (=0.34)$.

Henceforth we consider the reflected light modes $|{r_{{\uparrow}}}\rangle$, $|{r_{{\downarrow}}}\rangle$ and the loss modes $|{l_{{\uparrow}}}\rangle:=|{t_{{\uparrow}}}\rangle|{m_{{\uparrow}}}\rangle|{a_{{\uparrow}}}\rangle$ and $|{l_{{\downarrow}}}\rangle:=|{t_{{\downarrow}}}\rangle|{m_{{\downarrow}}}\rangle|{a_{{\downarrow}}}\rangle$, which have overlaps of

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We reflect the light when the atom is in an equal superposition of $|{{\uparrow}}\rangle$ and $|{{\downarrow}}\rangle$ and probe the atom after a consecutive $\pi/2$ rotation with phase $\theta$. The produced optical state immediately after the reflection and state detection of the atom is

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for the atom measured in $|{{\downarrow}}\rangle$, and similar with $\theta\rightarrow\theta+\pi$ for the atom in $|{{\uparrow}}\rangle$. Light in the loss modes will be dissipated in the environment. The remaining optical state $\rho$ is obtained by tracing out the losses:

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Here, the coherence terms are reduced due to optical losses by a factor $\exp(-2(1-\eta)\alpha_{0}^{2})$. The state $\rho$ is equivalent to a cat state of original amplitude $\alpha_{0}=\sqrt{\eta}\alpha$ that has undergone coherence-reducing intensity losses spagnolo2009 of $L_{\textnormal{eff}}=1-\eta\ (=0.19)$. Thus, only a part of the total cavity losses $L_{\textnormal{cav}}=1-\eta^{2}$ affects the coherences. In terms of optical depth, the coherence-reducing losses are exactly half the total losses.

The Wigner function $W$ of $\rho$, given for a generic lossy cat state in ref. spagnolo2009 , is

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It consists of two Gaussian peaks of separation $\sqrt{8}\eta\alpha$ and a fringe term in the centre whose amplitude is reduced by $\exp(-2(1-\eta)\alpha_{0}^{2})$ through the losses. The fringe centre $(q_{0},p_{0})=((r_{{\downarrow}}+r_{{\uparrow}})/\sqrt{2},0)$, is equal to $(0,0)$ in our experiment, because the reflection amplitudes $r_{{\downarrow}}$ and $r_{{\uparrow}}$ have the same magnitude. The interference fringe visibility defined by $V=\frac{\pi}{2}(W_{\textnormal{even}}(q_{0},p_{0})-W_{\textnormal{odd}}(q_{0},p_{0}))$ becomes

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taking into account the normalization of $W$. If the coherent contributions in the final cat states have little overlap, $\eta\alpha_{0}^{2}\gg 1$, the visibility decays exponentially with respect to the losses and to the cat size $\alpha_{0}^{2}$:

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The fringes essentially vanish when the number of lost photons $L_{\textnormal{eff}}\alpha_{0}^{2}$ exceeds one half. This effectively limits the achievable size of a cat state.

Finally we compute the fidelity of $\rho$ with respect to an ideal cat state of the same size

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which yields

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