Controllable generation of mixed two-photon states

Quantum information is a promising field that utilizes the nonclassical aspects of physical systems to perform sophisticated tasks of computations and communications. Quantum entanglement plays an important role in the implementation of these tasks. Arguably, the most famous entangled states are the Bell states, which are pure and maximally entangled. They are used in quantum cryptography [1], quantum teleportation [2] and the demonstration of various concepts of quantum mechanics [3, 4]. However, apart from the highly entangled pure states, there exists a vast uncharted region of state space, where states can be simultaneously mixed and entangled [5, 6]. Mixed states are useful in investigations of quantum computing [7], studies of the quantum-classical interface [8] and decoherence channels [9]. These applications motivate an interest in the generation and characterization of mixed states.

Up to now, several methods have been suggested for generating photonic mixed states. In [10], Werner states are produced with controlled addition of white noise to the Bell states. In [11], mixed states are produced by controlling the spectral degrees of freedom of the entangled photons. In [12], more sophisticated schemes for producing broad classes of states are presented. Such schemes employ an incoherent temporal mixing of state amplitudes, several decoherers or a hybrid technique. However, as commented in [12], such schemes require many additional optical components, and are challenging in practice.

In this paper, we report a controllable method for producing mixed states, which requires only a few additional optical components added to a Bell-state generation setup. We use variable polarization rotators (VPRs), placed in the pump and signal beams of a conventional spontaneous parametric down-conversion (SPDC) setup with two type-I crystals. Varying the parameters of the VPRs enables us to obtain mixed states of various purities and degrees of entanglement, which are then characterized by quantum state tomography. We remark that VPRs have been used earlier by Gogo et al [13] in studies of quantum eraser without, however, presenting a detailed study and systematic characterization of the generated states.

The basic principle underlying this method of producing mixed states is to generate incoherent mixtures of orthogonal Bell states with controllable weights. Consider an SPDC setup with two type-I crystals with orthogonal axes, pumped by a laser polarized at −45° [14]. The produced state is the Bell state $|\Phi _-\rangle = (|H\mathrm {_s}H_{\mathrm {i}}\rangle - |V\mathrm {_s}V_{\mathrm {i}}\rangle )/\sqrt {2}$, where H and V denote horizontal and vertical polarizations of down-converted photons, respectively, and the subscripts s—signal and i—idler (hereafter omitted for simplicity) label the spatial modes. If the pump is polarized at +45°, the orthogonal Bell state $|\Phi _+\rangle = (|HH\rangle + |VV\rangle )/\sqrt {2}$ is generated. By blending these two Bell states with different weights, one obtains the incoherent mixture

$$\begin{eqnarray} \rho&=&\alpha|\Phi_-\rangle\langle\Phi_-| + (1-\alpha)|\Phi_+\rangle\langle\Phi_+| \\ \nonumber &=&\frac{1}{2}(|HH\rangle\langle HH|+|VV\rangle\langle VV|)+\Big(\alpha-\frac{1}{2}\Big)(|HH\rangle\langle VV|+|VV\rangle\langle HH|), \end{eqnarray} \tag{ 1 }$$

where α (0 ⩽ α ⩽ 1) is the weight of the state |Φ₋〉〈Φ₋| in the mixture.

To generate such mixed states, one switches the polarization of the pump back and forth between −45° and +45° and averages over a sufficient interval of time to sample both polarization states. Such a switching can be realized by inserting a VPR with electrically driven retardance in the pump beam. By applying a square waveform across the VPR, one achieves fast flippings between the two orthogonal linear polarizations. The parameter α is equal to the duty cycle (DC = τ/T, where τ is the duration of the high voltage and T is the period of the waveform) of the applied square waveform.

The purity P = tr{ρ²} of such states is given by

$$\begin{equation} P=2\Big(\alpha-\frac{1}{2}\Big)^2+\frac{1}{2}. \end{equation} \tag{ 2 }$$

When α is zero or one, one of the Bell states is produced, and the purity is one. When α = 0.5, the two Bell states are mixed with the same proportion, and the purity has a minimum value of 0.5. Thus, changing α enables one to obtain states with different purities.

Different amounts of entanglement can also be obtained by varying the parameter α. The tangle T is a measure of the quantum-coherence properties of a quantum state [15, 16]. It has a value of zero for separable states, and one for maximally entangled states. For the states given by equation (1), the tangle is related to α by

$$\begin{equation} T = (1-2\alpha)^2. \end{equation} \tag{ 3 }$$

Generated states can be characterized by polarization correlation analysis. For the Bell states, the polarization correlation gives a visibility of 100% when measured in the ±45° basis. For the states given by equation (1), the visibility varies with α according to

$$\begin{equation} V = |1-2\alpha|=\sqrt{T}. \end{equation} \tag{ 4 }$$

In particular, when α = 0.5, the generated state is $\rho =\frac {1}{2}(|HH\rangle \langle HH|+|VV\rangle \langle VV|)$, which results in zero visibility of the polarization correlation in the ±45° basis. Note that for the state given by equation (1), $V=\sqrt {T}$; however, this relation is not true in general.

With a single VPR in the pump beam and α set to 0.5, one produces mixed states with minimal purity P = 0.5, see equation (2). However, for a completely mixed state, P = 1/d, where d is the dimension of the density matrix (d = 4 for a two-photon state). The completely mixed state can be written as

$$\begin{equation} \rho_{\mathrm{M}}=\frac{1}{4}(|HH\rangle\langle HH|+|VV\rangle\langle VV|+|HV\rangle\langle HV|+|VH\rangle\langle VH|)=\frac{\mathbb{1}}{4}. \end{equation} \tag{ 5 }$$

Thus, one has to include in the mixture the other two Bell states $|\Psi _-\rangle = (|HV\rangle - |VH\rangle )/\sqrt {2}$, and $|\Psi _+\rangle = (|HV\rangle + |VH\rangle )/\sqrt {2}$ with equal weights. These two Bell states can be generated in the same setup by inserting a VPR with a half-wave retardance in either the signal or idler beam. In this case the VPR transforms the states |Φ₋〉 → |Ψ₋〉 and |Φ₊〉 → |Ψ₊〉. By switching between zero and half-wave retardance of the VPR, incoherent mixtures of |Φ₋〉〈Φ₋| and |Ψ₋〉〈Ψ₋|, or |Φ₊〉〈Φ₊| and |Ψ₋〉〈Ψ₋|, are produced. Hence, by using two VPRs with duty cycles set to 0.5, one placed in the pump beam and the other in the signal or idler beam, one can obtain the state given by equation (5). A Werner state [17] can then be produced by mixing this completely mixed state with the singlet state. Notice that an important family of states, namely the Bell diagonal mixed states, can also be generated if the waveforms and duty cycles of the two VPRs are independently controlled. More generally, one can obtain mixtures of non-maximally entangled states |Ψ_θ〉 = cos θ|HH〉 + sin θ|V V 〉 by rotating the polarization of the pump beam to an arbitrary angle. This enables the generation of even more states such as the Collins–Gisin states [18], given by

$$\begin{equation} \rho_{\mathrm{CG}}(\alpha,\theta)=\alpha|\Psi_{\theta}\rangle \langle \Psi_{\theta}|+(1-\alpha)|HV\rangle\langle HV|. \end{equation} \tag{ 6 }$$

3.1. State preparation

The experimental setup is shown in figure 1. Two beta-barium borate (BBO) crystals of 0.8 mm thickness with orthogonal optic axes are cut at 29.5° for non-collinear frequency-degenerate phase matching. The BBOs are pumped by a frequency-stabilized continuous-wave diode laser at a wavelength of 404.6 nm (Ondax, LM series). The first crystal produces pairs of vertically polarized photons from the pump's horizontal polarization component, while the second crystal produces pairs of horizontally polarized photons from the pump's vertical polarization component. With non-collinear SPDC, the down-converted photons travel in different directions with a cone opening angle of 4°. Down-converted photons of narrow spatial bandwidth are collected into single-mode optical fibers using aspherical lenses (L) with a focal length of 11 mm, placed at a distance of 800 mm from the BBOs. The down-converted photons are filtered with interference filters centered at 810 nm, with 10 nm full-width at half-maximum. Polarization rotations in the optical fibers are compensated for by manual polarization controllers.

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With a HWP in the pump beam at 22.5°, the vertical polarization of the pump is rotated to −45°, and the resulting state is $(|HH\rangle -\mathrm {e}^{\mathrm {i}\phi }|VV\rangle )/\sqrt {2}$, where ϕ is the relative phase. This phase can be set to zero or integer multiples of 2π by manipulating the ellipticity of the pump with two quartz plates that are placed after the HWP, thus producing $|\Phi _-\rangle = (|HH\rangle - |VV\rangle )/\sqrt {2}$.

Two different types of polarization rotators have been used in the experiment. The first one is a photoelastic modulator (HINDS Instrument, I/FS50). It switches between zero and half-wave retardance at a frequency of 50 kHz, with a fixed DC of 0.5. The second one is a liquid crystal retarder (LCR) (Meadowlark, LRC-200). It is configured so that zero retardance is obtained at the 'low' voltage of V_L = 1.39 V. The pump polarization entering the crystal remains at −45°, and |Φ₋〉 is generated. For the 'high' voltage of V_H = 1.64 V, half-wave retardance is obtained. The pump polarization entering the crystal changes to +45°, and |Φ₊〉 is generated. A square waveform with the frequency of 1 Hz is applied to the LCR. The averaging is done over 3 min time intervals. By varying the duty cycle of the square waveform, different mixed two-photon states are obtained, see equation (1).

To produce the completely mixed state, given by equation (5), the photoelastic modulator is placed in the pump beam, and the LCR with DC = 0.5 is placed in the signal beam. The operation voltages of the LCR for down-converted photons are adjusted to V_L = 1.84 V and V_H = 6.95 V. The frequency of the LCR square waveform is kept at 1 Hz, and the averaging is done over 3 min time intervals.

3.2. State characterization

For the calibration of the setup, the polarization correlation analysis in the ±45° basis is performed with the Bell state |Φ₋〉. The constant 'low' voltage is applied across the LCR in the pump beam. The QWPs in the signal and idler arms are fixed at 0°. The HWP in the idler arm is fixed at −22.5° and the HWP in the signal arm is rotated. This causes the coincidence rate between the counts of two detectors in the transmitted ports of the PBSs to vary sinusoidally. The visibility is defined as $\displaystyle V=\Bigg |\frac {N_{\mathrm {+45^\circ }}-N_{\mathrm {-45^\circ }}}{N_{\mathrm {+45^\circ }}+N_{\mathrm {-45^\circ }}}\Bigg |$, where N_+45° and N_−45° are the coincidence rates when the HWP in the signal arm is oriented at +22.5° and −22.5°, respectively. The obtained value of visibility 97.3 ± 1.4% indicates high quality of the produced Bell state. The analysis is then extended to the mixed states prepared with the LCR operating at different DCs, and with the photoelastic modulator. The obtained experimental visibilities are in good agreement with the theoretical expectation of equation (3), see figure 2. One observes a degradation of the visibility as α increases from 0 to 0.5, after which the visibility is gradually restored as α increases from 0.5 to 1.

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Next, quantum state tomography on the generated states is performed. The matrix elements of reconstructed density matrices are shown graphically in figure 3, for α = 0.05, 0.25 and 0.5. The fidelities, F, of the reconstructed density matrices with the states given by equation (1) are calculated. Using the definition $\displaystyle F=\mathrm {tr}\big \{|\sqrt {\rho }\sqrt {\sigma }|\big \}$, where ρ is the maximum likelihood estimator and σ is the target state, fidelities F > 97% are consistently observed, see table. 1.

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Table 1. Fidelities of the reconstructed states with the target states for various DCs of the LCR. F > 97% are consistently obtained for all the states. The last two entries with PEM and $\mathbb{1}/4$ refer to the state generated using only the photoelastic modulator in the pump beam, and the completely mixed state generated using two VPRs, respectively.

α	Fidelity
0	0.9857±0.0005
0.05	0.9886±0.0005
0.25	0.9814±0.0006
0.35	0.9750±0.0012
0.45	0.9776±0.0007
0.50	0.9782±0.0006
0.55	0.9724±0.0012
0.65	0.9727±0.0008
0.75	0.9792±0.0007
0.95	0.9786±0.0012
1	0.9785±0.0003
PEM	0.9890±0.0005
$\mathbb{1}/4$	0.9942±0.0003

As an example, for α = 0.25, the reconstructed density matrix expressed in the HV basis is

$$\begin{eqnarray*}\fl \rho_{{ \alpha=0.25}}\widehat{=}\left( \begin{array}{cccc} \hphantom{-}0.545\hphantom{-0.012i_i} \,\, & \hphantom{-} 0.049+0.012\mathrm{i} \,\, & -0.013+0.038\mathrm{i} \,\, & -0.232+0.110\mathrm{i} \\ \hphantom{-} 0.049-0.012\mathrm{i} \,\, & \hphantom{-} 0.008\hphantom{+0.012i_i} \,\, & -0.005+0.008\mathrm{i} \,\, & \hphantom{-} 0.015+0.004\mathrm{i} \\ -0.013-0.038\mathrm{i} \,\, & -0.005-0.008\mathrm{i} \,\, & \hphantom{-} 0.013\hphantom{+0.012i_i} \,\, & -0.039+0.006\mathrm{i} \\ -0.232-0.110\mathrm{i} \,\, & \hphantom{-} 0.015-0.004\mathrm{i} \,\, & -0.039-0.006\mathrm{i} \,\, & \hphantom{-} 0.434\hphantom{+0.012i_i} \end{array} \right). \end{eqnarray*}$$

The reconstructed state has a tangle of T = 0.2476 ± 0.0049, a purity of P = 0.6295 ± 0.0022 and a fidelity with a target state of F = 0.9814 ± 0.0006.

As one can see from figure 3, an increase of α from 0 to 0.5 causes vanishing of the off-diagonal elements, which indicates the decreasing amount of entanglement of the mixed states. The calculated values of the tangle for the reconstructed states are in good agreement with equation (3), see figure 4(a). The state with almost zero tangle is generated with the photoelastic modulator. It performs better than the LCR at α = 0.5, as the wavelength of the pump laser is at the edge of the working range of the LCR.

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Lastly, one verifies that the method indeed produces states with different amounts of mixedness. The dependence of the purity of the states on α is shown in figure 4(b). The obtained results are close to those predicted by equation (2).

To demonstrate generation of the completely mixed state, the setup is modified by inserting VPRs in the pump and in the signal beams, see section 3.1. The reconstructed density matrix of the state is shown graphically in figure 5. The reconstructed state has a tangle of zero, a purity of P = 0.2615 ± 0.0007 and a fidelity with a target state of F = 0.9942 ± 0.0003. Such a state is separable and highly mixed.

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In summary, an accessible way of generating controllable mixtures of the Bell states with high fidelities is reported. There is no need for extra optical components, except for the VPRs. Furthermore, the method does not require any reconfiguration of the standard setup for Bell states generation, as the VPRs can be easily turned on and off, to generate mixed and pure states, respectively. In addition, it offers the important advantage of interferometric stability. The presented method will facilitate state generation and can be used in various quantum-information processing applications. We are planning to use it for an experimental realization of qubit-pair tomography with witness bases [21].

We thank Dmitry Kalashnikov for his assistance with the experiment and valuable discussions. This work is supported by the National Research Foundation and the Ministry of Education, Singapore, and is co-financed by the European Social Fund and the state budget of the Czech Republic, project no. CZ.1.07/2.3.00/30.0004 (POST-UP).