Large array of Schrödinger cat states facilitated by an optical waveguide

Quantum engineering using photonic structures offer new capabilities for atom-photon interactions for quantum optics and atomic physics, which could eventually lead to integrated quantum devices. Despite the rapid progress in the variety of structures, coherent excitation of the motional states of atoms in a photonic waveguide using guided modes has yet to be demonstrated. Here, we use the waveguide mode of a hollow-core photonic crystal fibre to manipulate the mechanical Fock states of single atoms in a harmonic potential inside the fibre. We create a large array of Schrödinger cat states, a quintessential feature of quantum physics and a key element in quantum information processing and metrology, of approximately 15000 atoms along the fibre by entangling the electronic state with the coherent harmonic oscillator state of each individual atom. Our results provide a useful step for quantum information and simulation with a wide range of photonic waveguide systems.


Introduction
The quantum engineering toolbox developed from free space atom-light interaction experiments over the decades has led to many exciting breakthroughs in the areas of quantum information, metrology, and simulation. Departing from free space interactions towards photonic system interfaces could provide new paradigms [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] , such as increased interaction strength, scalability of the devices, and novel design of functionalities. For instance, atoms trapped by the evanescent fields of nanofibres and photonic crystal slabs have been used to study long-range atom-atom interactions mediated by the coupling of superradiant emission into the structure 4,14 . In quantum communication applications, single collective excitation 6 and light storage [9][10][11] have been demonstrated in nanofibres and hollow-core fibres. Moreover, tight and diffraction-free confinement of atoms in photonic structures was used for optical switching 7,17 . In metrology, cold atoms in hollow-core fibres have also shown potential applications in timekeeping and sensing 8,12,13 . Regardless of these achievements, quantum control and manipulation of both external and internal degrees of freedom of atoms using photonic waveguide modes has not been realised.
Here, we demonstrate coherent excitation of the Fock states of atoms in an optical harmonic potential formed by the fundamental LP01 mode of a hollow-core photonic crystal fibre. We implement an anti-Jaynes-Cummings-type Hamiltonian to excite the coherent harmonic oscillator states 18 (denoted as a coherent state here) and create a Schrödinger cat (SC) state 18,19 using the LP01 mode. Our realisation of the SC states is based on a 3 mm array of atoms trapped in an optical lattice potential inside a hollow-core fibre, as shown in Fig. 1. The diffraction-free optical waveguide allows us to prepare an array of harmonic potentials with nearly identical axial trapping 3 frequency. The axial vibrational energy levels of the lattice form the Fock state basis |n> for our experiments.

Results
Ground state cooling to the Zeeman insensitive state. We first collect an ensemble of 85 Rb atoms by Doppler and sub-Doppler cooling 5 mm above a 4-cm-long open-end hollow-core photonic crystal fibre. The fibre is a hypocycloid-shaped photonic crystal fibre with 1/e 2 mode field radius of 22 μm [11][12][13] . Atoms are then transported into the fibre at a velocity of 2 cm s -1 by an optical conveyer belt using a moving optical lattice formed by a pair of counterpropagating beams 13 . The optical lattice has a period of 410.5 nm, with a power of 110 mW per beam. When atoms are in the fibre, we hold the atoms in a stationary lattice and optically pump them into the magnetic-field-insensitive |F=2, m=0> state to avoid the influence of inhomogeneous magnetic fields, where F denotes the hyperfine ground state of 85 Rb and m is the Zeeman state. The vibrational frequency of the harmonic-like trap formed by each lattice site is ω=2π×400 kHz in the axial direction and ωr=2π×3.5 kHz in the radial direction.
To prepare atoms in the axial vibrational ground state of the optical harmonic potential, we implement Raman sideband cooling (RSC) to cool atoms to the |F=2, m=0, n=0> state 20 . The cooling cycle starts with exciting atoms from |F=2, m=0, n> to |F=3, m=0, n-1> with a pair of linearly and orthogonally polarised Raman lasers (RB1 and RB2) at 821 nm, as shown in Fig. 2a, where a magnetic field of 2 G is applied along the fibre axis to define the quantisation axis and break the Zeeman degeneracy. A linearly polarised depump beam is used to bring atoms back to the |F=2, m, n-1> state in the Lamb-Dicke regime, where the coupling between the internal spin states and motional states is strongly suppressed during spontaneous emission, to preserve the vibrational quantum number. Finally, a π-polarised optical pump beam is applied to accumulate atoms in the |F=2, m=0, n-1> state to complete the cooling cycle. The cooling process continues until all the atoms are in the |F=2, m=0, n=0> state, which is the dark state for all the laser beams. Figure 2b shows the vibrational spectroscopy before and after cooling. The mean vibrational quantum number <n> is determined by taking the ratio of the areas of the first red sideband Arsb and first blue sideband Absb as <n>= (Arsb/Absb)/(1−(Arsb/Absb)). We achieve <n>0=0.25 after cooling from <n>=3.3 before cooling. An optical depth (OD) of one corresponds to approximately 1.5×10 4 atoms, which gives an average of 2 atoms per lattice site 12 .

Study of the coherence between Fock states.
To study the coherence of the spin-Fock states in our system, we use RB1 and RB2 detuned by 3 GHz−ω/2π to excite a two-photon transition between |F=2, m=0, n>≡|↓, n> and |F=3, m=0, n-1>≡|↑, n-1> through a π/2-π-π/2 spin-echo sequence. We observe 37% and 22% contrasts at 0.1 ms and 0.9 ms separation times between the π/2 and π laser pulses, respectively, as shown in Fig. 3a. In our differential light shift compensated optical lattice potential 13 , the spin coherence time of the two spin states |↑> and |↓> measured by a microwave π/2-π-π/2 spin-echo sequence exceeds tens of milliseconds. The decay of the contrast is, thus, mostly due to the amplitude noise of the optical lattice and the radial motion of atoms.
Preparation of a quantum coherent harmonic oscillator state. The oscillation amplitude of a classical harmonic oscillator increases when the oscillator is subject to an external sinusoidal force at its natural frequency. Similarly, for a quantum harmonic oscillator, resonant driving can be achieved when the oscillation frequency of the force is at the vibrational frequency of the quantum harmonic oscillator 21 . The oscillatory force in our experiment is created by the time-dependent dipole force from a pair of counterpropagating driving lasers whose relative frequency ω' is tuned near the vibrational frequency ω of the oscillator. In the quantum mechanical formulation, this corresponds to an initial |n=0> state displaced by displacement operator D(α) to a coherent state |α>. The coherent state created by applying the time-dependent dipole force for a time t can be written as α=(ηΩ/δ)sin(δt/2)e -iδt/2 , where η=√ωR/ω=0.20 is the Lamb-Dicke parameter, ωR is the recoil frequency of the driving lasers, Ω is the ac Stark shift of the atom internal state arising from the driving lasers, and δ= ω-ω' (see Methods).
The coherent state created by a harmonic oscillator can be written in the Fock state basis with a Poissonian distribution as |α>=exp(-|α| 2 /2)Σnα n /√n!|n>. Therefore, its amplitude can be determined by the mean phonon number <n> obtained from vibrational spectroscopy measurements. Figure  However, this amplitude measurement does not reveal information about the coherence between the superposition of Fock states in the coherent state 22 . We demonstrate the coherence in the coherent state by applying two consecutive driving lasers pulses with phase difference ϕ. The mean phonon number after the application of two successive displacement operators D(α) and D(αe iϕ ) is <n>=<n>0+2|α| 2 +2|α| 2 cosϕ, as shown in Fig. 3c. The mean phonon number returning to the initial value <n>0 when the phase ϕ is π indicates the coherence between different Fock states in the coherent state. The fitted data show the coherent state with an amplitude |α|=0.36.

Creation of an array of the SC states.
The SC state is a superposition of two spatially separated but localised classical states entangled with an auxiliary superposition state, where the projection measurement on the auxiliary state (or atomic decay) collapses the SC state into one or the other widely separated classical states (or live or dead cat) 19 . The SC state can be created by entangling a superposition of coherent states with the internal spin states of an atom. For trapped ions SC states, the coherent state α has been created in ions' motional Fock states basis 18,23-27 as large as |α|=12. In SC states with neutral atoms [28][29][30][31] and superconducting circuits 32 , the coherent states have been formed by the photonic Fock states in resonators with |α| around 1-3. These approaches, typically including one SC state in single apparatus, have been used for quantum information processing and for testing the boundary between classical and quantum physics 18,28 . For neutral atoms' motional Fock states, the coherent states have been demonstrated in a harmonic potential 33,34 . Here, we create the SC states in an optical lattice by entangling the coherent states and spin states in an optical waveguide.
The protocol of the SC state preparation is shown in Fig. 4a 23 . A microwave π/2 pulse of 7.5 μs brings the atoms from the vibrational ground state |↓, n=0> into a superposition state of spin up |↑, n=0> and spin down |↓, n=0>. The coherent state |↑, α> is prepared by the driving lasers, whose beat frequency is resonant on the vibrational frequency. The internal spin states are then flipped by a microwave π pulse, followed by another driving lasers pulse with a different phase ϕ to create a coherent state. The resulting SC state can then be written as Equation (1) is an entangled state between a superposition of coherent states and spin states. To confirm this, we measure the interference of the two separated coherent states by applying another microwave π/2 pulse with a phase shift δM relative to the first two microwave pulses, and the state becomes The population of atoms in state |↑> is then determined by the overlap between the two coherent states in phase space. We detect the population of the atoms in state |↑> by absorption detection (OD) with varying phase ϕ. The probability of finding the atoms in |↑> is (3) Figure 4b shows the experimental data of SC state interference fringes for different driving lasers pulse durations. The population of the atoms in |↑> is normalised to 0.5 when the driving lasers are off, and the relative phase is set at δM=π/2. When the microwave phase is set to δM=0 (π), the SC state is termed the odd (even) cat state. The odd (even) cat states only contain odd (even) Fock states, similarly to squeezed vacuum states. We use this approach to verify the validity of the SC state created in our experiment. Figure 4c shows a comparison of the SC states when (1-Ce -(2⟨n⟩ 0 +1)|β| 2 (1-cosθ) cos(δ M +|β| 2 sinθ)) , where β=(1+εe iϕ' )α, e iθ =(e iϕ' +ε)/(1+εe iϕ' ), ϕ'=ϕ−ϕf, and ϕf=ω'Δt is the phase advance of the driving lasers due to the increase in the pulse time Δt. For the residual driving, the parameter ε=0.19 is used to consider the off-resonant dipole force on |↓> (see Methods). The imperfect ground state preparation is taken into account by adding a factor of 2<n>0+1=1.5 into the exponent of Eq. (3).
The fitting parameters |α|, C, and ϕf are displayed in Fig. 4d. We study the effect of radial motion during the free evolution between driving lasers pulses by inserting an additional free evolution time τ, as shown in Fig. 4e. The sequence follows π/2˗ D (α)˗τ˗π˗D (αe iϕ )˗π/2. The data show that the radial motion does not significantly degrade the contrast at our experimental time scale. We also numerically integrate Eq. (4) over the radial distribution of atoms with radial position-dependent parameters of η, Ω, and δ, as shown in Fig. 4f (see Methods). Our results indicate that the radial distribution of atoms is not the main loss mechanism for the interference contrast. The anharmonicity of the optical lattice also causes deviation of the vibrational energy splitting between |n> and |n+1> from the pure harmonic potential as ωanh=ω−ωrec(1+n), where ωrec is the recoil frequency of the lattice. This gives an approximately few percent deviation at the centre of the fibre 35 . Other decoherence mechanisms, such as single-photon scattering of the driving lasers and amplitude fluctuation of the lattice potential 36 , are 10 and 100 times slower than the coherent interaction between the driving lasers and the atoms.

Discussion
Our results show the coherent interaction of single quantum harmonic oscillator states of matter with the light guided in a fibre. We use the fundamental waveguide mode to create entanglement between coherent states and spin states inside a hollow-core fibre. Such a state can enable quantum simulation of a truly one-dimensional system with the assistance of ground state cooling in the radial direction 1,37 . Other than the fundamental importance, the plethora of photonic structures of fibres 38 provides fruitful ways for manipulating atoms. For example, the higher-order modes can be tailored to create multiple lattice sites and different polarisations in the transverse plane, creating a three-dimensional atom array in a cylindrically symmetric photonic structure 39 . Strong atom-light interactions can be achieved by trapping atoms in the cladding modes, which can be patterned by engineering the geometry and thickness of the cladding structure 40  The two optical lattice beams are from an 821 nm Ti-sapphire laser. They separately pass through two independent acoustic-optical modulators (AOMs) to control their relative frequencies.
RB1 is formed by sending part of the Ti-sapphire laser to a 3 GHz electro-optical modulator, followed by a temperature-stabilised solid-state etalon that selects the +1 st order laser light. It then passes through an AOM for switching and frequency shifting. The driving lasers are from an extended cavity diode laser frequency locked at 0.744 GHz from F=3 to F'=2 on the D1 line. It is split into two paths with two independent AOMs before being coupled into the fibre from both ends.
Derivation of the displacement operator. The Hamiltonian of a forced harmonic oscillator in the interaction picture has a form similar to the classical harmonic oscillator where f(t) is a time-dependent force and â and â † are the annihilation and creation operators, respectively. The state after some interaction time t can be characterised by the time-evolution operator where we only consider the first term in the exponent and α is defined as The time-dependent force f(t) in our experiment is provided by a moving standing wave resulting from a pair of counterpropagating lasers whose frequency difference ω' is set to be near the vibrational frequency ω of the harmonic potential. The interaction Hamiltonian between the spin states |↑> of atoms and the lasers can be written as where Ω is the Rabi frequency of the driving lasers, keff is the effective wavenumber of the lasers, ω' is the relative frequency of the driving lasers, and ϕ is the relative phase between the two lasers.
The anti-Jaynes-Cummings-type Hamiltonian has the same form as a forced quantum harmonic oscillator, and α(t) can now be written as Schrödinger cat state interferometer. In the preparation of the coherent harmonic state, ideally, the driving lasers should only excite atoms in |↑, n=0>. However, off-resonant excitation of |↓, n=0> also occurs. We model this residual excitation by the coefficient ε=Δs/(Δs+fhf), where Δs= 744 MHz is the single-photon detuning of the driving lasers from the |F=3> to |F'=2> D1 transition, and fhf=3 GHz is the frequency splitting of |F=3> and |F=2>. After the first microwave π/2 pulse, the driving lasers pulse operates on |↓, n=0> and |↑, n=0> with εα and α, respectively, and the state becomes A π microwave pulse flips the spin, and the second driving lasers pulse operates on |↓, α> and |↑, εα> with εαe iϕ' and αe iϕ' , respectively, where ϕ'=ϕ+ω'Δt, ϕ is the relative phase of the driving lasers and ω'Δt is the phase advance during the extra waiting time between driving lasers pulses. The state then becomes The global phase term ε|α| 2 sinϕ can be ignored in our measurements. The final π/2 microwave pulse with phase δM relative to the first two microwave pulses creates the state where we define β≡(1+εe iϕ' )α and e iθ ≡(e iϕ' +ε)/(1+εe iϕ' ). The population of atoms in the |↑> state P↑ where the centre of the fibre is defined as r=0, wa 2 =W 2 kBTr/2/U is the 1/e radius of the spatial distribution of the atomic cloud in the radial direction, W=22 μm is the 1/e 2 mode field radius, kB is the Boltzmann constant, Tr is the radial atom temperature, and U is the radial trapping potential.
The results for different temperatures are shown in Fig. 4f.