Frequency domain Stern–Gerlach effect for photonic qubits and qutrits

1. INTRODUCTION

Nonlinear optical processes are widely used as platforms for the generation and manipulation of non-classical states of light. Entangled photon pairs and squeezing are the most common examples [1], though much attention has also been given to unitary operations on frequency-domain states [2–10]. Sum-frequency generation (SFG) [2], four-wave mixing [3–7], and electro-optic modulation [8–10] were employed for the dynamic coupling between different frequencies, allowing for high-fidelity spectral-to-spectral (i.e., where both the input and output are in the same spatial mode) quantum gates [10]. The recent interest in these photonic states as quantum information carriers is motivated mainly by their potentially high dimensionality and the ability to transmit them over long distances in optical fibers and in free space. In spite of the growing advancements in this field, quantum effects incorporating the spatial degree of freedom, in a manner that allows for a spatially separated projection on frequency-basis eigenstates, have been rarely explored to date. These spectral-to-spatial beam splitters are expected to benefit scalability, particularly when higher dimensional qudits are concerned, since they do not rely on additional optical elements such as frequency converters, waveplates, and dichroic prisms.

In this paper, we demonstrate how paraxial photons in quadratic nonlinear media are analogous to two-dimensional (2D) massive particles carrying internal angular momenta (either spin or orbital). These photons can interact with an external effective field, the components of which are given by the external optical pump fields coupling the different frequencies. For two-level photons, the dynamics coincides with the one described by the Pauli equation for spin-1/2 particles, instead of the usual Schrödinger-type analogy for paraxial propagation. Interestingly, the photon’s spectral degree of freedom can become entangled with the spatial ones through a spatial non-uniformity of the nonlinear coupling.

Here, we propose an analogous Stern–Gerlach (SG) effect for photons, which has been recently described for classical light [11]. Such an effect allows for the spatial separation of orthogonal frequency-superposition states, thereby realizing a projection of the quantum state on a different basis otherwise inaccessible within a single nonlinear interaction. We solve analytically for the dynamics and simulate the first-order correlation function of both qubit- and qutrit-photonic states undergoing the aforementioned effect. It is shown that the quantum nature of light is manifested in the form of coherent spatial superposition of the single-photon state as well as two-photon paraxial bunching of the Hong–Ou–Mandel (HOM) type. In contrast to the usual HOM case, here the bunching occurs for photons that are distinguishable owing to their different frequencies. The effect proposed herein allows for a realization, using a single optical element, of spectral-to-spatial beam splitters and tritters, all-optically controlled by the pump field, with applications to quantum information processing in the frequency domain.

2. SG EFFECT FOR PHOTONIC QUBITS

For the sake of simplicity, we first treat the case of two-level photonic qubits. The results of this section can, for the most part, be mapped onto the usual polarization subspace of photons that also exhibits SU(2) symmetry. However, it is of great conceptual merit to discuss them. The crucial difference is that the polarization subspace has only two dimensions, whereas the frequency subspace has unlimited dimensionality. The analysis of effects incorporating two frequencies is merely the most simplified subset of the vast variety of manipulations that can be done in this high-dimensional basis. As presented in the following section, these ideas can be readily generalized to effects that couple more than two frequencies.

Consider a quadratic nonlinear medium with a nonlinearity coefficient ${\chi }^{(2)}$, where a sum-frequency generation (SFG) process takes place. In such media, called nonlinear photonic crystals (NLPCs), the ${\chi }^{(2)}$ nonlinearity is periodically modulated along the propagation direction with a reciprocal lattice constant $2\pi /\mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is the modulation period. The magnitude of $\mathrm{\Lambda }$ is chosen to satisfy the momentum conservation (or phase matching) via an umklapp process for a specific three-wave mixing among the frequencies ${\omega }_1$, ${\omega }_2$, and ${\omega }_{12}={\omega }_2-{\omega }_1$ [12]. We denote the field at frequency ${\omega }_{12}$ as the pump field, coupling modes 1 and 2. This pump field will be considered a coherent state that is strong enough that it can be assumed constant throughout the nonlinear interaction. We shall denote $\hslash {\kappa }_{12}\propto {\chi }^{(2)}\sqrt{I_{12}}$ ($I_{12}$ being the pump intensity) as the typical coupling energy for the interaction.

The dynamics of the aforementioned model is handled within the paraxial quantum optics formalism [13]. This theory assumes that the electric field operator $E^{(+)}$ can be expressed as a product of a carrier wave $e^{i(k_{j}z-{\omega }_{j}t)}$ and a slowly varying envelope (SVE) operator given for each frequency ${\omega }_j$ by

The considered wavepackets have a typical temporal length $\tau$. We assume that their propagation is limited to lengths much smaller than the group velocity mismatch (GVM) distance, given by $L_{\mathrm{GVM}}=\tau /|v_{g1}^{-1}-v_{g2}^{-1}|$. Therefore, it is reasonable to neglect the effects of temporal walk-off and consider the different frequencies to travel at some mean group velocity ${\overline{v}}_g$. We now introduce a spinor notation $\mathrm{\varphi }={({\varphi }_1,{\varphi }_2)}^T$ and rewrite our Hamiltonian as

The system dynamics can be obtained by employing the Heisenberg equation of motion $i\hslash {\partial }_t{\varphi }_j=[{\varphi }_j,H_{\mathrm{SFG}}]$:

The key feature to consider here is that the effective magnetic field $\mathbf{B}$ may vary in space, i.e., $\mathbf{B}=\mathbf{B}(\mathbf{r})$. This is due to the spatial variation of the nonlinear coupling, either induced by the paraxial mode profile of the pump beam, or by spatially varying the nonlinear coefficient. Such variation gives rise to correlations between the spatial and spectral degrees of freedom for single-photon paraxial modes, as will be demonstrated below for the simpler cases of qubits and qutrits.

For two-frequency states, the Bloch sphere representation (Fig. 1) and the role of the SFG process as a rotation of state vectors have been well established [16,17]. Collinear SFG interactions, together with the dichroic beam splitter, therefore constitute the currently known possible manipulations that can be performed on a two-frequency state: state rotation (analogous to a spectral wave plate), and a projection along the $z$ axis of the Bloch sphere (analogous to a spectral filter), on which lie the pair of pure frequency states $|{\omega }_i⟩$ and $|{\omega }_s⟩$. It is therefore interesting to ask whether a projection on the Bloch sphere equator, hosting the frequency superposition states, is also possible within a single SFG interaction.

 

Fig. 1. (a), (b) A two-level photon may occupy two different frequencies, coupled by an external pump field, and have a certain spatio-temporal envelope. (c) The qubit state is represented by a point on the Bloch sphere: pure frequency states are situated on the poles and equal-superposition states lie on the equator. The magnetic field analog, $\mathbf{B}$, was chosen to point to the equator with an angle determined by the pump phase. Hence, the two eigenstates in the direction of $\mathbf{B}$ are superposition states, while the original state $|\psi ⟩$ can be projected onto each of them by employing the proposed SG effect.

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Recently, we have studied [11] the all-optical SG effect, in which a classical idler beam (${\omega }_1={\omega }_i$) enters into a nonlinear crystal with a transverse gradient in the magnitude of $\mathbf{B}$. This can be achieved in a quasi-phase-matched (QPM) interaction by varying the duty cycle of the periodic poling in the transverse direction, as illustrated in Fig. 2(e) [18]. When such a nonlinear crystal is pumped by a collinear pump wave (${\omega }_{12}={\omega }_p$), the incident idler beam is deflected into two composite classical states of the signal (${\omega }_2={\omega }_s$) and idler frequencies, at positive and negative deflection angles with respect to the optical axis. This is an analog of the SG experiment [19], where a beam of silver atoms carrying an internal spin-1/2 was deflected into two discrete angles by a magnetic field gradient.

 

Fig. 2. (a) Single-photon SG interference and (b) its representation on the Bloch sphere. A single-photon wavepacket state in the idler frequency $|{\psi }_i⟩$ is incident on a pumped nonlinear SG crystal and transformed into two spatially separated orthogonal frequency superposition single-photon states. (c) Two-photon SG-HOM interference and (d) its representation on the Bloch sphere. Two distinguishable photon wavepacket states are incident on the crystal, and are deflected into a frequency superposition 2002 state, where $|{\psi }_{\pm }⟩=(|{\psi }_i⟩\pm |{\psi }_s⟩)/\sqrt{2}$. (e) The nonlinearity gradient can be realized by transversely varying the QPM poling duty cycle, $D$; see Ref. [18].

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The all-optical SG effect is more than an aesthetic analogy between classical wave propagation and quantum mechanics. Recently, it has been proven that universal quantum computation schemes can be based on the ${\chi }^{(2)}$ interaction [20], where single-qubit gates can be readily realized in the undepleted-pump regime. In this context, a quantum version of the all-optical SG effect greatly simplifies experimental realizations of frequency-domain quantum state projection on the Bloch-sphere equator [Fig. 2(b)], as it requires only a single nonlinear crystal. It is therefore evident that the all-optical SG effect [11] merits a quantum formulation.

The analytical solution for the quantum case can be obtained in the following manner. In order to solve Eq. (4) for a linear gradient of the nonlinear coupling in the $y$ direction $\mathbf{B}=(B_0+B^{\prime }y)\widehat{\mathbf{B}}$, where $\widehat{\mathbf{B}}=\widehat{\mathbf{x}}\cos {\varphi }_{12}+\widehat{\mathbf{y}}\sin {\varphi }_{12}$ and $B^{\prime }\propto {\kappa }_{12}/W$ [$W$ being the poled region width as in Fig. 2(e)], it is most convenient to transform the spinor $\mathrm{\varphi }$ to the transverse momentum space. This is a Fourier transformation over the transverse coordinates ${\mathbf{r}}_T=(x,y)$, given by $\tilde{\mathrm{\varphi }}({\mathbf{k}}_T,z)=\int {\mathrm{d}}^2{r}_T\mathrm{\varphi }\exp (i{\mathbf{k}}_T·{\mathbf{r}}_T)$. The transformed field $\tilde{\mathrm{\varphi }}$ also happens to represent the far-field operator. In the limit of negligible spatial walk-off, the matrix $\mathbf{M}$ is replaced by a scalar $m\sim \overline{k}$ (where $\overline{k}$ represents a mean wavenumber), and Eq. (4) is diagonalized to the basis of the $\widehat{\mathbf{B}}$ direction. Consequently, there exist two eigenfields ${\tilde{\mathrm{\varphi }}}_{\pm }$ corresponding to the eigenvectors of $\mathrm{\Sigma }·\widehat{\mathbf{B}}$, which propagate according to their respective eigenvalue:

This SG effect for two-frequency photonic states is a simple manifestation of quantum correlations between the spatial degrees of freedom and the spectral (internal) ones. We proceed by writing the output field ${\tilde{\mathrm{\varphi }}}_{\mathrm{out}}$ in terms of the original input field ${\tilde{\mathrm{\varphi }}}_{\mathrm{in}}={({\tilde{\varphi }}_i,{\tilde{\varphi }}_s)}^T$ and the unitary transformation $\mathbf{U}$ that diagonalizes $\mathrm{\Sigma }·\widehat{\mathbf{B}}$, i.e., $\mathrm{\Sigma }·\widehat{\mathbf{B}}={\mathbf{U}}^{-1}{\sigma }_z\mathbf{U}$ [where ${\sigma }_z=\mathrm{diag}(1,-1)$ is the third Pauli matrix]. The general expression for ${\tilde{\mathrm{\varphi }}}_{\mathrm{out}}$ is given by

For simplicity’s sake we consider as a test case $\widehat{\mathbf{B}}=\widehat{x}$, i.e., the phase between the pump wave and the nonlinear modulation pattern is ${\varphi }_{12}=0$. (This is an arbitrary choice, whereas a different selection, such as ${\varphi }_{12}=\pi /2$, would have resulted in the projection of the state along the $\widehat{y}$ basis instead [see Fig. 1(c)]. In a practical experiment, two orthogonal transverse bases such as $\widehat{x},\widehat{y}$ can always be chosen by varying the pump phase by $\pi /2$.) An incident one-photon state in the idler frequency with a spatial envelope $\psi (r)$, written as $|{\psi }_{\mathrm{in}}⟩=|{\psi }_i⟩=|\psi (r)⟩\otimes |{\omega }_i⟩$, is then deflected into the output state:

As a proof of concept, we simulate the propagation of the photodetection amplitude ${\psi }_j(\mathbf{r})=⟨0|{\varphi }_j(\mathbf{r})|{\psi }_{\mathrm{in}}⟩$ ($j=i,s$) and the resulting photodetection probability density $G^{(1)}(\mathbf{r})$ (see Appendix A for details) under realistic experimental conditions. The idler, signal, and pump free-space wavelengths were chosen to be ${\lambda }_i=532\mathrm{nm}$, ${\lambda }_s=452\mathrm{nm}$, and ${\lambda }_p=3000\mathrm{nm}$, respectively. For the nonlinear medium, we chose periodically poled lithium niobate (PPLN) with $d_{33}=27\mathrm{pm}/\mathrm{V}$ and QPM period of 6.98 μm (larger periods can be used if the idler wavelength is in the near-infrared). The poled region was designed to induce a transverse gradient in the nonlinear coupling as in Fig. 2(e) (see Ref. [18] and, for further details, Ref. [11]), with a width of $W=400\mathrm{\mu m}$ and length along the propagation axis of $L=35\mathrm{mm}$. The pump peak power was $P_p=1\mathrm{MW}$ with a waist of 1 mm, giving a peak intensity of roughly $I_p\simeq 64\mathrm{MW}/{\mathrm{cm}}^2$. The simulated input states in the idler and signal frequencies were all Gaussian with a 50 μm waist. Figure 3 presents the propagation and far-field deflection of $G^{(1)}(\mathbf{r})$ for the different input states. The results agree well with the analytic predictions: pure eigenstates are deflected either to the left or to the right, whereas an idler frequency input is deflected into both angles, since it comprises a superposition of the two eigenstates.

 

Fig. 3. Simulated photodetection probability in PPLN with a SG poling pattern [with units ${\mathrm{mm}}^{-1}$ on the left figures and ${\mathrm{mm}}^{-2}$ on the right ones, normalized so that the total probability, by integrating over $y$ (left) or $xy$ (right), is 1]. Propagation along the $z$ axis is presented to the left, while the far-field pattern is given to the right. All input states are Gaussian with a waist of 50 μm. (a), (b) The input state is $|{\psi }_{\mathrm{in}}⟩=|{\psi }_i⟩$ and is deflected into two discrete angles. (c), (d) The input state is an eigenstate $|{\psi }_{\mathrm{in}}⟩=(|{\psi }_i⟩-|{\psi }_s⟩)/\sqrt{2}$, and is deflected only to the left. (e), (f) For the second eigenstate, $|{\psi }_{\mathrm{in}}⟩=(|{\psi }_i⟩+|{\psi }_s⟩)/\sqrt{2}$, the deflection is only to the right.

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An actual experimental design is expected to be sensitive to the quality and magnitude of the nonlinear coupling gradient. Two significant design considerations are worth mentioning. (i) Phase matching. As stated before, a phase mismatch $\mathrm{\Delta }k\ne 0$ between the wavevectors of the interacting photons introduces a $z$ component $B_z\propto \mathrm{\Delta }k$ to the effective coupling field $\mathbf{B}$, thus decreasing the magnitude of the transverse ($x$, $y$) gradient in $|\mathbf{B}|$. (ii) Nonlinear coupling strength. Stronger nonlinearities and pumps result in larger deflection angles; therefore, it is favorable to use crystals with a relatively high nonlinear coefficient, with PPLN being a good example. For crystals with lower nonlinearity, the design should be compensated, for example, by increasing the pump intensity. Moreover, we note that the duty-cycle-based gradient scheme [Fig. 2(e)] is easier to realize in PPLN than in PPKTP, as the strong anisotropy of the latter crystal tends to align the ferroelectric domain boundaries along the crystal’s $y$ axis.

More interesting phenomena involve two-photon interference. We consider an incident state at two different frequencies, $|{\psi }_{\mathrm{in}}⟩=|{\psi }_i{\psi }_s⟩$. The corresponding output state is

3. QUTRIT CASE

Next, we discuss the nontrivial case of paraxial photonic qutrits. The internal degree of freedom is now three dimensional, in contrast to the well-known two-state polarization subspace for light. Our analysis for qubits can be easily generalized by inserting a second interaction term between frequencies ${\omega }_i$ and ${\omega }_j$ and with coupling ${\kappa }_{ij}$ into Eq. (2):

The dynamics of paraxial qutrits is still given by Eq. (4), with $\mathrm{\varphi }={({\varphi }_1,{\varphi }_2,{\varphi }_3)}^T$ and the necessary adjustments for $\mathrm{\Sigma }$ and $\mathbf{B}$, to be discussed shortly. However, there is still freedom in the choice of coupling. The three possible choices are the $\mathrm{\Lambda }$ scheme, the $\mathrm{V}$ scheme, and the ladder scheme [see Figs. 4(a)–4(c)]. The operator $\mathrm{\Sigma }$ is again a three-component vector, this time of $3\times 3$ Hermitian matrices ${\mathrm{\Sigma }}_{jk}=i{L}_{jk}$ [with $(jk)=(12),(23)$ or $(31)$], where $L_{jk}$ are anti-Hermitian and given by

 

Fig. 4. Three possible configurations for frequency-domain photonic qutrits: (a) $\mathrm{\Lambda }$ scheme, (b) $V$ scheme, and (c) nearest-level (ladder) scheme. (d) Illustration of the SG deflection for a single $\mathrm{\Lambda }$ scheme photon in the lowest frequency $|{\psi }_{\mathrm{in}}⟩=|{\psi }_1⟩$. The expressions for the eigenstates $|{\psi }_{\pm }⟩$ and $|{\psi }_0⟩$ are given in Eq. (16).

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When a transverse coupling gradient is present, we expect that a SG effect [Eqs. (7)–(9)] is manifested in this case, as well. For example, if one chooses to work with the $\mathrm{\Lambda }$ scheme [Fig. 4(a)], the nonvanishing components of the coupling field $\mathbf{B}$ are $B_x={\kappa }_{23}/{\overline{v}}_g$ and $B_y={\kappa }_{31}/{\overline{v}}_g$. We again have $\widehat{\mathbf{B}}=\widehat{\mathbf{x}}\cos \theta +\widehat{\mathbf{y}}\sin \theta$, where this time $\tan \theta ={\kappa }_{31}/{\kappa }_{23}$. Each input photon is now projected on the three eigenstates of $\mathrm{\Sigma }·\widehat{\mathbf{B}}$ and, subsequently, either deflected to distinct angles on the right (R) or on the left (L), or remains undeflected in the middle (M). Consequently, Eq. (9) becomes

 

Fig. 5. Simulated photodetection probability in PPLN with a quasiperiodic SG poling pattern (with units ${\mathrm{mm}}^{-1}$ on the left figures and ${\mathrm{mm}}^{-2}$ on the right ones, normalized as in Fig. 3). All input states are Gaussian with a waist of 50 μm. (a), (b) The input state is the first eigenstate $|{\psi }_{\mathrm{in}}⟩=|{\psi }_{+}⟩$ and is deflected to the right. (c), (d) The second eigenstate $|{\psi }_{\mathrm{in}}⟩=|{\psi }_0⟩$ remains undeflected. (e), (f) The third eigenstate, $|{\psi }_{\mathrm{in}}⟩=|{\psi }_{-}⟩$, is deflected to the left. In (g), (h), the state $|{\psi }_{in}⟩=|{\psi }_3⟩$ is projected onto the eigenstates $|{\psi }_{+}⟩$ and $|{\psi }_{-}⟩$ [see Eq. (16)]. The resulting deflection is only to the right or to the left. Finally, (i), (j) show the projection of $|{\psi }_{in}⟩=|{\psi }_1⟩$ onto all three eigenstates. The resulting far field consists of all three possible deflections.

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Two-photon bunching can also be manifested in the qutrit case. For two $\mathrm{\Lambda }$ scheme photons in the initial pure state $|{\psi }_{\mathrm{in}}⟩=|{\psi }_1{\psi }_2⟩$, the output state is written as

4. CONCLUSIONS

In summary, we have shown that paraxial photons propagating in specially engineered nonlinear media exhibit the properties of 2D quantum particles with internal angular momentum of potentially arbitrary dimension. The couplings between the discrete set of frequencies that satisfy phase-matching constitute the analog of an external magnetic field with which the photons interact. When the coupling varies in space, quantum correlations appear between the spatial and spectral degrees of freedom. For the simpler cases of qubits and qutrits, we investigate the equivalent of an all-optical SG effect for photons, which allows the projection and spatial separation of orthogonal frequency-superposition states. Such an effect is also expected to produce, for example, two-photon entanglement in the frequency domain that resembles the familiar HOM 2002 state [2]. The possible applications for the proposed effect include frequency-basis quantum computing [20]; color-entanglement generation [27] for quantum information and communication protocols, without the necessity to use multiple spatial modes [25] or biphotons [26] for the case of qutrits; two-mode quantum state tomography [28]; and quantum key-distribution protocols, which have been proven to be more secure when using qutrits [24]. The SG deflector, comprising a single optical element, can be used in these schemes as a spectral-to-spatial beam splitter (or tritter), being all-optically controlled, as opposed to electronically controlled realizations [6,8,10]. Moreover, the resulting states in SG projection are expected to be robust against fluctuations in the pump pulse area (the latter will alter only the deflection angle), as compared to previous setups that used nonlinear state rotations [2,4].

The concept presented here can be further extended to higher dimensions, using nonlinear photonic crystals that will couple an arbitrary number of optical waves at different frequencies. Moreover, depending on the chosen element of the nonlinear tensor, the interacting modes may have either the same polarization or different polarizations. These generalizations can be done with efficient scalability, as the effect still requires only a single optical element, in contrast to what previous ${\chi }^{(2)}$ realizations [2,20] can offer. Since frequency-domain qudits are recently gaining attention as possible candidates for carrying quantum information [2–10,20], we are hopeful that this work will help promote such future advancements.