The role of coherence on two-particle quantum walks

TL;DR

This paper explores how coherence affects two-bosons quantum walks, providing analytical expressions and demonstrating that initial conditions and coherence influence photon correlations and propagation.

Contribution

It introduces a general analytical formula for two-bosons correlation functions in pure and mixed states and proposes a quantum-walk scheme affected by coherence.

Findings

Correlation functions depend on initial photon distribution and coherence

Photon average distance is influenced by initial phase and coherence

Propagation features align with analytical correlation predictions

Abstract

We investigate the dynamical properties of the two-bosons quantum walk in system with different degrees of coherence, where the effect of the coherence on the two-bosons quantum walk can be naturally introduced. A general analytical expression of the two-bosons correlation function for both pure states and mixed states is given. We propose a possible two-photon quantum-walk scheme with a mixed initial state and find that the two-photon correlation function and the average distance between two photons can be influenced by either the initial photon distribution, or the relative phase, or the degree of coherence. The propagation features of our numerical results can be explained by our analytical two-photon correlation function.

Equations

$$H=-\sum _q{T}_{q,q+1}({\hat{a}}_q^{†}{\hat{a}}_{q+1}+h.c.)+\sum _q{\beta }_q{\hat{a}}_q^{†}{\hat{a}}_q,$$

$${\beta }_q=T_{q+1,q}+T_{q-1,q}.$$

$$\mid 1{⟩}_q\mid 1{⟩}_r=\frac{1}{\sqrt{1+{\delta }_{q,r}}}{\hat{a}}_q^{†}{\hat{a}}_r^{†}\mid \mathrm{vac}⟩,$$

$$\mid \psi ⟩=\sum _{q,r}{c}_{q,r}\mid 1{⟩}_q\mid 1{⟩}_r,$$

$$\rho=\sum_{qr,q^{\prime}r^{\prime}}\rho_{qr,q^{\prime}r^{\prime}}\bigl{(}\mid\!1\rangle_{q}\mid\!1\rangle_{r}\bigr{)}\bigl{(}\langle 1\!\mid_{q^{\prime}}\langle 1\!\mid_{r^{\prime}}\bigr{)},$$

$$i\frac{\partial {\hat{a}}_q^{†}}{\partial t}={\beta }_q{\hat{a}}_q^{†}+T_{q,q+1}{\hat{a}}_{q+1}^{†}+T_{q,q-1}{\hat{a}}_{q-1}^{†},$$

$${\hat{a}}_q^{†}(t)=\sum _r{U}_{q,r}(t){\hat{a}}_r^{†}(0),U(t)=e^{-iHt},$$

$$\displaystyle\Gamma_{k,l}(t)=\sum_{q\neq r,q^{\prime}\neq r^{\prime}}\rho_{qr,q^{\prime}r^{\prime}}\Bigl{(}U_{kq^{\prime}}U_{lr^{\prime}}U_{lq}^{*}U_{kr}^{*}+U_{kq^{\prime}}U_{lr^{\prime}}U_{lr}^{*}U_{kq}^{*}+U_{kr^{\prime}}U_{lq^{\prime}}U_{lq}^{*}U_{kr}^{*}+U_{kr^{\prime}}U_{lq^{\prime}}U_{lr}^{*}U_{kq}\Bigr{)}$$

$$\displaystyle+\sum_{q,q^{\prime}\neq r^{\prime}}\sqrt{2}\rho_{qq,q^{\prime}r^{\prime}}\Bigl{(}U_{kq^{\prime}}U_{lr^{\prime}}U_{lq}^{*}U_{kq}^{*}+U_{kr^{\prime}}U_{lq^{\prime}}U_{lq}^{*}U_{kq}^{*}\Bigr{)}+\sum_{q\neq r,q^{\prime}}\sqrt{2}\rho_{qr,q^{\prime}q^{\prime}}\Bigl{(}U_{kq^{\prime}}U_{lq^{\prime}}U_{lq}^{*}U_{kr}^{*}+U_{kq^{\prime}}U_{lq^{\prime}}U_{lr}^{*}U_{kq}^{*}\Bigr{)}$$

$$+\sum _{q,q^{\prime }}2{\rho }_{qq,q^{\prime }{q}^{\prime }}{U}_{k{q}^{\prime }}{U}_{l{q}^{\prime }}{U}_{lq}^{\ast }{U}_{kq}^{\ast },$$

$$\rho ={\cos }^2\delta \mid {\psi }_1⟩⟨{\psi }_1\mid +{\sin }^2\delta \mid {\psi }_2⟩⟨{\psi }_2\mid .$$

$$\rho ={\rho }_{00,11}\mid 2{⟩}_0⟨2{\mid }_1+{\rho }_{11,00}\mid 2{⟩}_1⟨2{\mid }_0+{\rho }_{11,11}\mid 2{⟩}_1⟨2{\mid }_1+{\rho }_{00,00}\mid 2{⟩}_0⟨2{\mid }_0$$

$${\Gamma }_{q,r}=2\gamma \text{Re}(e^{i\varphi }{U}_{q0}{U}_{r0}{U}_{r1}^{\ast }{U}_{q1}^{\ast })+2\alpha |U_{r0}{U}_{q0}{|}^2+2(1-\alpha )|U_{r1}{U}_{q1}{|}^2,$$

$${\Gamma }_{q,r}(\tau )=-2\gamma \cos \varphi J_q(\tau )J_r(\tau )J_{r-1}(\tau )J_{q-1}(\tau )+2\alpha [J_q(\tau )J_r(\tau ){]}^2+2(1-\alpha )[J_{r-1}(\tau )J_{q-1}(\tau ){]}^2.$$

$$d$$

$$S=-\sum _i{\lambda }_i{\log }_2{\lambda }_i,$$

Full text

Keywords: two-particle quantum walk, degree of coherence, two-photon correlation function, pure state, mixed state

I Introduction

As the quantum mechanical counterparts of the classical random walk Ahar , the quantum random walk has been increasingly receiving attentions because of their potential applications range from quantum information to simulation of physical phenomena. For example, the quantum walk offers an advanced tool for building quantum algorithm Moh ; Shen ; Sal that is shown to be of the primitive for universal quantum computations und ; Lov ; chi ; and . We know that the quantum walk include two main classes that are discrete-time quantum walk and continuous-time quantum walk Jwa ; Far . The continuous-time quantum walk can evolve continuously with time through tunneling between neighbors sites and does not require quantum coin to generate superposition of states. This means the continuous-time quantum walk can be implemented via a constant tunneling of quantum particles in several possible lattice sites. So far, the quantum walks of single particles have been studied in experiments by using either classical waves peret , single photons Sch ; broo , or single atoms Kars ; Wei . Additionally, quantum walks of two correlated photons trapped in waveguide lattices were also studied in experiments Yar ; Alberto .

Note that many-particle quantum walks can exhibit more fascinating quantum features in contrast to single-particle quantum walks. The reason is that single-particle quantum walks can be exactly mapped to classical wave phenomena Knight but for quantum walks of more than one indistinguishable particle, the classical theory can not provide sufficient descriptions. In Ref. omar , the authors theoretically studied the discrete-time quantum walk of two particles and demonstrated the distinctly nonclassical correlations. Meanwhile, the effect of interactions between particles on quantum walk of two indistinguishable particles was theoretically studied in Ref. lahini ; Qin , where the system was assumed to be completely coherent and the influence of the degree of coherence was not studied. We know that except for the interaction between particles, the other factors, e.g., the initial states, the quantum-walk parameters and the degree of coherence of the system, can also affect the features of two-particle quantum walks. Especially, we know that the major challenge to experimentally realize quantum walks of correlated particles is to find a low-decoherence system that can preserves the nonclassical features of quantum walks Alberto , which implies that the influence of the degree of coherence of the system on two-particle quantum walks is important. Then it is worthwhile to investigate the properties of two-particle quantum walks with attention to different degrees of coherence since the decoherence effects in quantum walks have potential algorithmic applications kendon .

In this paper, we propose a density matrix formulism to study the two-particle quantum walk where the degrees of coherence can be naturally introduced. With the help of Heisenberg equation of motion, we derive a general analytical expression of the two-particle correlation function. As a concrete example, we propose a possible two-photon quantum-walk scheme with a mixed initial state to exhibit the quantum features of the two-particle quantum walk via the two-particle correlation and the average distance between the two particles. Our result exhibits that the propagation of the two particles depends not only on the initial distribution of the two particles but also on the relative phase and the degree of coherence of the system. Such a propagation feature can be explained by our analytical two-particle correlation function. In the next section, we present the model and derive the analytical expression of the two-particle correlation. In Sec. III, we propose a concrete scheme to show some dynamical features of two-particle quantum walks. Our main conclusions are summarized in Sec. IV.

II A general formulation

We consider a two-particle quantum walk in a one-dimensional lattice space. The propagation of the two particles is described by the evolution of the state of a tight-binding model,

[TABLE]

where the operators $a_{q}^{\dagger}$ and $a_{q}$ create and annihilate a bosonic particle at site $q$, respectively. Here the parameter $T_{q,q+1}$ refers to the tunneling strength of particles between the nearest neighbor sites, and

[TABLE]

Note that the above form of $\beta_{q}$ was picked to keep the probability conservation in the proposal of continuous-time quantum walk via decision tree Far . Now more generally, the value of $\beta_{q}$ can be arbitrary due to the probability conservation is naturally satisfied in quantum mechanics. If the tunneling strength $T_{q,q+1}$ is a constant, $\beta_{q}$ will become a constant for the periodical boundary condition. In this case, the value of $\beta_{q}$ does not affect the dynamical properties of the quantum-walk system. Whereas, for the open boundary condition, the values of $\beta_{q}$ for the two boundary sites are different from that for the other sites. This can naturally introduce two defects to the quantum-walk system. Note that the effect of defects on single-particle quantum walks was studied in Ref. Li .

Since we consider a two-bosons system, the Fock bases describing the system are

[TABLE]

where $\delta_{q,r}$ denotes the Kronecker delta. Equation (3) represents a two-particle state with one on the $q$th site and the other one on the $r$th site. Note that $\mid\!1\rangle_{q}\mid\!1\rangle_{r}$ is regarded as identical to $\mid\!1\rangle_{r}\mid\!1\rangle_{q}$ for indistinguishable particles that we considered. Meanwhile, the two particles can be in the same site (i.e., $q=r$) for bosonic particle that we considered. Thus the Hilbert space expanded by the aforementioned Fock bases is of $D=L(L+1)/2$ dimension. Here $L$ denotes the number of the sites. We know that the propagation of the two particles is determined not only by the property of the waveguide lattice but also by the two-particle input state. If the two particles are in a pure state at the initial time, the two-particle input state can be expressed as a wavefunction, namely, a coherent superposition of the Fock bases,

[TABLE]

where $\sum_{q,r}|c_{q,r}|^{2}=1$. However, if the two particles are in a mixed state at the initial time, the two-particle input state needs to be described by a density matrix rather than wavefunction. Such a density matrix is given by

[TABLE]

which is a $D\times D$ matrix. We know that $\textrm{Tr}\rho^{2}\leq(\textrm{Tr}\rho)^{2}$ where the equal sign holds only for pure states. In the following, we will focus on the two-particle quantum walk for the mixed input states.

Now we are in the position to study the propagation of the two particles with the help of Heisenberg equation of motion for the creation operators, namely,

[TABLE]

where we set $\hbar=1$ for simplicity in calculation. The creation operator $\hat{a}_{q}^{\dagger}$ at any time can be obtained with the help of Eq. (6),

[TABLE]

where $U_{q,r}(t)$ is the probability amplitude of a single particle transiting from the $q$th waveguide to the $r$th one. To exhibit the quantum behaviors of the two-particle quantum walk, let us firstly evaluate the two-particle correlation function $\Gamma_{k,l}(t)=\langle~{}\hat{a}_{k}^{\dagger}(t)\hat{a}_{l}^{\dagger}(t)\hat{a}_{l}(t)\hat{a}_{k}(t)~{}\rangle$ which manifests the probability that the two particles are coincident in the $k$th and the $l$th waveguide Yar ; Mattle . Since the two-particle input state can be described by the density matrix given in Eq. (5), the expectation value of any observable of the system can be calculated via $\langle~{}\hat{O}(t)~{}\rangle=\textrm{Tr}(\hat{O}(t)\rho)$. Then we obtain an expression of two-particle correlation function

[TABLE]

which presents a general form for either pure initial input states or mixed ones. One can obtain the two-particle correlation at any time as long as the density matrix corresponding to the input state is given. with the help of such an expression of two-particle correlation function, many dynamical features of two-particle quantum walk can be explained.

III Two-photon quantum walk for a concrete input state

In order to expose the quantum properties of two-particle quantum walks more clearly, we turn to a concrete example where the particles are assumed to be photons. We know that each beam can become two coherent beams after propagating through a grating sza and the pure two-photon input states can be experimentally realized via injecting two coherent beams into waveguide lattice Yar . Then we suppose that there are two incoherent light beams and the relation of their intensity is $\cos^{2}{\delta}$:$\sin^{2}{\delta}$. The two incoherent beams propagate through two gratings, respectively, and then simultaneously inject into the waveguide arrays. The two incoherent beams will create two pure two-photon states $\psi_{1}$ and $\psi_{2}$, respectively Yar ; sza . Because the initial two beams are not coherent, the initial state of the system needs to be described by the density matrix

[TABLE]

As an example, we take $\psi_{1}=\cos{\frac{\theta}{2}}\mid\!2\rangle_{1}+\sin{\frac{\theta}{2}}e^{i\phi}\mid\!2\rangle_{0}$ and $\psi_{2}=\mid\!2\rangle_{1}$, then the initial density matrix is

[TABLE]

with $\rho_{00,11}=\cos^{2}\delta\cos{\frac{\theta}{2}}\sin{\frac{\theta}{2}}e^{i\phi}$, $\rho_{11,00}=\rho_{00,11}^{*}$, $\rho_{11,11}=\cos^{2}\delta\cos^{2}{\frac{\theta}{2}}+\sin^{2}\delta$, and $\rho_{00,00}=\cos^{2}\delta\sin^{2}{\frac{\theta}{2}}$. Here $\mid\!2\rangle_{q}$ stands for $\mid\!1\rangle_{q}\mid\!1\rangle_{q}$ whose definition has been given in Eq. (3). We can find that the other matrix elements of $\rho$ are zeros except for the above four elements. Since the above four elements can be changed via $\delta$, $\theta$ and $\phi$, without losing the generality, we redefine them as $\rho_{00,00}=\alpha$, $\rho_{11,11}=1-\alpha$ and $\rho_{00,11}=\rho_{11,00}^{*}=e^{i\phi}\sqrt{\eta-1-4\alpha^{2}+4\alpha}/2$ with $0\leq\alpha\leq 1$. Here the parameter $\eta=2\textrm{Tr}(\rho^{2})-1$ $(0\leq\eta\leq 1)$ is introduced to characterize the degree of coherence lhlu . We have $\eta=1$ when the system is in a pure state, otherwise $\eta<1$. With the help of Eq. (II), the two-photon correlation of the system for the mixed state shown in Eq. (10) yields

[TABLE]

where $\gamma=\sqrt{\eta-1+4\alpha(1-\alpha)}$. This implies that the two-photon correlation depends not only on the initial probability distribution of the two photons but also on the degree of coherence and the relative phase of the system at the initial time. The first term in Eq. (11) is a coherent one that reveals well the quantum nature of two-photon quantum walk. Taking a pure initial state (i.e., $\eta=1$) as an example, the two-photon correlation function (11) becomes $\Gamma=|\sqrt{2\alpha}e^{i\phi}U_{r0}U_{q0}+\sqrt{2(1-\alpha)}U_{r1}U_{q1}|^{2}$, which implies that the two-photon correlation can take place when the two photons from the 0th site propagate to the $q$th and the $r$th sites, respectively, or when the two photons from the 1th site to the $q$th and the $r$th sites, respectively. Due to the two photons are indistinguishable, the two paths can interfere, which is essentially the Hanbury Brown Twiss(HBT) interference Han .

Now we investigate the quantum features of a two-photon quantum walk by considering a waveguide arrays consisting of (2$l$+1) identical waveguides. In this case, the tunneling strengths between nearest-neighbor arrays are all the same, i.e., $T_{q,r}=C$ with $C$ being a constant, and $\beta_{q}$ becomes a constant $2C$ for the periodical boundary condition we considered. Then $U_{q,r}(t)$ becomes $e^{i2Ct}i^{q-r}J_{q-r}(2Ct)$ where $J_{q}$ is the $q$th order Bessel function Led ; Yariv . With the help of Eq. (11), we can write out the two-photon correlation function in terms of Bessel functions.

[TABLE]

where $\tau=2Ct$.

In Fig. 1, we plotted the two-photon correlation matrix at time $t=4({1}/{C})$ for different initial conditions, where $1/C$ is the unit of time (i.e., the time $t$ is in the unit of the inverse of tunneling strength). We can see that each particle can be found on either side of origin after propagation, which is reflected in the four symmetric peaks in Fig. 1 (a). For this case (i.e., $\alpha=1$ and $\eta=1$), the system is in a pure state and Eq. (11) becomes $\Gamma_{q,r}=2|U_{r0}U_{q0}|^{2}$ which is the same as the result in Ref. Yar . Such a correlation function is just a product of the two classical probability distribution, so there is no interference and the photons propagate in the ballistic direction. From Fig. 1 (b), we can find that just like Fig. 1 (a), the two photons also favor to localize at the four corners of the correlation map. The reason is that there is no interference because the system is completely incoherent, which is confirmed by that the coherent term in Eq. (11) vanishes in the case of $\eta=0$. Whereas, with the increase of the degree of coherence, the coherent term emerges in the two-photon correlation function $\Gamma_{q,r}$, so the $\Gamma_{q,r}$ exhibits the properties of interference. Due to the existence of the Hanbury Brown-Twiss (HBT) interference, two local maximums emerge in the off-diagonal regions of the correlation matrix (see Fig. 1 (c) and (d)). That implies that the two photons favor to far from each other which is in contrast to the case of Fig. 1 (e) and (f) where except the initial relative phase $\phi$, the other parameters are the same as those in Fig. 1 (c) and (d), respectively. This is reasonable because the coherent term in Eq. (12) is in proportion to $\cos{\phi}$ for the periodical waveguide lattice we considered. Additionally, comparing the values of the maximums in Fig. 1 (c) and (d), it is easy to find that the larger the degree of coherence is, the more distinct the interference effect of the system will be.

To exhibit the propagation properties of the two photons, we also calculate the average distance between the two photons,

[TABLE]

Here the first term is a coherence one that is affected by the degree of coherence of the system due to $\gamma=\sqrt{\eta-1+4\alpha(1-\alpha)}$. We plot the time evolution of the distance between the two photons in Fig. 2, we can see that the distance between two photons is affected not only by the relative phase but also by the degree of coherence. In Fig. 3, we plot the dependence of the distance $d$ at time $t=4(1/C)$ on the degree of coherence and the initial relative phase. From the left panel of this figure, we can see that the distance between two photons becomes larger with the increase of the degree of coherence when $\phi=0$, which is contrast to the case of $\phi=\pi$. The reason for this phenomenon is that the HBT interference makes the two photons far from each when $0\leq\phi<\pi/2$, which can be confirmed by Fig. 1 (c) and (d) where there are two maximums in the off-diagonal regions. Therefore, the distance between the two photons becomes larger with the increase of the degree of coherence due to the fact that the increase of the degree of coherence makes the interference effect more significant. Whereas, the case of $\pi/2\leq\phi<\pi$ is in contrast to that of $0\leq\phi<\pi/2$ because the two photons favor to stay together when $\pi/2\leq\phi<\pi$, which can be confirmed by Fig. 1 (e) and (f). Note that when $\phi=\pi/2$, the interference term in the two-photon correlation function becomes zeros, so the degree of coherence does not affect the distance between two particles (see the dot-symbol line in the left panel of Fig. 3). The right panel of Fig. 3 exhibits that the relative phase of the system at the initial time can affect the distance between two photons in the case of $\eta>0$ but such an effect vanishes in the case of $\eta=0$. That is reasonable because the distance between two photons is in proportion to $\cos{\phi}\sqrt{\eta-1-4\alpha^{2}+4\alpha}$ which can be found in Eq. (13). Additionally, we calculate the von Neumann entropy to show the evolution of the entanglement of the system. We split the system into two halves, $L$ and $R$, in the center of the system, and build the reduced density matrix $\rho_{L}$ of the subsystem $L$ at any time Schach . Then we can calculate the von Neumann entropy of $\rho_{L}$ as

[TABLE]

where $\lambda_{i}$ are the non-zero eigenvalues of the matrix $\rho_{L}$. In Fig. 4, we plot the time evolution of the von Neumann entropy of the left half of the system for different initial conditions.

IV conclusion

We proposed a density matrix formulism to study the properties of two-particle quantum walks where the effect of coherence was introduced naturally. We gave the general analytical expression of the two-particle correlation function which is correct for systems in both mixed states and pure states. We suggested a possible two-photon scheme to exhibit the more fascinating quantum features of two-particle random walks with mixed initial states. For such a concrete scheme, we calculated the two-photon correlation and the average distance between the two photons. The corresponding results manifested that the propagation of the two photons depends not only on the initial distribution of the two photons but also on the relative phase and the degree of coherence of the system. Such propagation features of the two photons were explained with the help of the analytical expression of the two-particle correlation function we obtained.

The work is supported by the NBRP of China (2014CB921201), the NSFC (11104244 and 11274272, 11434008), and by the Fundamental Research Funds for Central Universities.