Quantifying quantum coherence of optical cat states

Miao Zhang; Haijun Kang; Meihong Wang; Fengyi Xu; Xiaolong Su; Kunchi Peng

doi:10.1364/PRJ.418417

Abstract

The optical cat state plays an essential role in quantum computation and quantum metrology. Here, we experimentally quantify quantum coherence of an optical cat state by means of relative entropy and the

l_{1}

norm of coherence in a Fock basis based on the prepared optical cat state at the rubidium D1 line. By transmitting the optical cat state through a lossy channel, we also demonstrate the robustness of quantum coherence of the optical cat state in the presence of loss, which is different from the decoherence properties of fidelity and Wigner function negativity of the optical cat state. Our results confirm that quantum coherence of optical cat states is robust against loss and pave the way for the application of optical cat states.

As a superposition of two coherent states, the optical cat state is an important quantum resource for quantum information processing, including quantum computation [1 - 3], quantum teleportation [4 - 6], and quantum metrology [7]. An optical cat state can be experimentally prepared by photon subtraction from a squeezed vacuum state [8 - 13]. It has been shown that the amplitude of optical cat states can be increased by time-separated two-photon subtraction [14] and three-photon subtraction [15]. Meanwhile, optical squeezed cat states are generated by different means [16 - 18]. Based on the prepared optical cat states, several applications have been experimentally demonstrated, including quantum teleportation of cat states [19], tele-amplification [20], preparation of hybrid entangled states and teleportation based on it [21 - 24], amplification of cat states [25], and the Hadamard gate [26].

Quantum coherence, which encapsulates the idea of superposition of quantum states, is a defining feature of quantum mechanics and plays a key role in applications of quantum physics and quantum information [27]. The resource theory of quantum coherence has attracted considerable interest recently [28 - 36]. The relations between quantum coherence and other quantum resources are extensively discussed, such as entanglement [32, 33], asymmetry [34, 35], and path information [36]. Several experiments related to quantum coherence have been demonstrated, including obtainment of maximal coherence via an assisted distillation process [37], observability and operability of robustness of coherence [38], wave-particle duality relation based on coherence measurement [39], the trade-off relation for quantum coherence [40], the relation between coherence and path information [41], the resilience effect of quantum coherence to transversal noise [42], the relation between quantum non-Markovianity and coherence [43], experimental control of the degree of non-classicality via quantum coherence [44], estimation of quantum coherence by collective measurements [45], and quantification of coherence of a tunable quantum detector [46]. The roles of quantum coherence in the Deutsch-Jozsa algorithm [47] and Grover quantum search algorithm [48] have also been discussed recently.

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\hat{ρ}

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Figure 1.(a) Illustration of the decoherence process of optical cat state in phase space. An ideal optical odd cat state with amplitude

α = 1

transmits through a lossy channel with transmission efficiency of 50%, and the Wigner functions are displayed before and after transmission. (b) Experimental setup. A lossy channel consists of a half-wave plate (HWP) and a polarization beam splitter (PBS). OPA, optical parameter amplification with cavity length of 480 mm; SHG, second harmonic generator with cavity length of 480 mm; IF, interference filter (0.4 nm); FC, filter cavity with cavity length of 0.75 mm; FC2, filter cavity with cavity length of 2.05 mm; HD, homodyne detector; LO, local oscillator; APD, avalanche photodiode.

\hat{ρ}

Figure 2.Experimental results. (a)–(d) Wigner functions

W (x, p)

, (e)–(h) projections, and (i)–(l) absolute values of the density matrix elements in the Fock basis when the transmission efficiencies are 100%, 80%, 60%, and 40%, respectively. All the results are corrected for 80% detection efficiency. Only the subspace up to seven photons is shown.

| α ⟩

Figure 3.Dependence of relative entropy,

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norm, fidelity, and negativity of cat states on the transmission efficiency. The red solid curve and black dashed curve correspond to the experimentally prepared cat state and the ideal cat state, respectively. The blue dots represent the experimental results.

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A solid-state titanium-sapphire continuous wave laser generates a 795?nm light source corresponding to the rubidium D1 line, which is divided into two parts. The first part is injected into a second harmonic generator (SHG) cavity with a cavity length of 480?mm, which contains two concave mirrors (

R = 50 ?? mm

), a high reflectivity mirror, a plane mirror with transmissivity of 8%, and a PPKTP (

1 ? mm \times 2 ? mm \times 10 ?? mm

) crystal. The second part passes through the model cleaner and is divided into the local oscillator of a homodyne detector and signal beam of a frequency degenerate OPA. The signal beam is periodically chopped by two acousto-optic modulators (AOMs), which defines distinct time bins by the presence and absence of the signal beam, and thus enables the alternate sequence of locking and photon counting without the signal beam. When the signal beam is absent, we generate a nearly pure squeezed vacuum state with

? 3 ?? dB

squeezing by an OPA with a cavity length of 480?mm, which contains two concave mirrors (

R = 50 ?? mm

), a high reflectivity mirror, a plane mirror with transmissivity of 12.5%, and a PPKTP crystal. The OPA works on the parametric amplification status in our experiment, which is guaranteed by locking the relative phase between the pump beam and signal beam to zero.

A beam splitter composed of a half-wave plate and a polarization beam splitter (PBS) transmits 5% of the squeezed vacuum state toward an APD through a filtering system for photon counting. The filter system consists of an interference filter (0.4?nm) and two filter cavities with fineness of 1200, whose cavity lengths are 0.75?mm and 2.05?mm.

The reflected beam by the beam splitter is measured by a homodyne detector when a photon is detected by the APD. The bandwidth of the homodyne detector is 30?MHz and the detection efficiency is about 80%, which includes four parts: interference efficiency between the signal light and local light (98.5%), quantum efficiency of photodiodes (92%, S3883), 19?dB clearance with the 16?mW local oscillator at 13?MHz of the homodyne detector (corresponding to equivalent efficiency 98.7%), and transmission efficiency (91%). Using the maximum likelihood algorithm, we obtain the density matrix and the associated Wigner function.

Optical cat states can be expressed as a quantum superposition of two coherent states:

| Ψ ? = \frac{1}{\sqrt{N \pm}} (| α ? \pm | ? α ?),

(B1)

where

N_{\pm} = 2 (1 \pm e^{? 2 {| α |}^{2}})

are the normalization factors, and

+

and

?

correspond to the even and odd cat states, respectively. Even and odd cat states can be expressed in a Fock basis as [55]

| Ψ_{+} ? = \frac{1}{\sqrt{\cosh {| α |}^{2}}} \sum_{n = 0}^{\infty} \frac{α^{n}}{\sqrt{n!}} δ_{0 n}^{[2]} | n ?,

(B2)

| Ψ_{?} ? = \frac{1}{\sqrt{\sinh {| α |}^{2}}} \sum_{n = 0}^{\infty} \frac{α^{n}}{\sqrt{n!}} δ_{1 n}^{[2]} | n ?,

(B3)

respectively.

δ_{0 n}^{[2]} = 0

δ_{1 n}^{[2]} = 1

when

n

is odd and

δ_{0 n}^{[2]} = 1

δ_{1 n}^{[2]} = 0

when

n

is even.

Although an ideal cat state is expressed in the infinite dimensional Fock basis, it can be expressed in the finite dimension when the density matrix can characterize the information of the cat state. For example, the odd cat state with amplitude of 1.3 is represented by a density matrix with 11 dimensions [10], and the even cat state with amplitude of 1.61 is represented by a density matrix with 10 dimensions [14]. For an odd cat state with amplitude of

α

, the density matrix elements are expressed as

\frac{1}{\sinh {| α |}^{2}} \frac{α^{m + n}}{\sqrt{m! n!}} δ_{1 m}^{[2]} δ_{1 n}^{[2]}

, and the probability of the

p_{| n ? ? n |}

\frac{1}{\sinh {| α |}^{2}} \frac{α^{2 n}}{n!} δ_{1 n}^{[2]}

. In the experiment, it is truncated to dimension

d

(corresponding to a photon-number cutoff of

d ? 1

) when the probability of higher photons (

d, \dots, \infty

) is small enough to be ignored (e.g.,??

10^{? 3}

Considering

d

-dimensional Hilbert space, the relative entropy of coherence of the even and odd cat states can be expressed as

C_{rel.ent.}^{even} (α) = \frac{1}{\cosh {| α |}^{2}} (\sum_{n = 0}^{d ? 1} \frac{α^{2 n}}{n!} δ_{0 n}^{[2]} \log_{2} ? \cosh {| α |}^{2} ? \sum_{n = 0}^{d ? 1} \frac{α^{2 n}}{n!} δ_{0 n}^{[2]} \log_{2} \frac{α^{2 n}}{n!}),

(B4)

C_{rel.ent.}^{odd} (α) = \frac{1}{\sinh {| α |}^{2}} (\sum_{n = 0}^{d ? 1} \frac{α^{2 n}}{n!} δ_{1 n}^{[2]} \log_{2} ? \sinh {| α |}^{2} ? \sum_{n = 0}^{d ? 1} \frac{α^{2 n}}{n!} δ_{1 n}^{[2]} \log_{2} \frac{α^{2 n}}{n!}) .

(B5)

The

l_{1}

norm of coherence of the even and odd cat states can be expressed as

C_{l_{1}}^{even} (α) = \frac{1}{\cosh {| α |}^{2}} \sum_{\binom{m, n = 0}{m \neq n}}^{d ? 1} \frac{α^{m + n}}{\sqrt{m! n!}} δ_{0 m}^{[2]} δ_{0 n}^{[2]},

(B6)

C_{l_{1}}^{odd} (α) = \frac{1}{\sinh {| α |}^{2}} \sum_{\binom{m, n = 0}{m \neq n}}^{d ? 1} \frac{α^{m + n}}{\sqrt{m! n!}} δ_{1 m}^{[2]} δ_{1 n}^{[2]} .

(B7)

The dependence of relative entropy and the

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norm of quantum coherence of an odd cat state on its amplitude with different dimensions is shown in Fig.?4. We can see that when the amplitude is less than two, the quantum coherences of cat states in the 16- or 12-dimensional Hilbert space are the same, which indicates the 12-dimensional Hilbert space we used in our experiment is enough to estimate the quantum coherence of cat states with small amplitudes.

Figure 4.Dependence of relative entropy and

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norm of quantum coherence of an optical cat state on its amplitude. The red solid and blue dashed curves correspond the results in the 16-dimensional and 12-dimensional Hilbert space, respectively.