Quantum State Transfer from a Single Photon to a Distant Quantum-Dot Electron Spin

TL;DR

This paper demonstrates quantum state transfer from a single photon to a distant quantum-dot electron spin with high fidelity, advancing quantum network capabilities through controlled frequency bin manipulation and entanglement measurement.

Contribution

It introduces a method for coherent control of photon frequency bins and achieves quantum state transfer with high fidelity between photonic and electron spin qubits over 5 meters.

Findings

Achieved spin-photon entanglement fidelity of 0.796

Demonstrated quantum state transfer with 78.5% fidelity

Controlled photon frequency bins using electro-optic modulators

Abstract

Quantum state transfer from flying photons to stationary matter qubits is an important element in the realization of quantum networks. Self-assembled semiconductor quantum dots provide a promising solid-state platform hosting both single photon and spin, with an inherent light-matter interface. Here, we develop a method to coherently and actively control the single-photon frequency bins in superposition using electro-optic modulators, and measure the spin-photon entanglement with a fidelity of $0.796\pm0.020$. Further, by Greenberger-Horne-Zeilinger-type state projection on the frequency, path and polarization degrees of freedom of a single photon, we demonstrate quantum state transfer from a single photon to a single electron spin confined in an InGaAs quantum dot, separated by 5 meters. The quantum state mapping from the photon's polarization to the electron's spin is demonstrated along three different axis on the Bloch sphere, with an average fidelity of $78.5\%$.

Equations

$$|\Psi ⟩=\frac{1}{\sqrt{2}}(|\downarrow ⟩|{\omega }_{red}⟩-|\uparrow ⟩|{\omega }_{blue}⟩)$$

$$|{\Psi }^{\prime }⟩=\frac{1}{\sqrt{2}}(|\downarrow ⟩|{\omega }_{red}⟩|H⟩|T⟩-|\uparrow ⟩|{\omega }_{blue}⟩|V⟩|R⟩).$$

$$|\Phi ⟩=\frac{1}{\sqrt{2}}[|\psi {⟩}_p\otimes (|\downarrow ⟩|{\omega }_{red}⟩|T⟩-|\uparrow ⟩|{\omega }_{blue}⟩|R⟩)].$$

$$|{\xi }^{\pm }⟩=(|H⟩|{\omega }_{red}⟩|T⟩\pm |V⟩|{\omega }_{blue}⟩|R⟩)/\sqrt{2},$$

$$|{\chi }^{\pm }⟩=(|H⟩|{\omega }_{blue}⟩|R⟩\pm |V⟩|{\omega }_{red}⟩|T⟩)/\sqrt{2}.$$

$$|\Phi ⟩=\frac{1}{2}[|{\xi }^{+}⟩{\sigma }_z+|{\xi }^{-}⟩-|{\chi }^{+}⟩i{\sigma }_y-|{\chi }^{-}⟩{\sigma }_x]\otimes |\psi {⟩}_s.$$

$$|{\xi }^{\pm }{⟩}_A=(|H⟩|{\omega }_{red}⟩\pm |V⟩|{\omega }_{blue}⟩)/\sqrt{2},$$

$$|{\chi }^{\pm }{⟩}_B=(|H⟩|{\omega }_{blue}⟩\pm |V⟩|{\omega }_{red}⟩)/\sqrt{2}.$$

$$|H⟩$$

$$|D^{+}⟩=(|H⟩+|V⟩)/\sqrt{2}$$

$$|{\sigma }^{+}⟩=(|H⟩+i|V⟩)/\sqrt{2}$$

Full text

Self-assembled semiconductor quantum dots (QDs) 1QD-review-1 ; 1QD-review-2 have received considerable attention for quantum information processing. They can serve as narrow-linewidth single-photon sources with a near-unity quantum efficiency, high photon indistinguishability, and high extraction efficiency in monolithic microcavities 2QD-photon-2 ; 2QD-photon-3 ; 2QD-photon-4 ; 2QD-photon-5 ; 2QD-photon-6 . Furthermore, QDs have been deterministically charged with single electrons or holes with long spin coherence time 3QD-coherence-1 . The confined spin state has been initialized by optical cooling 4QD-cooling-1 ; 4QD-cooling-2 ; 15Raman-2 and coherently controlled using picosecond laser pulses 5QD-rotation-1 ; 5QD-rotation-2 . The optical selection rules in a singly charged QD provides a high-fidelity quantum entanglement between the electron spin and the emitted photon s frequency and polarization. Previous demonstrations of QD spin-photon entanglement 7spinphotonentanglement-0 relied on fast photon detectors to resolve the frequency superposition passively 7spinphotonentanglement-1 ; 7spinphotonentanglement-2 ; 7spinphotonentanglement-3 ; 7spinphotonentanglement-4 , and the quantum teleportation from a single photon to a QD spin exploited two-photon interference on a beam splitter which was inherently probabilistic 14gao .

In this Letter, we develop a new technique for active measurement of single-photon frequency-bin superposition using a phase-locked electro-optic modulator ($p$-EOM). We also demonstrate quantum information transfer 6network from a single photon to a distant electron spin by Greenberger-Horne-Zeilinger (GHZ) state projection on the frequency, path and polarization degrees of freedom of the single photon. A layout of the experiment is depicted in Fig. 1a. Suppose Alice has a negatively charged single InGaAs QD housed in a 4.2 K bath cryostat. With an external magnetic field of 2.8 T applied in Voigt geometry, the spin ground states $|\downarrow\rangle$, $|\uparrow\rangle$ and one of the trion states $|\downarrow\uparrow\Downarrow\rangle$ form a $\Lambda$ system (see left inset of Fig. 1a). Bob, who is separated by 5 meter from Alice, aims to remotely prepare Alice’s QD spin in an arbitrary superposition state which Alice doesn’t know.

Firstly, Alice initializes her QD to $|\downarrow\rangle$ by optical cooling, and then near deterministically excite it to $|\downarrow\uparrow\Downarrow\rangle$ by a 400 ps $\pi$-pulse 15Raman-2 ; 15Raman-3 ; note (see Fig. 1b). The excited state $|\downarrow\uparrow\Downarrow\rangle$ decays via two possible channels, generating spin-photon entanglement. Two crossed polarizers in the confocal microscope are used to extinguish excitation laser leakage 16RF-2 , meanwhile projecting the photon polarization to be $(|H\rangle-i|V\rangle)/\sqrt{2}$, where $H$($V$) represents horizontal(vertical) polarization. After that, the generated spin-photon entangled state can be written as (see Supplemental Material suppinfor for details) 7spinphotonentanglement-1 ; 7spinphotonentanglement-2 ; 7spinphotonentanglement-3 :

[TABLE]

where $|\omega_{red}\rangle$ and $|\omega_{blue}\rangle$ are red and blue frequency bins from the two decay channels. This spin-photon entangled state can be directly characterized via active measurement of frequency superposition.

Alice then sends the photon to Bob through a 5-meter optical fiber. Out of the fiber, Bob prepares the photon polarization to be $(|H\rangle+|V\rangle)/\sqrt{2}$. The photon is then split by a polarizing beam splitter (PBS) into two paths, i.e., the $H$ is transmitted ($T$) whereas the $V$ is reflected ($R$), as shown in Fig. 1b. On the two paths, two etalons are placed, and temperature stabilized at the $|\omega_{red}\rangle$ and $|\omega_{blue}\rangle$ frequency bin for the $T$ and $R$ path, respectively. The bandwidth of the etalons are designed to be $\sim\,$1.0 GHz, larger than the single photon’s bandwidth ($\sim\,$0.7 GHz) but smaller than the separation of $|\omega_{red}\rangle$ and $|\omega_{blue}\rangle$ ($\sim\,$18.0 GHz).

Hence, the photon’s frequency, polarization and path qubits are correlated as: $|\omega_{red}\rangle\rightarrow|\omega_{red}\rangle|H\rangle|T\rangle$ and $|\omega_{blue}\rangle\rightarrow|\omega_{blue}\rangle|V\rangle|R\rangle$. Now, the spin-photon entanglement can be written in a four-qubit Greenberger-Horne-Zeilinger (GHZ) type state:

[TABLE]

After that, a half-wave plate (HWP) is inserted in the $R$ path to flip the $V$ polarization to $H$, disentangling the polarization from $|\Psi^{\prime}\rangle$. The target state to be transferred is encoded in the photon’s polarization. Both paths are then placed with a HWP and a quarter-wave plate (QWP) to prepare the polarization in arbitrary superposition: $|\psi\rangle_{p}=\alpha|H\rangle+\beta|V\rangle$. The composite quantum system can be written as:

[TABLE]

To achieve photon-to-spin state transfer, in a similar spirit to ref. 9Bouwmeester97-2 which is a variant of quantum teleportation scheme 8Bennett93 , a crucial step is carrying out joint measurement on the polarization, frequency and path degrees of freedom of the single photon, projecting them onto one of the four GHZ-type states:

[TABLE]

It is remarkable to note that the state $|\Phi\rangle$ can be written in the new basis of these four GHZ-type states,

[TABLE]

This means that, upon measuring the photon with an equal probability of $1/4$ at one of the four states $|\xi^{+}\rangle$, $|\xi^{-}\rangle$, $|\chi^{+}\rangle$, and $|\chi^{-}\rangle$, and applying simple Pauli corrections $\sigma_{z}$, $I$, $\sigma_{y}$ and $\sigma_{x}$, respectively, the initial state of the photon is transferred to the distant spin, which becomes $|\psi\rangle_{s}=\alpha|\downarrow\rangle+\beta|\uparrow\rangle$.

The above scheme requires a spin-photon entanglement as a quantum resource and two classical bits, which can in principle achieve remote preparation of arbitrary state with 100% efficiency. A simpler protocol would be to measure the photon state in arbitrary basis and project the spin in a corresponding state. Such protocol is, however, limited to a maximal success probability of 50% 9Bouwmeester97-2 .

To project and measure the photon in the GHZ-type states, the two paths are combined on the same PBS with a Sagnac-type interferometer. Out of the PBS, the four GHZ states can be separated into two groups: $|\xi^{\pm}\rangle$ exits through output port A, while $|\chi^{\pm}\rangle$ exists through port B, as shown in Fig. 1b. The photon state in port A and B becomes

[TABLE]

To further differentiate $|\xi^{+}\rangle_{A}$ ($|\chi^{+}\rangle_{B}$) with $|\xi^{-}\rangle_{A}$ ($|\chi^{-}\rangle_{B}$), one can analyze the polarization and frequency qubit in the superposition basis $(|H\rangle\pm|V\rangle)/\sqrt{2}$ and $(|\omega_{red}\rangle\pm|\omega_{blue}\rangle)/\sqrt{2}$. Therefore, the four GHZ-type states correspond to the detection events at four single-photon detectors 1, 2, 3, and 4, as shown in Fig. 1b.

The photon frequency qubit is coherently measured using a $p$-EOM and an etalon. As shown in Fig. 2a, the $p$-EOM is used to modulate the two frequency bins $|\omega_{red}\rangle$ and $|\omega_{blue}\rangle$ of the photon, where each bin is transformed into three peaks. When the modulation frequency is set at half of the two bins’ separation, the blue side band of $|\omega_{red}\rangle$ and the red side band of $|\omega_{blue}\rangle$ overlap with each other, which are then filtered out using an etalon. The intensity of this overlapped bin is proportional to the interference term of $|\omega_{red}\rangle$ and $|\omega_{blue}\rangle$, which thus reflects their relative phase. We control the phase of the driving RF field applied on the $p$-EOM to change the measurement basis of the frequency qubit. The coherent nature of this measurement method can be verified by observing a sinusoidal oscillation by measuring the photon intensity when the state of the spin and photon’s polarization is fixed at $(|\downarrow\rangle-|\uparrow\rangle)/\sqrt{2}$ and $(|H\rangle+|V\rangle)/\sqrt{2}$, respectively (see Fig. 2b and Supplemental Materials).

We verify the deterministically generated spin-photon entanglement state in Eq. (1) before performing the state transfer experiment. By replacing the combination of state encoding and GHZ-state measurement modules in Fig. 1b with the frequency qubit measurement module in Fig. 2a, correlation measurements on the spin and frequency qubits can be realized (see Supplemental Material suppinfor for setup details). While frequency qubit measurements are achieved by tuning RF field phase as shown in Fig. 2b, spin qubit measurements are accomplished by utilizing rotation pulse and Ramsey precession to transfer the target spin state population to spin $|\uparrow\rangle$. Then read spin $|\uparrow\rangle$ out with a 10-ns pulse where spin-dependent resonance fluorescence photons 19spinreadout ; 7spinphotonentanglement-1 ; 7spinphotonentanglement-2 ; 7spinphotonentanglement-3 ; 7spinphotonentanglement-4 are registered by a single-photon detector Ds, as shown in Fig. 1b. From the histogram of coincidence counts on ZZ basis given by Fig. 2c, we get ZZ basis fidelity $F_{ZZ}=0.942(28)$ , which is mainly degraded by the imperfection of the spin initialization/measurement pulse. Similarly, from the coincidence histograms presented in Fig. 2d and 2e, visibilities $V_{XX}=0.609(51)$ and $V_{YY}=0.690(31)$ are acquired for coherent basis XX and YY, respectively. These visibilities are mainly limited by a spin dephasing time $T_{2}^{*}=1.7(4)$ ns, where the major dephasing mechanism could be the hyperfine interaction of the electron with the nuclear spins 17nuclear . Except these aforementioned degrading factors, another common factor is QD re-excitation lead by the 400 ps pulse, which is estimated to degrade fidelity by $6.8\%$ note . Furthermore, based on these three axis correlation measurement results, we obtain a spin-photon entanglement fidelity $F=0.796(20)$, which exceeds the classical limit 0.5 by more than 14 standard deviations.

For the remote state transfer experiment, the electron spin coherence needs to be preserved till the photon propagates about five meters away and is measured. We utilize the optical spin echo technique 18Press to prolong the spin coherent time. The pulse sequence is shown in Fig. 3a. At the onset of each period, the spin is prepared to the superposition state $(|\downarrow\rangle$+$|\uparrow\rangle)/\sqrt{2}$ by a $\pi/2$ rotation pulse. After a 19-ns spin free precession and dephasing, a $\pi$-pulse reverses the spin precession direction and thus the spin rephases for another 19-ns symmetry evolution. We extract the visibilities of Ramsey interference fringes at different delay time, and obtain the spin decoherence time $T_{2}=2.7\pm 0.3$ $\mu$s (see Fig. 3b), which is prolonged for about three orders of magnitudes compared to $T{{}_{2}}^{*}$ and becomes sufficient for our experiment.

To test our scheme works for arbitrary spin superposition states, we prepare three mutually unbiased states along three different axis on the Bloch sphere. The aim is a faithful state mapping at a distance from the photon’s polarization to the spin (see also Fig. 4a):

[TABLE]

To evaluate the performance, we measure the state fidelity, i.e., the overlap of the transferred spin state and the ideal one. The transfer fidelity can be deduced from the coincidence counts of photon detection (detectors 1, 2, 3, or 4) and spin detection (Ds). The intrinsic randomness of the outcome of the projected four GHZ-type states is compensated by unitary operations in data post-processing (see Supplemental Materials) depending on the outcome of photon measurement.

Figure 3b-d present the experimental data for input states $|H\rangle$, $|D^{+}\rangle$, and $|\sigma^{+}\rangle$, respectively. The blue bars show the normalized events where the electron spin is measured to be in the correct transferred states, while the gray bars are when the spin ends up in the orthogonal states. We test and average over all the possible four outcomes of the GHZ-type projection (see Eqn. 4). From these data, we calculate the fidelities are $\mathcal{F}_{|H\rangle}=0.851\pm 0.017$, $\mathcal{F}_{|D^{+}\rangle}=0.756\pm 0.027$ and $\mathcal{F}_{|\sigma^{+}\rangle}=0.747\pm 0.027$.

We have demonstrated a new protocol of quantum state transfer from a single photon to a single solid-state spin, as a way to remotely prepare single electron spin in arbitrary superposition state. Although the protocol can in principle work deterministically as in Ref. 9Bouwmeester97-2 , the experimental realization still suffers from various loss, including photon extraction (\sim$$8\%), detection (\sim$$20\%), single-mode fiber coupling (\sim$$40\%), cross polarization (\sim$$50\%), wave plates and mirrors (\sim$$36\%), and frequency selection loss in the $p$-EOMs (\sim$$30\%). The efficiency can be enhanced in the future by, for instance, embedding the QD inside a micropillar cavity 2QD-photon-3 ; 2QD-photon-5 ; 2QD-photon-6 ; micropillar-11 . The loss only decreased the photon-to-spin state transfer success probability, however, doesn’t affect the spin state fidelity. Heralded upon the detection of a single photon after the GHZ-type projection, the distant spin states are demonstrated to be prepared in an arbitrary superposition state with a high fidelity. We expect our results can add a useful toolbox to the investigations of solid-state quantum networks spin-spin .